A methodology for dynamic parameters identification of 3-DOF parallel robots in terms of relevant parameters

Abstract

The identification of the dynamic parameters in mechanical systems is important for improving model-based control and performing realistic dynamic simulations. Generally, when identification techniques are applied, only a subset of parameters so-called base parameters can be identified. Moreover, some of these parameters cannot be properly identified if they make a small contribution to the robot dynamics. In this paper, a strategy for dynamic parameter identification of parallel robots in terms of relevant parameters is put forward. The proposed methodology starts from a dynamic model developed by means of the Gibbs–Appell equations. Afterwards, the model is simplified based on the considered geometry of each link and symmetry of the legs. The identification is done by Weighted Least Squares. With statistical considerations, the number of model parameters is reduced until the physical feasibility conditions are met. The strategy has been experimentally tested on two actual 3-DOF parallel robots. The response of the inverse and forward dynamics problem using the identified parameters agrees with experiments.

Introduction

Parallel robots have constituted a very active field of research over the last 20 years. Compared to serial robots, parallel robots essentially have two well-known advantages, namely greater precision in positioning and increased rigidity with respect to the relationship between size and workload limit. The main drawbacks of parallel robots are their small workspace and some specific problems related to the development of the control schemes. Although their implementation has been focused on academic circles, nowadays their application is transferring into industry [1]. Thus, the development of accurate dynamic models for this class of robots, particularly for those with less than 6-DOF, is of considerable current interest.

Accurate identification of the underlying dynamic parameters is carried out such that a realistic dynamic simulation of mechanical systems can be performed. On the other hand, the parameters identified can be used for the development of advanced model-based control schemes. The dynamic parameters of the robot to be identified are basically: mass, location of the center of gravity, the inertia tensor and friction parameters. Among the techniques that have been proposed for their determination, the identification of dynamic parameters through experimental methods is the one that has provided better results. However, the dynamic identification of robotic systems is far from being resolved [2].

The parameter identification process consists of fitting measured data to the response of the dynamic model. It is well-known that the dynamic model of a robotic system can be written in linear form with respect to the dynamic parameters to be identified, provided that the friction model considered is a linear one. Moreover, depending on the system configuration, not all the dynamic parameters contribute to the robot dynamics; therefore, only a subset of so-called base parameters can be identified. However, when a parameter identification process is performed experimentally, not even base parameters can be correctly identified due to noise in measurements and discrepancies in modeling. Parameters with an independent contribution which is less relevant than others are difficult to identify. For instance, in the identification of a simulated Puma-like serial robot it has been reported that for 36 base parameters, only 15 parameters were properly identified for a given noise level [3].

For parallel robots, not all the joints are active and they have a limited workspace compared to serial robots. These facts make the excitation of all the base parameters even more difficult when compared with serial robots. This fact was highlighted in reference [4] where the need is set out for an iterative methodology that considers what parameters are relevant for identification. The general idea is that a complete and complex dynamic model does not lead to improvements in the results when the parameters are not properly identified. Thus, it is preferable to use a reduced model with properly identified parameters.

An approach for finding an appropriate parameterization of the model identified for a 6-DOF robot was presented in [5]. Considerations of large amounts of noise in measurements for the dynamic identification of a 6-DOF robot were presented in [6]. In addition, some practical rules based on experiments have been presented for the experimental parameter identification of 6-DOF parallel robots [7]. Other authors have implemented bounded error techniques in which the solution is a set of parameter vectors consistent with measurement data, prior error bounds and modeling hypotheses [8]. Recently, a simplified model of a class of parallel robot was developed based on the structure of the considered robot [9].

Basically, two approaches have been presented for reducing the dynamic model. On the one hand, since the dynamic model can be written in linear form with respect to the parameters to be identified, the well-known linear techniques can be applied to evaluating which parameters are properly identified. For instance, relative standard deviation has been used as a reduction criterion [10]. On the other hand, the contribution of the estimated parameters to the joint torques can be used to guide the reduction process [11]. Regardless of the criteria used for reducing the model, in both approaches the reduction process stops by means of an empirical criterion. When reduction is carried out through the relative standard deviation, in reference [12], it is proposed to reduce it until relative standard deviation of the parameter with the maximum value reaches 10 times the one with the minimum value. On the other hand, when the reduction is carried out by considering the contribution of the parameter to the joint forces, the threshold is user-defined [11].

In the reduction strategies presented so far, physical feasibility was not considered, which is an aspect of paramount importance given the fact that a physically feasible set of parameters contributes to model-based control or motion simulation of robots [13]. Moreover, this fact is crucial when the values of parameters are needed for a physical understanding of a robot's characteristics. The identification of parallel robots could lead to identified parameters with unfeasible values. For instance, in the identification of a 6-DOF parallel robot, one of the parameters was unfeasible [14]. This fact was dealt with by eliminating the parameter with the unfeasible value. After that, the authors carried out a new identification with the reduced model. Physical feasibility was also considered in the identification of a 3-RPS parallel robot by using non-linear constrained optimization [15]. However, a complete and complex model was used, thus, a reduction process was not considered. In addition, identification was carried out by non-linear optimization techniques which can lead to a local minimum. Another important aspect which can be highlighted is that for fully parallel robots, considerations regarding symmetries and geometries of the robot parts can be used to simplify the dynamic model [16], [17].

In this paper, a methodology for identifying the dynamic parameters of parallel robots in terms of relevant parameters is put forward. Its application is experimentally tested on two different configurations of actual 3-DOF parallel robots. The objective of the proposed methodology is to start from a complete and complex dynamic model. Then, simplify due to the geometry of each link and the symmetry presented in the legs of fully parallel robots. After that, identification is made by Weighted Least Squares. Then, with statistical considerations, the model is reduced until the physical feasibility conditions are met. The parameters constituting the reduced model, obtained by the proposed methodology, are called relevant parameters. It is worth noting that to the author's best knowledge, obtaining the reduced model considering 1) simplification due to symmetries and geometries of the robot parts and 2) by using the physical feasibility conditions, has not been considered before.

The implementation of the proposed methodological strategy allows us to find a reduced model that has been verified by a comparison among the generalized forces from the identified models and the measured forces from the experiments. The forward dynamic problem in terms of the identified parameters is also addressed, which is a topic hardly dealt with in previous papers dealing with dynamic identification. To this end, an approach for writing the forward dynamic problem in terms of the identified parameters is put forward.

The paper is organized as follows. In Section 2, the linear dynamic model with respect to the dynamic parameters is developed and the steps of the proposed methodology, including a flowchart for its application, are presented. The results of applying the proposed methodology to an actual 3-RPS parallel robot are shown in Section 3 and for a 3-PRS robots in Section 4. In Section 5 the forward dynamic in terms of relevant parameters is developed. Finally, the main conclusions are presented.