Robust Locomotion in Legged Robotics

• Robust locomotion frameworks are algorithmic systems that combine planning, control, and learning to ensure stable and adaptive legged movement.
• They employ data-driven and predictive control methods, such as Hankel matrix-based optimization, to manage hybrid dynamics and contact events.
• These frameworks enable real-time replaning and effective disturbance recovery, as demonstrated on exoskeleton platforms in variable environments.

Robust locomotion frameworks comprise algorithmic and architectural solutions enabling legged robots and exoskeletons to execute stable, adaptable, and disturbance-resilient walking in real-world, dynamic environments. These frameworks integrate planning, control, and learning modules that systematically handle hybrid system dynamics, contact scheduling, trajectory optimization, and feedback adaptation. At their core, robust locomotion frameworks are distinguished by their ability to synthesize reliable, often real-time, strategies for contact events and whole-body motion in the presence of modeling uncertainties and external perturbations.

1. Hybrid System Foundations and Problem Setting

Robust locomotion for legged systems fundamentally involves hybrid, nonlinear, time-varying systems comprising discrete events (e.g., foot contact switches) coupled to high-dimensional continuous dynamics (body, joints). In exoskeletons and robots, this necessitates frameworks that simultaneously plan and control both the discrete sequence of contact transitions and the continuous evolution of body states.

The intrinsic hybridness arises from:

• Discrete contact events: Foot-strike and lift-off events defining walking subphases.
• Continuous phases: Dynamics of body and joints during stance and swing.

The challenge is increased by sensor/actuator noise, varying environments (e.g., terrain changes), model uncertainties, and unexpected disturbances. Performance metrics for robustness encompass: disturbance rejection (e.g., recovery from pushes), adaptability to unplanned events, and real-time closed-loop implementation on physical platforms.

2. Data-Driven and Predictive Control Methodologies

Innovations in robust locomotion frameworks are typified by the integration of data-driven representations and predictive control paradigms, enabling model-agnostic adaptation and online replanning.

Hankel Matrix-Based Data-Driven Predictive Control

In the Hybrid Data-Driven Predictive Control (HDDPC) paradigm (Li et al., 14 Aug 2025), system dynamics are encoded directly from exoskeleton sensor and actuator data with no explicit parametric identification, using block Hankel matrices. Given input-output trajectories, the block Hankel matrix $\mathcal{H}_L(w)$ is constructed: $$\mathcal{H}_L(w) = \begin{bmatrix} w_1 & w_2 & \ldots & w_{m-L+1} \ w_2 & w_3 & \ldots & w_{m-L+2} \ \vdots & \vdots & \ddots & \vdots \ w_L & w_{L+1} & \ldots & w_{m} \end{bmatrix}$$ where $w$ is input, output, or state, and $L$ is window length.

The evolution of future system behavior is realized as a linear combination of past data sequences, embedded as an equality constraint for an optimization variable $g$: $$\begin{bmatrix} U_p \ Y_p \ U_f \ Y_f \end{bmatrix} g = \begin{bmatrix} u_{\text{ini}} \ y_{\text{ini}} \ u_{\text{pred}} \ y_{\text{pred}} \end{bmatrix}$$ This setup directly incorporates measured exoskeleton responses across multiple continuous and discrete (contact) domains—a “hybrid time” representation.

Simultaneous Contact Scheduling and Trajectory Optimization

The HDDPC framework advances beyond conventional decoupled approaches by jointly optimizing:

• Discrete foot contact schedules (step-to-step timing and placements).
• Continuous domain trajectories (joint/body evolution within each support phase).

The resulting optimization at every control loop iteration: $$\min_{g, u_{\text{pred}}, y_{\text{pred}}, \text{sched}} J(y_{\text{pred}}, u_{\text{pred}}, \text{sched})$$ constrained by the Hankel embedding, input bounds, safety/stability, and switching/contact scheduling constraints (encoded via $\text{sched}$). This unified approach allows for dynamic replanning under nonstationary or perturbed conditions, leveraging real system data to counteract modeling inaccuracies.

3. Online Replanning, Adaptability, and Robustness Mechanisms

Modern robust locomotion frameworks must revisit and update planned motions in response to unforeseen disturbances or environmental transitions.

• HDDPC achieves this through real-time closed-loop optimization: both foot placement and continuous trajectory are decision variables, enabling the framework to replan after push disturbances, terrain changes, or sensor anomalies.
• Step-to-step (S2S) transitions, handled natively, enhance adaptation and recovery—crucial for robust walking on real hardware.

Significantly, the data-driven core facilitates continuous adaptation—as the exoskeleton gathers more operational data, the Hankel representation and thus the controller’s dynamics model evolve, further hardening the framework against time-varying robot dynamics or actuator drifts.

4. Experimental Validation and Performance Outcomes

The framework’s efficacy is quantitatively validated on the Atalante exoskeleton:

• Disturbance rejection: Simulated and real-world walking demonstrates that unexpected external pushes are countered through immediate sequence and trajectory replanning, with rapid return to stable gait.
• Online adaptability: Both step timing/placement and continuous trajectories are modified on-the-fly, outperforming rigid, precomputed gait controllers in agility and stability.
• Real-time execution: The current implementation delivers online feasible rates; ongoing work aims to further accelerate the trajectory planner and support Hankel matrix online updating.

Key reported improvements include:

• Enhanced rate of disturbance recovery and environmental response.
• Increased range of step viabilities and maneuverability in variable walking conditions.

5. Architecture and Algorithmic Summary

The architecture of advanced robust locomotion frameworks, as embodied by HDDPC, can be summarized as:

Module	Role
Data-driven dynamics (Hankel)	Real-time system modeling, capturing hybrid walking behavior
Contact scheduler (optimization)	Online adaptation of foot contact sequence and timing
Trajectory planner (optimization)	Joint evolution of reference kinematics and dynamics over horizons
Control constraints	Realistic bounds, stability, safety, and switching constraints enforced
Closed-loop & online updating	Iterative, real-time policy adjustment; data-driven model evolution

6. Context and Position within Broader Locomotion Research

HDDPC’s approach situates itself within a lineage of robust legged locomotion literature, sharing attributes with frameworks that:

• Unify hybrid (discrete-continuous) planning and control (Kasaei et al., 2019, Castillo et al., 2023, Gu et al., 2021).
• Employ data-driven or learning-based system representations (Castillo et al., 2023, Maslennikov et al., 14 Jul 2025, Kim et al., 24 Jul 2025).
• Emphasize real-time online replanning and closed-loop robustness over static precomputed policies.

The key distinction with HDDPC is its direct avoidance of explicit model identification, robust handling of both discrete and continuous locomotion aspects in a single optimization, and proven hardware applicability.

7. Future Directions and Implementation Considerations

Ongoing developments highlighted in (Li et al., 14 Aug 2025) include:

• Increasing real-time capability: Planned acceleration of the trajectory planner is vital for deployment in more dynamic or resource-constrained platforms.
• Continual data integration: Online updating of the Hankel matrix with operational data promises continual improvement in adaptation and robustness as the robot operates in diverse scenarios.

Potential limitations involve computation complexity scaling with data matrix size, and the need for persistently excited data for accurate system identification via Hankel matrices. However, as more exoskeleton sensor and actuator data are integrated, the model’s fidelity and the framework’s adaptivity are expected to improve further.

In summary, advanced robust locomotion frameworks such as HDDPC represent a significant evolution in hybrid, data-driven, and adaptive locomotion synthesis for bipedal exoskeletons, achieving both theoretical unification and hardware-validated robustness in real time.