Dynamic Walking on Slippery Surfaces: Demonstrating Stable Bipedal Gaits with Planned Ground Slippage * Wen-Loong Ma 1 , Yizhar Or 2 and Aaron D. Ames 3 Abstract— Dynamic bipedal robot locomotion has achieved remarkable success due in part to recent advances in tra- jectory generation and nonlinear control for stabilization. A key assumption utilized in both theory and experiments is that the robot’s stance foot always makes no-slip contact with the ground, including at impacts. This assumption breaks down on slippery low-friction surfaces, as commonly encountered in outdoor terrains, leading to failure and loss of stability. In this work, we extend the theoretical analysis and trajectory optimization to account for stick-slip transitions at point foot contact using Coulomb’s friction law. Using AMBER-3M planar biped robot as an experimental platform, we demonstrate for the first time a slippery walking gait which can be stabilized successfully both on a lubricated surface and on a rough no-slip surface. We also study the influence of foot slippage on reducing the mechanical cost of transport, and compare energy efficiency in both numerical simulations and experimental measurements. I. INTRODUCTION Tremendous progress in realizing robust bipedal robot locomotion has been achieved in the last decade. This is in part due to successful combination of theoretical modeling and analysis using the framework of hybrid systems [1], [2], application of advanced methods of nonlinear control [3], [4], as well as careful mechanical design and hardware implementation on various experimental platforms such as AMBER-3M [5], DURUS [6] and Cassie [7]. Underlying all of these results, along with successes for robots using other paradigms such as ZMP [8], [9] and spring-loaded inverted pendulum (SLIP) based models [10], [11], is the assumption that the foot does not slip. Thus, in all of these cases, the foot acts as a stationary pivot point. While this assumption may easily hold in sterile laboratory environments where the floors can be chosen with sufficiently high friction, it becomes impractical on natural outdoor terrains, wherein there are a plethora of slippery or slightly granulated irregular surfaces. Success in challenging the stationary contact point assumption include multi-contact walking [12] and bipedal running [13], [14]. 1 W. Ma is with Mechanical Engineering, California Institute of Technol- ogy, Pasadena, CA, USA. wma@caltech.edu 2 Y. Or is with the faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, Israel. izi@technion.ac.il 3 A. Ames is with the faculty of Mechanical Engineering and Control + Dynamical Systems, California Institute of Technology, Pasadena, CA, USA. aames@caltech.edu *This work is supported by NSF grant 1724464, 1544332, 1724457, and Disney Research LA. The work has been conducted while Y. Or was hosted by A. D. Ames and AMBER lab at Caltech during his sabbatical leave from the Technion. Fig. 1: Slippage in the beginning of a step: pre-slip on the left and post-slip on the right. The goal of this paper is to address this fundamental assumption of no slippage by embracing its violation while still being able to demonstrate the ability to achieve stable walking experimentally. In legged robots, foot slippage is often treated as an external disturbance which should be avoided at the gait planning stage [15], [16], or detected and recovered in real-time by feedback control at the experimen- tal implementation stage [17], [18]. Some of the most famous examples are Boston Dynamics’ robots BigDog [19] and SpotMini [20] successfully recovering from slippage. Con- versely, legged animals across a wide range of scales show impressive adaptability to slippery surface on natural terrains. Stick insects confronted with a slippery surface modulate their motor outputs to produce normal walking gaits, despite a drastic change in the loads that these limbs experience [21]. Slippage in bipedal running of Guinea fowl has been studied in [22], showing that falling on slippery surfaces is a strong function of both speed and limb posture at touchdown. Several works in human biomechanics literature study the conditions that cause slipping [23], its consequences [24] and dynamics [25]. Finally, [26] has measured feet motion in galloping gaits of horses on outdoor racing terrains and found significant phase of hoof slippage. Recent theoretical work has incorporated slippage into classic simple planar models of legged locomotion both in passive dynamic and actuated walking — the rimless wheel [27], compass biped [27], [28] and SLIP [29]. The models use Coulomb’s friction law and account for stick- slip transitions and friction-bounded inelastic impacts, which add complexity to the system’s multi-domain hybrid dynam- ics. Investigating the influence of friction on both passive dynamics down a slope and open-loop actuated walking, it has been found in [27], [28] that upon decreasing the friction coefficient, periodic solutions with stick-slip transitions begin to evolve while their orbital stability decreases until reaching arXiv:1812.04600v1 [cs.RO] 11 Dec 2018