Beispiel #1
0
RigidBodySystem::StateVector<double> RigidBodySystem::dynamics(const double& t, const RigidBodySystem::StateVector<double>& x, const RigidBodySystem::InputVector<double>& u) const {
  using namespace std;
  using namespace Eigen;
  eigen_aligned_unordered_map<const RigidBody *, Matrix<double, 6, 1> > f_ext;

  // todo: make kinematics cache once and re-use it (but have to make one per type)
  auto nq = tree->num_positions;
  auto nv = tree->num_velocities;
  auto num_actuators = tree->actuators.size();
  auto q = x.topRows(nq);
  auto v = x.bottomRows(nv);
  auto kinsol = tree->doKinematics(q,v);

  // todo: preallocate the optimization problem and constraints, and simply update them then solve on each function eval.
  // happily, this clunkier version seems fast enough for now
  // the optimization framework should support this (though it has not been tested thoroughly yet)
  OptimizationProblem prog;
  auto const & vdot = prog.addContinuousVariables(nv,"vdot");

  auto H = tree->massMatrix(kinsol);
  Eigen::MatrixXd H_and_neg_JT = H;

  { // loop through rigid body force elements and accumulate f_ext

    // todo: distinguish between AppliedForce and ConstraintForce
    // todo: have AppliedForce output tau (instead of f_ext).  it's more direct, and just as easy to compute.

    int u_index = 0;
    const NullVector<double> force_state;  // todo:  will have to handle this case
    for (auto const & prop : props) {
      RigidBodyFrame* frame = prop->getFrame();
      RigidBody* body = frame->body.get();
      int num_inputs = 1;  // todo: generalize this
      RigidBodyPropellor::InputVector<double> u_i(u.middleRows(u_index,num_inputs));
      // todo: push the frame to body transform into the dynamicsBias method?
      Matrix<double,6,1> f_ext_i = transformSpatialForce(frame->transform_to_body,prop->output(t,force_state,u_i,kinsol));
      if (f_ext.find(body)==f_ext.end()) f_ext[body] = f_ext_i;
      else f_ext[body] = f_ext[body]+f_ext_i;
      u_index += num_inputs;
    }
  }

  VectorXd C = tree->dynamicsBiasTerm(kinsol,f_ext);
  if (num_actuators > 0) C -= tree->B*u.topRows(num_actuators);

  { // apply contact forces
    const bool use_multi_contact = false;
    VectorXd phi;
    Matrix3Xd normal, xA, xB;
    vector<int> bodyA_idx, bodyB_idx;
    if (use_multi_contact)
      tree->potentialCollisions(kinsol,phi,normal,xA,xB,bodyA_idx,bodyB_idx);
    else
      tree->collisionDetect(kinsol,phi,normal,xA,xB,bodyA_idx,bodyB_idx);

    for (int i=0; i<phi.rows(); i++) {
      if (phi(i)<0.0) { // then i have contact
        double mu = 1.0; // todo: make this a parameter

        // todo: move this entire block to a shared an updated "contactJacobian" method in RigidBodyTree
        auto JA = tree->transformPointsJacobian(kinsol, xA.col(i), bodyA_idx[i], 0, false);
        auto JB = tree->transformPointsJacobian(kinsol, xB.col(i), bodyB_idx[i], 0, false);
        Vector3d this_normal = normal.col(i);

        // compute the surface tangent basis
        Vector3d tangent1;
        if (1.0 - this_normal(2) < EPSILON) { // handle the unit-normal case (since it's unit length, just check z)
          tangent1 << 1.0, 0.0, 0.0;
        } else if (1 + this_normal(2) < EPSILON) {
          tangent1 << -1.0, 0.0, 0.0;  //same for the reflected case
        } else {// now the general case
          tangent1 << this_normal(1), -this_normal(0), 0.0;
          tangent1 /= sqrt(this_normal(1)*this_normal(1) + this_normal(0)*this_normal(0));
        }
        Vector3d tangent2 = tangent1.cross(this_normal);
        Matrix3d R;  // rotation into normal coordinates
        R << tangent1, tangent2, this_normal;
        auto J = R*(JA-JB);  // J = [ D1; D2; n ]
        auto relative_velocity = J*v;  // [ tangent1dot; tangent2dot; phidot ]

        if (false) {
          // spring law for normal force:  fA_normal = -k*phi - b*phidot
          // and damping for tangential force:  fA_tangent = -b*tangentdot (bounded by the friction cone)
          double k = 150, b = k / 10;  // todo: put these somewhere better... or make them parameters?
          Vector3d fA;
          fA(2) = -k * phi(i) - b * relative_velocity(2);
          fA.head(2) = -std::min(b, mu*fA(2)/(relative_velocity.head(2).norm()+EPSILON)) * relative_velocity.head(2);  // epsilon to avoid divide by zero

          // equal and opposite: fB = -fA.
          // tau = (R*JA)^T fA + (R*JB)^T fB = J^T fA
          C -= J.transpose()*fA;
        } else { // linear acceleration constraints (more expensive, but less tuning required for robot mass, etc)
          // note: this does not work for the multi-contact case (it overly constrains the motion of the link).  Perhaps if I made them inequality constraints...
          static_assert(!use_multi_contact, "The acceleration contact constraints do not play well with multi-contact");

          // phiddot = -2*alpha*phidot - alpha^2*phi   // critically damped response
          // tangential_velocity_dot = -2*alpha*tangential_velocity
          double alpha = 20;  // todo: put this somewhere better... or make them parameters?
          Vector3d desired_relative_acceleration = -2*alpha*relative_velocity;
          desired_relative_acceleration(2) += -alpha*alpha*phi(i);
          // relative_acceleration = J*vdot + R*(JAdotv - JBdotv) // uses the standard dnormal/dq = 0 assumption

          cout << "phi = " << phi << endl;
          cout << "desired acceleration = " << desired_relative_acceleration.transpose() << endl;
//          cout << "acceleration = " << (J*vdot + R*(JAdotv - JBdotv)).transpose() << endl;

          prog.addContinuousVariables(3,"contact normal force");
          auto JAdotv = tree->transformPointsJacobianDotTimesV(kinsol, xA.col(i).eval(), bodyA_idx[i], 0);
          auto JBdotv = tree->transformPointsJacobianDotTimesV(kinsol, xB.col(i).eval(), bodyB_idx[i], 0);

          prog.addLinearEqualityConstraint(J,desired_relative_acceleration - R*(JAdotv - JBdotv),{vdot});
          H_and_neg_JT.conservativeResize(NoChange,H_and_neg_JT.cols()+3);
          H_and_neg_JT.rightCols(3) = -J.transpose();
        }
      }
    }
  }

  if (tree->getNumPositionConstraints()) {
    int nc = tree->getNumPositionConstraints();
    const double alpha = 5.0;  // 1/time constant of position constraint satisfaction (see my latex rigid body notes)

    prog.addContinuousVariables(nc,"position constraint force");  // don't actually need to use the decision variable reference that would be returned...

    // then compute the constraint force
    auto phi = tree->positionConstraints(kinsol);
    auto J = tree->positionConstraintsJacobian(kinsol,false);
    auto Jdotv = tree->positionConstraintsJacDotTimesV(kinsol);

    // phiddot = -2 alpha phidot - alpha^2 phi  (0 + critically damped stabilization term)
    prog.addLinearEqualityConstraint(J,-(Jdotv + 2*alpha*J*v + alpha*alpha*phi),{vdot});
    H_and_neg_JT.conservativeResize(NoChange,H_and_neg_JT.cols()+J.rows());
    H_and_neg_JT.rightCols(J.rows()) = -J.transpose();
  }

  // add [H,-J^T]*[vdot;f] = -C
  prog.addLinearEqualityConstraint(H_and_neg_JT,-C);

  prog.solve();
  //      prog.printSolution();

  StateVector<double> dot(nq+nv);
  dot << kinsol.transformPositionDotMappingToVelocityMapping(Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>::Identity(nq, nq))*v, vdot.value();
  return dot;
}
RigidBodySystem::StateVector<double> RigidBodySystem::dynamics(
    const double& t, const RigidBodySystem::StateVector<double>& x,
    const RigidBodySystem::InputVector<double>& u) const {
  using namespace std;
  using namespace Eigen;
  eigen_aligned_unordered_map<const RigidBody*, Matrix<double, 6, 1> > f_ext;

  // todo: make kinematics cache once and re-use it (but have to make one per
  // type)
  auto nq = tree->num_positions;
  auto nv = tree->num_velocities;
  auto num_actuators = tree->actuators.size();
  auto q = x.topRows(nq);
  auto v = x.bottomRows(nv);
  auto kinsol = tree->doKinematics(q, v);

  // todo: preallocate the optimization problem and constraints, and simply
  // update them then solve on each function eval.
  // happily, this clunkier version seems fast enough for now
  // the optimization framework should support this (though it has not been
  // tested thoroughly yet)
  OptimizationProblem prog;
  auto const& vdot = prog.AddContinuousVariables(nv, "vdot");

  auto H = tree->massMatrix(kinsol);
  Eigen::MatrixXd H_and_neg_JT = H;

  VectorXd C = tree->dynamicsBiasTerm(kinsol, f_ext);
  if (num_actuators > 0) C -= tree->B * u.topRows(num_actuators);

  // loop through rigid body force elements
  {
    // todo: distinguish between AppliedForce and ConstraintForce

    size_t u_index = 0;
    for (auto const& f : force_elements) {
      size_t num_inputs = f->getNumInputs();
      VectorXd force_input(u.middleRows(u_index, num_inputs));
      C -= f->output(t, force_input, kinsol);
      u_index += num_inputs;
    }
  }

  // apply joint limit forces
  {
    for (auto const& b : tree->bodies) {
      if (!b->hasParent()) continue;
      auto const& joint = b->getJoint();
      if (joint.getNumPositions() == 1 &&
          joint.getNumVelocities() ==
              1) {  // taking advantage of only single-axis joints having joint
                    // limits makes things easier/faster here
        double qmin = joint.getJointLimitMin()(0),
               qmax = joint.getJointLimitMax()(0);
        // tau = k*(qlimit-q) - b(qdot)
        if (q(b->position_num_start) < qmin)
          C(b->velocity_num_start) -=
              penetration_stiffness * (qmin - q(b->position_num_start)) -
              penetration_damping * v(b->velocity_num_start);
        else if (q(b->position_num_start) > qmax)
          C(b->velocity_num_start) -=
              penetration_stiffness * (qmax - q(b->position_num_start)) -
              penetration_damping * v(b->velocity_num_start);
      }
    }
  }

  // apply contact forces
  {
    VectorXd phi;
    Matrix3Xd normal, xA, xB;
    vector<int> bodyA_idx, bodyB_idx;
    if (use_multi_contact)
      tree->potentialCollisions(kinsol, phi, normal, xA, xB, bodyA_idx,
                                bodyB_idx);
    else
      tree->collisionDetect(kinsol, phi, normal, xA, xB, bodyA_idx, bodyB_idx);

    for (int i = 0; i < phi.rows(); i++) {
      if (phi(i) < 0.0) {  // then i have contact
        // todo: move this entire block to a shared an updated "contactJacobian"
        // method in RigidBodyTree
        auto JA = tree->transformPointsJacobian(kinsol, xA.col(i), bodyA_idx[i],
                                                0, false);
        auto JB = tree->transformPointsJacobian(kinsol, xB.col(i), bodyB_idx[i],
                                                0, false);
        Vector3d this_normal = normal.col(i);

        // compute the surface tangent basis
        Vector3d tangent1;
        if (1.0 - this_normal(2) < EPSILON) {  // handle the unit-normal case
                                               // (since it's unit length, just
                                               // check z)
          tangent1 << 1.0, 0.0, 0.0;
        } else if (1 + this_normal(2) < EPSILON) {
          tangent1 << -1.0, 0.0, 0.0;  // same for the reflected case
        } else {                       // now the general case
          tangent1 << this_normal(1), -this_normal(0), 0.0;
          tangent1 /= sqrt(this_normal(1) * this_normal(1) +
                           this_normal(0) * this_normal(0));
        }
        Vector3d tangent2 = this_normal.cross(tangent1);
        Matrix3d R;  // rotation into normal coordinates
        R.row(0) = tangent1;
        R.row(1) = tangent2;
        R.row(2) = this_normal;
        auto J = R * (JA - JB);          // J = [ D1; D2; n ]
        auto relative_velocity = J * v;  // [ tangent1dot; tangent2dot; phidot ]

        {
          // spring law for normal force:  fA_normal = -k*phi - b*phidot
          // and damping for tangential force:  fA_tangent = -b*tangentdot
          // (bounded by the friction cone)
          Vector3d fA;
          fA(2) =
              std::max<double>(-penetration_stiffness * phi(i) -
                                   penetration_damping * relative_velocity(2),
                               0.0);
          fA.head(2) =
              -std::min<double>(
                  penetration_damping,
                  friction_coefficient * fA(2) /
                      (relative_velocity.head(2).norm() + EPSILON)) *
              relative_velocity.head(2);  // epsilon to avoid divide by zero

          // equal and opposite: fB = -fA.
          // tau = (R*JA)^T fA + (R*JB)^T fB = J^T fA
          C -= J.transpose() * fA;
        }
      }
    }
  }

  if (tree->getNumPositionConstraints()) {
    int nc = tree->getNumPositionConstraints();
    const double alpha = 5.0;  // 1/time constant of position constraint
                               // satisfaction (see my latex rigid body notes)

    prog.AddContinuousVariables(
        nc, "position constraint force");  // don't actually need to use the
                                           // decision variable reference that
                                           // would be returned...

    // then compute the constraint force
    auto phi = tree->positionConstraints(kinsol);
    auto J = tree->positionConstraintsJacobian(kinsol, false);
    auto Jdotv = tree->positionConstraintsJacDotTimesV(kinsol);

    // phiddot = -2 alpha phidot - alpha^2 phi  (0 + critically damped
    // stabilization term)
    prog.AddLinearEqualityConstraint(
        J, -(Jdotv + 2 * alpha * J * v + alpha * alpha * phi), {vdot});
    H_and_neg_JT.conservativeResize(NoChange, H_and_neg_JT.cols() + J.rows());
    H_and_neg_JT.rightCols(J.rows()) = -J.transpose();
  }

  // add [H,-J^T]*[vdot;f] = -C
  prog.AddLinearEqualityConstraint(H_and_neg_JT, -C);

  prog.Solve();
  //      prog.PrintSolution();

  StateVector<double> dot(nq + nv);
  dot << kinsol.transformPositionDotMappingToVelocityMapping(
             Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>::Identity(
                 nq, nq)) *
             v,
      vdot.value();
  return dot;
}