13
http://pic.sagepub.com/ Engineering Science Engineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical http://pic.sagepub.com/content/228/4/599 The online version of this article can be found at: DOI: 10.1177/0954406213489067 599 originally published online 9 May 2013 2014 228: Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science Ozgur Baser and E Ilhan Konukseven Optimal posture control algorithm to improve the stability of redundant haptic devices Published by: http://www.sagepublications.com On behalf of: Institution of Mechanical Engineers can be found at: Science Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Additional services and information for http://pic.sagepub.com/cgi/alerts Email Alerts: http://pic.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: http://pic.sagepub.com/content/228/4/599.refs.html Citations: What is This? - May 9, 2013 OnlineFirst Version of Record - Mar 10, 2014 Version of Record >> at UNIV OF SOUTHERN CALIFORNIA on April 2, 2014 pic.sagepub.com Downloaded from at UNIV OF SOUTHERN CALIFORNIA on April 2, 2014 pic.sagepub.com Downloaded from

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http://pic.sagepub.com/Engineering Science

Engineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical

http://pic.sagepub.com/content/228/4/599The online version of this article can be found at:

 DOI: 10.1177/0954406213489067

599 originally published online 9 May 2013 2014 228:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

Ozgur Baser and E Ilhan KonuksevenOptimal posture control algorithm to improve the stability of redundant haptic devices

  

Published by:

http://www.sagepublications.com

On behalf of: 

  Institution of Mechanical Engineers

can be found at:ScienceProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical EngineeringAdditional services and information for

   

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Page 2: Optimal posture control algorithm to improve the stability of redundant haptic devices

Original Article

Optimal posture control algorithmto improve the stability of redundanthaptic devices

Ozgur Baser and E Ilhan Konukseven

Abstract

Stability is indispensable to haptic interfaces for the simulation of a large variety of virtual environments. On a multi-

degree of freedom (multi-DOF) haptic device, the passivity condition must be satisfied in both end-effector and joint

space to achieve stable interaction. In this study, a conservative passivity condition is utilized for the stability such that

guaranteeing the passivity at all joints is a sufficient condition for the passivity and then stability of the whole haptic

system. An optimal posture control algorithm is developed to satisfy this passivity condition and maximize the stability

performance of a redundant haptic device. The algorithm optimally adjusts the device postures, which are estimated by a

Golden Section Search algorithm. The proposed control algorithm was experimentally implemented on a virtual sphere

by using a 7-DOF redundant haptic device. Z-width stability metric was used to evaluate the performance of the

proposed algorithm. The results show that the optimal posture control approach significantly improves the stability

of the redundant haptic devices.

Keywords

Redundant haptic device, optimal posture control, stability, passivity condition, Z-width, transparency

Date received: 13 July 2012; accepted: 15 April 2013

Introduction

Stability is the major requirement in haptic interfacesfor a realistic simulation. Haptic interaction has a bi-directional energy flow between a device and a userthat cooperates with the device. The energy flowingtoward the user can lead to undesired vibrations.Smooth interaction without any vibration for any vir-tual environment is only possible provided that thepassivity of haptic interface is guaranteed. For a pas-sive haptic system, the energy flowing into the systemshould not be negative, which refers to the passivitycondition1:

Zt

0

FhðtÞVhðtÞdt5� E0 ð1Þ

for t4 0 and all admissible FhðtÞHaptic interfaces become active due to some trig-

gering factors such as the loss of information in thedata acquisition unit, computation delays, non-zeronoise/non-zero phase-lag in velocity–acceleration esti-mators and sensor signal filters, communicationdelays when a haptic device is connected to theremote environment by a network, and disturbancein compensation algorithms.

Stability of a haptic device can be evaluated by‘‘Z-width’’ plot which is defined as passive impedancerange that can be transmitted by a device.2 It can beillustrated by the displayable upper limits of stiffnessparameters (K) versus damping parameters (B) for avirtual wall collision. The limits shown on the K–Bplane indicates the stability region. The area underthe limit comprises the stable virtual environment par-ameters for a virtual wall collision (Figure 1).

The physical damping in a haptic system dissipatesthe excess energy resulting from the sensor signal fil-ters, communication delays, and disturbance in com-pensation algorithms, which change achievableimpedance range of a haptic system. Equation (2)explains the passivity criteria proposed by Colgateand Brown2 in terms of physical damping of hapticdevice (b), virtual model parameters (stiffness —K anddamping—B), and sampling time (Ts). This equationindicates that increasing the physical damping ‘‘b’’ in

Mechanical Engineering Department, Middle East Technical University,

Ankara, Turkey

Corresponding author:

E Ilhan Konukseven, Mechanical Engineering Department, Middle East

Technical University, Ankara 06531, Turkey.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

2014, Vol. 228(4) 599–610

! IMechE 2013

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the system improves the achievable impedance rangeof the haptic interface.

b5KTs

2þ B ð2Þ

Colgate et al.3,4 proposed passive and active elec-trical damping to satisfy the passivity condition. Theoccurrence of limit cycle oscillations can be reducedby electrical damping at high virtual impedances. Thedesign of frequency dependent electrical damping issimple and does not adversely affect a device’s lowimpedance range. Passive electrical damping generatesfrequency dependent damping by employing an elec-trical resistance and capacitance in parallel with themotor, which is limited by the resistance of the motorwinding. Another active electrical damping approachis implemented by Colgate et al.3,4 in which an analogcircuit is used in the amplifier to estimate the velocity.This analog velocity signal is fed back to the currentamplifier to provide active electrical damping for thehaptic display. Active electrical damping has certainadvantages since it can be controlled dynamically asan analog signal in the motor amplifier, which circum-vents the limitation of the rotor resistance and theneed to add mechanical complexity. An analoginput shaper has been proposed by Lim et al.5 toincrease the impedance range of haptic interfaces. Itis a low-pass filter and operates as a frequencydependent excess energy dissipative element. Analoginput shaper is independent from motor parametersand does not affect transparency in free motion.Adjusting output-limiter technique is proposed byLee and Lee6 for stable haptic interaction. The pro-posed adjusting output-limiter ensures stable inter-action with deformable objects of unknown and/orvarying impedance. The controller adjusts the max-imum allowable force to ensure the passivity of thehaptic system at every sampling instant. Adjustingoutput-limiter controls only the final force output,and it is independent on the properties of theemployed virtual model. This makes the proposedcontrol method applicable to haptic systems involving

deformable objects with unknown, nonlinear, and/ortime-varying impedance. However, this method suf-fers from oscillations of very stiff walls due to theaccumulation of past remaining dissipation in thememory. Ryu et al.7–9 proposed a time-domain pas-sivity algorithm composed of a passivity observer anda passivity controller. Fast sampling rate is requiredto guarantee the efficiency of this algorithm. Fastsampling rate for passivity observer/passivity control-ler means the sampling rate of the system is substan-tially faster than the dynamics of the haptic device,human operator, and virtual environment so that thechange in force and velocity with each sample is small.The sampling rate of 1 kHz, which many haptic inter-face systems have, is quite satisfactory since it is muchmore than the requirement for the dynamics of thehaptic device, human operator, and virtual environ-ment. Energy bounding algorithm proposed by Kimand Ryu10,11 restricts the energy generation of thesample and hold operator for stable haptic inter-action. Another method proposed by Ryu et al.12 isa force bounding approach that makes the interactionrobustly stable for any linear, nonlinear, delayed vir-tual environments. This method limits the desiredforce and sacrifices the transparency.

Principally, stability improvement studies in the lit-erature can be classified into two major techniques:increasing the level of damping and bounding thecommand force. However, these techniques sufferfrom the deterioration of transparency or they limitthe performance of the device. Furthermore, the sta-bility issue in multi-DOF haptic manipulators is morechallenging than the single-DOF devices. Milleret al.13 showed that the passivity of a multi-DOFhaptic system must be satisfied in both joint andend-effector space considering the system as a multi-input multi-output system. According to their studies,if a certain level of damping (�) in end-effector space isneeded to be displayed to interact with virtual envir-onment stably, then the damping (�m) in the jointspace must satisfy:

�m5JT�J ð3Þ

Equation (3) provides a conservative condition tomake the whole haptic system stable. Variable �m is amatrix of dissipation parameters for multi-DOFsystem and it becomes a scalar damping parameterb for a single-DOF system. In redundant hapticmanipulators, it is possible to increase the stabilityregion without changing the level of damping andbounding the command force. In this study, the pos-ture adjustment ability of a redundant haptic manipu-lator is used to satisfy the passivity condition in allactuators of the device. The posture of the redundantdevice is optimally controlled to minimize the torquerequirement on the critical actuator in terms of stabil-ity. Optimal postures are estimated applying a GoldenSection Search algorithm14 based on the passivity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

B (Nms/rad)

K (

Nm

/rad

)

Figure 1. Typical ‘‘Z-width’’ plot for a 1-DOF rotational

haptic interface.

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condition derived in section ‘‘Optimal posture controlalgorithm’’.

Haptic system where many users interact with acommon virtual environment is also a research sub-ject. Bianchini et al.15,16 proposed a passivity-basedstability criterion that takes into account the inter-action constraints where ‘‘virtual coupling-controller’’design is concerned. Unlike the virtual coupling,which constantly moderates the virtual environmentreferences, optimal posture control algorithmimproves stability by adjusting the posture of thedevice instead of moderating the virtual environmentforce requirements.

The rest of the paper is organized as follows: thenext section explains 7-DOF haptic device used in theexperiments, ‘‘Optimal posture control algorithm’’section presents the optimal posture control algorithmand derivation of passivity condition. Implementationof the proposed algorithm is explained in

‘‘Implementation and experiments’’ section. The lastsection is related to the discussion and conclusion partof the paper.

7-DoF haptic device

7-DOF haptic device used in this study is similar to6-DOF Phantom haptic device17,18 in terms of mech-anical structure, except that it has only one extraDOF that corresponds to the third joint of shoulderas shown in Figure 2. The device becomes similar to6-DOF Phantom device when this joint is locked athome position. The redundancy enables the device toreach any position in the workspace with various pos-tures. In order to improve the performance of aredundant device, optimal posture can be selectedusing appropriate optimization objectives such asmaximization of the stiffness, transparency, and

Figure 2. (a) 7-DOF haptic device, (b) kinematic model at home posture, and (c) various sample postures.

Figure 3. Overall configuration of a 1-DOF rotational haptic interface.

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stability, or minimization of the inertia, power con-sumption, singularity, and obstacle avoidance.19

The actuating units of the designed device areMaxon brushless DC motors for positioning stageand Faulhaber DC mini-motors for orientationstage. They are driven by Maxon 4QEC DES 50/5and LSC 4Q DC 30/2 servoamplifiers, respectively.The torque supplied by the motors is increasedusing capstan drive mechanisms in positioning stageand zero-backlash gearheads in orientation stage.There are two types of optical encoders attached tothe brushless DC motors 1000 pulse per revolution(ppr) and DC mini-motors with 512 ppr. A 6-DOFATI-NANO-17 force/torque transducer is employedat the handle of the device. The control algorithm isimplemented on the device by means of Real-TimeWindows Target Toolbox of MATLAB�.Humansoft MF-624 multifunction I/O card is usedfor the acquisition of the signals. Servo-loop fre-quency of the experimental set-up is 1 kHz.

Optimal posture control algorithm

In this section, the passivity condition and optimalposture control algorithm are presented. Haptic inter-faces produce the desired forces/torques to the usersdepending upon the virtual model. A haptic interfaceconsists of a haptic device, sample/hold unit, control-ler, and virtual environment. Particularly, hapticdevice includes mechanical components, actuators,and encoders. Figure 3 shows the simplified 1-DOFrotational haptic interface configuration.

The major factor affecting the stability is the loss ofinformation in sampling and zero-order hold (ZOH),which makes the haptic system active and causesvibration in haptic interaction. Note that ZOH is amathematical model that converts a sampled outputinto an output which is held constant between samplesat the last sampled value.

The basic passivity condition is briefly mentionedin the ‘‘Introduction’’ section, and it is represented byequation (1) which means that the energy-flow in ahaptic system should not exceed the initial energy inthe haptic interfaces. However, this equation needs tobe modified to develop a new passivity-based forcecontrol algorithm. This equation can be rewritten byconsidering dynamic equilibrium of the 1-DOF rota-tional haptic device configuration during 04 t4 nTthrough equations (4) to (6).

Dynamic equilibrium of a 1-DOF rotational hapticdevice can be expressed as follows:

ThðtÞ ¼ I €�hðtÞ þ b _�hðtÞ þ TaðtÞ ð4Þ

By substituting equation (4) into equation (1), thepassivity condition can be rewritten as:

ZnTs

0

I €�hðtÞ þ b _�hðtÞ þ TaðtÞ� �

_�hðtÞdt5� E0 ð5Þ

Since the velocities/accelerations of the deviceand human hand are the same in a haptic

0 5 10 15 20 25–0.1

–0.05

0

0.05

Time (s)

Rot

atio

n (r

ad.)

Inside the virtual wall

Stable interaction

K= 1 Nm/rad, B= 0 Nms/rad

0 5 10 15 20 25–0.1

–0.05

0

0.05

Time (s)

Rot

atio

n (r

ad.)

Inside the virtual wall

Stable interaction

K= 2 Nm/rad, B= 0 Nms/rad

0 5 10 15 20 25–0.1

–0.05

0

0.05

Time (s)

Rot

atio

n (r

ad.)

Inside the virtual wall

K= 7 Nm/rad, B= 0 Nms/rad

Unstable interaction

Virtual Wall

K

B

Elbow Joint(a)

(b)

Figure 4. Physical damping identification experiment for the elbow joint (a: the virtual wall location in elbow joint and b: collision

experiments for different virtual wall parameters K–B).

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interaction _�d ðtÞ ¼ _�hðtÞ, the passivity condition canbe expressed as:

ZnTs

0

I €�d ðtÞ _�d ðtÞdtþ

ZnTs

0

b _�2dðtÞdtþ

ZnTs

0

TaðtÞ _�d ðtÞdt5� E0

ð6Þ

The first term in equation (6) indicates energy stor-age by the inertia and it has finite nonnegative valuefor finite velocity motion. Therefore, it can be rewrit-ten as given in equation (7). Note that since the totalinitial energy E0 at t¼ 0 includes the initial kineticenergy by the inertia, the term of ðI _�2dð0Þ=2Þ was elimi-nated in equation (7).

ZnTs

0

I €�d ðtÞ _�d ðtÞdt ¼ I

ZnTs

0

_�d ðtÞd _�ddt

dt ¼ I

ZnTs

0

_�d ðtÞd _�d

¼1

2I _�2dðnÞ �

1

2I _�2dð0Þ ¼

1

2I _�2dðnÞ ð7Þ

The second term of equation (6) indicates energydissipation by the viscous damper and it can berewritten by Cauchy-Schwarz inequality20 as

ZnTs

0

b _�2dðtÞdt¼Xn�1k¼0

b

Zðkþ1ÞTs

kTs

_�2dðtÞdt

5Xn�1k¼0

b

Ts

Zðkþ1ÞTs

kTs

_�d ðtÞdt

24

35

2

¼b

Ts

Xn�1k¼0

��2dðkþ1Þ

ð8Þ

where D�d(k þ 1) ¼ [�d(k þ 1) ��d(k)]. Note that Bimplies energy dissipation capability between samples.

The last term of equation (6) indicates energy flowinto the subsystem which is composed of sample andhold, controller, and virtual environment. When ZOHis used for hold operator, it can be rewritten as givenin equation (9). It is assumed that the actuator torqueis equal to the desired actuator torque at the sample/hold unit for equation (9) (Td ¼ Ta).

ZnTs

0

TaðtÞ _�d ðtÞdt ¼Xn�1k¼0

Zðkþ1ÞTs

kTs

TaðtÞ _�d ðtÞdt

24

35

¼Xn�1k¼0

Td ðkÞ

Zðkþ1ÞTs

kTs

_�d ðtÞdt

24

35

¼Xn�1k¼0

Td ðkÞ��d ðkþ 1Þ ð9Þ

Note that the second term of the passivity condi-tion in equation (6) is much more dominant than thefirst term as the time passes.2,10–12 Therefore, the pas-sivity condition of the haptic system in joint space fora specific time period (04 t< nTs) can be expressed asfollows:

EðnÞ �b

Ts

Xn�1k¼0

��2dðkþ 1Þ

þXn�1k¼0

Td ðkÞ��d ðkþ 1Þ5� E0 ð10Þ

0 5 10 15–30

–28

–26

–24

–22

–20

Time (s)

Firs

t Joi

nt A

ngle

(de

g.)

Stable behaviour in the first joint

0 5 10 15–35

–30

–25

Time (s)

Sec

ond

Join

t Ang

le (

deg.

)

Stable behaviour in the second joint

0 5 10 1515

20

25

Time (s)

Elb

ow J

oint

Ang

le (

deg.

) Unstable behaviour in the critical elbow joint

0 5 10 150

20

40

60

80

100

Time (s)

Han

dle

Pos

ition

(m

m)

Inside the virtual sphere

6–DOF Phantom like posture

Unstable interaction at the handle

(a) (b)

(c) (d)

Figure 5. Unstable behavior of the elbow joint in a virtual sphere collision (K¼ 0.4 N/mm—B¼ 0 Ns/mm).

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The first term in equation (10) is always positiveand the second term is negative in displaying a virtualwall since the actuator torque and velocity are in thereverse direction. It means that the instability happensbefore the maximum wall penetration. Consequently,either physical damping should be maximized ordesired motor torque should be minimized for max-imum stability. The second approach is much morepreferable, since the increasing physical dampingdeteriorates the transparency. Hence, optimal posturecontrol algorithm aims to minimize the desired torqueon the critical joint in terms of stability. It is assumedthat if the passivity is ensured for the critical joint, thewhole system remains stable. However, it should bedecided that which joint is critical in 7-DOF device.According to the passivity condition given in equation(2), the joint which has minimum physical dampingcannot transmit the higher torques than the othersstably; therefore, it can be considered as a critical

joint. Experiments were conducted for each joint indi-vidually to determine their physical damping. In theseexperiments, each joint was considered as separate 1-DOF haptic device and they were tested on the rota-tional virtual environment (i.e. torsional spring—Kand rotational damping elements—B). Using onlystiffness parameter (K) in the virtual wall model,the passivity limit given in equation (2) can be simpli-fied as:

K42 b=Ts for Ts ¼ 1 ms ð11Þ

The physical damping (b) of all joints of the devicewere determined by an experiment using the virtualtorsional stiffness limit. In the experiments, theamount of virtual stiffness was increased incremen-tally until unstable interaction starts and the physicaldamping of the joint is calculated by using this tor-sional stiffness limit and simplified equation of

Figure 6. The proposed optimal posture control algorithm.

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passivity limit. The virtual stiffness limits of the joints(1, 2, and 3) and their corresponding physical damp-ing values are determined as (41, 29, and 7 Nm/rad)and (0.0205, 0.0145, and 0.0035 Nms), respectively.As an illustration, the physical damping identificationexperiment of the elbow joint is depicted in Figure 4.In the experiment, a virtual wall was located at thehome position of the elbow joint (Figure 4a). Thecollision experiment was repeated for each virtualwall stiffness parameter which was increased incre-mentally (Figure 4b). Note that the virtual wall inthe elbow joint is shown as a flat virtual wall for asimple illustration. In fact, this virtual wall corres-ponds to the virtual rotational stiffness and dampingelements.

Analyzing the results, it is concluded that the crit-ical joint of the device in the virtual wall collisions is

the elbow joint shown as Z2 in Figure 2(b). Similarresults were observed in a stiff virtual surface collisionexperiment implemented with 6-DOF Phantom likeposture of the device. In this experiment, the handleof the device was collided with a 50-mm radius stiffsphere in the virtual environment. During collision,joint encoder readings and the distance of thehandle tip point with respect to the sphere centerwere measured as shown in Figure 5(a) to (d). Theresults show that only elbow joint cause instabilityin the device and its unstable behavior (Figure 5c) issubstantially similar to the unstable behavior of thedevice handle (Figure 5d).

In the proposed algorithm, the redundant joint isemployed to adjust the device posture for the criticaljoint torque minimization. The remaining actuatingjoints are utilized for 6-DOF haptic feedback. Inorder to attain the desired minimum torque on thecritical joint, optimal redundant joint angles are esti-mated using Golden Section Search14 as an optimiza-tion method and critical elbow joint torque as a costfunction. The cost function is the critical elbow jointtorque transformed from the endpoint desired force.The problem can be defined as given below

Find �R to minimize

Tcð�RÞ ¼ JTð�RÞFd

� �41

ð12Þ

such that �L<�R<�UFigure 6 shows the detailed block diagram of the

proposed algorithm. It consists of two parts: redun-dant joint position controller and force feedback con-troller. Redundant joint adjusts the posture of thedevice for maximum stability while the force feedbackcontroller produces the required torques of theremaining joints. Closed-loop impedance type controlalgorithm is used for force control of the positioningstage.21 Since the 7-DOF haptic device cannot provide

Figure 7. The designed virtual environment for the

implementation of the proposed algorithm.

Figure 8. Redundant joint angle map on the virtual sphere for the optimal posture (a: initial; b: smoothed).

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stiff positioning, it is not suitable for admittance typecontrol algorithm. The estimated optimal redundantjoint angles are utilized in the position controller asthe input. These optimum redundant joint anglesdepend on the handle position and direction of the

desired force on the contact points of the virtualobject surface. In this paper, it was preferred to cal-culate optimal redundant joints offline before imple-mentation since the real-time calculation producesexcessive computational load. Therefore, an optimalredundant joint angle map was generated and a look-up table was formed to be used in position controllerfor displaying a virtual sphere surface in real time.The look-up table determines the desired redundantjoint angle according to the actual location of thehandle. It should be noted that different look-uptables are required for different virtual models. Anovel posture decision algorithm which can be per-formed online for different virtual models is thefuture aspect of this research. Figure 6 shows theblock diagram of the proposed algorithm and itcontains the virtual model, device dynamics, userdynamics, actuator models, force feedback controller,and redundant joint position controller. Proportionalgain can be preferred in the force controller since thederivative and integral terms affect the force controladversely.21,22 McJunkin22 explained the drawbacksof Proportional Derivative (PD) and ProportionalIntegral (PI) control laws with force/torque sensorsin haptic interfaces. In a PD control applicationusing a force/torque transducer, the derivative termmay create difficulty due to the noise of the sensorsignal. The numerical differentiation of the sensorsignal amplifies noise and thus the derivative term ofthe controller causes instability. On the other hand, PIcontrol has also practical difficulties in implementa-tion. The PI control law can be used to correct thesteady state error in a system. However, a user whoexplores a haptic environment generally may notmaintain static contact with an object consistently.All symbols used in Figure 6 are given in the nomen-clature. Note that Jacobian symbol (J) is used in thecontroller side of the block diagram to calculatethe desired actuator torque and in the plant side of

Figure 9. Schematic view for the reason of ridge on the redundant joint angle map.

Figure 10. Spherical offsets for the proposed optimal

posture control algorithm.

Figure 11. Stability test regions on the virtual sphere.

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the block diagram to show the Jacobian of the physicaldevice itself. The derivation of the Jacobian matrix forthe 7-DOF haptic device is given in Appendix 1.

Implementation and experiments

For the implementation of the proposed algorithm, asample virtual environment was designed using 3DAnimation Toolbox of the MATLAB Simulink�

(Figure 7). A 50-mm radius sphere with a coordinate(100, �100, �150mm) is located in the virtual envir-onment. The virtual sphere was modeled using spring-damper Voigt model lying from center to surface ateach contact point. When the user moves the handleinto the sphere, the device produces a force in thedirection of the surface normal depending upon thepenetration distance, velocity, and virtual modelparameters.

The redundant joint motor adjusts the device pos-ture for maximum stability using optimal redundantjoint angles. One of the MATLAB� optimizationfunctions, ‘‘fminbnd’’ serving as Golden SectionSearch algorithm, was used for the estimation of theoptimal redundant joint angles. Figure 8(a) shows theinitial map of redundant joint angles for the optimalpostures depending on the contact points. It can beseen from Figure 8(a) that there is a ridge on the

initial map. The reason of this ridge is that full rightposture gives the minimum torque requirement on lefthemisphere and the full left posture gives it on theright (Figure 9). This ridge may cause the suddenchange in the device posture which is not desiredsince it deteriorates the transparency and may beharmful for the user. Therefore, this ridge wassmoothed to avoid the sudden changes of the posture.Figure 8(b) shows this smoothed redundant jointangle map. The posture adjustment should startwith the movement of the handle toward the sphereand be completed before contacting the sphere.The device should transmit the contact forces withthe fixed optimal posture, since the posture adjust-ment during penetration may produce undesired iner-tia and deteriorate the transparent interaction.Therefore, some spherical offsets were constructedand different posture adjustment velocities wereassigned between the offsets as shown in Figure 10.The device adjusts the posture gradually whileapproaching the sphere surface with various postureadjustment speeds increasing from outer to inner off-sets. Note that � and � angles in Figure 10 denote thespherical coordinates to represent the handle positionwith respect to virtual sphere. The posture adjustmentduring surface contact (penetration) may cause unde-sired parasitic forces. These forces may prevent to

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

B (Ns/mm)

K (

N/m

m)

6-DOF Phantom like posuture

7-DOF Optimal posture

Collision exp.point (K)

0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

0.3

0.4

0.5

0.6

B (Ns/mm)

K (

N/m

m)

6-DOF Phantom like posuture

7-DOF Optimal posture

Collision exp.point (L)

0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

0.3

0.4

0.5

B (Ns/mm)

K (

N/m

m)

6-DOF Phantom like posuture

7-DOF Optimal posture

Collision exp.point (M)

(b)(a)

(c)

Figure 12. Z-width plots of 7-DOF optimal posture and 6-DOF Phantom like posture for three different regions (a, b, and c) of the

virtual sphere.

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Page 11: Optimal posture control algorithm to improve the stability of redundant haptic devices

transmit the virtual model impedances to the usertransparently due to the device dynamics. Therefore,the device adjusts its posture before penetration andthe innermost offset is considered as non-changingposture segment to preserve the transparency duringhaptic interaction. Since the innermost offset is con-sidered as non-changing posture segment, the postureadjustments should be made at outer offsets.Whenever the user wants to touch a point far awayfrom the actual contact point, he needs to get out ofthe innermost offset and reenter after the new postureadjustment. This strategy prevents the user from feel-ing the adjustment dynamics effects while penetratingthe virtual object surface. This strategy can be con-sidered to cause inconvenience for the haptic users;however, achieving stable and transparent interactionis more challenging in haptics.

Stability performance of a multi-DOF hapticdevice is not same on every points of a virtual surfacesince the joint torque requirements change dependingupon the force direction and location of the contactpoint. Hence, the stability performance of a multi-DOF haptic device should be evaluated on differentregions of the virtual model surface. In this study,three different regions were selected on the virtualsphere for the stability tests (Figure 11) and Z-widthplots were generated experimentally on these regions.The reason for the selection of these specific regions is

that they require different posture adjustments.Specifically, regions (a), (b), and (c) require the fullright posture, full left posture, and a posture which isclose to the home posture. Since the default homeposture of 7-DOF haptic device kinematics corres-ponds to 6-DOF Phantom haptic device kinematics,the proposed algorithm can be evaluated by compar-ing it with 6-DOF Phantom like posture of the device.Figure 12 shows the comparison of Z-width plotsbetween the optimal posture and 6-DOF Phantomlike posture with locked at default home posture.The results show that optimal posture control algo-rithm improves the achievable impedance ranges sig-nificantly on (a) and (b) sides and a little on (c) side ofthe sphere. The stable impedance range of 7-DOFoptimal posture and 6-DOF Phantom like posture isclose to each other on the test region (c) since theoptimal posture control algorithm requires a postureadjustment which is very close to 6-DOF Phantomlike posture (Figure 8).

In order to validate the proposed algorithm fur-ther, a set of stiff virtual surface collision experimentswere conducted on the same regions (a, b, and c) ofthe virtual sphere. In the experiments, the virtualsphere stiffness—K and damping–B parameters werereset to different values while testing the algorithm ondifferent test regions (a, b, and c in Figure 11) suchthat those were selected between the Z-width plots of

0 5 10 15 200

20

40

60

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100

Time (s)

Han

dle

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ition

(m

m)

Inside the virtual sphere Test region (a)6-DOF Phantom like posture

Unstable interaction

(a)

(b)

(c)

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ition

(m

m)

Test region (a)7-DOF optimal posture

Stable interaction

Inside the virtual sphere

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Inside the virtual sphere Test region (b)6-DOF Phantom like posture

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Inside the virtual sphere Test region (b)7-DOF optimal posture

0 5 10 15 200

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Inside the virtual sphere Test region (c)6-DOF Phantom like posture

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Inside the virtual sphere Test region (c)7-DOF optimal posture

Stable interaction

Figure 13. Virtual sphere collision experiments on the region (a) for K¼ 0.4 N/mm—B¼ 0.001 Ns/mm and region (b) for K¼ 0.3 N/

mm—B¼ 0.002 Ns/mm, and region (c) for K¼ 0.26 N/mm—B¼ 0 Ns/mm.

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6-DOF Phatom like posture and 7-DOF optimal pos-ture (K, L, and M points in Figure 12). Note that (K),(L), and (M) points in Figure 12 are selected in the Z-width regions where 6-DOF Phantom like posture isunstable and 7-DOF optimal posture is stable. In thestiff virtual sphere collision experiments, the distanceof the handle tip point with respect to the spherecenter was monitored. Ideally, during penetration,the haptic interface should present smooth distancemeasurements which are equal to the virtual sphereradius (50mm). According to the results shown inFigure 13, the 7-DOF optimal posture control algo-rithm provides stable interaction for the stiff virtualsurfaces where 6-DOF Phantom like posture isunstable (Figure 12).

Conclusion

Stability is a major requirement in haptic interactionsince unstable behavior impairs realistic interactionand may injure the user. In this paper, we propose anoptimal posture control algorithm to improve the sta-bility performance of the redundant haptic devices. Formulti-DOF haptic devices, the passivity condition mustbe satisfied for both end-effector space and joint space toachieve stable interaction. In the proposed algorithm,the desired torque of the critical actuator is minimizedby using posture adjustment ability of a redundanthaptic device. Golden Section Search algorithm basedon a conservative passivity condition is used to estimateoptimal redundant joint angles. The posture is adjustedautomatically using redundant actuator. This algorithmwas applied successfully with a 7-DOF redundanthaptic device on a virtual sphere. The performance ofthe proposed method evaluated using Z-width stabilitymetric. The results show that the optimal posture con-trol algorithm improves the stability significantly for theredundant haptic devices.

Conflict of interest

None declared.

Funding

The authors would like to thank TUBITAK (The Scientificand Technological Research Council of Turkey) for the

funding support to manufacture the experimental setup.

References

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Appendix 1

The Jacobian matrix for 7-DOF haptic deviceincludes seven columns and six rows as follows:

J ¼�Jr1 �Jr2 �Jr3 �Jr4 �Jr5 �Jr6 �Jr7�Ja1 �Ja2 �Ja3 �Ja4 �Ja5 �Ja6 �Ja7

� �

The Jacobian matrix components �Jrk and �Jak areone column and three row matrices as follows:

�Jr1 ¼ e~u3�1 ~u3e~u2�2 d3 �u3þd4e

~u3�3e~u2�4 �u3� �

�Jr2 ¼ e~u3�1e~u2�2 ~u2 d3 �u3þd4e~u3�3e~u2�4 �u3

� ��Jr3 ¼ e~u3�1e~u2�2d4e

~u3�3 ~u3e~u2�2 �u3

�Jr4 ¼ e~u3�1e~u2�2d4e~u3�3e~u2�2 ~u2 �u3,

�Jr5 ¼ �Jr6��Jr7 � 0 0 0� �T

�Ja1 ¼ �u3

�Ja2 ¼ e~u3�1 �u2

�Ja3 ¼ e~u3�1e~u2�2 �u3

�Ja4 ¼ e~u3�1e~u2�2e~u3�3 �u2

�Ja5 ¼ e~u3�1e~u2�2e~u3�3e~u2�4 �u3

�Ja6 ¼ e~u3�1e~u2�2e~u3�3e~u2�4e~u3�5 �u2

�Ja7 ¼ e~u3�1e~u2�2e~u3�3e~u2�4e~u3�5e~u2�6 �u3

where

�u1 ¼ 1 0 0� �T

, �u2 ¼ 0 1 0� �T

, �u3 ¼ 0 0 1� �T

e~u1�k ¼

1 0 0

0 cosð�kÞ �sinð�kÞ

0 sinð�kÞ cosð�kÞ

264

375,

e~u2�k ¼

cosð�kÞ 0 sinð�kÞ

0 1 0

�sinð�kÞ 0 cosð�kÞ

264

375,

e~u3�k ¼

cosð�kÞ �sinð�kÞ 0

sinð�kÞ cosð�kÞ 0

0 0 1

264

375

Notation

b physical damping of each jointB virtual environment damping parameter

di ith link offsetE0 initial energy of the haptic systemF force applied by the user handFd desired endpoint force for virtual

environmentFh user hand force in end-effector spaceFu resultant force applied on the device

handleim motor currentI physical inertia of each jointJ Jacobian matrix with respect to the end-

effectorK virtual environment stiffness parameterKemf motor back EMF constantKm motor torque constantKp proportional gain of force feedback

controllerL motor inductanceRm motor resistanceT0

7 transformation matrixTa hold device actuator torque by sample/

hold unitTc critical elbow joint torqueTd desired actuator torqueTe torque calculated in virtual environmentTh torque applied by the user handTs sampling timeva motor voltageVh user hand velocity in end-effector spacexh position of the haptic handlexu position of the user handZu impedance of the user handZv impedance of the virtual environment� desired damping matrix in the end-effector�m physical damping matrix in the joint space_� joint velocity of the device� joint position of the device_�d device handle angular velocity€�d device handle angular acceleration_�e transmitted velocity to virtual

environment_�h user hand angular velocity€�h user hand angular acceleration�L lower limit of redundant joint angle�R redundant joint angle�U upper limit of redundant joint angle� resultant joint torque�d desired torque command�m joint torque applied by motors�u joint torque applied by the user

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