Mathew Halm

Wanxin Jin, Alp Aydinoglu, Mathew Halm et al.

This paper investigates the learning, or system identification, of a class of piecewise-affine dynamical systems known as linear complementarity systems (LCSs). We propose a violation-based loss which enables efficient learning of the LCS parameterization, without prior knowledge of the hybrid mode boundaries, using gradient-based methods. The proposed violation-based loss incorporates both dynamics prediction loss and a novel complementarity - violation loss. We show several properties attained by this loss formulation, including its differentiability, the efficient computation of first- and second-order derivatives, and its relationship to the traditional prediction loss, which strictly enforces complementarity. We apply this violation-based loss formulation to learn LCSs with tens of thousands of (potentially stiff) hybrid modes. The results demonstrate a state-of-the-art ability to identify piecewise-affine dynamics, outperforming methods which must differentiate through non-smooth linear complementarity problems.

Bibit Bianchini, Mathew Halm, Nikolai Matni et al.

Inspired by recent strides in empirical efficacy of implicit learning in many robotics tasks, we seek to understand the theoretical benefits of implicit formulations in the face of nearly discontinuous functions, common characteristics for systems that make and break contact with the environment such as in legged locomotion and manipulation. We present and motivate three formulations for learning a function: one explicit and two implicit. We derive generalization bounds for each of these three approaches, exposing where explicit and implicit methods alike based on prediction error losses typically fail to produce tight bounds, in contrast to other implicit methods with violation-based loss definitions that can be fundamentally more robust to steep slopes. Furthermore, we demonstrate that this violation implicit loss can tightly bound graph distance, a quantity that often has physical roots and handles noise in inputs and outputs alike, instead of prediction losses which consider output noise only. Our insights into the generalizability and physical relevance of violation implicit formulations match evidence from prior works and are validated through a toy problem, inspired by rigid-contact models and referenced throughout our theoretical analysis.

Mathew Halm, Michael Posa

Robotic manipulation and locomotion often entail nearly-simultaneous collisions -- such as heel and toe strikes during a foot step -- with outcomes that are extremely sensitive to the order in which impacts occur. Robotic simulators commonly lack the accuracy to predict this ordering, and instead pick one with a heuristic. This discrepancy degrades performance when model-based controllers and policies learned in simulation are placed on a real robot. We reconcile this issue with a set-valued rigid-body model which generates a broad set of physically reasonable outcomes of simultaneous frictional impacts. We first extend Routh's impact model to multiple impacts by reformulating it as a differential inclusion (DI), and show that any solution will resolve all impacts in finite time. By considering time as a state, we embed this model into another DI which captures the continuous-time evolution of rigid body dynamics, and guarantee existence of solutions. We finally cast simulation of simultaneous impacts as a linear complementarity problem (LCP), and develop an algorithm for tight approximation of the post-impact velocity set with probabilistic guarantees. We demonstrate our approach on several examples drawn from manipulation and legged locomotion.

Mihir Parmar, Mathew Halm, Michael Posa

Frictional contact has been extensively studied as the core underlying behavior of legged locomotion and manipulation, and its nearly-discontinuous nature makes planning and control difficult even when an accurate model of the robot is available. Here, we present empirical evidence that learning an accurate model in the first place can be confounded by contact, as modern deep learning approaches are not designed to capture this non-smoothness. We isolate the effects of contact's non-smoothness by varying the mechanical stiffness of a compliant contact simulator. Even for a simple system, we find that stiffness alone dramatically degrades training processes, generalization, and data-efficiency. Our results raise serious questions about simulated testing environments which do not accurately reflect the stiffness of rigid robotic hardware. Significant additional investigation will be necessary to fully understand and mitigate these effects, and we suggest several avenues for future study.

Samuel Pfrommer, Mathew Halm, Michael Posa

Common methods for learning robot dynamics assume motion is continuous, causing unrealistic model predictions for systems undergoing discontinuous impact and stiction behavior. In this work, we resolve this conflict with a smooth, implicit encoding of the structure inherent to contact-induced discontinuities. Our method, ContactNets, learns parameterizations of inter-body signed distance and contact-frame Jacobians, a representation that is compatible with many simulation, control, and planning environments for robotics. We furthermore circumvent the need to differentiate through stiff or non-smooth dynamics with a novel loss function inspired by the principles of complementarity and maximum dissipation. Our method can predict realistic impact, non-penetration, and stiction when trained on 60 seconds of real-world data.

Mathew Halm, Michael Posa

Quasi-static models of robotic motion with frictional contact provide a computationally efficient framework for analysis and have been widely used for planning and control of non-prehensile manipulation. In this work, we present a novel quasi-static model of planar manipulation that directly maps commanded manipulator velocities to object motion. While quasi-static models have traditionally been unable to capture grasping and jamming behaviors, our approach solves this issue by explicitly modeling the limiting behavior of a velocity-controlled manipulator. We retain the precise modeling of surface contact pressure distributions and efficient computation of contact-rich behaviors of previous methods and additionally prove existence of solutions for any desired manipulator motion. We derive continuous and time-stepping formulations, both posed as tractable Linear Complementarity Problems (LCPs).

Mathew Halm, Michael Posa

Many fundamental challenges in robotics, based in manipulation or locomotion, require making and breaking contact with the environment. To represent the complexity of frictional contact events, impulsive impact models are especially popular, as they often lead to mathematically and computationally tractable approaches. However, when two or more impacts occur simultaneously, the precise sequencing of impact forces is generally unknown, leading to the potential for multiple possible outcomes. This simultaneity is far from pathological, and occurs in many common robotics applications. In this work, we propose an approach for resolving simultaneous frictional impacts, represented as a differential inclusion. Solutions to our model, an extension to multiple contacts of Routh's method, naturally capture the set of potential post-impact velocities.We prove that solutions to the presented model must terminate. This is, to the best of our knowledge, the first such guarantee for set-valued outcomes to simultaneous frictional impacts.