Drake
RadauIntegrator< T, num_stages > Class Template Referencefinal

Detailed Description

template<typename T, int num_stages = 2>
class drake::systems::RadauIntegrator< T, num_stages >

A selectable order (third- or first-order), fully implicit integrator with error estimation.

Template Parameters
TThe scalar type, which must be one of the default nonsymbolic scalars.
num_stagesThe number of stages used in this integrator, which must be either 1 or 2. Set this to 1 for the integrator to be implicit Euler and 2 for it to Radau3 (default).

A two-stage Radau IIa (see [Hairer, 1996], Ch. 5) method is used for propagating the state forward, by default. The state can also be propagated using a single-stage method, in which case it is equivalent to an implicit Euler method, by setting num_stages=1. Regardless of the order of propagating state, the local (truncation) error is estimated through the implicit trapezoid rule.

Radau IIa methods are known to be L-Stable, meaning both that applying it at a fixed integration step to the "test" equation y(t) = eᵏᵗ yields zero (for k < 0 and t → ∞) and that it is also A-Stable. A-Stability, in turn, means that the method can integrate the linear constant coefficient system dx/dt = Ax at any step size without the solution becoming unstable (growing without bound). The practical effect of L-Stability is that the integrator tends to be stable for any given step size on an arbitrary system of ordinary differential equations. Note that the implicit trapezoid rule used for error estimation is "only" A-Stable; whether this lesser stability has some practical effect on the efficiency of this integrator is currently unknown. See [Lambert, 1991], Ch. 6 for an approachable discussion on stiff differential equations and L- and A-Stability.

This implementation uses Newton-Raphson (NR). General implementation details were taken from [Hairer, 1996] Ch. 8.

  • [Hairer, 1996] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II (Stiff and Differential-Algebraic Problems). Springer, 1996.
  • [Lambert, 1991] J. D. Lambert. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, 1991.
See also
ImplicitIntegrator class documentation for information about implicit integration methods in general.
Radau3Integrator and Radau1Integrator alises for third- and first-order Template MetaProgramming with num_stages already specified.
Note
This integrator uses the integrator accuracy setting, even when run in fixed-step mode, to limit the error in the underlying Newton-Raphson process. See IntegratorBase::set_target_accuracy() for more info.

#include <drake/systems/analysis/radau_integrator.h>

Public Member Functions

 RadauIntegrator (const System< T > &system, Context< T > *context=nullptr)
 
 ~RadauIntegrator () final=default
 
bool supports_error_estimation () const final
 Derived classes must override this function to indicate whether the integrator supports error estimation. More...
 
int get_error_estimate_order () const final
 This integrator uses embedded second order methods to compute estimates of the local truncation error. More...
 
Does not allow copy, move, or assignment
 RadauIntegrator (const RadauIntegrator &)=delete
 
RadauIntegratoroperator= (const RadauIntegrator &)=delete
 
 RadauIntegrator (RadauIntegrator &&)=delete
 
RadauIntegratoroperator= (RadauIntegrator &&)=delete
 
- Public Member Functions inherited from ImplicitIntegrator< T >
virtual ~ImplicitIntegrator ()
 
 ImplicitIntegrator (const System< T > &system, Context< T > *context=nullptr)
 
int max_newton_raphson_iterations () const
 The maximum number of Newton-Raphson iterations to take before the Newton-Raphson process decides that convergence will not be attained. More...
 
void set_reuse (bool reuse)
 Sets whether the integrator attempts to reuse Jacobian matrices and iteration matrix factorizations (default is true). More...
 
bool get_reuse () const
 Gets whether the integrator attempts to reuse Jacobian matrices and iteration matrix factorizations. More...
 
void set_use_full_newton (bool flag)
 Sets whether the method operates in "full Newton" mode, in which case Jacobian and iteration matrices are freshly computed on every Newton-Raphson iteration. More...
 
bool get_use_full_newton () const
 Gets whether this method is operating in "full Newton" mode. More...
 
void set_jacobian_computation_scheme (JacobianComputationScheme scheme)
 Sets the Jacobian computation scheme. More...
 
JacobianComputationScheme get_jacobian_computation_scheme () const
 
int64_t get_num_derivative_evaluations_for_jacobian () const
 Gets the number of ODE function evaluations (calls to EvalTimeDerivatives()) used only for computing the Jacobian matrices since the last call to ResetStatistics(). More...
 
int64_t get_num_jacobian_evaluations () const
 Gets the number of Jacobian computations (i.e., the number of times that the Jacobian matrix was reformed) since the last call to ResetStatistics(). More...
 
int64_t get_num_newton_raphson_iterations () const
 Gets the number of iterations used in the Newton-Raphson nonlinear systems of equation solving process since the last call to ResetStatistics(). More...
 
int64_t get_num_iteration_matrix_factorizations () const
 Gets the number of factorizations of the iteration matrix since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_derivative_evaluations () const
 Gets the number of ODE function evaluations (calls to EvalTimeDerivatives()) used only for the error estimation process since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_derivative_evaluations_for_jacobian () const
 
int64_t get_num_error_estimator_newton_raphson_iterations () const
 Gets the number of iterations used in the Newton-Raphson nonlinear systems of equation solving process for the error estimation process since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_jacobian_evaluations () const
 Gets the number of Jacobian matrix computations used only during the error estimation process since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_iteration_matrix_factorizations () const
 Gets the number of factorizations of the iteration matrix used only during the error estimation process since the last call to ResetStatistics(). More...
 
- Public Member Functions inherited from IntegratorBase< T >
 IntegratorBase (const System< T > &system, Context< T > *context=nullptr)
 Maintains references to the system being integrated and the context used to specify the initial conditions for that system (if any). More...
 
virtual ~IntegratorBase ()=default
 
void Reset ()
 Resets the integrator to initial values, i.e., default construction values. More...
 
void Initialize ()
 An integrator must be initialized before being used. More...
 
StepResult IntegrateNoFurtherThanTime (const T &publish_time, const T &update_time, const T &boundary_time)
 (Internal use only) Integrates the system forward in time by a single step with step size subject to integration error tolerances (assuming that the integrator supports error estimation). More...
 
void IntegrateWithMultipleStepsToTime (const T &t_final)
 Stepping function for integrators operating outside of Simulator that advances the continuous state exactly to t_final. More...
 
bool IntegrateWithSingleFixedStepToTime (const T &t_target)
 Stepping function for integrators operating outside of Simulator that advances the continuous state using a single step to t_target. More...
 
const Context< T > & get_context () const
 Returns a const reference to the internally-maintained Context holding the most recent state in the trajectory. More...
 
Context< T > * get_mutable_context ()
 Returns a mutable pointer to the internally-maintained Context holding the most recent state in the trajectory. More...
 
void reset_context (Context< T > *context)
 Replace the pointer to the internally-maintained Context with a different one. More...
 
const System< T > & get_system () const
 Gets a constant reference to the system that is being integrated (and was provided to the constructor of the integrator). More...
 
bool is_initialized () const
 Indicates whether the integrator has been initialized. More...
 
const T & get_previous_integration_step_size () const
 Gets the size of the last (previous) integration step. More...
 
 IntegratorBase (const IntegratorBase &)=delete
 
IntegratorBaseoperator= (const IntegratorBase &)=delete
 
 IntegratorBase (IntegratorBase &&)=delete
 
IntegratorBaseoperator= (IntegratorBase &&)=delete
 
void set_target_accuracy (double accuracy)
 Request that the integrator attempt to achieve a particular accuracy for the continuous portions of the simulation. More...
 
double get_target_accuracy () const
 Gets the target accuracy. More...
 
double get_accuracy_in_use () const
 Gets the accuracy in use by the integrator. More...
 
const ContinuousState< T > * get_error_estimate () const
 Gets the error estimate (used only for integrators that support error estimation). More...
 
const T & get_ideal_next_step_size () const
 Return the step size the integrator would like to take next, based primarily on the integrator's accuracy prediction. More...
 
void set_fixed_step_mode (bool flag)
 Sets an integrator with error control to fixed step mode. More...
 
bool get_fixed_step_mode () const
 Gets whether an integrator is running in fixed step mode. More...
 
const Eigen::VectorXd & get_generalized_state_weight_vector () const
 Gets the weighting vector (equivalent to a diagonal matrix) applied to weighting both generalized coordinate and velocity state variable errors, as described in the group documentation. More...
 
Eigen::VectorBlock< Eigen::VectorXd > get_mutable_generalized_state_weight_vector ()
 Gets a mutable weighting vector (equivalent to a diagonal matrix) applied to weighting both generalized coordinate and velocity state variable errors, as described in the group documentation. More...
 
const Eigen::VectorXd & get_misc_state_weight_vector () const
 Gets the weighting vector (equivalent to a diagonal matrix) for weighting errors in miscellaneous continuous state variables z. More...
 
Eigen::VectorBlock< Eigen::VectorXd > get_mutable_misc_state_weight_vector ()
 Gets a mutable weighting vector (equivalent to a diagonal matrix) for weighting errors in miscellaneous continuous state variables z. More...
 
void request_initial_step_size_target (const T &step_size)
 Request that the first attempted integration step have a particular size. More...
 
const T & get_initial_step_size_target () const
 Gets the target size of the first integration step. More...
 
void set_maximum_step_size (const T &max_step_size)
 Sets the maximum step size that may be taken by this integrator. More...
 
const T & get_maximum_step_size () const
 Gets the maximum step size that may be taken by this integrator. More...
 
double get_stretch_factor () const
 Gets the stretch factor (> 1), which is multiplied by the maximum (typically user-designated) integration step size to obtain the amount that the integrator is able to stretch the maximum time step toward hitting an upcoming publish or update event in IntegrateNoFurtherThanTime(). More...
 
void set_requested_minimum_step_size (const T &min_step_size)
 Sets the requested minimum step size h_min that may be taken by this integrator. More...
 
const T & get_requested_minimum_step_size () const
 Gets the requested minimum step size h_min for this integrator. More...
 
void set_throw_on_minimum_step_size_violation (bool throws)
 Sets whether the integrator should throw a std::exception when the integrator's step size selection algorithm determines that it must take a step smaller than the minimum step size (for, e.g., purposes of error control). More...
 
bool get_throw_on_minimum_step_size_violation () const
 Reports the current setting of the throw_on_minimum_step_size_violation flag. More...
 
get_working_minimum_step_size () const
 Gets the current value of the working minimum step size h_work(t) for this integrator, which may vary with the current time t as stored in the integrator's context. More...
 
void ResetStatistics ()
 Forget accumulated statistics. More...
 
int64_t get_num_substep_failures () const
 Gets the number of failed sub-steps (implying one or more step size reductions was required to permit solving the necessary nonlinear system of equations). More...
 
int64_t get_num_step_shrinkages_from_substep_failures () const
 Gets the number of step size shrinkages due to sub-step failures (e.g., integrator convergence failures) since the last call to ResetStatistics() or Initialize(). More...
 
int64_t get_num_step_shrinkages_from_error_control () const
 Gets the number of step size shrinkages due to failure to meet targeted error tolerances, since the last call to ResetStatistics or Initialize(). More...
 
int64_t get_num_derivative_evaluations () const
 Returns the number of ODE function evaluations (calls to CalcTimeDerivatives()) since the last call to ResetStatistics() or Initialize(). More...
 
const T & get_actual_initial_step_size_taken () const
 The actual size of the successful first step. More...
 
const T & get_smallest_adapted_step_size_taken () const
 The size of the smallest step taken as the result of a controlled integration step adjustment since the last Initialize() or ResetStatistics() call. More...
 
const T & get_largest_step_size_taken () const
 The size of the largest step taken since the last Initialize() or ResetStatistics() call. More...
 
int64_t get_num_steps_taken () const
 The number of integration steps taken since the last Initialize() or ResetStatistics() call. More...
 
void add_derivative_evaluations (double evals)
 Manually increments the statistic for the number of ODE evaluations. More...
 
void StartDenseIntegration ()
 Starts dense integration, allocating a new dense output for this integrator to use. More...
 
const trajectories::PiecewisePolynomial< T > * get_dense_output () const
 Returns a const pointer to the integrator's current PiecewisePolynomial instance, holding a representation of the continuous state trajectory since the last StartDenseIntegration() call. More...
 
std::unique_ptr< trajectories::PiecewisePolynomial< T > > StopDenseIntegration ()
 Stops dense integration, yielding ownership of the current dense output to the caller. More...
 

Additional Inherited Members

- Public Types inherited from ImplicitIntegrator< T >
enum  JacobianComputationScheme { kForwardDifference, kCentralDifference, kAutomatic }
 
- Public Types inherited from IntegratorBase< T >
enum  StepResult {
  kReachedPublishTime = 1, kReachedZeroCrossing = 2, kReachedUpdateTime = 3, kTimeHasAdvanced = 4,
  kReachedBoundaryTime = 5, kReachedStepLimit = 6
}
 Status returned by IntegrateNoFurtherThanTime(). More...
 
- Protected Types inherited from ImplicitIntegrator< T >
enum  ConvergenceStatus { kDiverged, kConverged, kNotConverged }
 
- Protected Member Functions inherited from ImplicitIntegrator< T >
virtual int do_max_newton_raphson_iterations () const
 Derived classes can override this method to change the number of Newton-Raphson iterations (10 by default) to take before the Newton-Raphson process decides that convergence will not be attained. More...
 
bool MaybeFreshenMatrices (const T &t, const VectorX< T > &xt, const T &h, int trial, const std::function< void(const MatrixX< T > &J, const T &h, typename ImplicitIntegrator< T >::IterationMatrix *)> &compute_and_factor_iteration_matrix, typename ImplicitIntegrator< T >::IterationMatrix *iteration_matrix)
 Computes necessary matrices (Jacobian and iteration matrix) for Newton-Raphson (NR) iterations, as necessary. More...
 
void FreshenMatricesIfFullNewton (const T &t, const VectorX< T > &xt, const T &h, const std::function< void(const MatrixX< T > &J, const T &h, typename ImplicitIntegrator< T >::IterationMatrix *)> &compute_and_factor_iteration_matrix, typename ImplicitIntegrator< T >::IterationMatrix *iteration_matrix)
 Computes necessary matrices (Jacobian and iteration matrix) for full Newton-Raphson (NR) iterations, if full Newton-Raphson method is activated (if it's not activated, this method is a no-op). More...
 
bool IsUpdateZero (const VectorX< T > &xc, const VectorX< T > &dxc, double eps=-1.0) const
 Checks whether a proposed update is effectively zero, indicating that the Newton-Raphson process converged. More...
 
ConvergenceStatus CheckNewtonConvergence (int iteration, const VectorX< T > &xtplus, const VectorX< T > &dx, const T &dx_norm, const T &last_dx_norm) const
 Checks a Newton-Raphson iteration process for convergence. More...
 
virtual void DoImplicitIntegratorReset ()
 Derived classes can override this method to perform routines when Reset() is called. More...
 
bool IsBadJacobian (const MatrixX< T > &J) const
 Checks to see whether a Jacobian matrix is "bad" (has any NaN or Inf values) and needs to be recomputed. More...
 
MatrixX< T > & get_mutable_jacobian ()
 
void DoResetStatistics () override
 Resets any statistics particular to a specific integrator. More...
 
void DoReset () final
 Derived classes can override this method to perform routines when Reset() is called. More...
 
const MatrixX< T > & CalcJacobian (const T &t, const VectorX< T > &x)
 
void ComputeForwardDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xt, Context< T > *context, MatrixX< T > *J)
 
void ComputeCentralDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xt, Context< T > *context, MatrixX< T > *J)
 
void ComputeAutoDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xt, const Context< T > &context, MatrixX< T > *J)
 
void increment_num_iter_factorizations ()
 
void increment_jacobian_computation_derivative_evaluations (int count)
 
void increment_jacobian_evaluations ()
 
void set_jacobian_is_fresh (bool flag)
 
template<>
void ComputeAutoDiffJacobian (const System< AutoDiffXd > &, const AutoDiffXd &, const VectorX< AutoDiffXd > &, const Context< AutoDiffXd > &, MatrixX< AutoDiffXd > *)
 
- Protected Member Functions inherited from IntegratorBase< T >
const ContinuousState< T > & EvalTimeDerivatives (const Context< T > &context)
 Evaluates the derivative function and updates call statistics. More...
 
template<typename U >
const ContinuousState< U > & EvalTimeDerivatives (const System< U > &system, const Context< U > &context)
 Evaluates the derivative function (and updates call statistics). More...
 
void set_accuracy_in_use (double accuracy)
 Sets the working ("in use") accuracy for this integrator. More...
 
bool StepOnceErrorControlledAtMost (const T &h_max)
 Default code for advancing the continuous state of the system by a single step of h_max (or smaller, depending on error control). More...
 
CalcStateChangeNorm (const ContinuousState< T > &dx_state) const
 Computes the infinity norm of a change in continuous state. More...
 
std::pair< bool, T > CalcAdjustedStepSize (const T &err, const T &attempted_step_size, bool *at_minimum_step_size) const
 Calculates adjusted integrator step sizes toward keeping state variables within error bounds on the next integration step. More...
 
trajectories::PiecewisePolynomial< T > * get_mutable_dense_output ()
 Returns a mutable pointer to the internally-maintained PiecewisePolynomial instance, holding a representation of the continuous state trajectory since the last time StartDenseIntegration() was called. More...
 
bool DoDenseStep (const T &h)
 Calls DoStep(h) while recording the resulting step in the dense output. More...
 
ContinuousState< T > * get_mutable_error_estimate ()
 Gets an error estimate of the state variables recorded by the last call to StepOnceFixedSize(). More...
 
void set_actual_initial_step_size_taken (const T &h)
 
void set_smallest_adapted_step_size_taken (const T &h)
 Sets the size of the smallest-step-taken statistic as the result of a controlled integration step adjustment. More...
 
void set_largest_step_size_taken (const T &h)
 
void set_ideal_next_step_size (const T &h)
 

Constructor & Destructor Documentation

◆ RadauIntegrator() [1/3]

RadauIntegrator ( const RadauIntegrator< T, num_stages > &  )
delete

◆ RadauIntegrator() [2/3]

RadauIntegrator ( RadauIntegrator< T, num_stages > &&  )
delete

◆ RadauIntegrator() [3/3]

RadauIntegrator ( const System< T > &  system,
Context< T > *  context = nullptr 
)
explicit

◆ ~RadauIntegrator()

~RadauIntegrator ( )
finaldefault

Member Function Documentation

◆ get_error_estimate_order()

int get_error_estimate_order ( ) const
finalvirtual

This integrator uses embedded second order methods to compute estimates of the local truncation error.

The order of the asymptotic difference between the third-order Radau method and an embedded second order method is O(h³). The order of the asymptotic difference between the first-order Radau method and an embedded second order method is O(h²).

Implements IntegratorBase< T >.

◆ operator=() [1/2]

RadauIntegrator& operator= ( RadauIntegrator< T, num_stages > &&  )
delete

◆ operator=() [2/2]

RadauIntegrator& operator= ( const RadauIntegrator< T, num_stages > &  )
delete

◆ supports_error_estimation()

bool supports_error_estimation ( ) const
finalvirtual

Derived classes must override this function to indicate whether the integrator supports error estimation.

Without error estimation, the target accuracy setting (see accuracy settings) will be unused.

Implements IntegratorBase< T >.


The documentation for this class was generated from the following file: