Drake
Drake C++ Documentation
InitialValueProblem< T > Class Template Reference

## Detailed Description

### template<typename T> class drake::systems::InitialValueProblem< T >

A general initial value problem (or IVP) representation class, that allows evaluating the 𝐱(t; 𝐤) solution function to the given ODE d𝐱/dt = f(t, 𝐱; 𝐤), where f : t ⨯ 𝐱 → ℝⁿ, t ∈ ℝ, 𝐱 ∈ ℝⁿ, 𝐤 ∈ ℝᵐ, provided an initial condition 𝐱(t₀; 𝐤) = 𝐱₀.

The parameter vector 𝐤 allows for generic IVP definitions, which can later be solved for any instance of said vector.

By default, an explicit 3rd order RungeKutta integration scheme is used.

The implementation of this class performs basic computation caching, optimizing away repeated integration whenever the IVP is solved for increasing values of time t while both initial conditions and parameters are kept constant, e.g. if solved for t₁ > t₀ first, solving for t₂ > t₁ will only require integrating from t₁ onward.

Additionally, IntegratorBase's dense output support can be leveraged to efficiently approximate the IVP solution within closed intervals of t. This is convenient when there's a need for a more dense sampling of the IVP solution than what would be available through either fixed or error-controlled step integration (for a given accuracy), or when the IVP is to be solved repeatedly for arbitrarily many t values within a given interval. See documentation of the internally held IntegratorBase subclass instance (either the default or a user-defined one, set via reset_integrator()) for further reference on the specific dense output technique in use.

For further insight into its use, consider the following examples:

• The momentum 𝐩 of a particle of mass m that is traveling through a volume of a gas with dynamic viscosity μ can be described by d𝐩/dt = -μ * 𝐩/m. At time t₀, the particle carries an initial momentum 𝐩₀. In this context, t is unused (the ODE is autonomous), 𝐱 ≜ 𝐩, 𝐤 ≜ [m, μ], t₀ = 0, 𝐱₀ ≜ 𝐩₀, d𝐱/dt = f(t, 𝐱; 𝐤) = -k₂ * 𝐱 / k₁.
• The velocity 𝐯 of the same particle in the same exact conditions as before, but when a time varying force 𝐅(t) is applied to it, can be be described by d𝐯/dt = (𝐅(t) - μ * 𝐯) / m. In this context, 𝐱 ≜ 𝐯, 𝐤 ≜ [m, μ], 𝐱₀ ≜ 𝐯₀, d𝐱/dt = f(t, 𝐱; 𝐤) = (𝐅(t) - k₂ * 𝐱) / k₁.
Template Parameters
 T The scalar type, which must be one of the default nonsymbolic scalars.

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

## Public Types

using OdeFunction = std::function< VectorX< T >(const T &t, const VectorX< T > &x, const VectorX< T > &k)>
General ODE system d𝐱/dt = f(t, 𝐱; 𝐤) function type. More...

## Public Member Functions

InitialValueProblem (const OdeFunction &ode_function, const Eigen::Ref< const VectorX< T >> &x0, const Eigen::Ref< const VectorX< T >> &k=Vector0< T >{})
Constructs an IVP described by the given ode_function, using x0 as initial conditions, and parameterized with k. More...

VectorX< T > Solve (const T &t0, const T &tf) const
Solves the IVP from the initial time t0 up to time tf, using the initial state vector 𝐱₀ and parameter vector 𝐤 provided in the constructor. More...

std::unique_ptr< DenseOutput< T > > DenseSolve (const T &t0, const T &tf) const
Solves and yields an approximation of the IVP solution x(t; 𝐤) for the closed time interval between the given initial time t0 and the given final time tf, using initial state 𝐱₀ and parameter vector 𝐤 provided in the constructor. More...

template<typename Integrator , typename... Args>
Integratorreset_integrator (Args &&... args)
Resets the internal integrator instance by in-place construction of the given integrator type. More...

const IntegratorBase< T > & get_integrator () const
Gets a reference to the internal integrator instance. More...

IntegratorBase< T > & get_mutable_integrator ()
Gets a mutable reference to the internal integrator instance. More...

Does not allow copy, move, or assignment
InitialValueProblem (const InitialValueProblem &)=delete

InitialValueProblemoperator= (const InitialValueProblem &)=delete

InitialValueProblem (InitialValueProblem &&)=delete

InitialValueProblemoperator= (InitialValueProblem &&)=delete

## Static Public Attributes

static const double kDefaultAccuracy
Default integration accuracy in the relative tolerance sense. More...

static const T kInitialStepSize
Default initial integration step size. More...

static const T kMaxStepSize
Default maximum integration step size. More...

## ◆ OdeFunction

 using OdeFunction = std::function(const T& t, const VectorX& x, const VectorX& k)>

General ODE system d𝐱/dt = f(t, 𝐱; 𝐤) function type.

Parameters
 t The independent scalar variable t ∈ ℝ. x The dependent vector variable 𝐱 ∈ ℝⁿ. k The vector of parameters 𝐤 ∈ ℝᵐ.
Returns
The derivative vector d𝐱/dt ∈ ℝⁿ.

## ◆ InitialValueProblem() [1/3]

 InitialValueProblem ( const InitialValueProblem< T > & )
delete

## ◆ InitialValueProblem() [2/3]

 InitialValueProblem ( InitialValueProblem< T > && )
delete

## ◆ InitialValueProblem() [3/3]

 InitialValueProblem ( const OdeFunction & ode_function, const Eigen::Ref< const VectorX< T >> & x0, const Eigen::Ref< const VectorX< T >> & k = Vector0< T >{} )

Constructs an IVP described by the given ode_function, using x0 as initial conditions, and parameterized with k.

Parameters
 ode_function The ODE function f(t, 𝐱; 𝐤) that describes the state evolution over time. x0 The initial state vector 𝐱₀ ∈ ℝⁿ. k The parameter vector 𝐤 ∈ ℝᵐ. By default m=0 (no parameters).

## ◆ DenseSolve()

 std::unique_ptr > DenseSolve ( const T & t0, const T & tf ) const

Solves and yields an approximation of the IVP solution x(t; 𝐤) for the closed time interval between the given initial time t0 and the given final time tf, using initial state 𝐱₀ and parameter vector 𝐤 provided in the constructor.

To this end, the wrapped IntegratorBase instance solves this IVP, advancing time and state from t₀ and 𝐱₀ = 𝐱(t₀) to tf and 𝐱(tf), creating a dense output over that [t₀, tf] interval along the way.

Parameters
 tf The IVP will be solved up to this time, which must be ≥ t₀. Usually, t₀ < tf as an empty dense output would result if t₀ = tf.
Returns
A dense approximation to 𝐱(t; 𝐤) with 𝐱(t₀; 𝐤) = 𝐱₀, defined for t₀ ≤ t ≤ tf.
Note
The larger the given tf value is, the larger the approximated interval will be. See documentation of the specific dense output technique in use for reference on performance impact as this interval grows.
Exceptions
 std::exception if t0 > tf.

## ◆ get_integrator()

 const IntegratorBase& get_integrator ( ) const

Gets a reference to the internal integrator instance.

## ◆ get_mutable_integrator()

 IntegratorBase& get_mutable_integrator ( )

Gets a mutable reference to the internal integrator instance.

## ◆ operator=() [1/2]

 InitialValueProblem& operator= ( const InitialValueProblem< T > & )
delete

## ◆ operator=() [2/2]

 InitialValueProblem& operator= ( InitialValueProblem< T > && )
delete

## ◆ reset_integrator()

 Integrator* reset_integrator ( Args &&... args )

Resets the internal integrator instance by in-place construction of the given integrator type.

A usage example is shown below.

ivp.reset_integrator<RungeKutta2Integrator<T>>(max_step);
Parameters
 args The integrator type-specific arguments.
Returns
The new integrator instance.
Template Parameters
 Integrator The integrator type, which must be an IntegratorBase subclass. Args The integrator specific argument types.
Warning
This operation invalidates pointers returned by InitialValueProblem::get_integrator() and InitialValueProblem::get_mutable_integrator().

## ◆ Solve()

 VectorX Solve ( const T & t0, const T & tf ) const

Solves the IVP from the initial time t0 up to time tf, using the initial state vector 𝐱₀ and parameter vector 𝐤 provided in the constructor.

Exceptions
 std::exception if t0 > tf.

## ◆ kDefaultAccuracy

 const double kDefaultAccuracy
static

Default integration accuracy in the relative tolerance sense.

## ◆ kInitialStepSize

 const T kInitialStepSize
static

Default initial integration step size.

## ◆ kMaxStepSize

 const T kMaxStepSize
static

Default maximum integration step size.

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