template<typename T>
class maliput::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; π€), such that F(u; π€) = β«α΅₯α΅ f(x; π€) dx where f : β β β , u β β, v β β, π€ β βα΅. The parameter vector π€ allows for generic function definitions, which can later be evaluated for any instance of said vector. Also, note that π€ can be understood as an m-tuple or as an element of βα΅, 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(Ο; ΞΎ) = β«α΅ β(1 - ΞΎΒ² sinΒ² ΞΈ)β»ΒΉ dΞΈ becomes straightforward by defining f(x; π€) β β(1 - kβΒ² sinΒ² x)β»ΒΉ with π€ β [ΞΎ] and evaluating F(u; π€) at u = Ο.
- As the bearings in a rotating machine age over time, these are more likely to fail. Let Ξ³ be a random variable describing the time to first bearing failure, described by a family of probability density functions gᡧ(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 < Ξ³ β€ N mo.; l) = Gᡧ(N mo.; l) - Gᡧ(0; l), where Gᡧ(y; l) is the family of cumulative density functions, parameterized by bearing load l, and G'ᡧ(y; l) = gᡧ(y; l). Therefore, defining f(x; π€) β gᡧ(x; kβ) with π€ β [l] and evaluating F(u; π€) at u = N yields the result.
@tparam_nonsymbolic_scalar
Evaluates and yields an approximation of the definite integral F(u; π€) = β«α΅₯α΅ f(x; π€) dx for v β€ u β€ w, i.e.
the closed interval that goes from the lower integration bound v (see definition in class documentation) to the uppermost integration bound w, using the parameter vector π€ (see definition in class documentation) if present in values, falling back to the ones given on construction if missing.
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; π€) from v to w), creating a scalar dense output over that [v, w] interval along the way.
- Parameters
-
| w | The uppermost integration bound. Usually, v < w as an empty dense output would result if v = w. |
| values | The specified values for the integration. |
- Returns
- A dense approximation to F(u; π€) (that is, a function), defined for v β€ u β€ w.
- Note
- The larger the given
w 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.
- Precondition
- The given uppermost integration bound
w must be larger than or equal to the lower integration bound v.
-
If given, the dimension of the parameter vector
values.k must match that of the parameter vector π€ in the default specified values given on construction.
- Exceptions
-
| std::exception | if any of the preconditions is not met. |
