Centroidal Angular Momentum Dynamics

• Centroidal Angular Momentum Dynamics is a framework that computes total angular momentum about the CoM using a centroidal momentum matrix linking linear and angular momentum.
• It supports dynamic motion planning by enforcing conservation laws during flight phases and enabling agile maneuvers in legged and humanoid robots.
• Advanced methods including convexification, hierarchical control, and state estimation techniques provide practical, real-time execution in complex robotic systems.

Centroidal angular momentum dynamics governs the evolution and control of the total angular momentum of multi-link articulated robotic systems about their instantaneous center of mass (CoM). This framework is foundational for dynamically consistent locomotion, whole-body planning, robust control, and online estimation in legged and humanoid robots. Precisely modeling and exploiting centroidal angular momentum is essential for agile behaviors, highly dynamic maneuvers, and robust contact interaction.

1. Mathematical Foundations of Centroidal Angular Momentum

The fundamental construct is the centroidal momentum vector

$$h_G = \begin{bmatrix} \ell_G \ k_G \end{bmatrix} \in \mathbb{R}^6,$$

where $\ell_G$ is the linear momentum and $k_G$ is the angular momentum, both referenced at the CoM. For a floating-base robot with configuration $q$, the "centroidal momentum matrix" (CMM) $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$ linearly maps generalized velocities $\nu$ (6 base + $n$ joints) to centroidal momentum: $h_G(q, \nu) = A_G(q)\,\nu.$ Partitioning, $A_G = [A_l; A_k]$, where $A_l$ and $\ell_G$0 yield the linear and angular momentum mappings, respectively. Critically, the angular part further decomposes as $\ell_G$1 assigning contributions from base translation, base rotation, and joint rates:

• $\ell_G$2: translation-to-angular-momentum map ($\ell_G$3 by rigid-body kinematics)
• $\ell_G$4: base rotation-to-angular-momentum (invertible, equals $\ell_G$5)
• $\ell_G$6: joint velocity-to-angular-momentum.

Accordingly, the total angular momentum about the CoM is

$\ell_G$7

where $\ell_G$8 is the base angular velocity and $\ell_G$9 the vector of joint rates. The composite inertia $k_G$0 appears explicitly in $k_G$1. Alternatively, using linkwise summation,

$k_G$2

where $k_G$3 is the composite inertia about the CoM, $k_G$4 the position of link $k_G$5's CoM, and $k_G$6 its velocity (Sovukluk et al., 29 Jan 2025).

Time evolution (Newton--Euler form) is governed by the centroidal momentum balance: $k_G$7 with $k_G$8 the contact forces at $k_G$9 and $q$0 the CoM (Gazar et al., 2022, Ponton et al., 2017).

2. Physical Principles: Conservation and Nonholonomy

Conservation during Flight Phases

In contact-free (flight) motion, the absence of external moments yields

$q$1

where $q$2 is the angular momentum at contact lift-off. This strict conservation constrains the robot's reorientation and the set of achievable landing configurations—paramount for running, leaping, and acrobatic tasks (Sovukluk et al., 29 Jan 2025).

Nonholonomic Character

The mapping between shape velocities and $q$3 is generally nonintegrable: changes in internal joint states induce path-dependent (nonholonomic) changes in $q$4. The geometric structure is formalized via the mechanical connection $q$5, whose curvature

$q$6

encodes whether the centroidal frame’s orientation is determined solely by the instantaneous configuration or also by its path history. For generic robots with $q$7 internal DOF and nonzero CoM offsets, $q$8 and the centroidal orientation is not a function of configuration alone (Saccon et al., 2017).

3. Trajectory Optimization Incorporating Centroidal Angular Momentum

Centroidal angular momentum dynamics play a central role in trajectory optimization frameworks for legged and humanoid robots.

Direct Limb-Trajectory Optimization

For high-speed running, "Realtime Limb Trajectory Optimization for Humanoid Running Through Centroidal Angular Momentum Dynamics" formulates a nonlinear program over joint trajectories parameterized as polynomials, explicitly imposing conservation of $q$9 during flight. The instantaneous base angular rate $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$0 is algebraically recovered: $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$1 allowing the optimizer to coordinate limb and trunk motion for stable, minimal-tilt landings. Full nonlinear joint-space kinematics and CMM evaluation are handled efficiently, achieving sub-2ms solve times for bipeds and humanoids (Sovukluk et al., 29 Jan 2025).

Robust Optimization via Convexification

The non-convexities inherent in angular momentum coupling (cross-products, bilinear time terms) can be convexified through relaxation (trust regions, soft penalties), yielding tractable trajectory optimization even for time-adaptive, multi-contact scenarios (Ponton et al., 2017). Other methods decompose the original non-convex program into biconvex or block-coordinate schemes, alternately updating contact forces/wrenches and the centroidal states, enforcing centroidal dynamics at each step (Shah et al., 2021).

Stochastic Motion Generation

Uncertainties in contact locations and model parameters are integrated via stochastic trajectory optimization, modeling noise as Gaussians and enforcing chance constraints (converted into deterministic back-off inequalities) on centroidal state evolution: $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$2 enforced within the sequential convexification pipeline (Gazar et al., 2022).

4. Control and Estimation: From Hybrid Templates to State Observers

Hierarchical Whole-Body Control

Regulation of centroidal angular momentum is central in full-body task-space control. Recent extensions of reduced-order models (e.g., ALIP) to full humanoid systems require explicit regulation of $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$3 (centroidal angular momentum) to maintain template-level behaviors despite complex foot geometry and limb inertia. A prioritized optimization stack enforces (i) contact-consistent accelerations, (ii) $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$4 tracking, (iii) swing-foot placement, and (iv) posture, with centroidal momentum constraints formulated as

$A_G(q) \in \mathbb{R}^{6 \times (n+6)}$5

ensuring task-level stability and lift-off/landing consistency (Paredes et al., 2024).

Reduced-Order and Hybrid Models

Simplified models such as the prismatic inverted pendulum plus flywheel approximate centroidal angular momentum using a lumped flywheel torque, embedding the resulting (linear + angular) balance within hybrid automata for robust step planning and control under switches and disturbances (Zhao et al., 2017).

State Estimation via Torque Feedback

Centroidal angular momentum estimation leverages the structure of projected floating-base dynamics; by eliminating contact forces via a null-space projector, the process model for estimation depends only on measured joint torques and system kinematics: $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$6 with $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$7 the actuator map and $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$8 combining passive dynamics and geometric terms. Embedding this in an Extended Kalman Filter fuses torque sensing and proprioceptive state for improved $A_G(q) \in \mathbb{R}^{6 \times (n+6)}$9, $\nu$0, $\nu$1 estimation (Khorshidi et al., 2022).

5. Role in Acrobatic and High-Acceleration Motions

Explicit modeling and reward of centroidal angular momentum and angular velocity are essential for planning and learning highly dynamic tasks, such as aerial rotations and flips. In reinforcement learning settings, rewards based on the centroidal angular velocity

$\nu$2

incentivize true whole-body rotation and penalize spurious joint-level spinning (Kang et al., 18 May 2025). Incorporation of full-centroidal Newton–Euler dynamics and nonholonomic constraints enables task-space optimal control of maneuvers that exploit high-order nonlinearities of the system (Papatheodorou et al., 2023).

6. Computational Strategies and Performance

Efficient real-time computation of centroidal angular momentum derivatives, their inversion (for orientation recovery), and nonlinear optimization are supported by:

• Formulation of the CMM and its analytical blocks ($\nu$3, $\nu$4, $\nu$5)
• Symbolic and analytic differentiation for direct dynamics and optimal control
• Custom projected-gradient and line-search routines for full nonlinear programs
• Sequential convex programming and block coordinate descent for large-scale or robust problems (Sovukluk et al., 29 Jan 2025, Shah et al., 2021, Ponton et al., 2017). Performance metrics demonstrate millisecond-class solve times for typical gait optimization problems, enabling online re-planning and high-frequency feedback in control (Sovukluk et al., 29 Jan 2025).

7. Research Trends and Future Directions

Recent advances firmly establish centroidal angular momentum dynamics as a cornerstone for high-performance legged and humanoid locomotion. Ongoing directions include:

• Integration of flight-phase constraints into hierarchical whole-body planners and MPC
• Robust disturbance rejection and extension to terrain-adaptive and non-periodic behaviors
• Hardware validation of planning and RL pipelines leveraging centroidal dynamics
• Characterization of flatness/nonholonomy in connection-driven shape–momentum interplay
• Formal convergence, feasibility, and risk allocation analysis in stochastic settings A plausible implication is that efficient, multi-layered exploitation of centroidal angular momentum properties will be essential for closing the gap between model-based synthesis and robust, hardware-deployable dynamic behaviors in humanoid robotics (Sovukluk et al., 29 Jan 2025, Gazar et al., 2022, Paredes et al., 2024).