Modeling multi-legged robot locomotion with slipping and its experimental validation

TL;DR

The paper addresses the challenge of modeling multi-legged locomotion with slipping by employing a momentum-ignoring local connection, expressed as $v_b = A(r) \dot r$, and a bilinear viscous-Coulomb friction ansatz. It introduces a spring-supported contact model to identify which feet contact the ground and their gravity loading, followed by a linear force/moment balance that yields planar body velocity; the approach produces a linear, tractable system even with many slipping contacts. Experimental validation on a 6-DoF-per-leg BigANT hexapod with force-torque sensors demonstrates that the viscous-Coulomb model predicts foot forces and body velocity with accuracy comparable to Coulomb friction while enabling ~50× faster computation and easy parallelization. The method generalizes to multipods (6–12 legs) and a Ghost Spirit quadruped, showing strong scalability and potential for online planning and control in field robotics. Overall, the work provides a principled, fast, and scalable framework for multi-contact legged locomotion that leverages geometric mechanics and friction modeling to enable reliable predictions in slipping regimes.

Abstract

Multi-legged robots with six or more legs are not in common use, despite designs with superior stability, maneuverability, and a low number of actuators being available for over 20 years. This may be in part due to the difficulty in modeling multi-legged motion with slipping and producing reliable predictions of body velocity. Here we present a detailed measurement of the foot contact forces in a hexapedal robot with multiple sliding contacts, and provide an algorithm for predicting these contact forces and the body velocity. The algorithm relies on the recently published observation that even while slipping, multi-legged robots are principally kinematic, and employ a friction law ansatz that allows us to compute the shape-change to body-velocity connection and the foot contact forces. This results in the ability to simulate motion plans for a large number of potentially slipping legs. In homogeneous environments, this can run in (parallel) logarithmic time of the planning horizon

Figures (15)

• Figure 1: A 1-dimensional multilegged example. Consider two or more equal-height legs, with identical friction properties moving at different horizontal velocities relative to a body constrained to move only in the horizontal direction. The resulting body velocity under Coulomb friction with an odd number of legs is the median leg speed; with an even number of legs the answer is non-unique and any speed between the two median leg speeds will work.
• Figure 2: Visualization of a 2D case. Consider a 4-legged robot (right plot) whose feet move relative to the body with known velocities (arrows, right plot). We plotted contours of the magnitude of the total force on the body under the assumption that the body is moving at body velocity $(V_x, V_y)$ without rotating, both under Coulomb friction (top left) and our viscous-Coulomb ansatz (bottom left). For Coulomb friction, this function has point discontinuities at those velocities that put any of the feet into static friction (colored dots, color matched across sub-figures). For each friction model, we also indicated the body velocity at force-moment balance (red stars). The plots demonstrate that the viscous-Coulomb ansatz gives rise to a simple quadratic, whereas Coulomb friction produces an almost everywhere smooth function with point discontinuities, yet both produce very similar solutions.
• Figure 3: Visualization of the search for contact state in a 2D "robot". We indicated the height and pitch (${p}_{{z,{0}}}$,$\alpha$) states searched (labels 0-3 of (a)), and visualized the pose and contacts of the "robot" in each of these states (corresponding labels of plots in (b)). Each "robot" leg (zigzag lines in (b)) defines a corresponding codimension 1 plane (line here) in ($z_0$,$\alpha$) at which it contacts the ground (colored lines in (a) with color same as the leg in (b), (c)). At a ${p}_{{z,{0}}}$ above the plane, the leg is in the air; below it, the leg will be in contact and generate normal forces. With each state being searched (number label in (a)), there is a closed-form solution of the force equilibrium, which we connect to that state with a line interval (black in (a)). If the equilibrium lies in the same contact state the algorithm terminates (star; step 3). Otherwise, the portion of the line segment in another contact state is counter-factual (black dashed in (a)). Instead, we switch to the new contact state and solve it again. Each such transition between contact states lies on a plane corresponding to the leg that made contact (black dot in (a); circled leg in (b)).
• Figure 4: A tripod gait trajectory of BigANT. We plotted the trajectory of the BigANT robot (top row) measured from motion capture (red). We plotted roll, pitch, and yaw angles (second row, left to right), and body velocity components along the robot axis, across the robot, and rotational (third row, left to right). We used motion capture data as ground truth for kinematics (wide red lines). The pitch and roll are the same for both friction models and arise from the spring support model (purple lines), but other outputs are different for Coulomb friction (blue lines) and our viscous-Coulomb friction (orange lines). We indicated the body location obtained from motion tracking and by integrating the body velocity predictions (rectangles in "Robot Trajectory"). We further illustrated the motion by indicating the location of the robot body frame origin (crosses) at the beginning, halfway point, and end of motion (same sub-plot) To compare observed forces to our prediction we plotted details for the Front Right foot: the ground contact force in the $x,y$-axis, and the fraction of supporting force in the $z$-axis. We used force torque sensor measurements as ground truth (same colors and line types). (For all individual foot forces and torques refer to appendix figure \ref{['fig:tripod-all']}.)
• Figure 5: We plotted: (A) Metachronal gait phase vs. motor shaft angle for all six legs. (B) Magnitude of slipping velocity vs. magnitude of planar force divided by normal force, overlaying points from all six feet. (C) Ground contact forces at the front right foot. For each individual foot refer to appendix figure \ref{['fig:slipping-all']}.