Haptic Direct-Drive Robot Control Scheme in Virtual Reality

Journal of Intelligent and Robotic Systems 35: 247–264, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

247

Haptic Direct-Drive Robot Control Schemein Virtual Reality

MING-GUO HERDepartment of Mechanical Engineering, Tatung University, 40 Chung-Shang North Rd. 3rd. Sec.,Taipei, Taiwan 10451, Taiwan; e-mail: [email protected]

KUEI-SHU HSUDepartment of Automation Engineering, Kao Yuan Institute of Technology, 1821 Chung-Shan Rd.,Lu-Chu Hsiang, Kaohsiung, Taiwan; e-mail: [email protected]

TIAN-SYUNG LANDepartment of Mechanical Engineering, De Lin Institute of Technology, Tuchen, Taipei, Taiwan

M. KARKOUBMechanical Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

(Received: 29 January 2001; in final form: 3 December 2001)

Abstract. This paper explores the use of a 2-D (Direct-Drive Arm) manipulator for mechanismdesign applications based on virtual reality (VR). This article reviews the system include a userinterface, a simulator, and a robot control scheme. The user interface is a combination of a virtualclay environment and human arm dynamics via robot effector handler. The model of the VR systemis built based on a haptic interface device behavior that enables the operator to feel the actual forcefeedback from the virtual environment just as s/he would from the real environment. A primarystabilizing controller is used to develop a haptic interface device where realistic simulations of thedynamic interaction forces between a human operator and the simulated virtual object/mechanismare required. The stability and performance of the system are studied and analyzed based on theNyquist stability criterion. Experiments on cutting virtual clay are used to validate the theoreticaldevelopments. It was shown that the experimental and theoretical results are in good agreement andthat the designed controller is robust to constrained/unconstrained environment.

Key words: haptic device, virtual reality, direct-drive arm.

1. Introduction

Remote control with force freeback applications have been used in many fieldssuch as aviation, the medical field, and hazardous work places. Recently, the in-terest in virtual reality applications which is being extened from entertainmentgames to new airplanes, military training, etc., has led the development of sensitiveinterface devices that provide the tactile and force feedback [5]. However, puttinga human on a newly engineered airplane is very risky. Testing the plane remotelyputs at lest one human being out of harm’s way. In the medical field, precisionsurgery is often required to remove a tumor or perform biopsies and that requires

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248 M.-G. HER ET AL.

extremely skilled doctors in order not to endanger the life of the patient. Remotelycontrolled robots can be trained to perform this type of operation reducing thechance of human error. In hazardous areas, such as nuclear sites or war zones,remotely controlled machines can perform emergency procedures without puttingthe life of workers or soldiers at risk. The dire need for such machines and theever increasing speed and power of digital computers makes “virtual reality” (VR)techniques in tele-robotics with haptic techniques a very rewarding research area.

The interaction between a human operator and machines is achieved through anintermediary device known as “haptic interface device” [17, 21]. This device can beprogrammed to impose arbitrary trajectory-dependent forces on the operator arm.The haptic interface device has a force feedback characteristic allowing operatorsto feel the contact on the palm of the hand by grasping a handle or grip. Severalcontrol loops are associated with the handle and each loop can be independentlydesigned to perform a desired task. Moreover, the handle consists of an activehand controller and a passive force feedback. The elements of the force feedbackmechanism such as connecting rod, linkages, and encoders are mainly passivesince neither the controlled object nor the virtual environment provide any forcefeedback to the operator. However, the haptic interface device is power actuatedand provides a force feedback to the operator. Kazerooni and Her [11] used hapticinterface device for a manpower amplifier. In [10, 13, 22], the authors realized thetele-operation for the VR system with haptic characteristics that allows an operatorto probe and feel a remote virtual environment. It is seen that the performance canbe improved significantly by providing force feedback information of the remotesite of the virtual environment to the master. Hence, the operator can apply forceby the powered handle to maneuver the system of the remote site to achieve aspecific task. The use and utility of such devices have been on the rise due to theadvancements in various technological sectors.

Haptic interface devices come in different sizes and shapes; a mall pen-basedinterface arrangement has been used to deform free form surfaces [7] or simulatesurgical tools. Robot manipulators have been used as the force interface; Hamadaet al. [7] designed a haptic interface device that uses a specialized robot manip-ulator to apply force feedback between the user and a VR environment, whichis a computer generated immersive environment in which operators have real-time, multisensorial interactions to feel the actual environment or the conditionsof operation. Kanai and Takahashi [19] developed a model for a haptic interfacedevice which can give the operator a feel that s/he is maneuvering a mass, or push-ing onto a spring or a damper. Minsky and Ouh-Young [14] proposed a dynamicsimulator to create virtual textures via a powered joystick system with a forcefeedback device. Force feedback is becoming a routine task in the developmentof VR systems [18, 20].

This paper deals with the modeling and control of a virtual reality system witha haptic interface device. The system which includes the operator arm dynamics isused to perform a cutting procedure on a virtual clay that enables the operator to

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HAPTIC DIRECT-DRIVE ROBOT CONTROL 249

feel the actual force feedback from the virtual environment just as s/he would fromthe real environment [4, 8, 16]. In order to have interaction between the humanoperators and a virtual clay system, an active handle with a force feedback sensor isused. The active handle is power actuated by two direct-drive motors and provide aforce feedback to the operator. The advantage of the control scheme presented hereis the integration of the dynamics of the operator, handle actuator, and the simulatedsystems including the virtual environment in the control synthesis. The stabilityconditions of the closed-loop system are derived using the Nyquist stability crite-rion. The performance of the proposed control scheme is evaluated experimentally.The results of a series of experiments cutting a virtual clay system are presentedin this paper to show the effect of the control scheme on the system’s performanceand stability.

NOMENCLATURE

b: the coefficient of viscous damping of the virtual environment

C: the compensator transfer function

Cc: the stabilizing controller gain of the closed-loop G

fh: the force imposed on the handle actuator by the operator

fe: the force imposed on the handle actuator by the environment

G: the closed-loop transfer function of the handle actuator

G(q): the vector gravity

J (θ): Jacobian matrix

k: the stiffness of the spring

m: the handle mass

Sh: sensitivity function of the operator force to the handle position

Th: the dynamics of the operator arm

Te: the dynamics of the environment

ph: position of the handle

uh: the vector of the human muscle force initiating a maneuver

2. Modeling of VR System with Haptic Device

Figure 1 shows a diagram of a two-degree-of-freedom direct-drive robot systemwhere the computer simulates the dynamics of the environment with haptic inter-face device behavior. A general unstructured closed-loop model of the VR systemis shown in Figure 2. Since every physical system has its own unique behavior andproperties, a theoretical model should be built from the equations of motion in orderto simulate its dynamics. The VR system should include the human arm dynamics,handle actuator dynamics of the robot end-effector, and the environment dynamics,which makes it possible to explicitly derive the stability criterion of the closed-loop

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(a) (b)

Figure 1. A two-degree-of-freedom VR system with haptic device: (a) experimental setup,(b) schematic diagram of the system.

Figure 2. VR closed-loop control block diagram: arm and environment dynamics areincluded.

system. Moreover, the behavior of the VR system with the haptic interface deviceis felt by the operator. A generalized diagram for the control system, shown inFigure 2, is used to model the dynamics of the VR system (or VR object) with thehaptic interface device. The left side of Figure 2 represents the dynamics of thehuman arm while the right side represents the VR system simulating the physicalsystem having haptic interface device behavior. By examining the diagram shownin Figure 2, one can write

ph = Gδv + Sh(fh − fe), (1)

where

δv = C(fh − fe). (2)

Sh is the sensitivity function mapping the control force to the position of the handleof the robot end-effector, C is the compensator, and G is the closed-loop han-dle actuator system, i.e., the input-output relationship of the primary closed-loopcompliant controller.

The left side of Figure 2 represents a haptic interface device architecture dia-gram. The operator arm/handle interaction force, transmitted via a readily available

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hardware system, is assumed to be measurable using a force sensor. Our particularfocus is on the dynamic analysis of the handle of the robot end-effector. Sincethe dynamic behavior of the active end-effector is decoupled in its nominal con-figuration, separate control loops are needed for each motor. The block CC is theclosed-loop controller handling the position error. It is worth noting that if theclosed-loop transfer function with a stabilizing controller has a large loop-gain,then Sh is considered small compared to the loop gain, G. If the handle actuator isbackdrivable (i.e., system without transmission ratio), then, Sh is small regardlessof the choice of the stabilizing controller. Moreover, if the sensitivity function, Sh,is far away from the operating frequency range, the handle actuator may not drivethe handle to the desired position. This produces a large loop gain, G, in parallelwith Sh, which increases the effective sensitivity to the force imposed on the handleby the environment. It is desired that the operator feel the actual contact force whenmaneuvering the oil clay system using the haptic interface device. The force, fe, feltby the operator is due to the friction between the tool and the oil clay system.

In this paper, it is assumed that the actual system with the working environmentrepresented in the VR system can be approximated for modeling purposes withsprings and dampers. Therefore, the equation of motion of the cutting tool with theenvironment of the VR system can also be written as follows:

fe = mph + bph + kph. (3)

Taking the Laplace transform of (3) leads to:

Te = ms2 + bs + k, (4)

where Te denotes the transfer function from ph to fe (i.e., the environment dynam-ics). In general, the cutting tool actuator will drive the tool towards the positionindicated by the grasped handle. The enviroment dynamics Te is determined basedon the properties of the selected physical object. In our cutting clay system, Te

denotes the damping ratio bs only. However, as soon as the cutting tool comesinto contact with the VR environment, a force, fe, will be generated by the VRsystem. By examining the environment force, fe, one can conclude that if the hapticinterface device is at rest; i.e., ph = 0, then the force acting on the system istransmitted mainly by the nervous system of the operator. On the other hand, if thesystem is moving, and uh = 0, the force acting on the system is mainly a functionof the impedance of the operator, Th. Therefore, it appears that fe is resisting thereaction from the VR system.

3. Control Design and Stability Analysis

3.1. CONTROL DESIGN

Clay cutting is usually done manually; this is time consuming and irrecoverableerrors often take place. Therefore, it is proposed to simulate the process with a VR

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system having haptic behavior for safety and economic reasons. The telepresencewith haptic interface device behavior consists of a two-degree-of-freedom Direct-Drive Arm and a computer graphic interface which simulates the dynamics of theclay cutting system and the environment. The haptic interface device, shown inFigure 1, simulates the procedure for cutting clay. This procedure is shown in realtime on a computer monitor; the operator maneuvers the handle of the robot end-effector to control the VR system just as s/he is interacting with a real environment.In other words, by maneuvering the handle, the operator controls the motion of thevirtual cutter cutting into the virtual clay on the screen. Therefore, the operator isable to feel the necessary force to move the handle during the cutting process. Theblock diagram of the VR system device is shown in Figure 2. From Equations (1)and (2), the following equation is obtained

ph = (Sh + GC)(fh − fe), (5)

and

fe = Teph. (6)

Algebraic manipulations of (5) and (6) leads to:

fe(fh)−1 = (I + Te(Sh + GC)

)−1Te(Sh + GC). (7)

From (7), it is clear that the larger the term Te(Sh +GC) compared to I , the closerfe gets to fh. It is worth noting that the magnitude of the sensitivity function, Sh,is much smaller than that of the handle closed-loop transfer function, G. Also, theenvironment transfer function, Te, is related to the designed environment, and thecompensator, C, directly affects the stability of the system. In the absence of theenvironment dynamics, Te = 0, and uh = 0, the output, ph, of the system must bezero. This is equivalent to holding the handle in a stand still position. If the handlemoves, then C should be large enough to amplify fh such that the operator is ableto feel the reaction force of the environment continuously during the manipulationprocess. Therefore, the objective of the control design is to obtain an appropriatecompensator, C, which guarantees the stability of the closed-loop system.

3.2. STABILITY ANALYSIS

Referring to Equation (7), it can be concluded that since the dynamics of theoperator arm, Th, is not included in the closed-loop system, the performance ofthe system depends on the dynamics of the handle actuator (G, Sh), the environ-ment dynamics, Te, and the compensator transfer function, C only. Therefore, theperformance of the system will not be affected by the operator or operation maneu-ver. However, the stability of the VR system with haptic interface device behaviorshould be taken into account when the operator arm dynamics is included. As isusual the case, there is a trade off between the performance of the system and

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its stability. Therefore, a stability condition for the closed-loop system should bederived such that the closed-loop system is stable in the absence of the operatorarm dynamics, Th, and the environment dynamics, Te. Another sufficient stabilitycondition will be derived using the Nyquist theorem for the closed-loop systemwhen Th and Te are included in the closed-loop control system. From Figure 2, theoutput ph of the VR closed-loop system is given by:

ph = (I + GC(Te + Th) + Sh(Te + Th)

)−1(GC + Sh)uh, (8)

where the sensitivity, Sh, is comprised of Sx and Sy which represent the sensitivityof the operator arm in the x- and y-directions, respectively. Similarly, the transferfunction G consists of two parts: Gx and Gy which represent the handle actuatorin the x- and y-directions, respectively. The transfer function (Th + Te) can beconsidered as the closed-loop system resulting from the interaction between thehaptic interface device, the operator arm, and the environment.

Based on the Nyquist stability criterion, the net number of encirclements equalsthe number of zeros of I + GC(Th + Te) + Sh(Th + Te) in the RHP minus thenumber of open-loop poles of GC(Th + Te) + Sh(Th + Te) in the RHP. In orderto ensure the stability of the closed-loop system, the following conditions must besatisfied:1. I + GC(Th + Te) + Sh(Th + Te) is analytic in RHP;2. I + GC(Th + Te) + Sh(Th + Te) has a proper stable inverse transfer function.

Condition 2 implies that the encirclements of I + GC(Th + Te) + Sh(Th + Te) donot cross the jω-axis of the s-plane for all frequencies. That is,∣∣I + GC(Th + Te) + Sh(Th + Te)

∣∣ �= 0, ∀w ∈ [0,∞).

The norm of I + RGC + RS is the radius of the smallest circle that contains theNyquist plot of I +RGC+RS. Hence, I +RGC+RS has a proper stable inversetransfer function if and only if I + RGC + RS has no zeros in the RHP or

infω∈[0,∞)

∣∣I + RGC + RS∣∣ > 0. (9)

In other words, I + RGC + RS has a proper stable inverse transfer functionwhenever∣∣RGC + RS

∣∣ < 1, ∀w ∈ [0,∞). (10)

From Equation (10), one can write,∣∣GC(Th + Te) + Sh(Th + Te)∣∣ < 1, ∀w ∈ [0,∞). (11)

This leads to∣∣GC(Th + Te)∣∣ < ∣∣I + Sh(Th + Te)

∣∣, ∀w ∈ [0,∞), (12)

or

|GC| < ∣∣(I + Sh(Th + Te))(Th + Te)

−1∣∣, ∀w ∈ [0,∞). (13)

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Remark. From inequality (11), it can be seen that a small value for the sensitivityfunction, Sh, necessitates a small gain for the compensator C. At the limit, whenthe hand controller is infinitely stiff (i.e., Sh = 0), the magnitude of GC is boundedby (|Th|+|Te|)−1. Moreover, the more rigid the operator arm is, the smaller C mustbe. For the limit case, when the operator arm is stiff (i.e., Th = ∞), no C can befound to enable the interaction with the VR system. This means that the humanarm must possess some flexibility for the system to be stable.

4. Experiment

The experimental setup of the VR system is shown in Figure 3. The system consistsof three components: a control computer, an interface, and two actuating servocontrol systems (one for the x-axis handle actuator and the other for the y-axishandle actuator).

The characteristics of each component are given below.• The control computer provides programs with the following functions:

1. generation of the desired reference input,2. calculation of the feedback information and generation of the commanded

input for the x-axis and y-axis actuator servo systems.• The interface has three parts:

1. one PCL-816 A/D converter card supports two D/A output channels. Thereis only one output signal wire for each channel, i.e., the individual jointangles are driven by individual torques, τ , via a corresponding voltageoutput.

2. one PCL-816 A/D converter card including two low-pass filters whichconverts the analog signals from the force sensors.

3. one PCL-833 quadrature encoder card which encodes the position of theshaft of the motors for quadrature encoding, pulse/direction counting orup/down counting.

• The servo control systems for maneuvering the handle of the robot end-effector have the following characteristics:1. two direct drive motors with a two-link manipulator for driving the handles

in the x- and y-directions,

Figure 3. Layout of the experimental setup.

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2. one force sensor: one channel for the handle force input in the x-directionand another channel for the handle force input in the y-direction.

Motor 1 drives the endpoint in the x-direction while motor 2 drives the endpointin the y-direction. Based on the initial characteristics of the virtual object withwhich interaction forces are to be simulated, a desired trajectory can be calculated.The block J (q) is the Jacobian matrix which is a two-dimensional form of thederivatives relating joint velocities to task space velocities. The following equationis adopted to transform the local coordinates to the frame coordinates (x-axis andy-axis directions) of the planar robot end-effector trajectory:[

δx

δy

]=

[−l1s1 − l2s12 −l2s12

l1c1 + l2c12 l2c12

][δq1

δq2

], (14)

where s1 = sin(q1), c1 = cos(q1), s12 = sin(q1 + q2) and c12 = cos(q1 + q2). Inthe simulations, l1 = 0.24 m and l2 = 0.35 m, which represent the lengths of theDirect-Drive Arm configured as a two-link planar manipulator. The compensator Cshould be chosen to satisfy a specific range of Equation (11) provided that a virtualobject, Te, is selected. Conversely, the VR system, Te, is selected to satisfy a spe-cific range of Equation (11) if a constant gain, C, is selected. An input signal withvarying frequency is fed to the closed-loop system and its corresponding output isrecorded periodically every 0.36 milliseconds. Frequency record of the collecteddata is shown in Figure 4.

The transfer functions of the handle actuators for x- and y-directions are deter-mined as:

Gx = 910.19

s2 + 41.8s + 910.19(cm/cm) (15)

and

Gy = 765.5

s2 + 30.3s + 765.5(cm/cm), (16)

respectively. The motors of the robot are back drivable. Hence, the sensitivity trans-fer function, Sh, will be significant in comparison with G. Sh, the haptic interfacedevice sensitivity transfer function, maps the environment force, fh, onto the hapticinterface device position, δv. Frequency record of the collected data is shown inFigure 5. The transfer functions Sh of the handle actuators for x- and y-directionsare determined as:

Sx = 79.19

s2 + 41.1s + 432(cm/N) (17)

and

Sy = 13.1

s2 + 32.4s + 492.46(cm/N), (18)

respectively. The operator arm dynamics [15] do not affect the expected perfor-mance but do play an important role in the system stability. Several experiments

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Figure 4. Magnitude plot of the x-axis (top) and y-axis (bottom) closed-loop handle actuator:(∗) experimental data, (solid) theoretical approximation.

were conducted to obtain the operator arm dynamics, Th. To obtain Th, the fol-lowing is done: three operators’ hands take turn to grasp a handle mounted on apiezoelectric force sensor. The operators try to move his/her hand to follow thehandle so that zero contact force is maintained between the hand and the handle.An encoder, mounted on the motor, measures the orientation of the handle anda microcomputer records the input/output data every 0.36 ms and the results areshown in Figure 6. These data agree with previous results [9], in which passivewrist motion was shown to be second-order and dominated by the moment ofinertia. At low frequencies, however, the human can follow the large motions ofthe handle quite comfortable, but it is expected that some finite contact force ispresent. Therefore, the human arm dynamic Th approache a finite value at fre-quencies. Crossover frequencies in the experimental determination of the operatorarm dynamics, Th, were observed around 3-rad/s (see Figure 6). The crossover

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Figure 5. Magnitude plot of the x-axis (top) and y-axis (bottom) sensitivity functions: (∗)experimental data, (solid) theoretical approximation.

frequency is the maximum frequency at which the operator was able to accuratelycontrol the constrained movement. The transfer function of the operator arm isfound to be:

Th = 0.25s2 + 2s + 6 (N/m). (19)

4.1. CONSTANT CUTTING VOLUME IN VIRTUAL CLAY SYSTEM

Cutting with uniform plane of virtual clay is the process that removes the regularclay away from the virtual clay block. Figure 1 shows the operator cutting virtualclay using the handle on the robot end-effector via the DDArm manipulator drive.To perform the experiments, the following assumptions are made:1. A two-dimensional space is realizable and two dampers of equal value b1 = 10

N – s/cm in the x- and y-directions are used.

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Figure 6. Magnitude plot of the operator arm dynamics: (+) first operator, (◦) second operator,(∗) third operator, (—) theoretical approximation.

Figure 7. Handle path for constant cutting volume experiment.

2. The virtual clay is 10 cm in length located 5 cm in front of the cutting tool.3. The tool cuts through virtual clay with specific cutting volume and the damping

ratio of the tool in contact with the virtual clay is b2 = 90 N – s/cm (seeFigure 7).Using assumptions (1–3), the following can be deduced:

C = 1

s2 + 10s, (20)

Te = 90s. (21)

The sampling period is taken as 0.36 ms. The operator moves the virtual particlestarting from the (0,0) location by holding down the handle. The velocity–positionand (force/velocity)–position records in the x-direction are shown at the top andbottom of Figure 8, respectively. The (force/velocity)–position record shows thatin the two-dimensional space, there exist a damper whose value is 90 (N – s/m) inthe x-direction of the virtual clay (Te = 90 s) located between 5 cm and 15 cm.In the 0–5 cm and 18–20 cm regions, the measured approach zero and the gain ofthe controller C is maximum which is expected. In the 5–15 cm region, the envi-

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Figure 8. Velocity versus x-position (top). Force to velocity ratio versus x-position foruniform tangent plane cutting experiment (bottom).

ronment is proved to be of the assumed value of the damping ratio and the responseof the system is unaffected by the choice of the controller C. The velocity-positionrecord shows that in the 5–15 cm region, the tool velocity remains approximatelyconstant which is expected; when the velocity is constant, the force to velocity ratiois approximately equal to the value of the coefficient of viscous damping.

4.2. VARIABLE CUTTING VOLUME IN VIRTUAL CLAY SYSTEM

Variable cutting volume in virtual clay system is the process that removes theirregular clay away in the experimental process. For the operator to be affectedby a distinct reaction force from a differential cutting volume, the following isassumed:

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Figure 9. Handle path for varying volume clay cutting experiment.

Figure 10. Velocity versus position for varying tangent plane cutting experiment.

1. For the virtual clay contours shown in Figure 9, the value of the viscous damp-ing of the clay in the 0–5 cm region is 50 N – s/m in both the x- andy-directions.

2. The value of the viscous damping for the remainder of the clay is 100 N –s/m in the y-direction and (100 + 30t) N – s/m in the x-direction and t is thesampling period and it is equal to t = 0.36 ms.

3. The input force from the counterweight is equal to 1 N and makes a 45-degreeangle with the x- and y-axes.Based on these assumptions, the following is obtained:

C = 1

s2 + 10s(22)

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and the environment transfer function is defined as follows:

Tex = 50 s, 0 < x < 5 cm, (23)

Tey = 50 s, 0 < y < 5 cm, (24)

Tex = (100 + 0.15t) s, 5 cm < x, (25)

Tey = 100 s, 5 cm < y, (26)

where Tex and Tey are the x and y environment transfer functions, respectively.Figure 10 shows the cutting tool velocity as a function of the position in thex- and y-directions for varying volume cutting clay. The figure show that thevelocity remains constant in the 0–5 cm region, then the velocity drops abruptlyat 5 cm and starts decreasing slowly. The magnitude of the velocity decreases asthe operator cuts deeper into the workpiece.

This is expected since the value of the viscous damping increases with depth.

4.3. STABILITY TEST

The Nyquist theorem is used to develop a sufficient stability condition for theclosed-loop system. This sufficient condition results in a class of compensators C

that guarantees the stability of the closed-loop system. However, the stability con-dition derived in Section 3.2 does not give any indication on the system perfor-mance but only ensures its stability. In this section, it is assumed that a vary-ing boundary clay cut is performed; hence, the compensator C can be writtenas:

C = 1

s2 + 10s(27)

and the environment dynamics is given by:

Te = 100s. (28)

From inequality (11) and Equations (16), (19), (27), and (28), the value of thecompensator should not exceed 5C. Using compensator values of 3C and 25C, anarbitrary cut is performed on a virtual clay system and the results are shown inFigures 11. Figure 11 shows the time history of the force in the x-direction. This isthe force applied by the operator while maneuvering the device randomly and whenthe compensator is set at C1 = 3/(s2+10s) (top) and C2 = 25/(s2+10s) (bottom).It is clear that when compensator C2 is used, the stability condition is violatedwhich led to a large irregular operator force in the x-direction. In comparison,when the compensator C1 is used, the force follows an acceptable pattern and it ismuch smaller in magnitude than in the previous case.

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Figure 11. The human force, fh, versus the position, y, in the x-direction: (top) C =2/(s2 + 10s), (bottom) C = 25/(s2 + 10s).

5. Concluding Remarks

A closed-loop control scheme for a virtual manufacturing process is derived in thispaper. The dynamic system comprises the operator arm, a haptic interface device,and a virtual environment (workpiece). A procedure for a clay cutting system isdevised through the use of haptic interface devices. A sufficient stability conditionbased on the Nyquist criterion is derived to guarantee not only the stability ofthe closed-loop control system but also the performance of the VR system. It wasshown that a large sensitivity function Sh combined with a smaller gain C leadsto a more stable closed-loop system. If the gain C is large, this leads to a largeforce and a fast reaction; however, the stability of the system is compromised.Therefore, it can be concluded that the smaller the value of the gain C, the morestable the system is without being affected by the reaction of the environment.

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Moreover, when the structure of the system is rigid, a larger bandwidth for theposition controller G is achievable since the dynamic system is more stable.

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