Energy based Swing-up Control for a 3-Link Robot with Passive Last Joint: Design and Analysis

This paper addresses a swing-up control problem for a 3-link underactuated robot in the vertical plane, whose first and second joints are active (actuated) and the third (last) joint is passive (unactuated). The objectives of this paper are: 1) to design a control law under which the robot can be brought to any arbitrarily small neighborhood of the upright (up-up-up) equilibrium point, where all three links remain in their upright positions; 2) to attain a global analysis of the motion of the robot under the control law. By designing a coordinate transformation on the actuated joint variables of joints 1 and 2 and constructing a new Lyapunov function based on the transformation, this paper proposes an energy based swing-up control law. For any initial state of the robot, this paper provides a necessary and sufficient condition for non-existence of any singular point in the control law for all future time, and shows how to choose the control parameters such that the state of the robot eventually approach either any arbitrarily small neighborhood of the upright equilibrium point, or the up-up-down equilibrium point, where the links 1, 2 and 3 are in upright, upright and downward positions, respectively. Moreover, this paper shows that the up-up-down equilibrium point is unstable