Optimal Selection of Measurement Configurations for Stiffness Model Calibration of Anthropomorphic Manipulators

The paper focuses on the calibration of elastostatic parameters of spatial anthropomorphic robots. It proposes a new strategy for optimal selection of the measurement configurations that essentially increases the efficiency of robot calibration. This strategy is based on the concept of the robot test-pose and ensures the best compliance error compensation for the test configuration. The advantages of the proposed approach and its suitability for practical applications are illustrated by numerical examples, which deal with calibration of elastostatic parameters of a 3 degrees of freedom anthropomorphic manipulator with rigid links and compliant actuated joints. Introduction In the usual engineering practice, the accuracy of an anthropomorphic manipulator depends on a number of factors. Following [1-2], the main sources of robot positioning errors can be divided into two principal groups: geometrical (link lengths, assembling errors, errors in the joint zero values et al.) and non-geometrical ones (compliant errors, measurement errors, environment factors, control errors, friction, backlash, wear et al.). For the industrial manipulators, the most essential of them are related to the manufacturing tolerances leading to the geometrical parameters deviation with respect to their nominal values (the geometrical errors) as well as to the end-effector deflections caused by the applied forces and torques (the compliance errors). It is worth mentioning that these sources of errors may be either independent or correlated, but, in practice, they are usually treated sequentially, assuming that they are statistically independent. Usually, for the industrial applications where the external forces/torques applied to the endeffector are relatively small, the prime source of the manipulator inaccuracy is the geometrical errors. As reported by several authors [3], they are responsible for about 90% of the total position error. These errors are associated with the differences between the nominal and actual values of the link/joint parameters. Typical examples of them are the differences between the nominal and the actual length of links, the differences between zero values of actuator coordinates in the real robot and the mathematical model embedded in the controller (joint offsets) [4]. They can be also induced by the non-perfect assembling of different links and lead to shifting and/or rotation of the frames associated with different elements, which are normally assumed to be matched and aligned. It is clear that the geometrical errors do not depend on the manipulator configuration, while their effect on the position accuracy depends on the last one. At present, there exists various sophisticated calibration techniques that are able to identify the differences between the actual and the nominal geometrical parameters [5-9]. Consequently, this type of errors can be efficiently compensated either by adjusting the controller input (i.e. the target point coordinates) or by straightforward modification of the geometrical model parameters used in the robot controller. In some other cases, the geometrical errors may be dominated by non-geometrical ones that may be caused by influences of a number of factors [10-11]. However, in the regular service conditions, the compliance errors are the most significant source of inaccuracy. Their influence is particularly important for heavy robots and for manipulators with low stiffness. For example, the cutting forces/torques from the technological process may induce significant deformations, which are not negligible in the precise machining. In this case, the influence of the compliance errors on the robot position accuracy can be even higher than the geometrical ones. Generally, the compliance errors depend on two main factors: (i) the stiffness of the manipulator and (ii) the loading applied to it. Similar to the geometrical ones, the compliance errors highly depend on the manipulator configuration and essentially differ throughout the workspace [12]. So, in order to obtain correct prediction of the robot end-effector position, the maximum compliance errors compensation should be achieved [13]. One way to solve this problem is to improve the accuracy of the stiffness model by means of elastostatic calibration. This procedure allows to identify the stiffness parameters from the redundant information on the state of the robot endeffector position provided by the measurements, where the impacts of associated measurement noise on the calibration results have to be minimized. However, currently most of the efforts have been made for kinematic calibration, only few works directly address the issue of elastosatic calibration and its influences on the robot accuracy [14]. Besides, using various manipulator configurations for different measurements seems to be attractive and perfectly corresponds to some basic ideas of the classical design of experiments theory [15] that intends using the factors that are differed from each other as much as possible. In spite of potential advantages of this approach and potential benefits to improve the identification accuracy significantly, only few works addressed to the issue of the best measurement pose selection [16-19]. Hence, the problem of selection of the optimal measurement poses for elastostatic parameters calibration requires additional investigation. This problem can be treated as finding the strategy of determining a set of optimal measurement poses within the reachable joint space that minimize the effects of measurement noise on the estimation of the robot parameters. It should be mentioned that the end-effector location as well as its deflection under the loading are described by a non-linear set of functions. However, the classical results of the identification theory are mostly obtained for very specific models (such as linear regression), Therefore, they can not be applied directly and an additional enhancement is required. One of the key issues in the experiment design theory is comparison of the experimental plans. In the literature, in order to define the optimal experimental plan, numerous quantitative performance measures that reduce multi-objective optimization problem to a scalar factor have been proposed. Consequently, different factors that evaluate robot calibration performance have been defined as the objectives of optimization, associated with a set of measurement poses [20-24]. However, all the existing factors have their limitations that affect the calibration accuracy in different manners. As a result, they do not entirely correspond to the industrial requirements. This motivates a research direction of this work. In this paper, the problem of optimal design of the elastostatic calibration experiments is studied for the case of 3-link spatial anthropomorphic manipulator, which obviously does not cover all architectures used in practice. Nevertheless, it allows us to derive very useful analytical expressions and to obtain some simple practical rules defining optimal configurations with respect to the calibration accuracy. In contrast to other works, it is proposed a new criterion that evaluates the quality of compliance errors compensation based on the concept of manipulator test-pose. The proposed criterion has a clear physical meaning and directly related to the robot accuracy, and allows us essentially improving the efficiency of compliance errors compensation via proper selection of measurement poses. Problem statement The elastostatic properties of a serial robotic manipulator [12] are usually defined by Cartesian stiffness matrix C K , which is computed as 1 C θ T    K J K J (1) where J is the Jacobian matrix with respect to the joint angles q , and θ K is a diagonal matrix that aggregates stiffness of the joints. In order to describe the linear relation between the end-effector displacement and the external force, the stiffness model of this manipulator can be rewritten as follows θ T   p J k J F (2) where p is the robot end-effector displacement caused by the external loading, θ k is the joints compliance matrix; F is the external force/torque. It is assumed that the geometric parameters are well calibrated. So, for the unloaded mode ( 0  F ), the vector q is equal to the nominal value of the joint angles 0 q . However, for the case when the loading is not equal to zero 0  F , the joint angles include deflections, i.e. 0    q q q , where  q is the vector of joint displacements due to the external loading F . Thus, the elastostatic model (2) includes parameters of θ k that must be identified by means of calibration. It is assumed that each calibration experiment produces three vectors { , , } i i i p q F , which define the displacements of the robot end-effector, the corresponding joint angles and the external forces respectively, where i is the experiment number. So, the calibration procedure may be treated as the best fitting of the experimental data { , , } i i i p q F by using the stiffness model (2) that can be solved using the standard least-square technique. In practice, the calibration includes measurements of the end-effector Cartesian coordinates with some errors, which are assumed to be i.i.d (independent identically distributed) random values with zero expectation and standard deviation  . Because of these errors, the desired values of  k are always identified approximately. So, the problem of interest is to evaluate the identification accuracy for the desired parameters and to propose a technique for selecting the set of joint variables i q and external forces i F that leads to the accuracy improvement. Usually, the performance measures that evaluate the quality of the calibration plans are based on the analyses of the covariance matrix of the identified parameters, all elements of which should be as small as possible. However, in robots the stiffness parameters ( 2 1 , , ... k k ) have different influences on the end-effector displacements; moreover, their influence varies throughout the workspace. To overcome this difficulty, in this work it is assumed that the "calibration quality" is evaluated for the so-called test configuration 0 0 { , } q F , which is given by a user and for which it is required to have the best positioning accuracy under external loading. To solve this general problem, two sub-problems should be considered: (i) to propose a optimality criterion that is adapted to the elstostatic parameters calibration of the anthropomorphic manipulator; (ii) to find optimal configurations of the manipulator for elastostatic parameters calibration that provide the best compensation of errors. Influence of measurement errors For computational convenience, the linear relation (2) where the desired parameters are arranged in the diagonal matrix θ 1 2 ( , , ...) diag k k  k should be rewritten in the following form