Investigates robot calibration with planar constraints, and in particular the conditions for the parameters of the robot kinematic model to be observable. Mainly multiple-plane constraints for robot calibration are considered. It is first shown that a single-plane constraint is normally not sufficient to calibrate a robot. It is also proven that by using a three-plane constraint, the constrained system is equivalent to an unconstrained point-measurement system under certain conditions. The significance of this observation is that one can use the three-plane constraint setup to successfully calibrate a robot. Simulations have been conducted to verify the theory presented in the paper.

A single-stage linear method is devised in this paper to simultaneously fit rotation and translation (pose) parameters given two sets of 3-D point measurements. The necessary and sufficient conditions for the unique solution of the pose determination problem are stated. The computational complexity of the new algorithm is similar to the existing linear algorithms. However it offers a mechanism to incorporate the reliability of measurements and a procedure to implement the estimation recursively. Applications of the technique include localization of a robot in its environment and real-time estimation of object motion based on computer vision.

Robot calibration is an effective and economical means for enhancing the accuracy performance of a robot manipulator through modification of its control software. This paper reports some research results by applying Lenz and Tsai's approach (1989) to calibrate a SCARA arm equipped with a hand-mounted camera. In order to measure robot poses, a new technique was employed to calibrate the camera at various robot configurations. The camera calibration technique is singularity free even when the image plane is nearly-parallel to the camera calibration board. Second, the modified complete and parametrically continuous (MCPC) model was used to describe the geometry of the SCARA arms because there is no model singularity in the MCPC model for this type of robots. Experimental studies were conducted to demonstrate the feasibility of the present approach for calibrating SCARA arms. Some practical recommendations are also made for robot users who need to calibrate SCARA arms.

The solution method presented previously by us ( ibid. vol.9, no.2, p.174-85, 1993) employs the complete and parametrically continuous (CPC) model. In the first step of this approach, all CPC orientation parameters related to revolute joints, are sequentially determined. In the second step, CPC translation parameters, together with orientation parameters of prismatic joints, are simultaneously computed. While our approach eliminates propagation errors in the estimation of translation parameters, it has several drawbacks. In this paper we propose a modification to the linear approach of the previous method. This modification not only eliminates the two problems of the original method but also preserves the advantage of solving for the robot translation parameters simultaneously.

A popular configuration widely used in a variety of robotic applications is to mount a camera on the robot manipulator hand. Before performing a measurement task using such a system, both the camera and the robot need to be calibrated. In this paper, a procedure is developed for simultaneous calibration of a robot and a monocular camera. Unlike conventional approaches based on first calibrating the camera and then calibrating the robot, the algorithm solves for the kinematic parameters of the robot and camera in one stage, thus eliminating error propagation and improving noise sensitivity. Only two parameters are added to a robot calibration model to represent camera geometry. With this addition, different levels of calibration can be done under a unified framework. An error model relating-image measurement residuals to kinematic parameter deviations is derived. Simulation and experimental studies have been conducted to assess the effectiveness of the proposed procedure.>

Laser tracking systems based on interferometry have applications in robot and machine tool calibration. Relative distance measurements provided by laser interferometers have an extremely high resolution. However, accuracy errors of a coordinate measuring machine based on laser tracking are dominated by geometric errors in the tracking mirror system. Major geometric error sources include gimbal axis misalignments and mirror center offset. In this paper, a geometric model for a single-beam tracker is developed, in which a necessary and sufficient number of geometric parameters is used to represent these two types of error sources for arbitrary target positions. This model can be used for design, calibration and control of single-beam laser tracking measurement systems.>

Measuring robot positions and orientations is a crucial step in a robot calibration process. Off-line optimal selection of measurement configurations can significantly improve the accuracy of kinematic identification. Since the dimension of the parameter space is very large and the cost function is highly nonlinear, this selection process could be well beyond the capacity of today's computers if a global optimal solution is sought by an exhaustive search. On the other hand, gradient-based algorithms are often trapped into local minima. A simulated annealing (SA) approach is adopted in this paper to obtain optimal or near optimal measurement configurations for robot calibration. Simulated annealing is capable of overcoming local minimum points. It is also very convenient for the inclusion of joint travel limits. The SA algorithm is costly computationally; however, since optimal configuration selection can be performed off-line, this may not be a serious problem. To accelerate the convergence rate, a suitable cooling schedule is devised. Practical implementation considerations are discussed. Experimental results are presented to demonstrate the feasibility of the proposed approach.>

The paper presents a linear solution that allows a simultaneous computation of the transformations from robot world to robot base and from robot tool to robot flange coordinate frames. The flange frame is defined on the mounting surface of the end-effector. It is assumed that the robot geometry, i.e., the transformation from the robot base frame to the robot flange frame, is known with sufficient accuracy, and that robot end-effector poses are measured. The solution has applications to accurately locating a robot with respect to a reference frame, and a robot sensor with respect to a robot end-effector. The identification problem is cast as solving a system of homogeneous transformation equations of the form A/sub i/X=YB/sub i/,i=1, 2, ..., m. Quaternion algebra is applied to derive explicit linear solutions for X and Y provided that three robot pose measurements are available. Necessary and sufficient conditions for the uniqueness of the solution are stated. Computationally, the resulting solution algorithm is noniterative, fast and robust.>

A linear method for identifying the unknown kinematic parameters of a manipulator directly from the forward kinematic model is presented. The method requires the use of neither a nominal model nor a linearized error model of the robot. Such a solution is made possible by the use of a special robot kinematic modeling convention known as the complete and parametrically continuous (CPC) model, in which the independent CPC link parameters appear linearly in the system of equations to be solved, and the use of a particular sequence of robot pose measurements. The CPC orientation parameters of the revolute joints are first determined recursively under the condition that the pose measurements of the robot are taken while releasing each revolute joint one at a time and successively. The remaining CPC parameters are then computed in terms of the orientation parameters obtained earlier. Some practical issues related to kinematic parameter identification with the proposed approach are addressed through simulation studies.>

The problem of optimal design of robot accuracy compensators is addressed. Robot accuracy compensation requires that actual kinematic parameters of a robot be previously identified. Additive corrections of joint commands, including those at singular configurations, can be computed without solving the inverse kinematics problem for the actual robot. This is done by either the damped least-squares (DLS) algorithm or the linear quadratic regulator (LQR) algorithm, which is a recursive version of the DLS algorithm. The weight matrix in the performance index can be selected to achieve specific objectives, such as emphasizing end-effector's positioning accuracy over orientation accuracy or vice versa, or taking into account proximity to robot joint travel limits and singularity zones. The paper also compares the LQR and the DLS algorithms in terms of computational complexity, storage requirement, and programming convenience. Simulation results are provided to show the effectiveness of the algorithms.>

A linear solution which allows a simultaneous computation of the transformations from robot world to robot base and from robot eye to robot hand coordinate frames, is reported in this paper. It is assumed that the robot geometry, which is the transformation from the robot base frame to the robot flange frame, is accurately known, and that robot hand positions and orientations in world coordinates are measured. The solution has applications in accurate locating of the robot with respect to a reference frame, and of the robot sensor with respect to the robot end-effector. The identification problem is cast as the solution of a system of homogeneous transformation equations of the form A iX = Y Bi, i = 1, 2, ..., m. Quaternion algebra is applied to derive explicit linear solutions for X and Y provided that three robot pose measurements are available. Necessary and sufficient conditions for the uniqueness of the solution are stated. Computationally, the solution is noniterative, fast and robust. Simulation studies reveal that the method is a viable candidate for robot/world and eyehand calibration.

In the above-named work (ibid., vol.7, p.877-8, (Dec. 1991)), H. Zhuang and Z. S. Roth point out that a particular solution can be simplified by using quaternions to represent rotations. While it is true that this approach gives rise to an algorithm more efficient than the one considered, the commenter argues that it is incorrect to claim that a unique solution for R/sub X/ exists if and only if the axes of rotation of R/sub A1/ and R/sub A2/ are nonzero (theorem 1 of the original work). A counterexample is presented to prove this point. In replying, Zhuang and Roth note the error in the original work and provide an analysis leading to the revision of their original theorem 1.>

A kinematic modeling convention for robot manipulators is proposed. The kinematic model is named for its completeness and parametric continuity (CPC) properties. Parametric continuity of the CPC model is achieved by adopting a singularity-free line representation consisting of four line parameters. Completeness is achieved through adding two link parameters to allow arbitrary placement of link coordinate frames. The transformations from the world frame to the base frame and from the last link frame to the tool frame can be modeled with the same modeling convention used for internal link transformations. Since all the redundant parameters in the CPC model can be systematically eliminated, a linearized robot error model can be constructed in which all error parameters are independent and span the entire geometric error space. The focus is on model construction, mappings between the CPC model and the Denavit-Hartenberg model, the study of the model properties, and its application to robot kinematic calibration.>

A closed form solution for the unknown kinematic parameters of a robot manipulator obtained directly from the forward kinematic model is described. The method requires the use of neither a nominal model nor a linearized error model of the robot. Such a solution is possible because of (1) the use of a special robot kinematic modeling convention, namely the CPC model, and (2) the use of a particular sequence of robot pose measurements. The CPC orientational parameters of the revolute joints are determined recursively under the condition that the pose measurements of the robot are taken while moving the revolute joints one at a time and successively. The remaining CPC parameters are then computed in terms of the orientational parameters obtained earlier.>

The commenters point out that the derivation of the closed-form solution to the homogeneous transform equation AX=XB by Y.C. Shiu and S. Ahmad (see ibid., vol.5, no.1, p.16-27, Feb.1989), although containing many useful ideas, is somewhat lengthy. It can be presented much more compactly using quaternion algebra. Direct benefits of such an approach are presented. In their reply, Shiu and Ahmad admit that the commenters' method has significant advantages over the original solution for the rotational part. However, it does not provide the geometric insight that the solution for AX=XB has a rotational degree of freedom about k/sub A/ (the axes of rotation of A). The solution to the translational part of X discussed in the original paper is not affected by this discussion since its computation is not dependent on how the rotational part is computed.>

A kinematic modeling convention for robot manipulators is proposed. The kinematic model has complete and parametrically continuous (CPC) properties. The parametric continuity of the CPC model is achieved by adopting a singularity-free line representation. Completeness is achieved through adding two link parameters which allow arbitrary placement of link coordinate frames. The transformation from the base frame to the world frame and from the tool frame to the last link frame can be modeled with the same convention as that used for internal link transformations. These parameters make the CPC model particularly useful for robot calibration.>

The design of a kinematic accuracy compensator for a robot manipulator by using linear optimal control theory is discussed. The method is based on the assumption that either the actual kinematic parameters of the robot have been previously identified or that the pose errors of the manipulator can be measured online. A general mathematical framework is used, so that any linearized error model derived from the corresponding kinematic model can be used to construct an effective robot accuracy compensator. The additive corrections of joint commands are found by a linear quadratic regulator algorithm without explicitly solving the inverse kinematic problem for the actual robot. The weighting matrix and coefficients in the cost function can be chosen systematically to achieve specific objectives. It the poses of the manipulator can be measured online, a parameter identification phase of the robot calibration process can be eliminated, thus avoiding the need to identify all the error sources. A simplified algorithm is presented that accelerates significantly the process speed, making it suitable for real-time applications.>

An overview is given of the existing work on robot calibration, and some of the basic issues are identified in calibration and improvement of robot precision. Modeling, measurement, identification, and correction issues in robot calibration are discussed, and some of the unresolved questions are identified.