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From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation (1909.08560v4)

Published 18 Sep 2019 in cs.RO, cs.SY, and eess.SY

Abstract: This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot --- the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot --- the Vision 60, with 36 state variables and 12 control inputs --- both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds.

Citations (12)

Summary

  • The paper presents a novel dynamics decomposition method that breaks down complex quadruped robots into bipedal systems for rapid generation of dynamic walking gaits.
  • This approach optimizes virtual constraints for the decomposed biped systems using a Nonlinear Program, efficiently generating full-body dynamic trajectories.
  • Validated on a Vision 60 robot, the method rapidly generates gaits in seconds, achieving robust dynamic locomotion on hardware with simple controllers.

This paper presents a novel methodology for rapidly generating dynamic walking gaits for quadrupedal robots by systematically decomposing their full-body dynamics into two bipedal systems. The core idea is to leverage well-established gait generation techniques developed for lower-dimensional bipedal robots and apply them to the decomposed quadruped system.

The research addresses the challenge that generating gaits for complex, high-dimensional quadruped robots based on full-order dynamics is computationally intensive, often relying on simplified models. In contrast, bipedal gait generation using full dynamics and frameworks like Hybrid Zero Dynamics (HZD) has achieved formal guarantees and experimental success. This work bridges that gap.

Dynamics Decomposition:

The paper proposes decomposing a 3D quadruped robot, such as the 18-DOF, 12-input Vision 60 robot, into two identical 12-DOF bipedal systems (front and rear). This decomposition is exact and considers the full nonlinear, hybrid dynamics, including continuous dynamics governed by constrained differential algebraic equations (DAEs) and discrete dynamics modeling perfectly inelastic impacts.

The continuous dynamics of the full quadruped are represented by mass-inertia matrices, Coriolis and gravity terms, actuation matrices, and contact Jacobians, subject to holonomic toe constraints. The decomposition rewrites these dynamics as two separate sets of equations for the "front" and "rear" bipeds, coupled by an internal connection wrench (λc\lambda_c) and a kinematic constraint ensuring their body linkages move together.

The impact dynamics, modeling foot touchdown events, are similarly decomposed. The impulse effects on the full system are equivalent to the impulse effects on the decomposed bipeds, along with the effect of the connection impulse (Λc\Lambda_c).

Control Decomposition and Trajectory Generation:

The approach utilizes virtual constraints, which are time-dependent output functions defined on the robot's configuration space. By driving these outputs and their derivatives to zero, specific desired robot behaviors can be enforced. For the decomposed biped systems, the actuated joint angles ($y^a_\rmf, y^a_\rmr$) are chosen as outputs, which are constrained to follow desired trajectories ($\mathcal{B}_\rmf(t), \mathcal{B}_\rmr(t)$) parameterized by 5th-order Bezier polynomials.

A crucial step is defining the desired correlation between the front and rear biped trajectories. For diagonally symmetric gaits, the actuated joints of specific legs are related by a mirror mapping (MM) and offset (bb), i.e., $\mathcal{B}_\rmr(t) = M \mathcal{B}_\rmf(t) + b$. This allows generating a trajectory for one biped (e.g., the front) and inferring the trajectory for the other. The closed-loop dynamics of a single biped, subject to these virtual constraints and the coupling constraint with the other biped, can then be formulated independently of the other biped's state trajectory, depending only on its desired trajectory.

Decomposition-based Optimization:

Gait generation is formulated as a Nonlinear Program (NLP). The optimization problem seeks to find the coefficients of the Bezier polynomials (α\alpha) for the desired bipedal trajectory ($\mathcal{B}_\rmf(t)$) that satisfy the hybrid dynamics of the decomposed biped and additional physical constraints.

The objective function minimizes the body's vibration rate to promote stable, static torso movement. Constraints include:

  • Enforcing the decomposed continuous dynamics (CL-Dyn-f).
  • Satisfying collocation constraints (e.g., Hermite-Simpson) to accurately approximate the continuous dynamics between discretization nodes.
  • Satisfying the decomposed impact dynamics.
  • Imposing periodic continuity constraints to ensure that the post-impact state matches the initial state of the next step, leading to periodic gaits.
  • Including physical feasibility constraints such as torque limits, joint position/velocity limits, foot clearance requirements, and friction cone conditions.

The optimization problem is solved efficiently using the FROST toolbox, which leverages a direct collocation method with an implicit formulation of dynamics to avoid costly matrix inversions (O(n3)\mathcal{O}(n^3)). This decomposition-based approach drastically reduces the dimension of the system being optimized compared to optimizing the full quadruped dynamics directly.

Simulation and Experiments:

The optimized gaits are validated in the MuJoCo physics simulator and experimentally on the Vision 60 quadruped robot. A simple, time-based PD controller is used to track the optimized joint trajectories (u(qa,q˙a,t)=k1(y˙aB˙(t))k2(yaB(t))u(q_a, \dot q_a, t) = -k_1(\dot y_a - \mathcal{\dot{B}(t)) -k_2(y_a - \mathcal{B}(t))}). Event-based switching between gait phases is triggered by the step duration (TT) from the optimization.

The results demonstrate that the gaits generated using the decomposition method successfully produce dynamic quadrupedal locomotion, including stepping-in-place and ambling. A comparison between simulated and experimental joint trajectories and ground reaction forces shows good agreement. The computational performance benchmark shows that gaits can be generated in seconds (average 3.96s), which is an order of magnitude faster than previous methods optimizing the full quadruped model directly. Experiments conducted on an outdoor tennis court show that the robot can perform these gaits robustly without additional heuristics, suggesting that rigorous gait generation using full dynamics can translate well to hardware even with a simple tracking controller.

In conclusion, the paper provides a practical and computationally efficient method for generating dynamic gaits for quadrupedal robots by decomposing the complex full-body dynamics into simpler bipedal systems. This allows applying powerful bipedal gait generation techniques and achieving rapid iteration and realization of various quadrupedal behaviors on hardware using simple controllers.