Distributed Haptic Cooperation with Passive Multirate Wave Communications

ABSTRACT

Ramtin Rakhsha* Mechanical Engineering

Department University of Victoria,

BC, Canada

This paper investigates the use of passive multi rate wave commu­nications for distributed haptic cooperation between two users con­nected over a network with constant delay and low transmission rate. The paper develops the lifted state space model of the hap­tic cooperation system and uses it to compute the maximum stable gains for the users' feedback loops and for the coordination of the virtual object distributed across the network. The multi rate stabil­ity analysis predicts that: an order of magnitude larger coordination gains can be used when the two users are connected via passive wave communications than when they are connected via power­domain communications or via traditional wave-domain commu­nications; the maximum coordination gain stays unaffected by con­stant and symmetric network delays. A larger maximum coordi­nation gain independent of the network delays provides increased and robust coherency of the shared virtual object. Experiments in which two users manipulate a shared virtual cube together validate the multi rate stability analysis.

Index Terms: Haptic cooperation, Multi-rate stability analysis, Passive communications, Wave variables, Anti-aliasing filtering.

1 INTRODUCTION

Virtual environments haptically shared across computer networks can support physical interaction among remote users. Such con­nectivity is beneficial in applications like tele-therapy [24], virtual reality-based surgical training [19], haptically enabled multi-user online computer gaming [15]. However, computer networks suffer from limited transmission rates, communication delays, jitter, and packet loss, all of which adversely affect the stability and perfor­mance of the networked haptic cooperation [2, 14, 18, 11].

In recent years, haptic sharing of virtual environments over com­puter networks (first proposed in [10]) have attracted significant re­search effort. Several control approaches have been implemented and studied experimentally. For networks with constant delays but limited packet update rate like local area networks (LANs) and metropolitan area networks (MANs), [13] has investigated the sta­bility and transparency of the centralized and distributed multirate control of haptic cooperation. The results in [13] have shown that distributed controllers can render stiffer virtual contacts to users and are thus, more suitable for haptic interaction in rigid virtual envi­ronments. A centralized controller design methodology based on Linear Matrix Inequalities has been proposed in [7, 8, 9]. Guaran­teed stable multi-user haptic cooperation in the presence of variable delays has been proposed in [16] based on a passive integrator suit­able for point interaction in static virtual environments. All these

.e-mail: rrakhsha@uvic.ca

t e-mail: danielac@uvic.ca

IEEE Haptics Symposium 2012 4-7 March, Vancouver, BC, Canada 978-1-4673-0809-0/12/$31.00 ©2012 IEEE

Daniela Constantinescut Mechanical Engineering

Department University of Victoria,

BC, Canada

analytical investigations have focused on power domain communi­cations exclusively.

Haptic cooperation with wave-domain communications has been studied primarily experimentally [5,6,21,17]. Those experimental studies have confirmed the robustness of wave-domain communica­tions to transmission delays. An analysis of centralized haptic co­operation with wave-domain communications has been presented recently in [25, 26]. The analysis in [26] has demonstrated that passive wave-domain communications can be used to significantly increase the stiffness of the virtual environment that can be rendered to users who manipulate a centralized virtual object together.

This paper concerns with the integration of wave transformations into the communication stream for distributed multi-user haptic co­operation systems. The main contributions of the paper are: (i) a comprehensive multirate stability analysis of distributed control of the networked haptic cooperation with power-domain, wave­domain, and passive wave-domain communication types, and (ii) the experimental validation of the analytical results. In this pa­per, wave-based communications are integrated into the distributed haptic cooperation systems. In such architectures, symmetric wave transformations [22] reside between wave and power domains at each peer. The shared virtual object (SVO) copies are connected to each other through wave variable controllers over a LAN which in­volves two sampling rates, i.e, the control loop sampling time (Td and the network sampling time (Tn). This, thus, draws a multi­rate haptic feedback loop. Passivity conditions for the multi-rate wave-based communication medium are driven in [26] in a central­ized control framework. It is shown that by deploying low-pass filters with appropriate cutoff frequency (after each outgoing wave command and before the downsampling), the passivity is guaran­teed and the aliasing could be prevented. Here, by utilizing the lifting approach [4], the multi-rate analysis is performed in order to achieve the stability bounds for both power-domain and wave­domain communication types. The method derives the stability re­gions of distributed haptic cooperation between two peers based on the eigenvalue analysis of the closed-loop state transition matrix. However, having an accurate model of the haptic devices is crucial when deploying such multirate analysis. The results illustrate that deploying the anti-aliasing filter not only guarantees the passivity of the wave-based communication links but also, extends the sta­bility bounds. This allows for larger control gains and enables for substantially higher coordination gains which in turn, increases the realism of cooperative haptic interaction while staying unaffected under network delays.

In the remainder of this paper, Section 2 introduces the passive wave-based communications for distributed control of haptic coop­eration. Section 3 briefly overviews haptic cooperation with power­domain and wave-domain communications. The derivation of the multirate state space dynamics of haptic cooperation with passive multirate wave communications is presented in Section 4. Stability regions for haptic cooperation between two users connected with power-domain, wave-domain and passive wave-domain communi­cations across a network with low update rate and small and con­stant network delays are also presented in this section. The an-

117

118

alytical results are validated in Section 5 through experiments in which two networked users manipulate a shared virtual cube to­gether across a LAN. Section 6 summarizes the conclusions drawn from this work.

2 PASSIVE WAVE-BASED COMMUNICATIONS

Scattered [3] or wave-based [20] communications render the com­munications channel passive in the presence of constant time de­lays. Such communications have been introduced to guarantee the stability of the interaction of any passive user with any passive re­mote environment and for any constant network delay when the master and slave robots are rendered passive through local con­trol. To maintain the wave communications passive when the net­work transmission rate is lower than the rate of the force control loop, anti-aliasing low-pass filters need to be placed before the rate drop [26], as depicted in Figure l. In this figure: b is the wave impedance; Td is the network delay; FTi is the coordination force applied to the local copy of the shared virtual object at Peer i and also encoded in the wave signal sent to the other user; iOid is the desired velocity of the shared virtual object decoded from the wave signal received from the peer site; Uouti is the outgoing wave, sent by Peer i to the remote site; uini is the incoming wave, arriving at Peer i from the remote site; M is the downsampling factor from the rate of the force control loop to the network transmission rate; LP are anti-aliasing low-pass filters with cutoff frequency Qc = "1M which guarantee the passivity of the multirate wave-domain com­munications [26]; and Yfi is the output of the anti-aliasing filter at Peer i.

Peer side 1 Peer side 2

Figure 1: Multirate passive wave-based communications.

The state-space continuous-time dynamics of the anti-aliasing filter at Peer i are:

ifi = -Qcxfi + Uouti Yfi = Qcxfi

(1)

where Qc is the cutoff frequency of the LP filter. The following relationship holds between wave variables and filters' outputs:

(2)

where Td is the network delay. In this paper, the network delay is assumed constant and an integer multiple of the network packet update interval, Tn. 3 TWO-USERS DISTRIBUTED NETWORKED HAPTIC COOP-

ERATION

Distributed networked haptic cooperation is controlled through co­ordinating the local copies of the shared virtual object distributed at the peer users. During the cooperation, each user interacts with their own copy of the shared virtual object, and various control paradigms can be used to coordinate the copis of the shared vir­tual object. Virtual coupling control [12, 1] has typically been used to coordinate the copies of the shared virtual object [13, 22, 23]. The virtual coupling coordination of networked haptic cooperation

at Peer i is schematically depicted in Figure 2. In this architec­ture, the users exchange velocity data and the communications are power-domain communications, and FLCi is the interaction force at the contact between Peer i and their local copy of the shared virtual object.

Virtuol environment i

Figure 2: Control architecture at Peer i for distributed haptic coop­eration with power-domain communications and virtual coupling coordination of the shared virtual object.

In distributed haptic cooperation with power-domain communi­cations, the communication delay adversely affects the maximum allowable coordination control gains. Lowering the coordination gain lessens the coherency between the distributed copies of the shared virtual object and destroys the realism of the interaction. Wave-domain communications can be used to overcome the degra­dation of performance and the instability that may arise when the network delay increases, as shown in Figure 3.

Virtual environment i \

,-----------------------------;

Figure 3: Control architecture at Peer i for distributed haptic coop­eration with wave-based communications.

In the next section, the state space dynamics of distributed hap­tic cooperation (Figure 3) with the passive multirate wave-domain communications (Figure 1) are derived and used to determine the maximum coordination gain for the shared virtual object.

4 STABILITY ANALYSIS

The derivations in this section account for the fact that the haptic cooperation between two users is a discrete-time system with two sampling rates when the network transmission rate is lower than the rate of the force feedback loop. In this case, the fast sampling rate is due to the slow sampling interval of the force control loop Te, and the slow sampling rate is due to the large sampling interval of the packet updates across the network Tn. The derivations in this section use typical values [13] Te = 0.001 s and Tn = 0.008 s. Be­cause the derivations are based on the lifting approach introduced in [4] and applied to haptic cooperation with power-domain com­munications in [13], they are detailed only in as much as needed for the integration of the passive wave communications depicted in Figure l.

4.1 Open-loop Continuous-time State-space Represen-tation

To obtain the continuous-time state space dynamics of the open­loop two-users haptic cooperation with passive wave-based com­munications, the dynamics of the users, of the haptic interfaces, of the distributed copies of the shared virtual object, and wave vari­ables are grouped into fast and slow system inputs and outputs, hereafter denoted with the indices c and n respectively. Specifi­cally, the system inputs comprise the contact forces, updated at the fast haptic rate, and the coordination forces applied to the shared virtual object which include both fast and slowly updated compo­nents:

where:

and

uT = (uT

c

u; = (hCI FTl,

uT = (FTl n n

uTl n

hC2 FT2JT

FT2JT

FTic KTxOi + BTiOi

FTin -KTXOjd - BTiOh

(3)

(4) (5)

(6)

(7) The state vector comprises the states of all haptic interfaces and all copies of the shared virtual object:

(8) where:

(9) The output vector is:

yT = Yc T

== Xi T, (10) Hence, the continuous-time state-space model of the open-loop

two-user networked haptic cooperation is:

XSXl = A sxsx sx, +BSX6U6XI YSxl = CSXSXSXI (11)

4.2 Discrete-Time State-Space Representation

Following [4] and assuming that the network sampling interval is an integer multiple of the force control sampling interval, the discrete­time state-space representation of the open-loop system can be writ­ten in the following form:

Xo [k + 1] = AO Xo [k] +BO Uo [k] (Ax] 64x64 64x] 64x34 34xl YO"'XI [k] = C o",x", XO"'XI [k] + D O"'X34 U034Xl [k] (12)

where k is the k-th network update interval and more details about the derivations of the system matrices AO, BO, CO and DO can be found in [13].

The decoded desire velocity in discrete-time can be written as:

(13)

and the desired position is obtained after discrete-time integration of the velocity command from the wave signal and unwrapping of the typical algebraic loop of wave transformations:

(14) Desired position and velocities are integrated into the discrete-time state-space dynamics of the haptic cooperation system through aug­menting the state vector:

X [k]- (XO"'XI [k]) OW66Xl - xd [k] 2x I where:

and the input vector: (UOC32XI [k]) UOn2Xl [k] xd [k] 2xl uin [k] 2xI

where:

The augmented output matrix is then: (XOW66Xl [k]) k =

xd 2XI [k] YOWS6XI [ ] xd [k] 2x I Uoutl6XI [k]

(15)

(16)

(17)

(18)

(19)

where the outgoing waves are computed at the fast control rate:

u - (u I U 2 )T outl6x I - out 8x l out Sx 1 (20) After incorporating the wave dynamics, the discrete-time dynamics of the open-loop haptic cooperation system becomes:

xOw[k + 1] YOw[k]

AOw66X66 xOw [k] + BOW66X3S uOw [k] COWS6X66 xOw [k] + DOWS6X3S uOw [k]

(21) Since Tn = 8· Te, the discretized difference state equations of the anti-aliasing filters are:

XfDiSXl [k + 1] (YfDiCsxl [k]) YfDinlxl [k]

(22)

119

120

and the discrete time open-loop dynamics of the haptic cooperation system including the anti-aliasing filters become:

(XDI" [k + ll) XDf

�Xl [k + Il 16xl

(YDw 86"' ['I) Yf1 9x I [kl Yf29x1[kl

[A�IV�x� 16x66

+ [B�IV�X38 16x38

[C�1V86X� 18x 16

[D�1V86X38 + 18x38

O�XI6 ] (XDIV�XI [kl) ADfl6Xl6 XDfl6Xl [kl O�XI6 ] (UDW38Xl [kl )

BDfJ6Xl6 UOUtl6X1 [kl o ] 86x 16

CDfJSX16 o ] 86x 16

DDfJsXl6

(XDIV�XI [kl) XDfl6Xl [kl

(UDW38Xl [kl) UOUtl6x I [kl

(23)

By further augmenting the state vector with the delayed inputs [13], computational and communication delays are incorporated into Equation (23).

4.3 Stability Regions

For two-users networked haptic cooperation with passive wave­based communications, the feedback matrix FD comprises the con­tact and the coordination forces on the shared virtual object, and is computed using lifting [4]. Thereafter, the stability of the mul­tirate haptic cooperation system can be derived through eigenvalue analysis of the closed-loop state transition matrix Ag, calculated

via:

(24)

where AD ,BD ,CD and DD are the state transition ma-aug aug aug aug trices obtained after suitable augmentation with computational and communication delays. Thus, the two-users haptic cooperation is stable if and only if all eigenvalues of Ag are inside the unit circle:

(25)

The stability regions for two-users haptic cooperation with vari­ous communications are presented in Figure 4 for various values of one-way network communication delay. The calculations are performed using the following parameter values: rnHD; = 0.1 kg, bHD; = 1 Ns/m, BT; = 2 Ns/m, BLC = 2 Ns/m, rnO = 0. 6 kg, bO = 0 Ns/m, b = 10 Ns/m, Qc = 60 Hz, Tc = 0. 001 s, and Tn = 0.008 s. The analytical results in Figure 4 predict that wave­based communications increase the coordination stiffness by one order of magnitude compared to power-domain communications. Furthermore, the maximum coordination stiffness is unaffected by the network delay. Passive wave-based communications enlarge the coordination gain by another order of magnitude. Larger co­ordination gains result in lower position discrepancy between the distributed copies of the shared virtual object, as verified experi­mentally in the next section.

5 EXPERIMENTS

This section validates the comparative analysis presented in Sec­tion 4.3 through experimental one degree of freedom (DOF) haptic cooperations. The experiments contrast the performance of passive multi rate wave communications to that of multirate power-domain and traditional wave-domain communications. The experimental testbed comprises two Quanser 6 DOF haptic wands connected to two personal computers running Window XP on an Intel Core 2 Duo CPU at 2.67 GHz with 2 GB RAM (Figure 5). The two com­puters communicate over a local area network (LAN) via the UDP

protocol. The position sensing and force feedback rates for both haptic devices are set to 1 kHz. The network data transmission rate is 125 Hz.

Figure 5: The experimental setup.

In all experiments, proportional-derivative controllers constrain the 6-DOF haptic devices to move along the horizontal x-direction (parallel to the back wall in Figure 5). The virtual environment consists of a shared virtual cube whose dynamics are simulated at each user using forward Euler integration with fixed step equal to the sampling time of the force feedback loop. The cube has mass rno; = 0. 3 kg and no damping bo; = 0 Ns/m. The stiff­ness and damping of the virtual contacts are KLC; = 1000 N/m and BLC; = 2 Ns/m, respectively. The wave impedance is b = 10 Ns/m, and the coordination damping is BT; = 2 Ns/m. To allow meaning­ful comparison among successive experiments, the two users are replaced by controlled forces applied to the haptic devices as com­mands sent to the actuators.

5.1 Experimental Maximum Coordination Gains

In the first experiment, each user initially pushes the shared vir­tual cube towards their peer with a force equal to 0.1 N. During the cooperation, Peer 1 applies an additional force equal to 1 N force towards Peer 2 for 25 ms at intervals of 2. 5 s. The maxi­mum achievable coordination gains are used for each specific com­munications architecture, and they are as follows: KT = 310 N/m for power-domain communications (i.e., virtual coupling coordi­nation); KT = 1590 N/m for traditional wave communications; KT = 10620 N/m for passive wave communications. The experi­mental data recorded at Peer 2 are presented in Figure 6 for round trip network delay Td = 16 ms, and in Figure 7 for round trip net­work delay Td = 32 ms.

The results in Figures 6 and 7 validate that: (1) passive wave communications support much larger coordination gains than power domain communications and than traditional wave commu­nications; and (2) wave communications are robust to network de­lays, unlike power domain communications. The larger coordina­tion gain supported by the passive wave communications increases the coherency between the distributed copies of the shared virtual object, as can be observed in Figures 6 and 7, and is quantified in the next section.

5.2 Experimental Position Coherency

The second experiment investigates the position coherency between the local copies of the shared virtual cube. In this experiment, each user initially pushes the shared virtual cube towards their peer with a force equal to 1 N. During the cooperation, Peer 2 applies an addi­tional sinusoidal force with amplitude 0.25 N and frequency 3 rad/s towards Peer 1. The network round trip delay is Td = 0.016 s. The coordination stiffness is: KT = 310 N/m for power-domain com­munications (i.e., virtual coupling coordination); KT = 1590 N/m for traditional wave-domain communications; KT = 10620 N/m for passive wave-domain communications.

12000 �-�--�--�--�-�

10000

8000

-"1=0 '''''''"1 = 1 ...... nt = 2 ""'" nt = 3

Z� _ 6000

4000

o 500 1000 1500 2000 2500 KLc(N/m)

12000

� 10000 '''''''"1 = 1

...... nl = 2

8000 �

I 6000 I ,t ,t

4000

2500

12000

10000

8000

6000

4000 � '''''''"1 = 1

2000 ...... nj = 2 " " " ' "t = 3

�

00 500 1000 1500 KLC (N/m)

, { I \ . ;; ;1 ; ;, ! : , , ,

2000 2500

(a) Power-domain communications. (b) Traditional wave-based communications. (c) Passive wave-based communications.

Figure 4: Stability regions for two-user haptic cooperation for different communication delays Td = nl . Tn.

Figure 8 plots the coordination forces (FT ) applied to the shared virtual object when the two cooperating users are connected via each of the three communications architectures. As predicted in Section 4.3, passive wave communications maintain the interaction stable for higher coordination gains which, in turn, increase the po­sition coherency between the distributed copies of the shared virtual object. Furthermore, the passive wave communications improve co­herency while applying lower and smoother coordination forces.

The experimental average position discrepancy between the two copies of the shared virtual cube is plotted in Figure 9 for a round trip network delay Td = 0.016 s. Note in this figure that, compared to power-domain communications, wave-domain communications decrease the position discrepancy by one order of magnitude, and passive wave-domain communications decrease it by two orders of magnitude.

6 CONCLUSIONS

This paper has investigated the use of passive multirate wave com­munications for distributed haptic cooperation between two users connected over a network with constant delay and low transmis­sion rate. Passive communications have been obtained by sending wave signals free of aliasing over the network, as proposed in [26]. The paper has incorporated the anti-aliasing wave filters into the state-space model of the multirate haptic cooperation system and has used it to compute the maximum stable gains for the users' feedback loops and for the coordination of the virtual object dis­tributed across the network. The multi rate stability analysis has predicted that: an order of magnitude larger coordination gains can be used when two users are connected via passive wave communi­cations than when they are connected via power communications or via traditional wave communications; the maximum coordina­tion gain is unaffected by constant and symmetric network delays. A larger maximum coordination gain independent of the network delay provides increased and robust coherency of the shared virtual object. Experiments in which two users manipulate a shared virtual cube together have validated the multirate stability analysis.

The analysis in this paper has assumed constant communication delays that are equal in both directions, like those provided by lo­cal area networks and metropolitan area networks [13]. However, the availability and the low cost of Internet makes packet-switched communications most desirable for haptic cooperation applications. Future work will investigate the passivity of multi rate wave com­munications for distributed haptic cooperation across networks with

asymmetric and variable delays.

ACKNOWLEDGEMENTS

This work was supported through an NSERC Discovery Grant.

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0.05,-�======:o;;] -- Local SVO copy I - - - Remote SVO copy I

" , ,

� � 0.5 o LL C o � c '6 o -0.5 o

U -0.05L--- -�_,___-�-____:_�---"....., 34 34.5 35 35.5 36

_lL---_�_,___-��Lll�UULU 34 34.5 35 35.5 36

Time(s) Time (s) (a) Power-domain communications (Kr = 310 N/m)

0.031 ,-,====:==:=====;-] -- Local SVO copy

0.03

c :2 0.028 .(jj 8. 0.027

� 0.026 (fJ

0.025

- - - Remote SVO copy

0 02�L4 --3:': 5'--------:36::-----:3'= 7--3:': 8'-------" 39

Time(s)

� � 0.5 0 LL

C 0 � c � 0 0 U

-0.5 34 35 36 37

Time (s) 38

(b) Traditional wave-domain communications (Kr = 1590 N/m).

0.03 ,-r==========;-] -- Local SVO copy

0.029

§ 0.027 "" .� 0.026

� 0.025

iii 0.024

0.023

- - - Remote SVO copy

0.0223Ll--3�2-�33:-----:3� 4 --3�5,-------"

36 Time(s)

� Q) e 0 LL c 0 � c :e 0 0 U

0.5

-0.5 31 32 33 34 35 Time (s)

(c) Passive wave-domain communications (Kr = 10620 N/m).

39

36

Figure 7: Experimental haptic cooperation with the same coordi­nation gain KT as in Figure 6 for round trip network delay Td =

0.032 s.

1.8

� 16

�14 o LL

04

. _. _. Power-domain communications - - - Wave-domain communications

°:S�0----5�2----5�4----5�6---�58 Time(s)

Figure 8: Coordination force on the local copy of the shared virtual object (SVO) at Peer 2 for haptic cooperation across a network with round trip delay Td = 0.016 s.

0.04 -- Local SVO copy - - - Remote SVO copy

� 0.038 .§. c o 0.036 "" .(jj 0 CL 0.034

0 > (fJ 0.032

... '4.·,,,;

0.03 33 34 35 36 37 Time(s)

� Q) e 0 LL C 0 � c � 0 0 U

38

0.5

-0.5 33 34 35 36 37 Time (s)

(a) Power-domain communications (Kr = 310 N/m).

I 0.024

5 0.023 "" .� 0.022

o 0.021

iii 0.02

0.019

-- Local SVO copy - - - Remote SVO copy

0.0183:-2--3:':3-----:34::-----:3'=5--3:':6,-------"

37 Time(s)

� � 0 LL C 0 � c � 0 0 U

0.5

-0.5 32 33 34 35 36

Time (s)

(b) Traditional wave-domain communications (Kr = 1590 N/m).

0.024 '----;::===:====il -- Local SVO copy

I 0.022 c :2 0.021 .� a. 0.02

o > 0.019 (fJ

0.018

- - - Remote SVO copy � � o 0.5 LL c o � c :e 8 u

38

37

0.017 L---3�2-�33'-------"'3�4 --'3�5-�36---"

Time(s) -0.5 L---3�2-�33:-----'3�4 --3�5-�36

�

Time (s)

(c) Passive wave-domain communications (Kr = 10620 N/m).

Figure 6: Experimental haptic cooperation with maximum achiev­able coordination gain KT for round trip network delay Td =

0.016 s.

3.5

0.5

3.5

Power-domain communications

I 0.644

Wave-domain communications

0.099 Passive wave-domain

communications

Figure 9: Position discrepancy between the two copies of the shared virtual object (SVO) for haptic cooperation over a network with round trip delay Td = 0.016 s.

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