Experimental Robot Identification: Advantages of Combining Internal

Walter Verdonck e-mail: [email protected]

Jan Swevers Department of Mechanical Engineering, Katholieke Universiteit Leuven, 3001 Heverlee, Belgium

Jean-Claude Samin Department of Mechanical Engineering, Universite´ catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

1

Experimental Robot Identification: Advantages of Combining Internal and External Measurements and of Using Periodic Excitation This paper discusses the advantages of using periodic excitation and of combining internal and external measurements in experimental robot indentification. This discussion is based on the robot identification method developed by Swevers et al., a method that is recognized by industry as an effective means of robot identification that is frequently used, Hirzinger, G., Fischer, M., Brunner, B., Koeppe, R., Otter, M., Grebenstein, M., and Schafer, I, 1999, ‘‘Advances is Robotics: The DLR Experiment,’’ The International Journal of Robotics Research, Vol. 18, No. 11, pp. 1064–1087. Experimental results on a KUKA IR 361 show that the periodicity of the robot excitation is a key element of this method. Nonperiodic robot excitation complicates the signal processing preceding the parameter estimation, often yielding less accurate parameter estimates. An extension of this identification method combines internal and external measurements, Chenut, X., Samin, J. C., Swevers, J., and Ganseman, C., 2000, ‘‘Combining Internal and External robot Models for improved Model Parameter Estimation,’’ Mechanical Systems and Signal Processing. Vol. 14, No. 5, pp. 691–704, yielding robot models that allow to accurately predict the actuator torques and the reaction forces/torques of the robot on its base plate, which are both important for the path planning. This paper presents and critically discusses the first experimental results obtained with this method. 关DOI: 10.1115/1.1409936兴

Introduction

Robotics applications which require high path tracking accuracy and fast motions are currently becoming more and more important. Standard industrial controllers, however, disregard the nonlinearities of robot dynamics like dynamic coupling between the different links and friction, which results in deviations from the desired motion. To meet the requirements, the nonlinearities can be compensated by advanced control algorithms, like computed-torque, which require a dynamic model of the robot. Furthermore, a number of robots are dynamically overloaded. Optimization of robot trajectories with respect to cycle time is therefore an important issue, thereby taking into account physical limits of the robot such as: the workspace of the robot, limits on the actuator power and torques, and on the reaction forces/torques of the robot on the base plate. Both these limits are specified for industrial robots by the robot manufacturer. The limits on the base plate reaction forces/torques are extremely important and often very tight for robots that will be used on space stations. Too high reaction forces/torques may disturb the often very sensitive laboratory equipment present in these space stations. As a result, the most important aspect of the robot models for these applications is their ability to accurately predict the required actuator torques and the resulting base plate reaction forces/torques based on the desired robot motion. Experimental identification is the only time and effort efficient way to obtain these robot models, as well as indications on their accuracy, confidence, and validity. A typical experimental identification procedure consists of the following three steps: 共1兲 the generation of an identifiable dynamic robot model; 共2兲 the generation of optimized excitation trajectories; and 共3兲 the estimation of Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division February 17, 2001. Associate Editor: S. Fassois.

the model parameters. For each of these steps the user has to make an optimal choice between different options, depending on the application on the model. With respect to the model generation, several methods have been designed in the last decade. They can be divided into two categories according to the models and the type of sensors they use. In the classical identification approach 关1,5兴, the parameters are estimated from motion data and actuator torques or forces, both measured by ‘‘internal’’ measurement devices. The joint encoders are used for motion data, and the actuator torque data is obtained through actuator current measurements, i.e. no additional sensors are required. The dynamic model relating these inputs and outputs is called internal model. An alternative approach makes use of the so-called reaction or external model of the robot 关6,7兴. This model relates the motion of the robot to the reaction forces and torques on its base plate and is, therefore, totally independent from internal torques such as joint friction torques. The robot motion can be measured by means of joint encoders 共internal sensors兲 or by means of a high-precision visual position sensor 共external sensor兲. The reaction forces and torques are measured by means of an external sensor: a force/torque platform. Both models can also be combined to improve the accuracy of the parameter estimates 关4兴. It is well recognized that reliable, accurate, and efficient robot identification requires specially designed experiments. When designing an identification experiment for a robot manipulator, it is essential to consider whether 共1兲 the excitation is sufficient to provide accurate and fast parameter estimation in the presence of disturbances such as measurement noise and actuator disturbances, and 共2兲 the processing of the resulting data is simple and yields accurate and consistent results. This paper shows how both requirements are satisfied if the robot excitation is periodic. Section 2 describes the generation of dynamic robot models 共internal and external models兲, briefly discusses how both models can be combined, and the advantages associated with this. The use

Journal of Dynamic Systems, Measurement, and Control Copyright © 2001 by ASME

DECEMBER 2001, Vol. 123 Õ 1

of this combined model approach in robot identification was first presented in 关4兴, however without any experimental validation. The experiment design resulting in optimal periodic robot excitation and the parameter estimation are briefly discussed in Section 3. The main parts of this paper are Sections 4 and 5, which discuss the advantages of using a combined internal-external robot model and of using periodic excitation, respectively. Both discussions are based on experimental identification results obtained on a KUKA IR 361 industrial robot. The results presented in Section 4 are the first experimental results ever obtained with this combination of external and internal measurements. The importance of periodic excitation has never been illustrated experimentally.

2

Generation of Dynamic Robot Models

2.1 Combining Internal and External Robot Models. Chenut et al. 关4兴 show that a dynamic robot model based on barycentric parameters 关8兴, and combining internal measurements 共actuator torques and forces兲 and external measurements 共reaction forces and torques of the robot to its bed plate measured by a external base force/torque sensor兲 can be formulated as the following set of linear equations: ⌽ 共 q,q˙ ,q¨ 兲 ␪ ⫽ ␶ , with

␶⫽

(1)

冋册

␶i , ␶e

(2)

a column vector containing motor torque measurements ␶ i and base sensor force/torque data ␶ e . The parameter vector ␪ appearing in the combined model

␪⫽

冋 册 ␦i ␦fg ␦ e⶿ ␦ i

(3)

contains all inertial parameters, all friction parameters and gravity compensation parameters, and can be divided in three subsets: • a minimal set of barycentric parameters of the internal model: ␦ i . They appear in both internal and external model and are a subset of barycentric parameters of the external model ␦ e . • the remaining set of barycentric parameters of the external model: ␦ e / ␦ i . They appear only in the external model. • the parameters related to gravity compensation devices and joint friction: ␦ f g . They are coming from internal forces which only appear in the internal model. Remark: All parameter sets are assumed to be minimal, i.e., redundant parameters are removed 关8,9兴. The identification matrix ⌽ of the combined model consists of following submatrices: ⌽⫽

冋

⌿i

⌿fg

0

⌿ e1

0

⌿ e2

册

.

(4)

The matrix ⌿i is the rigid body regression matrix of the internal model defined in Eq. 共5兲. The columns of ⌿e1 and ⌿e2 correspond to the columns of the regression matrix ⌿e of the external model 共Eq. 共6兲.兲 related to the elements of ␦ i and ␦ e ⶿ ␦ i respectively. ⌿ f g models the friction torques and gravity compensation devices. Based on the type of measured inputs and outputs, this global model can also be split up into two different models: external and internal models. Internal models relate actuator torques or forces 共outputs兲 to the robot motion 共inputs兲: ⌿ i 共 q,q˙ ,q¨ 兲 ␦ i ⫽ ␶ i .

(5)

The external model relates robot motion 共inputs兲 to reaction forces and torques measured at the base of the robot 共outputs兲. It can be easily obtained by projecting the force and torque vectors at the 2 Õ Vol. 123, DECEMBER 2001

first joint on the axes of the inertial reference frame attached to the base plate. As a result, the external model equations have a form similar to Eq. 共5兲: ⌿ e 共 q,q˙ ,q¨ 兲 ␦ e ⫽ ␶ e .

(6)

It is common to model friction in robot joints as a torque ␶ f which consists of Coulomb and viscous friction, yielding the following friction model for joint j:

␶ f , j ⫽ f c, j sign共 q˙ j 兲 ⫹ f v , j q˙ j

(7)

where the parameters f c, j and f v , j represent the Coulomb and viscous friction coefficient respectively. Notice that this model is well suited only at ‘‘sufficiently high’’ velocities. Robot manipulators are often equipped with gravity compensation devices in order to reduce the load due to gravity of some links on their respective actuators 关1兴. As a consequence, the parameters related to the link’s center of mass, must be replaced by some other parameterization in the internal model. This is discussed in detail in 关4兴. We assume that the vector ␦ f g contains these additional parameters. 2.2 Effect of Rotor Inertia. The model reduction removes unidentifiable parameters and combines parameters that can only be estimated in combination. Within the robotics research field, the following simplification strategy is frequently applied. The rotor inertia 共including the inertia of the transmission and that of the brake兲 is replaced by an equivalent inertia which is added to the link inertia. As a result, the reduced dynamic equations correspond to dynamic equations of the link only and rotor and transmission dynamics are not identified separately. When combining internal and external models into a combined model, however, this simplification cannot be applied anymore because the rotor inertia has a different contribution to the internal model than to the external model. If the influence of these rotor inertia is significant, the mentioned simplification would yield that the combined inertial parameters of the internal model ␦ i is not subset anymore of the external model parameters ␦ e . To avoid this, the rotor inertia have to be included 共and estimated兲 as separate parameters in the combined model. However, knowledge of the values of the rotor inertias, usually provided by the manufacturers, simplifies this problem, simply by subtracting the torque contributions of these inertias from the actuator torque measurements. For the first three axes of the KUKA IR 361 robot 共described in Section 4.1兲, the torque calibration of the rotor inertia in the internal model is, respectively:

␶ 1 ⫽n 21 q¨ 1 I m 1

(8)

␶ 2 ⫽ 共 1⫹n 22 兲 q¨ 2 I m 2 ⫹ 共 q¨ 2 ⫹n 3 q¨ 3 兲 I m 3

(9)

␶ 3 ⫽ 共 n 3 q¨ 2 ⫹n 23 q¨ 3 兲 I m 3 .

(10)

For the external model, the following torques have to be subtracted from the measured robot reaction torques on the base plate in x, y, and z-direction, respectively: M x ⫽ 共 ⫺q¨ 2 共 1⫹n 2 兲 sin q 1 ⫺q˙ 1 q˙ 2 共 1⫹n 2 兲 cos q 1 兲 I m 2 ⫺ 共共 q¨ 2 ⫹n 3 q¨ 3 兲 sin q 1 ⫹q˙ 1 共 q˙ 2 ⫹n 3 q˙ 3 兲 cos q 1 兲 I m 3

(11)

M y ⫽ 共 q¨ 2 共 1⫹n 2 兲 cos q 1 ⫺q˙ 1 q˙ 2 共 1⫹n 2 兲 sin q 1 兲 I m 2 ⫹ 共共 q¨ 2 ⫹n 3 q¨ 3 兲 cos q 1 ⫺q˙ 1 共 q˙ 2 ⫹n 3 q˙ 3 兲 sin q 1 兲 I m 3 (12) M z ⫽n 1 q¨ 1 I m 1

(13)

with n i the transmission ratios of joint i 共n 1 ⫽94.147, n 2 ⫽103.235, n 3 ⫽51.441兲. I m i is the inertia of the motor of joint i. The inertia includes the inertia of the rotor, the brake and the tachometer. Transactions of the ASME

2.3 Advantages From Combining Internal and External Robot Model. The classical robot identification approach using the internal model suffers from an important drawback: the torques applied to the links are not directly available, so the accuracy of their estimates depends on friction torque modelling errors and the precision of the actuator torque constants. The external model is totally independent of internal torques such as joint friction torques. The reaction forces and torques are measured by means of an external sensor: a force/torque platform. This approach, however, suffers from the fact that joint friction parameters cannot be estimated. These parameters are important for accurate actuator torque prediction, which is used in advanced control algorithms and path optimization. Experimental robot identification benefits from combining internal and external robot models. The combined model allows taking into account more measurement data, i.e., joint torque data with base reaction force and torque data, in one parameter estimation problem, yielding more accurate robot model parameter estimates. Since the base sensor method is not influenced by friction, this method results in a more accurate estimation of the inertial parameters. By including the actuator torque measurements, however, this method is able to provide an estimation of the friction torque as well. The improved robot model parameter accuracy yields more accurate actuator torque predictions, as it is shown in 关4兴 by means of simulations. This is quite interesting because the design of an advanced robot controller, such as a computed torque controller, is based on the robot model, and its performance depends directly on the model accuracy. This also holds for trajectory optimization taking into account the physical limits of the robot. The combined model provides, in addition, estimates of the base plate reaction forces and torques, which is important for the path planning of e.g., space robots, as explained in the Introduction.

3

Experiment Design and Parameter Estimation

3.1 Optimal Periodic Robot Excitation. The second step in the identification procedure is the design and execution of a robot excitation experiment. Accurate robot identification requires specially designed experiments. In the design of an identification experiment, it is essential to consider whether the excitation is sufficient to provide accurate parameter estimation in the presence of disturbances, and whether the processing of the resulting data is simple, yielding consistent and accurate results. The generation of an optimal robot excitation trajectory involves nonlinear optimization with motion constraints. Several approaches have been presented 关10,11兴. They all use a different trajectory parameterization, none of them resulting in periodic trajectories. Swevers et al. 关1兴 present a robot excitation which is periodic. This approach is adopted here and the advantages related to the periodicity of the excitation are illustrated in Section 5. The excitation trajectory for each joint is a finite Fourier series, i.e., the angular position q j for each joint j is written as: Nj

q j共 t 兲⫽

兺a

p⫽1

j p

cos共 ␻ f pt 兲 ⫹b pj sin共 ␻ f pt 兲 ⫹q j0 ,

(14)

with w f the fundamental pulsation of the Fourier series. This Fourier series specifies a periodic function with period T f ⫽2 ␲ / ␻ f . The fundamental pulsation is common for all joints in order to preserve the periodicity of the overall robot excitation. Each Fourier series contains 2⫻N j ⫹1 parameters, that constitute the degrees of freedom for the optimization problem: a pj , and b pj , for p⫽1 to N j , which are the amplitudes of the cosine and sine functions, and q j0 which is the offset on the position trajectory. Appropriate values for these trajectory parameters can be selected by means of trial and error, as it is done in industry, or by solving a complex nonlinear optimization problem with motion constraints. A popular optimization criterion is based on a scalar meaJournal of Dynamic Systems, Measurement, and Control

sure log det共•兲 of the covariance matrix of the parameter estimates, P, yielding the so-called d-optimality criterion 关12兴. In the case of the Markov estimate 关1兴 estimate, this covariance matrix depends only on the robot trajectory via the identification matrix ⌽ 关5兴 and the variance of the noise on the force/torque measurements. It does not depend on the model parameters. The motion constraints are limitations on the joint angles, velocities, and accelerations, and on the robot end effector position in the cartesian space in order to avoid collision with the environment and with itself. This last type of constraint involves forward kinematics calculations. The advantages of using periodic excitation are discussed in Section 5, a discussion which is based on experimental results. 3.2 Parameter Estimation. The last step in the identification is the estimation of the robot parameters from the measured data 关2,11兴. The maximum likelihood parameter estimation method presented by Swevers et al. 关2兴 will be applied in this paper, since this approach is based on a statistical framework aiming at estimating the robot model parameters with minimal uncertainty. The maximum likelihood estimate of the parameter vector ␪ is given by the value of ␪ that maximizes the likelihood of the measurement. The minimization of such a likelihood function is a nonlinear least squares minimization problem. In general, this method assumes that the measured joint positions and actuator torques are both corrupted by independent zero-mean Gaussian noise. If the measured joint angles are free of noise, this minimization problem simplifies to the Markov estimate, i.e., a weighted linear least squares estimate, if the model is linear in the parameters, which is the case if barycentric parameters are used. The weighting function is the reciprocal of the standard deviation of the noise on the measured force/torque data 关2兴. This simplification is applied here, and is justified since the noise level on the joint position measurements is much smaller than the noise level on the force/torque measurements 关2兴. A consistent estimate of the noise variance is provided by the sample variance 关1兴, which can be applied without additional measurements if the robot excitation, and therefore also the measured joint torques, is periodic 共see Section 5兲. The models used by the above-mentioned identification methods are based on the so-called barycentric parameters 关8,13兴, which are combinations of the inertial parameters of the different bodies. The main advantage of this modelling approach is that the resulting models are linear in the unknown parameters, which simplifies significantly their estimation 关2兴.

4 Experimental Identification Results Using a Combined InternalÕExternal Model This section discusses the application of the above mentioned identification approach. Internal and external models are combined to improve the accuracy of the parameter estimates. 4.1 Description of Test Case and Robot Model. The considered test case is a KUKA IR 361 robot 共Fig. 1兲 placed on a KISTLER 9281 B21 force/torque platform 共Kistler Instrumente AG兲. Only the first three robot axes are considered. The robot is equipped with a internal spring which compensates the gravitation for the second link 关1兴. Parameter vector ␦ i contains 13 independent barycentric parameters. The subset ␦ e ⶿ ␦ i contains 6 additional inertial parameters, which only appear in the external model 关4兴. The friction model considers viscous and Coulomb friction, yielding 6 parameters. One parameter per joint is added to take offset on the joint torque measurements into account. This yields 15 parameters appearing only in the internal model and form the subset ␦ f g . The torque generated by the gravity compensation spring is a complex nonlinear function of the angle of the second joint q 2 and the spring parameters. This nonlinear dependency and the low DECEMBER 2001, Vol. 123 Õ 3

Fig. 1 Schematic representation of a KUKA IR 361 robot

sensitivity of spring force for changes in its parameters make the estimation of the real spring parameters from noisy data cumbersome 关14兴. In 关1兴, it is shown that for the case of rotational joints, a better modelling approach is to approximate the torque resulting from the combined effect of gravity and a gravity compensation spring by means of a series of N s harmonically related sine and cosine functions of q 2 . In the case of the KUKA IR 361 robot, the resulting torque on the second joint can then be modelled as: Ns

␶ res 2 ⫽

兺A l⫽1

l2

cos共 lq 2 兲 ⫹B l 2 sin共 lq 2 兲

(15)

Only the fundamental sine and cosine functions and their first and second harmonics have to be considered 共N s ⫽3 in Eq. 共15兲兲 for accurate modelling 关1兴, yielding 6 additional parameters: A l 2 . B l 2 for l⫽1,2,3. 4.2 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: 共1兲 identification using the internal model, which contains 28 parameters; 共2兲 identification using the external model, which contains 19 parameters; and 共3兲 identification using the combined model containing all 34 parameters. The fundamental frequency of the trajectories is 0.1 Hz, yielding a period of 10 seconds. The excitation trajectory consists of 5-term Fourier series, yielding 11 trajectory parameters for each joint, and a 0.5 Hz bandwidth for the excitation signal. The optimization of the trajectories is based on the d-optimality criterion. For comparison purposes, only one trajectory is optimized for the combined model and used for all three experiments. Figure 2 shows the optimized trajectory for the three robot axes. The trajectories are implemented on the COMRADE 关15兴 software platform and runs at a sampling rate of 150 Hz. After the transients have died out, 16 periods are measured and used for the identification. The signals measured during excitation are: 共1兲 the joint angles, measured by means of the encoders mounted on the actuator shafts; 共2兲 the actuator currents, which are considered proportional to the actuator torques; and 共3兲 the 6 reaction forces/torques measured at the base of the robot. The motor currents and reaction forces/torques are analog signals. In order to avoid aliasing, they are filtered using eighth-order analog low-pass butterworth 4 Õ Vol. 123, DECEMBER 2001

Fig. 2 Optimized robot excitation trajectory: axis 1 „full line…, axis 2 „dashed line… and axis 3 „dash-dotted line…

filters with a cut-off frequency of 40 Hz, before their A/Dconversion. The phase shift introduced by these filters is compensated digitally 关2兴. The variance of the noise on the averaged actuator torque and the reaction forces and torques measurements is estimated by calculating the sample variance: ␴ ␶ 1 ⫽3.55 Nm, ␴ ␶ 2 ⫽4.03 Nm, ␴ ␶ 3 ⫽1.26 Nm, ␴ f x ⫽10.54 N ␴ f y ⫽10.88 N, ␴ f z ⫽4.81 N, ␴ m x ⫽11.61 Nm, ␴ m y ⫽11.33 Nm and ␴ m z ⫽3.15 Nm. The estimation of the variance and the improvement of the signal to noise ratio through data averaging are only possible because of the periodicity of the excitation, as it will be explained in Section 5. The trajectory parameters q j0 , a pj , and b pj 共Eq. 共14兲兲 are 共re兲estimated using the discrete Fourier transform of the averaged encoder measurements, in order to account for tracking errors during the execution of the experiments due to robot controller limitations. The joint velocities and accelerations are then calculated analytically 共see Section 5 and 关2兴兲. 4.3

Discussion of Experimental Results

4.3.1 Parameter Accuracy. The model parameters are estimated using the Markov estimator. The variances on these parameters can be calculated explicitly based on knowledge of the model parameters and of the variance of the noise on the torque/ force data 关2兴. Comparison of these variances 共Table 1兲 shows that combining the internal and external models and measurements yields a significant improvement of the accuracy of the parameter Table 1 Selected set of estimated parameter values and standard deviations for the combined, internal, and external model parameter combined Kr 111 Kd 2 b 31 b 32 f v2 f c2 f v3 f c3 b 21 b 12 br 13 K 112

␴c

␴i

internal

external

␴e

18.2649 0.0917 18.0043 0.1148 18.2973 0.1073 ⫺10.2864 0.0895 ⫺10.1685 0.1113 ⫺10.2399 0.1051 0.6208 0.0585 0.6872 0.0627 0.5811 0.0917 2.8214 0.0400 18.4613 35.6298 6.4662 14.1962 16.7748

0.2412 0.1933 0.1516 0.1172 0.0873

2.8288 0.0423 18.9271 35.7410 6.4505 14.1673 -

0.2422 0.1937 0.1541 0.1185 -

3.1192 0.0990 16.5511 0.1096

0.4242 0.1171

-

-

0.2911 0.1233

19.1792 0.1157 ⫺0.8947 0.2293

-

-

19.0217 0.1189 ⫺0.8896 0.2304

Transactions of the ASME

Fig. 3 Joint angular position for the validation trajectory: axis 1 „full line…, axis 2 „dashed line… and axis 3 „dash-dotted line…

estimates: the accuracy of the parameters in the set ␦ i 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 关4兴. 4.3.2 Uncertainty on Actuator Torque Prediction. The uncertainty on the actuator torque predictions depends on the accuracy of the parameter estimates in ␦ i and ␦ f g , because of the linear dependency of the torque prediction on these model parameters 关16兴. Two covariance matrices for the actuator torque predictions can be constructed: one based on the covariance matrix of the model parameters resulting from the internal model identification, and one resulting from the combined model identification. The covariance matrix resulting from the combined model is smaller than that of the internal model, i.e., the difference is negative definite. 4.3.3 Experimental Model Validation. 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. Figure 4 compares the measured and predicted actuator torques and the corresponding prediction errors, obtained using the parameter estimates resulting from the combined model identification. Comparison of the prediction errors with the measured torques shows that the obtained model is capable of accurately predicting the actuator torque data. The peaks in the estimation residue occur at low joint angular velocity, which indicates that the assumed friction model, which includes viscous and Coulomb friction only, is too simple. Including more advanced friction and gear models results in smaller estimation residues 关17兴. Table 2 shows the root mean squared 共RMS兲 actuator torque prediction errors obtained with the different sets of estimated parameters for this 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. This improvement is rather limited and does not correspond to the significant improvement predicted by the simulation results presented in 关4兴. The reason for that is quite simple. The estimated noise levels on the measured reaction forces and torques are much higher than the levels that were assumed in simulation. In addition, the estimated noise levels on the measured Journal of Dynamic Systems, Measurement, and Control

Fig. 4 Measured motor torque, predicted torque and the corresponding torque prediction errors for the validation trajectory using the combined model

actuator torques are lower than the levels that were assumed in simulation, e.g., the estimated noise variance on reaction force f x equals 10.5 N, while the simulations presented in 关4兴 assumed a noise variance equal to 2 N. From this discussion, it can be concluded that combining internal and external measurements improves the accuracy of the actuator torque prediction of the robot model if the signal to noise ratio of the reaction force/torque measurements 共external measurements兲 is sufficient. An important advantage of this combined model is that it also allows predicting the robot base plate reaction forces/torques accurately 共see Table 3兲. Figure 5 compares the measured values, the predicted values, and the prediction errors for these reaction forces/torques. The prediction errors are small with respect to the measured values. It can be seen clearly that these predictions are sufficiently accurate. As a result, the obtained combined model can be used for path planning and optimization taking into account limits on actuator torques and base plate reaction forces and torques. The Canadian Space Agency 共CSA兲 has adopted this new combined experimental robot identification method. Their test setup is the SMT-robot mounted on a force/torque platform. The purpose of this hydraulic robot is to simulate on earth the behavior of the Special-Purpose Dexterous Manipulator 共SPDM兲 that will be mounted on the International Space Station. For that purpose, the CSA requires model that accurately predicts the actuator torque and base plate reaction forces/torques, i.e., an accurate combined Table 2 Root mean squared actuator torque prediction errors for the validation trajectory RMS prediction error

axis 1

axis 2

axis 3

internal model combined model

5.871 5.578

8.851 8.518

4.165 3.948

Nm Nm

Table 3 Root mean squared reaction forceÕtorque prediction errors for the validation trajectory RMS prediction error fx 5.631 N mx 11.785 NM

fy 16.419 N my 7.858 Nm

fz 5.657 N mz 3.108 Nm

DECEMBER 2001, Vol. 123 Õ 5

Fig. 5 Measured and predicted reaction forces and torques and the corresponding prediction errors for the validation trajectory using the combined model

model. Accurate prediction of the actuator torques is required to compensate gravity as accurately as possible, in order to simulate zero gravity conditions. Accurate prediction of the base plate reaction forces/torques is required for the path optimization taking into account tight restrictions on these reaction forces/torques. Remark: When using data from different measurement devices to estimate the same parameters, calibration of the measurement channels becomes important. The force platform can be calibrated very accurately. It is based on piezo crystals, which keep their properties for a very long time. Mostly a calibration sheet is provided by the manufacturer. The motor torques, however, are not directly available. They are measured indirectly by means of the motor current. For permanent magnet DC motors the torque constant K m relates the torque output ␶ i to the current I: ␶ i ⫽K m I. In practice, the actuator torque constant can vary considerably from the manufacturer’s specification. Torque constants may even show considerable variation between similar motor types on a single robot. In most cases, the measured torque constant is less than the manufacturer’s specification. Torque constants reduce gradually with time due to demagnetization. For all these reasons it is necessary to frequently recalibrate the actual torque constant for each motor in a robot, e.g., using 关18兴.

5 Advantages of Using Periodic Excitation for Robot Excitation One of the key elements of the presented robot identification method is the fact that the excitation is a finite Fourier series, yielding a periodic response, i.e., all measured signals are periodic, after the transient robot response has died out. The advantages of periodic excitation in robot identification have been briefly mentioned in 关2兴, but have never been demonstrated experimentally. This is the purpose of this section. The discussion is based on the experimental data presented in Section 4. The finite Fourier series that is used in this experiment contains 5 harmonically related sine and cosine functions, with a fundamental frequency of 0.1 Hz, yielding a bandwidth of 0.5 Hz. This periodic bandlimited trajectory is advantageous because it allows 共1兲 improving the quality of the measured signals, 共2兲 estimating the noise characteristics without performing additional measurements, and 共3兲 calculating accurate and noise-free estimates of the joint velocities and accelerations, which are required to calculated the identification matrix ⌽ 共Eq. 共4兲兲. The signal to noise ratio of the measured signals can be improved by data averaging, which is possible because of the peri6 Õ Vol. 123, DECEMBER 2001

Fig. 6 Accelerations obtained by analytical and numerical differentiation „using Tustin’s bilinear differentiation rule… of joint angle measurements: axis 1 „full line…, axis 2 „dashed line… and axis 3 „dash-dotted line…

odicity of the data. This is extremely important since torque/force measurements are usually very noisy. This technique is preferable to low-pass noise filtering because filtering colors the noise, and consequently, complicates a consistent and efficient, e.g., maximum likelihood, parameter estimation 关16兴. The variance of the noise on the measured signals can be calculated by means of the sample variance, without performing additional measurement: N

␴ 2x ⫽

1 共 M N⫺1 兲 k⫽1

M

兺 兺 共 x 共 k 兲 ⫺x˜ 共 k 兲兲 , j⫽1

j

2

(16)

where x presents any of the measured signals, M is the number of measured periods, Nis the number of samples per period, and x j (k) and ¯x (k) represent the kth sample of the jth period and the average of the signal, respectively. The calculation of the joint velocities and accelerations can be performed by analytical differentiation of the measured joint angles. This is possible because these signals are periodic, finite Fourier series. For the analytical differentiation, the measured encoder readings are first approximated in a least squares sense, as a finite sum of sine and cosine functions. The resulting sum, which corresponds to Eq. 共14兲, is then analytically differentiated once or twice to obtain velocity and acceleration, respectively. This approach corresponds to frequency domain differentiation combined with frequency domain windowing. First, the discrete Fourier transform of the averaged encoder readings is calculated and the excited frequency lines are selected by frequency domain windowing 共using a rectangular window兲. The selected frequency lines are then multiplied with the frequency response of a pure single and double differentiator, i.e., multiplied with j ␻ and ⫺ ␻ 2 , with ␻ the frequency in radians per second. The obtained frequency spectra are then transformed back into time domain using the inverse discrete Fourier transform, yielding joint velocities and accelerations. Notice that this approach can only be applied for periodic signals. The Fourier transform of nonperiodic signal would introduce leakage errors, which are systematic errors. The mentioned frequency domain windowing corresponding to ideal noise filtering, providing noise-free velocity and acceleration data. Figure 6 compares the joint accelerations obtained by analytical and nuTransactions of the ASME

Table 4 RMS actuator torque prediction errors using analytical and numerical differentiation for the validation trajectory differentiation method analytical numerical

axis 1

axis 2

axis 3

5.578 6.187

8.518 13.803

3.948 6.881

Nm Nm

measurements. More important, this combined approach yields models that allow the accurate predicted of actuator torques and base plate reaction forces and torques. Both of them are important for robot path planning/optimization, especially in space robotics applications.

Acknowledgment merical differentiation 共using Tustin’s bilinear differentiation rule兲 of the joint angles measured during the experiment discussed in Section 4. The noise on the velocity and acceleration data influences the accuracy of the robot identification. Noise on the elements of the identification matrix ⌽ 共Eq. 共4兲兲 introduces systematic estimation errors if it is not considered appropriately in the parameter estimation, which is the case for the linear least squares or Markov estimation. The maximum likelihood estimation method presented in 关19兴 is capable of handling this situation. However, this estimation method is complex because of the used iterative optimization scheme, and in addition, requires a good initial parameter estimate to avoid local minima. In order to show the mentioned loss of accuracy, the Markov parameter estimation method discussed in Section 3.2, has been applied to the experimental data presented in Section 4, using the combined model. Joint velocities and accelerations are obtained by means of the above mentioned numerical differentiation rule. Table 4 compares the accuracy of the models obtained using analytical and numerical differentiation of the joint angle data. The comparison is based on the RMS actuator torque prediction errors for the validation trajectory 共Fig. 3兲. A decrease of the accuracy up to 75% shows the importance of analytical differentiation, and consequently of periodic excitation. Notice that numerical differentiation combined with low-pass filtering reduces the noise level on the estimated velocities and accelerations. However, an appropriate choice of the cut-off frequency is crucial. This choice is quite difficult if the bandwidth of the excitation signal is not clearly specified, as it is the case for the nonperiodic excitation trajectories presented in 关10,11兴. The selection of the filter cut-off frequency is then a compromise between eliminating as much noise as possible without eliminating too much of the original useful signal.

6

Conclusion

Periodic excitation is a key element for accurate experimental robot identification, mainly because it allows calculating noisefree joint velocities and accelerations from joint angle measurements. New experimental results are provided and show that this approach yields robot models that are up to 75% more accurate in predicting actuator torques than models resulting from robot identification that do not take advantage of this periodicity, or that use nonperiodic excitation, therefore requiring the application of numerical differentiation techniques. Combining internal and external measurements in one robot identification can improve the model accuracy as well, compared to identification based on internal measurements only. The significance of the improvement depends on the quality of the external reaction force/torque

Journal of Dynamic Systems, Measurement, and Control

This paper presents research results supported by the Belgian program on Interuniversity Poles of attraction initiated by the Belgian State, Prime Minister’s Office-Science Policy Programming, and by K. U. Leuvens Concerted Research Action GOA/99/04. The scientific responsibility is assumed by its authors. The work of W. Verdonck is sponsored by a specialization grant of the Flemish Institute for Support of Scientific and Technological Research in Industry 共IWT兲.

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