Centroidal Steering Strategies

• Centroidal steering strategies are control methods that govern a robot's center-of-mass and angular momentum for precise turning and trajectory tracking.
• They combine model-based optimization, reinforcement learning, and geometric mechanics to plan and execute agile maneuvers in bipedal, quadrupedal, and multi-legged robots.
• Robust constraint handling and efficient real-time integration ensure stability and adaptability during complex maneuvers and dynamic environments.

Centroidal steering strategies constitute a suite of control and planning approaches that govern the heading and directional changes of legged and elongate robots at the level of their centroidal (center-of-mass and angular momentum) dynamics. These methodologies enable robots to execute precise and robust maneuvers, including turning, lane changes, and complex trajectory tracking, by manipulating centroidal states and harnessing advances in model-based optimization, data-driven policies, and geometric mechanics. The design and deployment of centroidal steering schemes are central to agile locomotion across bipedal, quadrupedal, humanoid, and multi-legged robotic platforms.

1. Centroidal Dynamics: Mathematical Foundation and Models

Centroidal dynamics describe the evolution of a robot’s total linear momentum $\vec{l}\in\mathbb{R}^3$ and angular momentum $\vec{k}\in\mathbb{R}^3$ about its center of mass (CoM), shaped by contact wrenches and external forces. In classical models, the linear and angular momentum rates are governed by the Newton–Euler equations: $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$

$$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$

where $m$ is the robot mass, $\vec{p}_\text{com}$ is the CoM position, $\vec{f}_{r,i}$ and $\vec{\tau}_{r,i}$ are contact force and torque at foot $i$, and $N_c$ is the number of contacts. In reduced-order models (e.g., the Prismatic Inverted Pendulum Model—PIPM), angular torques during single support can be neglected ($\vec{k}\in\mathbb{R}^3$0), and the system reduces to a second-order system per axis, parameterized by ground contact.

Centroidal models are ubiquitous across bipedal keyframe planning (Zhao et al., 2017), humanoid multi-contact trajectories (Murooka et al., 29 May 2025), quadrupedal RL-based frameworks (Xie et al., 2021), and data-driven planners (Viereck et al., 2020). In geometric-mechanics–inspired control for elongate robots, centroidal displacement arises from coordinated shape changes in low-dimensional body-wave spaces (Flores et al., 2024).

2. Steering Direction Encoding Across Robotic Morphologies

Steering is fundamentally achieved by regulating the robot’s orientation and the lateral bias of the CoM trajectories, as expressed relative to a local or global frame. In centroidal-momentum–based planners, a discrete heading variable $\vec{k}\in\mathbb{R}^3$1 is prescribed for each step $\vec{k}\in\mathbb{R}^3$2, defining a local sagittal-lateral coordinate transformation: $\vec{k}\in\mathbb{R}^3$3

$\vec{k}\in\mathbb{R}^3$4

where $\vec{k}\in\mathbb{R}^3$5 is the keyframe origin and $\vec{k}\in\mathbb{R}^3$6 are step-local sagittal/lateral coordinates. The step map

$\vec{k}\in\mathbb{R}^3$7

propagates foot placements and heading.

In whole-body motion networks, steering emerges as a learned allocation of yaw torques $\vec{k}\in\mathbb{R}^3$8 and lateral forces, either encoded implicitly via contact sequence or explicitly via command inputs such as desired yaw-rate or lateral velocity (Viereck et al., 2020). In RL frameworks, the commanded centroidal accelerations include both lateral translation $\vec{k}\in\mathbb{R}^3$9 and yaw acceleration $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$0, inducing turning via direct centroidal control (Xie et al., 2021). For elongate limbed systems, heading is realized through modulation of shape parameters (body undulation coefficients) that control net centroidal rotation per gait cycle (Flores et al., 2024).

3. Methodologies: Planning, Control Laws, and Constraints

Steering strategies typically integrate high-level planning, real-time feedback, and constraint enforcement.

Phase-Space Planning and Recovery

Hybrid automaton frameworks decompose planning into:

• Open-loop phase-space manifold generation: For each step, nominal sagittal and lateral phase-space trajectories are planned, mapped via $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$1 into global space.
• Lateral foot placement search: A Newton–Raphson iteration ensures the apex lateral velocity vanishes ($$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$2), situating the CoM for heading stability.
• Hybrid robust control: Deviation $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$3 from the nominal phase-space manifold $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$4 is monitored. When $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$5, a dynamic programming–based controller adjusts flywheel torques, or keyframes are re-planned (Zhao et al., 2017).

Preview and Model Predictive Control

In high-DoF humanoid steering, preview control replaces computationally heavy MPC. The centroidal state (linear and angular) is controlled via optimal jerk over a preview window $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$6, minimizing

$$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$7

with $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$8. Centroidal PD feedback closes the loop for disturbance rejection (Murooka et al., 29 May 2025).

Data-Driven and RL Approaches

Neural planners predict the next-step centroidal state and wrench based on the current state and future contact plan. Steering is induced by including desired footstep headings or explicit yaw/lateral commands in the plan window (Viereck et al., 2020). In centroidal-RL, the network directly outputs desired accelerations, which are realized by QP-based wrench/force distribution, subject to friction, unilateral, and reachability constraints (Xie et al., 2021).

Geometric Mechanics and Wave-Modulated Steering

For multi-legged elongate robots, steering is realized by modulating the amplitude and phase of a third "turning wave" in the body shape space: $$\dot{\vec{l}} = m\,\ddot{\vec{p}}_\text{com} = \sum_{i=1}^{N_c} \vec{f}_{r,i} - m\vec{g}$$9 where $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$0 are spatial basis functions and $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$1 in $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$2 parameterizes steering curvature. The net centroidal displacement and rotation are calculated from integrals of the local connection $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$3 over a shape-cycle (Flores et al., 2024).

4. Constraint Handling and Robustness Guarantees

Physical feasibility and robustness are maintained via explicit constraints and feedback mechanisms.

• Friction-cone and unilateral constraints are enforced in all platforms, typically via quadratic programs for force/wrench distribution:

$$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$4

where $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$5 encodes friction pyramid rays and $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$6 is the planned or feedback-corrected wrench (Murooka et al., 29 May 2025).

• Phase-space invariant and recoverability bundles are defined for bipedal steering: the tangent manifold $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$7 describes deviation from nominal phase, with the invariant bundle $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$8 and recoverability bundle $$\dot{\vec{k}} = \vec{\tau}_\text{com} = \sum_{i=1}^{N_c} (\vec{p}_\text{foot, i} - \vec{p}_\text{com}) \times \vec{f}_{r,i} + \sum_{i=1}^{N_c} \vec{\tau}_{r,i}$$9 constraining admissible states for robust heading maintenance (Zhao et al., 2017).
• Sensitivity metrics such as the phase-space norm $m$0 measure disturbance amplification during steering.
• Redundant contact and multi-contact support: Wrench projection and distribution ensure that feasible forces can be realized across arbitrary contact patches during complex maneuvers (Murooka et al., 29 May 2025).

In RL and learning-based controllers, disturbance rejection is addressed in the policy reward via penalty terms for orientation error and tracking, while feedback in the QP-based realization enforces instantaneous physical consistency (Xie et al., 2021).

5. Implementation and Real-Time Integration

Efficient centroidal steering strategies are designed for real-time operation.

• Feedforward and feedback structure: Preview-based planners precompute control gains for both centroidal CoM and orientation, requiring only matrix multiplications at run time (200 Hz control rate, <0.2 ms/cycle reported in (Murooka et al., 29 May 2025)).
• Hierarchical control: Data-driven and RL-based frameworks integrate the centroidal planner with inverse kinematics (IK) and whole-body torque control. The neural planner operates at 100 Hz, while low-level joint-torque loops close at >400 Hz (Viereck et al., 2020).
• Contact adaptation: In RL systems, Raibert-style heuristics adapt foot placements ($m$1), immediately biasing steering in response to new commands. Unfeasible contacts are projected to the nearest valid locations, as in stepping-stone traversal (Xie et al., 2021).
• Geometric phase-based control: For centipede-like robots, wave-parameter primitives for steering are updated every cycle ($m$2 amplitude/phase) by a table-lookup or proportional controller, validated in both indoor and field experiments (Flores et al., 2024).

6. Comparative Performance and Empirical Results

Key metrics and findings across platforms:

• Feasibility and stability: Robust tracking of non-periodic keyframes over diverse terrain, accurate heading tracking, and fast recovery from disturbances (Zhao et al., 2017).
• Computational efficiency: Preview-control planners achieve 0.2 ms per cycle for full centroidal and contact enforcement (Murooka et al., 29 May 2025); neural planners achieve 22–41× speedup over kino-dynamic trajectory optimization (Viereck et al., 2020).
• Tracking error: Neural planners on Solo12 quadruped achieve mean/max CoM position errors ≈0.01 m/0.03 m, base orientation errors ≈1.0°/4° (Viereck et al., 2020). RL-based centroidal controllers demonstrate heading errors <0.1 rad during turns on physical Unitree A1 (Xie et al., 2021).
• Adaptability: Learned planners and RL policies generalize to unseen turning gaits, multi-contact tasks, and variable terrain without explicit model changes (Viereck et al., 2020, Xie et al., 2021). For elongate robots, arc-following gaits achieve per-cycle headings up to $m$3 with $m$4 tangent deviation (Flores et al., 2024).
• Sim-to-real transfer: End-to-end centroidal steering policies in GLiDE demonstrate robust flat-ground, obstacle, and beam-following performance on real quadrupeds (Xie et al., 2021).

7. Extensions, Limitations, and Outlook

Current centroidal steering strategies accommodate a wide array of morphologies:

• Non-periodic and contact-rich motion: Hybrid planners (Zhao et al., 2017, Murooka et al., 29 May 2025) address non-repetitive stepping and transitions between contact modes.
• Learning-based steering: Neural networks robustly reproduce diverse motion primitives and quickly adapt to new directional commands (Viereck et al., 2020).
• Geometric-reduction for high-DoF systems: Geometric mechanics–based schemes compact complex body-shape coordination into low-dimensional primitives for steering in multi-legged platforms (Flores et al., 2024).

Limitations persist in the high-fidelity modeling of slip, compliance, and high-speed effects, especially for elongate robots where quasistatic assumptions may fail at larger amplitudes or rough terrain. Steering controllability is bounded by mechanical constraints (e.g., max $m$5 before self-collision), and heading-track errors may be underestimated due to simplified physical models (Flores et al., 2024).

A plausible implication is that future work will increasingly integrate direct model learning with constraint-aware optimization and invariant control, enabling scalable real-time steering for robot collectives and soft multilegged platforms across arbitrary environments.

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Key References:

• Robust Optimal Planning and Control of Non-Periodic Bipedal Locomotion with A Centroidal Momentum Model (Zhao et al., 2017)
• Centroidal Trajectory Generation and Stabilization based on Preview Control for Humanoid Multi-contact Motion (Murooka et al., 29 May 2025)
• Learning a Centroidal Motion Planner for Legged Locomotion (Viereck et al., 2020)
• Steering Elongate Multi-legged Robots By Modulating Body Undulation Waves (Flores et al., 2024)
• GLiDE: Generalizable Quadrupedal Locomotion in Diverse Environments with a Centroidal Model (Xie et al., 2021)