This paper presents Optimally Scaled Hip-Force Planning (OSHP), a novel approach to controlling the body dynamics of running robots. Controllers based on OSHP form the high-level component of a hierarchical control scheme in which they direct lower level controllers, each responsible for coordinating the motion of a single leg. An OSHP controller takes in the state of the runner at the apex of its primary aerial phase and returns desired profiles for the vertical and horizontal forces to be exerted at each hip during the subsequent stride. The hip force profiles returned by OSHP are scaled variants of nominal force profiles based on biological ground reaction force data. The OSHP controller determines the scaling parameters for these profiles through constrained nonlinear optimization on an approximate model of the runner's body dynamics. Evaluation of an OSHP controller for a quadruped model in simulation shows that even with very simple leg controllers, the OSHP controller can accelerate the runner from rest to steady-state running without a pre-defined footfall sequence.

This paper presents the relatively rich and interesting bifurcation structure that is present in the nature of optimal solutions to a multi-robot formation control problem. The problem considered is a two point nonlinear boundary-value problem that can only be solved numerically. Since common numerical solution techniques such as the shooting method are local in nature and hence are difficult to use to find multiple solutions, an alternative formulation of the problem is presented that can be solved through homotopy methods for polynomial systems. These methods are guaranteed to find all solutions within the resolution of the system description's discretization. Specifically, this paper studies a group of unicycle-like autonomous mobile robots operating in a 2-dimensional obstacle-free environment. Each robot has a predefined initial state and final state and the problem is to find the optimal path between two states for every robot. The path is optimized with respect to the control effort and the deviation from a desired formation. The bifurcation parameter is the relative weight given to penalizing the deviation from the desired formation versus control effort. It is shown that as this number varies, bifurcations of solutions are obtained. Considering the common use of optimization methods in robotic navigation and coordination problems, understanding the existence and structure of bifurcating and multiple solutions is of great importance in robotics.