Agile asymmetric multi-legged locomotion: contact planning via geometric mechanics and spin model duality

TL;DR

This work addresses the challenge of coordinating high-DOF, contact-rich locomotion in multi-legged robots by marrying geometric mechanics with spin-model duality to discover asymmetric, performance-enhancing gaits. The authors transform contact and shape-change planning into a graph optimization and exploit Potts-Ising duality via domain-wall methods to solve for optimal gaits efficiently. Their hexapod experiments reveal novel asymmetric gaits that outperform conventional baselines and even allow certain legs to be fixed as rigid parts without loss of speed. The framework is validated through simulations and robophysical experiments, demonstrating both higher forward speeds and insight into symmetry-breaking locomotion, with potential as priors for learning-based controllers and for morphology–control co-design. Overall, the paper provides a principled, scalable approach to leveraging additional legs for agile locomotion and sets directions for extending the method to uncertain environments and larger platforms.

Abstract

Legged robot research is presently focused on bipedal or quadrupedal robots, despite capabilities to build robots with many more legs to potentially improve locomotion performance. This imbalance is not necessarily due to hardware limitations, but rather to the absence of principled control frameworks that explain when and how additional legs improve locomotion performance. In multi-legged systems, coordinating many simultaneous contacts introduces a severe curse of dimensionality that challenges existing modeling and control approaches. As an alternative, multi-legged robots are typically controlled using low-dimensional gaits originally developed for bipeds or quadrupeds. These strategies fail to exploit the new symmetries and control opportunities that emerge in higher-dimensional systems. In this work, we develop a principled framework for discovering new control structures in multi-legged locomotion. We use geometric mechanics to reduce contact-rich locomotion planning to a graph optimization problem, and propose a spin model duality framework from statistical mechanics to exploit symmetry breaking and guide optimal gait reorganization. Using this approach, we identify an asymmetric locomotion strategy for a hexapod robot that achieves a forward speed of 0.61 body lengths per cycle (a 50% improvement over conventional gaits). The resulting asymmetry appears at both the control and hardware levels. At the control level, the body orientation oscillates asymmetrically between fast clockwise and slow counterclockwise turning phases for forward locomotion. At the hardware level, two legs on the same side remain unactuated and can be replaced with rigid parts without degrading performance. Numerical simulations and robophysical experiments validate the framework and reveal novel locomotion behaviors that emerge from symmetry reforming in high-dimensional embodied systems.

Figures (7)

• Figure 1: Multi-legged robot: a hexapod model. (a) Geometric and (b) robophysical hexapod model. (c) Two physics-informed alternative designs in which two legs are replaced by non-actuated rigid appendages, achieving forward locomotion performance identical to the fully actuated hexapod. (d) Field deployment under a tomato plant.
• Figure 2: Geometric mechanics of hexapod locomotion. (a) Geometric illustration of hexapod body bending angles $\alpha_1$ and $\alpha_2$ and body velocity ($\xi_x$, $\xi_y$, and $\xi_\theta$) (b) Shape space of the geometric model parameterized by $\alpha_1,\alpha_2$. (c.1) Local connection vector field mapping infinitesimal shape velocities to body-frame velocities. (c.2) Height function over shape space, given by the curl of the divergence-free component of the connection vector field; the net displacement for a closed gait is approximated by the signed area integral of the height function over the enclosed loop. (c.3) Potential function computed from the curl-free component of the connection vector field. The displacement associated with a shape change (e.g., from shape A to shape B) is gait-path-independent and determined by differences in function values between endpoints. An order-of-magnitude separation exists between the height function and the potential function over the same shape space. All panels share identical axes. (d) Experiment trajectory of a hexapod robot with body undulation evolving without contact changes. In the absence of contact modulation, the robot undergoes oscillatory body motion with peak-to-peak displacement of approximately $0.27$ body lengths (BL), while the net displacement over one cycle remains negligible ($<0.06$ BL).
• Figure 3: Gait path mapping. (a) We sample the shape space with a collection of shape variables $\{r_i\}$. $\chi_3$ is an example path connection of $r_3$ to $r_4$, only a fraction of a closed-loop gait path. (b) An example of a cheating gait path is colored in black. The unique non-cheating gait path is illustrated in red. (c) (top) Mapping between robot contact pattern and Potts model. (bottom) Duality between Potts model and Ising model.
• Figure 4: Comparison of hexapod forward locomotion. (Left) Time-lapse snapshots of the robot executing the prescribed gaits. (Right) Corresponding commanded leg–ground contact patterns. (a) Physics-informed asymmetric gait exhibiting slow, long-duration counterclockwise whole-body yaw (from $t=0$ to $\approx0.66T$) followed by rapid, short-duration clockwise yaw, resulting in net forward translation. Hind legs remain in contact throughout gait cycle. (b) Alternative physics-informed gait in which the front-left and hind-left legs remain in continuous contact throughout the cycle, yet still achieve high forward performance. Similar long-duration counterclockwise and short-duration clockwise whole-body yaw to (a). (c) Best bio-inspired gait based on an alternating tripod contact pattern. (d) Best learned open-loop gait obtained via reinforcement learning. (e) Best extended quadrupedal gait, in which the third body segment follows the same kinematics as the first two segments (effectively reducing the system to a quadruped). Corresponding to $\phi_{\alpha}=\pi$.
• Figure 5: Agile locomotion at high speed. (a) (a.1) Absolute speed and (a.2) step length (displacement per gait cycle) for the bio-inspired and physics-informed gaits as a function of gait frequency. Data above 1 Hz are excluded from quantitative analysis due to motor limitations that cause substantial deviation between commanded and executed position (see SI, Sec. S3); shown as hollow markers. We excluded outliers from each data set with studentized residuals $> 3$. Because step length increases approximately linearly with frequency, the absolute speed scales quadratically with frequency for both gaits. Panels share identical x-axes. (b.1) Mean standard deviation of pitch angle as a function of frequency for the physics-informed gait and (b.2) for the bio-inspired gait. Panels share identical axes. (c) Experimental measurements of whole-body pitch for the physics-informed gait at low and high frequencies over two cycles. At low frequencies, the robot exhibits large pitch oscillations due to loss of static stability near unstable configurations. At high frequencies, dynamically acquired stability compensates for these perturbations, resulting in substantially reduced pitch oscillations. (d) Illustration of the effect of gait frequency on duty factor: as frequency increases, duty factor decreases, which may partially account for the larger step length observed for the bio-inspired gait at higher frequencies.

Theorems & Definitions (1)

• Theorem 1