Optimization-based motion planning for legged robots

Legged machines have the potential to traverse terrain that wheeled robots cannot. These capabilities are useful in scenarios such as stairs in homes or debris-filled disaster scenes, such as earthquake areas. This thesis develops one of the algorithms necessary to achieve such a task, a motion-planner, which translates a high-level task into a desired motion-plan for a legged robot to execute. A core difficulty in legged locomotion is that a forward body movement cannot be directly generated, but results from contact forces between the end-effectors and the environment. Furthermore, there are various physical restrictions on the contact forces that can be generated. These can make hand-crafting valid motions for all the interdependent quantities such as body motion, end-effector motion and contact forces tedious and even infeasible for complex tasks. Alternatively, Trajectory Optimization can be utilized to generate motions in a more general, automated way. Only a high-level task is specified, while the algorithm determines the motions and forces taking into account the physical restrictions of legged locomotion. This approach is attractive because once the problem has been properly modeled, the algorithm would, in an ideal case, produce motions for any high-level task, solving legged locomotion planning on a general level. This thesis presents approaches to transcribe, using e.g. collocation, the legged locomotion problem into a mathematical optimization problem, then solvable by off-the-shelf software. The three developed motion-planning algorithms generate motion-plans for increasingly complex terrains and tasks. Simulation and experiments on the quadruped robots HyQ and ANYmal verify the physical correctness of the motion-plans. The final formulation and main contribution of this thesis holistically determines the gait- sequence, step-timings, footholds, contact forces, swing-leg motions and 6-dimensional body motion, given a desired goal state and 2.5D height map of the non-flat terrain. We model the robot as a single rigid body (SRBD) controlled by the contact forces at the end- effectors. Our novel phase-based parameterization of end-effector motion and forces allows optimizing over the discrete gait sequence using only continuous decision variables. The algorithm efficiently generates highly dynamic motion-plans with flight-phases for legged systems, such as monopeds, bipeds, and quadrupeds.