distance_approach_ipopt_relax_end_slack_interface.cc

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/******************************************************************************
 * Copyright 2019 The Apollo Authors. All Rights Reserved.
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 * http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *****************************************************************************/
 
/*
 * @file
 */
#include "modules/planning/planning_open_space/trajectory_smoother/distance_approach_ipopt_relax_end_slack_interface.h"
// #define ADEBUG AERROR
namespace apollo {
namespace planning {
 
DistanceApproachIPOPTRelaxEndSlackInterface::
    DistanceApproachIPOPTRelaxEndSlackInterface(
        const size_t horizon, const double ts, const Eigen::MatrixXd& ego,
        const Eigen::MatrixXd& xWS, const Eigen::MatrixXd& uWS,
        const Eigen::MatrixXd& l_warm_up, const Eigen::MatrixXd& n_warm_up,
        const Eigen::MatrixXd& s_warm_up, const Eigen::MatrixXd& x0,
        const Eigen::MatrixXd& xf, const Eigen::MatrixXd& last_time_u,
        const std::vector<double>& XYbounds,
        const Eigen::MatrixXi& obstacles_edges_num, const size_t obstacles_num,
        const Eigen::MatrixXd& obstacles_A, const Eigen::MatrixXd& obstacles_b,
        const PlannerOpenSpaceConfig& planner_open_space_config)
    : ts_(ts),
      ego_(ego),
      xWS_(xWS),
      uWS_(uWS),
      l_warm_up_(l_warm_up),
      n_warm_up_(n_warm_up),
      slack_warm_up_(s_warm_up),
      x0_(x0),
      xf_(xf),
      last_time_u_(last_time_u),
      XYbounds_(XYbounds),
      obstacles_edges_num_(obstacles_edges_num),
      obstacles_A_(obstacles_A),
      obstacles_b_(obstacles_b) {
  ACHECK(horizon < std::numeric_limits<int>::max())
      << "Invalid cast on horizon in open space planner";
  horizon_ = static_cast<int>(horizon);
  ACHECK(obstacles_num < std::numeric_limits<int>::max())
      << "Invalid cast on obstacles_num in open space planner";
 
  // TODO(RHe): use relax slacks from dual warm up
  // slack_warm_up_ = Eigen::MatrixXd::Zero(obstacles_num, horizon_ + 1);
 
  obstacles_num_ = static_cast<int>(obstacles_num);
  w_ev_ = ego_(1, 0) + ego_(3, 0);
  l_ev_ = ego_(0, 0) + ego_(2, 0);
  g_ = {l_ev_ / 2, w_ev_ / 2, l_ev_ / 2, w_ev_ / 2};
  offset_ = (ego_(0, 0) + ego_(2, 0)) / 2 - ego_(2, 0);
  obstacles_edges_sum_ = obstacles_edges_num_.sum();
  state_result_ = Eigen::MatrixXd::Zero(4, horizon_ + 1);
  dual_l_result_ = Eigen::MatrixXd::Zero(obstacles_edges_sum_, horizon_ + 1);
  dual_n_result_ = Eigen::MatrixXd::Zero(4 * obstacles_num_, horizon_ + 1);
  slack_result_ = Eigen::MatrixXd::Zero(obstacles_num_, horizon_ + 1);
  control_result_ = Eigen::MatrixXd::Zero(2, horizon_);
  time_result_ = Eigen::MatrixXd::Zero(1, horizon_);
 
  state_start_index_ = 0;
  control_start_index_ = 4 * (horizon_ + 1);
  time_start_index_ = control_start_index_ + 2 * horizon_;
  l_start_index_ = time_start_index_ + (horizon_ + 1);
  n_start_index_ = l_start_index_ + obstacles_edges_sum_ * (horizon_ + 1);
  slack_index_ = n_start_index_ + 4 * obstacles_num_ * (horizon_ + 1);
 
  distance_approach_config_ =
      planner_open_space_config.distance_approach_config();
  weight_state_x_ = distance_approach_config_.weight_x();
  weight_state_y_ = distance_approach_config_.weight_y();
  weight_state_phi_ = distance_approach_config_.weight_phi();
  weight_state_v_ = distance_approach_config_.weight_v();
  weight_input_steer_ = distance_approach_config_.weight_steer();
  weight_input_a_ = distance_approach_config_.weight_a();
  weight_rate_steer_ = distance_approach_config_.weight_steer_rate();
  weight_rate_a_ = distance_approach_config_.weight_a_rate();
  weight_stitching_steer_ = distance_approach_config_.weight_steer_stitching();
  weight_stitching_a_ = distance_approach_config_.weight_a_stitching();
  weight_first_order_time_ =
      distance_approach_config_.weight_first_order_time();
  weight_second_order_time_ =
      distance_approach_config_.weight_second_order_time();
  weight_end_state_ = distance_approach_config_.weight_end_state();
  weight_slack_ = distance_approach_config_.weight_slack();
 
  min_safety_distance_ = distance_approach_config_.min_safety_distance();
  max_steer_angle_ =
      vehicle_param_.max_steer_angle() / vehicle_param_.steer_ratio();
  max_speed_forward_ = distance_approach_config_.max_speed_forward();
  max_speed_reverse_ = distance_approach_config_.max_speed_reverse();
  max_acceleration_forward_ =
      distance_approach_config_.max_acceleration_forward();
  max_acceleration_reverse_ =
      distance_approach_config_.max_acceleration_reverse();
  min_time_sample_scaling_ =
      distance_approach_config_.min_time_sample_scaling();
  max_time_sample_scaling_ =
      distance_approach_config_.max_time_sample_scaling();
  max_steer_rate_ =
      vehicle_param_.max_steer_angle_rate() / vehicle_param_.steer_ratio();
  use_fix_time_ = distance_approach_config_.use_fix_time();
  wheelbase_ = vehicle_param_.wheel_base();
  enable_constraint_check_ =
      distance_approach_config_.enable_constraint_check();
  enable_jacobian_ad_ = true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::get_nlp_info(
    int& n, int& m, int& nnz_jac_g, int& nnz_h_lag,
    IndexStyleEnum& index_style) {
  ADEBUG << "get_nlp_info";
  // n1 : states variables, 4 * (N+1)
  int n1 = 4 * (horizon_ + 1);
  ADEBUG << "n1: " << n1;
  // n2 : control inputs variables
  int n2 = 2 * horizon_;
  ADEBUG << "n2: " << n2;
  // n3 : sampling time variables
  int n3 = horizon_ + 1;
  ADEBUG << "n3: " << n3;
  // n4 : dual multiplier associated with obstacle shape
  lambda_horizon_ = obstacles_edges_num_.sum() * (horizon_ + 1);
  ADEBUG << "lambda_horizon_: " << lambda_horizon_;
  // n5 : dual multipier associated with car shape, obstacles_num*4 * (N+1)
  miu_horizon_ = obstacles_num_ * 4 * (horizon_ + 1);
  ADEBUG << "miu_horizon_: " << miu_horizon_;
  // n6 : slack variables related to safety min distance
  slack_horizon_ = obstacles_num_ * (horizon_ + 1);
 
  // m1 : dynamics constatins
  int m1 = 4 * horizon_;
  // m2 : control rate constraints (only steering)
  int m2 = horizon_;
  // m3 : sampling time equality constraints
  int m3 = horizon_;
  // m4 : obstacle constraints
  int m4 = 4 * obstacles_num_ * (horizon_ + 1);
 
  num_of_variables_ =
      n1 + n2 + n3 + lambda_horizon_ + miu_horizon_ + slack_horizon_;
  num_of_constraints_ =
      m1 + m2 + m3 + m4 + (num_of_variables_ - (horizon_ + 1) + 2);
 
  // number of variables
  n = num_of_variables_;
  ADEBUG << "num_of_variables_ " << num_of_variables_;
  // number of constraints
  m = num_of_constraints_;
  ADEBUG << "num_of_constraints_ " << num_of_constraints_;
 
  generate_tapes(n, m, &nnz_jac_g, &nnz_h_lag);
 
  index_style = IndexStyleEnum::C_STYLE;
  return true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::get_bounds_info(
    int n, double* x_l, double* x_u, int m, double* g_l, double* g_u) {
  ADEBUG << "get_bounds_info";
  ACHECK(XYbounds_.size() == 4)
      << "XYbounds_ size is not 4, but" << XYbounds_.size();
 
  // Variables: includes state, u, sample time and lagrange multipliers
  // 1. state variables, 4 * [0, horizon]
  // start point pose
  int variable_index = 0;
  for (int i = 0; i < 4; ++i) {
    x_l[i] = -2e19;
    x_u[i] = 2e19;
  }
  variable_index += 4;
 
  // During horizons, 2 ~ N-1
  for (int i = 1; i < horizon_; ++i) {
    // x
    x_l[variable_index] = -2e19;
    x_u[variable_index] = 2e19;
 
    // y
    x_l[variable_index + 1] = -2e19;
    x_u[variable_index + 1] = 2e19;
 
    // phi
    x_l[variable_index + 2] = -2e19;
    x_u[variable_index + 2] = 2e19;
 
    // v
    x_l[variable_index + 3] = -2e19;
    x_u[variable_index + 3] = 2e19;
 
    variable_index += 4;
  }
 
  // end point pose
  for (int i = 0; i < 4; ++i) {
    x_l[variable_index + i] = -2e19;
    x_u[variable_index + i] = 2e19;
  }
  variable_index += 4;
  ADEBUG << "variable_index after adding state variables : " << variable_index;
 
  // 2. control variables, 2 * [0, horizon_-1]
  for (int i = 0; i < horizon_; ++i) {
    // steer
    x_l[variable_index] = -2e19;
    x_u[variable_index] = 2e19;
 
    // a
    x_l[variable_index + 1] = -2e19;
    x_u[variable_index + 1] = 2e19;
 
    variable_index += 2;
  }
  ADEBUG << "variable_index after adding control variables : "
         << variable_index;
 
  // 3. sampling time variables, 1 * [0, horizon_]
  for (int i = 0; i < horizon_ + 1; ++i) {
    x_l[variable_index] = -2e19;
    x_u[variable_index] = 2e19;
    ++variable_index;
  }
  ADEBUG << "variable_index after adding sample time : " << variable_index;
  ADEBUG << "sample time fix time status is : " << use_fix_time_;
 
  // 4. lagrange constraint l, [0, obstacles_edges_sum_ - 1] * [0,
  // horizon_]
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < obstacles_edges_sum_; ++j) {
      x_l[variable_index] = 0.0;
      x_u[variable_index] = 2e19;  // nlp_upper_bound_limit
      ++variable_index;
    }
  }
  ADEBUG << "variable_index after adding lagrange l : " << variable_index;
 
  // 5. lagrange constraint n, [0, 4*obstacles_num-1] * [0, horizon_]
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < 4 * obstacles_num_; ++j) {
      x_l[variable_index] = 0.0;
      x_u[variable_index] = 2e19;  // nlp_upper_bound_limit
 
      ++variable_index;
    }
  }
  ADEBUG << "variable_index after adding lagrange n : " << variable_index;
 
  // 6. slack variable constraint s, obstacles_num * (horizon_ + 1)
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < obstacles_num_; ++j) {
      x_l[variable_index] = 0.0;
      x_u[variable_index] = 2e19;
      ++variable_index;
    }
  }
  ADEBUG << "variable_index after adding slack s: " << variable_index;
 
  // Constraints: includes four state Euler forward constraints, three
  // Obstacle related constraints
 
  // 1. dynamics constraints 4 * [0, horizons-1]
  int constraint_index = 0;
  for (int i = 0; i < 4 * horizon_; ++i) {
    g_l[i] = 0.0;
    g_u[i] = 0.0;
  }
  constraint_index += 4 * horizon_;
 
  ADEBUG << "constraint_index after adding Euler forward dynamics constraints: "
         << constraint_index;
 
  // 2. Control rate limit constraints, 1 * [0, horizons-1], only apply
  // steering rate as of now
  for (int i = 0; i < horizon_; ++i) {
    g_l[constraint_index] = -max_steer_rate_;
    g_u[constraint_index] = max_steer_rate_;
    ++constraint_index;
  }
 
  ADEBUG << "constraint_index after adding steering rate constraints: "
         << constraint_index;
 
  // 3. Time constraints 1 * [0, horizons-1]
  for (int i = 0; i < horizon_; ++i) {
    g_l[constraint_index] = 0.0;
    g_u[constraint_index] = 0.0;
    ++constraint_index;
  }
 
  ADEBUG << "constraint_index after adding time constraints: "
         << constraint_index;
 
  // 4. Three obstacles related equal constraints, one equality constraints,
  // [0, horizon_] * [0, obstacles_num_-1] * 4
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < obstacles_num_; ++j) {
      // a. norm(A'*lambda) <= 1
      g_l[constraint_index] = -2e19;
      g_u[constraint_index] = 1.0;
 
      // b. G'*mu + R'*A*lambda = 0
      g_l[constraint_index + 1] = 0.0;
      g_u[constraint_index + 1] = 0.0;
      g_l[constraint_index + 2] = 0.0;
      g_u[constraint_index + 2] = 0.0;
 
      // c. -g'*mu + (A*t - b)*lambda + s > 0
      g_l[constraint_index + 3] = 0.0;
      g_u[constraint_index + 3] = 2e19;  // nlp_upper_bound_limit
      constraint_index += 4;
    }
  }
 
  // 5. load variable bounds as constraints
  // start configuration
  g_l[constraint_index] = x0_(0, 0);
  g_u[constraint_index] = x0_(0, 0);
  g_l[constraint_index + 1] = x0_(1, 0);
  g_u[constraint_index + 1] = x0_(1, 0);
  g_l[constraint_index + 2] = x0_(2, 0);
  g_u[constraint_index + 2] = x0_(2, 0);
  g_l[constraint_index + 3] = x0_(3, 0);
  g_u[constraint_index + 3] = x0_(3, 0);
  constraint_index += 4;
 
  for (int i = 1; i < horizon_; ++i) {
    g_l[constraint_index] = XYbounds_[0];
    g_u[constraint_index] = XYbounds_[1];
    g_l[constraint_index + 1] = XYbounds_[2];
    g_u[constraint_index + 1] = XYbounds_[3];
    g_l[constraint_index + 2] = -max_speed_reverse_;
    g_u[constraint_index + 2] = max_speed_forward_;
    constraint_index += 3;
  }
 
  // end configuration
  g_l[constraint_index] = -2e19;
  g_u[constraint_index] = 2e19;
  g_l[constraint_index + 1] = -2e19;
  g_u[constraint_index + 1] = 2e19;
  g_l[constraint_index + 2] = -2e19;
  g_u[constraint_index + 2] = 2e19;
  g_l[constraint_index + 3] = -2e19;
  g_u[constraint_index + 3] = 2e19;
  constraint_index += 4;
 
  for (int i = 0; i < horizon_; ++i) {
    g_l[constraint_index] = -max_steer_angle_;
    g_u[constraint_index] = max_steer_angle_;
    g_l[constraint_index + 1] = -max_acceleration_reverse_;
    g_u[constraint_index + 1] = max_acceleration_forward_;
    constraint_index += 2;
  }
 
  for (int i = 0; i < horizon_ + 1; ++i) {
    if (!use_fix_time_) {
      g_l[constraint_index] = min_time_sample_scaling_;
      g_u[constraint_index] = max_time_sample_scaling_;
    } else {
      g_l[constraint_index] = 1.0;
      g_u[constraint_index] = 1.0;
    }
    constraint_index++;
  }
 
  for (int i = 0; i < lambda_horizon_; ++i) {
    g_l[constraint_index] = 0.0;
    g_u[constraint_index] = 2e19;
    constraint_index++;
  }
 
  for (int i = 0; i < miu_horizon_; ++i) {
    g_l[constraint_index] = 0.0;
    g_u[constraint_index] = 2e19;
    constraint_index++;
  }
 
  for (int i = 0; i < slack_horizon_; ++i) {
    g_l[constraint_index] = 0.0;
    g_u[constraint_index] = 2e19;
    ++constraint_index;
  }
 
  ADEBUG << "constraint_index after adding obstacles constraints: "
         << constraint_index;
  ADEBUG << "get_bounds_info_ out";
  return true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::get_starting_point(
    int n, bool init_x, double* x, bool init_z, double* z_L, double* z_U, int m,
    bool init_lambda, double* lambda) {
  ADEBUG << "get_starting_point";
  ACHECK(init_x) << "Warm start init_x setting failed";
 
  CHECK_EQ(horizon_, uWS_.cols());
  CHECK_EQ(horizon_ + 1, xWS_.cols());
 
  // 1. state variables 4 * (horizon_ + 1)
  for (int i = 0; i < horizon_ + 1; ++i) {
    int index = i * 4;
    for (int j = 0; j < 4; ++j) {
      x[index + j] = xWS_(j, i);
    }
  }
 
  // 2. control variable initialization, 2 * horizon_
  for (int i = 0; i < horizon_; ++i) {
    int index = i * 2;
    x[control_start_index_ + index] = uWS_(0, i);
    x[control_start_index_ + index + 1] = uWS_(1, i);
  }
 
  // 2. time scale variable initialization, horizon_ + 1
  for (int i = 0; i < horizon_ + 1; ++i) {
    x[time_start_index_ + i] =
        0.5 * (min_time_sample_scaling_ + max_time_sample_scaling_);
  }
 
  // 3. lagrange constraint l, obstacles_edges_sum_ * (horizon_+1)
  for (int i = 0; i < horizon_ + 1; ++i) {
    int index = i * obstacles_edges_sum_;
    for (int j = 0; j < obstacles_edges_sum_; ++j) {
      x[l_start_index_ + index + j] = l_warm_up_(j, i);
    }
  }
 
  // 4. lagrange constraint m, 4*obstacles_num * (horizon_+1)
  for (int i = 0; i < horizon_ + 1; ++i) {
    int index = i * 4 * obstacles_num_;
    for (int j = 0; j < 4 * obstacles_num_; ++j) {
      x[n_start_index_ + index + j] = n_warm_up_(j, i);
    }
  }
 
  // 5. slack constraint s, obstacle_num * (horizon_+1)
  for (int i = 0; i < horizon_ + 1; ++i) {
    int index = i * obstacles_num_;
    for (int j = 0; j < obstacles_num_; ++j) {
      x[slack_index_ + index + j] = slack_warm_up_(j, i);
    }
  }
 
  ADEBUG << "get_starting_point out";
  return true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::eval_f(int n, const double* x,
                                                         bool new_x,
                                                         double& obj_value) {
  eval_obj(n, x, &obj_value);
  return true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::eval_grad_f(int n,
                                                              const double* x,
                                                              bool new_x,
                                                              double* grad_f) {
  gradient(tag_f, n, x, grad_f);
  return true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::eval_g(int n, const double* x,
                                                         bool new_x, int m,
                                                         double* g) {
  eval_constraints(n, x, m, g);
  // if (enable_constraint_check_) check_g(n, x, m, g);
  return true;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::eval_jac_g(
    int n, const double* x, bool new_x, int m, int nele_jac, int* iRow,
    int* jCol, double* values) {
  if (enable_jacobian_ad_) {
    if (values == nullptr) {
      // return the structure of the jacobian
      for (int idx = 0; idx < nnz_jac; idx++) {
        iRow[idx] = rind_g[idx];
        jCol[idx] = cind_g[idx];
      }
    } else {
      // return the values of the jacobian of the constraints
      sparse_jac(tag_g, m, n, 1, x, &nnz_jac, &rind_g, &cind_g, &jacval,
                 options_g);
      for (int idx = 0; idx < nnz_jac; idx++) {
        values[idx] = jacval[idx];
      }
    }
    return true;
  } else {
    return eval_jac_g_ser(n, x, new_x, m, nele_jac, iRow, jCol, values);
  }
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::eval_jac_g_ser(
    int n, const double* x, bool new_x, int m, int nele_jac, int* iRow,
    int* jCol, double* values) {
  AERROR << "NOT READY";
  return false;
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::eval_h(
    int n, const double* x, bool new_x, double obj_factor, int m,
    const double* lambda, bool new_lambda, int nele_hess, int* iRow, int* jCol,
    double* values) {
  if (values == nullptr) {
    // return the structure. This is a symmetric matrix, fill the lower left
    // triangle only.
 
    for (int idx = 0; idx < nnz_L; idx++) {
      iRow[idx] = rind_L[idx];
      jCol[idx] = cind_L[idx];
    }
  } else {
    // return the values. This is a symmetric matrix, fill the lower left
    // triangle only
 
    obj_lam[0] = obj_factor;
 
    for (int idx = 0; idx < m; idx++) {
      obj_lam[1 + idx] = lambda[idx];
    }
 
    set_param_vec(tag_L, m + 1, obj_lam);
    sparse_hess(tag_L, n, 1, const_cast<double*>(x), &nnz_L, &rind_L, &cind_L,
                &hessval, options_L);
 
    for (int idx = 0; idx < nnz_L; idx++) {
      values[idx] = hessval[idx];
    }
  }
 
  return true;
}
 
void DistanceApproachIPOPTRelaxEndSlackInterface::finalize_solution(
    Ipopt::SolverReturn status, int n, const double* x, const double* z_L,
    const double* z_U, int m, const double* g, const double* lambda,
    double obj_value, const Ipopt::IpoptData* ip_data,
    Ipopt::IpoptCalculatedQuantities* ip_cq) {
  int state_index = state_start_index_;
  int control_index = control_start_index_;
  int time_index = time_start_index_;
  int dual_l_index = l_start_index_;
  int dual_n_index = n_start_index_;
  int slack_index = slack_index_;
 
  // enable_constraint_check_: for debug only
  if (enable_constraint_check_) {
    ADEBUG << "final resolution constraint checking";
    check_g(n, x, m, g);
  }
  // 1. state variables, 4 * [0, horizon]
  // 2. control variables, 2 * [0, horizon_-1]
  // 3. sampling time variables, 1 * [0, horizon_]
  // 4. dual_l, obstacles_edges_sum_ * [0, horizon]
  // 5. dual_n, obstacles_num * [0, horizon]
  // 6. slack_, obstacles_num * [0, horizon]
  for (int i = 0; i < horizon_; ++i) {
    state_result_(0, i) = x[state_index];
    state_result_(1, i) = x[state_index + 1];
    state_result_(2, i) = x[state_index + 2];
    state_result_(3, i) = x[state_index + 3];
    control_result_(0, i) = x[control_index];
    control_result_(1, i) = x[control_index + 1];
    time_result_(0, i) = ts_ * x[time_index];
 
    for (int j = 0; j < obstacles_edges_sum_; ++j) {
      dual_l_result_(j, i) = x[dual_l_index + j];
    }
    for (int k = 0; k < 4 * obstacles_num_; ++k) {
      dual_n_result_(k, i) = x[dual_n_index + k];
    }
    for (int l = 0; l < obstacles_num_; ++l) {
      slack_result_(l, i) = x[slack_index + l];
    }
 
    state_index += 4;
    control_index += 2;
    time_index++;
    dual_l_index += obstacles_edges_sum_;
    dual_n_index += 4 * obstacles_num_;
    slack_index += obstacles_num_;
  }
  state_result_(0, 0) = x0_(0, 0);
  state_result_(1, 0) = x0_(1, 0);
  state_result_(2, 0) = x0_(2, 0);
  state_result_(3, 0) = x0_(3, 0);
  // push back last horizon for states
  state_result_(0, horizon_) = xf_(0, 0);
  state_result_(1, horizon_) = xf_(1, 0);
  state_result_(2, horizon_) = xf_(2, 0);
  state_result_(3, horizon_) = xf_(3, 0);
 
  for (int j = 0; j < obstacles_edges_sum_; ++j) {
    dual_l_result_(j, horizon_) = x[dual_l_index + j];
  }
  for (int k = 0; k < 4 * obstacles_num_; ++k) {
    dual_n_result_(k, horizon_) = x[dual_n_index + k];
  }
  for (int l = 0; l < obstacles_num_; ++l) {
    slack_result_(l, horizon_) = x[slack_index + l];
  }
 
  // memory deallocation of ADOL-C variables
  delete[] obj_lam;
  if (enable_jacobian_ad_) {
    free(rind_g);
    free(cind_g);
    free(jacval);
  }
  free(rind_L);
  free(cind_L);
  free(hessval);
}
 
void DistanceApproachIPOPTRelaxEndSlackInterface::get_optimization_results(
    Eigen::MatrixXd* state_result, Eigen::MatrixXd* control_result,
    Eigen::MatrixXd* time_result, Eigen::MatrixXd* dual_l_result,
    Eigen::MatrixXd* dual_n_result) const {
  // TODO(RHe): extract slack variables for further study
  ADEBUG << "get_optimization_results";
  *state_result = state_result_;
  *control_result = control_result_;
  *time_result = time_result_;
  *dual_l_result = dual_l_result_;
  *dual_n_result = dual_n_result_;
 
  if (!distance_approach_config_.enable_initial_final_check()) {
    return;
  }
  CHECK_EQ(state_result_.cols(), xWS_.cols());
  CHECK_EQ(state_result_.rows(), xWS_.rows());
  double state_diff_max = 0.0;
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < 4; ++j) {
      state_diff_max =
          std::max(std::abs(xWS_(j, i) - state_result_(j, i)), state_diff_max);
    }
  }
 
  // 2. control variable initialization, 2 * horizon_
  CHECK_EQ(control_result_.cols(), uWS_.cols());
  CHECK_EQ(control_result_.rows(), uWS_.rows());
  double control_diff_max = 0.0;
  for (int i = 0; i < horizon_; ++i) {
    control_diff_max = std::max(std::abs(uWS_(0, i) - control_result_(0, i)),
                                control_diff_max);
    control_diff_max = std::max(std::abs(uWS_(1, i) - control_result_(1, i)),
                                control_diff_max);
  }
 
  // 2. time scale variable initialization, horizon_ + 1
 
  // 3. lagrange constraint l, obstacles_edges_sum_ * (horizon_+1)
  CHECK_EQ(dual_l_result_.cols(), l_warm_up_.cols());
  CHECK_EQ(dual_l_result_.rows(), l_warm_up_.rows());
  double l_diff_max = 0.0;
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < obstacles_edges_sum_; ++j) {
      l_diff_max = std::max(std::abs(l_warm_up_(j, i) - dual_l_result_(j, i)),
                            l_diff_max);
    }
  }
 
  // 4. lagrange constraint m, 4*obstacles_num * (horizon_+1)
  CHECK_EQ(n_warm_up_.cols(), dual_n_result_.cols());
  CHECK_EQ(n_warm_up_.rows(), dual_n_result_.rows());
  double n_diff_max = 0.0;
  for (int i = 0; i < horizon_ + 1; ++i) {
    for (int j = 0; j < 4 * obstacles_num_; ++j) {
      n_diff_max = std::max(std::abs(n_warm_up_(j, i) - dual_n_result_(j, i)),
                            n_diff_max);
    }
  }
 
  ADEBUG << "state_diff_max: " << state_diff_max;
  ADEBUG << "control_diff_max: " << control_diff_max;
  ADEBUG << "dual_l_diff_max: " << l_diff_max;
  ADEBUG << "dual_n_diff_max: " << n_diff_max;
}
 
//***************    start ADOL-C part ***********************************
template <class T>
void DistanceApproachIPOPTRelaxEndSlackInterface::eval_obj(int n, const T* x,
                                                           T* obj_value) {
  // Objective is :
  // min control inputs
  // min input rate
  // min time (if the time step is not fixed)
  // regularization wrt warm start trajectory
  DCHECK(ts_ != 0) << "ts in distance_approach_ is 0";
  int control_index = control_start_index_;
  int time_index = time_start_index_;
  int state_index = state_start_index_;
 
  // TODO(QiL): Initial implementation towards earlier understanding and debug
  // purpose, later code refine towards improving efficiency
 
  *obj_value = 0.0;
  // 1. objective to minimize state diff to warm up
  for (int i = 0; i < horizon_ + 1; ++i) {
    T x1_diff = x[state_index] - xWS_(0, i);
    T x2_diff = x[state_index + 1] - xWS_(1, i);
    T x3_diff = x[state_index + 2] - xWS_(2, i);
    T x4_abs = x[state_index + 3];
    *obj_value += weight_state_x_ * x1_diff * x1_diff +
                  weight_state_y_ * x2_diff * x2_diff +
                  weight_state_phi_ * x3_diff * x3_diff +
                  weight_state_v_ * x4_abs * x4_abs;
    state_index += 4;
  }
 
  // 2. objective to minimize u square
  for (int i = 0; i < horizon_; ++i) {
    *obj_value += weight_input_steer_ * x[control_index] * x[control_index] +
                  weight_input_a_ * x[control_index + 1] * x[control_index + 1];
    control_index += 2;
  }
 
  // 3. objective to minimize input change rate for first horizon
  control_index = control_start_index_;
  T last_time_steer_rate =
      (x[control_index] - last_time_u_(0, 0)) / x[time_index] / ts_;
  T last_time_a_rate =
      (x[control_index + 1] - last_time_u_(1, 0)) / x[time_index] / ts_;
  *obj_value +=
      weight_stitching_steer_ * last_time_steer_rate * last_time_steer_rate +
      weight_stitching_a_ * last_time_a_rate * last_time_a_rate;
 
  // 4. objective to minimize input change rates, [0- horizon_ -2]
  time_index++;
  for (int i = 0; i < horizon_ - 1; ++i) {
    T steering_rate =
        (x[control_index + 2] - x[control_index]) / x[time_index] / ts_;
    T a_rate =
        (x[control_index + 3] - x[control_index + 1]) / x[time_index] / ts_;
    *obj_value += weight_rate_steer_ * steering_rate * steering_rate +
                  weight_rate_a_ * a_rate * a_rate;
    control_index += 2;
    time_index++;
  }
 
  // 5. objective to minimize total time [0, horizon_]
  time_index = time_start_index_;
  for (int i = 0; i < horizon_ + 1; ++i) {
    T first_order_penalty = weight_first_order_time_ * x[time_index];
    T second_order_penalty =
        weight_second_order_time_ * x[time_index] * x[time_index];
    *obj_value += first_order_penalty + second_order_penalty;
    time_index++;
  }
 
  // 6. end state constraints
  for (int i = 0; i < 4; ++i) {
    *obj_value += weight_end_state_ *
                  (x[state_start_index_ + 4 * horizon_ + i] - xf_(i, 0)) *
                  (x[state_start_index_ + 4 * horizon_ + i] - xf_(i, 0));
  }
 
  // 7. slack variables
  for (int i = 0; i < slack_horizon_; ++i) {
    *obj_value += weight_slack_ * x[slack_index_ + i];
  }
}
 
template <class T>
void DistanceApproachIPOPTRelaxEndSlackInterface::eval_constraints(int n,
                                                                   const T* x,
                                                                   int m,
                                                                   T* g) {
  // state start index
  int state_index = state_start_index_;
 
  // control start index.
  int control_index = control_start_index_;
 
  // time start index
  int time_index = time_start_index_;
 
  int constraint_index = 0;
 
  // 1. state constraints 4 * [0, horizons-1]
  for (int i = 0; i < horizon_; ++i) {
    // x1
    g[constraint_index] =
        x[state_index + 4] -
        (x[state_index] +
         ts_ * x[time_index] *
             (x[state_index + 3] +
              ts_ * x[time_index] * 0.5 * x[control_index + 1]) *
             cos(x[state_index + 2] + ts_ * x[time_index] * 0.5 *
                                          x[state_index + 3] *
                                          tan(x[control_index]) / wheelbase_));
 
    // x2
    g[constraint_index + 1] =
        x[state_index + 5] -
        (x[state_index + 1] +
         ts_ * x[time_index] *
             (x[state_index + 3] +
              ts_ * x[time_index] * 0.5 * x[control_index + 1]) *
             sin(x[state_index + 2] + ts_ * x[time_index] * 0.5 *
                                          x[state_index + 3] *
                                          tan(x[control_index]) / wheelbase_));
 
    // x3
    g[constraint_index + 2] =
        x[state_index + 6] -
        (x[state_index + 2] +
         ts_ * x[time_index] *
             (x[state_index + 3] +
              ts_ * x[time_index] * 0.5 * x[control_index + 1]) *
             tan(x[control_index]) / wheelbase_);
 
    // x4
    g[constraint_index + 3] =
        x[state_index + 7] -
        (x[state_index + 3] + ts_ * x[time_index] * x[control_index + 1]);
 
    control_index += 2;
    constraint_index += 4;
    time_index++;
    state_index += 4;
  }
 
  ADEBUG << "constraint_index after adding Euler forward dynamics constraints "
            "updated: "
         << constraint_index;
 
  // 2. Control rate limit constraints, 1 * [0, horizons-1], only apply
  // steering rate as of now
  control_index = control_start_index_;
  time_index = time_start_index_;
 
  // First rate is compare first with stitch point
  g[constraint_index] =
      (x[control_index] - last_time_u_(0, 0)) / x[time_index] / ts_;
  control_index += 2;
  constraint_index++;
  time_index++;
 
  for (int i = 1; i < horizon_; ++i) {
    g[constraint_index] =
        (x[control_index] - x[control_index - 2]) / x[time_index] / ts_;
    constraint_index++;
    control_index += 2;
    time_index++;
  }
 
  // 3. Time constraints 1 * [0, horizons-1]
  time_index = time_start_index_;
  for (int i = 0; i < horizon_; ++i) {
    g[constraint_index] = x[time_index + 1] - x[time_index];
    constraint_index++;
    time_index++;
  }
 
  ADEBUG << "constraint_index after adding time constraints "
            "updated: "
         << constraint_index;
 
  // 4. Three obstacles related equal constraints, one equality constraints,
  // [0, horizon_] * [0, obstacles_num_-1] * 4
  state_index = state_start_index_;
  int l_index = l_start_index_;
  int n_index = n_start_index_;
  int s_index = slack_index_;
 
  for (int i = 0; i < horizon_ + 1; ++i) {
    int edges_counter = 0;
    for (int j = 0; j < obstacles_num_; ++j) {
      int current_edges_num = obstacles_edges_num_(j, 0);
      Eigen::MatrixXd Aj =
          obstacles_A_.block(edges_counter, 0, current_edges_num, 2);
      Eigen::MatrixXd bj =
          obstacles_b_.block(edges_counter, 0, current_edges_num, 1);
 
      // norm(A* lambda) <= 1
      T tmp1 = 0.0;
      T tmp2 = 0.0;
      for (int k = 0; k < current_edges_num; ++k) {
        // TODO(QiL) : replace this one directly with x
        tmp1 += Aj(k, 0) * x[l_index + k];
        tmp2 += Aj(k, 1) * x[l_index + k];
      }
      g[constraint_index] = tmp1 * tmp1 + tmp2 * tmp2;
 
      // G' * mu + R' * lambda == 0
      g[constraint_index + 1] = x[n_index] - x[n_index + 2] +
                                cos(x[state_index + 2]) * tmp1 +
                                sin(x[state_index + 2]) * tmp2;
 
      g[constraint_index + 2] = x[n_index + 1] - x[n_index + 3] -
                                sin(x[state_index + 2]) * tmp1 +
                                cos(x[state_index + 2]) * tmp2;
 
      //  -g'*mu + (A*t - b)*lambda + d > 0
      T tmp3 = 0.0;
      for (int k = 0; k < 4; ++k) {
        tmp3 += -g_[k] * x[n_index + k];
      }
 
      T tmp4 = 0.0;
      for (int k = 0; k < current_edges_num; ++k) {
        tmp4 += bj(k, 0) * x[l_index + k];
      }
 
      g[constraint_index + 3] =
          tmp3 + (x[state_index] + cos(x[state_index + 2]) * offset_) * tmp1 +
          (x[state_index + 1] + sin(x[state_index + 2]) * offset_) * tmp2 -
          tmp4 + x[s_index];
 
      // Update index
      edges_counter += current_edges_num;
      l_index += current_edges_num;
      n_index += 4;
      constraint_index += 4;
      ++s_index;
    }
    state_index += 4;
  }
  ADEBUG << "constraint_index after obstacles avoidance constraints "
            "updated: "
         << constraint_index;
 
  // 5. load variable bounds as constraints
  state_index = state_start_index_;
  control_index = control_start_index_;
  time_index = time_start_index_;
  l_index = l_start_index_;
  n_index = n_start_index_;
  s_index = slack_index_;
 
  // start configuration
  g[constraint_index] = x[state_index];
  g[constraint_index + 1] = x[state_index + 1];
  g[constraint_index + 2] = x[state_index + 2];
  g[constraint_index + 3] = x[state_index + 3];
  constraint_index += 4;
  state_index += 4;
 
  // constraints on x,y,v
  for (int i = 1; i < horizon_; ++i) {
    g[constraint_index] = x[state_index];
    g[constraint_index + 1] = x[state_index + 1];
    g[constraint_index + 2] = x[state_index + 3];
    constraint_index += 3;
    state_index += 4;
  }
 
  // end configuration
  g[constraint_index] = x[state_index];
  g[constraint_index + 1] = x[state_index + 1];
  g[constraint_index + 2] = x[state_index + 2];
  g[constraint_index + 3] = x[state_index + 3];
  constraint_index += 4;
  state_index += 4;
 
  for (int i = 0; i < horizon_; ++i) {
    g[constraint_index] = x[control_index];
    g[constraint_index + 1] = x[control_index + 1];
    constraint_index += 2;
    control_index += 2;
  }
 
  for (int i = 0; i < horizon_ + 1; ++i) {
    g[constraint_index] = x[time_index];
    ++constraint_index;
    ++time_index;
  }
 
  for (int i = 0; i < lambda_horizon_; ++i) {
    g[constraint_index] = x[l_index];
    ++constraint_index;
    ++l_index;
  }
 
  for (int i = 0; i < miu_horizon_; ++i) {
    g[constraint_index] = x[n_index];
    ++constraint_index;
    ++n_index;
  }
 
  for (int i = 0; i < slack_horizon_; ++i) {
    g[constraint_index] = x[s_index];
    ++constraint_index;
    ++s_index;
  }
}
 
bool DistanceApproachIPOPTRelaxEndSlackInterface::check_g(int n,
                                                          const double* x,
                                                          int m,
                                                          const double* g) {
  int kN = n;
  int kM = m;
  double x_u_tmp[kN];
  double x_l_tmp[kN];
  double g_u_tmp[kM];
  double g_l_tmp[kM];
 
  get_bounds_info(n, x_l_tmp, x_u_tmp, m, g_l_tmp, g_u_tmp);
 
  const double delta_v = 1e-4;
  for (int idx = 0; idx < n; ++idx) {
    x_u_tmp[idx] = x_u_tmp[idx] + delta_v;
    x_l_tmp[idx] = x_l_tmp[idx] - delta_v;
    if (x[idx] > x_u_tmp[idx] || x[idx] < x_l_tmp[idx]) {
      AINFO << "x idx unfeasible: " << idx << ", x: " << x[idx]
            << ", lower: " << x_l_tmp[idx] << ", upper: " << x_u_tmp[idx];
    }
  }
 
  // m1 : dynamics constatins
  int m1 = 4 * horizon_;
 
  // m2 : control rate constraints (only steering)
  int m2 = m1 + horizon_;
 
  // m3 : sampling time equality constraints
  int m3 = m2 + horizon_;
 
  // m4 : obstacle constraints
  int m4 = m3 + 4 * obstacles_num_ * (horizon_ + 1);
 
  // 5. load variable bounds as constraints
  // start configuration
  int m5 = m4 + 3 + 1;
 
  // constraints on x,y,v
  int m6 = m5 + 3 * (horizon_ - 1);
 
  // end configuration
  int m7 = m6 + 3 + 1;
 
  // control variable bnd
  int m8 = m7 + 2 * horizon_;
 
  // time interval variable bnd
  int m9 = m8 + (horizon_ + 1);
 
  // lambda_horizon_
  int m10 = m9 + lambda_horizon_;
 
  // miu_horizon_
  int m11 = m10 + miu_horizon_;
 
  // slack_horizon_
  int m12 = m11 + slack_horizon_;
 
  CHECK_EQ(m12, num_of_constraints_);
 
  AINFO << "dynamics constatins to: " << m1;
  AINFO << "control rate constraints (only steering) to: " << m2;
  AINFO << "sampling time equality constraints to: " << m3;
  AINFO << "obstacle constraints to: " << m4;
  AINFO << "start conf constraints to: " << m5;
  AINFO << "constraints on x,y,v to: " << m6;
  AINFO << "end constraints to: " << m7;
  AINFO << "control bnd to: " << m8;
  AINFO << "time interval constraints to: " << m9;
  AINFO << "lambda constraints to: " << m10;
  AINFO << "miu constraints to: " << m11;
  AINFO << "slack constraint to: " << m12;
  AINFO << "total constraints: " << num_of_constraints_;
 
  for (int idx = 0; idx < m; ++idx) {
    if (g[idx] > g_u_tmp[idx] + delta_v || g[idx] < g_l_tmp[idx] - delta_v) {
      AINFO << "constratins idx unfeasible: " << idx << ", g: " << g[idx]
            << ", lower: " << g_l_tmp[idx] << ", upper: " << g_u_tmp[idx];
    }
  }
  return true;
}
 
void DistanceApproachIPOPTRelaxEndSlackInterface::generate_tapes(
    int n, int m, int* nnz_jac_g, int* nnz_h_lag) {
  std::vector<double> xp(n);
  std::vector<double> lamp(m);
  std::vector<double> zl(m);
  std::vector<double> zu(m);
  std::vector<adouble> xa(n);
  std::vector<adouble> g(m);
  std::vector<double> lam(m);
 
  double sig = 0.0;
  adouble obj_value;
  double dummy = 0.0;
  obj_lam = new double[m + 1];
  get_starting_point(n, 1, &xp[0], 0, &zl[0], &zu[0], m, 0, &lamp[0]);
 
  trace_on(tag_f);
  for (int idx = 0; idx < n; idx++) {
    xa[idx] <<= xp[idx];
  }
  eval_obj(n, &xa[0], &obj_value);
  obj_value >>= dummy;
  trace_off();
 
  trace_on(tag_g);
  for (int idx = 0; idx < n; idx++) {
    xa[idx] <<= xp[idx];
  }
  eval_constraints(n, &xa[0], m, &g[0]);
  for (int idx = 0; idx < m; idx++) {
    g[idx] >>= dummy;
  }
  trace_off();
 
  trace_on(tag_L);
  for (int idx = 0; idx < n; idx++) {
    xa[idx] <<= xp[idx];
  }
  for (int idx = 0; idx < m; idx++) {
    lam[idx] = 1.0;
  }
  sig = 1.0;
  eval_obj(n, &xa[0], &obj_value);
  obj_value *= mkparam(sig);
  eval_constraints(n, &xa[0], m, &g[0]);
  for (int idx = 0; idx < m; idx++) {
    obj_value += g[idx] * mkparam(lam[idx]);
  }
  obj_value >>= dummy;
 
  trace_off();
 
  if (enable_jacobian_ad_) {
    rind_g = nullptr;
    cind_g = nullptr;
    jacval = nullptr;
    options_g[0] = 0; /* sparsity pattern by index domains (default) */
    options_g[1] = 0; /*                         safe mode (default) */
    options_g[2] = 0;
    options_g[3] = 0; /*                column compression (default) */
 
    sparse_jac(tag_g, m, n, 0, &xp[0], &nnz_jac, &rind_g, &cind_g, &jacval,
               options_g);
    *nnz_jac_g = nnz_jac;
  }
 
  rind_L = nullptr;
  cind_L = nullptr;
  hessval = nullptr;
  options_L[0] = 0;
  options_L[1] = 1;
 
  sparse_hess(tag_L, n, 0, &xp[0], &nnz_L, &rind_L, &cind_L, &hessval,
              options_L);
  *nnz_h_lag = nnz_L;
}
//***************    end   ADOL-C part ***********************************
 
}  // namespace planning
}  // namespace apollo