Adaptive and Learning Systems pp 379-388 | Cite as

# Mathematical Theory of Learning with Applications to Robot Control

## Abstract

Fundamental forms of learning control law are proposed for linear and nonlinear dynamical systems which may be operated repeatedly at relatively low cost. Given a desired output *y* _{ d }(*t*) over a finite time duration [0, *T*] and an appropriate input *u* _{0}(*t*) for such a system, a general proposed law of learning control is described by a PID-type (Proportional, Integration, and Differentiation) iterative process: *u* _{ k } _{+1}(*t*) = *u* _{ k }(*t*) + {Φ + Γ*d*/*dt* + Ψ *∫* *dt*}(*y* _{ d }(*t*) − *y* _{ k }(*t*)), where *u* _{ k } denotes the input at the kth trial, *y* _{ k } the measured output when *u* _{ k } excites the system, and Φ, Γ and Ψ are constant gain matrices. For a class of linear mechanical systems where *x* and *y*(= *dx*/*dt*) stand for position and velocity vectors respectively, it is shown that a P-type or PI-type iterative learning control law with appropriate gain matrices Φ and Ψ is convergent in a sense that *y* _{ k }(*t*) approaches *y* _{ d }(*t*) pointwisely in *t* ∈ [0, *T*] and *x* _{ k }(*t*) does *x* _{ d }(*t*) uniformly in *t* ∈ [0, *T*] as *k* → ∞. In case of using a D-type or DP-type iterative learning control law, an analogous conclusion is also proved for a class of nonlinear dynamical systems. Finally, proposed learning methods are applied to some problems of trajectory or path tracking control of robot manipulators.

## Keywords

Nonlinear Dynamical System Robot Manipulator Learning Control Iterative Learning Control Gain Matrice## Preview

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## References

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*J. of Robotic Systems*, 1–2, pp. 123–140, 1984.CrossRefGoogle Scholar - [2]Ibid., “Can Mechanical Robots Learn by Themselves?”, in Robotics Research: The 2nd Inter. Symp., (H. Hanafusa and H. Inoue, ed.), pp. 127–134, MIT Press, Cambridge, Massachusetts, 1985.Google Scholar
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