Momentum-Based Balance Controller

• Momentum-based balance control is a method that regulates both linear and angular momentum to stabilize humanoid robots during dynamic locomotion and standing.
• Controllers employ reduced-order models like LIP and discrete LQR step-length modulation to ensure robust gait convergence and effective push recovery.
• Advanced approaches integrate full-body inverse dynamics, model predictive control, and capturability algorithms to achieve rapid, real-time disturbance rejection.

A momentum-based balance controller is a control architecture for humanoid and bipedal robots that stabilizes locomotion and standing by regulating system momentum—linear, angular, or both—thereby directly managing global dynamic quantities rather than only joint-level or position-based objectives. Originally motivated by the need to achieve robust, human-comparable balance during dynamic locomotion, these controllers unify feedback stabilization, push recovery, and complex behaviors such as walking on rough or compliant terrains within a momentum-centric formalism.

1. Foundations: Momentum Dynamics and Reduced-Order Models

Classically, momentum-based controllers are derived from the full rigid-body centroidal dynamics:

$\dot{h} = \sum_{i} W_i f_i + m g e_z$

where $$h = [h_{\text{lin}}; \, h_{\text{ang}}] \in \mathbb{R}^6$$ stacks linear and angular momentum about the Center of Mass (CoM), $f_i$ are contact forces, $W_i$ wrench maps, and $m$ is mass. The centroidal momentum can be further reduced to low-dimensional models for tractability (Pucci et al., 2016, Dafarra et al., 2017, Gong et al., 2021). A key insight is that, for walking, the system is often well-approximated by the Linear Inverted Pendulum (LIP) model or by nonlinear pendulum analogs that include angular momentum about the stance foot (ALIP), the CoM height dynamics, or flywheel effects (Gong et al., 2021, Ding et al., 2019).

Significantly, angular momentum about the ground contact is a superior indicator for foot-placement and step regulation compared to the CoM velocity-centric LIP models, as it captures whole-body rotational effects and admits smooth, low-noise state estimation (Gong et al., 2021).

2. Discrete-Time and Step-to-Step Limit Cycle Stabilization

Momentum-based controllers for bipedal walking often exploit the structure of the step-to-step dynamics. Standard LIP-based controllers model the discrete update of CoM state (position and velocity) across steps as

$X_{i+1}^0 = A X_i^0 + B L_c$

where $X_i^0$ is the CoM position/velocity state at step $i$ onset, $L_c$ is the nominal step length, and $A, B$ depend on the pendulum natural frequency and step time (Ghorbani et al., 2019). The stabilization objective is to drive the error $e_i = X_i^0 - X^*$ (where $X^*$ is the limit cycle) to zero via step-length modulation:

$L_i = L_c + u_i, \qquad u_i = -K e_i$

with feedback gain $K$ selected via discrete LQR or pole placement. This CoM-centric approach achieves robust recovery from disturbances and ensures exponential convergence of the gait to the desired limit cycle if eigenvalues of $A - BK$ are inside the unit circle. Simulated recovery times of ~1.2 s (3 steps) for moderate pushes have been demonstrated (Ghorbani et al., 2019).

In a distinct but related approach, momentum-based stabilization can also be formulated directly in the space of convergent/divergent modes (e.g., in CoM-centric or momentum-centric coordinates), or by using the "linear foot placement control" (LFPC), in which step location is chosen as a linear function of pre-impact velocity (Ye et al., 2022). Explicit bounds for the feedback gain guarantee stability and dead-beat control (one-step reconvergence) is possible for a precise gain selection.

3. Full-Body Momentum Feedback and Hierarchical Control

Beyond reduced-order models, full-body momentum-based controllers embed the centroidal dynamics within hierarchical whole-body inverse dynamics control architectures (Dafarra et al., 2017, Pucci et al., 2016, Herzog et al., 2015). The architecture typically includes:

• Momentum feedback regulation: Desired change in centroidal momentum $\dot{h}^*$ is tracked via optimal distribution of contact wrenches $f^*$, often using quadratic programming (QP) subject to contact, friction, and actuation constraints.
• Hierarchical task prioritization: Lower-priority tasks (e.g. swing foot trajectory, posture) are realized in null space of momentum tracking.
• Stability through zero dynamics: Postural/zero-dynamics controllers ensure that, once the global momentum reference is achieved, joint-space behavior converges to specified trajectories or equilibria.

Automatic gain tuning for these architectures can be performed via constrained optimization and online symmetric positive-definite matrix tracking to ensure closed-loop properties (desired natural frequencies, damping) for the joint-subsystem (Pucci et al., 2016).

4. Model Predictive Momentum-Based Control

Receding-horizon (MPC) frameworks greatly enhance momentum-based balance by predicting future centroidal momentum evolution, contact status, and force feasibility over a horizon (Dafarra et al., 2017, Dafarra et al., 2017, Ding et al., 2019). In MPC-based approaches:

• The controller solves, at each tick, a QP or QCQP to minimize deviation from target momentum, step placement, and contact wrench profiles over the horizon, while respecting ZMP, friction, and actuation constraints.
• Push recovery and step triggering are naturally incorporated by anticipating when imminent constraint violation (e.g. ZMP exiting support) requires a step, and planning accordingly.
• Extensions using nonlinear models (inverted pendulum plus flywheel with variable height and hip strategies) allow the controller to coordinate stepping, upper-body angular momentum, and CoM height—thereby achieving improved disturbance rejection and environmental adaptability (Ding et al., 2019).

Demonstrations on platforms such as iCub and COMAN show >20% improvements in maximum recoverable push and substantial reductions in joint peak torques versus simple reactive approaches (Dafarra et al., 2017, Ding et al., 2019).

5. Advanced Capturability and 3D Generalizations

Recent developments address the capturability problem by precisely characterizing the region of system states from which balance can be maintained under input constraints, notably for the Variable-Height Inverted Pendulum (VHIP) in 3D (Liu et al., 2021, Caron et al., 2017). Notable contributions include:

• Instantaneous Capture Input (ICI): A state feedback law rendering the ICI stationary, guaranteeing convergence to rest if all constraints are met (Liu et al., 2021). Online linear programs select the maximal admissible gains at each state to maintain capturability.
• Convex Boundedness Approach: Reformulates the 3D boundedness (capturability) constraints into a finite-dimensional convex QP, optimizing over the time-varying pendulum frequency $\omega(t)$ and CoP trajectories (Caron et al., 2017). This unifies traditional capture-point methods and CoM height variation in a globally optimal, real-time algorithm.

These approaches support push recovery from large disturbances and adaptation to uneven terrains, with sub-millisecond solve times.

6. Integration with Angular Momentum and Practical Implementation

Integration of angular momentum about the contact point improves prediction of foot placement and robustness to impacts (Gong et al., 2021). The controller tracks angular momentum using estimation (e.g., EKF) and stabilizes via a Poincaré map with closed-form updates, leveraging the ALIP model for superior prediction accuracy regarding step-end state.

Practical guidelines involve choosing model parameters (e.g., CoM height, nominal step period), computing momenta and velocities from state estimation, and tuning step placement/force feedback gains within analytically derived stability bounds. Modern implementations employ modular separation of high-level MPC/momentum planning and low-level torque/distributed control, on nested time scales.

7. Summary Table: Key Momentum-Based Control Paradigms

Model / Method	State Variables	Feedback Law	Disturbances Handled / Metrics
LIP-based LFPC (Ye et al., 2022)	CoM position, velocity	Step location: $p_{n+1} = -k v^-$	Recovers 0.3 m/s velocity change
Discrete LIP + length-shift (Ghorbani et al., 2019)	$x, \dot{x}$	$u_i = -K\,e_i$	20 N push, recovery in 1.2 s
Centroidal MPC (Dafarra et al., 2017)	$h$ (momenta)	$\dot{h}^*$ QP + CP/step logic	100 N·s push (simulation)
ALIP (Cassie) (Gong et al., 2021)	$(x_c, L)$	Step: dead-beat in $L$	Robust push recovery, <3% error
ICI for VHIP (Liu et al., 2021)	$(c, \dot{c})$, $\lambda$	Online gain LP on capture input	Maintains capturability
Full multi-contact MPC (Herzog et al., 2015)	$(r, l, \kappa)$	receding horizon LQR/QP	cm-scale tracking on rough terrain

• (Ghorbani et al., 2019) Stabilization of Bipedal Robot Motion based on Total Momentum
• (Dafarra et al., 2017) A Receding Horizon Push Recovery Strategy for Balancing the iCub Humanoid Robot
• (Pucci et al., 2016) Automatic Gain Tuning of a Momentum Based Balancing Controller for Humanoid Robots
• (Ding et al., 2019) Nonlinear Model Predictive Control for Robust Bipedal Locomotion: Exploring Angular Momentum and CoM Height Changes
• (Gong et al., 2021) Zero Dynamics, Pendulum Models, and Angular Momentum in Feedback Control of Bipedal Locomotion
• (Ye et al., 2022) The Simplest Balance Controller for Dynamic Walking
• (Liu et al., 2021) Instantaneous Capture Input for Balancing the Variable Height Inverted Pendulum
• (Dafarra et al., 2017) A Predictive Momentum-Based Whole-Body Torque Controller: Theory and Simulations for the iCub Stepping
• (Herzog et al., 2015) Trajectory generation for multi-contact momentum-control
• (Caron et al., 2017) Balance control using both ZMP and COM height variations: A convex boundedness approach
• (Ghobadi, 2019) Stability Control of Walking Biped Robots based on Total Momentum

These works collectively establish momentum-based balance control as a foundational methodology for robust, efficient, and adaptable humanoid locomotion, applicable from reduced-order models to full-body control architectures and advanced predictive and capturability-aware algorithms.