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Abstract—As the presence of friction in a haptic display device seriously affects its performance, proper compensation of the fric- tional effects in such a device is ...
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004

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Friction Modeling and Compensation for Haptic Display Based on Support Vector Machine D. Bi, Y. F. Li, Senior Member, IEEE, S. K. Tso, Senior Member, IEEE, and G. L. Wang, Member, IEEE

Abstract—As the presence of friction in a haptic display device seriously affects its performance, proper compensation of the frictional effects in such a device is of practical importance for advanced virtual reality applications where haptic display plays a critical role. This paper addresses the issue of friction modeling and compensation for haptic control system designs. A new method based on the Support Vector Machine (SVM) is developed in a controller design based on a two-port network to achieve accurate haptic display. The approximation model of friction is established offline through SVM learning and is used for online feed forward friction compensation. The advantages of this novel method are demonstrated through the experiments performed. Index Terms—Friction modeling and compensation, haptic display, Support Vector Machine (SVM), two-port-network-based controller.

I. INTRODUCTION

I

NCORPORATING haptic capability in a virtual reality system has become an important issue in recent research [1], [2]. The work in this area presents interesting challenges in allowing realistic and robust haptic interactions between a human and a complex dynamic virtual environment. Haptic interface is one of the most important elements for creating an immersive reality within the virtual environment. Haptic displays can be broadly classified into two categories: impedance display which measures motion and displays force, and admittance display which measures force and displays motion [3]. A type of controllers widely used in haptic display design is the controller based on a two-port network [3] originated from linear circuit theory. A two-port network model is a natural way for describing the stability and performance in the control of haptic display devices in which passivity is used as the tool for the analysis. A haptic interface with excessive net passivity can lower the system’s performance. To reduce excessive net passivity of the control system, Miller et al. [4] proposed a method for extracting the excessive passivity from the haptic interface to compensate for the active behavior of a nonlinear virtual environment to keep the whole system passive. Hannaford et al.

Manuscript received June 12, 2002; revised June 11, 2003. Abstract published on the Internet January 13, 2004. This work was supported by a Central Allocation Grant, Hong Kong SAR (CUHK1/OOC), by City University of Hong Kong (Project 7001431), and by Guangdong Natural Science Foundation (Project 970379/990796). D. Bi, Y. F. Li, and S. K. Tso are with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). G. L. Wang was with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. He is now with the Department of Electronic and Communication Engineering, Sun Yat-Sen University, Guangzhou 510275, China. Digital Object Identifier 10.1109/TIE.2004.825277

[5] moved a step further by proposing an adaptive energy dissipative element to absorb exactly the net energy output in real time. These attempts focused on stabilizing the virtual environment. For a mechanical haptic interface, friction is an important factor that can cause excessive net passivity. Therefore, an accurate friction modeling and compensation scheme is of vital importance for extracting the excessive passivity for high-peformance human perception during interactions. The desirable characteristics of a haptic display device include low friction as well as low inertia. Practically, however, more or less all mechanical haptic devices tend to have friction of some kind. If a haptic display requires multi-degree-offreedom motion, multiple joints with friction is unavoidable [6]. Friction is a complex nonlinear phenomenon that has many negative effects on control systems such as stiction, hysteresis, Stribeck effect, stick–slip, velocity dependence, and input frequency dependence. They become particularly significant when the motion is in low velocities, especially near zero crossings. In the last two decades there have been extensive studies on model-based friction compensation and various friction models have been proposed [7]. In addition to some well-known classic models, such as the Coulomb friction model and Dahl model, more elaborate dynamic friction model structures have been developed and improved behavior has been achieved [8]. The LuGre model can capture the most observed static and dynamic characteristics of friction [9]. The cost paid for the dynamic friction prediction of this model is the dependency of friction on an immeasurable internal state. This sophisticated dynamic model structure often requires complex and time-consuming identification processes that are difficult for practical applications. In addition, such dynamic friction model structure cannot be used in most adaptive control techniques which rely on the linear-in-the-parameters assumption [10]. To make the LuGre model viable, a variety of partially prior knowledge of the model has been exploited so as to facilitate the friction compensator design. A commonly used idea is to decouple the static friction model from the LuGre model by ignoring the instantaneous behavior in velocity changes. This indicates that we can approximate the LuGre model by its static friction model which takes velocity as its input. This leads to a velocity dependent friction model which has been in use in many studies [10], [11]. Some experimental work has proved that a good static friction model can approximate the real friction force with a high degree of confidence [12]. Model-free parameterization methods are commonly employed for adaptive static friction compensation. Neural networks provide a way for model-free parameterization with attractive properties, including online learning capability [11]. Whilst the advantage of neural adaptive techniques

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can be readily understood, some difficulties arise from the lack of knowledge on how to determine their structures. Recently, the Support Vector Machine (SVM) [13] has aroused research interests in data classification and regression. This mainly lies in the fact that SVM has a solid theoretic basis: Structure Risk Minimization principle originated from Statistical Learning Theory. In particular, an SVM for regression, like regularization networks [14], proves to be a potential tool for constructing the approximation of a function from sparse training data. The approximation of a function using SVM offers some attractive properties. For example, it does not suffer from the overfitting problem and it has good generalization ability. These features suggest that it may be a good candidate for constructing approximations for unknown nonlinear functions needed in control system designs. The feasibility of applying an SVM for learning feedforward control signals has been investigated in [15] but only with simulation studies. Up to now, using the SVM for friction modeling has not been exploited much. In this paper, we report our work in incorporating SVM in the controller design based on a two-port network [4] for haptic display. The approximation model of nonlinear friction is established offline by the SVM learning technique. We propose to use the SVM algorithm to learn the nonlinear friction model offline, and then implement it in feedforward compensation for online haptic control. A controller based on a two-port network is adopted to guarantee the stability of the haptic display process which is of vital importance for the safety of operators. The potential advantages arising from this novel strategy are presented. We demonstrate that the SVM learning technique used here can replace a sophisticated dynamic friction model in friction compensation. Using SVM learning technique in the controller for the haptic display can provide enhanced human perception during the interaction process. This paper is organized as follows. The SVM-learning-based friction modeling is formulated in Section II. In Section III, the experimental studies are presented in the context of a one-degree-of-freedom haptic display device. Potential advantages arising from the novel strategy proposed in this paper are reported with experimental verifications. Finally, in Section IV, some conclusions and perspective views are presented. II. SVM LEARNING FOR FRICTION MODELING AND COMPENSATION

, we can only use the know the probability measure that minknown training data sets to estimate the function . A possible solution consists in replacing the inimizes tegration by an empirical estimate to obtain the Empirical Risk Function (2) , the emIt is expected that if the training data number pirical risk would converge to the actual risk function (1). This function learning method is also called Empirical Risk Minimization (ERM). While the ERM learning method is useful if a large number of training data are available, the “overfitting” problem will arise if the training data is sparse. To overcome this problem, many new approaches have been studied in machine learning such as Regularization Networks [16], Gaussian Processes [17], where a Regularized Risk Function is proposed by adding a capacity control term to the Empirical Risk Function (3) In the above equation, is the capacity control term, denotes a reproducing kernel Hilbert space, is the correis regularization constant. The capacity sponding norm, control term controls the generalization error. Compared with the Empirical Risk Function, the Regularized Risk Function is closer to the real risk function if the regularization parameter is properly chosen. In Statistical Learning Theory, the minimization of (3) is also called Structural Risk Minimization (SRM) [13]. The Represented Theorem [18] states that if is the kernel of the corresponding reproducing kernel Hilbert space, the optimal solution through minimizing the Regularized Risk Function (3) can be described by (4) is a constant parameter. From Mecer’s Theorem, a where kernel function can be expressed as a dot product of some mapping function if it satisfies the Mecer’s condition. Thus, (4) can be expressed as (5)

A. SVM-Based Learning For a given set of input/output training data pairs , with an assumed proba, the goal of learning from this data set is bility function to find an approximating function to minimize a risk function [13] (1) is a cost function dictating how we will pewhere nalize the estimation errors based on the empirical training data set; , are the corresponding input vector and output vector; is the approximating function we choose. Since we do not

is a mapping function and repwhere resents the Hyperplane parameter. Equation (5) indicates that the optimal function can be expressed as a hyperplane in a mapping space. Equation (5) indicates that a large number of training data points would cause problems since the number of basis functions required for an optimal solution equals the number of samples. We would need a large memory with high computation cost when applying this identified model for control purpose. Based on minimizing the Regularized Risk Function, SVM regression overcomes this problem by introducing a special cost function that eliminates the contribution of the basis function corresponding to points close to their target values. This cost

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function is called - insensitive loss function. The mathematical expression for the -insensitive cost function is if otherwise (6) gives the absolute error tolerance. Using the -inwhere sensitive cost function in the Regularized Risk Function (3) yields the following equation: (7) where the capacity control term in the regularized risk for implementation by SVM. To function is represented by obtain the minimization result, the regularized risk function (7) can be minimized by another form (8) with the following constraints:

(9) . Here, and denote the distance from the for approximation function to the actual sample minus the allowed error tolerance . Minimizing (8) under the constraints (9) and applying Lagrangian optimization gives

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minimizing the Regularized Risk Function with -insensitive cost function is called SVM regression. SVM has the following characteristics: • no overfitting and curse of dimensionality problem; • only one global optimization solution; • sparse representation of the identified model. SVM for regression is a promising tool for offline friction identification and online feedforward compensation to achieve realistic haptic display. The kernel here plays an important role in implementing the SVM algorithm, as it determines the function structure in the function learning. Some useful kernels are as follows. • Polynomial Kernel: kernels which generate polynomials of order in the input space

• Spline Kernel: kernels which generate infinite splines with piecewise cubic in the input space

• RBF Kernel: kernels which generate radial basis functions with variance

B. Friction Modeling Via SVM Learning

(10) Subject to (11) . , are the Lagrangian multipliers and is the regularization parameter. Then, Lagrangian multipliers can be solved. The approximation function is identified as where

(12) is nonzero only when the distance of the training points from the target is larger than or equal to . These training points are called Support Vectors. This function identification method by

The SVM algorithm has been proved to be suitable for constructing approximations of unknown nonlinear functions. In what follows, we will apply the SVM learning in the nonlinear friction identification and feed forward compensation for haptic display. We will develop the method for friction identification in this subsection. The implementation using our experimental setup will be presented in Section III. As we know, friction exists in the mechanical components of a haptic device and it is highly nonlinear. To achieve high performance in the haptic display, modeling and compensation of the friction effects become necessary. Traditionally, friction is identified via a model-based approach. In such an approach, if the model structure chosen were too simple, the modeling errors would be large. If the model structure were too sophisticated, the model parameter identification process would be too complicated for implementation in a virtual reality system. Therefore a model-free approach would be desirable for adaptive friction compensation. SVM-based method belongs to a model-free method for identification and thus can offer its distinct advantages over a traditional neural-network-based approach. Friction is a complex phenomenon. In this study, we use a normally adopted approximate LuGre model by its static part which takes velocity as its input [10], [11]. In our experiment, the angular velocity is the input and friction force is the output in the model. Real friction contains discontinuity in the zero-velocity zone, as the static friction changes its sign there. Since the

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learning function chosen is generally smooth, the SVM learning process is divided into two parts: when the angular velocity and respectively. Combining these two identified parts of the model gives the entire friction learning model required. This method can retain the discontinuous properties of the original friction model. The implementation then consists of the following. • Obtain two groups of training data sets

and

from offline experiments, where , , , are the velocity values, and , , , are the friction torques. • Apply the SVM learning scheme to obtain the two parts of the friction model separately, based on (10) and (11). Solving the optimization problem, we can then obtain the identified friction model. III. EXPERIMENTAL STUDIES In this section, we present two sets of experiments conducted on the haptic display device. The first set of experiments was to obtain the friction model offline identified by SVM from the training data. Different kernels were attempted in the SVMbased learning process in order to arrive at the optimal model for this application. Then using the controller based on two-port network with the SVM-based friction compensation, a set of haptic display experiments was carried out. In the following, the setup of the haptic display device is firstly described. Then the SVM-based offline friction model learning process is presented. Finally, some results from the haptic display verification experiments are given. A. Experimental Setup Our experiments were performed on a planar one-degree-offreedom haptic display device as shown in Fig. 1. In this set of tests, we only dealt with the horizontal motion and force along y direction perpendicular to the link length direction. This display device consists of a mechanical link driven by a dc motor through a gearbox with a ratio of 1:80. A three-dimensional finger force sensor was installed at the tip of the link, which allows the interaction force between the operator and the haptic device to be measured along three directions. In each direction, the force sensor has a resolution of 0.005 N. The angular position was measured through an optical encoder with a resolution of 500 pulses per revolution. The armature current of the motor was also measured via a Hall sensor and was used to obtain the torque applied by the motor. As the link joint is vertical, the gravity torque is ignored when the haptic device is moved. The dynamics of this haptic device can be described by the following equation: (13)

Fig. 1. Photograph of the experimental setup.

where is the motor torque, is the frictional torque, is the mass moment of inertia, is the angular position, is a constant coefficient, and is the interaction force between the operator and the haptic interface along the direction. B. Offline Learning of Friction Model by SVM As the first step toward offline learning of the friction model, a number of experiments were performed to obtain the training data sets. According to the dynamics model (13) of the one-degree-of-freedom haptic device, assuming a simplified condition and controlling the haptic device to move with a constant angular velocity, then the motor driving torque equals the frictional torque. To obtain the friction torque values at different velocities, the experiments were repeated for a set of angular velocity values in the range rad s. The tests included two different directions of motions of the link. The corresponding friction torques were recorded to give two groups of angular velocity-friction torque training data sets for the two and respectively. cases with the angular velocity ) was measured by conducting The static friction (at “break-away” experiments in the two directions, as suggested by Armstrong [12]. Following a “warming up” process for the system, in an open-loop configuration, the motor current was increased until the joint moves (breaks away). The “break-away” current was recorded to calculate the static friction torque. The were obtained along the directions friction torques at for both the positive and negative velocities. Adding them to and ), we obtained the the two data groups (for training data for learning the discontinuous friction model. In the experiments, each data point of the friction torque was obtained by averaging 100 measurements. Finally, each training data group had 54 training data points with the velocity input rad s. Using the same method, we obtained 28 testing data points (different from the training data we obtained) to test the generalization ability for each part of the identified friction model.

BI et al.: FRICTION MODELING AND COMPENSATION FOR HAPTIC DISPLAY BASED ON SVM

Fig. 2.

Friction model learning (RBF kernel, _

 0).

Fig. 3.

Friction model learning (RBF kernel _

 0).

Next, we identified the friction model based on the obtained training data sets by applying the SVM algorithm. Before implementing the SVM algorithm, we had to select the value of the regularization coefficient , the error tolerance , and the kernel. As these three factors affect the characteristics of the SVM-based learning model significantly, we used some combinations of the parameter values in the model identification in order to experimentally find the most suitable set of parameter values for the learning. The evaluation of the SVM-based learning model is based on the number of Support Vectors and the modeling errors (including the approximation error and generalization error). We first chose an RBF kernel which is often used in neural networks, conducted the tests with different values of , , and kernel variance . 105 cases for each group of the friction training data were tested. After

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comparing the modeling errors and the number of Support Vectors of the model in each case, we selected two optimal , , , , the number cases: for of support Vectors was 29 and for , , , , the number of Support Vectors was 27. The learned models are shown in Figs. 2 and 3. In these two figures, the and solid lines represent the learned friction models for . The star points are the training data points that are not the Support Vectors. The dotted points are the training data points that are the Support Vectors. One shortcoming of the identified friction model is that the number of Support Vectors is relative large. Although we can reduce the number of Support Vectors by increasing the value of , experiments show that the generalization errors then tend to increase significantly.

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Fig. 4.

Friction model learning (spline kernel, _

 0).

Fig. 5.

Friction model learning (spline kernel, _

 0).

To compare the influence of different kernels on the learning accuracy, we chose another kernel: spline kernel for model identification. This time, the unknown parameters include only and . We chose 40 parameter-value combinations for learning each part of the friction model. The optimal results found were: , , with 10 Support Vectors, and for for , , with 9 Support Vectors. The learned models are given in Figs. 4 and 5, where the solid lines are the identified models, the star points are the training data points that are not the Support Vectors and the dot points are the training data points that are the Support Vectors. To verify the adequacy of the SVM-based model prediction ability with the chosen parameters, we compared the experimentally obtained 28 testing data with the predicted outputs by the

identified models. Tables I and II show the model error comparisons between results in the training and testing using the RBF stands kernel and Spline kernel separately. In the tables, the for the positive/negative-velocity regime with rad s; Mean Error stands for the absolute average error of all data points; Max Error stands for the maximum of the absolute error. From Tables I and II, it is observed that the Mean training error is nearly the same as the Mean prediction error. This means that the model error of the prediction output does not get worse compare with that of the training error. Thus the prediction ability using the SVM-based model with the chosen parameters does not change significantly whether the training data set or testing data is used as the model input. This also verifies the generalization ability of the SVM algorithm. In

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TABLE I COMPARISON OF TRAINING ERROR AND PREDICTION ERROR USING RBF KERNELS

TABLE II COMPARISON OF TRAINING ERROR AND PREDICTION ERROR USING SPLINE KERNELS

addition, we can further tune the parameters and to alter the model generation ability. For the Max Error values in the tables, as they are affected by the random noise applied to the system, the results do not have much implication on the performance of the model prediction ability. Comparing the results in Tables I and II, it can be seen that both the training error and prediction error in the spline-kernel-based models are smaller than those in the RBF-based models. This indicates that the spline kernels maybe more suitable for the friction model than the RBF kernels From the experiments, we have the following observations. • Kernel selection is of vital importance to the model learning. Different kernels would lead to different function structures. The closer the structure of the kernel to that of the task function we have to identify, the smaller the generalization errors and the number of Support Vectors in the learned model. For our friction model identification, the spline kernel seems more suitable than the RBF kernel due to its stronger generalization ability and smaller number of Support Vectors. has a twofold effect on • The generalization parameter the model learning. When increases, the approximation error using the training data becomes smaller. However, the generalization error tends to increase at the same time. The optimal value in is a tradeoff between these two kinds of errors. • The error tolerance parameter is influenced by both the spatial distribution of the training data and the accuracy needed for the model. It also affects the approximation error and the generalization error. The optimal SVM-based learning function is determined by the combined effects of the parameter value and kernel chosen. Any prior knowledge in the structure of the task function can be of a great help for selecting the kernel and parameter values. In this friction model learning case, we finally chose the spline , , for , and , kernel, and also for , corresponding to the models shown in Figs. 4 and 5. C. Haptic Display Experiments The friction model of the one-degree-of-freedom haptic device has been learned offline through SVM. In this section, we

will incorporate the identified model into controller based on a two-port network to test the performance of the haptic display interface. The platform for the experiments now include a virtual world which consists of graphics generated by another computer, in addition to the physical haptic device connected to the virtual world and acting as an interface between the human operator and the virtual world. In the following experiments, the operator grasps the handle (upper part of the finger force sensor) of the display device as shown in Fig. 1. As the inertia of the haptic display device is designed to be small, our experiments were conducted at low velocities and low rate of velocity changes. This allows us to neglect the inertia effect and focus on the friction effects. The first experiment we performed was on the comparison between the performances of the haptic displays in open loop free-motion mode with and without friction compensation. Our second experiment was on the haptic interactions with a virtual spring with friction compensation and without friction compensation. For the free-motion experiment, we will use to represent the velocity. The free-motion trajectory, i.e., the angular position of the link, is given in Fig. 6. Without friction compensation, interaction forces were felt by the operator although the virtual environment was simulating a free motion. Next, we incorporated the SVM-based friction compensation. Under the same condition of free motion in the virtual environment and with the same motion trajectory as shown in Fig. 6, the resistance forces felt by the operator becomes much smaller. The force trajectories for both cases are shown in Fig. 7. The SVM-based friction compensation has led to a much improved force output of the haptic display. From the results, we can also see that the compensated trajectory has no obvious improvement in the high frequency part which is mainly caused by the vibration of the gearbox. Thus the identified friction model cannot compensate for it. This effect is in fact much reduced by the human finger itself which acts as a low pass filter because of the force perception threshold [19]. Furthermore, the performance can be further improved by applying an additional low —pass filter to the force output. We then conducted another haptic display experiment. This time, in the virtual environment, we had an angular spring to push. The dynamics model of the angular spring is given as , where is the interaction force, is the initial angular position of the spring and is the angular po-

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Fig. 6.

Free-motion trajectory.

Fig. 7.

Interaction forces (dotted line: without friction compensation; solid line: with friction compensation).

sition after spring is pushed. is the spring constant and was N/rad in this experiment. In this experiment, set to the operator moved the haptic link to push against the virtual spring in the virtual environment. Using the controller based on a two-port network, we tested two cases for the controller: with and without friction compensation. For both cases, the motion trajectory is the same and is given in Fig. 8. Fig. 9 shows the interaction forces experienced by the operator in the two cases. It can be seen that with the friction compensated via our SVM-based method, the haptic output is significantly improved. In the current implementation, a simple experimental setup was used which served as a platform for a case study where the friction effects could readily be seen. We showed that the haptic output is improved through the friction compensation. For cases where there is no gearbox, the developed friction compensation scheme can be applied as well. In fact, a more accurate result could be achieved in such a case since the noise input (from the gearbox) to the learning model is greatly reduced then. Com-

pared with using a dedicated haptic device where tremendous efforts had to be made in the sophisticated mechanical design [20] to reduce the frictional effects, our method makes it possible for a simple device to be used for haptic display with acceptable performance. IV. CONCLUSION AND DISCUSSION In this paper, a model-free learning algorithm based on SVM was proposed for friction model identification and compensation for high fidelity haptic display. The SVM-based friction model is incorporated into controller based on a two-port network to improve the performance the haptic interface. The theoretical analysis and model identification experimental results using a simple haptic display device show that SVM is a very promising learning method. Compared with a neural network approach, SVM has a solid theoretical foundation, in the form of the Structural Risk Minimization

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Fig. 8. Motion trajectory.

Fig. 9. Interaction force (dotted line: desired force; solid line: with friction compensation; dashed-dotted line: without friction compensation).

Principle originated from the Statistical Learning theory. The SVM-based method has its special generalization ability. The learning model can be built up with sparse data and the curse of dimensionality problem can be overcome. As SVM has only one global optimal solution, the local minimum problem is avoided. Our study shows that the selection of the kernel and parameter values in SVM plays an important role in affecting the model accuracy. Experiments were conducted using the controller based on a two-port network incorporating SVM-based friction compensation. We tested the use of the haptic display device for both free motion and compliant motion in the virtual environment. The experimental results show that SVM-based friction compensation can enhance the haptic perception, enabling the operator to have a more precise feeling of what is happening in the virtual world. This is of practical importance since friction is existent in most mechanical haptic interfaces. The proposed method can be extended to a more complex and multi-degree-of-freedom

haptic display device. In such a case, the friction is not only velocity dependent, but also position/configuration dependent. So the learning process will be in a two-dimensional input space, with a two-dimensional kernel. Another consideration is that the gravity effect will also need to be compensated. REFERENCES [1] A. S. Mandayam and C. Basdogan, “Haptics in VE: Taxonomy, research status, and challenges,” Comput. Graph., vol. 21, no. 4, pp. 393–404, 1997. [2] Y. F. Li and D. Bi, “A method of dynamics identification for haptic display of the operating feel in a virtual environment,” IEEE/ASME Trans. Mechatron., vol. 8, pp. 476–482, Dec. 2003. [3] R. J. Adams and B. Hannaford, “Stable haptic interaction with virtual environment,” IEEE Trans. Robot. Automat., vol. 15, pp. 465–474, June 1999. [4] B. E. Miller, J. E. Colgate, and R. A. Freeman, “Guaranteed stability of haptic systems with nonlinear virtual environments,” IEEE Trans. Robot. Automat., vol. 16, pp. 712–719, Dec. 2000. [5] B. Hannaford and J. H. Ryu, “Time-domain passivity control of haptic interfaces,” IEEE Trans. Robot. Automat., vol. 18, pp. 1–10, Feb. 2002.

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[6] D.-S. Kwon and K. Y. Woo, “Control of the haptic interface with friction compensation and its performance evaluation,” in Proc. 2000 IEEE/RSJ Int. Conf. Intelligent Robots and Systems (IROS 2000), vol. 2, Takamatsu, Japan, 2000, pp. 955–960. [7] B. Armstrong-Helouvry, P. Dupont, and C. Canudas de wit, “A survey of models, analysis tools, and compensation methods for the control of machines with friction,” Automatica, vol. 30, pp. 1083–1183, 1994. [8] J. Swevers, F. Al-Bender, C. G. Ganseman, and T. Prajogo, “An integrated friction model structure with improved preshaping behavior for accurate friction compensation,” IEEE Trans. Automat. Contr., vol. 45, no. 4, pp. 675–686, 2000. [9] P. Dupont, V. Hayward, B. Armstrong, and F. Altpeter, “Single state elastroplastic friction models,” IEEE Transaction on Automatic Control, vol. 47, pp. 787–792, May 2002. [10] C. Canudas de wit, P. Noel, A. Aubin, and B. Brogliato, “Adaptive friction compensation in robot manipulators,” Int. J. Robot. Res., vol. 10, no. 3, pp. 189–199, 1991. [11] H. Du and S. S. Nair, “Low velocity friction compensation,” IEEE Contr. Syst. Mag., vol. 18, pp. 61–69, July 1998. [12] B. Armstrong-Helouvry, Control of Machines With Friction. Boston, MA: Kluwer, 1991. [13] V. Vapnik, Statistical Learning Theory. New York: Wiley, 1998. [14] T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Adv. Comput. Math., vol. 13, pp. 1–50, 2000. [15] B. J. de Kruif and T. J. A. d. Vries, “On using a support vector machine in learning feed-forward control,” in Proc. IEEE/ASME Int. Conf. Advanced Intelligent Mechatronics, Como, Italy, 2001, pp. 272–277. [16] F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectures,” Neural Comput., vol. 7, pp. 219–269, 1995. [17] C. K. I. Williams and C. E. Rasmussen, “Gaussian processes for regression,” in Advances in Neural Information Processing Systems, D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, Eds. Cambridge, MA: MIT Press, 1996, pp. 514–520. [18] D. Cox and F. O’Sullivan, “Asymptotic analysis of penalized likelihood and related estimators,” Ann. Stat., vol. 18, pp. 1676–1695, 1990. [19] S. J. Lederman and R. L. Klatzky, “Sensing and displaying spatially distributed fingertip forces in haptic interface for teleoperator and virtual environment systems,” Presence, vol. 8, pp. 86–103, 1999. [20] K. Y. Woo, B. D. Jin, and D. S. Kwon, “A 6 DOF force-reflecting hand controller using the fivebar parallel mechanism,” in Proc. IEEE Int. Conf. Robotics and Automation, San Diego, CA, 1994, pp. 3205–3210.

D. Bi received the B.S. degree in mechatronics engineering and the M.S. degree in mechanical engineering from Harbin Institute of Technology, Harbin, China, in 1992 and 1998, respectively. He is currently working toward the Ph.D. degree in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong. His research interests include mechatronics system design, haptic display, and modeling and control of dynamic systems.

Y. F. Li (M’91–SM’01) received the Ph.D. degree in robotics from the Department of Engineering Science, University of Oxford, Oxford, U.K., in 1993. From 1993 to 1995 he was a Postdoctoral Research Associate in the AI and Robotics Research Group, Department of Computer Science, University of Wales, Aberystwyth, U.K. In 1995, he joined City University of Hong Kong, Hong Kong, where he is currently an Associate Professor in the Department of Manufacturing Engineering and Engineering Management. His research interests include robotics, robot vision, robot sensing, and sensor-based control.

S. K. Tso (SM’81) received the B.Sc.(Eng.) degree from the University of Hong Kong, Hong Kong, and the Ph.D. degree from the University of Birmingham, Birmingham, U.K. He is currently a Professor of Mechatronics and Automation and Director of the Consortium of Intelligent Design, Automation and Mechatronics (CIDAM), City University of Hong Kong, Hong Kong. His research in the fields of industrial automation, system assessment, and intelligent control has resulted in more than 300 papers published in international journals and conference proceedings.

G. L. Wang (M’03) received the B.Sc., M.Sc., and Ph.D. degrees from Nankai University, Tianjin, China, in 1986, 1989, and 1992, respectively. In 1992, he joined the Department of Computer Science, Shantou University, where he was the Department Head from 1996 to 2000. As an Alexander von Humboldt Research Fellow, he was a Guest Scientist in the Control Engineering Laboratory, RuhrUniversity of Bochum, Germany, from 2000 to 2001, and the Institute for System Theory in Engineering, University of Stuttgart, Germany, in 2003. He was also a Visiting Research Fellow at City University of Hong Kong from 2001 to 2002. He is currently a Full Professor in the Department of Automation, Shanghai Jiaotong University, Shanghai, China. His research interests include sensor-based robot control and haptic interfaces for human–machine systems.