Linear controllability of two multi-link robotic systems with multiple unactuated joints

We deal with the linear controllability of two underactuated multi-link robotic systems in this paper. First, for a general n-link inverted pendulum mounted on a cart in a vertical plane, we prove that it is linearly controllable around the equilibrium point with the cart in a desired position and each link of the pendulum in the upright position regardless of its physical parameters, if and only if the cart is actuated. Second, for an n-link underactuated planar robot with its first joint mounted on a fixed base in a horizontal plane, we prove that it is linearly controllable around the trajectory where each link of the robot stretches straight out and rotates with a constant angular velocity, if and only if its first joint is actuated, regardless of its physical parameters and regardless of the actuation or underactuation of its rest joints. We illustrate the above results via a quintuple pendulum-cart system and a four-link planar gymnastic robot in literature. We give new insights into the linear controllability around an equilibrium point or a trajectory of underactuated robotic planar systems with multiple unactuated joints.