TAHRI et al DECOUPLED IMAGE BASED VISUAL SERVOING FOR CAMERAS OBEYING THE UNIFIED PROJECTION MODEL 685. of moments it becomes possible to create partitioned systems. with good decoupling and linearizing properties 25 26 For. instance by using such features the interaction matrix block. corresponding to the translational velocity can be a block di. agonal with no depth dependence However this approach is. limited to planar objects and conventional perspective cameras. A new decoupled image based control scheme using the projec. tion onto a unit sphere has been proposed in 27 which is based. on polynomials invariant to rotational motion computed from. a set of image points More recently a decoupled image based. control scheme based on the surface of triangle projection onto. a sphere has been proposed in 29 This paper synthesizes our Fig 1 Left Catadioptric camera and mirror geometry Right Unified image. contributions while developing the theoretical and experimental formation. results In particular the computation of the interaction matrix. related to the projection surface of triangles is detailed and its. X X Y Z in Fm The world point X is projected to, invariance to rotations is formally shown This paper also pro. vides a new and complete set of real experiments as well as new X Y. simulation results The proposed control schemes are not only m x y 1 1 1. compared between them but are also compared with an image. based control scheme using points coordinates as visual features and then mapped to the homogeneous image plane coordinate. As mentioned above the features we propose are computed p Km where K is a 3 3 matrix of camera and mirror. from the projection onto the unit sphere This means that the intrinsic parameters The matrix K and the parameter can. proposed method can work not only with classical perspective be obtained after calibration using for example the methods. cameras but can also be applied to wide angle cameras obeying proposed in 20 In the sequel the imaging system is assumed. the unified model 2 10 Wide angle cameras include cata to be calibrated In this case the inverse projection onto the unit. dioptric systems that combine mirrors and conventional cameras sphere can be obtained by. to create omnidirectional cameras providing 360 panoramic. Xs x y 1 2, views of a scene or dioptric fisheye lenses 1 It is highly de. sirable that such imaging systems have a single viewpoint 1 where. 24 i e there exists a single center of projection so that every. pixel in the sensed images measures the irradiance of the light 1 1 2 x2 y 2. passing through the same viewpoint in one particular direction 1 x2 y 2. The reason why a single viewpoint is so desirable is that it per. Note that the conventional perspective camera is nothing but. mits the extension of several results obtained for conventional. a particular case of this model where 0 The projection onto. cameras 11 In this paper we also take advantage of the prop. the unit sphere from the image plane is possible for all sensors. erties of such sensor system to develop control laws that are. obeying the unified model, valid for conventional catadioptric and fisheye cameras. In the next section we recall the unified camera model and. B Image Based Visual Servoing, the control law Moment definitions and the interaction matri. ces computation are also presented In Section III theoretical We define the vector of image features s and recall that its. details about feature selection are discussed and a new vector time variation. of features to control the 6 DOF camera is proposed Finally in. Section IV experimental results obtained using a conventional. camera and a fisheye camera mounted on a 6 DOF robot are is linear with respect to the relative camera object kinematics. presented to validate our approach screw V v and Ls is the interaction matrix related to s. The control scheme is usually designed to reach an exponential. II MODELING decoupled convergence of the visual features to their desired. A Camera Model value s 7 If we consider an eye in hand system observing a. static object the control law is defined as follows. Central imaging systems can be modeled using two consec. utive projections first spherical and then perspective This ge s s s. ometric formulation which is called the unified model was. proposed by Geyer and Daniilidis 10 Let us consider a vir where L s is a model or an approximation of Ls L s is the. tual unitary sphere centered on Cm and the perspective camera pseudoinverse of L s is a positive gain and Vc is the camera. centered on Cp see Fig 1 The frames attached to the sphere velocity sent to the low level robot controller Equation 4 is. and the perspective camera are related by a simple translation a linear approximation of the nonlinear mapping between 3 D. of along the Z axis Let X be a 3 D point with coordinates and image space and therefore valid for small displacements. 686 IEEE TRANSACTIONS ON ROBOTICS VOL 26 NO 4 AUGUST 2010. However for large displacements the approximation is not valid. and can lead to suboptimal robot trajectories, An important issue is therefore to determine those visual. features that will allow the system dynamics to be linear over. large displacements Furthermore using 4 local minima can. be reached when the number of features is not minimal There. fore one would like to choose a minimal representation i e the. number of features is equal to the number of DOFs but without. singularities and robust with respect to noise in the image. C Invariants to Rotational Motions From the Projection Onto. Fig 2 Triangle projection onto the unit sphere, the Surface of Unit Sphere. The shape of a planar object does not change under rotational. motions After a rotational motion of the sensor frame it can with respect to variation of object depth A comparison of the. easily be shown that the projected shape undergoes the same use of the two features will be made. rotational motion as the coordinates of the object 3 D points. This means that the invariants to rotation in 3 D space are also. D Interaction Matrix, invariant if the considered points are projected onto the unit. sphere The decoupled features we propose are based on this In the case of moments computed from a discrete set of points. invariance property It will be used to select features invariant the derivative of 6 with respect to time is given by. to rotations in order to control the three translational DOFs In. this way the following polynomial that is invariant to rotations. has been proposed in 27 to control the translational DOFs m i j k i xs h ys h zs h x s h j xis h ysj h 1 zskh y s h. I1 m200 m020 m200 m002 m2110 m2101, k xis h ysj h zskh 1 z s h 8. m020 m002 m2011 5, where mi j k is the 3 D moment of order i j k computed The interaction matrix LXs for a point on the unit sphere is well. from a discrete set of points defined by the following classical known 13 25 30 and is given by. LXs I3 Xs Xs Xs 9, mi j k xih yhj zhk 6, h 1 where r is the distance of the 3 D point to the sphere center For. where xh yh zh is the coordinates of the hth point and any set of points i e coplanar or noncoplanar we can combine. N is the number of points In our case these coordinates are that interaction matrix with that related to Lm i j k from 8 to. nothing but the coordinates of a point projected onto the unit obtain. sphere In this paper another kind of invariant is derived from. the projection onto the unit sphere More precisely the surface Lm i j k mv x mv y mv z mw x mw y mw z 10. that can be computed from the projection of three noncollinear. points onto the unit sphere will also be used In this case two where. kinds of surfaces that are invariants to rotations can be defined. ixi 1 j k i 1 j, s h ys h zs h d xs h ys h zs h, the surface defined by the triangle projected onto the unit sphere. which is defined by three circular arcs corresponding to the. projection of the triangle s edges onto the unit sphere and the. jxis h ysj h 1 zskh d xis h ysj h 1 zskh, surface of the triangle formed by projection onto the sphere. see Fig 2 The latter surface is computed by the well known. formula for triangle surface, kxis h ysj h zskh 1 d xis h ysj h zskh 1. Xs 2 Xs 1 Xs 3 Xs 1 7, where Xs 1 xs 1 ys 1 zs 1 Xs 2 xs 2 ys 2 zs 2 and Xs 3. mw x jmi j 1 k 1 kmi j 1 k 1, xs 3 ys 3 zs 3 are the coordinates of the triangle s vertices pro. mw y kmi 1 j k 1 imi 1 j k 1, jected onto the unit sphere. In the following it is this surface that will be used We mw z imi 1 j 1 k jmi 1 j 1 k. will show that after an adequate transformation new features. can be obtained from as well as from I1 given by 5 such and d i j k In the particular case of a coplanar set of. that the corresponding interaction matrices are almost constant points the interaction matrix related to mi j k can be determined. TAHRI et al DECOUPLED IMAGE BASED VISUAL SERVOING FOR CAMERAS OBEYING THE UNIFIED PROJECTION MODEL 687. 25 as follows Additionally it can easily be shown that. mv x A d mi 2 j k imi j k X21 X31 X31 X21 X21 X31 17. B d mi 1 j 1 k imi 1 j 1 k, Let us consider h X21 X31 which allows 16 to be written. C d mi 1 j k 1 imi 1 j k 1 as, mv y A d mi 1 j 1 k jmi 1 j 1 k. B d mi j 2 k jmi j k 4, i j 1 k 1 from which we immediately deduce. mv z A d mi 1 j k 1 kmi 1 j k 1, L 0 0 0 19, B d mi j 1 k 1 kmi j 1 k 1. which confirms the invariance of to rotations, C d mi j k 2 kmi j k. For translational velocity after tedious computation the in. mw x jmi j 1 k 1 kmi j 1 k 1 teraction matrix related to can be written as 20 shown at. the bottom of the page and after further computations it can be. mw y kmi 1 j k 1 imi 1 j k 1, shown that, mw z imi 1 j 1 k jmi 1 j 1 k. L v L v 1 L v 2 21, where A B C are the parameters defining the object. plane in the camera frame where, X 31 X 21 X 21 X 31 X 31 X 12. Xs Axs Bys Czs 12, The interaction matrix related to can be obtained in a L v 2 X31 X21 Xs 1 X32 Xs 1 Xs 1. similar way Let L be the 1 6 interaction matrix related to. Xs 2 X13 Xs 2 Xs 2, and is given by, Xs 3 X21 Xs 3 Xs 3 4. L L v L 13, In practice L v depends strongly on L v 1 because the nu. where L v and L are respectively two 1 3 matrices that merator of L v 2 is a polynomial of point projections with a. link the time variation of to the translational and the rotational higher order than the numerator of L v 1. velocities, Lemma 1 is invariant to rotations and III FEATURE CHOICE. L 0 0 0 In this section we detail our choice of image features First. we will explain how to obtain features to control the translational. Proof Let LXi be the interaction matrix related to the point DOFs with interaction matrices that are almost constant with. Xs i LXi j LXi LXj be the interaction matrix difference and respect to variation in object depth Then a vector of features. Xij Xs i Xs j be the coordinate vector difference The sur to control all six DOFs will be proposed. face can be written as, X X X X X X A Variation of the Interaction Matrix With Respect. X31 21 21 31 X21 31 31 21, to Camera Pose, 14 As mentioned above one of the goals of this study is to. Taking the time derivative of 14 we obtain decrease the system nonlinearity and coupling by selecting ad. equate features The invariance property for example results. X21 LX3 1 X31 X21 X31 LX2 1 in some interaction matrix entries being 0 thus removing cou. 4 pling between DOF as well as being constant during the servoing. task However other entries depend on the camera pose as will. Combining 15 with 9 it follows that, be shown next It will be also shown that the feature choice. X21 X31 X31 X21, It 1 I1 and s 1 leads to interaction matrices that. L 16 are almost constant with respect to the object depth variation. X31 X21 I3 Xs 2 Xs 2 Xs 2 Xs 1 Xs 1 Xs 1, X21 X31 I3 Xs 3 Xs 3 Xs 3 Xs 1 Xs 1 Xs 1. 688 IEEE TRANSACTIONS ON ROBOTICS VOL 26 NO 4 AUGUST 2010. 1 Variation With Respect to Rotational Motion Let us con features that scale as s Z where Z is the object depth so that. sider two frames F1 and F2 related to the unit sphere with the variation of their corresponding interaction matrices with. different orientations 1 R2 is the rotation matrix between the respect to depth is zero In the case where the object is defined. two frames but with the same center In this case the value of by an image region the following feature has been proposed to. It is the same for the two frames since it is invariant to rota control the motion along and around the optical axis 6 26. tion Let Xs and Xs 2 R1 Xs be the coordinates in the frame. F1 and F2 respectively of a projected point Let us consider sr. a function which is invariant to rotations f X1 XN that m00. can be computed from the coordinates of N points onto the unit where m00 is the moment of order 0 i e object surface in the. sphere The invariance condition between the frames F1 and F2 image using the conventional perspective projection model In. can be written as the case where the object is defined by a set of discrete points. f X1 XN f 2 R1 X1 2 R1 XN f X1 XN the selected optimal feature for rotation was. The interaction matrix that links the variation of the function sd 29. f with respect to translational velocities can be obtained as. f X1 T XN T where ij are the central moments computed from a set of. Lf v 23 discrete points for more details see 26 Unfortunately sr and. This combines a path planning process together with the servo ing one 4 5 9 21 23 A second way involves more Manuscript received August 28 2009 revised March 19 2010 accepted May 23 2010 Date of publication July 1 2010 date of current version August 10 2010 This paper was recommended for publication by Associate Editor D Kragic and Editor G Oriolo upon evaluation of

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