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The passivity property of a noncollocated single-link flexible manipulator with a parameterized output is studied. The system can be characterized by either the irrational transfer function of an infinite-dimensional model or its truncated rational transfer functions. Necessary and sufficient conditions for these transfer functions to be passive are found. It is also shown that a non-passive, marginal minimum-phase, truncated transfer function can be rendered passive by using either the root strain feedback or the joint angular acceleration feedback. For the noncollocated truncated passive transfer function, a PD controller suffices to stabilize the overall system. Numerical results are given to show the efficacy of the proposed approaches.

A noncollocated system has the potential of providing more precise tracking, improved disturbance rejection and increased bandwidth at the sensor location, but is considerably more difficult to stabilize than a collocated system due to its nonminimum phase nature. For a flexible manipulator, the problem becomes even more complicated because the system is inherently infinite dimensional. In this paper, a single-link flexible manipulator having a tip payload and a noncollocated actuator and sensor is investigated using a linear distributed parameter model. With the joint torque as the input and the joint angle plus a weighted value of tip deflection as the measured output, an exact transfer function involving transcendental functions is derived. Using the methods of infinite product expansion, the root locus, and the asymptotic property of the roots of the transcendental equation, a necessary and sufficient condition in terms of the weighting factor of tip deflection is obtained such that the transfer function does not have any open right-half plane zeros. This condition depends neither on the physical properties of the link nor on the mass properties of the tip payload. In order to correct the misinterpretation which has occurred in some closely-related works concerning the stability of the infinite-dimensional zero-dynamics, the equivalence of the zeros of the transfer function and the eigenvalues of zero-dynamics is also verified. Numerical results for the closed-loop performance of a single-link flexible arm using collocated and noncollocated PD controllers are given to show the efficacy of the proposed minimum phase transfer function approach.

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In order to correct the misinterpretation which has occurred in two recent works concerning the stability of infinite‐dimensional zero dynamics, the stability of zero dynamics of a noncollocated single‐link flexible manipulator is considered in this article for a class of outputs containing a constant parameter. The characteristic equation governing the eigenvalues of zero dynamics is derived. The distribution of these eigenvalues on the complex plane (which depends on the values of the constant parameter) can be found analytically by using the methods of infinite product expansion and root locus. The equivalence of the eigenvalues of zero dynamics and the zeros of the transfer function is also verified. © 2003 Wiley Periodicals, Inc.