Proceedings of 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems September 28 - October 2, 2004, Sendai, Japan
H2 Optimal Controller Design for Micro-Teleoperation with Delay Moussa Boukhnifer and Antoine Ferreira Laboratoire Vision et Robotique, ENSI de Bourges-Université d’Orléans, 10 Bld. Lahitolle, 18020 Bourges, France,
[email protected]
Abstract: This study aims to develop a force-reflecting macromicro teleoperator with different scaled worlds. A bilateral control system for scaled teleoperation provides the human operator with a feel of the task at the micro-scale. However, reliable and efficient tele-micromanipulation systems with haptic feedback over the Internet face to strong problems due to the nonlinear nature of microenvironment and time-varying delays in communication lines. A robust bilateral controller design framework using H2-optimal control approach is proposed. A comparative study for assessing the robustness against time-varying delays is performed through two different designs, i.e., a µ-synthesis framework and a Padé approximation. The proposed approaches allow a convenient means to tradeoff the robustness for a pre-specified time-delay margin. The validity of the proposed method is demonstrated by simulations. Keywords: Micro-teleoperation, H2-control, time delay. 1.
INTRODUCTION
These methods have proved their robustness in presence of time-delays less than one second. However, their application in presence of different-scaled worlds with scaling effects problems have never been validated. As stated above, the stability-performance trade-off is the main determinant of the control design for micro teleoperation systems. Recent controller designs using robust H∞ control theory and µ-synthesis/analysis are very effective for macro scale teleoperation [4],[5],[6], [7]. None of the above papers solves the problem of robust stability and performance of bilateral micromanipulation in the presence of a multiobjective problem: time-varying delay, variation of force scaling, and uncertainties in the master-slave models. In a previous study, the design of H∞ controller with on-line control of the scaling parameters for a network-based force-reflecting teleoperation has been examined [8]. In this work, we extend our previous approach to H2-optimal control which is directly obtained from the Riccati standard formulation [9],[10],[11]. Suboptimal H∞ controllers are more difficult to characterize (in the norm computation problem) than optimal ones. This is one major difference between the H∞ and H2 approaches which lead to difficulties in their practical implementation and synthesis. The paper is organized as follows. In section 3, the design of the H2 –optimal controller with compensation of time-delay is discussed. In Section 4, the developed micro-teleoperation system is introduced, and various simulations are given.
The field of micromanipulation is still in its initialisation stage and a wide variety of applications are emerging – ranging from high-precision assembly of mechanical microcomponents from MEMS industry to the handling of cells in medical or biological applications. However, to make these systems efficient and safe, multimedia information should be provided to the operator, which transfers human feeling to remote microenvironment through bilateral teleoperation. However, by their definition, teleoperation systems frequently experience significant time-delays in the communications between local and remote sites. Untreated, even small delays (in the order of 2. THE STANDARD H2 PROBLEM several hundred milliseconds) can lead to instabilities of current micro-teleoperation systems due to unwanted power Consider the system described by the block diagram: generation in the communications. In addition, the design of a bilateral controller for efficient and robust W Z micromanipulation requires the careful analysis of scaling G(s) between micro and macro environments’ forces. Actually, the main challenge of success of Internet-based remote U K sensing and manipulation in microenvironment concerns Y mainly the optimum design of robust bilateral controllers. The commonly proposed approaches to deal with bilateral Fig.1: Standard H2 Problem teleoperation with time-varying delays are mainly based on the scattering theory formalism [1], wave variable concept where G is the generalized plant and K is the controller. [2], or sliding mode controller [3].
0-7803-8463-6/04/$20.00 ©2004 IEEE
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Only finite-dimensional linear time invariant (LTI) sytems and controllers will be considered in this paper. The generalized plant G contains what is usually called the plant in a control problem plus all weithing functions. The signal w contains all external inputs, including disturbances, sensor noise, and commands; the output z is an error signal; y is the measured variables; and u is the control input. The diagram is also referred to as a linear fractional transformation (LFT) on K, and G is called the coefficient matrix for the LFT. The resulting closed-loop transfer function from w to z is denoted by Tzw. The problem of H2 standard is to synthesise a controller K which stabilises the system G and minimize the norm H2 of Tzw. The transfer matrix G is taken to be of the form [9] A C1 C2
B2 0 D12 D21 0
where X
and F2 =
= Ric ( H 2 ),
− B2' X 2 ,
= Gc B1
2 2
Y2C2' 0
(2)
xs
Micromanipulator
Force feedback
fs
fhd Gripper command
Fig.2: The video and force feedback control scheme of the internetbased microteleoperation system used in the experiment.
(6)
fh = n f fs
2
+ F2G f
2
where xm and xs are the position command from the master and the slave position, respectively; fh and fs are the operator force command and the external force from the slave to the master. The transfer functions of the master and the slave are represented by Pm(s) and Ps(s) such that:
L2 = −Y2C2'
I 0 ,
(4)
Pm (s ) =
A + L2 C 2 G f (s) = I
B1 + L2 D 21 0 The Hamilotonian matrices are defined as: A A' − B2 B2' , J 2 = H2 = − C1'C1 − A − B1B1'
Figure 3 illustrates the block diagram of the bilateral controller system with the varying time-delay parameters and fixed scaling factors. A generalized scaled bilateral manipulator is characterized by a fixed geometric/ kinematic scaling factor, np, and a fixed force scaling factor, nf. Using the scaling factors, the relationships between the master and slave in position and force are defined as: xm = n p xs
Y 2 = Ric ( J 2 )
A + B 2 F2 Gc (s) = C1 + D12 F2
Master command
I N T E R N E T
A. Bilateral System
− L2 = A + B2 F2 + L2C2 ' 0 − B2 X 2
2
Force feedback
fh
controller is :
2 2
xm
(1)
The problem is to find an admissible controller K which minimizes Tzw 2 . Recall that the unique H 2 optimal
min Tzw
Vision feedabck
Vision feedback
fm
B1 ' 0 D21 = I D21
^ K 2 = A2 F 2 and
Remote Operator
B1
the following assumptions are made i) (A,B1) is stabilizable and (C1,A) is detectable ii) (A,B2) is stabilizable and (C2,A) is detectable. ' iii) D12 [C1D12 ] = [0 I ]. iv)
feedback and video feedback. The controller to be used for the tele-micromanipulation system needs to satisfy some requirements such as stability under specified environment, tuning of scaling parameters, communication time-delay and modeling errors due to the non-linear behavior of nonidentical master and slave robots. This section analyzes H2 design framework for optimum bilateral controller design.
− C2' C2 (5) − A
3. DESIGN OF H2 -OPTIMIZATION AND µ-SYNTHESIS CONTROLLER The general structure of the developed bilateral microteleoperation scheme is shown in Fig.2, where the operator sends position commands and receives force
225
1 mm s 2 + k m s + bm
, Ps (s ) =
1
(7)
ms s 2 + k s s + bs
where mm , ms denote the mass of master and slave, km and ks the compliance coefficients and bm and bs denote the viscosity coefficients, The terms, fh and fs are the operation force and the reaction force while the output master and slave positions are, respectively, xm and xs. As the slave is in contact with the environment, we assume that Se ( Se = Ke / s ) is the nominal stiffness of a spring. Gs represents the slave which is in contact with the environment Se and the velocity of the slave is controlled by Ks. The time-delay from the master to the slave, and vice-versa, are represented by e-sT1 and e-sT2. Consider H2 optimal design of controllers for the master and slave for
fh
G m ( s) +
+
-
-
where Ws1,Ws2 , Ws3 and Ws4 are frequency dependent weighting matrices. Note that the design and implementation of Ks does not affect the performance of the master system. The weighting matrices for the design are as follows [12]: s s + 100 , Ws 2 = as 2 , Ws1 = a s1 2 s + 1000 s + 10s + 0.1 s Ws 2 = as 3 , Ws 4 = 1 0.001s + 1
xm
Pm Km
e
-s τ 1
e -s τ 2
Communication
nf
n p -1
Bilateral Controller +
K
+
Ks
fs
+
-
-
Ps
D. Time-Delay as External Perturbation
xs
As proposed by Leung et al. [7], the system of Fig.3 can be represented as Fig.4 where the blocks representing the time delay e-sT (master to slave to master) of T seconds in communication channel can be reconfigured as a perturbation ∆T to the bilateral system. Let :
Environment
S
e
G s (s)
Fig.3: Bilateral controller with communication channel and on-line varying scaling factors.
free motion. Let Km and Ks denote free motion controllers for the master and slave, respectively. B. Robust Controller Design of Master For the master controller design, the specifications are taken to be as follows: 1) The master position should track the reference fh so ( Xm- fh ) should be included in Z. 2) The plant Um should not exceed a prespecified saturation limit, so Um should be included in Z. fh W ( f − xm ) z = m1 h , w= Wm 2U m d y = xm + Wm3 d , u = Um
(8)
)
x + W s 3 d s1 y= m xs + Ws 4 d s 2
,
,
To reduce the conservatism, the maximum magnitude of e − jωT − 1 equals 2, that is, ∆ T = 2 , for every T >0. The ∞
system is conservative since it would be compensating for all perturbations of norm < 2, not just time delay. Then, a new perturbation ∆m is defined as:
fh
(11)
U
_
+
fh
_
+
m
U
m
Gm
Gm
xm ∆T
e -sT
C. Robust Controller Design of Slave For the slave controller design, the specifications are taken to be as follows: 1) The slave position Xs should track the master position Xm (so Xm- Xs should be included in Z). 2) The control Us should not exceed a pre-specified saturation limit, so Us should be included in Z. Noises ds1 and ds2 are introduced to regularize the problem.
(
In order to lump the delays into one block e-sT, the left delay block WT(s)=e-sT1 of Fig.3 is moved around the loop to the forward path of the loop. The block e-sT represents the time delay of T (T1+T2).
xm
s s + 10 , Wm 2 = am 2 , Wm1 = am1 2 0.001s + 1 s + 10 s + 0.01 s Wm3 = am3 0.001s + 1
W x − x s z = s1 m W s 2τ s1
(10)
∆ m = ∆T Gm
The H2 design procedure may be carried out using MATLAB. In order to satisfy the above mentioned conditions, Wm1, Wm2 and Wm3 are frequency dependent weighting matrices as follows [12]:
fh w = d s1 d s 2
∆ T (s ) = e − sT − 1 .
+
+
Fig.4: Perturbation model of time delay.
Since Gm is strictly proper by adequate design, there is a bandpass filter Wc such as: Wc −1∆ m
∞
