Adaptive Backstepping Control of the Bipedal Bead on a Rail Walking Model

Locomotion and control of legged robots is currently an issue of active research within the robotics community. Since the legs themselves are inspired by nature, one might also look to a biological example in order to develop a scheme for controlling the legged design. This ideology of using natural examples for engineering design is referred to as functional biomimesis [KKFK01]. Two biologically inspired possibilities for the control of such motion, not surprisingly, are feedforward and feedback. The purely feedforward approach forces the legs to simply track a signal generated by some clock (or central pattern generator). The motion each leg is completely independent of any environmental factors and of other legs (to the extent that they both might follow a common signal) in this feedforward design. Alternatively, the feedback design might sense environmental stimuli or use information from other legs to adjust the motion of an individual leg. The mathematical description of such a legged robot results in a hybrid dynamical system, since the legs make discrete transitions–say from protraction to retraction–but their dynamics between transitions are governed by some continuous time differential equations. If the focus is solely on developing and analyzing a feasible control for the system, obviously it is best to use the simplest possible model from which meaningful conclusions may be drawn. I will analyze the Bipedal Bead on a Rail (BBR) walking model in this paper. This is a simple one-dimensional walking model whose mechanics result in nonlinear differential equations that determine inter-transition behavior. Fortunately, the equations are in the chained integrator form so that the backstepping control design of [KKM91] and [ZFI03] may be employed. First, I examine the nonadaptive case in which all system parameters are known in order to verify that the backstepping method will work. Then, I relax this condition and use the adaptive design. Tracking performance as well as parameter convergence are achieved.