Linear strong structural controllability and observability of an n-link underactuated revolute planar robot with active intermediate joint or joints

For an n-link underactuated revolute planar robot with all the links moving in the same vertical plane, this paper studies the linear strong structural controllability and observability of the robot with only an active intermediate joint or active intermediate joints around the UEP (upright equilibrium point), where all the links are in the upright position. First, when the robot only has an active intermediate joint or active intermediate nonadjacent joints, we show that the robot is linearly strongly structurally uncontrollable and unobservable around the UEP via an illustrative example that there always exists a set of physical parameters that renders the robot linearly uncontrollable and unobservable around the UEP. Second, when the robot only has two active intermediate adjacent joints, without making any assumption on the physical parameters of the robot, we show that the robot is linearly strongly structurally controllable and observable around the UEP. Thus, the robot with its first and last joint being passive is linearly strongly structurally controllable and observable around the UEP if and only if there are at least two active adjacent joints among (n−2) intermediate joints.