Necessary and sufficient conditions for linear strong structural controllability and observability of n-link underactuated planar robot with multiple active intermediate links

This study concerns an n-link underactuated planar robot connected to a fixed base by n revolute joints in a vertical plane. The linear strong structural controllability and observability of such a robot around the upright equilibrium point, where all the links are in the upright position, are investigated. This study aims to solve an open problem regarding the linear strong structural controllability and observability of such a robot with only active intermediate links rather than the first or last link. This paper proves that it is linearly strongly structurally controllable and observable (linearly controllable and observable regardless of its mechanical parameters), if and only if there are at least two active adjacent links among the n - 2 intermediate links with measurable corresponding link angles, without making any other assumptions about any link of the robot, this paper proves that the robot is linearly strongly structurally controllable and observable. Second, when the robot only has two or more active intermediate links with measurable corresponding link angles, if there are no active intermediate adjacent links, this study proves that the robot is linearly strongly structurally uncontrollable and unobservable by presenting a constructive example to show that there always exists a set of mechanical parameters that renders the robot linearly uncontrollable and unobservable.