Analytical solution to transition function of state error in 1-DOF semi-passive dynamic walking

In this paper, we derive the analytical solution to the transition function of the state error in 1-DOF semi-passive dynamic walking for understanding how the gait stability changes according to acceleration or deceleration. We introduce the model of an active rimless wheel (RW) as the simplest walker for analysis and linearize the equation of motion incorporating a simple control torque. Through mathematical investigations, we finally derive the analytical solution to the transition function of the state error for the stance phase as a function only of the control parameters. We discuss the accuracy of the solution obtained through comparison with the values numerically-integrated in the linearized and the nonlinear walking models.