Angular Momentum Linear Inverted Pendulum Model

• Angular Momentum Linear Inverted Pendulum (AMLIP) is a reduced-order model for bipedal locomotion that uses contact-point angular momentum instead of linear velocity for state prediction.
• The model assumes constant center-of-mass height, point-foot contact, and negligible ankle torque, yielding linear unstable pendulum dynamics similar to the classical LIP.
• AMLIP underpins advanced foot-placement control across various scenarios by linking continuous dynamics with hybrid impact regulation and full-body momentum control.

Angular Momentum Linear Inverted Pendulum Model (AMLIP) is a reduced-order model for bipedal locomotion in which the horizontal center-of-mass (CoM) displacement and the angular momentum about the contact point are taken as the principal dynamic states. Under constant CoM height, pinned or point-foot contact, and negligible ankle-torque or centroidal-angular-momentum terms, AMLIP yields a linear unstable pendulum model whose continuous dynamics are formally identical to those of the classical Linear Inverted Pendulum (LIP), but with contact-point angular momentum replacing linear velocity as the state used for prediction and feedback. In the recent locomotion literature, closely related formulations also appear under the abbreviation ALIP, especially when the same state substitution is embedded in hybrid walking, constrained footholds, moving rigid surfaces, stair climbing, and full-body momentum regulation (Gong et al., 2020, Gong et al., 2021).

1. Historical emergence and terminology

The formulation entered the recent legged-locomotion literature through the argument that angular momentum about the contact point is a better indicator of robot state than linear velocity. “Angular Momentum about the Contact Point for Control of Bipedal Locomotion: Validation in a LIP-based Controller” reformulated a standard LIP controller in terms of angular momentum and validated the resulting feedback law on Cassie Blue (Gong et al., 2020). “Zero Dynamics, Pendulum Models, and Angular Momentum in Feedback Control of Bipedal Locomotion” then placed the same idea in the broader context of pendulum models and virtual-constraint-induced zero dynamics, arguing that an approximate model parameterized by angular momentum provides better predictions for foot placement on a physical robot than a related approximate model parameterized in terms of linear velocity (Gong et al., 2021).

The cited literature uses both “AMLIP” and “ALIP.” In both cases, the defining feature is that angular momentum about the stance or contact point is elevated to a first-class reduced-order state. In the planar sagittal setting, the state is commonly written as

$$\xi = \begin{bmatrix} x_n \ L \end{bmatrix},$$

where $x_n$ is the horizontal CoM displacement measured from the stance foot and $L$ is the sagittal-plane angular momentum about the stance foot (Gong et al., 2021). In 3D ALIP formulations, the same idea is applied separately in sagittal and frontal planes, with states $\mathbf{x}=[p_x,\;L_{c,y}]^\top$ and $\mathbf{y}=[p_y,\;L_{c,x}]^\top$ (Paredes et al., 2024).

The original reduced-order assumptions are narrow and explicit: the CoM is constrained to move at a constant height $H$ or $h$, the stance contact is pinned or point-foot and non-slipping, ankle torques are neglected or set to $u_a \simeq 0$, and other degrees of freedom are assumed to be regulated by high-bandwidth virtual-constraint or whole-body controllers so that their effect enters only through small neglected terms (Gong et al., 2021, Dai et al., 2021). Much of the subsequent literature can be read as a sequence of controlled relaxations of these assumptions.

2. Canonical continuous-time dynamics

The starting point is the exact contact-point angular momentum relation

$L = L_c + m(x_n \dot z_n - z_n \dot x_n),$

where $L_c$ is the angular momentum about the CoM, $x_n$0 is total mass, and $x_n$1 are CoM coordinates in the stance-foot frame (Gong et al., 2021). Under constant CoM height $x_n$2 and $x_n$3, this reduces to

$x_n$4

Neglecting the small CoM-about-CoM angular momentum term $x_n$5 gives the AMLIP kinematic equation

$x_n$6

Taking moments about the contact point and setting the ankle torque to zero yields

$x_n$7

The resulting linear time-invariant reduced model is therefore

$x_n$8

or, in second-order form,

$x_n$9

This is the canonical AMLIP system (Gong et al., 2021).

The formal relation to the traditional LIP is immediate. In the standard LIP, the state is $L$0 and the dynamics are $L$1. In AMLIP, the second state variable is replaced by $L$2, so the continuous dynamics are formally identical after the substitution $L$3 (Gong et al., 2021, Gong et al., 2020). In planar underactuated formulations, a mass-normalized momentum coordinate is also used: $L$4 which is algebraically equivalent after scaling by mass (Dai et al., 2021).

For 3D ALIP, the same constant-height approximation leads to decoupled planar systems,

$L$5

with closed-form step solutions expressed through hyperbolic functions (Paredes et al., 2024). This decoupled structure underlies much of the subsequent foot-placement and template-tracking design.

3. Hybrid step-to-step regulation and stability

AMLIP becomes a walking controller when the unstable continuous-time template is closed through discrete foot placement. In the fixed-step-time formulation with $L$6 the step duration and $L$7, the one-step-ahead predictor for contact-point angular momentum is

$L$8

Using impact continuity of $L$9 on flat ground and the reset of the horizontal coordinate by the landing location, the desired swing-foot landing position that achieves a target one-step-ahead momentum is

$\mathbf{x}=[p_x,\;L_{c,y}]^\top$0

This is the core AMLIP foot-placement law (Gong et al., 2021, Gong et al., 2020).

A more general asymptotic law introduces a gain $\mathbf{x}=[p_x,\;L_{c,y}]^\top$1, producing a step-to-step error recursion

$\mathbf{x}=[p_x,\;L_{c,y}]^\top$2

The corresponding discrete Poincaré map has eigenvalues $\mathbf{x}=[p_x,\;L_{c,y}]^\top$3, and the fixed point

$\mathbf{x}=[p_x,\;L_{c,y}]^\top$4

is exponentially stable for any $\mathbf{x}=[p_x,\;L_{c,y}]^\top$5 (Gong et al., 2021). In simulation on the planar Rabbit robot, the dominant eigenvalue of the full-order Poincaré map closely matches $\mathbf{x}=[p_x,\;L_{c,y}]^\top$6 predicted by the ALIP model across $\mathbf{x}=[p_x,\;L_{c,y}]^\top$7, supporting the interpretation of AMLIP as an approximation of the slow zero dynamics rather than only a heuristic template (Gong et al., 2021).

The physical interpretation is that AMLIP does not stabilize walking by canceling the unstable continuous-time pendulum; it stabilizes walking by making the impact map and step placement reshape the state from one support phase to the next. In the canonical formulation, no ankle torque is used to actively change $\mathbf{x}=[p_x,\;L_{c,y}]^\top$8 during a single support phase; all corrections to $\mathbf{x}=[p_x,\;L_{c,y}]^\top$9 occur via choosing the next foot placement (Gong et al., 2021).

4. Extensions to non-stationary, constrained, and time-varying walking

Several later variants retain the angular-momentum state but relax the assumptions that made the original model time-invariant and point-contact based.

Constrained footholds and impact momentum injection. “Bipedal Walking on Constrained Footholds: Momentum Regulation via Vertical COM Control” derives an underactuated AMLIP in which the discrete impact map couples pre-impact vertical CoM velocity to post-impact angular momentum: $\mathbf{y}=[p_y,\;L_{c,x}]^\top$0 When the swing foot strikes at ground level, $\mathbf{y}=[p_y,\;L_{c,x}]^\top$1, this reduces to

$\mathbf{y}=[p_y,\;L_{c,x}]^\top$2

The control strategy chooses a desired post-impact momentum from a viable LIP orbital energy, computes the required pre-impact vertical velocity, and realizes it through a small CoM-$\mathbf{y}=[p_y,\;L_{c,x}]^\top$3 QP combined with a task-space whole-body QP (Dai et al., 2021). Under the reported settings, AMBER achieved stable periodic walking speeds from $\mathbf{y}=[p_y,\;L_{c,x}]^\top$4 up to $\mathbf{y}=[p_y,\;L_{c,x}]^\top$5, traversing both fixed-pattern and randomly varying stepping stones for thousands of steps with no failures, while Cassie in simulation robustly ascended and descended staircases and crossed random stones at $\mathbf{y}=[p_y,\;L_{c,x}]^\top$6 (Dai et al., 2021).

Moving rigid surfaces and linear time-varying ALIP. “Time-Varying ALIP Model and Robust Foot-Placement Control for Underactuated Bipedal Robot Walking on a Swaying Rigid Surface” extends ALIP from stationary to horizontally moving surfaces with known trajectory $\mathbf{y}=[p_y,\;L_{c,x}]^\top$7. The continuous reduced-order dynamics become

$\mathbf{y}=[p_y,\;L_{c,x}]^\top$8

which is a linear time-varying, non-homogeneous ODE. Foot placement enters through the impact reset

$\mathbf{y}=[p_y,\;L_{c,x}]^\top$9

and the step-to-step controller takes the form

$H$0

With the monodromy operator $H$1, the cited theorem states that if all eigenvalues of $H$2 lie strictly inside the unit circle, then the zero solution of the homogeneous hybrid system is exponentially stable and the unique $H$3-periodic solution of the non-homogeneous hybrid system is exponentially stable (Gao et al., 2022).

Variable pendulum length and stair climbing. “Stair Climbing using the Angular Momentum Linear Inverted Pendulum Model and Model Predictive Control” replaces constant $H$4 by a known nominal $H$5 or $H$6, preserving linearity in the state while making the system linear time-varying. The key equations are

$H$7

or, in angular form,

$H$8

The paper combines virtual-constraint-based posture control with an MPC that acts on the stance ankle torque, using $H$9, prediction horizon $h$0, and ankle-motor limits $h$1 (Dosunmu-Ogunbi et al., 2023). In simulation on a 20 degree-of-freedom model of Cassie, turning off ankle torque causes falls within two steps on flat ground with fixed-step length, whereas with MPC ankle torque the robot remains stable despite large $h$2 torque-drop perturbations and climbs stairs for $h$3 steps while $h$4 and $h$5 track nominal periodic profiles (Dosunmu-Ogunbi et al., 2023).

5. Embedding AMLIP in full-body humanoid control

A central question is whether AMLIP survives once the robot has flat feet, non-passive ankles, nontrivial torso and limb inertia, and non-negligible centroidal angular momentum. “Moving past point-contacts: Extending the ALIP model to humanoids with non-trivial feet using hierarchical, full-body momentum control” addresses exactly this regime (Paredes et al., 2024). The paper starts from the floating-base rigid-body dynamics

$h$6

with no-slip contact constraints

$h$7

and the centroidal momentum mapping

$h$8

The control architecture uses a hierarchical task-space whole-body controller to regulate centroidal momentum so that $h$9, $u_a \simeq 0$0, and the horizontal CoM follows ALIP trajectories, while the desired foot placement is tracked as a lower-priority task (Paredes et al., 2024).

The significance of this construction is not that the original ALIP assumptions become true; rather, the full-order robot is coerced to match the reduced ALIP dynamics through contact wrench modulation and momentum regulation. In MuJoCo simulations on the Sarcos Guardian XO robot, which has mass $u_a \simeq 0$1, flat offset feet, and substantial limb and torso inertia, the measured CoM closely follows the ALIP-prescribed trajectory with RMS error $u_a \simeq 0$2, centroidal angular momentum is driven to near zero of order $u_a \simeq 0$3, predicted end-of-step contact angular momentum matches measured values within $u_a \simeq 0$4, and the $u_a \simeq 0$5 moving-average of $u_a \simeq 0$6 stays within $u_a \simeq 0$7 of the commanded speeds for stepwise changes up to $u_a \simeq 0$8 forward and $u_a \simeq 0$9 lateral (Paredes et al., 2024).

A related, but distinct, development expands the Divergent Component of Motion framework to include angular coordinates by combining classical 3D linear DCM with a 1D angular DCM derived from a flywheel-approximation SRBM. The resulting $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$0D linear and $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$1D angular DCM framework has open-loop unstable dynamics and decoupled exponentially stable closed-loop subsystems under VRP and VRO feedback. MATLAB simulations and TORO humanoid experiments report angular error $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$2, CoP excursion $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$3, ground reaction torque matching the predicted $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$4 within $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$5, and simulation-to-hardware agreement within $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$6 over a $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$7 trial (Herron et al., 2024). This suggests one route by which AMLIP-style angular-state reasoning can be connected to spatial DCM planning.

6. Advantages, common misconceptions, and limitations

A recurring misconception is that AMLIP is only the LIP written in different coordinates. Formally, the continuous equations are identical after the substitution $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$8, but the cited works emphasize several structural consequences of the momentum coordinate. In the planar constant-height setting, $L = L_c + m(x_n \dot z_n - z_n \dot x_n),$9 depends only on CoM position; $L_c$0 is invariant across flat impacts when $L_c$1; and the angular-momentum output has relative degree $L_c$2, yielding weak coupling to joint-torque peaks and swing-leg disturbances (Gong et al., 2020). On Cassie, one-step-ahead angular-momentum prediction error remained within $L_c$3 over each step at $L_c$4, whereas the standard LIP-based one-step velocity prediction would have exhibited prediction errors in excess of $L_c$5 under the same conditions in simulation comparisons (Gong et al., 2021).

A second misconception is that AMLIP necessarily implies direct ankle-torque regulation of the reduced state. In the canonical step-to-step AMLIP of (Gong et al., 2021), no ankle torque is used to actively change $L_c$6 during a single support phase; all regulation is accomplished through foot placement. Other papers deliberately break that restriction when the task requires it: stair climbing uses stance-ankle torque as the control input in an LTV ALIP-MPC controller (Dosunmu-Ogunbi et al., 2023); constrained-foothold walking uses pre-impact vertical CoM velocity to inject the desired angular momentum at impact (Dai et al., 2021); and flat-foot humanoid locomotion uses full contact wrenches and centroidal-momentum regulation to force a full-order robot to behave like an ALIP template (Paredes et al., 2024).

The model’s limitations are equally explicit in the literature. The original formulations assume constant CoM height, zero or negligible angular momentum about the CoM, pinned or point-foot contact, and zero vertical CoM velocity at impact; they neglect ankle torques in prediction and require reliable IMU or EKF estimation of $L_c$7 (Gong et al., 2020, Gong et al., 2021). Stairs violate both the constant-height assumption and unrestricted sagittal foot placement, motivating the variable-length and torque-actuated LTV model (Dosunmu-Ogunbi et al., 2023). Moving support surfaces introduce explicit time variation and non-homogeneous forcing into the reduced dynamics (Gao et al., 2022). Flat feet and non-centralized limb inertia invalidate the negligible-centroidal-momentum and zero-contact-moment assumptions unless a whole-body controller actively regulates them (Paredes et al., 2024). Impact-driven oscillations and spring dynamics can also generate brief torque spikes not predicted by the simplified ALIP model (Dosunmu-Ogunbi et al., 2023).

Within those bounds, AMLIP has served as a compact analytical template that links continuous pendulum dynamics, hybrid impact maps, foot-placement control, whole-body momentum regulation, and zero-dynamics interpretations. The common thread across its variants is not a single fixed equation set, but the use of angular momentum about the contact or stance point as the reduced coordinate through which walking balance, velocity regulation, and disturbance rejection are organized (Gong et al., 2020, Gong et al., 2021).