Terrain-Aware Foot Placement for Bipedal Locomotion Combining Model Predictive Control, Virtual Constraints, and the ALIP Grant Gibson, Oluwami Dosunmu-Ogunbi, Yukai Gong, and Jessy Grizzle Abstract— This paper draws upon three themes in the bipedal control literature to achieve highly agile, terrain-aware locomo- tion. By terrain aware, we mean the robot can use information on terrain slope and friction cone as supplied by state-of-the-art mapping and trajectory planning algorithms. The process starts with abstracting from the full dynamics of a Cassie 3D bipedal robot, an exact low-dimensional representation of its centroidal dynamics, parameterized by angular momentum. Under a piecewise planar terrain assumption, and the elimination of terms for the angular momentum about the robot’s center of mass, the centroidal dynamics become linear and has dimension four. Four-step-horizon model predictive control (MPC) of the centroidal dynamics provides step-to-step foot placement com- mands. Importantly, we also include the intra-step dynamics at 10 ms intervals so that realistic terrain-aware constraints on robot’s evolution can be imposed in the MPC formulation. The output of the MPC is directly implemented on Cassie through the method of virtual constraints. In experiments, we validate the performance of our control strategy for the robot on inclined and stationary terrain, both indoors on a treadmill and outdoors on a hill. I. I NTRODUCTION This paper contributes to the growing literature on terrain- aware, also known as terrain-adaptive, locomotion. Our ob- jective is to design a gait controller that enables an agile bipedal robot, such as Cassie in Fig. 1, to traverse terrain that is varying in height and surface friction at “speed”, meaning locomotion “as close to the planned velocity as the physical limits of the robot and terrain allow”. We will assume that the robot is provided a local map that specifies (a) terrain height, that is, z as a function of x and y in a region about the robot, and (b) a C 1 vector field of desired velocity (speed, heading, and yaw rate) as a function of the robot’s current pose and velocity. The integral curves of the vector field provide a family of paths that the robot may follow to reach a goal that is unknown to the local gait controller. The vector field may come from a reactive planner, as in [1], [2], or through a human operator and an RC-controller, as in this paper. We make a key simplifying assumption on the terrain, namely, that over distances on the order of the step length of the robot, it can be piecewise approximated by planes, with allowed jumps at the boundaries. This admittedly vague assumption will be made more precise in Sect. II, where we model the centroidal dynamics of the robot. The gait controller will plan N robot-steps ahead, where N is a small number such as four, and will assume that beyond this planning horizon, the terrain is flat with a nominal friction The authors are with the College of Engineering and the Robotics Institute, University of Michigan, Ann Arbor, MI 48109 USA {grantgib,grizzle}@umich.edu coefficient. More realistic assumptions on the terrain beyond the gait-controller’s planning horizon will be made in the near future when this controller is integrated with the reactive planner and a full perception and mapping system introduced in [3], [4]. The length of the gait-controller’s planning horizon is limited by the need for real-time computations with a model that contains more information on dynamics and constraints than is currently used in trajectory/motion planning methods. A. Related Work Blurring the Boundary between Gait Control and Tra- jectory Planning: As emphasized in [5], the separation of vehicle control into independently designed path planner and a low-level “speed and direction” controller inevitably results in performance degradation. The proposed remedy was an integrated system based on Model Predictive Control (MPC). Related work in the area of legged robots includes the quadrupedal robot Anymal B in [6], which achieves terrain-aware foothold planning by tightly coupling locally optimal footholds and center of mass velocity with a terrain height map. The authors show how to voxelize the terrain map and perform a real-time grid search to optimize foot placement. Switching Control Based on One-step Ahead Terrain Pro- file: Terrain aware locomotion of a simulated 3D humanoid is achieved in [7]. First, in an offline stage, a library is built that includes five periodic gaits and a set of transition gaits Fig. 1: Cassie Blue using a 3D-ALIP inspired MPC foot placement controller to walk sideways up a 22 o incline on wet grass. Lateral walking with Cassie is much more difficult than longitudinal walking due to tight workspace constraints, which are accounted for in our formulation. arXiv:2109.14862v1 [cs.RO] 30 Sep 2021