Rotary and radial forcing effects on center-of-mass locomotion dynamics

Rotary and radial forcing are two common actuation methods for legged robots. However, these two orthogonal methods of center-of-mass (CoM) forcing have not been compared as potentially alternative strategies of actuation. In this paper, we compare the CoM stability and energetics of running with rotary and radial actuation through the simulation of two models: the rotary-forced spring-loaded inverted pendulum (rotary-forced-SLIP), and the radially-forced-SLIP. We model both radial and rotary actuation in the simplest way, applying them as a constant force during the stance portion of the gait. A simple application of constant rotary forcing throughout stance is capable of producing fully-asymptotically stable motion; however, a similarly constant application of radial forcing throughout the stance is not capable of producing stable solutions. We then allow both the applied rotary and radial forcing functions to turn on or off based on the occurrence of the mid-stance event, which breaks the symmetry of actuation during stance towards a net forward propulsion. We find that both a rotary force applied in the first half of stance and a radial force applied in the second half of stance, are capable of stabilizing running. Interestingly, these two forcing methods improve the motion stability in different ways. Rotary forcing first reduces then greatly increases the size of the stable parameter region when gradually increased. Radial forcing expands the stable parameter region, but only in a moderate way. Also, it is found that parameter region stabilized by rotary and radial forcing are largely complementary. Overall, rotary forcing can better stabilize running for both constant and event-based forcing functions that were attempted. This indicates that rotary forcing has an inherent capability of stabilizing running, even when minimal time-or-event-or-state feedback is present. Radial forcing, however, tends to be more energy efficient when compared to rotary forcing. In addition, a balanced stability and energy efficiency can be achieved by combining both forcing methods. These results may be applied in the future study of how legged animals move.