Simbody
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F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this | document (the"Document") |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made at the cost of increased computation time Once an event has been localized to an acceptable the integrators | stepTo () method will return control to the time stepper at time tlow |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state | discontinuously (potentially updating both discrete and continuous variables) |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the | event (s) are defined to have occurred |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an | event (s) triggered condition |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt | termination (e.g.reached final time) Scheduled events are handled outside the integrator simply by allowing the time stepper to specify with each stepTo() call a maximum time to which the integrator may advance.When this time is reached |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and | again (SeeREF _Ref160937060\hFigure 2for an illustration.) This leads to the possibility of seeing a-1(0 sign transition while localizing a-1(1.This can cause difficulties since the next step will see a 0(1 transition which could trigger a spurious second event.SHAPE\*MERGEFORMATAn Integrator must take great care not to report two events in the above situation |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as and on the surface In either these triggers will have a significant zero state which is not an artifact of finite precision arithmetic In those circumstances the sign | transitions (0 and 0(1 can be meaningful and the caller may need to be notified at each transition.REF _Ref160937126\hFigure 3shows two examples.SHAPE\*MERGEFORMATTo allow the modeler to distinguish between accidental and purposeful zeroes, the System can optionally provide information about each of the event trigger functions.This information can include a specification of which sign transitions are considered significant for each trigger function.If only-1(1 and/or 1(-1 are specified, then we consider the event trigger function to be continuous and the integrator will only report a single event as the trigger passes through zero, even if it accidentally lands exactly on zero during localization.On the other hand, if the System tells us that zero is a significant value, then we assume that the trigger function has a discrete zero zone(or deadband) and we will report transitions to and from zero if they are seen during localization.REF _Ref160978458\hTable 1REF _Ref160978221\p\hbelowshows how observed transitions are treated based on whether rising or falling transitions have been specified, and whether zero is significant.Rising and falling transitions are disjoint so these specifications can be combined to allow a single event trigger to detect both rising and falling transitions.The first column inREF _Ref160978458\hTable 1lists the possible specifications that a System can make for the transitions of interest(rising/falling;continuous/discrete).The second column lists the transitions that might be observed initially, prior to localization.Typically, an initial step will be much wider than a localization interval so a1(1 or 1(1 transition is most likely to occur.The third column shows for each pre-localization observation the possible post-localization transitions(transitions to and from zero always localize to the same transition).The final column shows what transition will be reported back to the caller as having been seen.Note that although a wider set of transitions may be recognized during localization, the final report must come from the set that the system specified as interesting.For example, if a1(0 transition is seen after localization, but the system only watches for1(1, then well report1(0 as an instance of1(1.TableSEQ Table\*ARABIC1:handling of sign transitions Significant sign transitionsTransition seen before localizationLocalized transitionReported transitionRising-1(1continuous trigger-1(1any rising-1(1-1(0(unchanged)-1(10(1no eventdiscrete zero adds-1(0, 0(1-1(1any risingreport localized transition-1(0(unchanged)-1(00(1(unchanged)0(1Falling 1(-1continuous trigger1(-1any falling1(-11(0(unchanged)1(-10(-1no eventdiscrete zeroadds 1(0, 0(-11(-1any fallingreport localized transition1(0(unchanged)1(00(-1(unchanged)0(-1Notes:(1) For a continuous event trigger function that accidentally hits zero exactly, the transition to zero is treated as a zero crossing, but the subsequent transition from zero is ignored.(2) For a discrete-zero event trigger, if the function doesnt stay zero long enough then the zero transitions may be missed and a zero crossing(i.e.,-1(1 or 1(-1) will be reported instead.Accuracy, scaling, tolerances The time stepper is capable of achieving different levels of accuracy when simulating a particular system, withm o r ea c c u r a c yr e q u i r i n gm o r ec o m p u t a t i o nt i m e.O u rg o a li st oa l l o wa p p l i c a t i o np r o g r a m m e r st od e l i v e rt ot h e i re n du s e r st h ea b i l i t yt oc o n t r o lt h ea c c u r a c yl e v e lw i t has i n g l e, p h y s i c a l l ym e a n i n g f u ls c a l a rv a l u e, af r a c t i o ni nt h er a n g e0< d"1 . A t y p i c a l v a l u e i s = 0 . 0 0 1 ( 0 . 1 % ) m e a n i n g t h a t t h e i n t e g r a t o r s h o u l d p r o d u c e r e s u l t s w h i c h a r e w i t h i n 0 . 1 % o f t h e p e r f e c t a n s w e r . ( T h e s e p h r a s e s a r e i n q u o t a t i o n s b e c a u s e t h e y a r e m u c h e a s i e r t o s a y t h a n t o d e f i n e p r e c i s e l y ! ) E q u i v a l e n t l y , o n e c a n i n t e r p r e t as specifying the number of significant digits nd desired in the results, with EMBED Equation.DSMT4 . For example, EMBED Equation.DSMT4 . This goal is achievable only imperfectly, since in general accuracy is difficult to specify, to measure, to control, and even to define precisely. Reasonable users can disagree about exactly what they mean by 1% accuracy. Nevertheless we feel it is important to provide a single knob for a user to turn that delivers more accuracy at higher cost or less accuracy at lower cost, in a way that at least attempts to capture what at typical user might mean by these terms. The alternative of exposing the many complex issues involved to the end user assumes a kind of specialized expertise that would severely limit our intended audience. Instead, we expose these troubles to the application programmer in a way that allows us to collect information that can be used to define the Holy Grail control . T h e p r i m a r y d i f f i c u l t y w e e n c o u n t e r i s t h a t t h e v a r i a b l e s a n d e q u a t i o n s d e f i n i n g t h e u s e r s s y s t e m a r e n o t e v e n l y w e i g h t e d . W e e x p e c t t h a t i n m o s t c a s e s t h e p r o g r a m m e r w h o i s u s i n g S i m T K : : T i m e S t e p p e r w i l l k n o w s o m e t h i n g u s e f u l a b o u t t h e v a r i o u s w e i g h t s t h a t w i l l h e l p u s d e f i n e . T o a c c o m m o d a t e t h a t , w e a l l o w s p e c i f i c a t i o n o f f o u r k i n d s o f s c a l i n g i n f o r m a t i o n : t h e c h a r a c t e r i s t i c t i m e s c a l e a w e i g h t w i f o r e a c h c o n t i n u o u s s t a t e v a r i a b l e y i a t o l e r a n c e t i f o r e a c h o f t h e c o n s t r a i n t e r r o r s c i a l o c a l i z a t i o n t i m e w i n d o w w i d t h l i f o r e a c h e v e n t e i E a c h o f t h e s e q u a n t i t i e s i s d e f i n e d b e l o w t o b e a p r o p e r t y o f t h e s y s t e m ( m o d e l ) , i n d e p e n d e n t o f t h e a c c u r a c y w i t h w h i c h t h e s y s t e m i s b e i n g s i m u l a t e d . T h e i d e a i s t o d e f i n e f o r e a c h q u a n t i t y a u n i t e r r o r t o w h ich the accuracy requirement can then be applied. The integrator treats these values as constant during each integration step. However, we expect the state variable weights wi to change (slowly) as the trajectory (time and state) evolves, so the integrator will request updated weights from time to time during a continuous interval, according to some criteria. Time scale, constraint tolerances, and localization requirements, on the other hand, are expected to be independent of trajectory and are fixed once th e i n s t a n c e v a r i a b l e s a r e f r o z e n p r i o r t o t h e s t a r t o f a c o n t i n u o u s i n t e r v a l w i t h i n t h e s i m u l a t i o n . T i m e S c a l e T h e r e i s a c h a r a c t e r i s t i c t i m e s c a l e o f t h e s y s t e m w h i c h t h e p r o g r a m m e r m a y p r o v i d e t o c o n v e y t o t h e i n t e g r a t o r t h e s m a l l e s t t i m e o f i n t e r e s t f o r t h e s t u d y . T h i s s h o u l d b e t h e m i n i m u m t i m e i n t e r v a l o v e r w h i c h i n t e r e s t i n g c h a n g e s a r e e x p e c t e d t o o c c u r . T h e i n t e g r a t o r m a y u s e t h i s f o r s e l e c t i n g t h e i n i t i a l s t e p s i z e a n d d e f a u l t s f o r o t h e r s c a l i n g i n f o r m a t i o n m a y d e p e n d o n . W e i g h t s a n d c o n s t r a i n t t o l e r a n c e s W e a r e g i v e n a s e t o f w e i g h t s w i e"0f o re a c hy ia n das e to ft o l e r a n c e st i > 0f o re a c ho ft h ec o n s t r a i n te r r o r sc i.L e tWb ead i a g o n a lm a t r i xw i t ht h ew iso ni t sd i a g o n a l, a n dTb ead i a g o n a lm a t r i xw i t ht h er e c i p r o c a lt o l e r a n c e s1/t io ni t sd i a g o n a l(w ea l s oc a l lt h e s er e c i p r o c a lt o l e r a n c e sc o n s t r a i n tw e i g h t s).T h e ng i v e nt h ef r a c t i o n a la c c u r a c ys p e c i f i c a t i o n(e.g.=0.1%), t h ei n t e g r a t o ri sr e q u i r e dt os o l v ef o rt h et r a j e c t o r yy(t) s u c ht h a te a c hi n t e g r a t i o ns t e pm a i n t a i n st h el o c a le r r o r|W y|R M S d" a n d c o n s t r a i n t e r r o r | T c ( t , y ) | R M S d" a ta l lt i m e s.H e r eyi st h ev e c t o ro fe s t i m a t e da b s o l u t ee r r o r si ne a c hs t a t ev a r i a b l ey, a se s t i m a t e db yt h ei n t e g r a t o rf o rat r i a ls t e pu n d e rc o n s i d e r a t i o n.E v e n tl o c a l i z a t i o nw i n d o wE v e n tl o c a l i z a t i o nr e q u i r e sf o re a c he v e n tt r i g g e rf u n c t i o ne iau n i tl o c a l i z a t i o nr e q u i r e m e n tl i(i nu n i t so ft h es y s t e mst i m es c a l e) w h i c hi st h e nn a r r o w e db yt h ea c c u r a c yr e q u i r e m e n ts ot h a tt h ei n t e g r a t o rl o c a l i z e se v e n te it oat i m ei n t e r v a lo fw i d t h d" l i. Thus when the integrator reports that a set E of events has triggered in the event window (tlow,thigh], it guarantees that EMBED Equation.DSMT4 . Default accuracy, scaling and tolerances The defaults are as follows: EMBED Equation.DSMT4 EMBED Equation.DSMT4 We also support a number of different schemes for defining the weights. The conventional rtol, atol scheme is achieved by defining EMBED Equation.DSMT4 where ui represents one unit of error in yi (a typical value would b e 0 . 1 ) . T h i s i s t h e n e q u i v a l e n t t o r t o l = a n d a t o l i = u i . N u m e r i c a l o p t i m i z a t i o n ( S i m T K : : O p t i m i z e r ) T h e S i m T K : : O p t i m i z e r c l a s s c a n b e u s e d t o n u m e r i c a l l y s o l v e v a r i o u s o p t i m i z a t i o n p r o b l e m s . O p t i m i z a t i o n m e t h o d s c a n b e u s e d t o f i n d t h e o p t i m a l s o l u t i o n t o some problem. For example, what set of coordinates would give the minimum energy configuration for this molecule? The algorithms implemented in the Optimizer class search for points which are a minimum of a, user supplied, objective function. The algorithms start from some initial point and iteratively search for points which reduce the objective function. The algorithms terminate when the objective function stops decreasing. Note that the minimum found by the Optimizers is sometime a local minimum and not a global minimum. Therefore, the point which the algorithm starts searching from is important. This is illustrated in Figure 2 below. SHAPE \* MERGEFORMAT Note that if our goal was to find the maximum of some objective function f(x). We could still use the existing optimizers by using f(x) as our objective function. Sometimes the optimization problem has constraints on the objective function. Consider the simple example below: The following objective function: EMBED Equation.3 is subject to two constraints: C1: EMBED Equation.3 C2: EMBED Equation.3 Has an optimal solution at: X = (4, 2) Figure 3 shows a graph of this problem. The two constraints (C1, C2) bound the space of possible solutions to a feasible region which is shown in gray. The optimal solution to the problem must be a point within this feasible region which has the smallest value of the objective function. The two dashed circles indicate the constant value contours of our objective function. The smaller circle has a value of 1.0 and the larger circle has a value of 4.0. Optimization problems are solved in SimTK using two classes, the Optimizer class, and the OptimizerSystem class. The Optimizer class allocates an optimizer and sets the options for the optimizer. The OptimizerSystem class is used to describe the optimization problem. OptmizerSystem is an abstract class which has methods for computing the objective function, the gradient of the objective function, the constraints, and the constraint Jacobian. The application needs to define a concrete class that subclasses from OptimizerSystem. As a minimum the concrete class must supply an implementation of the objectiveFunction() method which computes the objective function for a given set of parameters. An instance of the concrete class is passed as an argument for the constructor of the Optimizer class. The Optimizer class uses the information from the OptimizerSystem concrete class to select the best optimizer algorithm and to allocate any workspace that the algorithm may need. The application can set various parameters for the optimizer such as convergence tolerance. Once the Optimizer has been instantiated and the options set the solution can be computed by calling the Optimizer classs optimize() method. Optimizer.optimize() takes a SimTK::Vector which sets the initial point the optimizer will begin searching from. If the optimizer is able to find a solution it will return the parameters for the optimal solution in the Vector. Optimizer.optimize() will also return the optimal value of the objective function. The code for using the Optimizer class to solve this problem is shown below. #include "SimTKmath.h" #include <iostream> static int NUMBER_OF_PARAMETERS = 2; static int NUMBER_OF_EQUALITY_CONSTRAINTS = 0; static int NUMBER_OF_INEQUALITY_CONSTRAINTS = 2; // user-written class class ProblemSystem : public OptimizerSystem { public: // Must provide this virtual function. int objectiveFunc( const Vector &coefficients, const bool new_coefficients, Real& f ) const { const Real *x; int i; x = &coefficients[0]; f = (x[0] - 5.0)*(x[0] - 5.0) + (x[1] - 1.0)*(x[1] - 1.0); return( 0 ); } int gradientFunc( const Vector &coefficients, const bool new_coefficients, Vector &gradient ) const{ const Real *x; x = &coefficients[0]; gradient[0] = 2.0*(x[0] - 5.0); gradient[1] = 2.0*(x[1] - 1.0); return(0); } /* ** Method to compute the value of the constraints. ** Equality constraints are first followed by the any inequality constraints */ int constraintFunc( const Vector &coefficients, const bool new_coefficients, Vector &constraints) const{ const Real *x; x = &coefficients[0]; constraints[0] = x[0] - x[1]*x[1]; constraints[1] = x[1] - x[0] + 2.0; return(0); } /* ** Method to compute the Jacobian of the constraints. ** */ int constraintJacobian( const Vector& coefficients, const bool new_coefficients, Matrix& jac) const{ const Real *x; x = &coefficients[0]; jac(0,0) = 1.0; jac(0,1) = -2.0*x[1]; jac(1,0) = -1.0; jac(1,1) = 1.0; return(0); } ProblemSystem( const int numParams, const int numEqualityConstraints, const int numInequalityConstraints) : OptimizerSystem( numParams ) { setNumEqualityConstraints( numEqualityConstraints ); setNumInequalityConstraints( numInequalityConstraints ); } }; main() { Real f; int i; /* create the system to be optimized */ ProblemSystem sys(NUMBER_OF_PARAMETERS, NUMBER_OF_EQAULITY_CONSTRAINTS, NUMBER_OF_INEQUALITY_CONSTRAINTS ); Vector results(NUMBER_OF_PARAMETERS); /* set initial conditions */ results[0] = 5.0; results[1] = 5.0; try { Optimizer opt( sys ); opt.setConvergenceTolerance( .0000001 ); /* compute optimization */ f = opt.optimize( results ); } catch (const std::exception& e) { std::cout << "ConstrainedOptimization.cpp Caught exception:" << std::endl; std::cout << e.what() << std::endl; } printf("Optimal Solution:f=%fparameters=%f%f\n", f, results[0], results[1]); return 0; } Output: Optimal Solution: f = 2.00000 parameters = 3.99998 1.999998 Sometimes the optimization problem has constraints on the parameters of the objective function. For example, the potential energy of a protein may be expressed as a function of the bond angles of the protein. Steric clashes may limit the bond angles of a protein to a range of valid values. Therefore minimum energy configuration of the protein would be constrained by these angles. It may be easier to express these constraints as limits on the values of our objective function parameters. This would cause the optimizer to search for a minimum only from within this feasible region. For example, if the parameter EMBED Equation.3 in our example had limits of EMBED Equation.3 our problem would look like the graph in Figure 4. These limits could be expressed by adding the following code: Vector lower_limits(NUMBER_OF_PARAMETERS); Vector upper_limits(NUMBER_OF_PARAMETERS); /* set limits on the parameters */ lower_limits[0] = 1.0; upper_limits[0] = 3.0; lower_limits[1] = -2e19; upper_limits[1] = 2e19; sys.setParameterLimits( lower_limits, upper_limits ); Optimal Solution: f = 4.000000 parameters = 3.000000 1.000073 Other Mathematical Tools SimTK provides a number of other tools for solving common mathematical problems. Examples include generating random numbers and finding the roots of polynomials. These tools are described below. Random Numbers (SimTK::Random) There are many cases where it is necessary to generate a set of random numbers, such as to drive a Monte Carlo simulation or to provide random initial conditions for a set of dynamics simulations. Most programming environments provide a random number generator, but they often are poorly suited to scientific applications. If the random number generator is not to bias the results of a simulation, it must have excellent statistical properties in terms of the distribution of values, correlation between successive values, and the length of the sequence it generates. 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"&7Ml5KPgi+JϻϻƵƞƵv)hfhi $B *CJOJQJ_HaJphhfhi $H *hhfhi $h\Zhi $B *OJQJphUh _HEhi $_HEhhi $B *phh\Zhi $_HEhi $%hghi $5OJQJ\mHnHu1hghi $5CJOJQJ\^JaJmHnHu+The sequences of numbers it generates may appear random, but if you reset it to its initial condition(or create a new random number generator instance), it will produce exactly the same sequence of numbers.If you need several random number generators that each produces a different random sequence, you can do this by initializing each one with a different seed value.Every possible seed value corresponds to a different sequence of random numbers that(in the case of a good generator) is independent of every other one.The SimTK::Random class is based on the SIMD-oriented Fast Mersenne Twister(SFMT) library.It provides excellent statistical properties, fast performance, and a very long sequence(219937-1).Never instantiate SimTK::Random directly.Instead, create an instance of one if its two subclasses, SimTK::Random::Uniform and SimTK::Random::Gaussian.These classes generate numbers according to uniform and Gaussian distributions, respectively.You can specify the minimum and maximum of a uniform distribution, and the mean and standard deviation of a Gaussian distribution.For example, to generate a sequence of numbers uniformly distributed between 0 and 100, you would write:Random::Uniform random(0.0, 100.0);Real nextValue=random.getValue();Roots of Polynomials(SimTK::PolynomialRootFinder) This class provides static methods for finding the roots of polynomials.There are specialized methods for quadratic and cubic polynomials, as well as general methods for polynomials of arbitrary degree.In each case, there are methods for polynomials with both real and complex coefficients.There are two different algorithms used by this class.The specialized methods for quadratic polynomials calculate the roots by explicit evaluation of the quadratic formula.They use the evaluation method described in section 5.6 of"Numerical Recipes in C++, Second Edition", by Press, Teukolsky, Vetterling, and Flannery.In addition, the method for quadratic polynomials with real coefficients performs an extra check to detect when the discriminant is zero to within machine precision.This helps to prevent round-off error from producing a tiny imaginary part in a multiple root.The methods for cubic and arbitrary degree polynomials use the Jenkins-Traub method, as implemented in the classic RPOLY and CPOLY functions.This is an iterative method that provides rapid convergence and high accuracy in most cases.For details, see Jenkins, M.A.and Traub, J.F.(1972), Algorithm 419:Zeros of a Complex Polynomial, Comm.ACM, 15, 97-99.Jenkins, M.A.(1975), Algorithm 493:Zeros of a Real Polynomial, ACM TOMS, 1, 178-189.As an example of using this class, the following code finds the roots of x3-6x2+11x-6:Vec4 coefficients(1.0,-6.0, 11.0,-6.0);Vec< 3, Complex > roots;PolynomialRootFinder::findRoots(coefficients, roots);cout<< "Roots: "<< roots<< endl;which produces the output Roots:~[(1, 0),(2, 0),(3, 0)] Notice that the coefficients are specified in order of descending powers.Also notice that the roots are always returned as complex numbers, even if the coefficients are real.This is because a polynomial with real coefficients can still have complex roots.ReferencesAPI:Application Programming Interface, i.e., a programming library.You can also select elements with the more C-like construct m[i][j], but that requires two operations(one to select the row and one to index that row).The m(i, j) form is substantially more efficient and should be used instead.Better yet, use vector and matrix operators rather than accessing individual elements whenever possible.The figure does not show the internal trial steps that an error controlled integrator would have taken and rejected along the way;only the accepted steps are shown.In practice this is not strictly true since an integrator may perform a little rounding off of internal steps to avoid having to interpolate over very small slivers of time.PAGE\*MERGEFORMATiiPAGE\*MERGEFORMATiiPAGE\*MERGEFORMAT32PAGE\*MERGEFORMAT31000(10(1FigureSEQ Figure\*ARABIC3:two intentionally-discrete event trigger functions.FigureSEQ Figure\*ARABIC2:a continuous event trigger function appears discrete in finite precision arithmetic.0(1-1(0-1(0Boolean trigger functionTrigger function with deadband0(1-1(0-1(1EMBED UnknownEMBED UnknownEMBED UnknownComputational function changes discretely;stays zero for finite intervalMathematical function changes continuously;crosses zero at a pointC2C1Optimal SolutionFigure 4:LimitsEMBED Equation.3are shown in green.The two constraints are shown in red and the feasible region is show in grey.The contours of the objective function are shown as dashed blue circles.EMBED Equation.3EMBED Equation.3C2C1Optimal SolutionFigure 3:Graph of problem to finding optimal solution to the problem described above.The two constraints are shown in red and the feasible region is show in grey.The contours of the objective function are shown as dashed blue circles.EMBED Equation.3EMBED Equation.3Figure 2:The optimizer begins searching along the objective function starting at the Initial point.It moves first to point P1 then point P2 which reduce the objective function.Eventually the optimizer terminates at the Local minimum when it cannot reduce the objective function any further.Objective functionP2P1Initial pointLocal maximumGlobal maximumLocal minimumGlobal minimumtn+4tn+1tn+2si+5si+4si+3si+2si+1sireporting timesinternal stepsteventtn+5An event occurs hereFigureSEQ Figure\*ARABIC1:Numerical integration over a continuous segment of a time-stepping trajectory.The continuous region starts at tn and terminates at tevent.Each tall green line represents a return of control to the time stepper:thin dotted ones are caller-requested reporting times;the thick dotted green line is an unrequested return of control just prior to event occurrence and 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A B B D D E !F C "G #H #H $I &K 'L &K 'L &J &K 'L &L (M (K (M (M 'M (L (M 'M &J 'M &J 'K (L (L (N (N ) M ) M (L ) O *N +L ) M ) M (L $J !F C > :6 4 1 0 0 ../ 2 4 7 8 9 8 5 7 3 1 0 (F n , L 3 6 5 4 4 5 5 6 :> @??9 7 5 5 7 7 ;:;;;;;< < =< < < ===?=> ?> => > > > > > > > > @?> =;> > > @@A B B B B D E !F "G #H #H $I %H &K %J 'L 'L 'L &K (M 'L 'J 'H (M (M (M 'K (K (M &J 'K (L 'K (N 'K )O (N (L )O )O )M )O )O (O *N *N *N )M 'M $I E @ = : 7 4 1 0 / - . 1 0 3 6 8 9 8 6 3 4 2 5 :R ȋ 4U 4 7 5 6 3 5 5 5 9 ; = ? > : 7 5 5 6 6 : : : ; : : ; ; < < > = > = < > ? = > > ? ? = > > > > ? > @ ? @ > @ ? ? > ? ? @ > @ A B B C C D "G !D "E $I #H %J &K $L &K &K &K &J 'L 'J 'L 'J (M &J (M (I (M (M (P (N )N )N *O (L 'K (N *O (L )M +Q )M )M )M *N *N *N *N *N *P *N (L &G #H F A < : 6 3 1 0 1 0 / 2 1 3 7 8 6 6 4 4 4 3 6 Nc ԥ ;[ 3 7 6 4 6 5 6 7 7 : ? @ > : 7 5 5 5 7 : : ; : < ; ; < < < < = > = ? = > > > > > @ ? ? ? = = ? @ ? ? ? ? ? @ ? @ ? ? > ? ? @ A A B D E E !F !F "G #H $I %J &K &K &K (M &K 'L &L 'L &J 'L (M (M (M (M (M 'J (K *O ) N (L (L +P ) L (N (L ) O ) O ) O *P *K (L +P ) M *N +O +N +O *N +O , P +O ) O &J !I E @< 8 5 2 1 /// 1 /2 6 8 8 8 5 6 4 3 2 =fy Ca 2 7 4 5 4 4 5 6 6 :??> ;8 7 6 5 7 :::;:;;;< < < =< =< ;< => ==> ?@A ?A @@@??@@@@@@?@> @?@A B B D D D !F "G "G #H $I %J %J %J %J &K 'L 'L 'L 'L 'L 'J (M &K 'J (K (M ) N ) N ) N (N (L (O +P +N +P *N *N ) M *N , Q *N *P *N *N +S *N *N *N *N *N +O +O , P +Q , P ) M &K !F C ;:8 5 2 0 1 / 1 . 3 5 6 9 8 9 7 7 4 3 2 +F À Pj 2 7 5 2 4 5 4 5 6 8 > @@< 8 6 5 6 6 8 ;:;;< ;< < < ;< =< ====> > => ??@@?@@@A A A A @A ?@@B A @A A A B C D D E !F !F "G #H #H $I %J %J %J &K &K %J %J 'L 'L 'L (M (M 'L 'J (M (K 'J (K *O (N )N (L *Q *O )N +P *O )M )M )M *N *P *P *N ,Q *Q *Q +Q *N +O +O +O +O ,P +O ,P +Q *N )O $H !F A > 9 7 3 2 0 / / 1 0 1 5 8 8 8 8 7 4 3 1 3 G_ Κ ]x 1 6 4 6 4 3 4 6 8 : = @ @ = : 7 6 5 7 9 ; ; : : 9 ; ; < < < < < ; = < = > ? > > > > ? ? ? @ > @ A @ A A A A A A A A A A B B A A C B C C E D !F !F "G $I $I $I $I $I &K %J %J %J &K &K 'L 'L &K (M (M (M 'L (M (P ) N *O ) N +M ) N *O ) M ) M *N +P , Q +P ) L *N (L , Q +N +O +Q *P , P -N , R , P +Q , P , P , P , P -S , P -Q , P ) M ) N %J E @> 9 6 4 2 0 1 //1 5 6 7 6 6 4 6 4 3 2 =`s ܶ h 0 6 6 6 4 6 4 5 5 9 < ?@> :7 6 6 7 :< 9 ::::;;;=< =< < ==> > > ====?@?> A @@@A A A @A A A A B A @B B B B B C C E E D !F #H #H $I %J $I %J %J %J %J %J 'L 'L 'L &K %J (M (M ) N (M (M (M ) N *O *O ) L ) Q *M +P *O +P *P +P *N ) J -R , O -R *N +O +O *N +O +O +Q , P , P , T , P , R , P , P .O -Q .O -Q -Q , P , Q ) N #H D @< 8 5 1 2 .. 1 0 1 6 7 8 7 8 7 6 5 3 3 , I y t 1 8 7 6 4 5 4 5 5 8 < ?@?;8 6 6 7 9 9 :9 8 ;:;< < < ;;=;< ==> > => ?> =????@A A @B A B B A B C B B D C C D C B C D E E E !F "G #H #H $I $I &K %J &K %J &K &K 'L (M %J 'L 'L (M 'J 'L (K )N )N )L )N *O )N *O *O +P *M )L )M +P )M +N *N ,O *N +O ,Q +O +O +O +O ,P ,P +Q ,P ,P +O -S -S -S -Q -S /S -Q .R .O .R -Q +O (M #H B ? ; 7 4 1 1 0 0 0 0 4 5 7 7 8 8 4 5 5 2 5 ?X ɓ ȴ ˃ 0 7 4 2 4 3 4 4 5 9 : = @ > < 8 6 6 6 8 : : : : ; : ; < ; ; ; < < < < < < > > < ? = ? < = @ ? @ > ? @ @ A A A A B B C C A C B D C D D C C C C "G E !F "G #H #H $I #H %J $I %J %J %J &K &K 'L %J %J (M (M (M 'J (M )N *O )N )N )N )N *M *O )N ,Q +P *O +P ,O +P +P +S .S -R ,Q *N +Q -R +Q -S .S .S ,P -Q +O -Q ,P ,P ,P -Q -S -Q -Q .R .R .R .T .R ,P *N &K "G B ?:7 2 2 1 2 0 0 3 4 5 7 7 8 7 6 5 4 3 ;Xj Yc_'`(2p=v} ё 3 9 5 6 5 4 4 4 8 6 9 ?@?< 9 7 6 7 7 8 8 ::9 8 ::;;< < ;;< ==> =?> > > > ??> @@@@@@A @A A C C C B B C C E C D E C D E D !F !F #H #H $I #H $I $I $I $I $I %J %J %J &K &K %J %J 'L &K ) N (M (M (M ) N ) N (N ) N (P (M *O +P *O +P +P *M +P , Q *R +N , O , O +O , P -R -N , P , R .S , Q , P .S .Q -N -Q .Q /R -Q -Q .T .R /U /S .R .R /P /S .R .R .R *O %J !F @=9 6 6 2 0 2 0 2 1 4 6 5 8 8 6 5 3 CB-`(`(_'b(_'b *@yL מ 5 6 9 6 7 5 3 4 5 6 :< B A =:7 7 8 9 9 :9 :::;;< ;< < < ;< < ======> > ????@A @@@A @@A B A B B B B C C D D D E E D E D E "G #H #H #H #H $I $I $I #H "G $I %J %J %J %J &K 'L &K &K &K $L 'L 'L (K 'O ) N ) L *O (M ) N *M ) L *O *O +P *M +P , Q +P +M -R , Q , Q -R -R .Q +Q -Q /T , P , P -Q .O .S -Q -Q /R /T -Q .T .T .T .T -Q /S .R .R 0T /S /R .R , R ) N $I C ?< 9 7 3 1 1 .0 0 2 6 6 7 7 T84b *b)a(a)`(_'`(`(_'g/U\í ܧ 7 6 7 4 5 4 4 4 5 6 ;?B A > ;7 6 5 8 9 9 :9 9 :;;;< ;< ;< < ====> < > > ?> =???@@@?@A A B A B B B B B B D C B C E E E E E E E !F !F "G #H #H #H #H #H $I $I $I %J %J %J %J &K %J 'L &K 'J 'L 'L 'L 'L (M )N )N )N *O )N )N +P +P *O *O *O +P -P +P ,Q -P ,Q ,O ,O -R ,O -P /T .S -Q -S +O ,P .Q /W /T .Q -Q -S ,P .S .O .R /S .T -Q .U .T .R .R /S .U .R .Q +M 'L "G B > ;7 3 3 1 3 2 0 2 2 k) 3`+c *b)b+c *b)_'`(_)`(a)^&_'&i3 ou :5 7 8 5 3 4 4 4 7 9 > @??;9 6 6 7 9 9 ::9 :;;:;;=< ;=< < ==?=< > ?> > ??@?@A @A B A A A @B B B C C D D D C D !F D E !F !F "G !F "G "G "G $I #H "G #H $I #H $I #H %J %J %J $I %J %J &K %M 'L &K 'J (M 'L 'L (M (M 'L )N )N *O *O *O *O +P *O +P +P +P ,Q +L ,Q ,O ,Q ,O -P -P ,O -R -R +L .S -P ,P -P .S .S ,P .Q -U ,P -Q -N ,P .O -Q -S .O -S /S 0T /S /S .R /U -Q ,O *O $I E = < 9 4 3 2 0 / x. `- d+ d* c* d+ c* d+ b+ a* `( a( `( `( a* a) `( `( 6tC > 6 6 8 4 5 3 4 5 7 9 < A @ ? < 8 7 6 8 9 9 9 9 8 ; ; ; ; : < ; ; < < < < = < = > = ? = > > ? @ > ? @ @ @ @ ? A A B B @ B B B B B C D C D E E E E "G E E E !F !F !F !F #H !F "G "G #H "G $I $I $I %J $I $I %J %J %J &K #K 'J &K 'L 'L 'L (M (M (M )N (M *O )N )N *O *O *O +P +P *R +N *O +P ,Q +P ,Q ,Q ,O -R -R ,Q -P -P -R -R ,O .S -N -P ,P ,P ,P -Q -N ,P .R ,P .O .R .T .R .R .R .R /P /U .R 0U /Q *O &K "G A ?;7 5 2([0h0g-g-f,f.d+d+c *d)c *`'b)b(`(`)`(a)`(_'d-KV !C 4 6 2 5 3 4 4 4 5 9 ;@A ?=9 7 6 7 9 9 9 9 9 9 :9 9 ::;;=< < < < =< > ===> > ??@?@?@A A @A A A B B B C A C C C C C D C E D C D D E E !F E !F !F !F !F !F "G "G #H $I #H $I $I %J $I %J %H %J &K %J &K %J &K &K &K &K &K 'L (M 'L (M (M (M ) N ) N ) L *M *O +P *O *O *O +P +P , Q *M +N +N -R , O , Q *N *N -R , Q -R +Q , O -R +O , P -P , P -P .S , P , P +O , P .S -Q /T -Q /P /P -Q .Q /R .O -P +S (K #H C @:?S7i0j0j0h.g-g-h-e,d,d+c.b)c *c *b)a(b)`)`(`(`(`(`(%j4 fo ˸ &E 4 9 2 4 3 4 4 4 4 9 ;?A ?;:8 6 7 8 9 9 9 :::9 :;:< ;< < ;< < < < =====?=??????@@@A @A A B B A B ?C B B C C C C C B C D D D C D D D !F E !F !F "G "G "G "G "G "G #H #H #H "G #H %J $I %H %J #H &K %J %J &K &K &K 'L &K 'L (M 'L (M )N )N 'O )L )N )L +P *O *M +P )L *O *M +N *N +P +P +P *N ,Q ,Q -R +S ,O ,Q -R -P -P .S ,R ,P ,P ,P ,R ,R +O /T .R -Q -Q .Q /T /T .S .S ,Q .S +P +R 'L &I TM; m3 o4 m3 j3 k1 k1 i/ g- g/ g, e, e- d+ c, d, d+ b* b+ a* a( a* a* `( `( _( _) a& 3q@ w~ -M 3 8 7 5 3 4 3 6 7 8 = = A @ > ; 8 7 7 9 8 : : 9 ; 9 : : ; : < ; ; ; < < < < < < = > > > = = ? ? ? @ @ ? @ @ ? @ @ ? @ A A @ A A A B A A A B B A B B B C C C C C C B D C E E !F !F E E !F "G "G $I "G "G #H $I #H $I $I $I %H %J %J $I %J %J 'L &K &K 'L &K (M 'L (K (M 'J )N )N )L )N *O *M )N )N +P *O *O )N *O ,Q +N ,Q )M )M +N *N ,Q -R ,M ,Q +O ,P -P -P +O ,Q -P ,P ,P .S ,P .S /T /R .S -P -R .S -R ,Q ,Q *O rIE n3 n3 m5 n6 n4 m5 n3 k3 k3 j2 i/ h. g, e. d, e, e- d- c* d) a) a( b) a( `( `( `) a' _' _' c, D}P 7U 3 6 6 4 5 4 3 4 6 8 ; = B @ > ; : 7 7 9 9 9 9 : : : : : : ; ; ; ; < : < < < = = < > > = = = > > > > ? ? > > ? ? ? ? ? @ @ @ @ @ @ @ @ ? A ? A A A @ A A B @ B B A A C C C B D D !F "G !F E !F !F !F "G "G #H #H #H $I $I #H $I $G $I $I $I %J %J &K %J &K (M 'L &I &K 'J 'L (M (M ) N (K ) N 'J ) L ) N ) L *O 'K *O +K *O +P *N +P ) M , Q *N *R , Q *N , Q *N *N +O +O *N +Q +O -R -R .S .S -P .Q -R /R .S .S , Q , Q , Q +R ?H!k6n8m3n6n6n4l4m5m3l4n5k1j0h3j2i-g,f.e+f,d+c *c *a+c *b+a *a(`(`)`)`(`(`("g1 Wa ű ?^ 1 ; 6 5 5 4 4 4 4 : : > @ A > < 9 7 7 7 8 9 9 9 9 : : : : ; : ; ; ; ; < < = = < < = > = < > = = > = > = = = = ? > ? > ? > > ? @ ? ? > ? ? ? ? @ @ ? @ @ A A A A B A A B @ C C C D D D D E E E !F !F !F E "E "G "G "G #F #H $I #H $I $I %J %J %J %J %J %J %J %J &K &K 'L 'L 'L 'L (M (M 'L (K 'L (P )N *O )N (K )L *O 'K )M )L )O +P )M (L +P (L +P *N )M )Q *N ,O ,Q ,R ,T -R ,P -R *N -P .Q .S +P -R ,Q *O 5J *h7 o4 l2 m5 o5 o4 k1 p6 n6 o7 n4 n4 l2 n5 j1 j0 i1 i/ h- h. f, f, f, d, e- c- d+ b+ c) c( a* `( `) `) _' `( `( `( 'j5 nv Ge 3 7 7 5 4 4 3 4 4 8 : ? A A ? < 9 8 7 8 8 : 9 9 : ; : : ; : : ; ; ; < ; ; ; ; < ; < ; < < < < < < = < > < = = = = < = = > > > > = > ? ? > ? ? @ ? @ > > ? ? @ @ @ @ A B A A A B B B B C C C D E C !F !F !F "G "G "G "G #H #H #H #H %J #F #H %H $G $I $I $I %J %J &K %H &I %J &K &I 'L 'L (M (M 'J 'O (K 'K 'K (P *O *O )L *O )N )L *R 'K *K )M *O (L ,Q )M )M )M +N +N ,Q ,O *N ,P +N +O *N ,O +S +P +P ,Q /L =a= o5 p5 o4 o4 m5 n8 m5 n4 n3 n3 m3 n4 l2 n3 m5 n5 m3 l2 j0 h0 i1 h. f. g- g/ f. e, d, c+ c* a( a* a( a* `( `( `' _' `* _' _( 8vE Xs 2 7 6 5 4 4 3 3 3 7 : = A A > = ; 9 8 8 8 9 : : : : : 9 9 9 9 : : : ; : ; ; ; ; ; < ; ; < : < < < < < < < < ; = < < = < < = = > > = = > > > > ? < = > < > > = ? ? ? @ @ A A @ A A A A A B C C D D D D E E !F !F !F "G !F "G "G #H #H #H #H #H #H $I $I $G $I %J #H %J %J %J %J &I (M %J $G 'L 'L (M 'L (M (M 'J 'K 'K (M ) L (P ) L (K ) N (L (L ) M *O +P (L , Q (L *R , Q +N *O +P , Q -R , Q +P +P , Q , Q *Q +Q KX >o4n6n4m5m3m5n5l2n4m3l3k6m5l4n4m3l4l2k6m5l4k1l1j/i/g/g/f.f,d *c.d+d+c+b)b)a(a(_'b *`(`(_ &_'c+S^ b{2 7 7 5 4 4 3 3 3 8 :< A A @=;9 8 7 8 9 9 9 :9 9 :9 9 9 :9 9 :9 :;:9 :;:9 ;:;;:< ;;;:;=< < ;< < < < < < < < ;< ;==< < ==< ==> ??> ??????@@@A ?A B B C A D C D E !F !F !F E "G "G "E "E "G #H "G $I #H #H "G #H #F #K #H #K $I $I $I &K &K &K %J &K &I &K &L 'L 'L &I (M (M (M &J (M (M (P (M (K (N )L )N )N )L )N +P ,Q *O +P +P ,Q +N *O *P +O *O +P +O bOA m3 m3 q9 o4 m3 l4 o7 p6 m2 m5 o4 m3 n6 o4 m3 k3 m2 l2 m3 l2 n3 m7 m3 n3 k1 j0 i1 j/ h1 g/ h. g/ e. d+ d+ d+ d+ c, c* a* b) `( `) _( `( `( `( ^' &i4 gp ξ q 1 5 7 5 6 4 3 3 3 6 9 > @ B A ? ; 9 8 8 9 9 9 8 8 8 8 9 9 7 9 9 : 9 9 9 9 9 9 9 9 : : 9 : : 9 8 9 : : : : : 9 ; 9 ; ; : ; : ; ; ; ; ; ; < ; < < < < < = < < = = = > > = = > > > ? ? @ @ > @ A ? A A A B C B C D E E E E !F D !D "E !F "G !F "G "G "E "G "G #F $I $I $I #H $I %H $I %J $G %H &K %J %J &K %I %I &K 'J 'L 'L 'L (M (M 'K 'K 'K (K ) N ) N ) L (L ) N ) N *O ) N ) L 'L *O ) L 'L *O *O (M ) N CHm7n6m3o7m5o7n4n6m2m5n3m3m5m3m5m3m5k2m5n4q6o4o5l2l2m3n4m5m2j2i/j0i/h-g-g/f/d+e,d,b)b+a)`)a(`*a *_'^&`(_'^'6tC w 0 6 7 5 6 5 4 3 3 6 9 > A @A ?< :8 9 9 9 9 9 :7 5 8 7 8 8 7 8 8 8 8 8 8 8 8 8 8 8 9 9 8 8 9 9 9 9 9 :9 ;9 :9 9 ::::9 :;;;:;;;;:< ;< < < < < < < ===< => ?=??> @> @@A A ?C B B C C C D E !F E !F !F !F !F !F E "G "G "G "G "G #F !F #H #H #H #H #H "C #H $G %H %H %J %F &K 'L 'J &K &K &I 'L 'J 'L &J &J (M (K ) L %K (M 'K ) L (K ) N (K ) L ) N (M (M ) N (M (M 'L 9I &j7n6o5s8o5m3o3n3o4m3o6n4o5n6l2n3l2m3m3n8o5o4n6n4n4n6l4l4n3n6m5l4n4l4i/i/g/g/e-e-e-e-d,c *c *b+b+a *`)`)_'_'_'_'_'d-JS χ 3 7 7 6 4 3 3 3 2 6 8 ;@C A > < 9 8 8 9 8 9 9 8 8 7 8 6 7 6 5 5 7 7 7 7 7 8 7 7 7 8 8 8 9 9 7 8 8 9 8 8 8 8 :9 9 9 9 8 8 8 9 8 9 :8 ::::;9 9 :8 9 ;;;:;< < < < ===?> > > < ?> > @?@@@A A C A B B C B D D !F E E E !F "G !F "E C !F !D "G #H "G $I #H #H $I $I $L $I $I $I &K $G %J %J &K %J %J &K &K 'L %I 'L 'L &I &N 'H 'J 'L &J (M (M (K 'L (M (M (M 'L 'L &K 0J 2e:p5p8n4q6m3o4q6o4n6o4o7n4n6m5n3o5n3o7n6l3o7m3o4o5m4l6l2p5m3n4m5k1n6l4l2l1j0i/h0h0f.f.e,e-c.c *c *b)a)a(`)a(`)_+_'_'_'_'$g2 Zf Ƴ ֗ 3 6 7 6 6 4 3 3 3 5 8 :?B A ?> ;9 :9 9 8 8 8 9 7 7 7 7 7 6 6 7 5 6 7 6 6 7 8 6 6 7 7 8 6 7 7 7 7 7 8 7 7 7 9 8 7 9 8 8 7 :8 8 ::8 8 8 :9 :9 9 8 9 9 ;::9 < ;< :;< < ===> ==> > > => ?> @@@A A A C B A A B C D D C D !F !F E C !F !F E !F "G "G !F #H "G "E $I $I $G $I $E #F $G %H %J %J %H &K %H $H %J $H &K &I 'L %F 'L &K 'J 'L 'J 'N 'L (L 'L 'L (L &K +I C^=o7n4o7p5p5o4r< r7m5n6n4m5m5o7q9o7p5n6n4p:p7n3m3m3n5m3o4n6o4m5l2m7m3n5p4n3l4m5l2k0k1i/i1f.g/g/f-d+e-d,b)b+a *a(`)`)`(_(`(_'^&^'.n< v| ڦ 5 3 7 6 4 4 1 3 2 4 8 :< A A A @;:9 9 9 8 9 9 8 8 8 8 8 7 7 7 6 6 6 6 6 6 6 6 6 6 5 6 5 7 6 6 6 6 7 5 8 7 6 7 7 6 8 8 8 7 8 8 8 8 8 8 8 9 8 7 :8 ;:7 7 9 :8 :::;:;:;;< ;< > ===> =????> ?@@A @A B A B B B C C C D E D E D D !F !F !F !F "G "G "E #H #H #H #H #F $I $I #H $I #F $I $G $G %J %J &K %J %J %H %H &K %H %K %H &K 'L &K %J 'K %J 'K &K %G (K WR> n4 q6 p8 p5 n6 o5 o6 n6 m3 o4 o4 n3 n6 m5 o4 o7 o7 n4 n6 n4 m3 n6 n6 n4 l2 o7 o7 n8 n6 m3 o4 o7 m3 n4 m1 m3 m5 m3 n6 m3 l2 l2 j2 j0 j0 i1 h. g- e0 e- e+ c+ d, d- c* b+ a* a) `) `' `( _' _' ^( _& _) ;vH 6 5 9 7 6 6 4 4 3 5 5 ; @ A A @ ? < : : 9 : 9 9 8 7 : 8 8 8 6 6 7 6 6 6 5 6 5 5 5 5 5 3 5 5 5 5 6 7 6 6 6 5 7 6 6 6 6 6 7 6 6 6 7 7 7 6 7 8 7 8 8 8 8 8 8 8 8 9 9 8 : 9 9 : : ; ; : ; : ; ; ; < < ; < < ; = = > ? = ? ? ? @ ? @ @ A A A A C B D C C C D D E D !F E !F !F !F "G "G !F "G "G "C #H #F "G "J #H $I $I $G #D #I $G $G $E &K %H %J %J %H %H &K $I %J %J $I $I &J &J 'K hE?n4p5o4o7q6o4m3m3o7n4r7n6p4q6o4o4n8n6p5m3o5o7n3o4n3m5o6n4n4p8m5o3o6o7o7m5n6m5n3l2l2n3l2n3l2j0j2j2k1h.g/g/g.d+e-e,e-b *b *`)a *`)`)_'`)^(]'_'^'b, T^® =5 6 7 6 3 3 3 2 3 6 9 > A B A ?< :9 9 :9 9 9 9 9 9 :9 6 8 7 7 7 7 6 6 6 7 4 5 4 4 4 4 5 5 5 5 6 5 5 5 5 7 5 5 5 5 6 7 5 6 5 5 7 7 6 7 7 6 6 6 7 6 8 8 8 8 8 8 9 8 9 9 9 9 9 9 ::8 ;;;< ;;< :< < ?===< > < ==?@@?@B B A D B C B D D B C D E !F E !F H !F E !D !F "G "G "G "G !F "E "E $I $I #H #H "G $I $G $G $I $G $I "H $G $I $I $I $I $I %I $H &J< Em5q6q6r7p5o4n6q9o4o7p6p6m3m5o5o9p7n4p8n4n6p5p5n6n6p4m4k2n6p5n4n4n3m5n6m3n4m4m2o4l1l2l2l3m3k3l2l4m3l4j0g-j2i/g/h0f.d,e-d,c,a-b)b *a *a(`)^(^(_'_'^&]&#\1 u{w A 4 7 8 5 4 3 3 3 3 6 9 > @B A ?=;:9 ::9 9 :9 9 9 8 8 8 7 8 7 6 6 6 7 6 5 5 5 6 4 4 4 4 4 4 3 4 5 5 4 4 4 4 4 5 6 6 6 5 5 6 5 4 6 6 6 6 5 7 5 9 6 5 7 7 6 7 7 7 8 7 8 8 9 9 9 8 :9 ::9 ::;;< < ;::=< =< ===> > ?> @@A A B A B B B C D C D D E E !F !F !F !F !D G !F !F !F E !F "E E "G "G "G "E "E #H "G #H #F #H #H #F $I #H #H "G "G "G #G #E 0C+i8o7o7n6q6n6n8q6p:o5p5p5n5n3o4n4o7n4q7p5n6n4o4n4m5o7p5o7m2n4n4p5o5o7o6p8p5p5o4m2l4n6l7m3l2m5m5m5m3m3o7n4l4m2i1h.h0i1f/e0g/d,g/c.b+c *a *`)`)a *`)`'`(Y%B<> &H 3 6 8 7 5 4 3 3 2 3 9 < @B B @> ;;::::8 9 9 9 9 9 8 7 9 8 8 7 7 7 6 7 7 6 6 5 4 5 4 5 4 5 4 4 3 4 4 4 5 4 5 5 5 4 4 5 5 5 5 3 5 3 5 6 5 5 5 5 6 4 6 6 5 7 6 7 7 8 7 8 8 9 9 7 9 8 8 9 9 8 9 ::;::;9 ::< < ==> =< > ?@???@A A B A B B C B C D C B D B !F E E E E E !F E !F !F "G !F #H "G "J !D #D #H "E "G #F "J !F "G "E #J #G #G $H $H *E 7a9o7p5o3o7o4o4o7q6t9q6o5p5p5m6o7n6p7o5p6p5q6m5n4n4n4o7o9m3o5o4p5q5o4n4p8n4n6m3l4m3m8n4n4m5l2l2m3m2l2m5m3m5k1m5k6l2k3l4j2h0g/f,g/e-e-e-e,b)c+b)a *`)`*/+s(~5 *I 4 8 6 7 5 4 3 3 4 5 5 < @B B @> =< ;;:9 9 9 :9 9 9 9 9 9 8 8 8 8 7 7 7 7 6 6 6 5 5 4 5 5 4 4 4 5 4 3 4 3 4 3 4 4 4 4 4 4 5 5 4 4 5 4 4 5 4 5 5 6 5 4 6 6 5 5 7 5 6 6 7 7 7 7 7 8 7 8 8 8 8 8 9 9 9 9 9 :::;;9 ;;< < < < ==> > > =?@@@@@A A B B C C C C B D D D D E D !F E C C !F C !D !F #H !D !D "G "G !F !F "G !G "G "G "F "F !E !E $H FV; p8 q6 p8 p5 r7 o7 p5 o7 q5 p6 n6 q8 p5 n6 n6 p5 o4 m4 p5 n4 q9 n3 o3 n4 o7 p5 l2 n6 o4 o7 o4 m5 p5 p5 n6 o4 o4 m3 m3 m3 l2 m3 m3 n6 m3 n6 l5 l2 l2 m3 n6 l4 n6 m5 l4 m3 l4 m3 k1 m5 k1 h1 j2 h0 f. h0 f. e- e- d, c* c+ b* =! p) 1 2 0 3 I_ ϡ 1Q 2 6 8 7 5 4 3 2 2 5 6 : ? A B A @ > < ; ; ; : ; : : 9 9 9 9 9 9 : 9 9 7 8 8 7 7 7 6 6 5 5 5 5 5 4 4 4 4 3 4 3 3 4 3 2 3 4 3 3 3 3 3 4 3 4 4 4 3 4 5 3 4 4 4 6 4 4 5 5 6 5 6 6 6 6 7 6 6 7 8 8 7 8 8 8 8 8 9 9 9 9 9 : 9 ; 8 < : ; < < < = < = = = > > > ? ? @ @ @ A A A @ B B B B E @ C D D D D F E E C A C !F H !F E E !D E E E !H !C E !E "F "F C XK; p6 s8 r7 q6 o5 o7 n4 n4 m3 n5 r7 q8 p5 p5 n4 o5 p5 o5 r7 p5 o7 n6 p5 o9 o7 p5 p5 n4 o7 q6 q9 o5 p5 p5 o5 l3 o3 o7 l4 n4 m3 m5 m3 m5 l2 m5 m5 l4 o5 m3 l4 m3 l2 m3 m5 l2 m5 n4 n4 m2 l2 o5 l2 l2 l1 i1 h0 f1 g/ g/ f. e- f, P& d' 4 3 3 2 2 2 . ; `t ۺ :X 3 7 8 7 6 3 2 3 2 5 7 8 > A C B @ > ; < ; : : ; : ; : 9 9 9 9 8 8 9 9 9 8 8 7 7 7 8 6 6 6 5 5 4 4 4 4 5 4 3 3 4 3 3 3 2 1 2 2 2 2 3 2 3 3 3 2 4 3 3 3 3 4 4 3 4 4 4 6 5 5 5 5 5 5 6 6 6 6 7 7 6 7 7 7 9 7 8 7 8 8 9 9 9 8 9 : : : : : : : < < < = < = = > > = ? ? ? @ @ @ @ A B A A B B B B C C B D B B D D C D B E C E E E E D B D D E !D A q?@ m4 r: n4 q6 o7 p6 n5 p5 o4 p5 p5 o5 o5 m3 o4 q7 q7 o5 q6 o5 p8 n4 o7 o7 o5 p5 o4 o7 p8 o5 o4 p5 o7 o7 n4 m3 n6 o4 n6 m3 k7 m3 o7 o5 m3 m3 l2 l4 m3 l2 o4 n4 m3 n4 o5 p5 o4 o6 m5 l2 l2 m5 n4 m5 n4 m2 l4 k3 j0 i1 i1 g/ f. _- M$ 8 8 8 7 6 4 4 2 2 1 / )E y Ea 1 8 7 7 5 4 3 3 4 4 6 9 ; @ C C > = = < ; ; ; ; ; : : : 9 9 9 8 9 9 8 8 8 8 7 8 8 7 7 6 6 5 5 6 4 4 4 5 4 4 3 4 4 3 3 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 4 3 3 4 4 4 4 4 5 5 5 4 5 5 6 6 5 5 5 5 7 7 6 6 7 7 6 8 8 8 9 9 7 8 : 9 : : 9 9 : : : : < < < < = = = > > = > > ? ? @ @ @ ? A A A A @ A C C C D C E D D C D D D D D D C D D C C C 1A 'm9 p6 p6 n4 n6 p8 p6 q6 p5 o5 q6 q6 o5 p6 m3 p6 o5 n4 n3 n3 m4 q6 n4 q7 n4 o7 n4 n3 p6 o7 n6 o4 l2 n6 n6 o4 n4 p5 m3 n6 m3 o7 m3 m5 l4 p; l2 n4 p6 n4 m3 o7 m5 n4 m5 m3 l3 l4 n6 n6 o7 n6 n4 l4 n4 l4 l1 m3 l2 l2 m3 k3 j/ i1 6% 2 6 9 = : : 8 7 6 4 3 2 3 0 / ;V ɗ So 0 7 8 7 6 4 3 2 2 2 4 8 ; ? B C A @ > > < < ; ; : ; : : : : 9 9 9 9 8 9 9 9 9 9 8 7 7 7 6 5 6 6 5 5 5 5 4 4 5 3 4 2 3 4 2 3 2 2 2 2 1 1 2 2 2 1 2 2 3 2 2 2 3 2 3 3 3 3 3 2 4 4 4 4 5 5 4 5 6 7 5 4 5 6 6 5 6 6 6 9 6 8 7 8 9 8 : 8 8 9 8 9 : : ; : ; ; ; < = < = = = = > > ? ? > ? ? ? @ ? @ @ A B B C C C C B C D A C C A C D B C C C B C (B -d: m3 o5 p8 q6 q6 s8 t9 q6 r6 n4 p8 q7 n6 q7 o5 r7 o5 m3 !r7 p8 n5 n4 o5 o7 q8 p5 o5 m3 p6 n6 n6 m3 n6 o5 m5 m3 p8 n4 n6 n4 m3 n4 q7 m5 l2 n6 m3 n6 n4 n6 l7 n4 m5 m2 m3 m3 m5 m3 j3 m3 m3 m5 n4 n4 m3 n6 l4 l2 m5 n4 o7 m5 m5 $:# y * 1 3 4 8 9 < 9 : 9 8 4 6 3 2 2 1 0 2 Ui ׯ ^w 0 5 7 7 6 4 3 3 4 2 4 7 ; ? A B @ @ > > = ; < < ; ; ; : : : 9 8 9 9 : 9 9 8 8 7 9 8 7 8 6 8 8 6 6 6 4 5 5 6 4 4 5 4 4 4 4 3 3 2 2 2 2 1 1 1 / 1 2 2 2 2 1 0 1 2 1 2 0 3 4 3 3 5 4 4 3 3 4 4 4 5 4 6 6 5 5 6 6 6 4 6 7 5 7 7 8 8 8 7 6 8 8 8 9 9 : : 8 9 : < < < < < < < = < > < > = > @ > ? ? 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d 8 | = I L I I J | @ A7 C7 C7 C7 C7 C7 C7 D< > ` C7 18 I ˟ 0 I I C7 Y# Y# I J U 8 - - - I Y# I 18 J A7 |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I | |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I | Website |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael | Sherman |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby | granted |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of | charge |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without | restriction |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to | use |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to | copy |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to | modify |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to | merge |
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F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the | Document |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do | so |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following | conditions |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY | KIND |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR | IMPLIED |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF | MERCHANTABILITY |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE | AUTHORS |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY | CLAIM |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER | LIABILITY |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF | CONTRACT |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR | OTHERWISE |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING | FROM |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK | http |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events | occurred |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the | window |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some | events |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is | required |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made | narrower |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made at the cost of increased computation time Once an event has been localized to an acceptable | tolerance |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems(SimTK::FactorLU)PAGEREF _Toc251069770\h5HYPERLINK\l"_Toc251069771"3.2 Linear Least Squares(SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made at the cost of increased computation time Once an event has been localized to an acceptable the integrators with a status indicating that the current state is the last one before an event occurs Generally that marks the end of a continuous interval The time stepper will next invoke the systems event handler on the state at thigh with an indication of which events occurred within the event window Note that this state at thigh is inconsistent in some | way |
that | is |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler | returns |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points | tlow |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event | occurrence |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next | interval |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been | altered |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be | terminated |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock | time |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event | handler |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event and reinitialize the integrator if continuity has been violated Time advanced events occur whenever the integrator has advanced time | irreversibly |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event and reinitialize the integrator if continuity has been violated Time advanced events occur whenever the integrator has advanced time that at the end of every successful internal integration step These are generally restricted to discrete variable updates which do not affect the continuous | system |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event and reinitialize the integrator if continuity has been violated Time advanced events occur whenever the integrator has advanced time that at the end of every successful internal integration step These are generally restricted to discrete variable updates which do not affect the continuous such as min max values used only for reporting Normally the integrator does not return control at the end of a | step |
however the hybrid system can request that if | necessary |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For | example |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the | clock |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly | Finally |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the | simulation |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some | point |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special | meaning |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window | However |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other | hand |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed | discretely |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as | above |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as | below |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as and on the surface In either | case |
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F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this document | ( | the"Document" | ) |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems (SimTK::FactorLU) PAGEREF _Toc251069770 \h 5 HYPERLINK \l "_Toc251069771" 3.2 Linear Least Squares (SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made at the cost of increased computation time Once an event has been localized to an acceptable the integrators stepTo | ( | ) |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state discontinuously | ( | potentially updating both discrete and continuous | variables | ) |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the event | ( | s | ) |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an event | ( | s | ) |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt termination | ( | e.g.reached final | time | ) |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and again | ( | SeeREF _Ref160937060\hFigure 2for an | illustration. | ) |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as and on the surface In either these triggers will have a significant zero state which is not an artifact of finite precision arithmetic In those circumstances the sign transitions | ( | 0 and 0(1 can be meaningful and the caller may need to be notified at each transition.REF _Ref160937126\hFigure 3shows two examples.SHAPE\*MERGEFORMATTo allow the modeler to distinguish between accidental and purposeful zeroes, the System can optionally provide information about each of the event trigger functions.This information can include a specification of which sign transitions are considered significant for each trigger function.If only-1(1 and/or 1(-1 are specified, then we consider the event trigger function to be continuous and the integrator will only report a single event as the trigger passes through zero, even if it accidentally lands exactly on zero during localization.On the other hand, if the System tells us that zero is a significant value, then we assume that the trigger function has a discrete zero zone(or deadband) and we will report transitions to and from zero if they are seen during localization.REF _Ref160978458\hTable 1REF _Ref160978221\p\hbelowshows how observed transitions are treated based on whether rising or falling transitions have been specified, and whether zero is significant.Rising and falling transitions are disjoint so these specifications can be combined to allow a single event trigger to detect both rising and falling transitions.The first column inREF _Ref160978458\hTable 1lists the possible specifications that a System can make for the transitions of interest(rising/falling;continuous/discrete).The second column lists the transitions that might be observed initially, prior to localization.Typically, an initial step will be much wider than a localization interval so a1(1 or 1(1 transition is most likely to occur.The third column shows for each pre-localization observation the possible post-localization transitions(transitions to and from zero always localize to the same transition).The final column shows what transition will be reported back to the caller as having been seen.Note that although a wider set of transitions may be recognized during localization, the final report must come from the set that the system specified as interesting.For example, if a1(0 transition is seen after localization, but the system only watches for1(1, then well report1(0 as an instance of1(1.TableSEQ Table\*ARABIC1:handling of sign transitionsSignificant sign transitionsTransition seen before localizationLocalized transitionReported transitionRising-1(1continuous trigger-1(1any rising-1(1-1(0(unchanged)-1(10(1no eventdiscrete zeroadds-1(0, 0(1-1(1any risingreport localized transition-1(0(unchanged)-1(00(1(unchanged)0(1Falling1(-1continuous trigger1(-1any falling1(-11(0(unchanged)1(-10(-1no eventdiscrete zeroadds 1(0, 0(-11(-1any fallingreport localized transition1(0(unchanged)1(00(-1(unchanged)0(-1Notes:(1) For a continuous event trigger function that accidentally hits zero exactly, the transition to zero is treated as a zero crossing, but the subsequent transition from zero is ignored.(2) For a discrete-zero event trigger, if the function doesnt stay zero long enough then the zero transitions may be missed and a zero crossing(i.e.,-1(1 or 1(-1) will be reported instead.Accuracy, scaling, tolerancesThe time stepper is capable of achieving different levels of accuracy when simulating a particular system, withm o r ea c c u r a c yr e q u i r i n gm o r ec o m p u t a t i o nt i m e.O u rg o a li st oa l l o wa p p l i c a t i o np r o g r a m m e r st od e l i v e rt ot h e i re n du s e r st h ea b i l i t yt oc o n t r o lt h ea c c u r a c yl e v e lw i t has i n g l e, p h y s i c a l l ym e a n i n g f u ls c a l a rv a l u e, af r a c t i o ni nt h er a n g e0< d"1 . A t y p i c a l v a l u e i s = 0 . 0 0 1 ( 0 . 1 % ) m e a n i n g t h a t t h e i n t e g r a t o r s h o u l d p r o d u c e r e s u l t s w h i c h a r e w i t h i n 0 . 1 % o f t h e p e r f e c t a n s w e r . ( T h e s e p h r a s e s a r e i n q u o t a t i o n s b e c a u s e t h e y a r e m u c h e a s i e r t o s a y t h a n t o d e f i n e p r e c i s e l y ! ) E q u i v a l e n t l y , o n e c a n i n t e r p r e t as specifying the number of significant digits nd desired in the results, with EMBED Equation.DSMT4 . For example, EMBED Equation.DSMT4 .This goal is achievable only imperfectly, since in general accuracy is difficult to specify, to measure, to control, and even to define precisely. Reasonable users can disagree about exactly what they mean by 1% accuracy. Nevertheless we feel it is important to provide a single knob for a user to turn that delivers more accuracy at higher cost or less accuracy at lower cost, in a way that at least attempts to capture what at typical user might mean by these terms. The alternative of exposing the many complex issues involved to the end user assumes a kind of specialized expertise that would severely limit our intended audience. Instead, we expose these troubles to the application programmer in a way that allows us to collect information that can be used to define the Holy Grail control . T h e p r i m a r y d i f f i c u l t y w e e n c o u n t e r i s t h a t t h e v a r i a b l e s a n d e q u a t i o n s d e f i n i n g t h e u s e r s s y s t e m a r e n o t e v e n l y w e i g h t e d . W e e x p e c t t h a t i n m o s t c a s e s t h e p r o g r a m m e r w h o i s u s i n g S i m T K : : T i m e S t e p p e r w i l l k n o w s o m e t h i n g u s e f u l a b o u t t h e v a r i o u s w e i g h t s t h a t w i l l h e l p u s d e f i n e . T o a c c o m m o d a t e t h a t , w e a l l o w s p e c i f i c a t i o n o f f o u r k i n d s o f s c a l i n g i n f o r m a t i o n : t h e c h a r a c t e r i s t i c t i m e s c a l e a w e i g h t w i f o r e a c h c o n t i n u o u s s t a t e v a r i a b l e y i a t o l e r a n c e t i f o r e a c h o f t h e c o n s t r a i n t e r r o r s c i a l o c a l i z a t i o n t i m e w i n d o w w i d t h l i f o r e a c h e v e n t e i E a c h o f t h e s e q u a n t i t i e s i s d e f i n e d b e l o w t o b e a p r o p e r t y o f t h e s y s t e m ( m o d e l ) , i n d e p e n d e n t o f t h e a c c u r a c y w i t h w h i c h t h e s y s t e m i s b e i n g s i m u l a t e d . T h e i d e a i s t o d e f i n e f o r e a c h q u a n t i t y a u n i t e r r o r t o w h ich the accuracy requirement can then be applied. The integrator treats these values as constant during each integration step. However, we expect the state variable weights wi to change (slowly) as the trajectory (time and state) evolves, so the integrator will request updated weights from time to time during a continuous interval, according to some criteria. Time scale, constraint tolerances, and localization requirements, on the other hand, are expected to be independent of trajectory and are fixed once th e i n s t a n c e v a r i a b l e s a r e f r o z e n p r i o r t o t h e s t a r t o f a c o n t i n u o u s i n t e r v a l w i t h i n t h e s i m u l a t i o n . T i m e S c a l e T h e r e i s a c h a r a c t e r i s t i c t i m e s c a l e o f t h e s y s t e m w h i c h t h e p r o g r a m m e r m a y p r o v i d e t o c o n v e y t o t h e i n t e g r a t o r t h e s m a l l e s t t i m e o f i n t e r e s t f o r t h e s t u d y . T h i s s h o u l d b e t h e m i n i m u m t i m e i n t e r v a l o v e r w h i c h i n t e r e s t i n g c h a n g e s a r e e x p e c t e d t o o c c u r . T h e i n t e g r a t o r m a y u s e t h i s f o r s e l e c t i n g t h e i n i t i a l s t e p s i z e a n d d e f a u l t s f o r o t h e r s c a l i n g i n f o r m a t i o n m a y d e p e n d o n . W e i g h t s a n d c o n s t r a i n t t o l e r a n c e s W e a r e g i v e n a s e t o f w e i g h t s w i e"0f o re a c hy ia n das e to ft o l e r a n c e st i > 0f o re a c ho ft h ec o n s t r a i n te r r o r sc i.L e tWb ead i a g o n a lm a t r i xw i t ht h ew iso ni t sd i a g o n a l, a n dTb ead i a g o n a lm a t r i xw i t ht h er e c i p r o c a lt o l e r a n c e s1/t io ni t sd i a g o n a l(w ea l s oc a l lt h e s er e c i p r o c a lt o l e r a n c e sc o n s t r a i n tw e i g h t s).T h e ng i v e nt h ef r a c t i o n a la c c u r a c ys p e c i f i c a t i o n(e.g.=0.1%), t h ei n t e g r a t o ri sr e q u i r e dt os o l v ef o rt h et r a j e c t o r yy(t) s u c ht h a te a c hi n t e g r a t i o ns t e pm a i n t a i n st h el o c a le r r o r|W y|R M S d" a n d c o n s t r a i n t e r r o r | T c ( t , y ) | R M S d" a ta l lt i m e s.H e r eyi st h ev e c t o ro fe s t i m a t e da b s o l u t ee r r o r si ne a c hs t a t ev a r i a b l ey, a se s t i m a t e db yt h ei n t e g r a t o rf o rat r i a ls t e pu n d e rc o n s i d e r a t i o n.E v e n tl o c a l i z a t i o nw i n d o wE v e n tl o c a l i z a t i o nr e q u i r e sf o re a c he v e n tt r i g g e rf u n c t i o ne iau n i tl o c a l i z a t i o nr e q u i r e m e n tl i(i nu n i t so ft h es y s t e mst i m es c a l e) w h i c hi st h e nn a r r o w e db yt h ea c c u r a c yr e q u i r e m e n ts ot h a tt h ei n t e g r a t o rl o c a l i z e se v e n te it oat i m ei n t e r v a lo fw i d t h d" l i. Thus when the integrator reports that a set E of events has triggered in the event window (tlow,thigh], it guarantees that EMBED Equation.DSMT4 .Default accuracy, scaling and tolerancesThe defaults are as follows: EMBED Equation.DSMT4 EMBED Equation.DSMT4 We also support a number of different schemes for defining the weights. The conventional rtol, atol scheme is achieved by defining EMBED Equation.DSMT4 where ui represents one unit of error in yi (a typical value would b e 0 . 1 ) . T h i s i s t h e n e q u i v a l e n t t o r t o l = a n d a t o l i = u i . N u m e r i c a l o p t i m i z a t i o n ( S i m T K : : O p t i m i z e r ) T h e S i m T K : : O p t i m i z e r c l a s s c a n b e u s e d t o n u m e r i c a l l y s o l v e v a r i o u s o p t i m i z a t i o n p r o b l e m s . O p t i m i z a t i o n m e t h o d s c a n b e u s e d t o f i n d t h e o p t i m a l s o l u t i o n t o some problem. For example, what set of coordinates would give the minimum energy configuration for this molecule?The algorithms implemented in the Optimizer class search for points which are a minimum of a, user supplied, objective function. The algorithms start from some initial point and iteratively search for points which reduce the objective function. The algorithms terminate when the objective function stops decreasing. Note that the minimum found by the Optimizers is sometime a local minimum and not a global minimum. Therefore, the point which the algorithm starts searching from is important. This is illustrated in Figure 2 below. SHAPE \* MERGEFORMAT Note that if our goal was to find the maximum of some objective function f(x). We could still use the existing optimizers by using f(x) as our objective function.Sometimes the optimization problem has constraints on the objective function. Consider the simple example below:The following objective function: EMBED Equation.3 is subject to two constraints:C1: EMBED Equation.3 C2: EMBED Equation.3 Has an optimal solution at: X = (4, 2)Figure 3 shows a graph of this problem. The two constraints (C1, C2) bound the space of possible solutions to a feasible region which is shown in gray. The optimal solution to the problem must be a point within this feasible region which has the smallest value of the objective function. The two dashed circles indicate the constant value contours of our objective function. The smaller circle has a value of 1.0 and the larger circle has a value of 4.0.Optimization problems are solved in SimTK using two classes, the Optimizer class, and the OptimizerSystem class. The Optimizer class allocates an optimizer and sets the options for the optimizer. The OptimizerSystem class is used to describe the optimization problem. OptmizerSystem is an abstract class which has methods for computing the objective function, the gradient of the objective function, the constraints, and the constraint Jacobian. The application needs to define a concrete class that subclasses from OptimizerSystem. As a minimum the concrete class must supply an implementation of the objectiveFunction() method which computes the objective function for a given set of parameters. An instance of the concrete class is passed as an argument for the constructor of the Optimizer class. The Optimizer class uses the information from the OptimizerSystem concrete class to select the best optimizer algorithm and to allocate any workspace that the algorithm may need. The application can set various parameters for the optimizer such as convergence tolerance.Once the Optimizer has been instantiated and the options set the solution can be computed by calling the Optimizer classs optimize() method. Optimizer.optimize() takes a SimTK::Vector which sets the initial point the optimizer will begin searching from. If the optimizer is able to find a solution it will return the parameters for the optimal solution in the Vector. Optimizer.optimize() will also return the optimal value of the objective function. The code for using the Optimizer class to solve this problem is shown below. #include "SimTKmath.h"#include <iostream>static int NUMBER_OF_PARAMETERS = 2;static int NUMBER_OF_EQUALITY_CONSTRAINTS = 0;static int NUMBER_OF_INEQUALITY_CONSTRAINTS = 2;// user-written classclass ProblemSystem : public OptimizerSystem {public: // Must provide this virtual function. int objectiveFunc( const Vector &coefficients, const bool new_coefficients, Real& f ) const { const Real *x; int i; x = &coefficients[0]; f = (x[0] - 5.0)*(x[0] - 5.0) + (x[1] - 1.0)*(x[1] - 1.0); return( 0 ); } int gradientFunc( const Vector &coefficients, const bool new_coefficients, Vector &gradient ) const{ const Real *x; x = &coefficients[0]; gradient[0] = 2.0*(x[0] - 5.0); gradient[1] = 2.0*(x[1] - 1.0); return(0); } /* ** Method to compute the value of the constraints. ** Equality constraints are first followed by the any inequality constraints */ int constraintFunc( const Vector &coefficients, const bool new_coefficients, Vector &constraints) const{ const Real *x; x = &coefficients[0]; constraints[0] = x[0] - x[1]*x[1]; constraints[1] = x[1] - x[0] + 2.0; return(0); } /* ** Method to compute the Jacobian of the constraints. ** */ int constraintJacobian( const Vector& coefficients, const bool new_coefficients, Matrix& jac) const{ const Real *x; x = &coefficients[0]; jac(0,0) = 1.0; jac(0,1) = -2.0*x[1]; jac(1,0) = -1.0; jac(1,1) = 1.0; return(0); } ProblemSystem( const int numParams, const int numEqualityConstraints, const int numInequalityConstraints) : OptimizerSystem( numParams ) { setNumEqualityConstraints( numEqualityConstraints ); setNumInequalityConstraints( numInequalityConstraints ); }};main() { Real f; int i; /* create the system to be optimized */ ProblemSystem sys(NUMBER_OF_PARAMETERS, NUMBER_OF_EQAULITY_CONSTRAINTS, NUMBER_OF_INEQUALITY_CONSTRAINTS ); Vector results(NUMBER_OF_PARAMETERS); /* set initial conditions */ results[0] = 5.0; results[1] = 5.0; try { Optimizer opt( sys ); opt.setConvergenceTolerance( .0000001 ); /* compute optimization */ f = opt.optimize( results ); } catch (const std::exception& e) { std::cout << "ConstrainedOptimization.cpp Caught exception:" << std::endl; std::cout << e.what() << std::endl; } printf("Optimal Solution:f=%fparameters=%f%f\n", f, results[0], results[1]); return 0;}Output:Optimal Solution: f = 2.00000 parameters = 3.99998 1.999998Sometimes the optimization problem has constraints on the parameters of the objective function. For example, the potential energy of a protein may be expressed as a function of the bond angles of the protein. Steric clashes may limit the bond angles of a protein to a range of valid values. Therefore minimum energy configuration of the protein would be constrained by these angles. It may be easier to express these constraints as limits on the values of our objective function parameters. This would cause the optimizer to search for a minimum only from within this feasible region. For example, if the parameter EMBED Equation.3 in our example had limits of EMBED Equation.3 our problem would look like the graph in Figure 4. These limits could be expressed by adding the following code: Vector lower_limits(NUMBER_OF_PARAMETERS); Vector upper_limits(NUMBER_OF_PARAMETERS); /* set limits on the parameters */ lower_limits[0] = 1.0; upper_limits[0] = 3.0; lower_limits[1] = -2e19; upper_limits[1] = 2e19; sys.setParameterLimits( lower_limits, upper_limits );Optimal Solution: f = 4.000000 parameters = 3.000000 1.000073Other Mathematical ToolsSimTK provides a number of other tools for solving common mathematical problems. Examples include generating random numbers and finding the roots of polynomials. These tools are described below.Random Numbers (SimTK::Random)There are many cases where it is necessary to generate a set of random numbers, such as to drive a Monte Carlo simulation or to provide random initial conditions for a set of dynamics simulations. Most programming environments provide a random number generator, but they often are poorly suited to scientific applications. 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@QUVW\9Pl$d7 $8 $H $a $gdi $56]b9, gdi $$d7 $8 $H $a $gdi $1YZ\`a9nwϼϼϼώώσtYϼ4h-2hi $B *CJOJQJ^JaJmHnHph uh-2hi $0J-B *aJphh-2hi $^JaJ%h!\O CJOJQJ^JaJmHnHu4h'h' B *CJOJQJ^JaJmHnHph u%h' CJOJQJ^JaJmHnHu+h-2hi $CJOJQJ^JaJmHnHu4h-2hi $B *CJOJQJ^JaJmHnHph u" = @ ֑֑֑֬֬~l_XN h6 hi $ 5\ hi $ 5\ hxZ/ hi $ B*_HEph #h' h' CJ ^J aJ mH nH u%hi $ CJ OJ QJ ^J aJ mH nH u4h-2 hi $ B*CJ OJ QJ ^J aJ mH nH ph u 4h-2 hi $ B*CJ OJ QJ ^J aJ mH nH ph u h-2 hi $ 0J- B*aJ ph +h-2 hi $ CJ OJ QJ ^J aJ mH nH u%h' CJ OJ QJ ^J aJ mH nH u # 9 : C D a b S } $ d 7$ 8$ H$ a$ gdi $ 8 9 l m "gdi $gdi $$&F" a$ gdi $ gdi $ , gdi $ 7 8 9 k l m q w 랑mbbW hT% hi $ B*ph hT% hi $ B*ph %hi $ CJ OJ QJ ^J aJ mH nH u h.| hi $ CJ OJ QJ ^J aJ j h.| hi $ EHU%j`Ihi $ CJ OJ QJ UVaJ j hJ hi $ EHU_HE%j+_Ihi $ CJ OJ QJ UVaJ j hi $ U_HEj hi $ Uhi $ _HE hi $ h6 hi $ 5\ hi $ 5\ - ! "&7Ml5KPgi+JϻϻƵƞƵv)hfhi $B *CJOJQJ_HaJphhfhi $H *hhfhi $h\Zhi $B *OJQJphUh _HEhi $_HEhhi $B *phh\Zhi $_HEhi $%hghi $5OJQJ\mHnHu1hghi $5CJOJQJ\^JaJmHnHu+The sequences of numbers it generates may appear random, but if you reset it to its initial condition(or create a new random number generator instance), it will produce exactly the same sequence of numbers.If you need several random number generators that each produces a different random sequence, you can do this by initializing each one with a different seed value.Every possible seed value corresponds to a different sequence of random numbers that(in the case of a good generator) is independent of every other one.The SimTK::Random class is based on the SIMD-oriented Fast Mersenne Twister(SFMT) library.It provides excellent statistical properties, fast performance, and a very long sequence(219937-1).Never instantiate SimTK::Random directly.Instead, create an instance of one if its two subclasses, SimTK::Random::Uniform and SimTK::Random::Gaussian.These classes generate numbers according to uniform and Gaussian distributions, respectively.You can specify the minimum and maximum of a uniform distribution, and the mean and standard deviation of a Gaussian distribution.For example, to generate a sequence of numbers uniformly distributed between 0 and 100, you would write:Random::Uniform random(0.0, 100.0);Real nextValue=random.getValue();Roots of Polynomials(SimTK::PolynomialRootFinder) This class provides static methods for finding the roots of polynomials.There are specialized methods for quadratic and cubic polynomials, as well as general methods for polynomials of arbitrary degree.In each case, there are methods for polynomials with both real and complex coefficients.There are two different algorithms used by this class.The specialized methods for quadratic polynomials calculate the roots by explicit evaluation of the quadratic formula.They use the evaluation method described in section 5.6 of"Numerical Recipes in C++, Second Edition", by Press, Teukolsky, Vetterling, and Flannery.In addition, the method for quadratic polynomials with real coefficients performs an extra check to detect when the discriminant is zero to within machine precision.This helps to prevent round-off error from producing a tiny imaginary part in a multiple root.The methods for cubic and arbitrary degree polynomials use the Jenkins-Traub method, as implemented in the classic RPOLY and CPOLY functions.This is an iterative method that provides rapid convergence and high accuracy in most cases.For details, seeJenkins, M.A.and Traub, J.F.(1972), Algorithm 419:Zeros of a Complex Polynomial, Comm.ACM, 15, 97-99.Jenkins, M.A.(1975), Algorithm 493:Zeros of a Real Polynomial, ACM TOMS, 1, 178-189.As an example of using this class, the following code finds the roots of x3-6x2+11x-6:Vec4 coefficients(1.0,-6.0, 11.0,-6.0);Vec< 3, Complex > roots;PolynomialRootFinder::findRoots(coefficients, roots);cout<< "Roots: "<< roots<< endl;which produces the outputRoots:~[(1, 0),(2, 0),(3, 0)]Notice that the coefficients are specified in order of descending powers.Also notice that the roots are always returned as complex numbers, even if the coefficients are real.This is because a polynomial with real coefficients can still have complex roots.ReferencesAPI:Application Programming Interface, i.e., a programming library.You can also select elements with the more C-like construct m[i][j], but that requires two operations(one to select the row and one to index that row).The m(i, j) form is substantially more efficient and should be used instead.Better yet, use vector and matrix operators rather than accessing individual elements whenever possible.The figure does not show the internal trial steps that an error controlled integrator would have taken and rejected along the way;only the accepted steps are shown.In practice this is not strictly true since an integrator may perform a little rounding off of internal steps to avoid having to interpolate over very small slivers of time.PAGE\*MERGEFORMATiiPAGE\*MERGEFORMATiiPAGE\*MERGEFORMAT32PAGE\*MERGEFORMAT31000(10(1FigureSEQ Figure\*ARABIC3:two intentionally-discrete event trigger functions.FigureSEQ Figure\*ARABIC2:a continuous event trigger function appears discrete in finite precision arithmetic.0(1-1(0-1(0Boolean trigger functionTrigger function with deadband0(1-1(0-1(1EMBED UnknownEMBED UnknownEMBED UnknownComputational function changes discretely;stays zero for finite intervalMathematical function changes continuously;crosses zero at a pointC2C1Optimal SolutionFigure 4:LimitsEMBED Equation.3are shown in green.The two constraints are shown in red and the feasible region is show in grey.The contours of the objective function are shown as dashed blue circles.EMBED Equation.3EMBED Equation.3C2C1Optimal SolutionFigure 3:Graph of problem to finding optimal solution to the problem described above.The two constraints are shown in red and the feasible region is show in grey.The contours of the objective function are shown as dashed blue circles.EMBED Equation.3EMBED Equation.3Figure 2:The optimizer begins searching along the objective function starting at the Initial point.It moves first to point P1 then point P2 which reduce the objective function.Eventually the optimizer terminates at the Local minimum when it cannot reduce the objective function any further.Objective functionP2P1Initial pointLocal maximumGlobal maximumLocal minimumGlobal minimumtn+4tn+1tn+2si+5si+4si+3si+2si+1sireporting timesinternal stepsteventtn+5An event occurs hereFigureSEQ Figure\*ARABIC1:Numerical integration over a continuous segment of a time-stepping trajectory.The continuous region starts at tn and terminates at tevent.Each tall green line represents a return of control to the time stepper:thin dotted ones are caller-requested reporting times;the thick dotted green line is an unrequested return of control just prior to event occurrence and just after event 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F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Sherman |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby granted |
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F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to modify |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to merge |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to publish |
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F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to sublicense |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the Document |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do so |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following conditions |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY KIND |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR IMPLIED |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE AUTHORS |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER LIABILITY |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF CONTRACT |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR OTHERWISE |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING FROM |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK http |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has occurred |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems (SimTK::FactorLU) PAGEREF _Toc251069770 \h 5 HYPERLINK \l "_Toc251069771" 3.2 Linear Least Squares (SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the window |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard events |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems (SimTK::FactorLU) PAGEREF _Toc251069770 \h 5 HYPERLINK \l "_Toc251069771" 3.2 Linear Least Squares (SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is required |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems (SimTK::FactorLU) PAGEREF _Toc251069770 \h 5 HYPERLINK \l "_Toc251069771" 3.2 Linear Least Squares (SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made narrower |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems (SimTK::FactorLU) PAGEREF _Toc251069770 \h 5 HYPERLINK \l "_Toc251069771" 3.2 Linear Least Squares (SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made at the cost of increased computation time Once an event has been localized to an acceptable tolerance |
F I A7 e2 6 J 8 q2 L F F 6 F 6 p I I I I I I I C7 C7 I I I I I I I F I I I I I I I I I Michael Peter Eastman Permission is hereby free of to any person obtaining a copy of this to deal in the Document without including without limitation the rights to and or sell copies of the and to permit persons to whom the Document is furnished to do subject to the following WITHOUT WARRANTY OF ANY EXPRESS OR INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY DAMAGES OR OTHER WHETHER IN AN ACTION OF TORT OR ARISING OUT OF OR IN CONNECTION WITH THE DOCUMENT OR THE USE OR OTHER DEALINGS IN THE DOCUMENT Acknowledgment SimTK software and all related activities are funded by the HYPERLINK vectors and matrices PAGEREF _Toc251069768 h HYPERLINK l _Toc251069769 Linear algebra PAGEREF _Toc251069769 h HYPERLINK l _Toc251069770 Solving Linear Systems (SimTK::FactorLU) PAGEREF _Toc251069770 \h 5 HYPERLINK \l "_Toc251069771" 3.2 Linear Least Squares (SimTK otherwise the best guess is that tevent is in the middle of the window Finite width localization windows create a likelihood that multiple events will occur within the same window We cannot say with certainty in what order these events so for precise treatment they must be considered simultaneous If an approximate ordering is acceptable then the integrators tevent guesses can be used to order the events within the but even those may be identical for some and in any case the integrator cannot guarantee that the events actually occurred in the order they appear when sorted by estimated time of occurrence If more precise information is then the localization window must be made at the cost of increased computation time Once an event has been localized to an acceptable the integrators with a status indicating that the current state is the last one before an event occurs Generally that marks the end of a continuous interval The time stepper will next invoke the systems event handler on the state at thigh with an indication of which events occurred within the event window Note that this state at thigh is inconsistent in some way |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That is |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler returns |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points tlow |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event occurrence |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next interval |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been altered |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be terminated |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock time |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event handler |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event and reinitialize the integrator if continuity has been violated Time advanced events occur whenever the integrator has advanced time irreversibly |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event and reinitialize the integrator if continuity has been violated Time advanced events occur whenever the integrator has advanced time that at the end of every successful internal integration step These are generally restricted to discrete variable updates which do not affect the continuous system |
that it contains events that have triggered but have not been processed Thus it is not a legitimate point along the systems trajectory and should never be returned to the caller For discussion we label this improper state EMBED Equation DSMT4 The event handler will correct the state creating a modified state with time still at thigh but not triggering any events We label the modified state EMBED Equation DSMT4 Once the systems handler the state EMBED Equation DSMT4 can be output as part of the trajectory and used as the initial condition for the next continuous interval Thus the time stepper generates the sequence of legitimate trajectory points just prior to event immediately followed by EMBED Equation DSMT4 which is the time at which the but after the event handler has modified the state to deal with those events Consecutive intervals I and I will consist of trajectory points EMBED Equation DSMT4 with round brackets indicating that EMBED Equation DSMT4 is not part of the trajectory Event handlers Event handlers are solvers which are able to take a state in an resolve the set up appropriate event triggers for the next and report back to the time stepper the degree to which the system continuity has been so that the integrator can be reinitialized appropriately An event handler can also indicate that the simulation should be in which case the time stepper will return the final state to its caller and disallow further time stepping Other event types Not all events have to be localized There are several special case clock user interrupt control is returned to the time stepper The time stepper can then declare that a scheduled event has call the systems event and reinitialize the integrator if continuity has been violated Time advanced events occur whenever the integrator has advanced time that at the end of every successful internal integration step These are generally restricted to discrete variable updates which do not affect the continuous such as min max values used only for reporting Normally the integrator does not return control at the end of a step |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For example |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the clock |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly Finally |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the simulation |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some point |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special meaning |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window However |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other hand |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed discretely |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as above |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as below |
however the hybrid system can request that if in which case end of step is treated like any other event External events are typically handled through the end of step event mechanism For the end of step function can look at the keyboard etc and respond accordingly a termination event occurs either when the integrator advances to a designated final time for the or when an event handler indicates that the simulation must be terminated for some other reason Event trigger transition details This section probably covers more than you really want to know about event triggers But if you run into some mysterious behavior regarding your event trigger functions at some you may want to come back to this section Normally event triggers are used to detect zero crossings That the event in question occurs when the triggers value goes from negative to positive or vice versa at some moment in time Zero itself has no special and in a mathematical world it would never actually be encountered when localizing to a finite width time window in a computational world landing exactly on zero during localization is not only possible but certain to happen now and since there is only a single zero crossing despite appearances On the other many event trigger functions are most naturally expressed often as boolean functions which toggle between false and true when an event occurs Others may be designed to have three states such as and on the surface In either case |
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