Stabilizing Bipedal Locomotion Using a Hybrid Lyapunov-Based Feedback Controller with Reference Spreading

Alapfogalmak

A novel hybrid formulation of the linear inverted pendulum model is proposed to stabilize bipedal locomotion, using a Lyapunov-based feedback controller and a reference spreading mechanism to handle the non-synchronized contact times between the reference and the actual robot motion.

Kivonat

The paper presents a novel hybrid formulation of the linear inverted pendulum model (LIPM) for bipedal locomotion, where the foot switches are triggered based on the center of mass position, removing the need for pre-defined footstep timings.

The key highlights are:

1. A hybrid modification of the reference dynamics, similar to the "reference spreading" concept, is adopted to handle the non-synchronized contact times between the reference and the actual robot motion.

2. A saturated linear feedback controller is designed by solving a convex optimization problem to stabilize the hybrid error dynamics. The gains are selected to ensure local asymptotic stability of the tracking error, with a certified estimate of the basin of attraction.

3. The stability proof is based on an optimized selection of a quadratic Lyapunov function, where the decrease along flows is ensured by the linear feedback, and the increase across jumps is bounded using an extended class K∞ function of the position error.

4. The proposed framework is shown to have advantages over a standard model predictive control formulation, as it can better describe the natural interplay between stepping frequency and walking speed.

5. The theoretical results are validated through simulations on a full-body model of a real humanoid robot, demonstrating the practical applicability of the proposed approach.

Statisztikák

The paper provides the following key figures and metrics:

The linear inverted pendulum model (LIPM) dynamics are described by the equations:
¨xc = ω^2(xc - u)
where xc is the center of mass position, u is the center of pressure, and ω = sqrt(g/zc) with g the gravity acceleration and zc the constant CoM height.

The half-step size is denoted as r̄, corresponding to half the longitudinal length of a step.

The periodic reference motion has a period T that depends on the half-step size r̄ and the peak forward speed v̄ according to the relations:
v̄ = (cosh(ωT) + 1) / (ω sinh(ωT)) * r̄
T = 1/ω * ln((v̄/ω + r̄) / (v̄/ω - r̄))

The control gains K and L are obtained by solving the convex optimization problem (24), where the parameter α > ω is selected to ensure feasibility.

Idézetek

"Contrary to the classical approach, where the contact timings are fixed a priori, we assume that contacts occur when the CoM reaches a given position, using hybrid jumps to shift the reference frame at each step, so that it corresponds to the center of the foot on the ground."
"Our solution presents two main advantages with respect to the classical fixed-contact approach. First, it well describes the natural interplay between the stepping frequency and the walking speed. As the robot walks faster, its stepping frequency automatically increases. As empirically shown in our results in Section 6, this can allow the robot to keep walking in cases where a classic approach would lead to a fall."
"Secondly, in view of a well-posed formulation, it allows proving rigorous robust stability guarantees (see [16, Chapter 7]). In particular, our stability proof is based on an optimized selection of a quadratic Lyapunov function for which we can ensure asymptotic stability of the error dynamics despite a mismatch in the impact times of the reference and actual motion."

Főbb Kivonatok

Hybrid Lyapunov-based feedback stabilization of bipedal locomotion based on reference spreading

by Riccardo Ber... : arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.02184.pdf

Mélyebb kérdések

How could the proposed hybrid formulation be extended to also stabilize the lateral and vertical motions of the bipedal robot, beyond just the longitudinal direction

To extend the proposed hybrid formulation to stabilize the lateral and vertical motions of the bipedal robot, we can incorporate additional dynamics and control strategies.
For the lateral motion, we can introduce a similar hybrid model for the lateral inverted pendulum dynamics, where the foot switches are triggered based on the lateral position of the center of mass. By defining nontrivial tracking error coordinates for the lateral direction and stabilizing them using a saturated feedback controller, we can ensure stability in the lateral motion. The control gains can be selected through convex optimization, similar to the longitudinal direction, to achieve local asymptotic stability.
For the vertical motion, we can consider the height of the center of mass as a dynamic variable and incorporate it into the hybrid model. By defining appropriate error coordinates for the vertical direction and designing a feedback controller to stabilize them, we can extend the stability framework to include vertical motion. This would involve considering constraints on the vertical position, velocity, and acceleration to ensure robust stability in the vertical direction.
By integrating these additional dynamics and control strategies into the hybrid formulation, we can create a comprehensive framework that stabilizes the bipedal robot's motion in all three dimensions, providing a more complete and robust control solution for bipedal locomotion.

What are the potential limitations or drawbacks of the reference spreading mechanism used in this approach, and how could it be further improved or generalized

The reference spreading mechanism used in the proposed approach has certain limitations and drawbacks that could be addressed for further improvement and generalization.
One limitation is that the reference spreading technique may lead to drift in the reference trajectory if the robot is not synchronized with the reference. This drift can affect the overall stability and tracking performance of the system. To mitigate this issue, advanced synchronization methods or adaptive control strategies could be implemented to ensure better alignment between the robot's motion and the reference trajectory.
Another drawback is the reliance on pre-defined footstep positions and durations in the lateral and vertical directions. This can restrict the adaptability of the system to changing terrain or walking conditions. To enhance the flexibility and robustness of the approach, real-time adjustment of footstep positions and durations based on feedback from the environment could be integrated into the control framework.
To further improve the reference spreading mechanism, incorporating predictive modeling or learning algorithms to anticipate variations in the reference trajectory and adapt the control strategy accordingly could enhance the system's performance and adaptability in dynamic environments.

Given the focus on local stability guarantees, what strategies could be explored to achieve more global stability and robustness for the bipedal locomotion problem

To achieve more global stability and robustness for the bipedal locomotion problem beyond the local stability guarantees provided by the proposed approach, several strategies can be explored:

Nonlinear Control Techniques: Implementing advanced nonlinear control methods such as nonlinear model predictive control or adaptive control can enhance the system's robustness to uncertainties and disturbances, providing a more globally stable control solution.

Optimal Control Strategies: Utilizing optimal control strategies like reinforcement learning or trajectory optimization can help in finding globally optimal control policies that maximize stability and performance metrics over a wider range of operating conditions.

Hybrid Control Architectures: Integrating multiple control strategies, such as combining the proposed hybrid Lyapunov-based feedback stabilization with reinforcement learning for adaptive control, can offer a more comprehensive and robust control architecture that ensures stability across various scenarios.

Robustness Analysis: Conducting thorough robustness analysis, including worst-case scenario analysis and sensitivity analysis, can help identify potential vulnerabilities and improve the system's resilience to uncertainties and disturbances.

Experimental Validation: Performing extensive experimental validation on physical bipedal robots in diverse environments can provide valuable insights into the real-world performance and robustness of the control system, enabling iterative improvements for enhanced global stability.