affine visual servoing directly at the control level。 This is obtained without losing the advantages of the image-based control formulation, since we show that all control parameters can be estimated in the im-age plane without requiring camera calibration。 An-other important contribution arises from using non-linear control theory [12,19] to synthesize a control law ensuring global asymptotic stability using the Lya-punov direct method。 We also show that the control law fulfills robustness requirements, since in the pres-ence of bounded modeling and measurement uncer-tainties the tracking error is also bounded。

The paper is organized as follows。 In Section 2, visual modeling issues are addressed。 In Section 3。1 we derive a state space representation of robot camera–object interaction; in Section 3。2, the control system is synthesized, and stability and robustness analysis are carried out; in Section 3。3, it is shown how to estimate 3D state variables from image plane measurements。 Section 4。1 presents results of simula-tions, while experimental results with an eye-in-hand robotic system are presented in Section 4。2。 Finally, in Section 5 the major contribution of the paper is summarized and future work is outlined。

2。 Models and measurements

Assume that the visible object's surface is a smooth manifold of equation 。x; y/ D z in terms of the camera frame hci D fic ; j c ; kc g。 The region of the visible surface is mapped into a patch & of the image plane in which visual analysis is carried out。 Let c Pc D [xc ; yc ; zc ]T be the point belonging to the

Fig。 2。 Ambiguity of weak perspective projection。

visible surface whose projection in the image plane sentation of orientation commonly used in computer

is the centroid of & 。 We consider a local approxima- vision [6]。

tion ofaround c Pc , resulting in the tangent plane Eq。 (3) provides two different solutions for   (and

equation: '), which differ by   。 This results in a pose ambigu-论文网

。x; y/   px C qy C c; (1) ity which is typical of any perspective linearization:

there are two distinct object poses sharing the same

where p  D  。@  。x; y/=@ x/jxc ;yc , q  D 。@  。x; y/= visual appearance [6]。 In the case of weak perspec-

tive, such ambiguity can be written as T 。 ;  ;  ; '/ D

@y/ xc ;yc , and c D 。xc ; yc / − pxc − qyc 。 The plane

j T 。 ;  C  ; ; ' C / (see also Fig。 2)。 Given two ob-

coefficients determine, up to a degree of freedom, the 1 2 object point, and

relative pose between the camera and the visible sur- ject views  p g and fp g of the same