The Exponential Stable Research Of The Underactuated Manipulator

Underactuated robot is a kind of manipulator systems with one or more passive joints, and these passive joints have not steering forces. The major character is that the dimension of input space (control space) is less than dimension of configuration space. At the same time, the constraint equations of the passive joints generally are un-integral, so they are belong to second order nonholonomic constraints. The research on controlling the underactuated robot systems has been a challenging and popular focus because of the decrease of actuated joints.The research object of this paper is the planar underactuated manipulators and the main contents are generalized as the following:To the 2R and 3R planar underactuated manipulators with one unactuated joint respectiely, this paper sets up the dynamical models, analysies their nonholonimic propertys based on the differential geometry, and proves that they all belong to the second-order nonholonimic constraints. This characteristic is very important for the controllability and the stabilization of the systems. Then, this paper analyses the accessibility and controllabliliy of the underactuated systems, and proves the systems are all accessible. To the planar underactuated manipulators, this paper proves that the 2R planar underactuated manipulator cannot satisfy the sufficient condition of the small-time local controllability (STLC), but could be control by the structural methods; and proves that the 3R planar underactuated manipulator is controllability even more the STLC, so the system could realize not only the trajectory planning but also the trajectory tracking. This paper also proves that there is not smooth time-invariant control law to stability the equilibrium point.Besides, this paper gives an example of a single inverted-pendulum to compare the inverted-pendulum systems and the planar underactuated manipulators systems , and gets their discrimination and relationship. Besides, this paper sets up the inverted-pendulum modle and proves that the inverted-pendulum systems are also second-order nonholonomic systems, and even more after made the linearization around the unstable equilibrium point, the governing equation is controllability. Basing on these, this paper gets the linearizing governing equation of the single and double inverted-pendulums,designs the LQR to stabilize the systems exponentially and makes the simulation. At the end, this paper makes the real object verification using the single and second inverted-pendulums producted by the Googol Company.