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Underactuated systems are those possessing fewer actuators than degrees of freedom. The double inverted pendulum is a particular underactuated system and a well-known benchmark case for which many solutions have been offered in the literature. The control objective is to bring the system to its unstable top equilibrium point. The underactuated horizontal double pendulum is a two-link planar robot with only one actuator either at the shoulder or the elbow. The fundamental difference between a double inverted pendulum and an underactuated horizontal double pendulum is that in the latter gravity effects do not exist. Gravity is important to the controllability of the system. Therefore, we added springs in the underactuated horizontal double pendulum in order to create a source of potential energy. Two different types of such systems are analyzed: spring coupled underactuated horizontal double pendulums and underactuated horizontal double pendulums with spring-loaded sliding bar constraint. The main contribution of this work is in proving that the zero state of the spring coupled systems is globally asymptotically stabilizable. Explicit control laws were developed.
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