On the Emergence of Pendular Structure in Multi-Contact Locomotion

Legged robot control has long relied on the Linear Inverted Pendulum Model (LIPM) as a simplifying assumption for gait planning, but until now it was unclear why that model works so well. In a new paper on arXiv (cs.RO/2605.05707), researchers Lingxue Lyu and Zihui Liu provide a rigorous explanation by analyzing a centroidal optimal control problem that penalizes the rate of change of angular momentum. They show that under this cost, the optimal force patterns naturally converge to a pendular structure, effectively proving that the LIPM emerges from basic control principles rather than being arbitrarily imposed.

The authors derive three key results. First, with full-rank stance (all feet on ground), the optimal forces drift toward a pendular pattern at a rate determined by the singular value decomposition of the moment Jacobian, with the constant set by foot-span geometry—matching experimental data to within 16%. Second, for two-contact stances like trotting, the friction cone introduces a fundamental lower bound on the angular momentum rate that no amount of weight tuning can fix, along with a non-smooth feasibility kink at a critical horizontal acceleration. Third, adding a task term requesting nonzero angular momentum moves the optimum away from the pendular set in a predictable way. The authors test their claims on a point-mass quadruped and the Unitree Go1 in MuJoCo, using both open-loop QP and torque-level closed-loop controllers. The work not only demystifies why LIPM is such a good approximation but also provides a theoretical foundation for designing more efficient and stable locomotion controllers.