[Paper Review] DiffPills: Differentiable Collision Detection for Capsules and Padded Polygons

Kevin Tracy, Taylor A. Howell+1 more|arXiv (Cornell University)|Jul 1, 2022

Robotic Path Planning Algorithms4 citations

TL;DR

This paper introduces DiffPills, a differentiable collision detection framework for capsules and padded polygons by formulating collision queries as differentiable convex quadratic programs. The method computes proximity values and closest points with full gradient support, enabling integration into gradient-based robotics workflows such as trajectory optimization, state estimation, and reinforcement learning.

ABSTRACT

Collision detection plays an important role in simulation, control, and learning for robotic systems. However, no existing method is differentiable with respect to the configurations of the objects, greatly limiting the sort of algorithms that can be built on top of collision detection. In this work, we propose a set of differentiable collision detection algorithms between capsules and padded polygons by formulating these problems as differentiable convex quadratic programs. The resulting algorithms are able to return a proximity value indicating if a collision has taken place, as well as the closest points between objects, all of which are differentiable. As a result, they can be used reliably within other gradient-based optimization methods, including trajectory optimization, state estimation, and reinforcement learning methods.

Motivation & Objective

• Address the lack of differentiable collision detection in robotics, which limits the use of gradient-based optimization in simulation, planning, and learning.
• Enable accurate and differentiable contact modeling for complex geometries using capsules and padded polygons as primitives.
• Formulate collision detection as a differentiable convex optimization problem to support backpropagation through proximity and closest-point computations.
• Support gradient computation with respect to object configurations (position, orientation) for both capsules and padded polygons.
• Demonstrate applicability in real-world robotics tasks such as trajectory optimization with collision avoidance constraints.

Proposed method

• Model capsules and padded polygons as geometric primitives where collision is determined by proximity between their closest points.
• Formulate the collision detection problem as an inequality-constrained convex quadratic program (QP), parameterized by object configurations.
• Use primal-dual interior-point methods with Mehrotra predictor-corrector steps to solve the QP, enabling differentiable optimization via automatic differentiation.
• Derive gradients of the proximity value and closest points using the KKT conditions and implicit function theorem, propagating derivatives through the QP solution.
• Recover closest points on the capsule and polygon from the QP solution variables using analytical expressions based on parametric representations.
• Integrate the differentiable collision detection into trajectory optimization pipelines using gradient-aware solvers like ALTRO.

Experimental results

Research questions

• RQ1 Can collision detection between capsules and padded polygons be made fully differentiable with respect to object configurations?
• RQ2 How can proximity and closest-point information be computed differentially using convex optimization?
• RQ3 To what extent can differentiable collision detection improve the performance and feasibility of gradient-based motion planning?
• RQ4 What is the computational and numerical stability of the proposed differentiable QP formulation in robotics workloads?
• RQ5 Can the framework be used to enforce smooth, differentiable collision avoidance constraints in trajectory optimization?

Key findings

• The proposed method successfully computes differentiable proximity values and closest points between capsules and padded polygons using differentiable convex QP solving.
• Gradients of the proximity value with respect to capsule and polygon configurations are computed via implicit differentiation through the QP solution, enabling backpropagation.
• The method was successfully integrated into the ALTRO trajectory optimizer, producing a feasible and optimal trajectory that avoids collision with a stationary capsule.
• The open-source implementation in Julia (DiffPills.jl) enables reproducibility and integration into robotics research pipelines.
• The differentiable collision detection enables the use of expressive primitives like capsules and padded polygons in gradient-based learning and control, overcoming limitations of spherical keep-out zones.
• The approach supports smooth, continuous gradients even at collision boundaries, enabling stable optimization in non-smooth contact scenarios.