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

## Detailed Description

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

A thin wrapper of the ScalarInitialValueProblem class that, in concert with Drake's ODE initial value problem solvers ("integrators"), provide the ability to perform quadrature on an arbitrary scalar integrable function.

That is, it allows the evaluation of an antiderivative function F(u; 𝐤), such that F(u; 𝐤) = ∫ᵥᵘ f(x; 𝐤) dx where f : ℝ → ℝ , u ∈ ℝ, v ∈ ℝ, 𝐤 ∈ ℝᵐ. The parameter vector 𝐤 allows for generic function definitions, which can later be evaluated for any instance of said vector. Also, note that 𝐤 can be understood as an m-tuple or as an element of ℝᵐ, the vector space, depending on how it is used by the integrable function.

See ScalarInitialValueProblem class documentation for information on caching support and dense output usage for improved efficiency in antiderivative function F evaluation.

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

• Solving the elliptic integral of the first kind E(φ; ξ) = ∫ᵠ √(1 - ξ² sin² θ)⁻¹ dθ becomes straightforward by defining f(x; 𝐤) ≜ √(1 - k₀² sin² x)⁻¹ with 𝐤 ≜ [ξ] and evaluating F(u; 𝐤) at u = φ.
• As the bearings in a rotating machine age over time, these are more likely to fail. Let γ be a random variable describing the time to first bearing failure, described by a family of probability density functions gᵧ(y; l) parameterized by bearing load l. In this context, the probability of a bearing under load to fail during the first N months becomes P(0 < γ ≤ N mo.; l) = Gᵧ(N mo.; l) - Gᵧ(0; l), where Gᵧ(y; l) is the family of cumulative density functions, parameterized by bearing load l, and G'ᵧ(y; l) = gᵧ(y; l). Therefore, defining f(x; 𝐤) ≜ gᵧ(x; k₀) with 𝐤 ≜ [l] and evaluating F(u; 𝐤) at u = N yields the result.
Template Parameters
 T The scalar type, which must be one of the default nonsymbolic scalars.

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

## Public Types

using IntegrableFunction = std::function< T(const T &x, const VectorX< T > &k)>
Scalar integrable function f(x; 𝐤) type. More...

## Public Member Functions

AntiderivativeFunction (const IntegrableFunction &integrable_function, const Eigen::Ref< const VectorX< T >> &k=Vector0< T >{})
Constructs the antiderivative function of the given integrable_function, parameterized with k. More...

Evaluate (const T &v, const T &u) const
Evaluates the definite integral F(u; 𝐤) = ∫ᵥᵘ f(x; 𝐤) dx from the lower integration bound v to u using the parameter vector 𝐤 specified in the constructor (see definition in class documentation). More...

std::unique_ptr< ScalarDenseOutput< T > > MakeDenseEvalFunction (const T &v, const T &w) const
Evaluates and yields an approximation of the definite integral F(u; 𝐤) = ∫ᵥᵘ f(x; 𝐤) dx for v ≤ u ≤ w, i.e. More...

template<typename Integrator , typename... Args>
Integratorreset_integrator (Args &&... args)
Resets the internal integrator instance. 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
AntiderivativeFunction (const AntiderivativeFunction &)=delete

AntiderivativeFunctionoperator= (const AntiderivativeFunction &)=delete

AntiderivativeFunction (AntiderivativeFunction &&)=delete

AntiderivativeFunctionoperator= (AntiderivativeFunction &&)=delete

## ◆ IntegrableFunction

 using IntegrableFunction = std::function& k)>

Scalar integrable function f(x; 𝐤) type.

Parameters
 x The variable of integration x ∈ ℝ . k The parameter vector 𝐤 ∈ ℝᵐ.
Returns
The function value f(x; k).

## ◆ AntiderivativeFunction() [1/3]

 AntiderivativeFunction ( const AntiderivativeFunction< T > & )
delete

## ◆ AntiderivativeFunction() [2/3]

 AntiderivativeFunction ( AntiderivativeFunction< T > && )
delete

## ◆ AntiderivativeFunction() [3/3]

 AntiderivativeFunction ( const IntegrableFunction & integrable_function, const Eigen::Ref< const VectorX< T >> & k = Vector0< T >{} )

Constructs the antiderivative function of the given integrable_function, parameterized with k.

Parameters
 integrable_function The function f(x; 𝐤) to be integrated. 𝐤 ∈ ℝᵐ is the vector of parameters. The default is the empty vector (indicating no parameters).

## ◆ Evaluate()

 T Evaluate ( const T & v, const T & u ) const

Evaluates the definite integral F(u; 𝐤) = ∫ᵥᵘ f(x; 𝐤) dx from the lower integration bound v to u using the parameter vector 𝐤 specified in the constructor (see definition in class documentation).

Parameters
 v The lower integration bound. u The upper integration bound.
Returns
The value of the definite integral.
Exceptions
 std::exception if v > u.

## ◆ 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.

## ◆ MakeDenseEvalFunction()

 std::unique_ptr > MakeDenseEvalFunction ( const T & v, const T & w ) const

Evaluates and yields an approximation of the definite integral F(u; 𝐤) = ∫ᵥᵘ f(x; 𝐤) dx for v ≤ u ≤ w, i.e.

the closed interval that goes from the lower integration bound v to the uppermost integration bound w, using the parameter vector 𝐤 specified in the constructor (see definition in class documentation).

To this end, the wrapped IntegratorBase instance solves the integral from v to w (i.e. advances the state x of its differential form x'(t) = f(x; 𝐤) from v to w), creating a scalar dense output over that [v, w] interval along the way.

Parameters
 v The lower integration bound. w The uppermost integration bound. Usually, v < w as an empty dense output would result if v = w.
Returns
A dense approximation to F(u; 𝐤) (that is, a function), defined for v ≤ u ≤ w.
Note
The larger the given w value is, the larger the approximated interval will be. See documentation of the specific dense output technique used by the internally held IntegratorBase subclass instance for more details.
Exceptions
 std::exception if v > w.

## ◆ operator=() [1/2]

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

## ◆ operator=() [2/2]

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

## ◆ reset_integrator()

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

Resets the internal integrator instance.

A usage example is shown below.

antiderivative_f.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 AntiderivativeFunction::get_integrator() and AntiderivativeFunction::get_mutable_integrator().

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