Versatile, Robust, and Explosive Locomotion with Rigid and Articulated Compliant Quadrupeds

TL;DR

The paper introduces TD{-}aSLIP, a general template model that captures parallel elasticity and trunk rotation to enable versatile, robust, and explosive quadrupedal locomotion. It combines a dual-layer trajectory optimization scheme with a singularity-free quaternion representation and a hierarchical whole-body control pipeline that accounts for parallel compliance, delivering dynamic motions such as pronking, froggy jumping, and hop-turn without hand-tuned references. Simulation and hardware experiments on rigid and articulated compliant quadrupeds demonstrate that parallel elasticity enhances explosive performance, reduces peak torque and power, and improves robustness against disturbances and modeling errors, with hardware validation on an E-Go-V2 platform showing up to 50 cm froggy jumping and improved landing accuracy. The framework generalizes to multiple locomotion modalities and provides a pathway toward energy-efficient, momentum-aware, and agile dynamic locomotion in legged robots.

Abstract

Achieving versatile and explosive motion with robustness against dynamic uncertainties is a challenging task. Introducing parallel compliance in quadrupedal design is deemed to enhance locomotion performance, which, however, makes the control task even harder. This work aims to address this challenge by proposing a general template model and establishing an efficient motion planning and control pipeline. To start, we propose a reduced-order template model-the dual-legged actuated spring-loaded inverted pendulum with trunk rotation-which explicitly models parallel compliance by decoupling spring effects from active motor actuation. With this template model, versatile acrobatic motions, such as pronking, froggy jumping, and hop-turn, are generated by a dual-layer trajectory optimization, where the singularity-free body rotation representation is taken into consideration. Integrated with a linear singularity-free tracking controller, enhanced quadrupedal locomotion is achieved. Comparisons with the existing template model reveal the improved accuracy and generalization of our model. Hardware experiments with a rigid quadruped and a newly designed compliant quadruped demonstrate that i) the template model enables generating versatile dynamic motion; ii) parallel elasticity enhances explosive motion. For example, the maximal pronking distance, hop-turn yaw angle, and froggy jumping distance increase at least by 25%, 15% and 25%, respectively; iii) parallel elasticity improves the robustness against dynamic uncertainties, including modelling errors and external disturbances. For example, the allowable support surface height variation increases by 100% for robust froggy jumping.

Figures (25)

• Figure 1: The compliant Go1 with parallel springs (we call it E-Go-V2) performs versatile and explosive motions. From top to bottom, the robot performs 1)$40\,$cm forward pronking, 2)$40\,$cm forward froggy jumping, 3)$145^{\circ}$ clockwise hop-turn, and 4) robust froggy jumping from the uneven surface. At the bottom, the robot jumps from the non-coplanar surface ($10\,$cm height variation) without knowing the terrain information in advance. The red lines in the first two rows mark the desired landing positions for the rear legs.
• Figure 2: Optimization-base motion planning and control for versatile, robust, and explosive quadrupedal locomotion. The singularity-free formulation is incorporated into trajectory optimization and compliant control.
• Figure 3: The TD-aSLIP model with parallel elasticity, (a) temple model, and (b) quadrupedal motion mapping in the homing pose.
• Figure 4: The equivalent spring stiffness regarding leg length. The raw data (marked by the blue colour) is sampled from a pre-designed workspace. The green solid curve shows the fitness result, while the purple dash curve shows varying stiffness regarding the vertical height. The red star in the purple curve marks the equivalent stiffness when the robot stands in the homing pose.
• Figure 5: Contact sequence for (a) pronking and (b) froggy jumping .