Lyapunov-based control of underactuated mechanical systems

Underactuated systems are mechanical systems with fewer controls than degrees of freedom. In recent years, there has been extensive interest among the researchers in control of underactuated mechanical systems due to their broad applications. Many real-life control systems including aircraft, spacecraft, helicopters, underwater vehicles, surface vessels, mobile robots, walking robots, and flexible-link robots are examples of underactuated systems.; A novel control method based on Lyapunov stability theory is presented. The positive definite candidate Lyapunov function, Vq,q&d2; =12q&d2; TKDq&d2;+F q, is defined with that KD ∈ reals nxn is a symmetric positive definite matrix and represents the kinetic part and phi( q) is a positive definite function representing the potential part of the Lyapunov function. For convenience, KD is defined as KD = P( t)M(t), where P( t) ∈ realsnx n is a nonsingular matrix which satisfies the conditions that KD is symmetric and positive definite and M(t) ∈ realsn xn is the mass matrix. In order to show that the Lyapunov candidate function decreases in time, three matching conditions are derived. The first matching condition produces linear differential equations for the each unknown element of the P matrix which is evaluated as a function of time. The control signal is introduced as being the sum of two quantities, F1 and F 2. The first part of the control signal, F1, is determined from the second matching condition. In the case of viscous damping being introduced into the system, the second matching condition will change accordingly. The potential phi is obtained from the third matching condition. The second part of the control, F2, is also obtained from the third matching condition.; To guarantee the stability of the system, two lemmas were developed. The first lemma shows that KD will remain symmetric if it starts off symmetric. The second lemma demonstrates that K D will stay positive definite as long as the initial P matrix can produce a positive definite initial KD . The system performance depends on the values of the elements of the P matrix. In general, the P matrix will not return to its initial values after a period of stabilizing control. So that the system performance is not repeatable. A third control law contribution is developed. This third contribution produces control of the elements of P and drives the P matrix back to the initial values once the equilibrium of the mechanical system is achieved. It has been noticed in some examples that the determinant of the P matrix is constant. In general, it was demonstrated that it is not true.; Three applications are presented to show the effectiveness of the proposed control method. The inverted pendulum-cart with viscous damping was used as the first example. The system of the rotary inverted pendulum is relatively complicated which increases the complexity of the P matrix while the control signal remains simple. The Lyaponov based control is demonstrated to work with the ball & beam system.