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
-
| 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.
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
-
