Real-Time-Feasible Collision-Free Motion Planning For Ellipsoidal Objects

TL;DR

The paper tackles real-time collision-free motion planning for ellipsoidal objects within an MPC framework. It introduces a differentiable collision-avoidance constraint based on a parametric over-approximation of the Minkowski sum $\mathcal{E}(0,M_1) \oplus \mathcal{E}(0,M_2)$, controlled by a direction-dependent parameter $\gamma$, and shows improved computational efficiency over separating-hyperplane methods. A fixed-$\gamma$ variant and a safe, robust bounding strategy for $\gamma$ reduce nonlinearity and enhance real-time solvability, with modest suboptimality. The approach is validated through challenging simulations and real-world experiments on a differential-drive robot, demonstrating real-time feasibility and practical effectiveness in cluttered environments.

Abstract

Online planning of collision-free trajectories is a fundamental task for robotics and self-driving car applications. This paper revisits collision avoidance between ellipsoidal objects using differentiable constraints. Two ellipsoids do not overlap if and only if the endpoint of the vector between the center points of the ellipsoids does not lie in the interior of the Minkowski sum of the ellipsoids. This condition is formulated using a parametric over-approximation of the Minkowski sum, which can be made tight in any given direction. The resulting collision avoidance constraint is included in an optimal control problem (OCP) and evaluated in comparison to the separating-hyperplane approach. Not only do we observe that the Minkowski-sum formulation is computationally more efficient in our experiments, but also that using pre-determined over-approximation parameters based on warm-start trajectories leads to a very limited increase in suboptimality. This gives rise to a novel real-time scheme for collision-free motion planning with model predictive control (MPC). Both the real-time feasibility and the effectiveness of the constraint formulation are demonstrated in challenging real-world experiments.

Figures (6)

• Figure 1: Illustrative example
• Figure 2: MPC simulation results. The over-approximations are optimized within the OCPs. The solid-line ellipsoids depict the differential-drive robot at different time steps, and the ellipsoids colored in brown depict the obstacles to avoid.
• Figure 3: Computation time for solving OCPs in MPC simulation. The time for computing the value of $\hat{\gamma}$ and $\hat{\eta}$ is excluded. The maximum number of iterations is sufficiently big for the SQP to converge.
• Figure 4: Relative additional cost due to fixing the over-approximation parameters $\hat{\gamma}$ and fixing the separating hyperplane parameters $\hat{\eta}$. The lines show the percentage of OCPs whose relative additional cost exceeds certain values.
• Figure 5: Computation time in real-world experiments (maximum two QP iterations). The white circle is the median. The black bar goes from the lower to the upper quartile.

Theorems & Definitions (7)

• Definition 1: Supporting function and supporting halfspace
• Definition 2: Minkowski sum
• Lemma 1
• Lemma 2
• Lemma 3
• Corollary 4
• Remark 5