Energy-preserving control of a passive one-legged running robot

Spatio-temporal optimization of multi-phase movements: Dealing with contacts and switching dynamics

“…• Update the vector of temporal scaling factor β and corresponding sampling time ∆t 0 , · · · , ∆t K in (27) via gradient descent.…”

“…We augment the hopping robot model in [27] with variable compliance elements in the hip and the leg actuators. In this hopping example, we demonstrate the feasibility of the proposed approach on an increasingly challenging task that includes switching of different mode of dynamics (flight and stance) and more complex discontinuous state transition arising from impact at touch-down.…”

“…where q 1 = [x com , y com , θ, φ] T and q 2 = [θ, φ, r] T are the partial configuration vector for the flight and stance phase, respectively (for the specific form of the dynamics in each phase, please refer to [26], [27]). …”

“…The specific form of the velocity transition map is given in the study of passive running with an additional analysis of energy dissipation at impact [27]. At lift-off, the velocity of the leg is reset to zero asṙ = 0 at r = r 0 .…”

“…Note that in order to minimize this term, it is important to find an appropriate flight time T 1 as the trajectory of the center of mass cannot be directly controlled in the flight phase for a given initial lift-off condition. The term Q T2,2 (µ − ) 2 minimizes energy loss at touch-down where µ − is called the energy dissipation coefficient [27] …”