We propose two novel algorithms for constructing convex collision-free polytopes in robot configuration space. Finding these polytopes enables the application of stronger motion-planning frameworks such as trajectory optimization with Graphs of Convex Sets [1] and is currently a major roadblock in the adoption of these approaches. In this paper, we build upon IRIS-NP (Iterative Regional Inflation by Semidefinite & Nonlinear Programming) [2] to significantly improve tunability, runtimes, and scaling to complex environments. IRIS-NP uses nonlinear programming paired with uniform random initialization to find configurations on the boundary of the free configuration space. Our key insight is that finding near-by configuration-space obstacles using sampling is inexpensive and greatly accelerates region generation. We propose two algorithms using such samples to either employ nonlinear programming more efficiently (IRIS-NP2) or circumvent it altogether using a massively-parallel zero-order optimization strategy (IRIS-ZO). We also propose a termination condition that controls the probability of exceeding a user-specified permissible fraction-in-collision, eliminating a significant source of tuning difficulty in IRIS-NP. We compare performance across eight robot environments, showing that IRIS-ZO achieves an order-of-magnitude speed advantage over IRIS-NP. IRIS-NP2, also significantly faster than IRIS-NP, builds larger polytopes using fewer hyperplanes, enabling faster downstream computation.

Recent reinforcement learning approaches have shown surprisingly strong capabilities of bang-bang policies for solving continuous control benchmarks. The underlying coarse action space discretizations often yield favourable exploration characteristics while final performance does not visibly suffer in the absence of action penalization in line with optimal control theory. In robotics applications, smooth control signals are commonly preferred to reduce system wear and energy efficiency, but action costs can be detrimental to exploration during early training. In this work, we aim to bridge this performance gap by growing discrete action spaces from coarse to fine control resolution, taking advantage of recent results in decoupled Q-learning to scale our approach to high-dimensional action spaces up to dim(A) = 38. Our work indicates that an adaptive control resolution in combination with value decomposition yields simple critic-only algorithms that yield surprisingly strong performance on continuous control tasks.

Many computations in robotics can be dramatically accelerated if the robot configuration space is described as a collection of simple sets. For example, recently developed motion planners rely on a convex decomposition of the free space to design collision-free trajectories using fast convex optimization. In this work, we present an efficient method for approximately covering complex configuration spaces with a small number of polytopes. The approach constructs a visibility graph using sampling and generates a clique cover of this graph to find clusters of samples that have mutual line of sight. These clusters are then inflated into large, full-dimensional, polytopes. We evaluate our method on a variety of robotic systems and show that it consistently covers larger portions of free configuration space, with fewer polytopes, and in a fraction of the time compared to previous methods.

However, recent results have shown that competitive performance can be achieved with strongly reduced, discretized versions of the original action space (Tavakoli et al., 2018; Tang & Agrawal, 2020; Seyde et al., 2021). This opens the question whether tasks with complex high-dimensional action spaces can be solved using simpler critic-only, discrete action-space algorithms instead. A potential candidate is Q-learning which only requires learning a critic with the policy commonly following via ϵ-greedy or Boltzmann exploration (Watkins & Dayan, 1992; Mnih et al., 2013). While naive Q-learning struggles in high-dimensional action spaces due to exponential scaling of possible action combinations, the multi-agent RL literature has shown that factored value function representations in combination with centralized training can alleviate some of these challenges (Sunehag et al., 2017; Rashid et al., 2018), further inspiring transfer to single-agent control settings (Sharma et al., 2017; Tavakoli, 2021). Other methods have been shown to enable application of critic-only agents to continuous action spaces but require additional, costly, sampling-based optimization (Kalashnikov et al., 2018).

Understanding the geometry of collision-free configuration space (C-free) in the presence of task-space obstacles is an essential ingredient for collision-free motion planning. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing C-free regions with rigorous certificates due to the complexity of mapping task-space obstacles through the kinematics. In this work, we present the first to our knowledge rigorous method for approximately decomposing a rational parametrization of C-free into certified polyhedral regions. Our method, called C-IRIS (C-space Iterative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parameterization of the configuration space which are rigorously certified to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion planning. Based on convex optimization, our method works in arbitrary dimensions, only makes assumptions about the convexity of the obstacles in the task space, and is fast enough to scale to realistic problems in manipulation. We demonstrate our algorithm's ability to fill a non-trivial amount of collision-free C-space in several 2-DOF examples where the C-space can be visualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa, a 6-DOF UR3e and 12-DOF bimanual manipulators. An implementation of our algorithm is open-sourced in Drake. We furthermore provide examples of our algorithm in interactive Python notebooks.