A bilevel optimal motion planning (BOMP) model with application to autonomous parking (2312.00314v1)

Abstract: 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.

• The paper introduces a bilevel optimal motion planning model that unifies nonlinear vehicle dynamics with linear collision avoidance strategies.
• It employs modified J2 functions (MJ and APMJ) to optimize path feasibility and reduce computational complexity by nearly two orders of magnitude.
• Simulations with Turtlebot3 validate the model's performance, highlighting its potential for scalable and robust autonomous parking solutions.

Optimizing Autonomous Parking with Bilevel Programming: A Fresh Approach

Autonomous vehicles have the potential to radically transform how we think about personal transportation. One area of particular interest is the automation of parking maneuvers, which can be especially challenging in congested, urban areas with limited space. The proposed bilevel optimal motion planning (BOMP) model tackles this challenge head-on by treating vehicle motion planning and collision avoidance as a unified optimization problem.

What is the BOMP Model?

The BOMP model approaches autonomous parking as an optimal control problem. At its heart, it consists of two levels of optimization problems:

• The upper-level problem focuses on vehicle nonlinear dynamics, ensuring that the vehicle moves according to the laws of physics and its own mechanical constraints.
• The lower-level problem is a linear programming problem that primarily deals with avoiding collisions with obstacles.

A significant aspect of the BOMP model is its use of the 𝐽2-function. This function serves as a constraint within the upper-level problem and acts as a distance measure between the vehicle and potential obstacles, which are represented by convex polyhedrons. Essentially, ensuring that the 𝐽2-function maintains a value above zero keeps the vehicle a safe distance from collisions during parking.

Advancements and Performance

The paper introduces novel modifications to the 𝐽2-function, referred to as MJ and APMJ functions, which reduce the number of variables and time complexity. This results in an iterative two-stage algorithm that provides dynamical feasibility and collision-free trajectories for autonomous vehicles:

• The initial stage uses the MJ function to map out a connected, feasible path and identify "active points" or critical areas for navigation.
• The final stage applies the APMJ function, optimizing the trajectory based on the information gathered earlier.

Simulations and experiments with the Turtlebot3 robot validate the BOMP model and boast a significant leap in computation speed—almost two orders faster compared to previous methods.

Implications and Future Research

The paper has opened the door to efficient and intelligent autonomous parking solutions. Although the BOMP model currently only considers static environments, the potential for adaptation to dynamic settings with moving obstacles remains a promising avenue for future research. Additionally, further work can explore how to integrate the BOMP approach into cluttered environments and address issues such as the smoothness of control variables and terminal angle adjustment. Overall, the BOMP model represents a scalable and robust framework that is applicable across various autonomous systems, from vehicles to robotic arms.

Takeaway

Autonomous parking technology has taken a notable step forward with the bilevel optimal motion planning model. By optimizing vehicle dynamics and collision avoidance simultaneously, this approach not only makes autonomous parking more precise and efficient but also showcases the potential of optimal control methodologies in advanced robotics applications. The success of this model hints at an exciting road ahead, where autonomous vehicles could adeptly navigate and negotiate even the trickiest of parking scenarios.