arXiv subjects

Kevin Tracy

Publications and source records attributed to Kevin Tracy.

11 recordsLinked to original sources

arXiv2026

A Differentiable Interior-Point Method in Single Precision

Primal-dual interior-point methods solve constrained convex optimization problems to tight tolerances with speed and robustness. Their solutions are also efficiently differentiable with respect to the problem data through the implicit function theorem. However, the standard treatment of primal-dual complementarity makes the underlying linear systems increasingly ill-conditioned near the solution. While this ill-conditioning is often benign in double precision, it can be catastrophic in single precision, preventing interior-point methods from fully exploiting the accelerated hardware that underpins modern machine learning. This paper introduces a differentiable interior-point method designed for low-precision arithmetic. By using an alternative complementarity representation, we ensure that the underlying linear systems remain spectrally bounded -- even near the solution -- a property that is essential for computing accurate gradients and avoiding arithmetic exceptions. As a result, our method enables interior-point solvers to reliably solve and differentiate optimization problems in single precision that were previously confined to double precision. We demonstrate the approach through an ablation study against the standard interior-point formulation and applications in bilevel and end-to-end learning settings where differentiating through constrained optimization is essential. The source code is available at https://github.com/qpax-solver/qpax.

math.OC↗

arXiv2025

The Trajectory Bundle Method: Unifying Sequential-Convex Programming and Sampling-Based Trajectory Optimization

We present a unified framework for solving trajectory optimization problems in a derivative-free manner through the use of sequential convex programming. Traditionally, nonconvex optimization problems are solved by forming and solving a sequence of convex optimization problems, where the cost and constraint functions are approximated locally through Taylor series expansions. This presents a challenge for functions where differentiation is expensive or unavailable. In this work, we present a derivative-free approach to form these convex approximations by computing samples of the dynamics, cost, and constraint functions and letting the solver interpolate between them. Our framework includes sample-based trajectory optimization techniques like model-predictive path integral (MPPI) control as a special case and generalizes them to enable features like multiple shooting and general equality and inequality constraints that are traditionally associated with derivative-based sequential convex programming methods. The resulting framework is simple, flexible, and capable of solving a wide variety of practical motion planning and control problems.

math.OC↗

arXiv2024

On the Differentiability of the Primal-Dual Interior-Point Method

Primal-Dual Interior-Point methods are capable of solving constrained convex optimization problems to tight tolerances in a fast and robust manner. The derivatives of the primal-dual solution with respect to the problem matrices can be computed using the implicit function theorem, enabling efficient differentiation of these optimizers for a fraction of the cost of the total solution time. In the presence of active inequality constraints, this technique is only capable of providing discontinuous subgradients that present a challenge to algorithms that rely on the smoothness of these derivatives. This paper presents a technique for relaxing primal-dual solutions with a logarithmic barrier to provide smooth derivatives near active inequality constraints, with the ability to specify a uniform and consistent amount of smoothing. We pair this with an efficient primal-dual interior-point algorithm for solving an always-feasible $\ell_1$-penalized variant of a convex quadratic program, eliminating the issues surrounding learning potentially infeasible problems. This parallelizable and smoothly differentiable solver is demonstrated on a range of robotics tasks where smoothing is important. An open source implementation in JAX is available at github.com/kevin-tracy/qpax.

math.OC↗

arXiv2023

Efficient Online Learning of Contact Force Models for Connector Insertion

Contact-rich manipulation tasks with stiff frictional elements like connector insertion are difficult to model with rigid-body simulators. In this work, we propose a new approach for modeling these environments by learning a quasi-static contact force model instead of a full simulator. Using a feature vector that contains information about the configuration and control, we find a linear mapping adequately captures the relationship between this feature vector and the sensed contact forces. A novel Linear Model Learning (LML) algorithm is used to solve for the globally optimal mapping in real time without any matrix inversions, resulting in an algorithm that runs in nearly constant time on a GPU as the model size increases. We validate the proposed approach for connector insertion both in simulation and hardware experiments, where the learned model is combined with an optimization-based controller to achieve smooth insertions in the presence of misalignments and uncertainty. Our website featuring videos, code, and more materials is available at https://model-based-plugging.github.io/.

cs.RO↗

arXiv2023

ReLU-QP: A GPU-Accelerated Quadratic Programming Solver for Model-Predictive Control

We present ReLU-QP, a GPU-accelerated solver for quadratic programs (QPs) that is capable of solving high-dimensional control problems at real-time rates. ReLU-QP is derived by exactly reformulating the Alternating Direction Method of Multipliers (ADMM) algorithm for solving QPs as a deep, weight-tied neural network with rectified linear unit (ReLU) activations. This reformulation enables the deployment of ReLU-QP on GPUs using standard machine-learning toolboxes. We evaluate the performance of ReLU-QP across three model-predictive control (MPC) benchmarks: stabilizing random linear dynamical systems with control limits, balancing an Atlas humanoid robot on a single foot, and tracking whole-body reference trajectories on a quadruped equipped with a six-degree-of-freedom arm. These benchmarks indicate that ReLU-QP is competitive with state-of-the-art CPU-based solvers for small-to-medium-scale problems and offers order-of-magnitude speed improvements for larger-scale problems.

cs.RO↗

arXiv2022

Data-Efficient Model Learning for Control with Jacobian-Regularized Dynamic-Mode Decomposition}

We present a data-efficient algorithm for learning models for model-predictive control (MPC). Our approach, Jacobian-Regularized Dynamic-Mode Decomposition (JDMD), offers improved sample efficiency over traditional Koopman approaches based on Dynamic-Mode Decomposition (DMD) by leveraging Jacobian information from an approximate prior model of the system, and improved tracking performance over traditional model-based MPC. We demonstrate JDMD's ability to quickly learn bilinear Koopman dynamics representations across several realistic examples in simulation, including a perching maneuver for a fixed-wing aircraft with an empirically derived high-fidelity physics model. In all cases, we show that the models learned by JDMD provide superior tracking and generalization performance within a model-predictive control framework, even in the presence of significant model mismatch, when compared to approximate prior models and models learned by standard Extended DMD (EDMD).

cs.RO↗

arXiv2022

A Square-Root Kalman Filter Using Only QR Decompositions

The Kalman filter operates by storing a Gaussian description of the state estimate in the form of a mean and covariance. Instead of storing and manipulating the covariance matrix directly, a square-root Kalman filter only forms and updates a triangular matrix square root of the covariance matrix. The resulting algorithm is more numerically stable than a traditional Kalman filter, benefiting from double the working precision. This paper presents a formulation of the square root Kalman filter that leverages the QR decomposition to dramatically simplify the resulting algorithm.

eess.SY↗

arXiv2022

DiffPills: Differentiable Collision Detection for Capsules and Padded Polygons

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.

cs.RO↗

arXiv2022

Differentiable Collision Detection for a Set of Convex Primitives

Collision detection between objects is critical for simulation, control, and learning for robotic systems. However, existing collision detection routines are inherently non-differentiable, limiting their applications in gradient-based optimization tools. In this work, we propose DCOL: a fast and fully differentiable collision-detection framework that reasons about collisions between a set of composable and highly expressive convex primitive shapes. This is achieved by formulating the collision detection problem as a convex optimization problem that solves for the minimum uniform scaling applied to each primitive before they intersect. The optimization problem is fully differentiable with respect to the configurations of each primitive and is able to return a collision detection metric and contact points on each object, agnostic of interpenetration. We demonstrate the capabilities of DCOL on a range of robotics problems from trajectory optimization and contact physics, and have made an open-source implementation available.

cs.RO↗

arXiv2022

CALIPSO: A Differentiable Solver for Trajectory Optimization with Conic and Complementarity Constraints

We present a new solver for non-convex trajectory optimization problems that is specialized for robotics applications. CALIPSO, or the Conic Augmented Lagrangian Interior-Point SOlver, combines several strategies for constrained numerical optimization to natively handle second-order cones and complementarity constraints. It reliably solves challenging motion-planning problems that include contact-implicit formulations of impacts and Coulomb friction and state-triggered constraints where general-purpose non-convex solvers like SNOPT and Ipopt fail to converge. Additionally, CALIPSO supports efficient differentiation of solutions with respect to problem data, enabling bi-level optimization applications like auto-tuning of feedback policies. Reliable convergence of the solver is demonstrated on a range of problems from manipulation, locomotion, and aerospace domains. An open-source implementation of this solver is available.

cs.RO↗

arXiv2021

Practical limits on Nanosatellite Telescope Pointing: The Impact of Disturbances and Photon Noise

Accurate and stable spacecraft pointing is a requirement of many astronomical observations. Pointing particularly challenges nanosatellites because of an unfavorable surface area to mass ratio and proportionally large volume required for even the smallest attitude control systems. This work explores the limitations on astrophysical attitude knowledge and control in a regime unrestricted by actuator precision or actuator-induced disturbances such as jitter. The external disturbances on an archetypal 6U CubeSat are modeled and the limiting sensing knowledge is calculated from the available stellar flux and grasp of a telescope within the available volume. These inputs are integrated using a model-predictive control scheme. For a simple test case at 1 Hz, with an 85 mm telescope and a single 11th magnitude star, the achievable body pointing is predicted to be 0.39 arcseconds. For a more general limit, integrating available star light, the achievable attitude sensing is approximately 1 milliarcsecond, which leads to a predicted body pointing accuracy of 20 milliarcseconds after application of the control model. These results show significant room for attitude sensing and control systems to improve before astrophysical and environmental limits are reached.

astro-ph.IM↗