High Dimensional Motion Planning for Dynamic Robot Locomotion

State of the art robot controllers typically work by canceling the natural dynamics of a system and introducing some desirable (controlled) dynamics. Locomotion occurs through unilateral contact (push-not-pull) at the ground, and this naturally introduces underactuated degrees of freedom which inherently constrain the set of achievable motions. Consider for example a large humanoid robot, such as Asimo. The robot's walking controller maintains at least one flat foot on the ground at all times. This is done to approximate a fully-actuated regime of operation. However, if the foot was allowed to roll over the toe during a step, this would increase the range and speed of the gait, and would produce more natural (and likely more efficient) human-like walking. Such motions are not currently possible with Asimo at the moment, however. Similar control strategies have been applied to quadruped robots, by maintaining 3 feet in contact with the ground at all times to produce a stable support polygon. but limits the range of dynamic behaviors. LittleDog is a small position-controlled quadruped robot that has 2 motors at each shoulder and 1 at each knee; this robot has 18 Degrees of Freedom (3 for each leg, and 6 for the body pose). This means the overall system has 36 state variables (each DoF has a position and velocity that must be considered), and 12 actuators. Pure feedback control is not possible here because future constraints need to be considered carefully before executing a motion. Even the best motion planning algorithms cannot handle such a high dimensional problem. Recently, we have made progress in controlling this quadruped robot, even when it does not have a support polygon to rely on (for example, when it is on two of the four legs). The robot is then truly underactuated, because there is a passive joint at the ground contact (the robot has point feet and no ankles). Our approach is to consider the Euler-Lagrange equations of motion for the system. These governing equations turn out to have a lot of structure, which can be exploited by considering simplifications (e.g. symmetry) and dimensionality reduction. Our first approach allows us to reduce the dynamics to a simple model that relates angular acceleration of the body to Cartesian acceleration of the Center of Mass. The reduced equations can be solved to produce interesting controlled behavior, including a ""dynamic lunge"", in which the robot stands up on its hind (point-footed) legs, and then throws its front feet forward, allowing it to put its front feet on large obstacles. Our second approach is to exploit dimensionality reduction in motion planning by applying randomized search on the full system, in a space which is constrained by actions defined by a low dimensional task space. The task space is a carefully chosen mapping from state space to a lower dimensional space; we utilize center of mass of the whole robot in polar coordinates (e.g. angle and radius) as the task space. We treat the corresponding low dimensional system as if it is fully actuated, and develop a mapping from accelerations in the low dimensional space to torques in the robot actuator-space. The problem then becomes a motion planning problem, in which we must find a suitable trajectory in the 2-dimensional task space, which maps to feasible (bounded) torques of the 12 actuators of the robot. In addition, trajectories are checked to ensure that constraints in the 36-dimensional state space, such as avoiding collisions with obstacles, joint limits, and velocity limits of the actuators, are not violated. This approach results in a narrow set of solutions requiring search. However, because the search is only done over 2 degrees of freedom of the task space, it is much faster than planning over the full system. Thus, we are able to apply a Rapidly-exploring Randomized Tree (RRT) to search this space efficiently, and produce fast motion plans. During execution, the plans themselves can be feedback stabilized using Partial Feedback Linearization in Task Space.