Combining Internal and External Robot Models to Improve Model Parameter Estimation*

Transcription

1 Proceedings of the 2001 IEEE International Conference on Robotics (L Automation Seoul, Korea. May 21-26, 2ClO1 Combining Internal and External Robot Models to Improve Model Parameter Estimation* W. Verdonck! J. Swevers Dept. of Mechanical. Engineering Katholieke Universiteit Leuven Leuven (Heverlee), Belgium kuleuven.ac. be X. Chenut. J.C. Samin Dept. of Mechanical Engineering Universitk catholique de Louvain Louvain-la-Neuve, Belgium Abstract Experimental robot identification techniques can principally be divided into two categories, based on the type of models they use: internal or external. Internal models relate the joint torques or forces and the motion of the robot; external models relate the reaction forces and torques on the bedplate and the motion data. This paper describes how internal and external robot models can be combined into one identifiable minimal model. This model allows to combine joint torque/force and reaction torque/force measu:-ements in one parameter estimation scheme. This ccmbined model estimation will yield more accurate parameter estimates, and consequently better actuator tonpe predictions, which is shown experimentally on an industrial robot (KUKA IR 361). 1 Introduction The ever increasing market demands on price and quality force machine and robot manufacturers to integrate more mechatronic principles in the design and operation of their products. Improvement of the accuracy and faster motion calls for the inclusion of system knowledge in the design of controllers. The knowledge of the dynamical behaviour is of particular interest when switching from analog, joint velocity controllers to integrated digital current (computed torque) con- * Acknowledgement This text presents research results of the Belgian programme on InteruniversiQi Poles of attraction initiated by the Belgian State, Prime Minister s Office, Science Policy Programming. The scientific responsibility is assumed by its authors. t His work is sponsored by a specialization grant of the Flemish Institute for Support of Scientific and Technological Research in Industry (IWT) trol. The accuracy of the current controller directly depends on the model accuracy. Experimental robot identification techniques estimate dynamic robot parameters based on motion and force/torque data measured during well-designed identification experiments, i. e. robot motions along optimized trajectories [5]. It is the only time and effort efficient way to obtain accurate robot models, as well as indications on their accuracy, confidence and validity. Several methods have been designed in the last decade, and they can be divided into two categories according to the models and the type of sensors they use. In the classical identification approach [3, 81, the parameters are estimated from motion data and actuator torques or forces, both measured by internal measurement devices (i. e. joint encoders for motion data and actuator current measurements for forces and torques). The dynamical model relating these inputs and outputs will be called here internal model. The classical approach suffers however from an important drawback: the torques applied to the links are not directly available so that they are corrupted by friction torque modeling errors and by the low precision of the actuator torque constants. An alternative approach makes use of the so-called reaction or external model of the robot [7]. This model relates the motion of the robot to the reaction forces and torques on its bedplate and is, therefore, totally independent from internal torques such as joint friction torques. The robot motion can be measured by means of joint encoders (internal sensors) or by means of a high-precision visual position sensor (external sensor). The reaction forces and torques are measured by means of an external sensor: a force/torque platform. This approach suffers from the fact that joint friction parameters cannot be estimated. These parameters are important for accurate actuator torque /01/$ IEEE 2846

2 prediction which is used in computed torque control. This paper describes how both models can be combined and investigates the benefits associated with this. It is shown that the combined model yields more accurate model parameters, resulting in much more accurate actuator torque predictions, which allows more accurate computed torque control. The external model consists in a reformulation of the system dynamics relating the motion of the robot to the reaction forces and torques on its bedplate. It can be easily obtained by projecting the force and torque vectors at the first joint on the axes of the inertial reference frame attached to the bedplate. The external model equations have thus a form similar to equation (2) : 2 Generation of dynamic robot models The generatioil of a dynamic robot model is based on the kinematic structure of the robot (e.g. link lengths, transformation matrices between the local coordinate systems of the different links,...). The model relates system inputs and outputs. Based on the type of inputs and outputs, two different models can be distinguished : external and internal models. External models relate robot motion (iiiputs) to reaction forces and torques measured at the base of the robot (outputs). Internal models relate actuator torques or forces (outputs) to the robot motion (inputs). 2.1 Internal and External Models The robot is considered as a set of n rigid bodies interconnected by one d.0.f. joints which can be either prismatic or revolute. We assume that the structure is a topological tree, excluding the presence of any closed loops in the system. If we denote the n joint coordinates by q, the equations of motion, obtained by the Potential Power Principle, take the form : M(q)ti + C(q, 479) = 7 2 (1) where M(n x 7%) is the symmetric positive definite mass matrix. C(n x 1) includes the Coriolis, centrifugal, gyroscopic and gravity terms, ~,(n x 1) is the vector of the generalized forces associated with q. i. e. the forces and torques applied at the joints by the actuators. In these equations, the internal model of the robot, as any multibody system, is characterized by a set of ineitial parameters describing the mass distribution of the links. These parameters could be considered individually for each body, but it is well-known that some of them combine together [GI to form the so-called barycentric parameters. It can be shown [l] that equation (1) (znternal model) can be written as: *L(q, 4, ti)& = 7, (2) where SZ(n, x 1) is the vector containing then, barycentric parameters, Q,(n x n,) is the associated identification matrix. where de(n, x 1) contains the ne barycentric parameters associated with the external model, Qe(6 x ne) is the associated identification matrix, ~,(6 x 1) is the vector of the reaction forces and torques. Note here that the dimension G in Q, and T, represents the 3 components of forces and torques at the base of the robot. It is shown in [2] that all the barycentric parameters appearing in equation (2) are also present in equation (3). This means that the barycentric parameter set 6i is a subset of the parameters 6, appearing in the external model: 6i c 6,. In order to ensure the identifiability of the system, it is mandatory to make use of a minimal set of parameters in the internal and external models. This is however not always the case with the barycentric quantities. In fact, depending on the nature (prismatic or revolute) of the joints, additional combinations do appear between the barycentric parameters. Systematic recursive rules have been developed in order to determine this minimal parameterization [2] which is crucial to ensure the identifiability of the system. Applying these rules, we finally obtain the equations describing the internal and external models : *5(q, q, $65 = Ti (4) q(q, 414)s: = Te (5) where the superscript * indicates the use of the minimum sets of parameters. The parameters of Sf and 6: are linear combinations of the parameters of 6i and 6, respectively. Moreover, 6: is still a subset of Joint friction and gravity compensation Besides inertial parameters, parameters that model joint friction and the effect of gravity compensation devices, if they are present, have to be considered in the internal model. It is common to model friction in robot joints by means of viscous and Coulomb friction, yielding the following friction model for joint j: 2847

3 where fc, and fuj represent the Coulomb and viscous friction parameters respectively. Often, robot manipulators are equipped with gravity compensation devices in order to reduce the load due to gravity on some of their actuators [8]. As a consequence, the parameters related to the involved link's center of mass, must be replaced by some other parameterization in the internal model. This has been discussed in more detail in [l]. Anyway, the modeling of the effect of a gravity compensation device on a joint always yields parameters that have to be added to the set of robot internal model parameters. The friction parameters are also part of that set. These parameters however do not influence the external model. We consider that the vector 6f, contains all these additional pa1 ameters. 3 Combining internal and external robot models Combining the internal and external models into one single identification scheme, requires the determination of the parameter set involved in such combined model. For this purpose, one can divide the set of all parameters appearing in the internal and external models in two subsets: 0 the manamum set of barycentrac parameters of the external model: 6:. They include the minimum set of barycentric parameters of the internal model 6:. e the parameters whach only appear an the anternal model. They are the parameters related to internal forces coming from gravity compensation devices and friction: 6fg. The total combined robot model can be formulated as follows: $(q, q, q)8 == T (7) with an (n + 6) x 1 column vector. The total parameter vector 8 consists of 3 subvectors: [ 8: 8= 6!:%;] (9) The total identification matrix ing submatrices: consists of follow- corresponds to matrix \ET (equation (4)). Qi12 describes, in combination with 6j,, the joint friction and gravity compensation effects : a126fg = T,,, +.~f, with Tf and T,,, being n x 1 column vectors of which the j-th elements is the friction torque and resulting torque of gravity and the gravity compensation device on joint j respectively. The elements T,,, corresponding to links that are not equipped with a gravity compensation device are zero. The columns of a21 correspond to the columns of (equation (5)) which are related to the elements of 6:. The columns of $23 correspond to the columns of matrix 9: (equation (5)) which are related to the elements of 6: \ 6;. 4 Experimental robot identification A robot identification experiment consists of two important steps: (1) the design and execution of a robot excitation experiment, and (2) the estimation of the robot parameters from the measured data [4,9]. In this section, the maximum likelihood approach presented in Swevers et al. [9] will be applied, because this approach is based on a statistical framework aiming at estimating the robot model parameters with minimal uncertainty, and because their experiment design is more directly applicable than other existing design approaches [9]. The maximum likelihood estimate ~ M of L the parameter vector 19 is given by the value of 8 which maximizes the likelihood of the measurement. The minimization of such a likelihood function is a nonlinear least squares minimization problem. If the measured joint angles are free of noise, this minimization problem simplifies to the Markov estimate, i.e. the weighted linear least squares estimate for which the weighting function is the reciprocal of the standard deviation of the noise on the measured force/torque data [9]. This simplification is applied here, and is justified since the noise level on the joint angle measurements is much smaller than the noise level on the force/torque measurements [9]. The excitation trajectory for each joint is a finite Fourier series, i.e. the angular position q3 for each joint j is written as: NJ q3 (t) = a; cos(wfpt) + b"p sin(wfpt) + qj 01 (11) p=l with wf the fundamental pulsation of the Fourier series. This Fourier series specifies a periodic function with period Tf = 27r/wf. The fundamental pulsation is common for all joints in order to preserve the periodicity of 2848

4 the overall robot excitation. Each Fourier series contains 2 x N3 + 1 parameters. that constitute the degrees of freedom for the optimization problem: a;, and bjl, for p = 1 to N3, which are the amplitudes of the cosine and sine functions, and qj0 which is the offset on the position trajectory. The design criterion for these trajectory parameters is the uncertainty on the parameter estimates, which is represented by the covariance matrix of the parameter estimates, P. In the case of the Markov estimate, this covariance matrix depends only on the robot trajectory via the identification matrix [3] and the variance of the noise on the force/torque measurements. It does not depend on the model parameters. For this optimization, the covariance matrix is replaced by a representative scalar measure, its determinant, yielding the so called d-optamality criterion. ' This optimization is a complex nonlinear optimization problem with motion constraints. 5 Experimental verification 5.1 Description of test case and robot model The considered test case is a KUKA IR 361 robot (figure 1) placed on a KISTLER B21 force/torque platform (Kistler Instruinente AG). Only the first three robot axes are considered. The robot is equipped with a internal spring which compensates the gravitation for the second link [8]. Figure 1: Schematic representation of a KUKA IR 361 robot Parameter vector Sf contains 13 independent barycentric parameters. The torque generated by the gravity compensation spring is a complex nonlinear function of the joint angle q2 and the spring parameters. This nonlinear dependency and the low sensitivity of spring force for changes in its parameters makes the estimation of the spring parameters from noisy data cumbersome. In [8], it is shown that for the case of rotational joints, a better modeling approach is to approximate the torque resulting from the combined effect of gravity and a gravity compensation spring by means of a series of Ns harmonically related sine and cosine functions of the joint angle. In the case of the KUKA IR 361 robot, the resulting torque on joint 2 can then be modeled as: N, Tresz = Ai, cos(kz) + Bi, sin(kz) (12) 1=1 Only the fundamental sine and cosine functions and their first and second harmonics have to be considered (N, = 3 in equation (12)) for accurate modeling [8], yielding G parameters: Al,, Bi, for 1 = 1,2,3. Friction consists of viscous and Coulomb friction, yielding G Parameters. One parameter per joint is added to take offset on the joint torque measurements into account. This yields 15 parameters which appear only in the internal model and form the subset 6fg. The subset 6: \ 6: contains G additional inertial parameters which only appear in the external model. 5.2 Description of the experiments The experiments investigate how the accuracy of the parameter estimates and of the actuator torque prediction can be improved by combining internal and external models. Therefore, three different identification experiments are considered: (1) identification using the internal model, which contains 28 parameters, (2) identification using the external model, which contains 19 parameters and (3) identification using the combined model, which defines the relationship between the motion, the 3 driving motor torques and the 6 reaction forces/torques at the base.of the robot. This model contains all 34 parameters. The excitation trajectory consists of 5-term Fourier series, yielding 11 trajectory parameters for each joint. The fundamental frequency of the trajectories is 0.1 Hz, yielding a period of 10 seconds. The optimization is based on the d-optimality criterion. For comparison purposes, only one trajectory is optimized for the combined model and is used for all three experiments. The trajectories are implemented on the COM- RADE [lo] software platform and runs at a sampling rate of 150 Hz. Figure 2 shows the optimized trajectory for the three robot axes. The outputs are joint actuator torques for joints 1 to 3 and 6 reaction forces/torques measured at the base of the robot. 2849

5 resulting from the combined model covariance matrix equals lo-". Comparison of these results already indicate that actuator torque prediction based on the parameter estimates obtained from the combined model is more accurate than predict,ion based on the parameter estimates obtained from the internal model only Model validation Figure 2: Optimized robot excitaiion trajectory: axis 1 (full line), axis 2 (dashed line) and axis 3 (dash-dotted line) Figure 3 shows the validation trajectory. It goes through 20 points randomly chosen in the workspace of the robot. The robot moves with maximum acceleration and deceleration between these points, and comes to full stop at each point. 5.3 Comparison of results Parameter accuracy The parameters are estimated using the Markov estimator. Comparison of the variances of these parameters shows that combining the internal and external models and measurements yields a significant improvement of the accuracy of the parameter estimates: the accuracy of the parameters in the set Sf is highly improved by the base platform measurements since more measurements are taken into account in one parameter estimation problem. Actually, the combined model yields the best accuracy for each parameter in comparison to the internal and the external model Actuator torque prediction The accuracy of the actuator torque predictions depends on the accuracy of the parameter estimates in S,* and Sf,, since the covariance matrix of the actuator torque predictions depends cm the covariance matrix of these parameter estimates. Two covariance matrices for the actuator torque predictions can be constructed: one based on the covxiance matrix of the model parameters resulting from the internal model identification, and one resulting from the combined model identification. An appropriate measure of the accuracy of the actuator torque predictions is the determinant of the covariance matr IX since these determinants are related to the volume of highest probability density region for the parameters [5]. The determi- nant of the covariance matrix resulting from the internal model equals lo-"' and the determinant, Figure 3: Position for validation trajectory: axis 1 (full line), axis 2 (dashed line) and axis 3 (dash-dotted line) Table 1 shows the root mean squared actuator torque prediction errors for the different sets of estimated parameters over the validation trajectory. The RMS prediction errors for axes 1, 2 and 3 resulting from the combined model are respectively 5%, 4% and 5% smaller than those resulting from the internal model. RMS prediction error I axis 1 I axis 2 I axis 3 I internal model I I I I Nm 1 combined model I Nm Table 1: Root mean squared prediction error for the validation trajectory Figure 4 compares the measured values, predicted values and the prediction error for the motor torques with the parameters of the combined model. The peaks in the estimation residu occur at low joint angular velocity, which indicates that the assumed friction model, 2850

6 z E Measured torque axis 1 Estimated torque axis 1 Estimation resid axis s o 0 0 c Measured torque axis 2 Estimated torque axis 2 estimation resid axis I, 2oor I e z f F Measured torque axis 3 Estimated torque axis 3 Estimation resid axis z I Time (s) Time (s) Time (5) Figure 4: Measured motor torque, predicted torque and torque prediction error for the validation trajectory using the combined model which includes viscous and Coulomb friction, is too simple. It can he expected that including more advanced friction arid gear models results in smaller estimation rcsidus. Remark: Whm using data from different measurement devices to estimate the same parameters, calibration of the measurement channels becomes important. Whereas the force platform measurements can be calibrated very acciirately, the motor torques are not directly available. For DC motors the torque constant relates the currciit to the torque. In practice this constant can vary considerably from the manufacturer s specification. For this reason, it is necessary to determine the actual torque constant for each motor in a robot. Although the combined model already gives better torque predictions, we expect even better results after calibration of the torque constants. 6 Conclusion This paper discusses the combination of internal and external models and measurements for the identification of the dynamic parameters of a robot. It can be concluded that combining both models considerably improves the a.ccuracy of the estimation of the inertial parameters that appear in both models, and of the actuator torque prediction. Robot manufacturers may therefore consider the inclusion of base platform measurements during the dynamical calibration of their I robots in order to improve the control performance. Experimental identification and validation on an industrial robot indicate the improvement of the actuator torque prediction even when modeling errors are present during the identification. Further improvement can be expected after calibration of the motor torque constants and by using a more elaborate friction model. References CHENUT, X.; SAMIN, J.C.: SWEVERS: J.: GANSE- MAN, C., 2000, Combining internal and external robot models for improved model parameter estimation, Mechanical Systems and Signal Processing, Vol. 14, NO. 5, FISETTE, P.) RAUCENT, B., SAMIN, J.C., 1996, Minimal dynamic characterization of tree-like multibody systems, Nonlinear Dynamics, Vol. 9, pp GAUTIER, M., 1986, Identification of robot dynamics, Proc. IFAC Symposium on Theory of Robots, Vienna, page 351. GAUTIER: M., KHALIL, W., 1992, Exciting trajectories for the identification of base inertial parameters of robots, Int. J. Robotics Reseach, Vol. 10, pp GOODWIN, G.C., AND PAYNE: R.L., 1977, Dynamic system identification : experiment design and data analysis, Academic Press; New-York. MAES, P.: SAMIN, J.C., WILLEMS, P.Y., 1989, Linearity of multibody systems with respect to barycentric parameters : dynamic and identification models obtained by symbolic generation, Mechanics of Structures and Machines, vol. 17, No. 2. RAUCENT, B., SAMIN, J.C., 1993, Modelling and identification of dynamic parameters of robot manipulators, Robot Calibration, R. Bernhardt and S. Albright (eds.), Chapman & Hall, London, pp SWEVERS: J., GANSEMAN, C., DE SCHUTTER? J.: VAN BRUSSEL, H., 1996, Experimental robot identification using optimized periodic trajectories, Mechanical Systems and Signal Processing, Vol. 10, No. 5, SWEVERS, J., GANSEMAN: C., BILGIN, D., DE SCHUTTER, J., VAN BRUSSEL, H., October 1997, Optimal robot excitation and identification, IEEE Trans. on Robotics and Automation, Vol. 13, No. 5, VAN DE POEL; P., WITVROUW, W.; BRUYNINCKX, H.: AND DE SCHUTTER, J., An environment for developing and optimising compliant robot motion tasks, In 6th Int. IEEE Conference on Advanced Robotics, Atlanta, 1993, pp