Stable and Robust SLIP Model Control via Energy Conservation-Based Feedback Cancellation for Quadrupedal Applications

TL;DR

This work addresses stable, dynamic quadrupedal locomotion by leveraging a reduced-order SLIP model and an energy-conservation principle to shape a parabolic reference trajectory for stance. A two-module controller combines stance-leg-extension regulation with a flight-leg-angle servo, and uses a feedback-cancellation tracking law to ensure convergent error dynamics in the presence of sensor noise. The approach is validated through MATLAB simulations parameterized for the Ghost Minitaur, demonstrating reliable bouncing gaits and robustness to up to 10% state estimation errors. The results indicate a lightweight, computationally efficient framework with practical resilience to noisy measurements, with future goals including 3D locomotion extension and physical robot validation.

Abstract

In this paper, we present an energy-conservation based control architecture for stable dynamic motion in quadruped robots. We model the robot as a Spring-loaded Inverted Pendulum (SLIP), a model well-suited to represent the bouncing motion characteristic of running gaits observed in various biological quadrupeds and bio-inspired robotic systems. The model permits leg-orientation control during flight and leg-length control during stance, a design choice inspired by natural quadruped behaviors and prevalent in robotic quadruped systems. Our control algorithm uses the reduced-order SLIP dynamics of the quadruped to track a stable parabolic spline during stance, which is calculated using the principle of energy conservation. Through simulations based on the design specifications of an actual quadruped robot, Ghost Robotics Minitaur, we demonstrate that our control algorithm generates stable bouncing gaits. Additionally, we illustrate the robustness of our controller by showcasing its ability to maintain stable bouncing even when faced with up to a 10% error in sensor measurements.

Figures (8)

• Figure 1: (a) A quadruped robot with center of mass (COM) depicted. (b) A reduced-order spring-loaded inverted pendulum (SLIP) model of the quadruped robot. The SLIP model provides a computationally tractable yet reliable representation of the quadruped dynamics during pronking motion. We use the SLIP representation of the quadruped in our controller to enable real-time stable and robust running.
• Figure 2: Depiction of a SLIP model of a quadruped robot with a COM of mass $m$ exhibiting dynamic running. The SLIP model is a hybrid dynamical system, switching between two dynamical modes: flight and stance. The flight phase is defined by projectile dynamics, while the stance phase is characterized by spring-loaded inverted pendulum dynamics. The touchdown and liftoff events define the guard surfaces in the state space that control the switching of these dynamical modes.
• Figure 3: The relationship between apex height and compression length used to calculate the parabolic stance trajectory for energy-shaping control. Here $\Delta h_{\text{apex}} = \Delta y_{\text{fl}}$ and $\Delta y_{\text{st}} = \Delta y_{\text{min}}$
• Figure 4: Depiction of a SLIP model during stance phase. Here $r$ measures the distance from the foot to the center of mass, and $\theta$ measures the angular displacement w.r.t. the ground.
• Figure 5: Overview of our control architecture. Here, $v^0_{\text{des}}$ denotes the desired initial forward velocity and $y_{\text{des}}$ denotes the desired vertical height.