Stable Walking for Bipedal Locomotion under Foot-Slip via Virtual Nonholonomic Constraints

1. Problem Statement

Bipedal robots face significant instability when walking on low-friction or uncertain terrain due to foot slip.

• Limitation of Current Methods: Standard control frameworks for bipedal locomotion (e.g., Hybrid Zero Dynamics) assume ideal no-slip contact (the stance foot is fixed relative to the ground). When tangential forces exceed friction limits, these models degrade because they treat slip merely as a disturbance to be rejected rather than a dynamic phenomenon to be modeled.
• The Challenge: When slip occurs, the stance foot moves tangentially. Enforcing a zero-velocity constraint is physically impossible and leads to control failure. The system requires a control strategy that explicitly allows and regulates tangential motion while maintaining the stability of the walking gait.

2. Methodology

The authors propose a control framework that integrates slip directly into the locomotion model using Virtual Nonholonomic Constraints (VNHCs).

A. Hybrid Locomotion Model

The system is modeled as a hybrid dynamical system with:

• Continuous Dynamics: Governed by Euler-Lagrange equations during the swing phase. Unlike standard models, the stance foot is not fixed; it is allowed to slide tangentially while maintaining normal contact ($v_n = 0, v_t \neq 0$).
• Discrete Dynamics: Modeled by impact events when the swing foot touches the ground, governed by an impulse-momentum reset map.

B. Virtual Constraints

The control design combines two types of constraints:

1. Virtual Holonomic Constraints (VHCs): These coordinate the robot's joint configurations to generate a nominal walking gait (e.g., using Bézier polynomials). They are enforced via outputs $y(q)$ with relative degree two.
2. Virtual Affine Nonholonomic Constraints (VNHCs): These are introduced to regulate the tangential velocity of the stance foot. Instead of forcing $v_t = 0$, the controller enforces a desired slip profile:
   $$v_t(q, \dot{q}) = b(q)$$
   where $b(q)$ is a smooth map specifying the desired slip behavior. This is an affine constraint on the velocity space ($A(q)\dot{q} = b(q)$).

C. Control Design

A nonlinear feedback law is derived to simultaneously enforce both the holonomic gait constraints and the nonholonomic slip constraints.

• The system is treated as a mixed relative-degree system.
• The control input $u$ is calculated to drive the slip output error ($\eta_s$) and the gait output error ($y$) to zero exponentially.
• This results in a Slip-Compatible Hybrid Zero Dynamics (HZD) manifold ($Z_{int}$), which is invariant under both continuous dynamics and impact events.

D. Stability Analysis

• Orbital Stability: The stability of the periodic walking gait is analyzed using the Poincaré return map restricted to the invariant manifold $Z_{int}$.
• Theorem: If the reduced Poincaré map has a fixed point and its linearization (Jacobian) has eigenvalues strictly inside the unit circle, the full hybrid system exhibits local exponential orbital stability.

3. Key Contributions

1. Explicit Slip Modeling: The paper moves beyond treating slip as a disturbance. It explicitly incorporates tangential slip into the hybrid dynamics model.
2. Virtual Nonholonomic Constraints: It introduces a class of velocity-based virtual constraints specifically designed to regulate stance-foot slip while remaining compatible with the geometric structure of standard gait generation.
3. Unified Control Law: A single nonlinear feedback controller is derived that simultaneously stabilizes the walking gait (via holonomic constraints) and regulates slip (via nonholonomic constraints).
4. Theoretical Guarantees: The authors provide a rigorous proof of the existence of a slip-compatible hybrid zero dynamics manifold and characterize the orbital stability of the resulting periodic gaits.

4. Numerical Results

The authors validated the approach using a 7-DOF planar biped model with variable slip conditions.

• Simulation Setup: The robot walked over 50 steps with piecewise-constant slip levels ($s_k \in \{0, 0.015, 0.03, \dots\}$) to simulate changing ground friction.
• Open-Loop vs. Controlled:
  • Open-Loop (No Slip Regulation): The robot failed after 29 steps. The forward velocity degraded progressively, and the gait became unstable.
  • Controlled (Proposed Method): The robot successfully completed 50 steps. The controller successfully reorganized the gait after each change in slip level.
• Performance Metrics:
  • Slip Output Error: Under control, the slip output magnitude ($|\eta_s|$) was reduced from a mean of 0.65 (open-loop) to 0.0041, demonstrating precise regulation.
  • Velocity: The controlled robot maintained a higher and more stable pre-impact forward velocity ($0.5964$ vs. $0.2565$).
  • Trajectory: The hip trajectory remained coherent and geometrically consistent throughout the simulation, whereas the open-loop trajectory diverged.

5. Significance

This work represents a significant advancement in legged robotics by bridging the gap between idealized no-slip models and the reality of low-friction terrain.

• Robustness: It enables bipedal robots to walk stably on surfaces where friction is uncertain or insufficient, a critical requirement for real-world deployment (e.g., icy roads, wet floors).
• Theoretical Framework: By defining a "slip-compatible" hybrid zero dynamics manifold, it extends the powerful HZD framework to non-ideal contact conditions without abandoning the geometric control structure.
• Future Impact: The methodology provides a foundation for future work on 3D locomotion, experimental validation on physical robots, and adaptive slip laws that respond dynamically to terrain changes.