Mathematical modeling and analysis for robotic control (Q2717290)

This paper presents an application of advanced mathematical analysis to robotics. Its contribution is twofold. First, a mathematical model of the dynamics of a 1 d.o.f. flexible manipulator arm is studied, containing both bending and torsional effects. Second, a general geometrical perspective is sketched for investigating the dynamics and control of flexible manipulators through geometric and topological properties of the configuration manifold. The first contribution of the paper seems to be much more material than the second one. The dynamics model of a flexible arm includes 6 equations: two PDEs describing transverse and torsional vibrations of the arm, three Euler-Lagrange ODEs characterizing the dynamics of motion of the end effector, and an ODE involving the arm turning angle and a control torque at the joint. All these equations are represented as a nonhomogeneous evolution equation on a Hilbert space of the form NEWLINE\[NEWLINE{du\over dt}= Au+ F(t, U).NEWLINE\]NEWLINE A series of theorems (2.4, 3.1-3.6) refers to properties of the operator \(A\) and characterizes existence and stability of solutions of the evolution equation. They review results published in the paper ``Control and stability of a torsional elastic robot arm'', coauthored by the author of this paper along with \textit{X. Hou}, that appeared in J. Math. Anal. Appl. 243, 140-162 (2000; Zbl 0979.93055). A control law of proportional (P) type with compensation is proposed (Theorem 3.7) providing a set point regulation of the angular position of the arm.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00047].