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3.1. PreliminariesThe main objective of this paper is to introducea powerful new estimation algorithm in this Þeld. Underthe following assumption, attention is restricted to 2-Dvisual servoing, in order to discuss the robustness againstdisturbances for the linear system of the planar model(e.g. (Kelly, 1996)) of the visual servoing.Assumption (Fig. 2(A)): The target moves in a plane,called A, and the camera moves in another plane, calledB. The target moving plane, A, runs parallel to thecameramoving plane, B. The optical axis of the camera isperpendicular to each plane. Further, plane B containsthe focal point of the camera. The distance Z0betweenplane A and plane B is constant and known.Now, consider the camera frame, with its origin at thefocal point, its Z-axis along the optical axis, and appro-priate X- and ½-axes on plane B. Here a pinhole cameramodel, is assumed as shown in Fig. 2(B). Let f denote thefocal length, and sxand sythe cameraÕs pixel dimensionsin the x and y directions, respectively. The camera canrotate around the Z-axis, and can translate in any of thedirections in plane B. All the feature points Þxed on thetarget are located in plane A. Hence, without any loss of generality, two feature points on the target are assumedin these experiments. These provide su¦cient informa-tion to determine the motion of the center of mass of thetarget.Let Pj, j"1, 2, be the feature points with the coordi-nates [Xj, ½j, Z0]T in the camera frame. Each featurepoint Pj, j"1, 2, projects onto a point pj, j"1, 2, withthe image coordinates [xj, yj]T, respectively. Let Pgbethe center of mass of these feature points, with the trans-lational positions X, ½ and with the rotational positionH around the Z-axis. In addition, let X, ½, and H denotethe corresponding velocities of Pg. Finally, let ¹ox, ¹oy,and Rozdenote the velocities of Pginduced by the targetmotion, and ¹cx, ¹cyand Rczdenote the velocities ofinduced by the camera motion (see Fig. 2(a)).3.2. Observation equationNext, consider the observation equation. The relation-ships between the coordinates Pj"[Xj, ½j, Z0]T andpj"[xj, yj]T (see Fig. 2(B)) can be directly written(Hutchinson et al., 1996) asxj"fsxXjZ0, yj"fsy½jZ0, j"1, 2. (1)The time-dependence of each variable will be eliminatedin an obvious way. From the above equations, one cangeometrically derivex1#x22" fsxZ0X#wx,y1#y22" fsyZ0½#wy(2)atan2(y2!y1, x1!x2)"H#wh(3)where atan2(), )) stands for the arc-tangent function,which is popular in the robotic literature, and the termswx, wy, and whrepresent the e¤ects of various uncertain-It is noted that, in these experiments, the output vectoryk can be obtained directly as the output of the visionsystem. Further, if the disturbance terms wk due to theuncertainties are ignored, the state variables xk can becomputed, since C is nonsingular.3.3. State equationFinally, consider the state equation, which describesthe dynamics of the system. Under the assumptions,clearlyX"¹ox!¹cx, ½"¹oy!¹cy, H"Roz!Rcz. (8)Thus, deÞning the vectors asu :"[!¹cx,!¹cy,!Rcz]T, q :"[¹ox, ¹oy, Roz]T, (9)(8) yields x"u#q. Under the discretization, the systembecomesxk`1!xk"¹suk#¹sqk#¹2s2vk(10)where vkdenotes the e¤ect of the uncertainties arisingfrom the discretization process and/or the variousmodeling errors (see, e.g., (Papanikolopoulos et al.,1993)).
Hencexk`1"Axk#Buk#Eqk#Hvk(11)whereA:"I3, B:"TsI3, E :"¹sI3, H:"¹2s2I3and I3is a 3]3 identity matrix.It is noted that, in (11) and (7), the disturbance termsvkand wkdue to the uncertainties a¤ect the system. Here,these terms will not be treated as a class of stochasticprocesses, but rather regarded as a class of unknown-but-bounded signals. This treatment of the uncertainty signalswill naturally lead to an estimation problem in theH= setting.