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Switching Time Domain Passivity Control for Multilateral Teleoperation Systems Usman Ahmad, Student Member, IEEE, Ya-Jun Pan*, Senior Member, IEEE, Anees ul Husnain, Member, IEEE.
Abstract—Time domain passivity control, a well known control scheme widely used for teleoperation systems, normally works under the constraint of zero division. Small force or velocity signals can cause the occurrence of zero division which ultimately leads the system to be unstable. This paper presents a novel switching time domain passivity control scheme for multilateral teleoperation systems which not only ensures the stability of the system but also avoids zero division. In contrast to bilateral teleoperation systems, the multilateral teleoperation system is much more complex as it involves increased number of master and slave hardware, multiple operators and transmission of multiple signals over the communication network. A new framework for the communication channel has been proposed which incorporates the use of weighting coefficients to give the masters and slaves authority depending upon the requirements of the operation. As the switching time domain passivity control keeps the system passive all the time, the stability is guaranteed. The proposed control scheme is valid for n masters and n slaves. Simulations with two masters and two slaves are carried out to verify the effectiveness of the proposed scheme. Index Terms—Multilateral teleoperation, time domain passivity control, zero division, switching dissipation.
I. I NTRODUCTION Teleoperation, as the name depicts, is a loop of actions in which a human operates a master hardware which in turn operates a slave device at a distance over a communication network. Applications of teleoperation can be listed as underwater exploration, mining, robotic surgery [1], tele-therapy, education, multi-user online computer gaming [2], manipulations in hazardous environments such as outer space and nuclear plants [3], medical training, telemaintenance and entertainment [4]. In multilateral teleoperation systems, a human operates n master devices to send position or velocity signals to n slave devices and the slave devices send back the force signals to the master side which a human operator feels. Motivation of the multilateral teleoperation comes from the idea of a perfect collaboration of the masters and slaves for the proper execution of a required task in a remote environment. Multilateral teleoperation systems enable humans to collaboratively perform a task with coordination depending upon the needs but the cooperative nature of multilateral teleoperation systems is difficult to realize to achieve desired performance criteria. A widely used approach for teleoperation systems is the time domain passivity control (TDPC) [5], [6]. This scheme is basically based on varying damping injection which maintains the system passivity. The energy based time domain passivity Usman Ahmad and Ya-Jun Pan are with the Department of Mechanical Engineering, Dalhousie University, Halifax, NS B3H 4R2, Canada. Anees ul Husnain is with the Department of Computer Systems Engineering, The Islamia University of Bahawalpur, Pakistan. *Corresponding Author:
[email protected]
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control (ETDPC) observes overall energy of the system. As teleoperation systems sometime encounter large force signals from the environment so to eliminate this, TDPC with reference energy scheme has already been proposed [7]. [6], [8] proposed a power based time domain passivity control (PTDPC) which was applied to teleoperation system and in [9] with application to haptic control. Less complex and model free structure of time domain passivity control schemes makes them easier to implement but as mentioned earlier the only drawback of these schemes is the zero division which should be avoided at any cost for the stability of the teleoperation system and the safety of the human operator. One mechanism that has been put forward in [10] naturally avoids this problem of zero division by removing the noisy behavior due to zero velocity. Another approach to avoid zero division has been presented in [11] which takes the advantage of deactivating the passivity controller and outputs a zero in low force or velocity. However, the deactivation of passivity controller at an instant of time causes the loss of passivity at that instant and the activeness may accumulate leading to instability of the system. Although several schemes for teleoperation systems, especially for multilateral teleoperation systems [12]–[15], have appeared in literature, but initially the notion of switching time domain passivity control was proposed by [16] to overcome the zero division but it was found that the scheme was not valid where the varying time delays were under consideration. This paper proposes a simple yet practical time domain passivity control based on the same switching idea with application to multilateral teleoperation systems. The main contributions of the paper are: • To propose a new switching TDPC scheme for multilateral teleoperation systems with constant time delays which ensures the stability of the system. • To introduce a novel communication structure incorporating the weighting coefficients for the proper weight selection of the masters and slaves. • To verify the effectiveness of the proposed scheme with simulation results. II. T IME D OMAIN PASSIVITY C ONTROL The power P of a system can be expressed as dE + Pdiss , (1) dt where f is the output force, v is the input velocity, E is the low bounded stored energy and the dissipation is Pdiss . It is well known that if (2) Pdiss ≥ 0,
69
P = fv =
2
then the system is passive. The integration of Eq. (1) can be expressed as t
P(τ)dτ 0
= =
t 0
vsci (t) =
f (τ)v(τ)dτ
E(t) − E(0) +
t 0
Pdiss (τ)dτ ≥ −E(0) (3)
0
f (τ)v(τ)dτ ≥ 0,
fmci (t) =
0
P(τ)dτ =
(4)
(6)
∑ fmci j (t),
fmci j (t) = α f i j fs j (t − Tsi j ).
(7)
n
P = ∑ [vmi (t) fmci (t) − vsci (t) fsi (t)]
(8)
i=1
t m
( ∑ fi (τ)vi (τ))dτ ≥ −E(0),
vsci j (t) = αvi j vm j (t − Tmi j ),
j=1 n
where αvi j and α f i j describe the weighted effects of weighting coefficients on masters and slaves (i= j=1,· · · ,n). Tmi j and Tsi j represent the delay from the jth master to the ith slave and from the jth slave to the ith master respectively. The power flow of the multilateral communication is,
ensuring that for all time the overall energy of the system must be positive. Similarly, the m-port network is passive iff t
n
∑ vsci j (t),
j=1
E(0) is the initial energy and is typically zero so Eqn. (3) can be reduced to t
vmi (t) and fsi (t) are sent with delays Tmi and Tsi , are governed by
(5)
0 i=1
A positive constant bi is introduced as in [17], and (8) is rewritten as P =
where fi (t) and vi (t) represent the force and velocity for the ith port. The idea of TDPC [5] is derived from the inequality (4).
1
bi
1
i=1
1 2 bi 1 f (t) + v2sci (t) − ( fsi + bi vsci )2 (t)] 2bi si 2 2bi n 1 2 1 (t) − ( fmci − bi vmi )2 (t) ∑ [ bi fmci 2b i i=1 1 1 2 ( fsi + bi vsci )2 (t) + f (t) +bi v2sci (t) − 2bi 2bi si bi bi 1 2 f (t) + v2mi (t) − v2sci (t)] − 2bi mci 2 2 +
= III. S WITCHING D ISSIPATION FOR M ULTILATERAL T ELEOPERATION S YSTEMS The complete multilateral teleoperation system can be shown as in Fig.1 in which b is the damping parameter, B and K are the proportional and derivative gains of the slave PD controller. [9] proposed the use of a passivity observer (PO) to calculate the power condition in (2). Passivity observers are designed at the master and slave sides to observe the power at every time instant and a combination of a switch and a damper is utilized to assure the stability of the system.˙ In contrast to bilateral teleoperation, the system for multilateral teleoperation is much more complex as it involves more masters and slaves and multiple operators which increase the number of transmitted signals over the communication network. The masters and slaves are represented by effective masses of m. A human operates the masters with force fh . The force that the slave controllers generate is fs . The force that an environment exerts on the slaves is fe . As in Fig.2, signals
n
2 (t) + v2mi (t) − ( fmci − bi vmi )2 (t) ∑ [ 2bi fmci 2 2bi
(9)
Using Eqns. (6) and (7), the last four terms in (9) become n
1
1
2 (t)] ∑ [ 2bi fsi2 (t) − 2bi fmci
i=1
n
n 1 2 [ fsi (t) − ( ∑ α f i j fs j (t − Tsi j ))2 ] i=1 2bi j=1
=∑ n
n 1 2 [ fsi (t) − n ∑ α 2f i j fs2j (t − Tsi j )] i=1 2bi j=1
≥∑ n
= ∑[ i=1
n
bi
n 1 2 2 1 2 fsi (t) − n ∑ α f ji fsi (t − Ts ji )] 2bi 2b j j=1
(10)
bi
∑ [ 2 v2mi (t) − 2 v2sci (t)]
i=1
n
n bi 2 [vmi (t) − ( ∑ αvi j vm j (t − Tmi j ))2 ] i=1 2 j=1
=∑ n
n bi 2 [vmi (t) − n ∑ αvi2 j v2m j (t − Tmi j )] i=1 2 j=1
≥∑
n n bj bi = ∑ [ v2mi (t) − n ∑ αv2ji v2mi (t − Tm ji )] 2 i=1 j=1 2
(11)
if Ts ji and Tm ji are constants. In the proposed framework, the selection of the weighting coefficients in (6) and (7) is based on a generalized form to satisfy Fig. 1. Multilateral teleoperation system with switching TDPC
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n 1 1 2 =n∑ α f ji , 2bi 2b j j=1
n bj bi = n ∑ αv2ji 2 j=1 2
(12)
3
This selection method of weighting coefficients helps us to further simplify (12) when all the damping coefficients are equal, we get n
n
n ∑ α 2f ji = 1
and
j=1
n ∑ αv2ji = 1 j=1
For specific medical applications, the selection of these weights is based on different factors as listed in [18]. The right hand side of the inequalities (10) and (11) is represented as, dE dt
=
n
∑[
i=1
n 1 2 1 2 2 fsi (t) − n ∑ α f ji fsi (t − Ts ji )] 2bi j=1 2b j
n n bj bi + ∑ [ v2mi (t) − n ∑ αv2ji v2mi (t − Tm ji )] (13) 2 i=1 j=1 2
E
n
n
∑ ∑[
=
t
i=1 j=1 t
+
t−Ts ji
t−Tm ji
n 2 2 α f (τ)dτ 2b j f ji si
nb j 2 2 α v (τ)dτ] 2 v ji mi
And the control laws for PCmi and PCsi , (passivity controllers at the master and slave sides), are defined as mi < 0 v b i f Pobs f pci = mi i mi ≥ 0 0 i f Pobs f si si < 0 i f Pobs (22) v pci = bi si ≥ 0 0 i f Pobs It can be noted that these control laws naturally avoid zero division situation. The proposed switching TDPC approach mi is negative, the shown in Fig.1 clearly shows that when Pobsv mi dampers bi are activated; when Pobsv is positive, the dampers bi si is negative, the dampers 1 are acare deactivated. When Pobsv bi si is positive, the dampers 1 are deactivated. tivated; when Pobsv bi To better show the detailed scheme for a simplified case with n = 2, the novel communication channel that is used for the switching TDPC for two masters and two slaves is shown in Fig.2 which can be extended to any number of masters and slaves without any loss of generality.
(14)
Substituting (10),(11) and (13) in (9), the power flow of the communication channel can be given as, P
≥
n
1
1
2 (t) − ( fmci − bi vmi )2 (t) ∑ [ bi fmci 2bi
i=1
+bi v2sci (t) −
1 dE ( fsi + bi vsci )2 (t)] + 2bi dt
(15)
The comparison of (1) and (15) gives us the total power dissipation of multilateral teleoperation with constant time delays as 1 2 1 fmci (t) − ( fmci − bvmi )2 (t) + bi v2sci (t) bi 2bi 1 − ( fsi + bi vsci )2 (t) (16) 2bi
Fig. 2. Multilateral teleoperation structure with two masters and two slaves
Pdiss < 0 shows the non-passiveness of the system, an indication that the action of passivity controller is needed. For realtime checking, (16)≥0 is divided in two sufficient conditions,
The multilateral teleoperation system shown in Fig.1 is simulated for the 2 masters 2 slaves (2M/2S) with the simulation parameters given in Table I. bi is the parameter that changes the effect of damping. A higher value of bi will make the system to be more damped. In contrast, a lower value of bi makes the system less damped. The selection of this parameter depends on the application. [19] and [20] suggest to choose this value equal to the derivative gain of PD controller of the slave (SCi ) side for proper impedance matching. In our simulations, we simply follow the same principle for impedance matching. The human operator model is taken from
Pdiss
≥
1 2 1 f (t) − ( fmci − bi vmi )2 (t) ≥ 0, bi mci 2bi 1 bi v2sci (t) − ( fsi + bi vsci )2 (t) ≥ 0. 2bi
(17) (18)
The passivity observers for the masters and the slaves are designed the similar way as in [6], 1 2 1 f (t) − ( fmci − bi vmi )2 (t), bi mci 2bi 1 si Pobsv = bi v2sci (t) − ( fsi + bi vsci )2 (t). 2bi
mi Pobsv =
IV. S IMULATION R ESULTS
(19)
TABLE I S IMULATION PARAMETERS
(20) Time delays Ratio bi Slave P gain
Thus, the power flow of the communication channel (15) can be rewritten as n dE mi si + Pobs ]+ P ≥ ∑ [Pobs i=1 ©2016 IEEE dt 978-1-5090-4059-9/16/$31.00
(21)
Value 0.5, 1 s 2.5 Ns/m 370 N/m
Endpoint mass m Slave D gain Sampling rate
Value 0.1 Kg 2.5 Ns/m 1 KHz
[21], which is a PD controller with gains 75 N/m and 50 Ns/m
71
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A. Without Switching Action The system is unstable without switching actions when a contact is made. Fig.3(a) clearly show that without the switching action oscillations are produced in the positions of the masters and slaves. Force values of the first pair of master and slave are shown in Fig.3(b). The contact force of first slave with the environment is shown in Fig.3(c). Such higher contact forces are not desirable in teleoperation systems. The energy values are given in Fig.3(d) and negative energy values clearly show that the system is unstable without switching dissipation. The simulation results are same for the second master slave pair so are not presented here. 0DVWHUDQG6ODYH3RVLWLRQV
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happens. Velocity signals for both the masters and slaves side are recorded in Figs.4(c) and 4(d). Passivity observers values for both the masters and their corresponding dissipation values are shown in Figs.5(a), 5(b), 5(c) and 5(d). It can be observed that in the start of the simulation, as the passivity observers at the master side observe the negative power values, the dampers get activated and start injecting the damping in the system to maintain the passivity. And as soon as the passivity observers start observing the positive power values, the dampers go in an inactive state as the switching action is no longer needed. Similarly, passivity observer values for both the salves and their corresponding dissipation values are shown in Figs.6(a), 6(b), 6(c) and 6(d).
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Fig. 7. Case 2: M/S positions and contact forces with delay of 1s and different αi
C. With Switching Action - Case 2 In this case, we simulated the multilateral teleoperation system with two masters and two slaves in the presence of a delay of 1s with a human input as a sinusoid. The sinusoidal input is simulated as sin(t) which is then fed to a PD controller as discussed earlier. The weighting coefficients are chosen to obey (12).
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V. C ONCLUSION
αv11 = α f 11 = αv22 = α f 22 = 0.7
Figs.7(a) and 7(b) show the positions of both the masters and slaves. It can be seen that the position tracking is satisfactory even in the presence of a large delay. The slaves track the positions of the masters. Environmental forces for both the slaves are presented in Figs.7(c) and 7(d) which are almost the same for both the slaves except the first significant spike as they move in a similar manner. Passivity observer values for the masters and their corresponding dissipation values are shown in Figs.8(a), 8(b), 8(c) and 8(d) respectively. Similarly, the passivity observer values for the slaves and their corresponding dissipation values are given in Figs.9(a), 9(b), 9(c) and 9(d) respectively. It is clear that the switching dissipation scheme is working perfectly as the passivity is guaranteed by the use of damping injection as needed. Force and velocity signals for both the masters and slaves transmitted in communication channel are shown in Figs.10(a), 10(b), 10(c) and 10(d) and the effect of non-matching weighting coefficients is evident from the results. The total energy of the system is given in Fig.11 which is positive all the time affirming the stability of the system.
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of zero division. We propose a switching TDPC approach to resolve the zero division making overall control scheme simpler and better. A novel communication channel is used with the weighting coefficients for two masters and two slaves. This communication channel structure can be extended to any number of masters and slaves without any loss of generality. The passivity of the system is ensured by the use of switching action to inject damping in the system. Numerical simulations results verify the successful implementation of the proposed scheme.
TDPC is considered a reliable scheme for the stability of the teleoperation systems incorporating the only drawback 73 978-1-5090-4059-9/16/$31.00 ©2016 IEEE
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This work was supported by NSERC, Canada.
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