Stable haptic interaction using a damping model to implement a ...

Virtual Reality (2008) 12:99–106 DOI 10.1007/s10055-008-0094-x

ORIGINAL ARTICLE

Stable haptic interaction using a damping model to implement a realistic tooth-cutting simulation for dental training Guanyang Liu Æ Yuru Zhang Æ Dangxiao Wang Æ William T. Townsend

Received: 29 May 2006 / Accepted: 14 March 2008 / Published online: 16 April 2008 Ó Springer-Verlag London Limited 2008

Abstract It is difficult to implement a stable and realistic haptic simulation for cutting rigid objects that is based on a damping model because of an inevitable conflict between stability and high output force. This paper presents passivity techniques to show that an excessive damping coefficient causes the output stiffness to exceed the maximum output stiffness of the haptic device, leading to instability. By analysing the damping model of a haptic dental-training simulator, we construct a relationship among the damping coefficient, position resolution, sampling frequency, human operation, and the maximum achievable device stiffness that will still maintain device stability. A method is also provided to restrict the output stiffness of the haptic device to ensure stability while enabling the realistic haptic simulation of cutting rigid objects (teeth) that is based in a damping model. Our analysis and conclusions are verified by a damping model that is constructed for a dental-training haptic display. Three types of haptic devices are used in our analysis and experiments. Keywords Haptic display system Output stiffness Damping model Stability criterion

G. Liu (&) Y. Zhang D. Wang State Key Lab of Virtual Reality Technology and System, Beihang University, Beijing, China e-mail: [email protected] W. T. Townsend Barrett Technology Inc., Cambridge, USA e-mail: [email protected]

1 Introduction Dental students use a haptic dental-training display to simulate dental operations. Dentists can move the tip of haptic device to operate on virtual teeth with a level of haptic-visual feedback that is similar to those of physical operations (Fig. 1). The haptic display system is capable of artificially recreating the forces that simulate the interaction between a dental pin and a virtual tooth. The computational engine of the haptic display, namely the force model, computes and provides realtime force feedback, which is critical in building the dental-training system. The damping model, which is used to compute the response force based on human’s operating speed, is a kind of force model in haptic simulations. In nearly all mechanical operations, the cutting force is computed based on the operating speed (Zhang 1990). Some researchers also use damping models to compute the force feedback in haptic simulations. Tanaka et al. (1998) and Basdogan et al. (1999) implemented haptic interactions based on damping models. Using the damping model to implement a haptic rendering during the tooth-cutting operation in dental training is more efficient and simpler than other methods. However, applying a damping model in haptic simulations frequently leads to instability and low output force. For example, the three degree-of-freedom haptic Phantom Desktop device will oscillate and lose stability when the damping coefficient exceeds 0.002 N s/mm with the sampling frequency of haptic loop set 1 kHz. The output force can reaches only 0.3 N, which does not meet the needs of dental simulator. In real tooth cutting, the force between the tooth and the dental pin can reach 2.8 N. Therefore, the goal of this paper is to analyse why a haptic simulation

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The paper is organized as follows. A damping model for dental simulation is constructed in Sect. 2 and applied in the dental-training system. The conservative criterion of stable haptic interaction is presented in Sect. 3, and the cause of instability when applying the damping model is discussed. In Sect. 4, a method to realize high output force stably is presented. The factors affecting the damping coefficient—operating speed, position resolution, and sampling frequency—are discussed. Finally, the dental training system is built to test our analysis and method.

2 Damping model and implementation

Fig. 1 The haptic-visual system for dental training

based on a damping model initially fails the needs of the dental simulator and then overcoming this failure by finding a solution that provides both stability and high output forces. Some authors have considered the issues of stability in the haptic simulations. Minsky et al. (1990) noted a critical trade-off among the simulation rate, the virtual wall stiffness, and the device viscosity and provided insights into the role of the human operators in the stability concerned. Salcudean and Vlaar (1997) found that very low device friction significantly limited the achievable stiffness of the virtual environment. Virtual coupling and time domain passivity methods have been used to analyse the stability of haptic system (Adams and Hannaford 1999; Hannaford and Ryu 2002). Colgate and Brown (1994) presented the dynamic range of achievable impedances ‘‘Z-Width’’ which should satisfy a robust property, such as passivity, and the factors affecting Z-Width—sample and hold, inherent interface dynamics, displacement sensor quantization and velocity filtering. They focused on the design of haptic device and did not elaborate how to implement the damping model in order to increase the output force stably. The large error of computed operating speed has been presented as the basic cause for the instability of haptic simulations that are based on a damping model. Some authors have attempted to improve the computed operating speed and force feedback to implement high output force stably (Mahvash and Hayward 2003; Adachi et al. 1995; Mark et al. 1996). Although more-precise encoders can solve the problem, doing so sharply increases the cost of the haptic device and so is not a practical option. The paper presents a stability principle to analyse above problems and presents a method which implements the damping model while enabling high output force stably.

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In the process of tooth preparation, dentists move the dental holders (rotary dental pins) to cut or grind teeth while sensing the forces between the dental pin and a tooth. The force between the dental pin and a tooth is composed of resistance to the movement imposed by the dentist and a reactive friction force that opposes the rotation of dental pin. Based on the results measurements described later in this section, the value of friction component (\0.25 N) is much smaller than that of resistance component (up to 2.8 N). The paper focuses on the stability analysis of the damping model and not the accurate force model for dental training. Therefore, we do not consider the friction component in the force model. We first analyse the tooth cutting operation with cylindrical dental pin. When a dentist uses the cylindrical dental pin to cut into the tooth interior, the tooth will react to the dental pin by a force Fe whose direction is opposite to the direction of the human’s operation Fh (Fig. 2). Therefore, in the haptic simulation of cutting into the tooth interior, we compute the force feedback Fe which should be imposed against operators based on human’s operating speed:

Fig. 2 Force analysis in tooth cutting


F e ¼ BV c

101

ð1Þ

Vc is the computed operating speed of the human in the haptic simulation; and B is the damping coefficient, which is decided by the stiffness of tooth and the contact geometry between the dental pin and the tooth. In order to determine the value of the damping coefficient B, we construct a measuring system which is designed to have multiple capabilities such as: (a) measuring the cutting forces, (b) recording the positions of human’s operation, (c) computing the orientations of dental pin in the process of tooth preparation. The equipment consists of a dental engine to generate high-speed pin rotation (5 9 105 rpm) and two subsystems: a force-measuring subsystem and an opticaltracking subsystem (Figs. 3, 4). The first subsystem consists of a 6-axis force/torque sensor (manufactured by ATI Inc.), gypsum to fix the sensor on the test-bed and a mechanism to fix teeth on the force sensor. The optical-tracking subsystem consists of two digital cameras (manufactured by Panasonic) and an image acquisition board. The subsystems acquire and process digital data by computer control. When a dentist cuts a tooth from the surface to the interior, cutting forces and the positions of the dental pin can be measured with a sampling frequency at 40 Hz.

Fig. 3 Measuring system and dental engine

Table 1 Maximum damping coefficient and output force when the haptic simulation remains stable Haptic device

Omega 3DOF

Phantom desktop

BHhd 3DOF

Position resolution (mm)

0.009

0.01

0.02

Maximum achievable stiffness (N/mm)

12

3.5

5

Sampling frequency of position signal (Hz)

1,000

1,000

1,000

Maximum damping coefficient (N s/mm)

0.0051

0.0016

0.003

Maximum output force (N)

0.28

0.13

0.15

Twelve dentists take part in measuring and cutting operations on forty teeth that were previously removed from patients. The measuring results show that the maximum cutting force can reach 2.8 N in the enamel layer of tooth and 0.9 N in the hard-dentin layer. The maximum operating speed among these 12 dentists is measured to be 0.48 mm/s. The computed value of B is between -11.3 and -6.1 N s/mm in the enamel layer (tooth attachment) and between -6.5 and -3.8 N s/mm in the layer of hard dentin. We first choose the damping coefficient B to be 6.5 N s/mm with a of haptic-loop frequency of 1 kHz to simulate cutting teeth (Eq. (1)). When dentists move the tip of the haptic device to cut the virtual tooth, the Phantom oscillates badly enough that they can hardly maintain their grasp of the stylus. In all three types of haptic devices that we tested (Phantom Desktop, Omega and BHhd manufactured by us) the phenomenon is the same. Operators cannot finish cutting the virtual tooth at all. The only way to avoid oscillation is to decrease the sampling frequency of the position signal or decrease the damping coefficient. Although doing so can prevent oscillation of haptic device, the output force decreases significantly and cannot reach 0.3 N, well below the requirements for realistic dental simulation. Table 1 shows the relation among maximum damping coefficient, sampling frequency of position signal, and maximum output force required to keep the haptic simulation stable. What causes the instability when the damping model is applied in haptic simulation? How do we improve the damping model to increase the output force stably and meet the requirements of dental simulation? We attempt to solve these two problems from a stability analysis.

3 Stability analysis

Fig. 4 Force measuring and the dental holder with marks

The phenomenon of oscillation is that the output force changes so acutely that the dentist almost loses grasp of the haptic device. In the process of oscillation, the haptic

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102


Va Human Operator

Vic ¼ ðpci pci 1 Þf

Vc Haptic Interface

Fh

Virtual Environment

F

e

Fig. 5 Network of haptic display system

device transfers energy to the dentist. It violates a stability principle of haptic simulation in which the operator can never extract energy from the haptic display. Figure 5 shows the network model of haptic simulation. The haptic display system is composed of three elements: (1) the operator, (2) the haptic interface, and (3) the virtual environment. The operator exerts a force Fh on the haptic interface to change the physical character of the virtual environment, while the virtual environment generates a force Fe through the haptic interface to resist the actions of the dentist. For a static environment (a tooth model), if an operator uses the haptic display system to simulate cutting (not touching) the deformable tooth model stably, the haptic device will not transfer energy to the dentist: e a Fi Vi 0 ð2Þ Fih Via 0 i is the sampling period of haptic loop and Vai is the accurate dentist operating speed with no error at each sampling period. A haptic simulation of tooth cutting which does not satisfy Eq. (2) will lose stability and oscillate. What causes violation of stability principle Eq. (2) in haptic simulations? We consider that the primary reason is that the output stiffness exceeds the maximum achievable stiffness of the haptic device. In fact, the haptic device generates variational stiffness against the dentist in haptic simulations. The output stiffness should not exceed the maximum achievable stiffness which is provided by the manufacturer of haptic device. In the haptic simulation of touching a virtual wall, if the simulated virtual stiffness exceeds the maximum achievable output stiffness of haptic device, the above stability principle (Eq. (2)) will be violated and oscillation is inevitable. If the simulated virtual stiffness is smaller than the maximum achievable stiffness of haptic device, the haptic simulation of touching a virtual wall can remain stable. In the next step, we analyse the instability of haptic simulation by using a damping model with the above conclusion: output stiffness of haptic device should not exceed the maximum achievable output stiffness in order to to keep the haptic simulation stable. In the haptic simulation of tooth cutting, the speed of the dentist is computed:

123

ð3Þ

pci is the measured position of the dentist by the encoders in haptic device at each sampling period and f is the sampling frequency of position signal. The accurate operating speed Vai can be defined: Via ¼ ðpai pai1 Þf

ð4Þ

Pai is the accurate position of the dentist at each sampling period with no error. In order to simplify the equation, we define d as the minimum resolution of each axis of the haptic interface in Cartesian coordinates. d ¼ ½ d d d T (mm) is defined as the position resolution of the haptic interface in Cartesian coordinates. Therefore, the error of the measured position and the accurate position can be determined: 8 a < pix d pcix paix þ d pa d pciy paiy þ d ð5Þ : iya piz d pciz paiz þ d The position change of the dentist between two sampling periods can be determined as 8 a a c c a a < pix pði1Þx 2d pix pði1Þx pix pði1Þx þ 2d a a c c a p pði1Þy 2d piy pði1Þy piy paði1Þy þ 2d : iya piz paði1Þz 2d pciz pcði1Þz paiz paði1Þz þ 2d ð6Þ Based on Eqs. (3), (4) and (6), the error of accurate operating speed Vai and the computed operating speed Vci can be determined: 8 a < Vix 2df Vixc Vixa þ 2df V a 2df Viyc Viya þ 2df ð7Þ : iya Viz 2df Vizc Viza þ 2df From Eqs. (1) and (7), for a measured damping coefficient B, the error w ¼ ½ wx wy wz T of force Fe computed by the damping model in haptic simulation and a realistic cutting force Fea can be determined: 8 e e e < Fax 2Bdf Fx Fax þ 2Bdf e F e 2Bdf Fye Fay þ 2Bdf : ay e e e Faz 2Bdf Fz Faz þ 2Bdf 8 < 2Bdf wx 2Bdf 2Bdf wy 2Bdf : 2Bdf wz 2Bdf

ð8Þ

ð9Þ

We define the computed force change DFei between two sampling periods as e DFie ¼ Fie Fi1

ð10Þ

From Eqs. (9) and (10), the computed output force change DFei between two sampling periods of the haptic loop can be determined as:


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8 a a e a a > < 4Bðf d þ jVix Vði1Þx jÞ DFix 4Bðf d þ jVix Vði1Þx jÞ a a e a a 4Bðf d þ jViy Vði1Þy jÞ DFiy 4Bðf d þ jViy Vði1Þy jÞ > : 4Bðf d þ jV a V a e a a iz ði1Þz jÞ DFiz 4Bðf d þ jViz Vði1Þz jÞ ð11Þ Based on the above analysis, the output stiffness of haptic device should be smaller than its maximum achievable stiffness to keep the haptic simulations stable. Therefore, 8 > D Fixe K > > c c > > pix pði1Þx > > > > > > D Fe > > : pc pciz K iz

ði1Þz

K is the maximum achievable stiffness of haptic interface. Here, we assume that the position signal of the tip of haptic device has the largest error in haptic simulation: 8 c a < pix ¼ pix þ d pc ¼ paiy þ d ð13Þ : iyc piz ¼ paiz þ d i represents the odd sampling periods of the encoderposition signals. 8 c a > < pjx ¼ pjx d c pjy ¼ pajy d ð14Þ > : pc ¼ pa d jz jz j represents the even sampling periods of the encoderposition signals. Based on above assumption and substituting Eqs. (11, 13) and (14) into Eq. (12), we can obtain the sufficient condition of Eq. (12): 8 a e 4Bðf dþVixa Vði1Þx > Þ > D Fix > > D pc ¼ pa pa þ2d K > > ix > ix ði1Þx > > > > > > < e 4Bðf dþViya V a Þ ði1Þy D Fiy ð15Þ D pc ¼ pa pa þ2d K iy iy > ði1Þy > > > > > > 4Bðf dþV a V a Þ > > iz > ði1Þz > D Fize ¼ K > c a pa > p D p þ2d : iz iz ði1Þz

Table 2 Comparison between measured maximum achievable damping coefficient and computed value by Eq. (16)

B

4df 2

KðVika þ 2df Þ KðVika þ 2df Þ ¼ ; a a 4df 2 þ aaik þ f ðVik Vði1Þk Þ

k ¼ x; y; z ð16Þ

where aaik is the acceleration of the dentist at each sampling period in the haptic simulation (Wang and Zhang 2004). From Eqs. (15) and (16), we can obtain the relationship among the damping coefficient B, the maximum achievable output stiffness of haptic device K, the position resolution d, the sampling frequency of position signal f (the frequency of haptic loop is 1 kHz), and the dentist which should be satisfied to keep the haptic simulation stable. In haptic simulation, the damping coefficient is decided by the haptic device, the operation of the human, and the sampling frequency of the position signal. For the same operation and the same sampling frequency, the haptic device that has the largest position resolution and largest achievable stiffness can output the largest damping coefficient. For the same haptic device, if the operator can keep a high operating speed in the whole cutting process with small acceleration, the damping coefficient can be increased stably. However, it is difficult for an operator to keep a high and constant operating speed in the whole simulation. Equation (16) reveals that the damping coefficient is nearly inverse proportional to the sampling frequency. However, the increasing the damping coefficient in combination with decreasing of sampling frequency does not increase the output force [Eqs. (2) and (3)]. The output force is proportional to the product of the damping coefficient and the sampling frequency. If the sampling frequency is set at 1 kHz, the value of aaik is much smaller than that of 4df2 (Eq. (16), Table 1) and the effect of acceleration can be ignored. Therefore, for the tooth-cutting simulation, we can assume that the operating speed is 0.5 mm/s (maximum operating speed in tooth cutting) and use Eq. (16) to compute the lowest damping coefficient with which the haptic simulation can keep stable. Table 2 shows experimental results which verify our analysis. The computed damping coefficient is close to the value obtained by haptic simulations. In other words, the damping coefficient B and maximum output force are determined simultaneously by the maximum output stiffness, the position resolution, the dentist operating speed, and the sampling frequency. For the same sampling

Haptic Device

Omega 3DOF

Phantom desktop

BHhd 3DOF

Position resolution (mm)

0.009

0.01

0.02

Maximum achievable stiffness (N/mm)

12

3.5

5

Sampling frequency of position signal (Hz)

1,000

1,000

1,000

Maximum damping coefficient (measured) (N s/mm)

0.0051

0.0022

0.003

Maximum damping coefficient (computed) (N s/mm)

0.006

0.00175

0.0025

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104


frequency, the haptic device, which has a larger position resolution and maximum achievable output stiffness, also has a larger damping coefficient when the haptic simulation remains stable.

4 Method and experiment We can change the damping coefficient and the sampling frequency to keep the haptic simulations stable. However, the output force of haptic device cannot meet the requirements of simulating cutting stiff teeth. For the damping model of haptic simulation, increasing the damping coefficient and the sampling frequency of position signal simultaneously is a way to realize high output force. However, it may result in too large of an error for computed operating speed and output force which may lead the output stiffness to exceed the maximum stiffness of haptic device. Oscillation and instability are inevitable. If any other encoder with higher resolution is selected, the position error will be decreased and the damping coefficient and output force can be increased accordingly. However, it is difficult to replace the encoders of an existent haptic device and the cost of more precise encoders is much higher. Therefore, other methods are required to solve the conflict between stability and high output force when the damping model is applied in haptic simulations. From the above analysis, we have drawn the conclusion that large error of computed output force by a force model that results in the output stiffness exceeding the maximum stiffness is the basic source of instability. If the output stiffness can be restricted to a stable range which the haptic device can endure, the haptic display will remain stable. Therefore, before outputting force feedback to operators, we can deal with the computed force by leveraging the damping model to make the output stiffness smaller than the maximum achievable stiffness of haptic device. If the computed output stiffness exceeds the maximum achievable stiffness of haptic device, the force computed by the damping model should be modified by our proposed method to satisfy Eq. (12). We propose the method: e 8 ðF F e Þ ðpc pc Þ > ic i1 i i1 e > \Kc Fic > > < (pc pc )2 i i1 e Fie ¼ e > Fi1 Þ ðpci pci1 Þ > > F e + Kc (pc pc ) ðFic > Kc : i1 i i1 (pc pc )2 i i1 ð17Þ where Feic is the computed output force according to the damping-force model at each period of the haptic loop and Fei is the actual force which is outputted to the operator by

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Fig. 6 Dental training system and its virtual environment

the haptic device (Wang et al. 2005). In Eq. (17), the output stiffness can always be restricted to be smaller than the maximum achievable stiffness of the haptic device. The effect of this method is verified in the dental training system. The dental training system consists of a single-processor PIV/2 GHz with 512 MB PC133 RAM, a NVIDIA GeForce 4 Ti4600 and three types of haptic devices for haptic interaction (Fig. 6). In the virtual environment, a tooth triangle mesh and a cylindrical dental pin mesh are constructed for cutting simulations. Dentists can move the stylus of the haptic device to cut the virtual tooth and can sense the cutting force generated by the simulation. Cutting-force feedback is computed by the damping model which is produced by our proposed method. In the first step of the experiment, we require the dentists to try to cut the virtual tooth as quickly as possible. The damping coefficient of force model is also increased to the maximum value at which the haptic simulation remains stable. The maximum output force can be seen in Table 3. The maximum output force of the Omega haptic device can reach 3.7 N, which is much larger than those of the Phantom and Bhhd haptic devices (Fig. 9). The larger achievable stiffness and position resolution of the Omega explains why it can generate the highest stable forces against dentists.

Table 3 Maximum output force when the method is applied Haptic device

OMEGA 3DOF

Phantom desktop

BHhd 3DOF [2

Operating speed (mm/s)

[2

[2

Kc (N/mm)

10

2.5

4

Sampling frequency of haptic loop (Hz) Maximum continuous output force (N)

1,000

1,000

1,000

3.7

0.8

1.1


105

Fig. 7 Output force of OMEGA in haptic simulation when this method is applied

Fig. 8 Measured force and computed (output) force of OMEGA at the same operating speed in measurement and simulation Fig. 9 Maximum output force of OMEGA in haptic simulation with damping model

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106

Figure 7 shows the maximum output force by haptic device when a dentist moves the tip of the Omega haptic device to cut the virtual tooth with an operating speed of 0.5 and 0.2 mm/s respectively. The computed force values are close to those of real physical dentist operations at the same operating speed. Figure 8 shows the force comparison between the measured force in real physical operation and force generated by each haptic device in haptic simulation when the same dentist finishes the same operation (cutting the enamel of tooth with the operating speed at 0.4 mm/s). The computed force values are close to that of physical operations with the same operating speed. The error between measured value and computed value is smaller than 0.1 N. Therefore, the Omega haptic device with our damping model and proposed method meets the requirements for a dental training system: realistic and stable force feedback. (Fig. 9) With the damping model and our proposed method, we are able to simulate tooth-cutting operations. Four dentists (two professors and two students) of Beijing stomatological hospital have used the prototype to simulate tooth cutting. They first point out that the value and direction of force feedback are close to those of physical operations. The relation between the force imposed by operator and the sensed force feedback could reflect the physical phenomenon.

5 Conclusion This paper introduces an explanation for instability when applying the damping model in haptic simulation. The damping coefficient is decided not only by the simulated operations but also by the sampling frequency, the position resolution of the haptic device, the operation of the human (dentist), and the maximum achievable output stiffness of the haptic device. If the relationship among them is violated, oscillation and instability are inevitable. A method to restrict the output stiffness of a haptic device is proposed to deal with the damping model to realize high output forces that are stable. The experimental results show that the three-degree-of-freedom Omega haptic device with a damping model can meet the requirements of dental training.

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Our future work will focus on realizing high-fidelity force feedback stably in the haptic simulations for other operations, such as grinding and drilling operations. Acknowledgments This research received support from the National Science Foundation of China under the grant no. 50575011.

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