On the Emergence of Pendular Structure in Multi-Contact Locomotion

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.

• The paper shows that penalizing angular-momentum rate drives full-rank multi-contact systems toward pendular motion at an O(1/α) rate, with simulations producing a 227× residual reduction and an LIPM fit of R² = 0.954.
• Two-contact stances cannot fully cancel angular-momentum residuals because rank deficiency and friction constraints create an α-independent floor, matching the predicted Go1 geometric residual of 0.2835 m²/s² in static tests.
• A nonzero angular-momentum tracking objective shifts the optimum away from the pendular manifold by λ/(α+λ), while closed-loop experiments reveal additional errors from impacts, contact transitions, compliance, and actuator limitations.

The linear inverted pendulum model (LIPM) underpins most legged-locomotion control stacks, yet it is almost always introduced as a modeling assumption rather than derived from the controller's own objective. The paper "On the Emergence of Pendular Structure in Multi-Contact Locomotion" (2605.05707) addresses this gap directly: it asks what cost structure makes a pendular force pattern optimal, and where that structure necessarily breaks. The answer is developed through a small centroidal optimal control problem (OCP) with a quadratic penalty on angular momentum rate $\dot{L}$, and is validated on a point-mass quadruped and a Unitree Go1 in MuJoCo.

Problem formulation

The setting is centroidal dynamics in the sense of Orin et al.: Newton–Euler equations relate CoM acceleration and net wrench to contact forces $f_i$ at points $r_i$, each constrained to its friction cone. The OCP minimizes, over trajectories,

$$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$

subject to centroidal dynamics, friction constraints, and boundary conditions. The pendular manifold $\mathcal{P}(n,h)$ collects states where $\dot{L}=\bm{0}$ and the net force is co-linear with the foot-to-CoM vector, $F=(mg/h)(c-p)$. The analysis proceeds by cases: full-rank stance ($N=4$), rank-deficient stance ($N=2$), and task-dominated costs ($\lambda>0$).

Asymptotic emergence under full-rank stance

The central result is that with full-rank moment Jacobian, the optimum collapses onto the pendular set at rate $f_i$0, with matching upper and lower bounds. The proof has three parts: a pointwise certificate showing the pendular force uniquely minimizes $f_i$1 for prescribed horizontal acceleration; an asymptotic bound showing $f_i$2 and pendular deviation bounded by $f_i$3; and a tightness argument showing any trajectory with pendular deviation incurs at least proportional angular-momentum cost.

A refinement makes the constant explicit via the SVD of the moment Jacobian $f_i$4: for $f_i$5, the residual scales as $f_i$6 with $f_i$7, where $f_i$8 are excitation projections of gravity and inertia. For a rectangular four-foot polygon, the singular values are exactly the geometric foot spans ($f_i$9, $r_i$0, $r_i$1). This yields a practical weight-selection rule, $r_i$2, requiring only one calibration sweep.

Two classical objects then fall out for free on the pendular manifold: the ZMP equals the virtual pivot $r_i$3, and the DCM satisfies the orbital equation $r_i$4. The implication is that ZMP/DCM controllers are implicitly solving this OCP at high $r_i$5—the pendular structure is what the cost prefers, not merely an imposed template.

Lower bounds under N=2 stance

With two contacts, the moment operator drops to rank one, and the picture changes qualitatively. A residual along the foot-separation direction $r_i$6 is uncancellable by any force redistribution, giving an $r_i$7-independent floor:

$r_i$8

For Go1 diagonal trot, the pendular direction tilts $r_i$9 from vertical while the friction cone half-angle is only $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$0 at $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$1, so the QP saturates at the cone boundary. Empirically the $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$2-projection captures 55–98% of the residual across excitation directions, capping cancellation at $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$3–$$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$4 versus roughly $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$5 for four-foot stance.

A second obstruction appears in the perpendicular channel: the min-norm canceller $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$6 may violate the friction cone, in which case even the geometric floor is unreachable. The transition occurs at a closed-form critical fore–aft acceleration $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$7, producing a non-smooth kink ($$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$8 but not $$\int_0^T \Big[\alpha^2\|\dot{L}\|^2 + \lambda\|L - L^\mathrm{task}\|^2 + \beta(n^\top\ddot{c})^2 + \gamma\sum_i\|f_i\|^2\Big]dt,$$9) in the achievable infimum. For Go1 parameters this gives $\mathcal{P}(n,h)$0. The authors are appropriately cautious here: they note this value sits below speeds where real quadrupeds visibly struggle, so they do not read the match quantitatively, and the transition does not appear when sweeping $\mathcal{P}(n,h)$1 at fixed acceleration—only along the parameter driving the perpendicular moment across the cone.

Task-dominated regime

Adding a tracking term for nonzero angular momentum yields a pointwise result: the optimal residual is $\mathcal{P}(n,h)$2, sliding smoothly off the pendular set as $\mathcal{P}(n,h)$3 grows. A corollary shows that if the task demands $\mathcal{P}(n,h)$4 with $\mathcal{P}(n,h)$5, no feasible trajectory lies on the pendular manifold. This formalizes what is otherwise a fuzzy switch between balance-dominated and task-dominated locomotion: the same robot can appear LIPM-like standing and manifestly non-pendular during agile motion, with only the ratio $\mathcal{P}(n,h)$6 moving.

Numerical validation

Five experiments test the theory:

Test	Setting	Prediction	Outcome
A	Point-mass OCP	$\mathcal{P}(n,h)$7 rate	Slope $\mathcal{P}(n,h)$8; $\mathcal{P}(n,h)$9 reduction
B	Go1 $\dot{L}=\bm{0}$0 QP	Geometric constant $\dot{L}=\bm{0}$1	$\dot{L}=\bm{0}$2 vs. $\dot{L}=\bm{0}$3 (~16%)
C	Go1 $\dot{L}=\bm{0}$4 QP	Geometric floor	$\dot{L}=\bm{0}$5 vs. $\dot{L}=\bm{0}$6 (four digits)
D	Go1 trot MPC	Floor survives actuation	$\dot{L}=\bm{0}$7 (~$\dot{L}=\bm{0}$8 static floor)
E	Point-mass ZMP	ZMP = pivot	5.3 mm plateau

Tests A–C track predictions closely. In Test A, at $\dot{L}=\bm{0}$9 the optimal forces are co-linear with $F=(mg/h)(c-p)$0 to within 5.9% of $F=(mg/h)(c-p)$1, and an LIPM fit gives $F=(mg/h)(c-p)$2—the LIPM emerges without being imposed. In Test B, the singular values match foot spans to four decimals, which the authors identify as the part of the scaling proposition they trust most; the 16% gap in $F=(mg/h)(c-p)$3 is not fully accounted for (windowing differences explain part but not all).

Test D is the most informative discrepancy. Closed-loop torque-level MPC at 100 Hz reproduces the qualitative behavior—an $F=(mg/h)(c-p)$4-independent floor that no weight tuning breaks—but the empirical floor of $F=(mg/h)(c-p)$5 sits roughly $F=(mg/h)(c-p)$6 above the quasi-static prediction of $F=(mg/h)(c-p)$7. Swing impacts, contact transitions, and servo-tracking error all contribute, none obviously dominant. A single outlier at $F=(mg/h)(c-p)$8 (CoM oscillation of 2.9 cm) suggests a high-$F=(mg/h)(c-p)$9 tracker fighting the floor and losing on height stability—a regime outside the open-loop analysis. In Test E, the ZMP–pivot deviation plateaus at ~5 mm rather than continuing to fall; the authors suspect numerics (SLSQP tolerances, discretization) rather than a theoretical obstruction, but have not run the experiment that would settle it.

Limitations and open questions

The analysis is explicitly quasi-static: it treats $N=4$0 as the controlled quantity and ignores coupling between contact transitions, swing-leg inertia, and finite control rate. On hardware, compliance, foot slip, and friction heterogeneity appear large enough to overwrite static residuals—the Test D gap is consistent with this. The $N=4$1 analysis assumes ideal point contacts; finite contact patches or ankle torque may relax the floor. Theorem 4 is pointwise, leaving horizon trade-offs open. Flight phases and large height variations are outside the model entirely, and there is no closed-loop counterpart to the emergence theorem—which is precisely what would be needed to bound the Test D gap. Whether the $N=4$2 ratio serves as a useful runtime signal for mode switching remains unchecked.

Conclusion

This paper converts the LIPM from an assumed template into a consequence of a specific cost structure. Under full-rank stance, pendularity emerges at $N=4$3 with a constant readable from foot-span geometry; under two-contact stance, friction imposes an unavoidable floor with a non-smooth feasibility kink; and a task term shifts the optimum off the pendular set by the transparent factor $N=4$4. The static numerical tests validate the theory tightly, while the closed-loop results honestly expose where the asymptotic account ends—and thereby delineate exactly which questions remain open.