Tip position control of a two-link flexible robot manipulator based on nonlinear deflection feedback. (Q1419359)

This useful paper presents the tip position control of a two-link flexible robot manipulator based on nonlinear deflection feedback. In the dynamic modelling of the flexible manipulator, it is assumed that the links satisfy the assumptions of the Euler-Bernoulli beam theory. Each link has a uniform density \(\rho_i\) and constant flexural rigidity \((EI)_i\). The authors consider the following system of the partial differential equations for unknown state functions \(y_i(x, t)\) (deflection) \[ (EI)_i {\partial^4 y_i(x_i,t)\over\partial x^4_i}+ \rho_i{\partial^2 y_i(x,t)\over\partial t^2}= 0,\qquad i= 1,2,\dots,n\tag{1} \] with boundary conditions at the base and the end of each link, \(y_i (0, t)= 0\), \(y_i'(0, t)= 0\). Moreover, for the tip of the links, mass boundary conditions are considered (representing balance of moment and shear force) \[ E_i I_i{\partial^2 y_i(x_i, t)\over\partial x^2}\Biggl|_{x_i= I_i}= -J_{L_1}\Biggl({\partial y_i(x_i, t)\over\partial x_i}\Biggr|_{x_i= I_i}\Biggr)- (MD)_i{d^2\over dt^2} (y_i(x_i, t)|_{x_i= I_i}) \] and \[ (EI)_i {\partial^3y_i(x_i, t)\over\partial x^3_i}\Biggl|_{x_i= I_i}= M_{L_1} {d^2\over dt^2} (y_i(x_i, t)|_{x_i= I_i})+ (MD)_i {d^2\over dt^2} \Biggl({\partial y_i(x_i,t)\over\partial x_i}\Biggr|_{x_i= I_i}\Biggr), \] where \(M_{L_i}\) and \(J_{L_i}\) are the actual mass and moment of inertia at the end of link \(i\), \((MD)_i\) represents the contributions of masses of distal links, noncolocated at the end of link \(i\). Main result: Equation (1) is solved using a modal technique and the method of separation of variables. Due to the Lagrange approach, a closed-form equation for the two-link flexible manipulator is obtained. The resulting equations of motion have the form \[ B(q)\ddot q+ h(q,\dot q)+ Kq=Qu, \] where \(q\) is the \(N\)-vector of generalized coordinates, \(u\) is the \(n\)-vector of joint (actuator) torques, \(B(q)\) is the positive-definite symmetric inertia matrix, \(h(q,\dot q)\) is the vector of Coriolis and centrifugal forces, \(K\) is the stiffness matrix, \(Q\) is the input weighting matrix.