Y. Shiu and S. Ahmad. Calibration of Wrist Mounted Robotic Sensors by Solving Homogeneous Transform Equations of the Form AX = XB. IEEE Trans. on Robotics and Automation, 5(1):16--29, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... using the expression equation (23) the motion equation of the plane is M (n id) M = Rn R i( GammaRn Rn R) id = Rn R i(d Gamma Rn Rt) Rn R i(d Gamma RnRt ) 32) 4 The Hand Eye Problem The well known hand eye equation firstly formulated by Shiu and Ahmad [8] and Tsai and Lenz [10] reads AX = XB (33) where A = A 1 A 2 and B = B 1 B 2 express the elimination of the transformation hand base to world. From the expression equation (33) the following matrix and a vector equations can be derived RARX = RXRB and (RA Gamma I) t X = RX t B ....

Shiu Y.C. and Ahmad S. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Trans. Robotics and Automation, 5:16--27, 1989. 11

....motion process determines both the camera motion and the Euclidean structure of the scene. The camera mo tion parameters thus extracted are combined with the known robot motions to estimate the hand eye trans formation. This method has several advantages when compared to the standard approaches [19, 4, 8, 10, 16, 5, 20, 15]. The first one is that there is no need to use a calibrated object with known geometry. One does not either need to perform 2D to 3D matching between each one of the images and the object points. On the opposite, structure from motion requires only image to image point tracking. Moreover, since ....

....the corresponding camera poses Pi can be computed, providing the rigid motions of the camera Ai. With the associated recorded robot motions Bi, the hand eye transformation can be computed. Section 6 concludes this work. 2. Classical hand eye calibration We present here the usual approach [19, 4, 8, 10, 16, 5, 20] which states that when the camera undergoes a rigid motion A = Ra, ta) and the corresponding robot motion is B = Rb, tb) then they are conjugated by the hand eye transformation X = Rx, tx) AX: XB (1) In the prior work, correspondences are established between the 3D points on the calibration ....

Y. Shiu and S. Ahmad. Calibration of wrist mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Transactions on Robotics and Automation, 5(1):16-29, Febru- ary 1989.

....must be calibrated on site, the amount of free robot workspace is limited and the motions have therefore to be of small amplitude. Therefore, the self calibration method must be able to handle a large variety of motions, including small ones. Hand eye calibration was first studied a decade ago [30, 27]. It was shown that any solution to the problem requires to consider both euclidean end effector motions and camera motions 1. While the end effector motions can be obtained from the encoders, the camera motions are to be computed from the images. It was also shown, both algebraically [30] and ....

....It was also shown, both algebraically [30] and geometrically [4] that a sufficient condition to the uniqueness of the solution is the existence of two calibration motions with non parallel rotation axes. 1Notice that this requirement may be implicit as in [24] Several methods were proposed [30, 8, 15, 27, 5, 31] to solve for hand eye calibration under the assumption that both end effector and camera motions were known. They differ by the way they represent Euclidean motions, but all have two points in common: i) rotation is represented by a minimal pa rameterization and (ii) all proposed methods use ....

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Y. C. Shiu and S. Ahmad. Calibration of wrist mounted robotic sensors by solving homogeneous transform equations of the form AX=XB. IEEE Trans. on Robotics and Automation, 5(1):16 29, February 1989.

....We will furthermore assume that the camera is calibrated, i.e. that the intrinsic parameters are known. The problem of both recovering the hand eye calibration and the robot hand calibration has been treated in [3, 13, 12] Hand eye calibration has been treated by many researchers, e.g. [10, 11, 2, 4]. The standard approach relies on (i) a known reference object (calibration object) and (ii) the possibility to reliably track points on this reference object in order to obtain corresponding points between pairs of images. This approach leads to the study of the equation AX = XB, where A, X and B ....

Y. C. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form ax = xb. IEEE Trans. Robotics and Automation, 5(1):16--29, 1989.

....equivalent of the robot s gripper pose while the X ray imaging system takes the place of the attached camera. This observation lets us use several algorithms, which have been proposed to solve the hand eye calibration problem. Di#erent authors proposed to solve it by linear and non linear systems [14, 16, 3, 8, 4]. These algorithms use motion equations between pairs of image frames to solve for the two unknown transformations separately. The motion of C arm between two arbitrary image frames is described by E 1 i E ref = V L 1 i Q 1 Q L ref V 1 = V L 1 i L ref V 1 . 6) The left side of ....

....rotation. Using special properties of rotation matrices, Equation 9 can be transformed into rE (i) R V rL (i) 11) with rE (i) and rL (i) representing the eigenvectors (rotation axes) of RE (i) and RL (i) This equation system can be solved from two or more equations, either directly [14], or representing the unknown rotation matrix R V as a quaternion [8] Equation 9 directly can be transformed into a quaternion equation which is then solved using the constraint of a resulting unit quaternion [3] The translation is estimated afterwards using a linear least square method solving ....

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Y. C. Shiu and S. Ahmad. Calibration of wristmounted robotic sensors by solving homogeneous transform equations of the form AX=XB. IEEE Transactions on Robotics and Automation, 5(1):16--29, 1989.

....We will furthermore assume that the camera is calibrated, i.e. that the intrinsic parameters are known. The problem of both recovering the hand eye calibration and the robot hand calibration has been treated in [3, 12, 11] Hand eye calibration has been treated by many researchers, e.g. [9, 10, 2, 4]. The standard approach relies on (i) a known reference object (calibration object) and (ii) the possibility to reliably track points on this reference object in order to obtain corresponding points between pairs of images. This approach leads to the study of the equation AX = XB, where A, X and B ....

Y. C. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form ax = xb. IEEE Trans. Robotics and Automation, 5(1):16--29, 1989.

....(in the camera system estimated from point correspondences) respectively and X denotes the transformation between the hand coordinate system and the camera coordinate system, i.e. the handeye calibration. This approach has been investigated in a number of articles by different researchers, c.f. [11, 13, 2]. An active vision approach which uses specially designed camera motions is presented by Ma in [7] This makes it possible to cope without a calibration grid but it still uses the possibility to be able to detect features and track them through a motion sequence. To solve the translational part ....

Y. C. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form ax = xb. IEEE Trans. Robotics and Automation, 5(1):16--29, 1989.

....B Z Figure 3.7: Robot world (Z) and hand eye (X) calibration. The camera is mounted in the gripper and camera motions are determined using a calibration pattern. The calibration pattern is aligned with world coordinate system. solutions were proposed in the past to solve for X [Wan92] PM94] [SA89] as well as a non linear optimization method [HD94] Recently, Zhuang et al. ZRS94] proposed a method that allows the simultaneous estimation of both transformations (1) from the world centred to the robot base and (2) from gripper to camera. The identification problem is changed into the ....

Y. C. Shiu and S. Ahmad. Calibration of wrist mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Journal on Robotics and Automation, 5(1):16-- 29, June 1989.

.... and second position of the robot hand (in the robot coordinate system) and the camera (in the camera system estimated from point correspondences) respectively and X denotes the transformation between the hand coordinate system and the camera coordinate system, i.e. the hand eye calibration, see [11, 12, 2, 4]. The hand eye calibration problem can be simplified considerably by using the possibility to move the robot in a controlled manner and invstigating the arising motion field of points in the images. In [9] this fact has been exploited by first only translating the camera and using the focus of ....

Y. C. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form ax = xb. IEEE Trans. Robotics and Automation, 5(1):16--29, 1989.

....particular motion invariant. Recall that the rotors and translators are bivectors and you can commute and associate them without acting the sign. 4. 2 Application 2: Motors for hand eye calibration as a case of motion of lines The well known hand eye equation firstly formulated by Shiu and Ahmad [27] and Tsai and Lenz [28] reads AX = XB (29) where A = A 1 A Gamma1 2 and B = B 1 B Gamma1 2 express the elimination of the transformation hand base to world. Here matrices are represented in bold. The geometry of the hand eye system is depicted in Figure 1. From the expression (29) the ....

Shiu Y.C. and Ahmad S. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Trans. Robotics and Automation, 5:16--27, 1989.

....camera [BMV93] or for visual servoing (using a sensor inside a control servo loop) ECR92] In the past some solutions were proposed in the particular case of the sensor being a TV camera. With almost no exception, all existing solutions attempt to solve a homogeneous matrix equation of the form ([SA89], TL89] CK91] Che91] Wan92] AX = XB (1) This paper has the following main contributions. First we show that there are two possible formulations of the hand eye calibration problem. One formulation is the classical one that we just mentioned. A second formulation takes the form of the ....

....RARX nB = RXRBnB = RX nB and we conclude that RX nB is equal to nA , the eigenvector of RA associated with the unit eigenvalue: nA = RX nB (18) To conclude, solving for AX = XB is equivalent to solving for eq. 18) and for eq. 16) Solutions were proposed, among others, by Shiu Ahmad [SA89], Tsai Lenz [TL89] Chou Kamel [CK91] and Wang [Wan92] All these authors noticed that at least three positions are necessary in order to uniquely determine X, i.e. RX and t X . Shiu Ahmad cast the rotation determination problem into the problem of solving for 8 linear equations in 4 ....

Y. C. Shiu and S. Ahmad. Calibration of wrist mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Journal of Robotics and Automation, 5(1):16--29, February 1989.

.... R 0 n b R Rm b R) 2) The motion of a plane can be seen as the motion of the dual of the point M (n id) M = Rn R i(d (Rn R) Delta t) 3) 3 Motors for hand eye calibration as a case of motion of lines The well known hand eye equation firstly formulated by Shiu and Ahmad [5] reads AX = XB (4) where A = A 1 A Gamma1 2 and B = B 1 B Gamma1 2 express the elimination of the transformation hand base to world. Equation (4) can be reformulated as A Gamma1 2 A 1 Y = YB. Now if A Gamma1 2 A 1 is written as a function of the projection parameters it is possible ....

Shiu Y.C. and Ahmad S. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Trans. Robotics and Automation, 5:16--27, 1989.

.... specifically, hand eye calibration consists in determining the rigid transformation (i.e. a rotation and a translation) between a robot reference frame (usually, the end effector frame) and the visual system reference frame (usually, in monocular systems, the camera frame) The current approaches [1, 2, 6, 7, 9, 10, 11, 12] operate within a standard off line calibration scheme. However, we would like to free ourselves from the human intervention required by such approaches. The goal of this paper is therefore to propose an on line calibration method, i.e. without a priori knowledge or any calibration object ....

Y. C. Shiu and S. Ahmad. Calibration of wrist mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Journal of Robotics and Automation, 5(1):16--29, February 1989.

....robot end effector gripper frame and camera frame has to be determined by the calibration procedure which logically connects the two Cartesian spaces, i.e. measurements in one space can be referred to the other. Many approaches have been proposed to find the corresponding transformation matrix, [7, 9, 1, 11, 4]. They all try to solve a transform equation of the form AX=XB, where A is a relative transform between two different endeffector frames known from the robots kinematic model and joint measurements, and B is the corresponding relative transform between the two camera frames known from camera ....

Y. C. Shiu and S. Ahmad. Calibration of wristmounted robotic sensors by solving homogeneous tranform equations of the form AX=XB. IEEE Transactions on Robotics and Automation, 5(1):16-- 29, February 1989.

....RR 1 = R 2 R (48) 12 This idea is considerably developped in [35] I Gamma R 2 )t = 2 t 2 Gamma 1 Rt 1 (49) where 1 and 2 are unknown scale factors associated to D 1 and D 2 , respectively. The rst equation has been much studied in the framework of hand eye calibration [4] [44], 52] 3] Thus the reader is refered to those references for a more detailed analysis of unicity and sensitivity to noise. We just indicate bellow how, if we perform two displacements of the stereo rig, we can solve the two resulting matrix equations (48) to compute R, and point out to some ....

Y.S. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Transactions on robotics and automation, 5(1):1629, 1989.

....with unknown relative positions. The capability of utilizing multiple points is important in improving robustness and reducing the number of robot motions required. For the camera on hand configuration, earlier work on hand eye calibration assume that the cameras have been calibrated in advance [21], 24] By moving the robot hand to at least three stations, the hand eye calibration problem was shown to be equivalent to solving equations of the form A i X = XB i [21] 24] 30] 25] In this approach, the robot motion matrix A i is calculated from the known robotic kinematics; while the ....

.... the camera on hand configuration, earlier work on hand eye calibration assume that the cameras have been calibrated in advance [21] 24] By moving the robot hand to at least three stations, the hand eye calibration problem was shown to be equivalent to solving equations of the form A i X = XB i [21], 24] 30] 25] In this approach, the robot motion matrix A i is calculated from the known robotic kinematics; while the camera motion matrix B i is determined by camera extrinsic calibration in terms of a known control field [24] Recently, Zhuang [31] et. al calibrated a hand camera, the ....

[Article contains additional citation context not shown here]

Y.C. Shiu, S. Ahmad, "Calibration of wrist-mounted robotic sensors by solving homogeneous transformation equations of the form AX = XB," IEEE Trans. Robotics and Automat., Vol.RA-5, No.1, pp.16-29, 1989.

....can be decomposed in the following two matrix equations: RR 1 = R 2 R (26) I Gamma R 2 )t = 2 t 2 Gamma 1 Rt 1 (27) where 1 and 2 are unknown scale factors associated to D 1 and D 2 , respectively. The first equation has been much studied in the framework of hand eye calibration [4] [42], 46] 3] Thus the reader is refered to those references for a more detailed analysis of unicity 22 Quang Tuan LUONG Olivier FAUGERAS and sensitivity. We just show below that if we do two displacements of the stereo rig, we can solve the two resulting matrix equations (26) to compute R. The ....

Y.S. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Transactions on robotics and automation, 5(1):16--29, 1989.

....and orientations by the sensors. Then, the residual errors of the plane equations are minimized with respect to the laser position parameters and the plane s coordinates. As for hand eye calibration, i.e. the determination of the mounting parameters of a sensor on a robot hand, Shiu and Ahmad [8] proposed to solve a homogeneous equation of the kind A i X = XB i by making sensor measurement at different robot stations, where A i is the the robot motion matrix for the ith station, which can be calculated from the known robotic kinematics; and B i is the corresponding sensor motion matrix ....

....system. What we need is only a flat plane, which is arbitrarily oriented in the 3D space. Of course, assumptions are made of the known robot motions fR g0j ; t g0j g. This has been an assumption adopted in most previous work on hand eye calibration, e.g. Tsai and Lenz [9] Shiu and Ahmad [8], to name a few. Since (7) is highly nonlinear in the unknowns, its solution may need good initial guesses for the unknown parameters. With the initial values, standard techniques like the Newton Raphson method can be used to iteratively adjust the unknown parameters until convergence is reached. ....

Y.C. Shiu, S. Ahmad, "Calibration of wrist-mounted robotic sensors by solving homogeneous transformation equations of the form AX = XB," IEEE Trans. Robotics Automat., Vol.RA-5, No.1, pp.16-29, 1989.

.... equation (23) the motion equation of the plane is M (n id) M = Rn R i( GammaRn R t 2 t 2 Rn R) id = Rn R i(d Gamma Rn Rt) Rn R i(d Gamma RnRt ) 32) 4 The Hand Eye Problem The well known hand eye equation firstly formulated by Shiu and Ahmad [8] and Tsai and Lenz [10] reads AX = XB (33) where A = A 1 A Gamma1 2 and B = B 1 B Gamma1 2 express the elimination of the transformation hand base to world. From the expression equation (33) the following matrix and a vector equations can be derived RARX = RXRB and (RA Gamma I) t X = ....

Shiu Y.C. and Ahmad S. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Trans. Robotics and Automation, 5:16--27, 1989.

No context found.

Y. Shiu and S. Ahmad. Calibration of Wrist Mounted Robotic Sensors by Solving Homogeneous Transform Equations of the Form AX = XB. IEEE Trans. on Robotics and Automation, 5(1):16--29, 1989.

No context found.

Y. Shiu and S. Ahmad. Calibration of Wrist Mounted Robotic Sensors by Solving Homogeneous Transform Equations of the Form AX = XB. IEEE Trans. on Robotics and Automation, 5(1):16--29, February 1989.

No context found.

Y. Shiu and S. Ahmad. Calibration of Wrist Mounted Robotic Sensors by Solving Homogeneous Transform Equations of the Form AX = XB. IEEE Trans. on Robotics and Automation, 5(1):16--29, February 1989.

No context found.

Y. Shiu and S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Transactions on Robotics and Automation, 5(1):16--29, February 1989.

No context found.

Y. C. Shiu and S. Ahmad. Calibration of wristmounted robotic sensors by solving homogeneous tranform equations of the form AX=XB. IEEE Transactions on Robotics and Automation, 5(1):16-- 29, February 1989.

No context found.

Y. Shiu and S. Ahmad. Calibration of Wrist Mounted Robotic Sensors by Solving Homogeneous Transform Equations of the Form AX = XB. IEEE Trans. on Robotics and Automation, 5(1):16--29, February 1989.

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