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Int J Adv Manuf Technol (2002) 19:743–751Ownership and Copyright 2002 Springer-Verlag London Limited

Analysis and Design of a Haptic Control System: Virtual RealityApproach

M.-G. Her, K.-S. Hsu and W.-S. YuDepartment of Mechanical Engineering, Tatung University, 40 Chung-Sang North Road 3rd Sec, Taipei, Taiwan

In this paper, the analysis and design of telerobotics based onthe haptic virtual reality (VR) approach for simulating the claycutting system is proposed. The main components of theapproach include a user interface, networking, simulation, anda robot control scheme. The telerobotics for the clay cuttingsystem and the environment is simulated by a haptic virtualsystem that enables operators to feel the actual force feedbackfrom the virtual environment just as they would from the realenvironment. The haptic virtual system integrates the dynamicsof the cutting tool and the virtual environment whereas thehandle actuator consists of the dynamics of the handle andthe operator on the physical side. The control scheme employsa dynamical controller which is designed considering both theforce and position that the operator imposes on the handleand feedforward to the cutting tool, and the environmentalforce imposed on the cutting tool and the feedback to thehandle. The stability robustness of the closed-loop system isanalysed based on the Nyquist stability criterion. It is shownthat the proposed control scheme guarantees global stabilityof the system, with the output of the cutting tool approachingthat of the handle when the ratios of the position and theforce are selected correctly. Experiments in the virtual environ-ment on cutting a virtual clay system are used to validate thetheoretical developments.

Keywords: Clay cutting system; Haptic virtual system; x-ytype active handle

1. Introduction

Many industries (aviation, medicine, nuclear) employ remotedistant control systems for manufacturing in hazardous areassuch as nuclear sites, for training pilots with in-flight simu-lators, etc. [1] and latest experiences have shown that thesymbiosis of virtual reality (VR) techniques can work satisfac-

Correspondence and offprint requests to: M.-G. Her, Departmentof Mechanical Engineering, Tatung University, 40 Chung-SangNorth Road 3rd Sec, Taipei, 10451 Taiwan. E-mail: d8601002�mail.ttu.edu.tw

torily. VR with a haptic property offers the chance to buildsimulation models for the operator for controlling the remotesystems. The interaction between the operators and teleroboticsfor remote and distant control processes is achieved byemploying some specified telemechanism that can copy humanactions at the end effector to carry out a task and vice versa[2–4]. An example of such processes is telerobotic systems ina virtual environment with a haptic property, which is of greatuse in environments where direct operator contact is deemedto be lethal or risky and the operator cannot keep watch onthe distant end-effector. The telerobotics and the environmentare simulated by the haptic virtual system that enables theoperator to feel the actual force feedback from the virtualenvironment just as she/he would from the real environment.The reaction forces from the virtual environment are felt bythe operator through a feedback control scheme. This allowsoperators to be located in a safe area and to control the distantmanipulator with a virtual image and duplicate on-line themotion of the distant processes quickly and accurately, so thatthey can feel the actual environment or the conditions ofoperation. However, it is well known that operator adaptabilityto the environment and to the conditions of operation is vitalfor a smooth operation. One way to cope with the problem isto let the operator intervene in the system from a safe distanceby means of a controller, and to be part of the overall closed-loop control system. Takahashi and Ogata [5] studied theteleoperation and human–machine interfaces and simulated arobotic system using VR technology. In general, robot manipu-lators have been used by the manufacturing industries forperforming certain automation tasks, or used as a master/slavetype manipulator in teleoperation. They require an accuratemodel description for designing the stabilising controllers. Inparticular, hand controllers for creating various forces havebeen designed for manoeuvring the remote side of the slavemanipulator [6]. However, by this approach it is difficult toobtain the respective ratios of the position/position andforce/force between the master manipulator and the slave.Hirata and Sato [7] and Lee et al. [8] realised the teleoperationof a VR system with haptic characteristics that allows anoperator to probe and feel a remote virtual environment. It isseen that the performance can be improved significantly byproviding force feedback information from the remote site of

744 M.-G. Her et al.

the virtual environment to the master. The operator can applyforce by a powered handle to manoeuvre the system at theremote site to achieve a specific task. Satoshi and Hidetomo[9] developed a model for haptic behaviour which can giveoperators the feel that they are manoeuvring a mass, or pushingonto a spring or a damper. Minsky and Ouh-young [10]proposed a dynamic simulation to create virtual textures by apowered joystick system with force feedback devices. Forcedisplay is widely used in developing VR systems [11]. Liand Wang [12] modelled force/torque sensing for a workingenvironment using physical based components (e.g. mass andspring/damper) in the VR simulating system. However, an in-depth study of system stability analysis and the implementationof stabilising controllers in a VR system are still lacking.

This paper presents telepresence/telerobotics with a hapticproperty based on virtual reality for simulating a clay cuttingsystem. The telerobotics and the environment for the cuttingtool system are simulated by the haptic virtual system whichenables operators to feel the actual force feedback from thevirtual environment just as they would from the real environ-ment. The main components of the haptic virtual system includea user interface, networking, simulation, and a robot controlscheme. In order to provide interaction between a human beingand a haptic virtual system, an x-y type active handle with aforce feedback sensor is required. The active handle is poweractuated and provides the force feedback to the operator. Thehaptic virtual system is the integration of the dynamics of thecutting tool and the environment, and the simulating systems,including the dynamics of the cutting tool actuator and theenvironment of the closed-loop control system. The controlscheme employs a dynamical controller which is designedconsidering both the force and position that the operatorimposes on the handle and the feedforward to the cutting tool,and the force from the environment imposed on the cuttingtool and the feedback to the handle. The stability condition ofthe closed-loop system is derived based on the Nyquist stabilitycriterion. The performance of the proposed control scheme isevaluated experimentally. The results of a series of experimentsperformed on cutting a virtual clay system are presented inthis paper to show the effect of the control scheme on thesystem performance and stability. It was shown that the virtualexperimental and the theoretical results are in good agreement,and that the designed controller is robust in a constrained/unconstrained environment.

2. The Haptic VR Simulation System

A VR system shown in Fig. 1 was developed to simulate thetelepresence of the clay cutting system with haptic properties.A block diagram of the system is shown in Fig. 2. Since theclay cutting system is usually manual, it is time consuming totrain a new operator, and irrecoverable errors often take placeduring the training processes. Therefore, it is proposed tosimulate the process with the haptic VR system for safety andeconomic reasons.

The telepresence with a haptic property consists of a two-degree-of-freedom x-y type active handle and a computer inter-

Fig. 1. A haptic VR system with x-y type active handle: experi-mental setup.

Fig. 2. The diagram of the VR system with both physical and virtualsides.

face which simulates the dynamics of the clay cutting systemand the environment.

fh = uh − Thyh (1)

yh = Ghvh + Shfh (2)

where yh, uh, and fh and respectively, are the handle position,the force imposed on the force sensor supplied by the operatorarm, the force imposed on the handle actuator and the feedfor-ward to the virtual side of the system. Gh is the transferfunction of the active handle actuator, and Sh is the transferfunction from fh to yh. For the virtual side, we have

ft = ut − Ttyt (3)

yt = Gtyt + Stft (4)

where yt, ut, and ft, respectively, are the cutting tool position,the environment force, the force imposed on the cutting toolactuator and the feedback to the physical side of the system.Gt is the transfer function of the virtual cutting tool actuator,and St is the transfer function from ft to yt. The transferfunctions Sh and St can be regarded as the stiffness of thehandle and that of the virtual cutting tool, respectively. It is

Analysis and Design of a Haptic Control System 745

assumed that the actual system and the working environmentexpressed in the VR system can be approximated for modellingpurposes by using a spring and damper. Therefore, thedynamics of the cutting tool actuator with the environment ofthe VR system can be taken as:

Gt = 1/Mts2 + Bts + Kt (5)

where Mt, Bt, and Kt are, respectively, the virtual cutting toolmass, the coefficient of viscous damping, and the stiffness ofthe spring which is related to the modulus of elasticity of thematerial. The cutting tool actuator can drive the tool back andforth and to the desired position that the human operator wantsby controlling the handle.

Since there are cross-feedback positions and forces betweenthe physical and virtual sides, the operator can feel the reactionforce between the cutting tool and the virtual clay system. Thecontrol procedures are monitored with the user interface inreal-time when the operators use the x-y type active handle tocontrol the cutting tool in the haptic VR system just as if theyare interacting with the real environment. Hence, by properlyselecting the compensators Ch1, Ch2, Ct1, and Ct2, the operatorcan feel the amount of necessary feedback force and move thehandle appropriately toward the desired position during thecutting process. The compensators work as a compliant controlfor the handle and the virtual cutting tool. Smaller compensatorgains increase stiffness of the handle and cutting tool. Wecannot choose Ch1, Ch2, Ct1, and Ct2 with very much largermagnitudes because the stability of the closed-loop system maybe compromised. In what follows, we first derive the relation-ship of the output positions of the handle and the cutting toolin the system.

From Fig. 2, the control signals of the handle and the cuttingtool actuators can be obtained as

yh = Ch1fh − Ct1ft (6)

vt = −Ch2fh + Ct2ft (7)

respectively.Hence, the respective outputs of the positions for the handle

and the cutting tool can be given by

yh = Gh(Ch1fh − Ct1ft) + Shfh (8)

yt = Gt(− Chfh + Ct2ft) + Stft (9)

A matrix Ah1 exists such that the following are satisfied:

GtCh2 = Ah1(GhCh1 + Sh) (10)

GtCt2 + St = Ah1(− GhCt1) (11)

Hence, we have from Eqs (10) and (11)

yt = Ah1yh (12)

where the elements of the diagonal matrix Ah1 denote the ratiosof the positions between the handle and the cutting tool in thex- and y-directions, respectively, and this is taken as theidentity matrix so that the handle and the cutting tool movethe same distance in the x- and y-directions, respectively.

Thus, from Eq. (12), the positions of the handle and thecutting tool can influence each other with a one to one ratioby virtue of the haptic property.

Next, we will derive the net force that the operator imposeson the cutting tool via the position of the handle to achieve acompliant control property. First, it is assumed that there isno external force from the environment, i.e. ut = 0 and thisleads to

yh = Gh(− Ct1 (I + TtGtCt2 + StTt)−1 (TtGtCh2) (13)+ Ch1)fh + Shfh

Then, suppose that the virtual side does not exist, i.e. the forceis imposed only on the handle by the human operator. There-fore, the handle position can be obtained by

yh = GhCh1f 0h + Shf 0

h (14)

where f 0h is the force imposed on the handle actuator, as the

virtual side does not exist.However, it is seen that the net force, f e

h, that moves thecutting tool purely on the virtual side can be defined by

f eh = fh − f 0

h (15)

From Eqs (14) and (15) this leads to

yh = GhCh1(fh − f eh) + Sh(fh − f e

h) (16)

Then, manoeuvring the handle with the same trajectory, withand without considering virtual environment, we have fromEqs (13) and (16)

f eh = (GhCh1 + Sh)−1Gefh (17)

where

Ge = GhCt1(I + TtGtCt2 + ShTr)−1 (TtGtCh2) (18)

Furthermore, the relationship between ft and fh can be directlyobtained from Fig. 2 as follows:

ft = (I + TtGtCt2 + ShTt)−1 (GtTtCh2)ft (19)

From Eqs (17)–(19), we have

ft = (Ct1Gh)−1 (Ch1Gh + Sh)f eh = Ah2f e

h (20)

where the elements of the diagonal matrix Ah2 denote therespective ratios of the forces between the handle and thecutting tool in the x- and y-directions.

Note that the force f eh is identical to ft if the matrix Ah2 is

chosen as an identity matrix. This guarantees that the operatorcan feel the same amount of reaction force when the cuttingtool is cutting the oil clay system. Thus, the forces on thehandle and the cutting tool can influence each other at a ratioof one to one by the virtue of haptic property. If Ah2 and Ch1

are selected properly, Ct1 can be obtained from Eq. (20) as:

Ct1 = (GhCh1 + Sh) (GhAh2)−1 (21)

Similarly, using Eqs (10) and (11), Ch2 and Ct2 and aregiven by:

Ch2 = (− Gt)−1 Ah1(GhCh1 + Sh) (22)

Ct2 = G−1t (−Ah1A−1

h2(GhCh1 + Sh) − St) (23)

respectively.Hence, we should select the compensators Ch1, Ch2. Ct1 and

Ct2 properly such that the closed-loop VR system is guaranteedto be stable. In this paper, a sufficient stability condition will


be derived using the Nyquist theorem for the closed-loopsystem when taking the operator and the environmentaldynamics into consideration in the closed-loop control system.

3. Stability Analysis

Since the motion of the x-y type active handle and the cuttingtool are independent in the x- and y-directions, respectively,we can let Gh = diag{Ghx, Ghy}, Gt = diag{Gtx, Gty}, Sh =diag{Shx, Shy}, St = diag{Stx, Sty}, Th = diag{Thx, Thy}, Tt =diag{Ttx, Tty}, and

C1 = C2 = �Ch1 Ct1

Ch2 Ct2�

It is seen that the output y of the VR system is given by thefollowing compact form Fig. 3:

y = (I + RGC + RS)−1 (GC + S)u = Pu (24)

It is seen that if P is a proper rational matrix and �I + RGC+ RS� � 0, ∀ � � (0, �), then the closed-loop VR system isstable. Note that the stability condition does not give anyindication of the system performance but ensures only thestability of the system. Also the stability condition is only asufficient condition. The purpose of the paper is to select acompensator, C, such that the stability of the closed-loopsystem is guaranteed and the required performance is achieved.The stability of the closed-loop system is derived based onthe Nyquist stability criterion.

In order to ensure the stability of the closed-loop system,the following conditions must be satisfied:

1. I + RGC + RS is analytic in RHP.2. I + RGC + RS has a proper stable inverse transfer function.

Condition 2 implies the encirclements of I + RGC + RS donot cross the j�-axis of the s-plain for all frequencies, i.e.�I + RGC + RS� � 0, ∀� � [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 aproper stable inverse transfer function if and only if I + RGC+ RS has no zeros in the RHP or

inf��[0,�) �I + RGC + RS� � 0 (25)

From Eq. (25), we have

Fig. 3. A compact block diagram of the telerobotic system.

�I + RGC + RS� = ThxTtx�U + U22Ttx + ThxU11 + 1 (26)

where,

�U = U11U22 − U12U21

U11 = GhxCh1 + Shx

U12 = GhxCt1 (27)U21 = GtxCh2

U22 = GtxCt2 + Stx

From Eqs (10), (11), (21) and (27), we obtain

Ct1 = U11(Ah2Ghx)−1

Ch2 = Ah1G−1tx U11 (28)

Ct2 = G−1tx ((−Ah1A−1

h2)U11 − Stx)

Comparing Eqs (26), (27) and (28), we have

�I + RGC + RS� = ThxTtx [(GhxCh1

+ Shx) (GtxCt2 + Stx) − GhxGtxCt1Ch2] (29)+ (GhxCt2 + Stx)Ttx + Thx (GhxCh1 + Shx) + 1= GhxCh1 (Ah1A−1

h2Ttx + Thx) + Shx (Ah1A−1h2Ttx + Thx) + 1

From Eq. (25), we have

�GhxCh1� � |Shx + (Ah1A−1h2Ttx + Thx)−1| ∀� � [0, �) (30)

for achieving the stability of the closed-loop system in the x-direction. Therefore, if the stability criteria (30) is satisfied,then the closed-loop system of the VR clay cutting system inboth the x- and y-directions can be guaranteed to be stable.By proper selection of Ch1, the stability margin of the systemcan be obtained if Ah1, Ah2, Tt and Th on the righthand sideof the inequality (30) are known a priori. It should be notedthat the larger the magnitude of Ch1 in (30) is selected, thesmaller is the stability margin of the system, which will resultin a less stable system. Also, a small magnitude of Sh willrequire a small magnitude of Ch1 accordingly, i.e. the largeris the transmission of the mechanism, the smaller the magnitudeof Ch1 required. Moreover, when sh has zero magnitude, i.e. anon-back-drivable case, then the magnitude of GhCh1 is restric-ted by �(Ah1A−1

h2Tt + Th)−1�. When the actuator dynamics incorpor-ates the operator dynamics and it has rigid properties, then themagnitude of the actuator dynamics Gh should be selected tobe large and Ch1 small. When Gh, ∀� � [0, �) approachesto �, no Ch1 exists, and the mechanism of the virtual reality


clay cutting system does not operate. Thus, in order to achievethe control design objective, the operator should possesssome adaptability.

4. Experiment

The experimental set-up of the VR system for simulating theclay cutting system, the environment, and the handle is shownin Fig. 4. The VR system consists of four components: userinterface, networking, simulation, and robot control scheme.

The characteristics of each component are given below:

1. The user interface provides programs with the followingfunctions:Generation of the desired reference input.Calculation of the feedback information and generation ofthe command input for the x- and y-axes actuator servosystems.

2. The networking has three parts:An I/O card which includes a programmable interrupt con-troller 8259, a programmable peripheral interfacing chip8255 and a programmable interval timer 8254 to executethe motion-control commands, i.e. the joint angles and jointvelocities for motion in the x- and y-directionss.A PCL-816 A/D converter card including two low-passfilters which convert the analogue signal from the forcesensors to a digital signal.A PCL-833 quadrature encoder card which encodes theposition of the shaft of the motors for encoding,pulse/direction counting or up/down counting.

3. The robot control scheme for manoeuvring the handle hasthe following characteristics:Two a.c. servo motors with screw gears for driving thehandle in the x- and y-directions.A force sensor with two channels: one channel is for thehandle force input in the x-direction and the other one isfor the y-direction.

To begin with, we should derive the transfer functions ofthe handle actuators in the the x- and y-directions, whichincludes the operator dynamics. Since the motions of the handlein the x- and y-directions are independent, Gh can be selected

Fig. 4. Layout of the experimental setup.

as a 2 × 2 diagonal transfer function matrix in which itselements represent the x- and y-axes handle actuator dynamics.To obtain the dynamics of Ghx and Ghy, the operator shouldhold the handle with zero contact force and follow the motionof the handle until the operator can no longer follow thehandle motion when the input of the handle actuator is asinusoid signal of increasing frequency from zero. Then, wecan obtain the frequency response of the compounded handleactuator and the operator dynamics in the x- and y-directionsshown in Figs 5 and 6, respectively, and their transfer functionsfor the x- and y-directions are given by:

Fig. 5. The response of the x-direction of the active handle actuator.

Fig. 6. The response of the y-direction of the active handle actuator.


Ghx =18s + 1

0.005s3 + 0.4s2 + 18s + 1(mm mm−1) (31)

Ghy =8s + 1

0.002s3 + 0.17s2 + 8s + 1(mm mm−1) (32)

Owing to the small lead angle of the lead-screw mechanism,the x-y type active handle used here is a non-back-drivablemechanism, that is, the mechanism has a large transmissionrate and Sh and St are selected as zero. Therefore, the handlecannot be moved by the force exerted by the operator but canonly be controlled by way of the force sensor. Moreover, toobtain the transfer function of Th, the following is done: threeoperators take it in turn to grasp the handle. An encoder,mounted on the motor, measures the orientation of the handle.A microcomputer is used to record the input position/outputforce data every 0.44 ms and the results are shown in Fig. 7.

Further, the transfer functions Th and Tt along the x-directioncan be obtained as:

Thx = Ttx = 0.25s2 + 2s + 6 (N m−1) (33)

4.1 Constant Cutting Rate Experiments

Constant cutting rate for a virtual clay system shown in Fig.8 is the process in which the cutting depth is deemed fixedand the cutting width is kept constant so that the clay systemis cut in a regular way. Figure 1 shows an operator cuttingthe virtual clay system, where the handle actuator is equippedwith two lead-screw type a.c. servo motors for controlling the

Fig. 7. The responses of the operator dynamics: (+) the 1st operator, (o)the 2nd operator, (*) the 3rd operator, (---) theoretical approximation.

Fig. 8. The profile of the tool path for constant material removal rate.

x- and y-directions. Each axis of the actuator has a built-inPID position controller to regulate the cutting tool motion inthe prespecified direction. To perform the experiments, thefollowing assumptions are made:

1. The weight of the cutting tool is simulated as 1 kg wt.2. The virtual clay system is 10 cm in length.3. When the cutting tool cuts the virtual clay system, the

damping ratio of the virtual environment is set as Bt = 100N s m−1.

From assumptions 1–3, and letting the virtual cutting tool cutthe clay system along the x-direction, we have

Ch1 =1

s2 + 10s(34)

Gtx =1

s2 + 100s(35)

It is seen that the stability condition (26) is satisfied. Thesampling period is taken as 0.54 ms. The responses of theforce versus time, the velocity versus position and the forceversus position are shown in Figs 9, 10, and 11, respectively.Figure 9 shows that for time t � 0.4 s, the cutting force willsteadily increase until the cutting force reaches a thresholdvalue of 3.6 N, where the cutting tool starts making contactwith the clay system. Soon after that, i.e. for t � 0.4 s, thecutting force hovers around 3.6 N owing to the reaction andcorrection of the applied force by the operator. From Fig. 10,the velocity is almost constant between x = 2 cm and x = 12

Fig. 9. The response of the cutting force with constant material removalrate.


cm and it is approximately equal to 0.038 m s−1. This meansthat the experiments are in good agreement with the assumptionthat the virtual clay system behaves like a damper of value100 m N−1 s−1. Note also from Fig. 11 that the damper willbe kept constant during the cutting process since the velocityis kept constant.

4.2 Variable Cutting Rate Experiments

In this subsection, we use the same experimental set-up as inSection 4.1, except that the shape of the workpiece shown inFig. 12 is irregular.

In this experiment, the following assumptions are made:

1. The weight of the cutting tool is 1 kg wt.2. The virtual clay system is 15 cm in length.3. The cutting force is kept 3 N.4. The virtual clay volume is simulated by decrementing every

2 cm in length.5. The damping ratio of the virtual clay environment is p N

s−1 m−1 for the outermost clay system.

From the above assumptions and letting the virtual cutting toolcut the clay system along the x-direction, we have

Fig. 10. The response of the cutting velocity versus position withconstant material removal rate.

Fig. 11. The response of the cutting force versus position with constantmaterial removal rate.

Ch1 =1

s2 + 10s(36)

Gtx =1

s2 + ps(37)

where p is taken as 100, 80, 60, 40 and 20 in turn when thecutting tool cuts the clay system every 2 cm. The samplingperiod is taken as 0.54 ms. The cutting profile shown in Fig.12 has a staircase shape; each step has a length of 3 cm.Figure 13 shows the response of the cutting velocity withrespect to the position when a constant cutting force 3 N isapplied. Compared to the given profile, the velocity can bekept constant along each step, except for the changing betweeneach step. That is because the cutting velocity is proportionalto the amount of the clay cut.

4.3 Performance Analysis

Since we use the Nyquist theorem to develop a sufficientstability condition for the closed-loop system, this sufficientcondition results in a class of compensators C that guaranteesthe stability of the closed-loop system. However, it should benoted that the stability condition derived in Section 3 does notgive any indication on the system performance. To investigateexperimentally the stability of the system, we consider the caseof the constant cutting rate experiment when the compensatorC is given by:

Ch1 =k

s2 + 10s(38)

where k � 0 and

Gtx =1

s2 + 100s(39)

From inequality (26) and Eqs (33), (34), (38), and (39), themagnitude of the compensator should not exceed k = 5. Usingthe values of k = 3, k = 13, and k = 19, an arbitrary cut isperformed on the virtual clay system. Data from the experi-mental set-up are collected every 0.54 ms and the results areshown in Figs 14, 15, and 16. Figure 14 shows that the cuttingforce does not exceed 6 N when k = 3. When the compensatorgain is selected as k = 13, the cutting force remained less than7 N up to time 1.5 s. For time t � 1.5 s, the cutting forcebecame large and reached 20 N at time t = 1.8 s approximately(see Fig. 15). When the magnitude of the compensator wasset as k = 19, the cutting force became larger than for k = 3and k = 13, and reached 44 N at time t = 1.3 s (see Fig. 16).

5. Concluding Remarks

In this paper, we have presented a VR system with hapticproperties to simulate a clay cutting system. The main compo-nents of the system include a user interface, networking,simulation, and a robot control scheme. The user interface isimplemented as a combination of a virtual environment and astandard graphical user interface. The telerobotics for the cut-ting tool system and the environment is simulated by the haptic


Fig. 12. The profile of the tool path for variable material removal rate.

Fig. 13. The response of the cutting velocity versus position for vari-able material removal rate for clay system with a constant cuttingforce of 3 N.

Fig. 14. The response of the cutting force for an arbitrary clay cutwhere the gain of the compensator is k = 3.

virtual system that enables the operators to feel the actualforce feedback from the virtual environment just as they wouldfrom the real environment. The control scheme incorporatesthe dynamics of the human arm, the actuators, and the virtualenvironment in the closed-loop control system for stabilityanalysis. A sufficient stability condition based on the Nyquistcriterion is derived to guarantee not only the stability, but alsothe performance of the closed-loop control system. Experimentson the VR system on the clay cutting system with constantand variable cutting rates and performance analysis are used



to validate the theoretical developments. It was shown that theexperimental and theoretical results are in good agreement andthat the designed controller is robust in constrained/unconstrained environments.

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