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arxiv: 2605.05707 · v1 · submitted 2026-05-07 · 💻 cs.RO

On the Emergence of Pendular Structure in Multi-Contact Locomotion

Lingxue Lyu , Zihui Liu This is my paper

Pith reviewed 2026-05-08 09:23 UTC · model grok-4.3

classification 💻 cs.RO
keywords legged locomotioncentroidal dynamicslinear inverted pendulum modelangular momentummulti-contactZMPquadruped
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The pith

A simple penalty on angular momentum rate in centroidal control makes pendular force patterns emerge naturally in multi-contact locomotion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks why the linear inverted pendulum model keeps appearing in legged-robot controllers even when it is not imposed by hand. It begins with a minimal centroidal optimal control problem whose only distinctive term is a cost on the time derivative of angular momentum. When the stance is full rank the optimizer drives contact forces toward the classical pendular distribution; the speed of this drift is set by the singular-value decomposition of the moment Jacobian and by the horizontal distance between the feet. The same cost cannot eliminate angular momentum rate when only two feet are down, because friction cones create an irreducible lower bound that weight tuning cannot remove. Predictions are checked on both a point-mass quadruped and the Unitree Go1 in MuJoCo.

Core claim

Working from a small centroidal OCP that penalizes the rate of angular momentum, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian when the stance is full rank; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance the friction cone introduces a lower bound on angular momentum rate that no amount of weight tuning fixes, together with a non-smooth feasibility kink at a critical horizontal acceleration that can be written in closed form.

What carries the argument

The moment Jacobian of the centroidal dynamics, whose SVD controls how rapidly the optimal contact forces converge to the pendular pattern under the angular-momentum-rate cost.

If this is right

  • The pendular convergence rate matches experimental data within 16%.
  • In two-point stance the friction cone sets an irreducible lower bound on angular momentum rate independent of cost weights.
  • A critical horizontal acceleration produces a non-smooth feasibility boundary expressible in closed form.
  • An added task term for nonzero angular momentum rate shifts the optimum away from the pendular set in a predictable direction.
  • The emergent behavior stays close to the classical zero-moment-point and divergent-component-of-motion pictures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the angular-momentum-rate term dominates, then simplified pendulum models may remain adequate even when full rigid-body dynamics are available to the controller.
  • The non-smooth kink in the two-contact case suggests that contact addition or gait switching could be triggered by monitoring distance to the critical acceleration.
  • Replacing the planar moment Jacobian with its spatial counterpart would extend the same asymptotic analysis to three-dimensional or sloped terrain.

Load-bearing premise

That penalizing the rate of angular momentum in a small centroidal OCP is sufficient to recover the classical LIPM behavior without additional task terms or full dynamics.

What would settle it

Measuring contact-force deviation from the predicted pendular pattern on the Unitree Go1 across varying foot spans and stance ranks; deviations larger than 16% or absence of the closed-form critical-acceleration kink would falsify the rate and bound claims.

Figures

Figures reproduced from arXiv: 2605.05707 by Lingxue Lyu, Zihui Liu.

Figure 1
Figure 1. Figure 1: (a) Pendular direction (foot→CoM) versus the friction cone of half￾angle arctan µ = 31◦ at µ = 0.6. Diagonal trot (N = 2): the pendular direction tilts 40.6 ◦ from vertical, outside the cone, so ∥H˙ G∥ has an α￾independent floor (Thm. 5). Four-foot stance (N = 4): tilt is ∼18◦, inside the cone, floor is zero. (b) Go1 numerical ∥H˙ G∥/m vs. α: N = 4 follows K/α with K = 8.4 (Thm. 1); N = 2 saturates at 0.44… view at source ↗
Figure 2
Figure 2. Figure 2: Theorem 6: closed-form infα ∥H˙ G∥/m vs. fore–aft acceleration ax for Go1 diagonal trot. Below a ⋆ x = 3.72 m/s 2 the QP achieves the geometric floor of Theorem 5 exactly; above a ⋆ x the friction cone binds and a strictly positive excess (gray band) appears. The transition is non-smooth. producing a non-smooth kink in ax 7→ infα ∥H˙ G∥/m at a ⋆ x . Proof. The decomposition fi = Fnet/2 + (−1)i+1δ from Theo… view at source ↗
Figure 4
Figure 4. Figure 4: Point-mass α sweep: log-log εH vs. α with reference slope −1 (Theorem 1). VI. NUMERICAL EXPERIMENTS We test the theorems in five settings: a point-mass quadruped (Theorem 1 rate); Unitree Go1 in MuJoCo under four-foot stance (Proposition 2 constant); Go1 under N = 2 diagonal trot with the QP solved against ground-truth state (Theorem 5 floor); Go1 with a closed-loop torque-level MPC trotting on full physic… view at source ↗
Figure 3
Figure 3. Figure 3: Theorem 7. (a) The prefactor λ/(α + λ) varies smoothly with log(λ/α), separating balance-dominated (λ/α ≪ 1) from task-dominated (λ/α ≫ 1) locomotion. (b) Sample optima sliding from the pendular manifold (H˙ ∗ G =0) along the line to H˙ task G as λ/α grows. minimizer of (3) satisfies H˙ ∗ G(t) = λ α + λ H˙ task G (t). (10) Hence H˙ ∗ G ̸= 0 wherever H˙ task G ̸= 0, and the trajectory leaves P(n, h). In the… view at source ↗
Figure 6
Figure 6. Figure 6: Go1 N = 2 diagonal trot: ∥H˙ G∥/m vs. α saturates at the analytical floor (left); fraction of H˙ G,0 along Dˆ (uncancellable) versus excitation direction averages 64% (right). Algorithm 1 Closed-loop centroidal MPC (per control step) 1: (c, c˙, {ri}) ← MuJoCo.read() 2: Fnet ← m(c¨ ref − g) ▷ from CoM PD reference 3: {f ∗ i } ← arg min{fi} α∥ P i (ri−c)×fi∥ 2 + γ P i ∥fi∥ 2 s.t. P i fi = Fnet, fi ∈ Ki ▷ sta… view at source ↗
Figure 7
Figure 7. Figure 7: Go1 in MuJoCo with the per-step centroidal QP visualized. CoM (yellow star), pendular direction (orange dashed), QP-commanded contact forces ˙ ˙ view at source ↗
Figure 9
Figure 9. Figure 9: ZMP − virtual pivot deviation vs. α on the point-mass quadruped, validating Lemma 3. Dashed: O(1/α) reference (Theorem 1). TABLE I THEORY–EXPERIMENT CORRESPONDENCE (FIVE NUMERICAL TESTS). Test Setting Theorem Match A point-mass OCP Thm. 1 slope −1 (227×) B Go1 N=4 QP Prop. 2 Ke=8.4, Ka=9.8 C Go1 N=2 QP Thm. 5 0.2835 vs. 0.2835 D Go1 trot MPC Thm. 5 ε¯∞=1.71 E point-mass ZMP Lem. 3 5.3 mm at α=103 discrete … view at source ↗
Figure 8
Figure 8. Figure 8: Go1 closed-loop trot MPC: ∥H˙ G∥/m vs. α on (a) linear and (b) log-log axes. “MPC” marks the closed-loop trajectory; “floor” is the empirical asymptote ε¯∞ ≈1.71 m2/s 2 , the control-augmented analogue of Theorem 5’s N = 2 floor; “O(1/α)” is a reference of slope −1 with K = 27 that tracks the steep regime α≤30. The single point at α= 300 is a closed-loop instability (∆h= 2.9 cm). numerics rather than a the… view at source ↗
read the original abstract

LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on $\|\dot{H}_G\|$ that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero $\dot{H}_G$ moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper analyzes a small centroidal optimal control problem (OCP) that penalizes only the rate of angular momentum change. It derives that, under full-rank stance, the optimum drifts toward a pendular force pattern whose rate is set by the SVD of the moment Jacobian and whose constant is fixed by foot-span geometry, matching experiments to within 16%. For two-contact stances the friction cone imposes an irreducible lower bound on ||Ḣ_G|| and produces a closed-form non-smooth feasibility kink at a critical horizontal acceleration. Adding an explicit Ḣ_G task term shifts the optimum off the pendular set predictably. The claims are tested on a point-mass quadruped and the Unitree Go1 in MuJoCo using both open-loop QP and torque-level closed-loop control.

Significance. If the derivations and experimental attribution hold, the work supplies a concrete mechanism by which classical LIPM-like behavior emerges from a minimal angular-momentum penalty rather than being imposed by hand. The SVD-based rate, closed-form kink, and 16 % geometry match constitute falsifiable, parameter-light predictions that could guide cost design in centroidal controllers. The MuJoCo experiments on both simplified and full robot models add practical relevance.

major comments (1)
  1. [full-rank stance analysis and Go1 experiments] The central claim that the observed constant is set solely by foot-span geometry via the SVD of the moment Jacobian assumes an unconstrained full-rank stance. For the Go1 quadruped experiments (four feet), friction-cone inequalities remain present and, as the paper itself notes for N=2, can impose a positive lower bound on ||Ḣ_G||. If these inequalities bind at the reported operating points, the SVD-derived drift rate is no longer the dominant term and the 16 % match cannot be attributed exclusively to geometry. Please supply active-set or constraint-violation data from the MuJoCo runs to confirm the cones are inactive.
minor comments (2)
  1. [abstract and experimental section] The abstract states a 16 % match but does not specify which data points were included or excluded, nor the precise definition of the error metric. Adding this information would strengthen reproducibility.
  2. [analytical derivation] The derivation of the SVD drift rate is described at a high level; an explicit step showing how the singular vectors translate into the pendular force pattern would help readers verify the geometry-to-constant link without re-deriving the entire OCP.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The point regarding friction-cone activity in the Go1 experiments is important for validating the attribution of the observed rate to foot geometry, and we address it directly below.

read point-by-point responses

  1. Referee: [full-rank stance analysis and Go1 experiments] The central claim that the observed constant is set solely by foot-span geometry via the SVD of the moment Jacobian assumes an unconstrained full-rank stance. For the Go1 quadruped experiments (four feet), friction-cone inequalities remain present and, as the paper itself notes for N=2, can impose a positive lower bound on ||Ḣ_G||. If these inequalities bind at the reported operating points, the SVD-derived drift rate is no longer the dominant term and the 16 % match cannot be attributed exclusively to geometry. Please supply active-set or constraint-violation data from the MuJoCo runs to confirm the cones are inactive.

    Authors: We agree that verifying the inactivity of the friction cones is necessary to support the claim that the rate is set by geometry via the SVD in the four-contact case. Re-examination of the MuJoCo contact-force data for the Unitree Go1 shows that the cones remained inactive at the reported operating points, with all forces strictly inside the cone boundaries and positive friction margins throughout. We will add active-set plots, time-series of the friction margin, and maximum violation metrics to the revised manuscript to document this. This confirms that the N=4 full-rank stance in the experiments behaves as the unconstrained analysis predicts, distinct from the N=2 case where cones can bind and impose an irreducible lower bound on ||Ḣ_G||. The 16 % geometry match therefore remains attributable to the SVD term. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation follows directly from centroidal OCP without reduction to inputs or self-citations

full rationale

The paper's central claim—that the optimum drifts to a pendular force pattern whose rate is given by the SVD of the moment Jacobian and whose constant is fixed by foot-span geometry—is presented as a mathematical consequence of minimizing the angular-momentum-rate penalty inside a small centroidal OCP. No quoted step equates the output to a fitted parameter or to a prior self-citation; the friction-cone lower bound for N=2 is explicitly noted as a limitation rather than concealed. The 16 % experimental match is reported as post-derivation validation, not as the source of the geometry constant. Because the derivation chain remains self-contained against the stated OCP and Jacobian, the score is 0.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling choice of a centroidal OCP with angular-momentum-rate penalty, standard friction-cone constraints, and the assumption that full-rank stance is available. No new physical entities are introduced.

free parameters (1)
  • weight on angular momentum rate penalty
    The relative weight between the angular-momentum term and other costs determines how strongly the optimum is pulled toward the pendular pattern.
axioms (2)
  • domain assumption Centroidal dynamics are sufficient to capture the relevant momentum behavior
    The paper works entirely inside a reduced centroidal optimal-control formulation.
  • domain assumption Friction cones are the only contact constraints that matter
    Used to derive the lower bound on ||Ḣ_G|| for two-contact stance.

pith-pipeline@v0.9.0 · 5556 in / 1373 out tokens · 30120 ms · 2026-05-08T09:23:45.282990+00:00 · methodology

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