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This paper presents a motion control technique for flexible robots and manipulators. It takes into account both joint and link flexibility and can be applied in adaptive form if robot parameters are unknown. It solves the main problems that are related to the fact that the number of degrees of freedom exceeds both the number of actuators and the number of output variables. The proposed method results in trajectory tracking while all state variables remain bounded. Global, asymptotic stability is ensured for all values of the stiffnesses of joints and links. To show the characteristics of the proposed control law, some simulation results are presented.

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2) Remark 3 is unnecessary because the stability criterion in this case will have a range of frequency between 0 and 2-rr. VII Concluding RemarksA numerical approach to model-matching controller synthesis has been introduced. The method is applicable when the system is open-loop stable and redundantly actuated. The major merits of the method are as follows.1 APPENDIX The Existence of R t (s) and R 2 (s)Because G(s) = g(i)8(*), (4) implies that Since ffi m is square when m = n -1, a full rank (R j; _ l is guaranteed. And for any m>n -1, an additional 2(m -n + 1) columns will be added on to SB m . At the same time, m -n + 1 linearly independent rows will also be introduced into the matrix. Thus, the full rank of S8 m will be maintained for all m > n -1 and the arbitrary placement of B^R^s) + B^R^s) is ensured. Two final comments should be made. First, if desired @(s) happens to be inside the column space of SB m> (A.l) can also be satisfied for m < n -1. In general, then, m > n -1 is a sufficient but not a necessary conditions for (A.l) to be satisfied. Second, the lowest possible value of m in (A.l) will occur when Q(s) = 1 and thus when m = n -n. In general, then, it is necessary that m > max(n -n, n -1). B^R^s) + B 2 {s)R 2 (s) = A(s)@(k). (A.l) To show that the pair R { (s) and R 2 (s) always exists for any arbitrary §(s), express B x (s) and