Robust control of nonlinear mechanical systems using linear feedback (Q1881904)

The paper is concerned with the control of rigid and flexible joint robotic manipulators using basically the PID controller. The following results have been proved. (1) For a rigid manipulator with dynamics model \[ A(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)=u \] it is proved that the control of the form \[ u=-K_p(q-q_d)-K_vs+v,\;\;v=const, \;\;\tau\dot{s}=\dot{q}-s \] stabilizes the equilibrium point \(q=q(v),\) \(\dot{q}=0,\) \(s=0\). After adding an integral action \[ \dot{v}=-K_i(q-q_d), \] the equilibrium \(q=q_d\), \(\dot{q}=0,\) \(s=0\) is stabilized. (2) For a flexible joint manipulator, with the so-called reduced Spong model, \[ \begin{aligned} &A(q_l)\ddot{q}_l+C(q_l,\dot{q}_l)\dot{q}_l+g(q_l)=\Gamma(q_m-q_l)\\ &J\ddot{q}_m+\Gamma(q_m-q_l)=u, \end{aligned} \] where \(q_l\), \(q_m\) refer, respectively, to the link and the motor position, the control law of the form \[ u=-K_p(q_l-q_d)-K_v\dot{q}_m+v,\;\;\dot{v}=-K_i(q_l-q_d), \] stabilizes the equilibrium point \(q_l=q_d=\text{const}\), \(q_m=\text{const}\) and \(v=\text{const}\). Furthermore, a modified control law \[ u=-\gamma(-\theta(q_l-q_d)+\dot{q}_m+v),\;\;\dot{v}=\beta(q_l-q_d)+\alpha\dot{q}_m \] has been proposed containing a high gain coefficient \(\gamma\) that stabilizes the equilibrium point \(q_l=q_d\). Above, \(\alpha,\gamma>0\) and \(\beta,\theta\) are symmetric matrices. The proof of the last result is based on a corollary of the Hoppenstead theorem from singularly perturbed systems. Two simple examples illustrate the theory.