Angular Momentum Linear Inverted Pendulum Model

• The AMLIP is a dynamic model that extends the classical LIPM by explicitly incorporating angular momentum to regulate nonzero momentum and enable independent end-effector force and torque planning.
• It integrates full momentum dynamics—including CoM, linear momentum, and angular momentum—into trajectory optimization and receding horizon LQR feedback for enhanced stability and predictive foot placement.
• Variants such as ALIP and time-varying AMLIP demonstrate practical robustness in managing disturbances and adapting to non-coplanar, multi-contact environments in humanoid robotics.

The Angular Momentum Linear Inverted Pendulum Model (AMLIP) extends the classical linear inverted pendulum model (LIPM) by introducing angular momentum as an explicit or implicit state variable—either for predictive modeling, feedback, or both—enabling enhanced control authority in humanoid and underactuated robot locomotion tasks. This class of models fundamentally addresses the key limitations of LIPM by enabling momentum regulation, independent end-effector force/torque planning, and support for multi-contact and non-coplanar environments. The AMLIP concept manifests in several modeling, planning, and control paradigms, ranging from explicit angular momentum feedback laws to trajectory optimization using full momentum dynamics.

1. Motivations and Limitations of the LIPM

The LIPM is a foundational template for bipedal control, modeling the motion of a single-mass "center of mass" (CoM) atop a massless leg, with assumptions of a fixed CoM height, co-planar contacts, and zero angular momentum. While adequate for flat-ground walking and compliant with simple feedback policies (e.g., ZMP stabilization), the LIPM cannot:

• Control individual end-effector forces independently (only the net contact wrench is managed).
• Model or regulate nonzero angular momentum generated by limb motions (e.g., leg swings or upper-body inclinations).
• Accommodate non-coplanar or multi-contact situations, e.g., rough or stepped terrain, climbing tasks, or hands-in-contact scenarios.

These limitations become critical as the task complexity or environmental complexity increases, motivating the move toward models that incorporate angular momentum and explicit force/torque contact planning (Herzog et al., 2015).

2. Formulations Incorporating Angular Momentum

2.1 Full Momentum Dynamics

The full momentum formulation for a floating-base humanoid robot includes:

• Center of mass position $\mathbf{r}$
• Linear momentum $\mathbf{l}$
• Angular momentum $\boldsymbol{\kappa}$

The dynamics are governed by: $$M\dot{\mathbf{r}} = \mathbf{l} \qquad \dot{\mathbf{l}} = Mg + \sum_i \mathbf{f}_i \qquad \dot{\boldsymbol{\kappa}} = \sum_i \boldsymbol{\tau}_i + \sum_i (\mathbf{p}_i - \mathbf{r}) \times \mathbf{f}_i$$ where $\mathbf{f}_i$, $\boldsymbol{\tau}_i$ are the contact forces and torques at end-effector location $\mathbf{p}_i$.

This model accounts for the generation of angular momentum through contact forces and explicitly allows for arbitrary, possibly multi-point, non-coplanar contact configurations. Angular momentum can be planned, tracked, and regulated, enabling motion patterns such as leg swings and trunk rotations critical for dynamic tasks (Herzog et al., 2015).

2.2 Angular Momentum as a State Variable in LIP Models

Recent AMLIP frameworks substitute or augment the traditional state $\{x, \dot{x}\}$ with $\{x, L\}$, where $L$ is the angular momentum about the stance foot contact. In planar contexts: $$\dot{x}_c = \frac{L}{mH}, \qquad \dot{L} = mgx_c + u_a$$ This so-called ALIP model (Angular Linear Inverted Pendulum) captures the system state with a higher degree of robustness against actuator noise and disturbances, due to the higher relative degree of $L$ (angular momentum is the integral of force/torque, while velocity is a direct derivative), and yields superior predictive foot placement accuracy compared to velocity-state LIP variants (Gong et al., 2020, Gong et al., 2021).

3. Trajectory Optimization and Planning with Angular Momentum

Trajectory generation frameworks based on the full momentum model seek to plan CoM, linear momentum, angular momentum, and contact force/torque profiles as a solution to a constrained optimal control problem. The resulting trajectories are expressed via basis function parameterizations (e.g., polynomials), with optimization objectives penalizing deviation from nominal or desired momentum and position profiles. Task-space constraints (such as friction cone, contact location, and ZMP/COP limits) are embedded into the optimization, enabling physically consistent solutions in arbitrary contact configurations and over rough terrains (Herzog et al., 2015).

An illustrative objective function takes the form: $$J = \sum_t \left[ \| \mathbf{l}(t)-\mathbf{l}_\mathrm{des}(t)\|^2_{W_1} + \| \mathbf{r}(t)-\mathbf{r}_\mathrm{des}(t)\|^2_{W_2} + \|\boldsymbol{\kappa}(t)-\boldsymbol{\kappa}_\mathrm{des}(t)\|^2_{W_3} \right]$$ where the weighting matrices $W_i$ shape the tracking and regularization trade-offs (Herzog et al., 2015).

This approach also supports the formation of non-trivial angular momentum trajectories, such as the swing-phase tasks that would be infeasible under a zero-angular-momentum LIPM assumption.

4. Feedback Control: LQR Applied to Momentum Dynamics

State feedback around planned momentum trajectories is accomplished via discrete-time, receding horizon LQR design. The momentum dynamics are linearized about the reference trajectories, yielding the classic linear system form: $\delta \dot{x} = A(t) \delta x + B(t) \delta u$ LQR gain matrices $K_t$ are computed along the trajectory, providing optimal feedback: $\lambda = \lambda^\ast - K_t(x - x^\ast)$ where the feedback structure naturally couples linear and angular momentum errors, yielding robust tracking, especially in regimes involving underconstrained dynamics, contact switching, and dynamic multi-contact (Herzog et al., 2015).

A critical property observed is that off-diagonal gain terms in $K_t$ reflect physically meaningful couplings between CoM, linear momentum, and angular momentum. Fixed gain or PD-like strategies (which neglect these couplings) are suboptimal, particularly on terrains where the support geometry and contact configuration are time-varying.

5. Model Variants and their Scope

The AMLIP concept spans a set of models and implementations:

Full Momentum Models:

• Incorporate all degrees of freedom in momentum evolution, enable arbitrary torque/force/contact planning, and apply to scenarios with significant coupling between linear and angular modes (e.g., rough terrain, multi-contact) (Herzog et al., 2015).

ALIP (Angular Momentum LIP):

• Uses $\{x, L\}$ as the reduced-order state. Key for bipedal robots with significant leg mass, the ALIP has been shown to produce more accurate one-step-ahead foot placement predictions and to be more robust to disturbances and impact events (Gong et al., 2021).

Hybrid and Time-Varying AMLIP:

• Models that allow for time-varying "pendulum" length (to model, e.g., stair climbing with large CoM height variations) and actuated torque inputs as in $\{L, \theta_c\}$ (Dosunmu-Ogunbi et al., 2023).

These model variants are unified by acknowledging that foot placement alone becomes inadequate when kinematic constraints (e.g., stair geometry, non-coplanar support) prevent sufficient control authority, so explicit angular momentum regulation—often via stance ankle torque—is introduced (Dosunmu-Ogunbi et al., 2023).

6. Practical Implementation and Application

Trajectory Generation and Control Loop:

• Trajectory optimization yields desired CoM, linear, and angular momentum trajectories and time-varying contact forces.
• LQR-based receding horizon feedback executes the planned motion, tracking both linear and angular momentum closely.
• Hierarchical whole-body control schemes integrate these references at the operational space or joint torque level.

Empirical Evidence:

• Simulations on full humanoid models (e.g., Sarcos robot) verify the model's validity for complex terrain navigation, with good matching of planned and tracked momentum trajectories and adaptive gains contingent on contact configuration (Herzog et al., 2015).
• Hardware demonstrations on 20-DOF robots (Cassie Blue) achieve walking at up to 2.1 m/s, rapid turning, rough terrain traversal, and robust disturbance rejection under an ALIP-based feedback policy (Gong et al., 2020, Gong et al., 2021).
• Stair climbing in the presence of significant CoM height variation is achieved by combining time-varying ALIP and predictive control of stance ankle torque (Dosunmu-Ogunbi et al., 2023).

Quantitative Results Table

Model/Application	Angular Momentum State Regulated	Key Feedback/Feedforward Functionality
Full Momentum Planning	Yes	LQR on linearized momentum dynamics
ALIP on Cassie	Yes	Foot placement from L prediction
Stair-Climbing ALIP	Yes (w/ time-varying $r_c$)	Ankle torque MPC for sagittal stabilization
Rough Terrain (Sarcos)	Yes, plus independent forces	Trajectory optimization plus receding LQR

7. Theoretical and Experimental Implications

The AMLIP unifies and extends the LIP paradigm by:

• Providing direct, dynamically consistent prediction of momentum and thus future CoM and body states, outperforming linear velocity-based LIP predictions, particularly on robots with non-negligible leg mas or impacts (Gong et al., 2021).
• Enabling physically meaningful and tractable feedback and feedforward design in scenarios requiring nonzero angular momentum (e.g., leg swing, trunk maneuvers, stair climbing) (Dosunmu-Ogunbi et al., 2023).
• Validating a hierarchy of controllers (planned momentum, receding LQR, joint-level torque) under realistic noise, delay, and model mismatch.
• Offering a flexible modeling philosophy: from highest-fidelity full momentum trajectory optimization to reduced-order AMLIP approaches suitable for online computation.

A plausible implication is that as robot tasks and designs move away from the restrictive regimes assumed by LIPM (constant height, flat ground, zero angular momentum), the AMLIP class and its associated planning and control tools become foundational elements in the design of robust locomotion controllers.

References to Major Contributions

• Full momentum-based multi-contact trajectory planning and receding LQR feedback: (Herzog et al., 2015)
• ALIP formulation and Cassie Blue experimental validation: (Gong et al., 2020, Gong et al., 2021)
• Model extensions to stair climbing and time-varying pendulum length: (Dosunmu-Ogunbi et al., 2023)
• Integration of AMLIP with whole-body and underactuated robot control hierarchies: (Herzog et al., 2015, Gong et al., 2021)