Abstract— This paper considers the robust stability of uncertain teleoperation systems. Sufficient stability conditions are derived in terms of LMI by representing the teleoperation scheme in retarded form of time-delay systems. By choosing Lyapunov-Krasovski functional, a delay-independent robust stability criterion is presented. We show that the teleoperation system is stable and has good performance under specific LMI condition. With the given controller parameters, stability of system is guaranteed in the presence of any value of delay and admissible uncertainty. To evaluate the theoretical analysis, Numerical simulations are performed.
Key words: Teleoperation, Delay, Robust Stability, LMI I. INTRODUCTION
HE Internet is now serving billions of users worldwide.
It is now primarily used for communicating, information gathering, online shopping and other financial transactions, etc. But looking deeply, it is a good media to control physical systems from distance. This can be done in an open-loop arrangement in which the reference signal and not the control signal propagates through the net, or in a closed-loop arrangement for bilateral teleoperation, which requires transmitting the control signal through the network.
This exposes the control loop to the time delay of packets in the network. Time delay, which is inseparable from the Internet, may be not noticeable in open-loop control, because it may just oppose a delay in action. Yet in closed-loop configuration it can render the system unstable or degrade its performance, as in the case of teleoperation [1]. Several techniques have been proposed to compensate for this effect, such as a time forward observer developed for a supervisory control over the Internet ([2] and [3]), a position-based force-feedback scheme [4], scattering theory by Anderson and Spong [5] and a wave variable based technique developed by Niemeyer and Slotine [6].
A bilateral teleoperation system consists of the master which is manipulated by a human operator and the slave which is designed to track the master in a remote environment.
Information is transmitted between master and slave via communication channels. Internet is the most common communication channel used in this field. An overall block
Manuscript received November 23, 2010.
M. Azadegan is in Control of Communication Networks Lab, Tarbiat Modarres University, Tehran, Iran (e-mail: azadegan@ modares.ac.ir).
S. Ozgoli is with the Department of Control, Tarbiat Modarres University, Tehran, Iran (corresponding author; e-mail:
H. R. Taghirad is with the Department of Control, K. N. Toosi University of Technology, Tehran, Iran (e-mail: [email protected]).
diagram of teleoperation system is shown In Fig. 1. So far many researchers have employed position, velocity, force or impedance information to propose a variety of control structures, but most of these controllers can’t ensure both stability and transparency independent of time delay, as there is a tradeoff between these two goals. In an extensive survey presented by [14] and [15], a large amount of control architectures are reviewed.
In this paper, we present a delay-independent robust stability criteria for teleoperation system which is robust against network’s time-delay and any admissible uncertainties. With the given controller parameters, stability of system is guaranteed in the presence of any value of delay and admissible uncertainties.
This paper is organized as follows: In section 2 an overall description and modeling of teleoperation system is introduced. In Section 3, we represent the system equations in state-space model and present a robust controller against time-delay of network by using LMI. In section 4, we extend the proposed criteria to the robust case. Simulation results are presented in section 5 to validate properties of the proposed framework. Section 6 contains summary and concluding remarks.
II. TELE-OPERATION MODELING
For the sake of simplicity, the master and the slave have been modeled linear, as mass-damper systems, as shown in Fig. 2. This model is popular in this literature and is used in several articles (to name a few [19] and [20]).
Figure 1. Dynamic of master and slave
ݑǡ ݑ௦are the control signals,݂ is the force applied to the master by the operator and ݂ is the force exerted on the slave by the environment. The master and slave dynamics can be described by
ܯݒሶ ܤݒൌ ݑ ݂ (1) ܯ௦ݒሶ௦ ܤ௦ݒ௦ ൌ ݑ௦െ ݂ (2) Whereܯ,ܤ and ݒ denote inertia, damping coefficient
and velocity, respectively. Subscript '݉ ' and 'ݏ ' denote
Delay-Independent Robust Stability Analysis of Teleoperation
Masoumeh Azadegan, Sadjaad Ozgoli, and Hamid Reza Taghirad, Member, IEEE
T
Figure 2. Dynamic of master and slave
master and slave, respectively. The forces݂ and ݂ are given by
݂ൌ ݂כെ ܼݒ (3)
݂ൌ ܼݒ௦ (4) Where ݂כis the operator exogenous force. ܼand ܼ are
human and environment impedances that are supposed to be as mass, damper and spring
ܼൌ ܯݏ ܤ
௦
ܼൌ ܯݏ ܤ
௦ (5)
A block diagram of the bilateral teleoperation system is shown In Fig. 3. This framework has been first introduced by Spong in [17].
us
usd
T
T
master Cs slave
fh vm vmd vs
Figure 3. Block diagram of a bilateral control system
In the above structure, position of master (ݒሻ is transmitted to the slave and at slave side, we use a PD controller for control position. Also, the force information from slave is transmitted to the master side. So the dynamic characteristics of the master is described as follows
ܯݒሶ ܤݒൌ ݂െ ݑ௦ሺݐ െ ߬ሻ 6 Where ݑ௦ is the control input of the slave given by a PD controller as
ݑ௦ൌ ݇ሺݔௗെ ݔ௦ሻ ݇ሺݒௗെ ݒ௦ሻ 7 In which ݔௗ ൌ ݔሺݐ െ ܶሻǡ ݒௗൌ ݒሺݐ െ ܶሻ and ݇ǡ ݇ are controller gains.
III. CONTROLLER DESIGN
The goal of our design is to find gains of the PD controller using LMI framework such that the position and the force of master and slave track each other, in the presence of time-delay.
Substituting (3) to (7) into the master and slave dynamics equations, (1) and (2), closed-loop state equations of the system become as follows:
ݔሶሺݐሻ ൌ ܣݔሺݐሻ ܣௗݔௗሺݐሻ ܤݓ 8 where
ܣ ൌ ۏ ێ ێ ێ
ۍ Ͳ ͳ Ͳ Ͳ
െ
ெାெ െ
ெାெ Ͳ Ͳ
Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ െ ು
ெೞାெ െ ವ
ெೞାெے ۑ ۑ ۑ ې
ǡ
ܣௗൌ ۏ ێ ێ ێ
ۍ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ െ ು
ெାெ െ ವ
ெାெ
Ͳ Ͳ Ͳ Ͳ
െ ು
ெೞାெ െ ವ
ெೞାெ Ͳ Ͳ ے
ۑ ۑ ۑ ې
ǡ
ܤ ൌ ۏ ێ ێێ ۍ Ͳ
ଵ
ெାெ Ͳ
ଵ
ெାெے ۑ ۑۑ ې
Where ݓ ൌ ݂כ and the state-space vector ݔሺݐሻ א ܴis defined as:
ݔሺݐሻ ൌ ሾݔሺݐሻݒሺݐሻݔሺݐሻݒሺݐሻሿ (9) In which ݔൌ ݔௗെ ݔ௦
Theorem1: The teleoperation system (8) with control gains given at (11) is stable and has good performance for any constant delay of communication channel, if there exists symmetric positive definite matrices ܳᇱǡ ܵ and ܭ such that the LMI shown in (10) holds
ܴ ൌ ቈܣܳᇱ ܤܭ ܳᇱܣ் ܭ்ܤ் ܵ ܣଵܭ ܭ்ܣଵ் െܵ ൏ Ͳ 10
where
ܭ෩ ൌ ܭሺܳᇱሻିଵൌ ݇ ݇ Ͳ Ͳ
Ͳ Ͳ ݇ ݇൨ 11 Proof. Consider the following Lyapunov-Krasovski functional
ܸሺݔሻ ൌ ݔ்ሺݐሻܲݔሺݐሻ ௧ି௧ ݔ்ሺ߬ሻܳݔሺ߬ሻ݀߬ 12 It is clear that the Lyapunov function candidate is Positive Definite (PD). Taking the derivative of this function we have
ௗሺ௫ሻ
ௗ௧ ൌ ݔሶሺݐሻܲݔሺݐሻ ݔ்ሺݐሻܲݔሶሺݐሻ 13
ݔ்ሺݐሻܳݔሺݐሻ െ ݔ்ሺݐ െ ݄ሻܳݔሺݐ െ ݄ሻ For nominal conditions (݂כൌ Ͳ), substituting (8) in (13) yields
ܸሶ ൌ ݔ்
ݔ்ሺݐ െ ݄ሻ൨ ܴ ቂ ݔ
ݔሺݐ െ ݄ሻቃ 14 where
ܴ ൌ ܣ்ܲ ܲܣ ܳ ܲܣௗ
ܣௗ்ܲ െܳ൨
If ܴ ൏ Ͳ, which guarantees ܸሶ ൏ Ͳ, then tele-operation system is asymptotically stable, but the inequality ܴ ൏ Ͳ is not linear, so in order to decorate it in a linear format, by multiplying pre and post ܲିଵ , we can rewrite it in its dual form [16]
ܴᇱൌ ܣܳᇱ ܳᇱܣ் ܵ ܣௗܳᇱ
ܳᇱܣௗ் െܵ ൨ ൏ Ͳǡ 15 where
ܳᇱൌ ܲିଵǡ ܵ ൌ ܲିଵܳܲିଵ
To separate the matrices including controller gains which should be calculated, we factorized matrices ܣǡ ܣௗ as
ܣ ൌ ۏ ێێ
ۍ Ͳ ͳ Ͳ Ͳ
െெ
ାெ െெ
ାெ Ͳ Ͳ
Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳے
ۑۑ ې
ۏێ ێێ
ۍͲ Ͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ െெ ଵ
ೞାெےۑۑۑې
݇ ݇ Ͳ Ͳ
Ͳ Ͳ ݇ ݇൨ ൌ ܣ ෩16
ܣௗൌ ۏ ێێ ێ
ۍ Ͳ Ͳ
Ͳ െெ ଵ
ାெ
Ͳ Ͳ
െெ ଵ
ೞାெ Ͳ ے
ۑۑ ۑ ې
݇ ݇ Ͳ Ͳ Ͳ Ͳ ݇ ݇൨
$.෩ 17 By substituting (16), (17) in (15) we have
ܴᇱൌ
ܣܳᇱ ෩ܳᇱ ܳᇱܣ்
ܳᇱ൫෩൯் ܵ ଵ෩ܳᇱ
ܳᇱ൫ଵ෩൯ െܵ ൩ ൏ Ͳ 18
By choosing ܭ ൌ ܭ෩ܳᇱ
ܴᇱൌ
ቈܣܳᇱ ܤܭ ܳᇱܣ் ܭ்ܤ் ܵ ܣଵܭ
ܭ்ܣଵ் െܵ ൏ Ͳǡ 19 This inequality is linear and can be solved by LMI toolbox.
The gains of the controller then can be found from
ܭ෩ ൌ ܭሺܳᇱሻିଵ 20 This proves the theorem.
IV. ROBUST STABILITY ANALYSIS
Now, we consider uncertain teleoperation system as follows
ݔሶሺݐሻ ൌ ሺܣ ȟܣሻݔሺݐሻ ሺܣௗ ȟୢሻݔௗሺݐሻ ܤݓ (21 Where matrices ȟܣ and ȟୢ characterize the uncertainties in the system and satisfy the following assumption.
Assumption 1:
ȟܣ ൌ ܪܨሺݐሻܧǡ ȟୢൌ ܪௗܨௗሺݐሻܧௗ 22 Where ܪǡ ܧǡ ܪௗǡ and ܧௗ are known real constant matrices, and ܨሺݐሻ, ܨௗሺݐሻ are unknown matrix functions with Lebesgue-measurable elements and satisfy ܨሺݐሻ்ܨሺݐሻ ܫ andܨௗሺݐሻ்ܨௗሺݐሻ ܫ, in which ܫ is the identity matrix.
Before proceeding, we recall following lemma that will be used for the proof of the next theorem.
Lemma 1 [18]: For constant matrices ܪ andܧ, and scalarɂ Ͳ, the following inequality holds:
ܪܨܧ ܧ்ܨ்ܪு ߝܪܪ் ߝିଵܧ்ܧ 23
Where ܨ satisfiesܨሺݐሻ்ܨሺݐሻ ܫ.
Theorem2: The uncertain teleoperation system (21) with given control gains is stable and has good performance for any constant delay of communication channel and any uncertainty satisfying assumption 1, if there exists symmetric positive definite matrices ܳᇱ and ܵ, and positive scalars ߝ and ߝௗsuch that the LMI shown in (24) holds
ۏ ێێ ێ
ۍܳԢ ܳԢܶ ܵ ߝܪܪܶ ߝ݀ܪ݀ܪ݀ܶ ܣ݀ܳԢ ܧܳԢ Ͳ
ܳԢܣ݀ܶ െ Ͳ ܧ݀ܳԢ
ܳԢܧܶ Ͳ െߝ Ͳ
Ͳ ܳԢܧ݀ܶ Ͳ െߝ݀ےۑۑۑې
൏ Ͳ
24 Proof. By replacing ܣ and ܣௗ with ܣ ȟܣ and ܣௗ ȟୢ in (15) we have
ቈሺܣȟܣሻܳᇱ ܳᇱሺ ȟሻ ܵ ሺܣௗ ȟୢሻܳᇱ
ܳᇱሺୢ ȟୢሻ െܵ ൏ Ͳǡ 25
We can write (25) as follows
ቈሺܣȟܣሻܳᇱ ܳᇱሺ ȟሻ ܵ ୢܳᇱ
ܳᇱୢ െܵ ቈ Ͳ ȟୢԢ
ᇱȟୢ Ͳ ᇣᇧᇧᇧᇧᇤᇧᇧᇧᇧᇥ
൏ Ͳǡ 26
Substituting ȟୢ from (22), we have Ƚ ൌ Ͳ
ᇱୢ൨ ୢሾୢ Ͳሿ ቂୢ
Ͳቃ ୢሾͲ ୢԢሿ 26 By applying lemma 1 we have
Ƚ ߝ݀ୢ
Ͳ൨ ሾୢ Ͳሿ ߝ݀െͳ Ͳ
ᇱୢ൨ ሾͲ ୢԢሿ 27 in the other hand by substituting ȟ from (22) and applying lemma 1 we have
ȟܣܳᇱ ܳᇱȟ ɂ ɂିଵᇱ 28 By combination of (25) to (28) and applying Schur complement, we can write (24) and the theorem will be proved. The rest stages for designing controller, is similar to the previous section.
V. NUMERICAL EXAMPLES
To illustrate the validation of our proposed method, consider the teleoperation system with the following parameters,
ܯൌ ͳǡ ܤൌ ͳǤͷǡ ܯ௦ൌ ͳǡ ܤ௦ൌ ͳǤͷ 29 ܯൌ ͳǡ ܤൌ ͳǡ ܭൌ ʹͷ
ܯൌ ͳǡ ܤൌ ͳǡ ܭൌ ʹͷ ܪ ൌ ሾͲͲǤͳͲͲሿǡ ܧ ൌ ሾͳͲͲͲሿ ܪௗൌ ሾͲͲǤͳͲͲሿǡ ܧௗ ൌ ሾͳͲͲͲሿ
In the following simulation, we assumeܨሺݐሻ ൌ ݏ݅݊ݐ, and it can be seen that ԡܨሺݐሻԡ ͳǤUsing the LMI Toolbox of the MATLAB, the gains are calculated as follows
ܭൌ Ͳǡ ܭൌ ʹͲ 30 First assume that communication time delay is 100 ms.
Fig.4 and Fig.5 show the simulation results of the master and slave responses. As shown in Fig.4, by applying the designed control signal to the system, the slave position track the delayed master position. Fig.5 shows the slave and delayed master forces. From Fig.5, we can see that the force tracking is very good too. So we have achieved acceptable transparency
Figure 4. Position of the master with determined delay and slave, (߬ ൌ ͳͲͲ݉ݏሻ
Figure 5. Force of the master with determined time delay and slave, (߬ ൌ ͳͲͲ݉ݏሻ
Fig. 6 shows that position tracking is yet good, even when the time-delay is considered to be as much as 1000 ms. the same is true for force, shown in Fig. 7.
Figure 6. Position of the master with determined delay and slave, (߬ ൌ ͳͲͲͲ݉ݏ݁ܿሻ
0 1 2 3 4 5 6 7 8 9 10
-0.04 -0.02 0 0.02 0.04
Time(s)
X (m)
master delayed slave
0 1 2 3 4 5 6 7 8 9 10
-0.4 -0.2 0 0.2 0.4
Time(s)
F (N)
slave master delayed
0 1 2 3 4 5 6 7 8 9 10
-0.1 -0.05 0 0.05 0.1
Time(s)
X (m)
master delayed slave
Figure 7. Force of the master with determined time delay and slave, (߬ ൌ ͳͲͲͲ݉ݏ݁ܿሻ
It can be seen that the teleoperation system is stable and has good performance against any value of time delay and admissible uncertainties. The simulation results demonstrate the validity of the methodology presented by this paper.
VI. CONCLUSION
The problem of stabilizing an uncertain teleoperation system in the presence of time delay in the communication channel is addressed. To achieve robust stability while acceptable tracking performance, a controller is designed via LMI. The LMI condition is independent of the time-delay of the network. Numerical example is used to show feasibility and performance of the proposed controller.
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0 1 2 3 4 5 6 7 8 9 10
-2 -1 0 1 2
Time(s)
F (N)
slave master delayed
