A bilevel optimal motion planning model with application to autonomous parking

In this paper, we present a bilevel optimal motion planning (BOMP) model for autonomous parking. The BOMP model treats motion planning as an optimal control problem, in which the upper level is designed for vehicle nonlinear dynamics, and the lower level is for geometry collision-free constraints. The significant feature of the BOMP model is that the lower level is a linear programming problem that serves as a constraint for the upper-level problem. That is, an optimal control problem contains an embedded optimization problem as constraints. Traditional optimal control methods cannot solve the BOMP problem directly. Therefore, the modified approximate Karush-Kuhn-Tucker theory is applied to generate a general nonlinear optimal control problem. Then the pseudospectral optimal control method solves the converted problem. Particularly, the lower level is the $J_2$-function that acts as a distance function between convex polyhedron objects. Polyhedrons can approximate vehicles in higher precision than spheres or ellipsoids. Besides, the modified $J_2$-function (MJ) and the active-points based modified $J_2$-function (APMJ) are proposed to reduce the variables number and time complexity. As a result, an iteirative two-stage BOMP algorithm for autonomous parking concerning dynamical feasibility and collision-free property is proposed. The MJ function is used in the initial stage to find an initial collision-free approximate optimal trajectory and the active points, then the APMJ function in the final stage finds out the optimal trajectory. Simulation results and experiment on Turtlebot3 validate the BOMP model, and demonstrate that the computation speed increases almost two orders of magnitude compared with the area criterion based collision avoidance method.