Analysis of the Fixed Point Transformation Based Adapive Robot Control J´ ozsef K. Tar Budapest Tech John von Neumann Faculty of Informatics Inst. of Intelligent Engineering Systems B´ ecsi ´ ut 96/B, Budapest, H-1034, Hungary tar.jozsef@nik.bmf.hu Imre J. Rudas Budapest Tech John von Numann Faculty of Informatics Inst. of Intelligent Engineering Systems B´ ecsi ´ ut 96/B, Budapest, H-1034, Hungary rudas@bmf.hu Abstract— In this paper the properties of a novel adaptive non- linear control recently developed at Budapest Tech for “Multiple Input-Multiple Output (MIMO) Systems” is comapred with that of the sophisticated “Adaptive Control by Slotine & Li” widely used in robot control literature. While this latter traditional method utilizes very subtle details of the structurally and formally exact analytical model of the robot in each step of the control cycle in which only the exact values of the parameters are unknown, the novel approach is based on simple geometric considerations concerning the method of the “Singular Value Decomposition (SVD)”. Furthermore, while the proof of the asymptotic stability and convergence to an exact trajectory tracking of Slotine’s & Li’s control is based on “Lyapunov’s 2 nd Method”, in the new approach the control task is formulated as a Fixed Point Problem for the solution of which a Contractive Mapping is created that generates an Iterative Cauchy Sequence. Consequently it converges to the fixed point that is the solution of the control task. Besides the use of very subtle analytical details the main drawback of the Slotine & Li method is that it assumes that the generalized forces acting on the controlled system are eaxctly known and are equal with that exerted by the controlled drives. So unknown external perturbations can disturb the operation of this sophisticated method. In contrast to that, in the novel method the computationally relatively costly SVD operation on the formally almost exact model need not to be done within each control cycle: it has to be done only one times before the control action is initiated. In the control cyle the inertia matrix is modeled only by a simple scalar. In a more general case the SVD of some approximate model can be done only in a few typical points of the state space of a Classical Mechanical System. To illustrate the usability of the proposed method adaptive control of a Classical Mechanical paradigm, a cart plus crane plus hamper system is considered and discussed by the use of simulation results. I. THE ESSENCE OF THE SLOTINE-LI METHOD The adaptive version of Slotine & Li control [[1]] strongly utilizes that the Lagrangian of a robot has the form as follows: L = 1 2  i,j H ij (q)˙ q i ˙ q j - V (q) (1) in which q ∈ℜ n denotes the generalized coordinates of the robot, H ij = H ji is the symmetric inertia matrix of the robot depending only on q, and V (q) is the potential energy also depending only on q. Via analyzing the symmetries in the terms obtained in the Euler-Lagrange equations by substituting the above expression into the appropriate operations they arrived in the conclusion that the equations of motion have the general form as H(q)¨ q + C(q, ˙ q)˙ q + g(q)= Q (2) in whic the genious idea is also incorporated that though the originally obtained expressions are quite symmetric regarding the positions of the components that are quadratic in ˙ q i , they can be treated in an “asymmetric” manner by including them partly in the matrix C(q, ˙ q), too. [This decomposition is unambiguous since C must be linear in the ˙ q i components.] Term g(q) originates from the gravitation. The second great idea in the Slotine & Li Method is that the available, formally exact but numerically inexact model marked by the caret ( ˆ ) symbol can be used for asymmetrically calculating a feedback force that contains PD-type terms plus an additional one that resembles to the “Error Metrics” S := ˙ e +Λe normally used in the robust “Variable Structure / Sliding Mode (VS/SM)” controllers as follows: Q = ˆ H ˙ v + ˆ Cv +ˆ g - K D [˙ e +Λe] (3) in which the tracking error is e = q - q N , q and q N denote the actual and the nominal coordinates in the given time instant, respectively, v := ˙ q N - Λe is used for tracking correction, and K D is a positive definite matrix. Since its is assumed that the so calculated Q is the only contribution to the generalized forces and no additional external perturbations are present, it also is related to the actual state propagation of the system as follows: Q = H(q)¨ q + C(q, ˙ q)˙ q + g (4) that is a kind of “weak point” of this sophisticated aproach. By combining (3) and (4) an equation can be obtained in which at one side terms containg the error metrics S, at the othre side the modeling errors [marked by the tilde symbol ( ˜ )] are present: H(q) ˙ S +CS +K D S = ˜ H(q)˙ v + ˜ Cv +˜ g := Y (q, ˙ q, v, ˙ v)˜ p (5) 978-1-4244-2083-4/08/$25.00 ©2008 IEEE