QuinticHermiteSpline.hpp

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// Copyright (c) FIRST and other WPILib contributors.
// Open Source Software; you can modify and/or share it under the terms of
// the WPILib BSD license file in the root directory of this project.
 
#pragma once
 
#include "wpi/math/linalg/EigenCore.hpp"
#include "wpi/math/spline/Spline.hpp"
#include "wpi/util/SymbolExports.hpp"
#include "wpi/util/array.hpp"
 
namespace wpi::math {
/**
 * Represents a hermite spline of degree 5.
 */
class WPILIB_DLLEXPORT QuinticHermiteSpline : public Spline<5> {
 public:
  /**
   * Constructs a quintic hermite spline with the specified control vectors.
   * Each control vector contains into about the location of the point, its
   * first derivative, and its second derivative.
   *
   * @param xInitialControlVector The control vector for the initial point in
   * the x dimension.
   * @param xFinalControlVector The control vector for the final point in
   * the x dimension.
   * @param yInitialControlVector The control vector for the initial point in
   * the y dimension.
   * @param yFinalControlVector The control vector for the final point in
   * the y dimension.
   */
  QuinticHermiteSpline(wpi::util::array<double, 3> xInitialControlVector,
                       wpi::util::array<double, 3> xFinalControlVector,
                       wpi::util::array<double, 3> yInitialControlVector,
                       wpi::util::array<double, 3> yFinalControlVector)
      : m_initialControlVector{xInitialControlVector, yInitialControlVector},
        m_finalControlVector{xFinalControlVector, yFinalControlVector} {
    const auto hermite = MakeHermiteBasis();
    const auto x =
        ControlVectorFromArrays(xInitialControlVector, xFinalControlVector);
    const auto y =
        ControlVectorFromArrays(yInitialControlVector, yFinalControlVector);
 
    // Populate first two rows with coefficients.
    m_coefficients.template block<1, 6>(0, 0) = (hermite * x).transpose();
    m_coefficients.template block<1, 6>(1, 0) = (hermite * y).transpose();
 
    // Populate Row 2 and Row 3 with the derivatives of the equations above.
    // Then populate row 4 and 5 with the second derivatives.
    for (int i = 0; i < 6; i++) {
      // Here, we are multiplying by (5 - i) to manually take the derivative.
      // The power of the term in index 0 is 5, index 1 is 4 and so on. To find
      // the coefficient of the derivative, we can use the power rule and
      // multiply the existing coefficient by its power.
      m_coefficients.template block<2, 1>(2, i) =
          m_coefficients.template block<2, 1>(0, i) * (5 - i);
    }
    for (int i = 0; i < 5; i++) {
      // Here, we are multiplying by (4 - i) to manually take the derivative.
      // The power of the term in index 0 is 4, index 1 is 3 and so on. To find
      // the coefficient of the derivative, we can use the power rule and
      // multiply the existing coefficient by its power.
      m_coefficients.template block<2, 1>(4, i) =
          m_coefficients.template block<2, 1>(2, i) * (4 - i);
    }
  }
 
  /**
   * Returns the coefficients matrix.
   * @return The coefficients matrix.
   */
  Matrixd<6, 6> Coefficients() const override { return m_coefficients; }
 
  /**
   * Returns the initial control vector that created this spline.
   *
   * @return The initial control vector that created this spline.
   */
  const ControlVector& GetInitialControlVector() const override {
    return m_initialControlVector;
  }
 
  /**
   * Returns the final control vector that created this spline.
   *
   * @return The final control vector that created this spline.
   */
  const ControlVector& GetFinalControlVector() const override {
    return m_finalControlVector;
  }
 
 private:
  Matrixd<6, 6> m_coefficients = Matrixd<6, 6>::Zero();
 
  ControlVector m_initialControlVector;
  ControlVector m_finalControlVector;
 
  /**
   * Returns the hermite basis matrix for quintic hermite spline interpolation.
   * @return The hermite basis matrix for quintic hermite spline interpolation.
   */
  static constexpr Matrixd<6, 6> MakeHermiteBasis() {
    // Given P(i), P'(i), P"(i), P(i+1), P'(i+1), P"(i+1), the control vectors,
    // we want to find the coefficients of the spline
    // P(t) = a₅t⁵ + a₄t⁴ + a₃t³ + a₂t² + a₁t + a₀.
    //
    // P(i)    = P(0)  = a₀
    // P'(i)   = P'(0) = a₁
    // P''(i)  = P"(0) = 2a₂
    // P(i+1)  = P(1)  = a₅ + a₄ + a₃ + a₂ + a₁ + a₀
    // P'(i+1) = P'(1) = 5a₅ + 4a₄ + 3a₃ + 2a₂ + a₁
    // P"(i+1) = P"(1) = 20a₅ + 12a₄ + 6a₃ + 2a₂
    //
    // [P(i)   ] = [ 0  0  0  0  0  1][a₅]
    // [P'(i)  ] = [ 0  0  0  0  1  0][a₄]
    // [P"(i)  ] = [ 0  0  0  2  0  0][a₃]
    // [P(i+1) ] = [ 1  1  1  1  1  1][a₂]
    // [P'(i+1)] = [ 5  4  3  2  1  0][a₁]
    // [P"(i+1)] = [20 12  6  2  0  0][a₀]
    //
    // To solve for the coefficients, we can invert the 6x6 matrix and move it
    // to the other side of the equation.
    //
    // [a₅] = [ -6.0  -3.0  -0.5   6.0  -3.0   0.5][P(i)   ]
    // [a₄] = [ 15.0   8.0   1.5 -15.0   7.0  -1.0][P'(i)  ]
    // [a₃] = [-10.0  -6.0  -1.5  10.0  -4.0   0.5][P"(i)  ]
    // [a₂] = [  0.0   0.0   0.5   0.0   0.0   0.0][P(i+1) ]
    // [a₁] = [  0.0   1.0   0.0   0.0   0.0   0.0][P'(i+1)]
    // [a₀] = [  1.0   0.0   0.0   0.0   0.0   0.0][P"(i+1)]
    return Matrixd<6, 6>{{-06.0, -03.0, -00.5, +06.0, -03.0, +00.5},
                         {+15.0, +08.0, +01.5, -15.0, +07.0, -01.0},
                         {-10.0, -06.0, -01.5, +10.0, -04.0, +00.5},
                         {+00.0, +00.0, +00.5, +00.0, +00.0, +00.0},
                         {+00.0, +01.0, +00.0, +00.0, +00.0, +00.0},
                         {+01.0, +00.0, +00.0, +00.0, +00.0, +00.0}};
  }
 
  /**
   * Returns the control vector for each dimension as a matrix from the
   * user-provided arrays in the constructor.
   *
   * @param initialVector The control vector for the initial point.
   * @param finalVector The control vector for the final point.
   *
   * @return The control vector matrix for a dimension.
   */
  static constexpr Vectord<6> ControlVectorFromArrays(
      wpi::util::array<double, 3> initialVector,
      wpi::util::array<double, 3> finalVector) {
    return Vectord<6>{{initialVector[0]}, {initialVector[1]},
                      {initialVector[2]}, {finalVector[0]},
                      {finalVector[1]},   {finalVector[2]}};
  }
};
}  // namespace wpi::math
 
#include "wpi/math/spline/proto/QuinticHermiteSplineProto.hpp"
#include "wpi/math/spline/struct/QuinticHermiteSplineStruct.hpp"