Optimization-Based Trajectory Planning for Tractor-Trailer Vehicles on Curvy Roads: A Progressively Increasing Sampling Number Method

TL;DR

The paper tackles on-road trajectory planning for tractor-trailer systems by formulating a single-phase OCP with a Cartesian representation of lateral and orientation errors, along with a coarse trajectory to identify the homotopy class. It introduces the PISNO framework to manage computational complexity by solving a sequence of intermediate OCPs with progressively finer sampling numbers, warming each solution with the previous one and adapting collision constraints via wide/narrow segmentation. Simulation and real-world experiments demonstrate that PISNO achieves faster planning times and robust obstacle avoidance while maintaining trajectory quality compared to several baselines. The approach offers practical benefits for safe, efficient tractor-trailer operation on curvy roads and lays groundwork for future work in uncertainty handling and multi-vehicle coordination.

Abstract

In this work, we propose an optimization-based trajectory planner for tractor-trailer vehicles on curvy roads. The lack of analytical expression for the trailer's errors to the center line pose a great challenge to the trajectory planning for tractor-trailer vehicles. To address this issue, we first use geometric representations to characterize the lateral and orientation errors in Cartesian frame, where the errors would serve as the components of the cost function and the road edge constraints within our optimization process. Next, we generate a coarse trajectory to warm-start the subsequent optimization problems. On the other hand, to achieve a good approximation of the continuous-time kinematics, optimization-based methods usually discretize the kinematics with a large sampling number. This leads to an increase in the number of the variables and constraints, thus making the optimization problem difficult to solve. To address this issue, we design a Progressively Increasing Sampling Number Optimization (PISNO) framework. More specifically, we first find a nearly feasible trajectory with a small sampling number to warm-start the optimization process. Then, the sampling number is progressively increased, and the corresponding intermediate Optimal Control Problem (OCP) is solved in each iteration. Next, we further resample the obtained solution into a finer sampling period, and then use it to warm-start the intermediate OCP in next iteration. This process is repeated until reaching a threshold sampling number. Simulation and experiment results show the proposed method exhibits a good performance and less computational consumption over the benchmarks.

Figures (9)

• Figure 1: The schematic of a tractor-trailer vehicle system.
• Figure 2: The geometric formulation to characterize the lateral and orientation errors from center line. The reference point $(x_i^{\rm ref}, y_i^{\rm ref})$ is the projection of the initial guess point $(x_i^{\rm init}, y_i^{\rm init})$ on the center line.
• Figure 3: The process of generating coarse path.
• Figure 4: The flow diagram of the proposed PISNO framework.
• Figure 5: The optimization process starts with a sampling number $N_{\rm curr} = 25$, and then refines the trajectory by progressively increasing the sampling number until $N_{\rm curr} = 200$ in (c)-(f).

Theorems & Definitions (2)

• Remark 1
• Remark 2