An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems

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1 An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Sstems Mili Shah Loola Universit Marland 4501 North Charles Street Baltimore, MD, Roger D. Eastman Loola Universit Marland 4501 North Charles Street Baltimore, MD, Tsai Hong National Institute of Standards and Technolog 100 Bureau Drive Gaithersburg, MD, ABSTRACT In this paper, an overview of methods that solve the robotsensor calibration problem of the forms A B and A YB is given. Each form will be split into three solutions: separable closed-form solutions, simultaneous closedform solutions, and iterative solutions. The advantages and disadvantages of each of the solutions in the case of evaluation of perception sstems will also be discussed. Categories and Subject Descriptors C.4 [Performance of Sstems]: Performance attributes; B.8.2 [Performance and Reliabilit]: Performance Analsis and Design Aids; G.1.6 [Optimiation]: Global optimiation; I.4.8 [Scene Analsis]: Motion, Tracking; I.5.4 [Applications]: Computer Vision General Terms Computer Vision, Robot-Sensor Calibration, Hand-Ee Calibration, Performance Evaluation 1. INTRODUCTION Robot-sensor calibration has been an active area of research for man decades. The most common mathematical representations for the robot-sensor calibration problem consist of two forms: A B and A YB. Eamples for each of the forms can be seen in Figure 1. Specificall in Figure 1a, A i represents robot motion, B i represents camera motion, and the unknown represents the fied homogeneous transformation between the robot base and camera. Following the arrows, it can easil be seen that A i B i A B, where A A i and B B i. Similarl in Figure 1b, A i represents the transformation from robot base to gripper, B i represents the transformation from camera to object, and c 2012 Association for Computing Machiner. ACM acknowledges that this contribution was authored or co-authored b a contractor or affiliate of the U.S. Government. As such, the Government retains a noneclusive, roalt-free right to publish or reproduce this article, or to allow others to do so, for Government purposes onl. PerMIS 12, March 20-22, 2012, College Park, MD, USA. Copright c 2012 ACM /22/12...$ the unknown represents the fied homogeneous transformation between gripper and camera. Following the arrows A 1B 1 A 2B 2 A 1 2 A 1 B 2B 1 1 A B, where A A 1 2 A1 and B B2B 1 1. Finall in Figure 1c, A i represents the transformation from target to sensor, B i represents the transformation from camera to object, the unknown represents the fied homogeneous transformation between sensor and object, and the unknown Y represents the fied homogeneous transformation between target and camera. Following the arrows A i YB i A YB, where A A i and B B i. In this paper, we will give an overview of methods to solve A B and A YB. Notice that for A B RA t A R t R t RA R R A t + t A Thus, R A R R R B, RB t B R R B R t B + t which we will define as the orientational component, and R A t + t A R t B + t., which we will define as the positional component for A B. The orientational component and positional component R A R R Y R B, R A t + t A R Y t B + t Y for A YB can similarl be constructed. The methods to solve A B and A YB consist of three forms: separable closed-form solutions, simultaneous closed-form solutions, and iterative closed-form solutions. The separable closed-form solutions arise from solving the orientational component separatel from the positional component, the simultaneous closed-form solutions arise from simultaneousl solving the orientational component and the positional component, while the iterative solutions arise from solving both the orientational component and positional component iterativel using optimiation techniques. Details of each of the

2 b separating the problem into its orientational component Robot Base Ai Robot Base Bi a A B where the unknown represents the transformation from robot base to camera. A 1 B1 Gripper Object B 2 A 2 b A B where the unknown represents the transformation from gripper to camera. Y Target Sensor Aj Bj Object c A YB where the unknown represents the transformation from sensor to object and the unknown Y represents the transformation from target to camera. Figure 1: Different eperimental setups for robotsensor calibration. solutions will be discussed in the following sections. Specificall for A B, separable closed-form solutions will be discussed in Section 2.1, simultaneous closed-form solutions will be discussed in Section 2.2, and iterative solutions will be discussed in Section 2.3. Following, in Section 3, will be a section discussing the different solutions for A YB. Finall, concluding remarks, which will include the advantages and disadvantages for each of the solutions in the evaluation of perception sstems, will be discussed in Section AB SOLUTIONS 2.1 Separable Solutions for AB The robot-sensor calibration problem of the form A B was introduced in the work of Shiu and Ahmad [21]. In this paper, the solve the robot-sensor calibration problem and positional component R A R R R B R A t + t A R t B + t. The solve the orientational component b utiliing the angleais formulation of rotation; i.e., let R Rotk R, θ, where k R is the ais of rotation of R and θ is the angle. Specificall, the state that the general solution where R Rotk Ai, β ir Pi, R Pi Rotv, ω v k Bi k Ai ω atan2 k Bi k Ai, k Bi k Ai and β i is calculated b solving a 9 2n linear sstem of equations where the number of frames n 2. The also prove for uniqueness at least two of the aes of rotation of R Ai cannot be parallel. Once R is formulated, the positional component R A1 I R t B1 t A1. R An I t. R t Bn t An can be solved using standard linear sstem techniques. This is the general technique of separable solutions for A B: first calculate R using some technique and then use that R to solve for t using standard linear sstem techniques. Thus, for the rest of this section concentration will be placed solel on calculating the optimal rotation R. A problem with the Shiu and Ahmad method is that the sie of the linear sstem doubles each time a new frame is added to the sstem. An alternative method b Tsai and Len [23] solves the robot-sensor calibration method using a fied sie linear sstem. The derivation is simpler than the Shiu and Ahmad method and computationall more efficient. Specificall, Tsai and Len solve the orientational component b again considering the angle-ais formulation R Rotk R, θ for rotation. The find the ais of rotation k R for R b solving Sk k RAi + k RBi k R k RAi k RBi 1 k R 2k R 1 + k R 2 where the skew-smmetric matri Sk 3 0 1, and the angle of rotation θ for R b setting θ 2atan k R. Another formulation that utilies the angle-ais formulation was presented b Wang in [24]. The solve the orientational component b considering the properties of the aes of rotation of R Ai, R Bi, R Ai+1, and R Bi+1 for i 1, 2,... n 1. Wang compares his method with the Shiu and

3 Ahmad method [21] and the Tsai and Len method [23]. He concludes that of the three methods, the Tsai and Len method is the best on average. The angle-ais methods for calculating the solution of the robot-sensor calibration problem up to this point can be cumbersome. In order to simplif the problem, Park and Martin formed a solution for R b taking advantage of Lie group theor to transform the orientational component into a linear sstem [17]. Specificall, the take advantage of the propert that for a given rotation matri R log R θ R R T Skr. 2 sin θ Here, r θk R where θ is the angle of rotation of R and k R is the ais of rotation of R. For this paper, r is the shorthand notation of log R. Using this formulation, R Ai R R R Bi R a i b i where a i and b i are the shorthand logarithms of A i and B i, respectivel. In the presence of noise, Park and Martin calculate the solution of the robot-sensor problem b solving n min R a i b i 2, R i1 whose closed-form solution can be calculated efficientl as R UV 1/2 U 1 M T where M n i1 biat i and the eigendecomposition of M T M UVU 1. Chou and Kamel introduce quaternions into the robotsensor calibration problem in [4, 5]. The notice that the orientational component R A R R R B q A q q q B where q is the quaternion representation of the rotation matri R. Using the matri form of quaternion multiplication, the orientational component can be restructured into a linear sstem since q A q q q B q A q q B q q A q B q 0 0 q q B T b0 0I + Sk b 0b 0 T b b 0 + 0I + Sk b b 0 0 b T b 0 + b 0I Skb b0 b T 0 b b 0I Skb q B q. Chou and Kamel solve the linear sstem using the singular value decomposition. Horaud and Dornaika form another closed-form solution for R via quaternions in [11]. Specificall, the find that the quaternion representation q for R can be found as the eigenvector associated with the smallest positive eigenvalue of n A A T i A i i1 where A i a i a i a i 0 a i + b i a i b i 0 a i b i b i a i a i + b i b i a i a i + b i 0 a i b i a i + b i 0 + b i + b i b i and a i a i, a i, a i T is the ais of rotation for A i and b i b i, b i, b i T is the ais of rotation for B i. Zhuang and Roth also appl quaternions to the robotsensor calibration problem in [28] to get a closed-form solution that is ver similar in formulation to the angle-ais formulation 1 of Tsai and Len [23]. Liang et al. appl the Kronecker product to the orientational component of the robot-sensor problem to solve for R in [14]. As a result, the orientational component becomes the linear sstem R A1 I I R T B 1. vec R 0. 2 R An I I R T B }{{ n } L Here the Kronecker product a 1,1B a 1,nB A B , a m,1b a m,nb where a i,j is the i, j-th element of A, and veca vectories a matri A column-wise. Liang et al. solve sstem 2 b Here, 1. Calculating the eigenvector corresponding to the smallest eigenvalue of L 2. Forming Y vec 1 3. Setting R UV T where the singular value decomposition of Y USV T A { A if deta 0 A if deta < 0. For all these separable solutions, errors in the calculation of the optimal rotation R get carried into the calculations of the optimal translation t. In order to minimie these errors, simultaneous solutions for A B were created. However, these solutions have their own problems as will be discussed. 2.2 Simultaneous Solutions for AB Chen in [3] believes that separating the orientational component from the positional component, which implies that one has nothing to do with the other, is invalid. Thus, Chen creates a new solution, based on screw theor, that simultaneousl solves the orientational component with the positional component. Specificall, he finds that the A B problem can be reduced to an absolute orientation problem of finding the best rigid transformation R and t that transforms the camera screw ais to the robot screw ais. Daniilidis and Baro-Corrochano describe an algebraic interpretation of Chen s screw theor method via dual quaternions in [6, 7]. Specificall, the use the vector portions from

4 the dual-quaternion representations a i + a i and b i + b i of A i and B i respectivel to create the matri T T S T 1 S T 2... S T n ai b i Sk ai + b i 0 0 S i a i a b i Sk i + b ai i b i Sk ai + b i Using the singular value decomposition on T, Daniilidis and Baro-Corrochano show that the dual-quaternion representation for the unknown can be calculated as a linear combination of the last two right singular vectors of T. It should be noted that the authors developed a similar method through the use of Clifford Algebra in [2]. Zhao and Liu also develop a similar method through the algebraic properties of screw theor in [27]. Lu and Chou [15] appl the quaternions via the eight step method to solve the robot-sensor calibration problem simultaneousl. Specificall, b the use of quaternions, the can simplif the problem to a single linear sstem which the solve using Gaussian elimination and Schur decomposition. Andreff et al. are the first to appl the Kronecker product to simultaneousl solve the robot-sensor problem in [1]. The reformulate the robot-sensor problem into a linear sstem of the form I RBi R Ai 0 t T B i I I R Ai vecr 0. t t Ai Andreff et al. prove that at least two independent general motions with non-parallel aes are needed to have a unique solution to the linear sstem. A problem with this method is that due to noise the solution for R ma not necessaril be an orthogonal matri. Thus, an orthogonaliation step for the orientational component has to be taken. However, the corresponding positional component is not recalculated, which causes errors in the solution. Therefore, Andreff et al. suggest separating the orientational and positional components as was shown in the work of Liang et al. see Section 2.1 in [14]. 2.3 Iterative Solutions for AB Simultaneous solutions were developed to solve the problem of orientational errors propagating into the positional errors. Another option to solve this problem is to create an iterative solution for A B. Zhuang and Shiu propose a one-step iterative method, based on minimiing A B with the Levenberg-Marquardt algorithm in [30]. The iterative method solves both the orientational and positional components simultaneousl. Furthermore, the method is not dependent on robot orientation R Bi information. Fassi and Legnani propose a similar algorithm in [9]. This paper also provides a geometric interpretation of the hand-ee calibration problem. Wei et al. [25] create an efficient iterative method that is optimied b the sparse structure of the corresponding normal equations. Horaud and Dornaika in [11] also propose to solve the orientational and positional components simultaneousl using an iterative method. However, their method is based on using the quaternion representation for the orientational component. Mao et al. [16] appl the Kronecker product in their iterative formulation. An issue with the Mao et al. optimiation problem is that the solution is based on the initial condition. Therefore, different initial conditions could result in varing solutions. A remed to this problem is to use conve optimiation as shown in the work of Zhao [26]. Zhao claims that his Kronecker product algorithm is ver fast and not dependent on an initial condition. However, their setup gives no guarantee that the orientational component R of the solution is a rotation matri. Therefore, his algorithm ma cause errors that are similar to the errors of Andreff et al. [1]. Shi et al. [20] have a similar formulation to Zhao thus similar problems, but their iterative algorithm optimies motion selection to improve accurac and to avoid degenerate cases. Strobl and Hiringer create an iterative method that is based on a parameteriation of a stochastic model in [22]. This iterative method is novel since it creates an inherent algorithm to weight the orientational and positional components to optimie the accurac of the method. Kim et al. etend this formulation in [12] with the use of the Minimum Variance method. These iterative methods get rid of the propagation of orientational errors into the positional component. However, solving the robot-sensor calibration method in this manner can be computationall taing since these methods often contain comple optimiation routines. In addition, as the number of equations n gets larger, the differences between iterative solutions and closed-form solutions often get smaller. Thus, one has to decide whether the accurac of an iterative solution is worth the computational costs. 3. AYB SOLUTIONS In this section we will give an overview of techniques to solve A YB. The methods for solving this sstem are ver similar to the A B problems, i.e., the methods can be organied into three groups: separable solutions, simultaneous solutions, and iterative solutions. Wang proposes the A YB problem in [24], though he assumes that one of the unknowns is given. Zhuang et al. were the first to give a separable closed-form solution via quaternions in [29]. Dornaika and Horaud etend Zhuang et al. s separable solution to give a more accurate separable closed-form solution via quaternions in [8]. Shah creates a formulation based on Kronecker product in [19]. Li et al. look at simultaneous closed-form solutions via dual-quaternions and Kronecker products in [13]. Their formulations follow the methodolog of the A B formulation of dual quaternions of Daniildis [7] and the formulation of Kronecker product of Andreff et al. [1]. Iterative solutions for the A YB problem were first introduced in the work of Rem et al. [18]. Here the define a nonlinear optimiation problem and use the Levenberg- Marquardt method to solve it. Hirsh et al. develop an iterative method in [10] that optimies the orientational and positional components separatel, while Strobl and Hiringer create an iterative method [22] that simultaneousl solves the orientational and positional components. Their method is based on a parameteriation of a stochastic model which is identical to their A B model. Kim et al. also use a model [12] identical to their A B model to simultaneousl solve A YB using the Minimum Variance method. 4. CONCLUSION In this paper, we give an overview of methods to solve the

5 robot-sensor calibration problem of the forms A B and A YB for the evaluation of perception sstems. Each form s solutions can be split into three categories: separable solutions, simultaneous solutions, and iterative solutions. The separable solutions are simple and fast solutions; however, errors calculated from the orientational component get carried over to the positional component. As a result, simultaneous solutions were developed. However, these solutions produce variable results depending on the scaling of the positional component. To weight the orientational and positional components, iterative methods were created. However, though these solutions are often more accurate, the solutions are often comple and generall depend on starting criteria. In addition, there is generall no guarantee that the convergent solution is the optimal solution. Thus, users must decide which tpe of method to use for evaluation which is dependent on their desired accurac and compleit. 5. REFERENCES [1] N. Andreff, R. Horaud, B. Espiau On-line Hand-Ee Calibration. In Second International Conference on 3-D Digital Imaging and Modeling 3DIM 99, pages , [2] E. Baro-Corrochano, K. Daniilidis, G. Sommer Motor Algebra for 3D Kinematics: The Case of the Hand-Ee Calibration. In Journal of Mathematical Imaging and Vision, 13: , [3] H. H. Chen A screw motion approach to uniqueness analsis of head-ee geometr. In IEEE Proceedings of Computer Vision and Pattern Recognition CVPR 91, pages , [4] J. C. K. Chou, M. Kamel Quaternions approach to solve the kinematic equation of rotation, A aa A A b, of a sensor-mounted robotic manipulator. In IEEE International Conference on Robotics and Automation, pages , [5] J. C. K. Chou, M. Kamel Finding the Position and Orientation of a Sensor on a Robot Manipulator Using Quaternions. In The International Journal of Robotics Research, 103: , [6] K. Daniilidis, E. Baro-Corrochano The dual quaternion approach to hand-ee calibration. In Proceedings of the 13th International Conference on Pattern Recognition, pages , [7] K. Daniilidis Hand-Ee Calibration Using Dual Quaternions. In The International Journal of Robotics Research,183: , [8] F. Dornaika, R. Horaud, Simultaneous robot-world and hand-ee calibration, IEEE Transactions on Robotics and Automation, 144: , [9] I. Fassi, G. Legnani Hand to Sensor Calibration: A Geometrical Interpretation of the Matri Equation AB. In Journal of Robotic Sstems, 229: , [10] R. L. Hirsh, G. N. DeSoua, A. C. Kak An Iterative Approach to the Hand-Ee and Base-World Calibration Problem. In Proceedings of the 2001 IEEE International Conference on Robotics and Automation, 3: , [11] R. Horaud, F. Dornaika Hand-Ee Calibration In International Journal of Robotics Research, 143: , [12] S. Kim, M. Jeong, J. Lee, J. Lee, K. Kim, B. You, S. Oh Robot Head-Ee Calibration Using the Minimum Variance Method. In Proceedings of the 2010 IEEE International Conference on Robotics and Biomimetics, pages , [13] A. Li, L. Wang, D. Wu Simultaneous robot-world and hand-ee calibration using dual-quaternions and kronecker product. InInternational Journal of the Phsical Sciences, 510: , [14] R. Liang, J. Mao Hand-Ee Calibration with a New Linear Decomposition Algorithm. In Journal of Zhejiang Universit, 910: , [15] Y. Lu, J. C. K. Chou Eight-Space Quaternion Approach for Robotic Hand-ee Calibration. In Sstems, IEEE International Conference on Man and Cbernetics, 4: , [16] J. Mao,. Huang, L. Jiang A Fleible Solution to AB for Robot Hand-Ee Calibration. In Proceedings of the 10th WSEAS International Conference on Robotics, Control and Manufacturing Technolog, pages , [17] F. Park, B. Martin Robot Sensor Calibration: Solving A B on the Euclidean Group. In IEEE Transactions on Robotics and Automation, 105: , [18] S. Rem, M. Dhome, J. M. Lavest, N. Daucher Hand-ee Calibration. In Proceedings of the 1997 IEEE/RSJ International Conference on Intelligent Robots and Sstems, 2: , [19] M. I. Shah. Solving the Robot-World/Hand-Ee Calibration Problem using the Kronecker Product. Technical report, Department of Mathematics and Statistics, Loola Universit in Marland, [20] F. Shi, J. Wang, Y. Liu An Approach to Improve Online Hand-Ee Calibration. In Pattern Recognition and Image Analsis, 3522: , [21] Y. Shiu, S. Ahmad Calibration of Wrist-Mounted Robotic Sensors b Solving Homogeneous Transform Equations of the Form A B. In IEEE Transactions on Robotics and Automation, 51:16 29, [22] K. H. Strobl, G. Hiringer Optimal Hand-Ee Calibration. In Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Sstems, pages , [23] R. Tsai, R. Len A New Technique for Full Autonomous and Efficient 3D Robotics Hand/Ee Calibration. In IEEE Transactions on Robotics and Automation, 53: , [24] C. Wang Etrinsic Calibration of a Vision Sensor Mounted on a Robot. In IEEE Transactions on Robotics and Automation, 82: , [25] G. Wei, K. Arbter, G. Hiringer Active self-calibration of robotic ees and hand-ee relationships with model identification. In IEEE Transactions on Robotics and Automation, 141: , [26] Z. Zhao Hand-Ee Calibration Using Conve Optimiation. In 2011 IEEE International Conference on Robotics and Automation, pages , [27] Z. Zhao, Y. Liu Hand-Ee Calibration Based on Screw

6 Motions. In 18th International Conference on Pattern RecognitionICPR 06, pages , [28] H. Zhuang, Z. S. Roth Comments on Calibration of Wrist-Mounted Robotic Sensors b Solving Homogeneous Transform Equations of the Form A B. In IEEE Transactions on Robotics and Automation, 76: , [29] H. Zhuang, Z. S. Roth, R. Sudhakar Simultaneous Robot/World and Tool/Flange Calibration b Solving Homogeneous Transformation Equations of the form AYB. In IEEE Transactions on Robotics and Automation, 104: , [30] H. Zhuang, Y. C. Shiu A Noise-Tolerant Algorithm for Robotic Hand-Ee Calibration with or without Sensor Orientation Measurement. In IEEE Transactions on Sstems, Man, and Cbernetics, 234: , 1993.