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Decoupling Collision Avoidance in and for Optimal Control using Least-Squares Support Vector Machines

Dries Dirckx, Wilm Decré, Jan Swevers

TL;DR

The paper tackles the non-convexity of collision-avoidance constraints in optimal control by reformulating the separating hyperplane condition as a least-squares classification problem, then decoupling the hyperplanes from the OCP as parameters updated between NLP iterations. This yields linear collision-avoidance constraints in the OCP and significantly reduces the number of constraints and decision variables, enabling scalable trajectory optimization in cluttered environments. A two-phase algorithm combines LS-SVM/QP-SVM hyperplane computation with inter-iteration updates, along with broad-phase and trust-region filters to stabilize convergence. Empirical results show 50–90% speedups in 2D planning and around 51% speedups in 3D manipulator tasks, with negligible losses in optimality and preserved differentiability for gradient-based solvers.

Abstract

This paper details an approach to linearise differentiable but non-convex collision avoidance constraints tailored to convex shapes. It revisits introducing differential collision avoidance constraints for convex objects into an optimal control problem (OCP) using the separating hyperplane theorem. By framing this theorem as a classification problem, the hyperplanes are eliminated as optimisation variables from the OCP. This effectively transforms non-convex constraints into linear constraints. A bi-level algorithm computes the hyperplanes between the iterations of an optimisation solver and subsequently embeds them as parameters into the OCP. Experiments demonstrate the approach's favourable scalability towards cluttered environments and its applicability to various motion planning approaches. It decreases trajectory computation times between 50\% and 90\% compared to a state-of-the-art approach that directly includes the hyperplanes as variables in the optimal control problem.

Decoupling Collision Avoidance in and for Optimal Control using Least-Squares Support Vector Machines

TL;DR

The paper tackles the non-convexity of collision-avoidance constraints in optimal control by reformulating the separating hyperplane condition as a least-squares classification problem, then decoupling the hyperplanes from the OCP as parameters updated between NLP iterations. This yields linear collision-avoidance constraints in the OCP and significantly reduces the number of constraints and decision variables, enabling scalable trajectory optimization in cluttered environments. A two-phase algorithm combines LS-SVM/QP-SVM hyperplane computation with inter-iteration updates, along with broad-phase and trust-region filters to stabilize convergence. Empirical results show 50–90% speedups in 2D planning and around 51% speedups in 3D manipulator tasks, with negligible losses in optimality and preserved differentiability for gradient-based solvers.

Abstract

This paper details an approach to linearise differentiable but non-convex collision avoidance constraints tailored to convex shapes. It revisits introducing differential collision avoidance constraints for convex objects into an optimal control problem (OCP) using the separating hyperplane theorem. By framing this theorem as a classification problem, the hyperplanes are eliminated as optimisation variables from the OCP. This effectively transforms non-convex constraints into linear constraints. A bi-level algorithm computes the hyperplanes between the iterations of an optimisation solver and subsequently embeds them as parameters into the OCP. Experiments demonstrate the approach's favourable scalability towards cluttered environments and its applicability to various motion planning approaches. It decreases trajectory computation times between 50\% and 90\% compared to a state-of-the-art approach that directly includes the hyperplanes as variables in the optimal control problem.
Paper Structure (12 sections, 1 theorem, 4 equations, 6 figures, 4 tables, 3 algorithms)

This paper contains 12 sections, 1 theorem, 4 equations, 6 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction: Autonomous Motion Planning
  2. Related Work
  3. Contributions
  4. Collision Detection as Least-Squares Classification
  5. (Least-Squares) Support Vector Machines
  6. Decoupled Collision Avoidance in Optimal Control
  7. Broad-phase and Trust-region Filter
  8. Two-step Optimisation
  9. Experimental Validation
  10. Two-dimensional Holonomic Motion Planning
  11. Configuration-to-Point Manipulator Planning
  12. Conclusion and Future Work

Key Result

Theorem 1

Suppose $A$ and $B$ are nonempty disjoint convex sets, i.e., $A \cap B$ = $\emptyset$. Then there exist $\boldsymbol{w} \neq 0$ and $w_{b}$ such that $\boldsymbol{w}^{\top}\boldsymbol{y} + w_{b} \geq 0$ for all $\boldsymbol{y} \in A$ and $\boldsymbol{w}^{\top}\boldsymbol{y} + w_{b} \leq 0$ for all $

Figures (6)

  • Figure 1: (left) Separating hyperplane for two disjoint, nonempty convex sets A and B Boyd2004. (right) Illustration of the set of vertices for a capsular robot ($V_{r}$) and two obstacles, one triangular ($V_{o1}$) and one rectangular ($V_{o2}$).
  • Figure 2: Illustration of two linearly separable classes, the hyperplane normal $w$, the classification margin $\frac{1}{w^{\top}w}$ and a misclassification characterised by the error $e_{k}$.
  • Figure 3: Visualisation of an example of the broadphase filter thresholds and the trust-region filter described in Algorithm 3 and employed in the experiments.
  • Figure 4: Two randomly generated environments with five obstacles, where the AMR has to navigate through both in the holonomic case and spline-based case. The start and goal positions are indicated by the green and checkered circles, respectively. Ten different start and goal combinations are constructed per environment by varying them along the dotted and dash-dotted line.
  • Figure 5: Total wall time (top) and wall time per iteration (bottom) required to solve the holonomic robot OCP with an increasing number of obstacles for both the coupled and decoupled approach.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1: Separating hyperplane theorem Boyd2004