euler_converter.cc

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#include <towr/variables/euler_converter.h>

#include <cassert>
#include <cmath>

namespace towr {


EulerConverter::EulerConverter (const NodeSpline::Ptr& euler)
{
  euler_ = euler;
  jac_wrt_nodes_structure_ = Jacobian(k3D, euler->GetNodeVariablesCount());
}

Eigen::Quaterniond
EulerConverter::GetQuaternionBaseToWorld (double t) const
{
  State ori = euler_->GetPoint(t);
  return GetQuaternionBaseToWorld(ori.p());
}

Eigen::Quaterniond
EulerConverter::GetQuaternionBaseToWorld (const EulerAngles& pos)
{
  Eigen::Matrix3d R_WB = GetRotationMatrixBaseToWorld(pos);
  return Eigen::Quaterniond(R_WB);
}

Eigen::Vector3d
EulerConverter::GetAngularVelocityInWorld (double t) const
{
  State ori = euler_->GetPoint(t);
  return GetAngularVelocityInWorld(ori.p(), ori.v());
}

Eigen::Vector3d
EulerConverter::GetAngularVelocityInWorld (const EulerAngles& pos,
                                           const EulerRates& vel)
{
  return GetM(pos)*vel;
}

Eigen::Vector3d
EulerConverter::GetAngularAccelerationInWorld (double t) const
{
  State ori = euler_->GetPoint(t);
  return GetAngularAccelerationInWorld(ori);
}

Eigen::Vector3d
EulerConverter::GetAngularAccelerationInWorld (State ori)
{
  return GetMdot(ori.p(), ori.v())*ori.v() + GetM(ori.p())*ori.a();
}

EulerConverter::Jacobian
EulerConverter::GetDerivOfAngVelWrtEulerNodes(double t) const
{
  Jacobian jac = jac_wrt_nodes_structure_;

  State ori = euler_->GetPoint(t);
  // convert to sparse, but also regard 0.0 as non-zero element, because
  // could turn nonzero during the course of the program
  JacobianRow vel = ori.v().transpose().sparseView(1.0, -1.0);
  Jacobian dVel_du  = euler_->GetJacobianWrtNodes(t, kVel);

  for (auto dim : {X,Y,Z}) {
    Jacobian dM_du = GetDerivMwrtNodes(t,dim);
    jac.row(dim) = vel*dM_du + GetM(ori.p()).row(dim)*dVel_du;
  }

  return jac;
}

EulerConverter::Jacobian
EulerConverter::GetDerivOfAngAccWrtEulerNodes (double t) const
{
  Jacobian jac = jac_wrt_nodes_structure_;


  State ori = euler_->GetPoint(t);
  // convert to sparse, but also regard 0.0 as non-zero element, because
  // could turn nonzero during the course of the program
  JacobianRow vel = ori.v().transpose().sparseView(1.0, -1.0);
  JacobianRow acc = ori.a().transpose().sparseView(1.0, -1.0);

  Jacobian dVel_du  = euler_->GetJacobianWrtNodes(t, kVel);
  Jacobian dAcc_du  = euler_->GetJacobianWrtNodes(t, kAcc);


  for (auto dim : {X,Y,Z}) {
    Jacobian dMdot_du = GetDerivMdotwrtNodes(t,dim);
    Jacobian dM_du    = GetDerivMwrtNodes(t,dim);

    jac.row(dim) = vel                               * dMdot_du
                   + GetMdot(ori.p(), ori.v()).row(dim)* dVel_du
                   + acc                             * dM_du
                   + GetM(ori.p()).row(dim)           * dAcc_du;
  }

  return jac;
}

EulerConverter::MatrixSXd
EulerConverter::GetM (const EulerAngles& xyz)
{
  double z = xyz(Z);
  double y = xyz(Y);

  // Euler ZYX rates to angular velocity
  // http://docs.leggedrobotics.com/kindr/cheatsheet_latest.pdf
  Jacobian M(k3D, k3D);

       /* - */           M.coeffRef(0,Y) = -sin(z);  M.coeffRef(0,X) =  cos(y)*cos(z);
       /* - */           M.coeffRef(1,Y) =  cos(z);  M.coeffRef(1,X) =  cos(y)*sin(z);
  M.coeffRef(2,Z) = 1.0;          /* - */            M.coeffRef(2,X) =  -sin(y);

  return M;
}

EulerConverter::MatrixSXd
EulerConverter::GetMdot (const EulerAngles& xyz,
                         const EulerRates& xyz_d)
{
  double z  = xyz(Z);
  double zd = xyz_d(Z);
  double y  = xyz(Y);
  double yd = xyz_d(Y);

  Jacobian Mdot(k3D, k3D);

  Mdot.coeffRef(0,Y) = -cos(z)*zd; Mdot.coeffRef(0,X) = -cos(z)*sin(y)*yd - cos(y)*sin(z)*zd;
  Mdot.coeffRef(1,Y) = -sin(z)*zd; Mdot.coeffRef(1,X) =  cos(y)*cos(z)*zd - sin(y)*sin(z)*yd;
              /* - */              Mdot.coeffRef(2,X) = -cos(y)*yd;

 return Mdot;
}

EulerConverter::Jacobian
EulerConverter::GetDerivMwrtNodes (double t, Dim3D ang_acc_dim) const
{
  State ori = euler_->GetPoint(t);

  double z = ori.p()(Z);
  double y = ori.p()(Y);
  JacobianRow jac_z = GetJac(t, kPos, Z);
  JacobianRow jac_y = GetJac(t, kPos, Y);

  Jacobian jac = jac_wrt_nodes_structure_;

  switch (ang_acc_dim) {
    case X: // basically derivative of top row (3 elements) of matrix M
      jac.row(Y) = -cos(z)*jac_z;
      jac.row(X) = -cos(z)*sin(y)*jac_y - cos(y)*sin(z)*jac_z;
      break;
    case Y: // middle row of M
      jac.row(Y) = -sin(z)*jac_z;
      jac.row(X) = cos(y)*cos(z)*jac_z - sin(y)*sin(z)*jac_y;
      break;
    case Z: // bottom row of M
      jac.row(X) = -cos(y)*jac_y;
      break;
    default:
      assert(false);
      break;
  }

  return jac;
}

EulerConverter::MatrixSXd
EulerConverter::GetRotationMatrixBaseToWorld (double t) const
{
  State ori = euler_->GetPoint(t);
  return GetRotationMatrixBaseToWorld(ori.p());
}

EulerConverter::MatrixSXd
EulerConverter::GetRotationMatrixBaseToWorld (const EulerAngles& xyz)
{
  double x = xyz(X);
  double y = xyz(Y);
  double z = xyz(Z);

  Eigen::Matrix3d M;
  //  http://docs.leggedrobotics.com/kindr/cheatsheet_latest.pdf (Euler ZYX)
  M << cos(y)*cos(z), cos(z)*sin(x)*sin(y) - cos(x)*sin(z), sin(x)*sin(z) + cos(x)*cos(z)*sin(y),
       cos(y)*sin(z), cos(x)*cos(z) + sin(x)*sin(y)*sin(z), cos(x)*sin(y)*sin(z) - cos(z)*sin(x),
             -sin(y),                        cos(y)*sin(x),                        cos(x)*cos(y);

  return M.sparseView(1.0, -1.0);
}

EulerConverter::Jacobian
EulerConverter::DerivOfRotVecMult (double t, const Vector3d& v, bool inverse) const
{
  JacRowMatrix Rd = GetDerivativeOfRotationMatrixWrtNodes(t);
  Jacobian jac = jac_wrt_nodes_structure_;

  for (int row : {X,Y,Z}) {
    for (int col : {X, Y, Z}) {
      // since for every rotation matrix R^(-1) = R^T, just swap rows and
      // columns for calculation of derivative of inverse rotation matrix
      JacobianRow jac_row = inverse? Rd.at(col).at(row) : Rd.at(row).at(col);
      jac.row(row) += v(col)*jac_row;
    }
  }

  return jac;
}

EulerConverter::JacRowMatrix
EulerConverter::GetDerivativeOfRotationMatrixWrtNodes (double t) const
{
  JacRowMatrix jac;

  State ori = euler_->GetPoint(t);
  double x = ori.p()(X);
  double y = ori.p()(Y);
  double z = ori.p()(Z);

  JacobianRow jac_x = GetJac(t, kPos, X);
  JacobianRow jac_y = GetJac(t, kPos, Y);
  JacobianRow jac_z = GetJac(t, kPos, Z);

  jac.at(X).at(X) = -cos(z)*sin(y)*jac_y - cos(y)*sin(z)*jac_z;
  jac.at(X).at(Y) =  sin(x)*sin(z)*jac_x - cos(x)*cos(z)*jac_z - sin(x)*sin(y)*sin(z)*jac_z + cos(x)*cos(z)*sin(y)*jac_x + cos(y)*cos(z)*sin(x)*jac_y;
  jac.at(X).at(Z) =  cos(x)*sin(z)*jac_x + cos(z)*sin(x)*jac_z - cos(z)*sin(x)*sin(y)*jac_x - cos(x)*sin(y)*sin(z)*jac_z + cos(x)*cos(y)*cos(z)*jac_y;

  jac.at(Y).at(X) = cos(y)*cos(z)*jac_z - sin(y)*sin(z)*jac_y;
  jac.at(Y).at(Y) = cos(x)*sin(y)*sin(z)*jac_x - cos(x)*sin(z)*jac_z - cos(z)*sin(x)*jac_x + cos(y)*sin(x)*sin(z)*jac_y + cos(z)*sin(x)*sin(y)*jac_z;
  jac.at(Y).at(Z) = sin(x)*sin(z)*jac_z - cos(x)*cos(z)*jac_x - sin(x)*sin(y)*sin(z)*jac_x + cos(x)*cos(y)*sin(z)*jac_y + cos(x)*cos(z)*sin(y)*jac_z;

  jac.at(Z).at(X) = -cos(y)*jac_y;
  jac.at(Z).at(Y) =  cos(x)*cos(y)*jac_x - sin(x)*sin(y)*jac_y;
  jac.at(Z).at(Z) = -cos(y)*sin(x)*jac_x - cos(x)*sin(y)*jac_y;

  return jac;
}

EulerConverter::Jacobian
EulerConverter::GetDerivMdotwrtNodes (double t, Dim3D ang_acc_dim) const
{
  State ori = euler_->GetPoint(t);

  double z  = ori.p()(Z);
  double zd = ori.v()(Z);
  double y  = ori.p()(Y);
  double yd = ori.v()(Y);

  JacobianRow jac_z  = GetJac(t, kPos, Z);
  JacobianRow jac_y  = GetJac(t, kPos, Y);
  JacobianRow jac_zd = GetJac(t, kVel, Z);
  JacobianRow jac_yd = GetJac(t, kVel, Y);

  Jacobian jac = jac_wrt_nodes_structure_;
  switch (ang_acc_dim) {
    case X: // derivative of top row (3 elements) of matrix M-dot
      jac.row(Y) = sin(z)*zd*jac_z - cos(z)*jac_zd;
      jac.row(X) = sin(y)*sin(z)*yd*jac_z - cos(y)*sin(z)*jac_zd - cos(y)*cos(z)*yd*jac_y - cos(y)*cos(z)*zd*jac_z - cos(z)*sin(y)*jac_yd + sin(y)*sin(z)*jac_y*zd;
      break;
    case Y: // middle row of M
      jac.row(Y) = - sin(z)*jac_zd - cos(z)*zd*jac_z;
      jac.row(X) = cos(y)*cos(z)*jac_zd - sin(y)*sin(z)*jac_yd - cos(y)*sin(z)*yd*jac_y - cos(z)*sin(y)*yd*jac_z - cos(z)*sin(y)*jac_y*zd - cos(y)*sin(z)*zd*jac_z;
      break;
    case Z: // bottom Row of M
      jac.row(X) = sin(y)*yd*jac_y - cos(y)*jac_yd;
      break;
    default:
      assert(false);
      break;
  }

  return jac;
}

EulerConverter::JacobianRow
EulerConverter::GetJac (double t, Dx deriv, Dim3D dim) const
{
  return euler_->GetJacobianWrtNodes(t, deriv).row(dim);
}

} /* namespace towr */