URDF+: An Enhanced URDF for Robots with Kinematic Loops

Designs incorporating kinematic loops are becoming increasingly prevalent in the robotics community. Despite the existence of dynamics algorithms to deal with the effects of such loops, many modern simulators rely on dynamics libraries that require robots to be represented as kinematic trees. This requirement is reflected in the de facto standard format for describing robots, the Universal Robot Description Format (URDF), which does not support kinematic loops resulting in closed chains. This paper introduces an enhanced URDF, termed URDF+, which addresses this key shortcoming of URDF while retaining the intuitive design philosophy and low barrier to entry that the robotics community values. The URDF+ keeps the elements used by URDF to describe open chains and incorporates new elements to encode loop joints. We also offer an accompanying parser that processes the system models coming from URDF+ so that they can be used with recursive rigid-body dynamics algorithms for closed-chain systems that group bodies into local, decoupled loops. This parsing process is fully automated, ensuring optimal grouping of constrained bodies without requiring manual specification from the user. We aim to advance the robotics community towards this elegant solution by developing efficient and easy-to-use software tools.

Probabilistic Homotopy Optimization for Dynamic Motion Planning

We present a homotopic approach to solving challenging, optimization-based motion planning problems. The approach uses Homotopy Optimization, which, unlike standard continuation methods for solving homotopy problems, solves a sequence of constrained optimization problems rather than a sequence of nonlinear systems of equations. The insight behind our proposed algorithm is formulating the discovery of this sequence of optimization problems as a search problem in a multidimensional homotopy parameter space. Our proposed algorithm, the Probabilistic Homotopy Optimization algorithm, switches between solve and sample phases, using solutions to easy problems as initial guesses to more challenging problems. We analyze how our algorithm performs in the presence of common challenges to homotopy methods, such as bifurcation, folding, and disconnectedness of the homotopy solution manifold. Finally, we demonstrate its utility via a case study on two dynamic motion planning problems: the cart-pole and the MIT Humanoid.

CusADi: A GPU Parallelization Framework for Symbolic Expressions and Optimal Control

The parallelism afforded by GPUs presents significant advantages in training controllers through reinforcement learning (RL). However, integrating model-based optimization into this process remains challenging due to the complexity of formulating and solving optimization problems across thousands of instances. In this work, we present CusADi, an extension of the CasADi symbolic framework to support the parallelization of arbitrary closed-form expressions on GPUs with CUDA. We also formulate a closed-form approximation for solving general optimal control problems, enabling large-scale parallelization and evaluation of MPC controllers. Our results show a ten-fold speedup relative to similar MPC implementation on the CPU, and we demonstrate the use of CusADi for various applications, including parallel simulation, parameter sweeps, and policy training.

Integrating Model-Based Footstep Planning with Model-Free Reinforcement Learning for Dynamic Legged Locomotion

In this work, we introduce a control framework that combines model-based footstep planning with Reinforcement Learning (RL), leveraging desired footstep patterns derived from the Linear Inverted Pendulum (LIP) dynamics. Utilizing the LIP model, our method forward predicts robot states and determines the desired foot placement given the velocity commands. We then train an RL policy to track the foot placements without following the full reference motions derived from the LIP model. This partial guidance from the physics model allows the RL policy to integrate the predictive capabilities of the physics-informed dynamics and the adaptability characteristics of the RL controller without overfitting the policy to the template model. Our approach is validated on the MIT Humanoid, demonstrating that our policy can achieve stable yet dynamic locomotion for walking and turning. We further validate the adaptability and generalizability of our policy by extending the locomotion task to unseen, uneven terrain. During the hardware deployment, we have achieved forward walking speeds of up to 1.5 m/s on a treadmill and have successfully performed dynamic locomotion maneuvers such as 90-degree and 180-degree turns.

Tailoring Solution Accuracy for Fast Whole-body Model Predictive Control of Legged Robots

Thanks to recent advancements in accelerating non-linear model predictive control (NMPC), it is now feasible to deploy whole-body NMPC at real-time rates for humanoid robots. However, enforcing inequality constraints in real time for such high-dimensional systems remains challenging due to the need for additional iterations. This paper presents an implementation of whole-body NMPC for legged robots that provides low-accuracy solutions to NMPC with general equality and inequality constraints. Instead of aiming for highly accurate optimal solutions, we leverage the alternating direction method of multipliers to rapidly provide low-accuracy solutions to quadratic programming subproblems. Our extensive simulation results indicate that real robots often cannot benefit from highly accurate solutions due to dynamics discretization errors, inertial modeling errors and delays. We incorporate control barrier functions (CBFs) at the initial timestep of the NMPC for the self-collision constraints, resulting in up to a 26-fold reduction in the number of self-collisions without adding computational burden. The controller is reliably deployed on hardware at 90 Hz for a problem involving 32 timesteps, 2004 variables, and 3768 constraints. The NMPC delivers sufficiently accurate solutions, enabling the MIT Humanoid to plan complex crossed-leg and arm motions that enhance stability when walking and recovering from significant disturbances.

Learning Quadruped Locomotion Using Differentiable Simulation

While most recent advancements in legged robot control have been driven by model-free reinforcement learning, we explore the potential of differentiable simulation. Differentiable simulation promises faster convergence and more stable training by computing low-variant first-order gradients using the robot model, but so far, its use for legged robot control has remained limited to simulation. The main challenge with differentiable simulation lies in the complex optimization landscape of robotic tasks due to discontinuities in contact-rich environments, e.g., quadruped locomotion. This work proposes a new, differentiable simulation framework to overcome these challenges. The key idea involves decoupling the complex whole-body simulation, which may exhibit discontinuities due to contact, into two separate continuous domains. Subsequently, we align the robot state resulting from the simplified model with a more precise, non-differentiable simulator to maintain sufficient simulation accuracy. Our framework enables learning quadruped walking in minutes using a single simulated robot without any parallelization. When augmented with GPU parallelization, our approach allows the quadruped robot to master diverse locomotion skills, including trot, pace, bound, and gallop, on challenging terrains in minutes. Additionally, our policy achieves robust locomotion performance in the real world zero-shot. To the best of our knowledge, this work represents the first demonstration of using differentiable simulation for controlling a real quadruped robot. This work provides several important insights into using differentiable simulations for legged locomotion in the real world.

FLD: Fourier Latent Dynamics for Structured Motion Representation and Learning

Motion trajectories offer reliable references for physics-based motion learning but suffer from sparsity, particularly in regions that lack sufficient data coverage. To address this challenge, we introduce a self-supervised, structured representation and generation method that extracts spatial-temporal relationships in periodic or quasi-periodic motions. The motion dynamics in a continuously parameterized latent space enable our method to enhance the interpolation and generalization capabilities of motion learning algorithms. The motion learning controller, informed by the motion parameterization, operates online tracking of a wide range of motions, including targets unseen during training. With a fallback mechanism, the controller dynamically adapts its tracking strategy and automatically resorts to safe action execution when a potentially risky target is proposed. By leveraging the identified spatial-temporal structure, our work opens new possibilities for future advancements in general motion representation and learning algorithms.

Learning Emergent Gaits with Decentralized Phase Oscillators: on the role of Observations, Rewards, and Feedback

We present a minimal phase oscillator model for learning quadrupedal locomotion. Each of the four oscillators is coupled only to itself and its corresponding leg through local feedback of the ground reaction force, which can be interpreted as an observer feedback gain. We interpret the oscillator itself as a latent contact state-estimator. Through a systematic ablation study, we show that the combination of phase observations, simple phase-based rewards, and the local feedback dynamics induces policies that exhibit emergent gait preferences, while using a reduced set of simple rewards, and without prescribing a specific gait. The code is open-source, and a video synopsis available at https://youtu.be/1NKQ0rSV3jU.

Recursive Rigid-Body Dynamics Algorithms for Systems with Kinematic Loops

We propose a novel approach for generalizing the following rigid-body dynamics algorithms: Recursive Newton-Euler Algorithm, Articulated-Body Algorithm, and Extended-Force-Propagator Algorithm. The classic versions of these recursive algorithms require systems to have an open chain structure. Dealing with closed-chains has, conventionally, required different algorithms. In this paper, we demonstrate that the classic recursive algorithms can be modified to work for closed-chain mechanisms. The critical insight of our generalized algorithms is the clustering of bodies involved in local loop constraints. Clustering bodies enables loop constraints to be resolved locally, i.e., only when that group of bodies is encountered during a forward or backward pass. This local treatment avoids the need for large-scale matrix factorization. We provide self-contained derivations of the algorithms using familiar, physically meaningful concepts. Overall, our approach provides a foundation for simulating robotic systems with traditionally difficult-to-simulate designs, such as geared motors, differential drives, and four-bar mechanisms. The performance of our library of algorithms is validated numerically in C++ on various modern legged robots: the MIT Mini Cheetah, the MIT Humanoid, the UIUC Tello Humanoid, and a modified version of the JVRC-1 Humanoid. Our algorithms are shown to outperform state-of-the-art algorithms for computing constrained rigid-body dynamics.

How to Model Brushless Electric Motors for the Design of Lightweight Robotic Systems

A key step in the development of lightweight, high performance robotic systems is the modeling and selection of permanent magnet brushless direct current (BLDC) electric motors. Typical modeling analyses are completed a priori, and provide insight for properly sizing a motor for an application, specifying the required operating voltage and current, as well as assessing the thermal response and other design attributes (e.g.transmission ratio). However, to perform these modeling analyses, proper information about the motor's characteristics are needed, which are often obtained from manufacturer datasheets. Through our own experience and communications with manufacturers, we have noticed a lack of clarity and standardization in modeling BLDC motors, compounded by vague or inconsistent terminology used in motor datasheets. The purpose of this tutorial is to concisely describe the governing equations for BLDC motor analyses used in the design process, as well as highlight potential errors that can arise from incorrect usage. We present a power-invariant conversion from phase and line-to-line reference frames to a familiar q-axis DC motor representation, which provides a ``brushed'' analogue of a three phase BLDC motor that is convenient for analysis and design. We highlight potential errors including incorrect calculations of winding resistive heat loss, improper estimation of motor torque via the motor's torque constant, and incorrect estimation of the required bus voltage or resulting angular velocity limitations. A unified and condensed set of governing equations is available for designers in the Appendix. The intent of this work is to provide a consolidated mathematical foundation for modeling BLDC motors that addresses existing confusion and fosters high performance designs of future robotic systems.