Real-time Coupled Centroidal Motion and Footstep Planning for Biped Robots

TL;DR

The paper tackles real-time planning for dynamic bipedal robots navigating uneven terrain by coupling centroidal CoM motion with footstep planning using a SLIP-like model. It introduces a convex approximation that represents potential footholds as forces applied to the CoM, enforces at most two contacts per time step, and uses iteratively reweighted l1 minimization to approximate cardinality while accounting for terrain costs. The framework achieves fast solve times (sub-40 ms per iteration for 1.5 s horizons) and demonstrates automatic gait discovery, terrain versatility, and adaptability to different path geometries in simulation. This approach enables real-time planning that can warm-start higher-fidelity trajectory optimization and operational space control for complex terrains, with ongoing work focused on stability in receding-horizon setups and tighter integration with full-body models.

Abstract

This paper presents an algorithm that finds a centroidal motion and footstep plan for a Spring-Loaded Inverted Pendulum (SLIP)-like bipedal robot model substantially faster than real-time. This is achieved with a novel representation of the dynamic footstep planning problem, where each point in the environment is considered a potential foothold that can apply a force to the center of mass to keep it on a desired trajectory. For a biped, up to two such footholds per time step must be selected, and we approximate this cardinality constraint with an iteratively reweighted $l_1$-norm minimization. Along with a linearizing approximation of an angular momentum constraint, this results in a quadratic program can be solved for a contact schedule and center of mass trajectory with automatic gait discovery. A 2 s planning horizon with 13 time steps and 20 surfaces available at each time is solved in 142 ms, roughly ten times faster than comparable existing methods in the literature. We demonstrate the versatility of this program in a variety of simulated environments.

Figures (5)

• Figure 1: Example trajectory computed by the proposed program. Given a terrain cost map and a high-level desired trajectory, a CoM trajectory and footstep plan is generated for a SLIP-like robot model jumping over a chasm. The 4.8s horizon is solved in 0.544 s with 20 available contact locations at each time step.
• Figure 2: Methodology overview: foothold selector calculates a set of forces to apply, phase assignment groups the forces into contact phases and assigns a side to the steps. The output goes to a trajectory optimizer that calculates CoM and foot trajectories, which can then enter an operational space controller apgar2018fast to calculate motor torques.
• Figure 3: Conceptual depiction of the cumulative sum of forces to follow the desired CoM trajectory: the green patches represent traversable terrain; the red line is the desired path bounded by the gray dashed lines; the blue dot and dashed line represent the CoM trajectory; and the red arrows are the forces applied.
• Figure 4: Trials in a range of environments with different desired trajectories. The maximum CoM path deviation is constrained to 0.1 m (transparent blue circles), and the CoM path and force arrow colors relate to time. Two arrows of the same color indicate a double support phase. (a) 3s horizon, straight path on flat ground; (b) 3.4s horizon, step stones of varying heights with a gap; (c) 5.1s horizon, climbing up a staircase; (d) 5.1s horizon, bend in the path; (e) 6.1s horizon, discrete step stones. We found that the more challenging environments (b, c, and e) required 2-3 reweighting iterations to meet the cardinality constraint.
• Figure 5: Trend in solve time with increasing number of time steps for the flat ground environment with k=20, dt=0.15 s. On a log-log plot the slope is 1.7, indicating almost quadratic growth in computation time with planning time horizon. Each trial required one reweighting iteration.