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On the Emergence of Pendular Structure in Multi-Contact Locomotion

This paper establishes a theoretical link between angular momentum minimization in optimal control and the emergence of pendular locomotion patterns, deriving closed-form conditions for their optimality and validating these findings through simulations and experiments on a quadruped robot while highlighting the limitations of asymptotic approximations in closed-loop control.

Original authors: Lingxue Lyu, Zihui Liu

Published 2026-05-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lingxue Lyu, Zihui Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot dog how to walk. For years, engineers have used a simple mental model called the "Linear Inverted Pendulum Model" (LIPM). Think of this model like a child on a swing: the robot's body is the seat, and the legs are the chains. The theory assumes the robot's center of mass naturally wants to swing back and forth in a smooth, pendulum-like arc.

For a long time, engineers just assumed this pendulum behavior was the best way to walk and built their controllers on top of that assumption. But this paper asks a different question: Does the math actually want the robot to swing like a pendulum, or did we just force it to?

The authors ran a series of experiments to see what happens when they let a robot solve a walking problem from scratch, without forcing the pendulum shape, but simply by asking it to "stay balanced" and "not spin out of control."

Here is what they found, broken down into everyday concepts:

1. The "Perfect" Walk (When All Four Feet Are Down)

Imagine a robot standing still or walking slowly with all four paws on the ground. The researchers gave the robot a goal: "Minimize the effort required to keep your body from spinning."

They found that when the robot has a solid, four-footed stance, the math naturally pushes the robot toward that pendulum shape. It's not because they told it to; it's because that shape is the most efficient way to stay balanced.

The Analogy: Think of a tightrope walker. If they have a long pole (representing the four feet), they naturally find a balance point that feels like a pendulum. The paper proves that if you have enough "grip" (four feet), the most efficient path is almost exactly the pendulum path we've been using for years. The more you prioritize balance, the closer the robot gets to this perfect swing.

2. The "Two-Foot" Problem (The Trot)

Now, imagine the robot starts trotting, lifting two diagonal legs off the ground so only two feet are touching the floor at a time. This is where the magic breaks.

The paper discovered that when a robot is on only two feet (like a diagonal trot), it hits a "hard floor" on how well it can balance. No matter how much you tune the robot's settings, it cannot achieve that perfect, zero-spin pendulum motion.

The Analogy: Imagine trying to balance a broom on your hand while standing on one foot versus standing on two feet. On two feet, you have a limit to how much you can lean before you slip. The robot's "friction cone" (the limit of how hard it can push against the floor without slipping) creates a barrier.
The Result: There is a specific speed and direction where the robot cannot stop spinning, no matter how smart the controller is. It's like trying to turn a car on a patch of ice; eventually, physics says you will slide. The paper calculates exactly where that "slip point" happens.

3. The "Task" vs. "Balance" Switch

Sometimes, a robot needs to do more than just balance; it might need to jump, turn sharply, or recover from a push. The paper looked at what happens when you add a "task" (like "move your body this way") to the "balance" goal.

They found a simple rule: The robot's behavior is a smooth blend between "just balancing" (pendulum) and "doing the task" (non-pendulum).

The Analogy: Think of a dimmer switch. If you turn the "balance" knob up high, the robot acts like a pendulum. If you turn the "task" knob up high, the robot stops swinging and starts doing exactly what you told it to do, even if it looks awkward or unbalanced. The robot doesn't need to switch modes; it just slides along a spectrum based on which goal is more important at that moment.

4. The Real-World Test

The authors tested these theories on a real robot (the Unitree Go1) in a simulation.

The Good News: When the robot had all four feet down, the simulation matched their math perfectly. The robot naturally found the pendulum shape.
The Bad News: When the robot was trotting (two feet down), the real robot couldn't get as balanced as the simple math predicted. The "floor" was higher than expected.
Why? The paper suggests that in the real world, things like the robot's joints moving, the impact of feet hitting the ground, and the robot's legs swinging add extra "noise" that the simple math didn't account for. It's like the difference between a perfect marble rolling on a glass table and a real ball rolling on a bumpy sidewalk.

Summary

This paper confirms that the "pendulum" way of walking isn't just a convenient guess engineers made; it is actually the mathematically optimal way to walk if you have a solid, four-footed stance. However, once you start trotting on two feet, physics puts a hard limit on how balanced you can be, and you can't tune your way out of it.

The authors essentially wrote a manual that explains why robots walk the way they do, showing us exactly when the "pendulum" rules apply and when the "friction limits" take over.

Technical Summary: On the Emergence of Pendular Structure in Multi-Contact Locomotion

Problem Statement

Linear Inverted Pendulum Model (LIPM) is a ubiquitous assumption in legged locomotion control, often imposed as a modeling constraint rather than emerging naturally from the controller's objective. While extensions exist for variable height or angular momentum, the pendular form is typically added "by hand." This paper investigates a narrower question: What specific objective function causes the pendular force pattern to emerge as the optimal solution, and where does this picture break down?

The authors analyze a small centroidal Optimal Control Problem (OCP) that penalizes the rate of change of angular momentum ( $\|\dot{H}_G\|$ ) alongside standard terms for contact forces and normal acceleration. The goal is to determine if minimizing angular momentum rate naturally drives the system toward the pendular manifold (where $\dot{H}_G = 0$ ) and to characterize the limits of this behavior under different stance configurations (full-rank vs. $N=2$ ) and task requirements.

Methodology

The paper formulates a Net-Wrench OCP (Definition 1) minimizing a cost function composed of:

Angular Momentum Penalty: $\alpha \|\dot{H}_G\|^2$
Task Tracking: $\lambda \|\dot{H}_G - \dot{H}_{task}\|^2$ (for non-balance tasks)
Normal Acceleration Penalty: $\beta (n^\top \ddot{c})^2$
Force Regularization: $\gamma \sum \|f_i\|^2$

The analysis proceeds in three theoretical stages, validated by numerical experiments on a point-mass quadruped and the Unitree Go1 robot in MuJoCo (both open-loop QP and closed-loop torque-level MPC):

Full-Rank Stance Analysis: Using Singular Value Decomposition (SVD) of the moment Jacobian, the authors derive the asymptotic behavior as the angular momentum weight $\alpha \to \infty$ .
$N=2$ Stance Analysis: Investigating the rank-deficient case (e.g., diagonal trot), where the moment operator has rank 1, leading to unavoidable residuals and friction cone constraints.
Task-Driven Analysis: Examining how adding a task-tracking term ( $\lambda > 0$ ) shifts the optimum away from the pendular manifold.

Key Contributions and Theoretical Results

1. Emergence of Pendular Structure in Full-Rank Stance

Theorem 1: For full-rank stances, the optimal solution asymptotically collapses toward the pendular force pattern ( $\dot{H}_G \to 0$ ) at a rate of $O(1/\alpha)$ .
Geometric Constant: The constant of proportionality is determined by the SVD of the moment Jacobian, which corresponds directly to the foot-span geometry.
Connection to Classical Models: On this pendular manifold, the Zero Moment Point (ZMP) naturally equals the virtual pivot point, and the Divergent Component of Motion (DCM) orbital equation emerges as a consequence, rather than an assumption.

2. Fundamental Limits in $N=2$ Stance (Diagonal Trot)

Theorem 5 (Geometric Floor): In $N=2$ stances, the moment Jacobian is rank-1. A residual angular momentum rate remains along the direction of the line connecting the two feet ( $\hat{D}$ ), regardless of force redistribution. This creates a non-zero lower bound (floor) for $\|\dot{H}_G\|$ .
Theorem 6 (Friction Cone Kink): A second obstruction arises from the friction cone. If the force required to cancel the perpendicular moment component falls outside the friction cone, a "non-smooth kink" occurs at a critical horizontal acceleration ( $a_x^*$ ). Beyond this point, the achievable $\|\dot{H}_G\|$ strictly exceeds the geometric floor.
Implication: In diagonal trot, the pendular direction often tilts outside the friction cone (e.g., $40.6^\circ$ vs. $31^\circ$ for $\mu=0.6$ ), making the pendular solution infeasible and creating an $\alpha$ -independent floor.

3. Task-Driven Deviation

Theorem 7: When a task term ( $\lambda > 0$ ) is added, the optimal angular momentum rate becomes a smooth interpolation: $\dot{H}_G^* = \frac{\lambda}{\alpha + \lambda} \dot{H}_{task}^*$ .
Phase Diagram: The system transitions smoothly from a "balance-dominated" regime (pendular, $\lambda/\alpha \ll 1$ ) to a "task-dominated" regime (non-pendular, $\lambda/\alpha \gg 1$ ). This explains why the same robot can appear LIPM-like in static balance but non-pendular during agile maneuvers without switching controllers.

Experimental Results

The authors validated these claims using the Unitree Go1 and a point-mass model:

Rate Verification (Point-Mass): As $\alpha$ increased, the deviation from the pendular force pattern dropped by a factor of 227, with a log-log slope of $-1$, confirming the $O(1/\alpha)$ rate (Theorem 1).
Geometric Constant (Go1 N=4): In a four-foot stance, the empirical scaling constant ( $K_e = 8.4$ ) matched the analytical prediction derived from foot spans ( $K_a = 9.8$ ) within $\sim 16\%$ . The singular values of the moment Jacobian matched the physical foot spans to four decimal places.
Floor Verification (Go1 N=2): In diagonal trot, $\|\dot{H}_G\|$ saturated at a predicted geometric floor of $0.2835 \, \text{m}^2/\text{s}^2$ , matching the analytical bound to four digits.
Closed-Loop Reality: In a closed-loop torque-level MPC, the floor persisted but was higher ( $\approx 1.71 \, \text{m}^2/\text{s}^2$ ) than the static prediction. The authors attribute this gap to dynamic effects (swing impacts, contact transitions, servo errors) not captured in the quasi-static analysis.
ZMP Identity: The numerical experiments confirmed that on the pendular manifold, the force-based ZMP converges to the virtual pivot, validating the theoretical link between LIPM and the OCP optimum.

Significance and Claims

The paper claims to make the link between LIPM and optimal control explicit and checkable. It argues that:

LIPM is a consequence, not just a choice: The pendular pattern emerges naturally when minimizing angular momentum rate in full-rank stances, provided the weight $\alpha$ is sufficiently high.
Limits are predictable: The breakdown of the pendular assumption in $N=2$ stances is not a failure of the controller but a fundamental geometric and frictional limitation. The paper provides closed-form expressions for the lower bounds and the critical acceleration where friction constraints bind.
Unified View of Balance and Task: The ratio $\lambda/\alpha$ provides a continuous parameter explaining the transition between balance-centric (LIPM-like) and task-centric (agile) behaviors.

The authors remain modest regarding the closed-loop results, noting that while the qualitative behavior (the existence of a floor) holds in dynamic simulations, the quantitative gap between static theory and closed-loop reality is significant and likely due to unmodeled dynamics (compliance, slip, finite control rates). They do not claim to solve these dynamic gaps but rather to define the static boundaries within which those dynamics operate.