A Comprehensive Numerical Study on the Number of Identifiable Kinematic Parameters of Parallel Mechanisms

Kinematic error model plays an important role in improving the positioning accuracy of robot manipulators by kinematic calibration. The identifiability of kinematic parameters in the error model directly affects the positioning accuracy of the mechanism. And the number of identifiable kinematic parameters determines how many parameters can be accurately identified by kinematic calibration, which is one of the theoretical basis of kinematic error modeling. For serial mechanisms, a consensus has been reached that the maximum number of identifiable kinematic parameters is 4R + 2P + 6, where R and P represent the numbers of revolute and prismatic joints, respectively. Due to complex topologies of parallel mechanisms, there is still no agreement on the formula of the maximum number of identifiable parameters. In this paper, a comprehensive numerical study on the number of identifiable kinematic parameters of parallel mechanisms is conducted. The number of identifiable parameters of 3802 kinds of limbs with different types or actuation arrangements are analyzed. It can be concluded that the maximum number of identifiable kinematic parameters is Σ i = 1 n 4Ri + 2Pi + 6 − Ci − 2(PP)i/3(PPP1)i/(2Ri + 2Pi)(PPP)i, where Ci represents the number of joints whose motion cannot be measured and n denotes the number of limbs in a parallel mechanism; (PP)i, (PPP1)i, and (PPP)i represent two consecutive unmeasurable P joints, three consecutive P joints in which two of them cannot be measured, and three unmeasurable P joints, respectively.

A New Kinematic Error Modelling and Identification Method of the Parallel Manipulator

Parallel manipulators have broad application prospects on hybrid machine tools. Kinematic error modelling and identification are two key processes to improve the accuracy of parallel manipulators. The traditional kinematic error modelling method adopts the partial differentiation of the ideal kinematic model. However, the partial differentiation method is pure mathematical calculation, which ignores physical meaning of error terms corresponding to each link. In the process of error identification, the Jacobian matrix obtained from the partial differentiation method is usually ill-conditioned, which leads to non-convergence of the identification process. In order to solve the above problems, this paper proposes a new kinematic error modelling method and an error identification model. Firstly, the basic error terms for single link are analyzed. Based on basic error terms, the kinematic error model is established by using the practical connection point of two adjacent links. Then, a new error identification model is derived from the kinematic error model. Finally, as a study case, a 3-DOF parallel tool head is used to verify the correctness of the proposed method. The numerical results show that the proposed method is effective and the accuracy of the 3-DOF parallel tool head improves significantly after compensation of error terms.

Characterization of Kinematic Error Model Consistency for Five-Axis Machine Tools

New metrology tools, such as laser trackers, are enabling the rapid collection of machine tool geometric error over a wide range of the workspace. Error models fit to this data are used to compensate for high-order geometric errors that were previously challenging to obtain due to limited data sets. However, model fitting accuracy can suffer near the edges of the measurable space where obstacles and interference of the metrology equipment can make it difficult to collect dense data sets. In some instances, for example when obstacles are permanent fixtures, these locations are difficult to measure but critically important for machining, and thus models need to be accurate at these locations. In this paper, a method is proposed to evaluate the model accuracy for five-axis machine tools at measurement boundaries by characterizing the statistical consistency of the model fit over the workspace. Using a representative machine tool compensation method, the modeled Jacobian matrix is derived and used for characterization. By constructing and characterizing different polynomial order error models, it is observed that the function behavior at the boundary and in the unmeasured space is inconsistent with the function behavior in the interior space, and that the inconsistency increases as the polynomial order increases. Also, the further the model is extrapolated into unmeasured space, the more inconsistent the kinematic error model behaves.

Accuracy enhancement of kinematic error model of three-axis computer numerical control machine tools

Accurate estimation of volumetric errors is an important issue in machining operations. For this purpose, a kinematic error model is used to characterize machine tool’s related errors on its workspace. In this research, it is shown that when measuring the linear and positioning errors using a laser interferometer, part of the angular errors are converted to linear and positioning errors and their magnitudes are overestimated. These values are calculated twice in the models which use homogeneous transformation matrix since Abbe’s principle is not considered. In this article, a kinematic error model is proposed which eliminates this overestimation. This model’s methodology is based on rigid body kinematic and errors measurement by laser interferometer and can be generalized for all three-axis machine tools. A software package is developed to integrate the kinematic errors with the NC-codes. A workpiece is machined in the virtual environment and compared with a workpiece machined in real environment. It is shown that the kinematic error model developed in this research predicts the kinematic errors more accurately.

Test of Robot Distance Error and Compensation of Kinematic Full Parameters

In order to improve the positioning accuracy of robots in the workspace, the maximum cube of robots is solved according to ISO 9283:1998 standard. In addition, in order to efficiently test and identify the kinematic parameters of robots, a mapping from robot's distance error onto kinematic parameter errors is presented based on Hayati's modified D-H model. Then, by analyzing the condition number of distance error matrix, it is concluded that parameter d i can be deleted when parameter β i is added in the joint of the modified model. Furthermore, by analyzing the relationship among the parameters of the distance error model, it is found that the deletion of some unidentified kinematic parameters may not result in the accuracy decrease of kinematic error model. Finally, some compensation experiments of the proposed model without unidentified kinematic parameters are carried out by using a laser tracker system. The results show that the proposed method effectively reduces the distance error and greatly improves the positioning accuracy of robots.

Mathematical Model for the Effect of Dynamic Parameter Posed on Free Floating Space Manipulator’s Kinematic Accuracy

Aiming at kinematic accuracy and its’ error sources of a free floating space robot, a mathematical kinematic error model based on the concept of virtual manipulator and screw theory is proposed in this paper for a free-floating space robot. Based on screw theory, structural parameters in the form of motion screw and their error expressions derived from various error sources are deduced. The effect of mass error, CM (Center of Mass) error and structural parameter error on the kinematic accuracy of the free-floating space manipulator is analyzed. A simulation is demonstrated for verifying the correctness of the kinematic error model and the effect of various error sources on the free-floating space robot. The error model and the result deriving from analyzing are vital for studying the kinematic accuracy of the space manipulator when it is under a free-floating mode, and for controlling and assigning various errors when a space robot is developed.

Identifiable Parameter Analysis for the Kinematic Calibration of a Hybrid Robot

In this paper, a statistical method for the determination of the identifiable parameters of a hybrid serial-parallel robot IWR (Intersector Welding Robot) is presented. This method is based on the Markov Chain Monte Carlo (MCMC) algorithm to analyze the posterior distribution and correlation of the error parameters. Differential Evolution algorithm is employed to search a global optimizer as initial values for the random sampling of MCMC. The robot under study has ten degrees of freedom (DOF) and will be used to carry out welding, machining, and remote handing for the assembly of vacuum vessel of the international thermonuclear experimental reactor (ITER). In this paper, a kinematic error model which involves assembling and manufacturing error parameters is developed for the proposed robot. Based on this error model, the mean values of the unknown parameters are statistically analyzed and estimated using the proposed method. Computer simulations reveal that all the reduced independent kinematic parameters can be identified with the complete pose measurements. Results also demonstrate that the identification method is robust and effective with the given measurement noise.