������������������� ��������������� ��������������������������������������������������������������������������������������������������������������������ICARCV2010

Identification of the Parameters of Robot Manipulators Dynamics about an Operating Point

using Perturbed Dynamics Azeddien Kinsheel, Member,IEEE (Author)

Department of Engineering Design and Manufacturing Faculty of Engineering, University of Malaya

Kuala Lumpur, Malaysia azkensh@um.edu.my

Zahari Taha Faculty of Manufacturing Engineering and Management

Technology, University Malaysia Pahang, Malaysia zaharitaha@ump.edu.my

Abstract— In this paper, a new approach to identify the

parameters of robot manipulator dynamics is presented. The approach is based on exciting the robot manipulator with multiple identification trajectories that pass through the operating point with specific speeds and accelerations. The position, velocity and acceleration of the robot joints along with the corresponding torque at the operating point are recorded. The relative change of the joint motion and the corresponding torque compared to a reference operating point trajectory is used to identify the robot dynamics parameters. By using perturbed rather than the absolute motion values, several modeled and un-modeled effects can be eliminated from the identification equations. This is very useful when an accurate model of the robot dynamics at specific operating point(s) of the workspace is required.

Keywords—Robot, Dynamics, Modeling, Identifcation, Operating Point, Trajectory.

I. INTRODUCTION The advanced application of robotic systems such as Robot-Assisted Surgery requires an accurate model of the robotic system for preoperative planning, surgical operation simulations and training. However, obtaining such models requires sufficiently accurate knowledge of the parameters of the robot dynamics. Robot technical manuals typically do not provide all of the required parameters. Alternatively, experimental identification methods are used. An extensive description of the methods used for experimental identification of the parameters of the robot manipulator dynamics are presented in [1] and [2]. In general, the presented methods can be classified as integral or differential methods. Differential methods which depend on generalized accelerations have several advantages compared to integral methods which depend on energy [1]. Differential identification methods in turn can be either direct or indirect [3]. In the indirect differential identification process, system parameters are individually identified in multiple steps. The identification trajectories usually consist of one or two joints motions at a time. Such trajectories can be generated by simple controllers, normally the manufacturer supplied

controller. Examples of such identification methods can be found in [3], [4] and [5]. In general, indirect identification methods provide, relatively, accurate torque prediction at low angular velocities and accelerations [3]. The main drawbacks of the indirect schemes are error propagation and the large number of experiments required [2]. In contrast, direct differential identification methods are based on an optimized single motion trajectory that excites all the parameters defining the manipulator dynamics. Joint positions, velocities and accelerations and the corresponding measured torques are used to estimate the inverse model using linear or nonlinear regression. The process is carried out in one step which makes it superior to the indirect method in terms of the number of experiments required for the identification process. The challenging part of this method is the design of the identification trajectory that leads to an optimum estimation of the system parameters. The trajectory optimization problem is addressed in several research works such as [6], [7] and [8]. A an extensive presentation of a direct identification method based on a periodic band-limited excitation trajectory is presented in [9]. The presented method allows the integration of the experiment design, signal processing, and parameter estimation. This integration simplifies the identification procedure and yields accurate models. The result of the method is a global linearized model of the robot dynamics over the whole workspace of the robot manipulator. The global linearized model estimated by direct methods provides an optimum model with minimum error over the part of the workspace covered by the identification trajectory. However, the method is sensitive to measurement errors and suffers from the rank deficiency problem associated with the regression matrix. In this paper a new method to identify the parameters of the robot manipulator dynamics is presented. The aim of the method is to accurately identify those parameters at a specific operating point of the workspace. The approach in this method uses the perturbed motion around the operating point to identify the actual parameters of the robot manipulator dynamics at that point. The approach is suitable for cases where an accurate model of the robot is required at a certain area of the robot workspace. In the next section the

2010 11th Int. Conf. Control, Automation, Robotics and VisionSingapore, 7-10th December 2010

144

linearization of the robot dynamics about an operating point is presented. Sections III and IV describes the identification process and the trajectory generation, respectively. Section V presents the performance of the method compared to that of the direct identification in simulation.

II. LINEARIZATION OF THE ROBOT DYNAMICS AROUND AN OPERATING POINT

The nonlinear and coupled inverse dynamics model of a robot manipulator with N degree of freedom can be represented in the general form as [10]: ����� � ���� ��� � ��� � � (1)

where D(�) is the (N x N) inertia matrix (acceleration

coefficients), H(�� �� �is the (N x N) Coriolis, centrifugal, and frictional torque vector (velocity coefficients) , G(�) is the (N

x 1) gravity loading vector, and T is the N x 1 joint torque vector. �(t), �(t) and � (t) are the N x 1 vectors of joint angular

positions, velocities, and accelerations, respectively. In direct identification methods the linear form of the robot dynamics is derived from [1] and is written as:

� � ���� � � � �� (2)

where � ��������barycentric parameter vector and, � is the observation or identification matrix. Note that � depends only on the motion data. Formulating the equation of motion in the linear form allows the use of linear regression techniques, such as the linear least squares method. The result of this method is a linearized global model that covers all the workspace. Alternatively, the inverse dynamics model given in (1) can also be linearized around any operating point in the workspace -������ ��� �� � �� �� where Tn is the vector of joints torques at the operating point and���� �� � ��� are the nominal position, velocity and acceleration, respectively.

� � ��� � ��� � �� (3) When the robot moves with a trajectory �(t), about the operating point, then the change in joint torques Γ with respect to the nominal torque Tn can be related to the change in the robot motion or “perturbation” from the operating point. Thus, (3) can be written as: � � ��� � ��� � �� (4) where, � � �� � ���� � �� (5)

and, � � �� � ���� � �� (6) �� � �� � � ��� � � � (7)

Note that:

� � �! � and �� � � ! � Dn, Hn and Gn are the coefficients of the robot dynamic estimated at operating point Pn with nominal acceleration and velocity. Writing the equation of the robot dynamics in the relative form as in (4), allows the elimination of the common mode effects, such as friction and gravity, thus improving the signal to noise ratio. The linearized model coefficients of (4) can be estimated using the classical direct method with a single optimized trajectory as described in [9]. The accuracy of globally linearized model depends significantly on the identification trajectory and varies from one point to another in the workspace. Therefore, there is no guarantee of an accurate prediction of the actual robot torques at critical operating points. In addition, the estimated coefficients may not have any physical meaning; therefore, they cannot be used to identify the actual physical parameters of the manipulator dynamics such as joint inertia. If accurate coefficients that can provide accurate system dynamics at a specific point of the workspace are desired, then (4) must be solved at that specific point “the operating point”. This will also allow the identification of the actual system parameters. In order to accurately identify the coefficients of (4) around an operating point, the identification trajectory must be designed to pass through that operating point several times and collect the data at that operating point. The collected motion data, i. e. position velocity and acceleration and the corresponding torques can be used to solve (4) and obtain the physical parameters if a parametric model is available. In order to avoid an effect from other terms, each term coefficients of (4) should be identified separately. Identifying those coefficients separately will improve the model’s accuracy and provide flexible selection of the regression matrices.

III. DESCRIPTION OF THE RELATIVE IDENTIFICATION METHOD

In this method, the identification trajectories are designed so that each trajectory excites only one term of (4). That means three different trajectories are needed for identifying D, H and G in addition to the nominal torque measurement. The summary of the identification trajectory specification and corresponding regression vectors for separate identification are shown in Table 1. Last column of Table 1 shows the robot dynamic equation under each trajectory. This method differs from the indirect identification methods. The identification process here is carried out for all joints simultaneously, and consequently the number of experiments is less.

145

TABLE 1: SUMMARY OF THE IDENTIFICATION TRAJECTORY SPECIFICATION AND CORRESPONDING REGRESSION VECTORS

Trajectory Specification

Regression vectors

Torque

Param. " (t) " (t) "(t) � � �� # Tn �� �� �� 0 0 0 0 D � �� �� 0 0 �� D� H �� � �� 0 � 0 H� G 0 0 � � 0 0 G � The identification process is carried out in four steps:- Step 1: The nominal torque is measured at the operating point by exciting the robot joints with a motion trajectory that passes through the operating point with the nominal velocity and acceleration for each joint. The corresponding torques of the robot joints are measured at that point only. Step 2: The robot joints are moved to several positions in the application workspace including the operating point. While the robot is held stationary at that configuration, the joint torques are measured. Since the joints velocities and accelerations are zeros, Equation (4) in this case is simplified to:

� � �� (8) The regression vector q is obtained from the relative displacement of the robot joints from the nominal joint positions ��$ Step 3: Similarly, the velocity coefficients H are obtained by moving the robot joints through the operating point several times with the same acceleration and different speeds each time. The difference between the instantaneous torque acting on the robot joints due to such a trajectory, which is measured at the operating point �n, can be written as:

� � ��� (9) Step 4: The matrix Dn, is obtained by moving the robot joints with a trajectory that passes the operating point several times with the same speed and different accelerations. In this case (4) is simplified to:

� � ��� (10) Among the motion trajectories that can satisfy the above condition is the quadratic trajectory;

���� � %�& � '� � ( (11)

The overall trajectory consists of several segments of the quadratic motion all designed with the following conditions as follows:

� ���)*+*, � %- (12)

����)*+*, � �� (13)

���.�/-01 � ���2�/- (14)

and ���.�/-01 � ���.�/- (15)

Note that the acceleration at the operating point ak is changing in each quadratic segment. The last two conditions ensure the smooth transition of the motion from the final position at time tf of the kth segment to the starting position at time ts of segment k+1 of the identification trajectory.

IV. TRAJECTORY PARAMETERIZATION AND OPTIMIZATION The choice of the operating point and its parameters depend basically on the application at hand; there could be more than one critical operating point where an accurate model of the system is required, e.g. drilling at several locations or spot welding or assembling sophisticated parts at different places. Once the operating point along with its velocity and acceleration are set, the corresponding torque can be measured, most probably through the actuator currents. The identification trajectory parameters such as positions velocities and accelerations are chosen to cover the application space and provide full rank regression matrices. Arbitrary selection of those parameters may lead to a rank deficient regression matrix. Appropriate values for the trajectory parameters that can ensure full rank regression matrices can be selected either by trial and error or by solving a nonlinear optimization problem. In both cases, finding the appropriate regression metrics with a low condition number for the proposed method is much easier compared to the absolute identification method based on periodical trajectory. This is because the regression matrices in the relative identification method are totally independent.

V. SIMULATION RESULTS The performance of the proposed method compared to that of the direct identification method presented in [9] is studied in simulation. The study is carried out using an approximate model of the CRS A465 arm. The robot arm parameters are obtained from [11], [12] and [13]. The dynamic equations for the arm are developed using L-E formulation. The operating point specifications are set to suit the application at hand, i.e. Robot Assisted Orthopedic Surgery. In this application the CRS A465 is used to carry on the drilling part of the bone fixation procedure. The simulation tests are carried out assuming accurate compensation of friction and gravity and a noise free system. This is to ensure that the tests and the comparison of the methods are carried out under ideal conditions. The operating point specifications for the proposed identification method are shown in Table 2.

146

TABLE 2: OPERATING POINT POSITION, VELOCITY AND ACCELERATION Joint � (rad) " (rad/sec) " (rad /sec2) Tn (Nm) 1 0.7854 0.0890 -0.1550 -1.1883 2 0.3927 -0.1060 -0.5000 -3.4797 3 -0.5236 0.0220 0.0340 -1.7304 4 0.0 0.0560 -0.2910 -0.1377 5 1.0472 0.0110 -0.0080 -0.4466 6 0.0 0.0140 0.1710 0.0539 The perturbed identification vectors �� �� � are generated separately as described above from the random chosen positions, velocities and accelerations close to the nominal values. A sample of the selected values is shown in Fig. 1 a, b and c.

Figure 1.a: Randomly selected position for joints 2, 3 and 4.

Figure 1.b: Randomly selected speeds for joints 2, 3 and 4

Figure 1.c: Sample identification trajectory consisting of several quadratic motions with different accelerations for D-matrix identification. The direct identification process is carried out as described in [9] using the band-limited periodic trajectory as follows:

�3��� � �3�4 �5�%3�- ��6�782

9

-+1��

�'3�- :;��782 ��� (16) where 82 , is the excitation base frequency. The Fourier series coefficients a and b are obtained by solving a non-linear optimization problem with constraints imposed on the robot motion such as maximum acceleration, maximum speed and workspace limits. Optimization of the input trajectory is required to reduce the estimation error. The identification trajectories for the CRS465 arm are presented in [13] and shown in Fig. 2.

Figure 2: Direct identification trajectories based on periodical functions. The estimated coefficients of the linearized model of the robot manipulator in both methods are calculated by the linear least square method as mentioned above. The torque estimated by both models for joints 2, 3 and 4 are shown in Figs. 3, 4 and 5, respectively. It can be seen that both models produced a high accuracy estimate of the joint torques. However, as expected, the torque estimated with the direct method has a minimum error over the whole trajectory. In contrast, the torque estimated with the proposed perturbation motion identification method is closer to the actual torque near the operating point. At the boundaries, the estimation error is high for both methods. This is can be explained by the nature of the robot dynamics at high speed where nonlinear effects such as Coriolis and centrifugal torques are significant.

Figure 3: Comparison of estimated torques for joint 2.

147

Figure 4: Comparison of estimated torques for joint 3.

Figure 4: Comparison of estimated torques for joint 4.

The parameters estimated by the direct identification method are based on error minimization and cannot be related to the actual robot parameters. Therefore, physical robot parameters such as the joint inertia cannot be derived from the estimated parameters. In contrast, the D matrix of (4) estimated by the perturbed motion identification method provides, theoretically, the actual instantaneous acceleration coefficients at the operating point. With the knowledge of the robot parametric model, the actual joints inertial parameters can easily be calculated. In order to prove that the coefficients of the D(�) matrix of (1) were recorded during the simulation, the study of the coefficients of the D(�) matrix, which holds the robot joints inertia, shows that the perturbation method provides accurate estimation of the inertial coefficients compared to the direct identification method. Table 3 shows the actual and estimated acceleration coefficients for joint 1 (first column of D matrix). The accurate estimation of this matrix allows accurate identification of the actual joints inertias using the parametric model of the robot dynamics. TABLE 3: JOINT 1 INERTIA’S COEFFICIENTS CALCULATED FROM THE ESTIMATED TORQUE AT THE SAME OPERATING POINT.

D11 D21 D31 D41 D51 D61Actual 9.956

0.00 0.00 -1.37 0.00 -0.27

Perturbation Method

9.95

0.00 0.00 -1.37 0.00 -0.27

Direct method

4.73 -0.26 1.68 -0.87 0.39 -1.95

The simulation results presented earlier are obtained under ideal test conditions assuming accurate compensation for friction and gravity. Therefore, the actual performance of the proposed method depends significantly on the precision of the robot manipulator and the efficiency of the measurement system.

VI. CONCLUSION In this paper, an operating point based identification method for robot manipulator dynamics is presented. The simulation results of this method prove that this method is efficient when an accurate dynamic model of the robotic system at specific locations of the workspace is desired. The method uses the perturbed dynamics to identify the system parameters in multiple steps. Although the number of experiments required in this method is quite large compared to the classical direct identification method, the accuracy of the model around the operating point is higher. In addition, adopting perturbed dynamics instead of absolute dynamics improves the signal to noise ratio and consequently the model accuracy. The method can be extended to obtain several sets of robot dynamics parameters at different operating points.

REFERENCES [1] Kozlowski, K., Modelling and Identification in Robotics. Springer-Verlag, London. 1998 [2] Wisama Khalil , Etienne Dombre, Modeling, Identification and Control of Robots, Taylor & Francis, Inc., Bristol, PA, 2002. [3] Francesc Benimeli, Vicente Mata and Francisco Valero,”A comparison between direct and indirect dynamic parameter identification methods in industrial robots”, Robotica , vol 24, pp. 579–590, 2006. [4] K. Otani and T. Kakizaki, “Motion Planning and Modeling for Accurately Identifying Dynamic Parameters of an Industrial Robotic Manipulator,” Proceedings of 24th ISIR, Tokyo , pp. 743–748, 1993. [5] M. Grotjahn, B. Heimann and H. Abdellatif, “Identification of Friction and Rigid-Body Dynamics of Parallel Kinematic Structures for Model-Based Control,” Multibody System Dynamics 11, 273–294 ,2004. [6] P. K. Khosla, “Categorization of Parameters in the Dynamic Robot Model,” IEEE Trans. Robot. Automat., vol. 5, no. 3, 1989. [7] B. Armstrong, “On Finding Exciting Trajectories for Identification Experiments Involving Systems with Nonlinear Dynamics,” Int. J. Robot. Res., vol. 8, no. 6,pp. 28–48, 1989. [8] Fernando Reyes and Rafael Kelly On Parameter Identification of Robot Manipulators”, Proceedings of the 1997 IEEE lntemational Conference on Robotics and Automation Albuquerque, New Mexico, pp1910-1915, 1997 [9] Jan Swevers, Walter Verdonck, and Joris De Schutter, "Dynamic Model Identification for Industrial Robots." Control Systems Magazine, IEEE ,Vol 27,no. 5, pp. 58-71,2007. [10] A. Swarup and M. Gopal,” Comparative study on linearized robot models”, Journal of Intelligent and Robotic Systems, vol 7, no3 ,pp 287-300, 1993. [11] A465 Robot Arm User Guide, CRS Robotics Cooperation, 2000. [12] Jorge Gudiño-lau and Marco A. Arteaga, "Dynamic model and simulation of cooperative robots: a case study." Robotica 23(5): 615-624 2005. [13] Radkhah, K., D. Kulic and E. A. Croft, "Dynamic Parameter Identification for the CRS A460 Robot". IEEE/RSJ International Conference on Intelligent Robots and Systems, San Diego, CA. pp: 3842-3847, 2007

148