三种常用坐标系的拉梅系数（Cartesian与Frenet坐标系转换公式推导）

void CartesianFrenetConverter::cartesian_to_frenet( const double rs, const double rx, const double ry, const double rtheta, const double rkappa, const double rdkappa, const double x, const double y, const double v, const double a, const double theta, const double kappa, std::array<double, 3>* const ptr_s_condition, std::array<double, 3>* const ptr_d_condition) { const double dx = x - rx; const double dy = y - ry; const double cos_theta_r = std::cos(rtheta); const double sin_theta_r = std::sin(rtheta); const double cross_rd_nd = cos_theta_r * dy - sin_theta_r * dx; ptr_d_condition->at(0) = std::copysign(std::sqrt(dx * dx dy * dy), cross_rd_nd); const double delta_theta = theta - rtheta; const double tan_delta_theta = std::tan(delta_theta); const double cos_delta_theta = std::cos(delta_theta); const double one_minus_kappa_r_d = 1 - rkappa * ptr_d_condition->at(0); ptr_d_condition->at(1) = one_minus_kappa_r_d * tan_delta_theta; const double kappa_r_d_prime = rdkappa * ptr_d_condition->at(0) rkappa * ptr_d_condition->at(1); ptr_d_condition->at(2) = -kappa_r_d_prime * tan_delta_theta one_minus_kappa_r_d / cos_delta_theta / cos_delta_theta * (kappa * one_minus_kappa_r_d / cos_delta_theta - rkappa); ptr_s_condition->at(0) = rs; ptr_s_condition->at(1) = v * cos_delta_theta / one_minus_kappa_r_d; const double delta_theta_prime = one_minus_kappa_r_d / cos_delta_theta * kappa - rkappa; ptr_s_condition->at(2) = (a * cos_delta_theta - ptr_s_condition->at(1) * ptr_s_condition->at(1) * (ptr_d_condition->at(1) * delta_theta_prime - kappa_r_d_prime)) / one_minus_kappa_r_d; }

bool ReferenceLine::XYToSL(const common::math::Vec2d& xy_point, SLPoint* const sl_point) const { double s = 0.0; double l = 0.0; if (!map_path_.GetProjection(xy_point, &s, &l)) { AERROR << "Cannot get nearest point from path."; return false; } sl_point->set_s(s); sl_point->set_l(l); return true; }