710 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 11, NO. 5, OCTOBER 1995 An Implicit Loop Method for Kinematic Calibration and Its Application to Closed-Chain Mechanisms Charles W. Wampler, Member, IEEE, John M. Hollerbach, Member, IEEE, and Tatsuo Arai, Member, IEEE Abstract— A unified formulation for the calibration of both serial-link robots and robotic mechanisms having kinematic closed-loops is presented and applied experimentally to two 6- degree-of-freedom devices: the RSI 6-DOF Hand Controller and the MEL “Modified Stewart Platform.” The unification is based on an equivalence between end-effector measurements and con- straints imposed by the closure of kinematic loops. Errors are al- located to the joints such that the loop equations are satisfied ex- actly, which eliminates the issue of equation scaling and simplifies the treatment of multi-loop mechanisms. For the experiments reported here, no external measuring devices are used; instead we rely on measurements of displacements in some of the passive joints of the devices. Using a priori estimates of the statistics of the measurement errors and the parameter errors, the method esti- mates the parameters and their accuracy, and tests for unmod- elled factors. I. I NTRODUCTION T HE implicit loop method for kinematic calibration is founded on an equivalence between displacement mea- surements and kinematic closed-loops. For example, docking a robot’s end-effector into a mechanical fixture that determines its position and orientation is equivalent to a measurement of the location of the end-effector by other means, such as theodolites or laser interferometry. Clearances between mat- ing surfaces of the fixture and end-effector yield uncertain- ties that correspond to measurement error. If we consider the end-effector to have a (typically) 6 degree-of-freedom “joint” with respect to ground, measurements of the end-effector lo- cation may be regarded as joint measurements. With this con- vention, the kinematic model of any mechanism—open-chain or closed-chain—becomes a statement that the displacements around a closed loop must sum to zero. Whenever the total number of joint measurements (includ- ing the end-effector “joint”) exceeds the number of degrees of freedom of motion, the kinematic equations predict dependen- cies between the measured quantities. A set of measurements Manuscript received June 24, 1994; revised January 6, 1995. This work was supported in part by a Japan Science and Technology Agency (STA) Fellowship, by NSF under Grant INT-9120186, by the Natural Sci- ences and Engineering Research Council (NSERC) Network Centers of Ex- cellence Institute for Robotics and Intelligent Systems (IRIS), and by the NSERC/Canadian Institute for Advanced Research (CIAR) Industrial Re- search Chair in Robotics. C.W. Wampler is with the Mathematics Department, General Motors Re- search and Development, 30500 Mound Road, Box 9055, Warren, MI 48090. J.M. Hollerbach is with the Department of Computer Science, University of Utah, 3335 Merrill Engineering Bldg., Salt Lake City, UT 84112. T. Arai is with the Mechanical Engineering Laboratory, Namiki 1-2, Tsukuba, Ibaraki 305, Japan. IEEE Log Number 9411918. taken at various poses of the mechanism is not likely to sat- isfy these equations exactly; such discrepancies must be ex- plained by some combination of measurement error and error in the parameters of the kinematic model (including also the sensor parameters, such as gains or offsets). We look for the most likely combination of such errors that satisfy the kine- matic equations exactly. The answer depends upon statistical models of the distribution of both measurement errors and er- rors in the parameters of the mechanism. The result will be our best estimate of the true values of the parameters in light of the given measurements. The term “implicit loop method” emphasizes that the er- rors enter the kinematic loop equations implicitly, rather than being explicit outputs of a conventional input-output formula- tion. By removing the requirement to express errors explicitly, the formulation allows the analyst to concentrate on correctly attributingall sources of error. For example, a typical formula- tion for a serial-link robot finds kinematic parameters that min- imize the difference between the measured end-effector loca- tion and the prediction of the model. But the differences may in fact be due to errors in the joint angle measurements (“in- put noise,” in statistical parlance). By not acknowledging this potentially significant source of error, the parameter estimates may be biased. In contrast, the implicit loop method puts joint and end-effector measurements on equal footing, with weights assigned according to the accuracy of each. A more complete picture of the implicit loop method’s place in the context of current practice is available in [1]. The method is illustrated in experiments on two in-parallel mechanisms. Overviews of the robot calibration literature [2]- [4] reveal that until recently almost all investigations con- sidered only the serial-link case. Calibration of robots with closed loops was considered in [5] and 6-in-parallel RRPRRR platforms were studied in [6]. Bennett and Hollerbach [7] con- sidered serial-link arms that form a closed loop by interact- ing with their environment. This work is similar to ours in that the calibration is performed using only measurements of the robot’s internal joint motions, with no external devices re- quired. These simulation studies have been followed by ex- periments by several researchers [8], [10]. This paper presents a re-analysis of the Hollerbach and Lokhurst experiments and also describes a new experiment using the MEL Modified Stewart Platform. We begin by reviewing our method of analysis, previously outlined in [11]. We use a statistical maximum-likelihood cri- terion, formulated to handle the kind of implicit measurement equations that arise in closed-chain calibration experiments.