TRACKING CONTROL OF MULTIPLE MOBILE ROBOTS: A CASE STUDY OF INTER-ROBOT COLLISION-FREE PROBLEM

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1 265 Asian Journal of Control, Vol. 4, No. 3, pp , September 22 TRACKING CONTROL OF MULTIPLE MOBILE ROBOTS: A CASE STUDY OF INTER-ROBOT COLLISION-FREE PROBLEM Jurachart Jongusuk an Tsutomu Mita ABSTRACT Tracking control problem of multiple mobile robots is consiere. Our system is compose of a reference an two follower robots of unicycle type, which have their own priority numbers. The purpose is to control the two followers so that the reference is tracke with arbitrary esire clearance an also to avoi an inter-robot collision, which can occur ranomly uring the control process subjecte to communication range limitation. We introuce two switching controllers an appropriately using their convergence properties leaing to the collision-free movement. Simulation results prove efficiency of our control techniques. KeyWors: Multiple mobile robots, feeback, collision, tracking control, terminal attractor. I. INTRODUCTION Control of multiple mobile robots has been subject of a consierable research effort over the last few years. The rapi progress in this fiel has been the interplay of systems, theories an problems [1]. Three examples are introuce as the recent research trens, namely foraging, box-pushing an traffic control, foraging aresses the problem of learning an istribute algorithm; box-pushing focuses particularly on cooperative manipulation; an traffic control aresses the problem occurring when multiple robots move in a common environment that typically attempt to avoi collisions. Despite the vast amount of papers publishe on the collision avoiance motivate by the case of traffic control, the majority has concentrate on centralize path an motion plannings, which mostly utilize fee-forwar control in orer to accomplish a goal, leaing also to a computational time consumption, see for examples Arkin [2] an Lavalle [3], while less attention has been pai to control of the multiple mobile robots by a concept of feeback control. Desai [5] has stuie the problem of control an formation where a notation of l-l control is originally presente, however, the issue of sensor capability is absent. l-l control gives a motivation of using relative ynamics in the system, which will be further extene in our paper. In this paper, we focus on tasks in which the multiple mobile robots are require to follow a given trajectory in Manuscript receive January 3, 21; accepte July 18, 21. The authors are with Department of Control an Systems Engineering, Tokyo Institute of Technology, Japan. leaer-follower formation while avoiing collision among themselves. In aition, because of the high-banwith of inter-robot communication, consieration of communication capability becomes necessary. Here, a limite range of communication is consiere. The problem of control an coorination for multiple mobile robots is formulate using ecentralize controllers. We assume that there are no obstacle in the work space, which simplifies the problem to avoiing inter-robot collision. We also propose a simple technique that helps in ecision process of collision avoiance base on geometric analysis. This proposition correspons to the issue of Latombe [4] state that collision avoiance is an intrinsically geometric problem in configuration space. The paper is organize as follows: Section 2 escribes moel of the wheele mobile robot of a unicycle type an formulates the tracking problem for a group of three mobile robots without the presence of obstacle. Section 3 introuces a simple collision etection moel, virtual robot tracking control (VR control) as a main controller an collision avoiance control (l-l control), which is the extene version of the original work of Desai [5]. Analysis of each controllers properties an combination is also provie, which leas to the main issue. Section 4 contains simulation results that illustrate the performance of the propose control strategies. The paper conclues with the summary of results an recommenations for further research. II. PROBLEM FORMULATION A unicycle-type mobile robot, of which kinematic moel is efine by

2 Asian Journal of Control, Vol. 4, No. 3, September q i = B i u i = cos θ i sin θ i 1 u i (1) where q i =[x i, y i, θ i ] T an u i =[υ i, ω i ] T is consiere. This moel captures the characteristic of a class of restricte mobile robot which is what we nee to investigate. The robot are assume to satisfy pure rolling an non-slipping conitions, which lea to nonholonomic velocity constraints, x i sin θ i y i cos θ i =, (2) x i cos θ i + y i sin θ i = υ i (3) III. CONTROLLER DESIGN AND ANALYSIS We arrive here with assumptions efine in the last section. We will first apply a VR tracking control approach 1 to the system to complete the requirement in item (1) of the problem while aing an ability of collision checking. The check is one in a sufficiently small sampling time. When a robot etects possibility to collie with another, it checks if its priority number is higher (small value) or not. If not, it has to change the control to l-l control 2 in orer to avoi the collision an complete the requirement of item (2) of the problem. It is easy to unerstan the algorithm above by the flowchart shown in Fig Assumptions We summarize here all the assumptions use in this paper. (i) System is homogeneous, i.e., all robots are of the same moel escribe in (1) an satisfy the velocity constraints, (2) an (3). (ii) The workspace (xy-plane) is flat an contains no obstacle. This comforts us in fining solution focusing on only inter-robot collision problem. (iii) The reference robot follows a smooth trajectory, an maintains positive velocity. We nee this assumption in orer to guarantee that our problem is tracking not stabilizing control, see [7]. (iv) Each follower robot is inexe by ifferent priority number. Then, system becomes hierachy an we can manage the behavior of each robot uring switching control as will be seen in algorithm in Section 3. (v) Each robot can extract any necessary information from its communication equipment with negligible elay. In practical case, high-spee ceiling camera shoul be use to track the absolute position of each robot. 2.2 Problem statement To keep our investigation on the way, the problem is preliminarily state as follows. Statement 1. Giving initial positions an orientations, q i () for the follower robot i, an the motion of reference robot, it is require that control law u i gives a motion satisfying assumptions (i)-(v) such that, as t, 1 formation is establishe, 2. no collision occurs among robot i an any robot j, an 4. a whole motion satisfies the limitation of communication range. 1 See etails in Section See etails in Section 3.3. Fig. 1. Algorithm flowchart. Referring to this algorithm, we nee to esign a simple collision etecting technique an introuce two feeback control approaches, VR tracking control an l-l control, that also fulfill the requirement in item (3) of the problem, which are the keys of the main issue. 3.1 Detecting collision A simple technique use to etect the collision between any two robots can be realize by moeling both

3 267 J. Jongusuk an T. Mita: Tracking Control of Multiple Mobile Robots: A Case Stuy of Inter-Robot robots to be covere with ouble circles centering at the control points, referring to Fig. 2. The soli ones cover the entire robots with raius D while the outer otte circles with raius D + is esigne for the require clearances between robots. For convenience, we efine the istance between two robots in the following efinition. Definition 1. Let (x i, y i ) an (x j, y j ) enote the control points of robot i an robot j, respectively. Distance between robots is efine by ρ ij = (x i x j ) 2 +(y i y j ) 2 (4) Then, we efine a function that can be use to check the collision between robots i an j as follows. f ij = ρ ij2 4(D + ) 2 f ij > safety collision (5) Here, f ij mathematically represents a conition for item (2) of the problem statement. Note that the choice of shoul be esigne with appropriate value. A large increases safety level but ecreases the smoothness of the robot because it etects the possibility of collision more often, which yiels more switching of control inputs. 3.2 Virtual robot tracking control This approach offers a tracking control with arbitrary clearance between robots, which is, in fact, necessary for navigating the multiple mobile robots while maintaining the formation as we nee. However, the collision that can occur uring transient state of the control is not consiere. We give the etail of this approach in this section. To easily separate many robot configuration symbols, we will use the following efinitions thereafter. q r = [x r, y r, θ r ] T refers to a reference robot, q i = [x i, y i, θ i ] T, i = 1, 2,, refers to follower robot i an q υi =[x υi, y υi, θ υi ] T is for VR of follower robot i. In orer to separate follower from reference robot avoiing collision, a concept of VR is efine with aitional objective to force error parameters to zero when system approaches final time, as follows. Definition 2. Virtual robot (VR): a hypothetical robot whose orientation is ientical to that of its corresponing follower robot, but position is place apart from it by the preefine r l clearances. The symbols l, r enote longituinal clearance an clearance along rear wheel axis, respectively. The relation between VR an the follower robot is written as follows. x υi = x i r sin θ i + l cos θ i y υi = y i + r cos θ i + l sin θ i θ υi = θ i (6) Note that the above equation is satisfie when the follower is on VR s right-han sie. From (6) an assumptions (2), (3), we can erive the kinematic moel of VR as follows. q υi = cos θ i r cos θ i l sin θ i sin θ i r sin θ i + l cos θ i u i = 1 B vi 1 u i (7) See proof in appenix A. Figure 3 escribes the parameters of VR on xy-plane. A stanar technique of I/O linearization is use to generate a control law for each follower [9]. A property inherite in this controller is escribe by the following statement. Property 1. VR tracking control gives solution that exponentially converges in an internal shape x e an y e. D Fig. 2. Collision avoiance moel. Fig. 3. VR tracking moel.

4 Asian Journal of Control, Vol. 4, No. 3, September Proof. An error moel is formulate as z ei = z υi z r = B vi u i b r υ r (8) where B vi is the matrix efine in (7), υ r enotes reference robot s translational velocity an u i enotes follower robot s velocity vector, an z υi = x υi y υi, z r = x r y r, b r = cos θ r sin θ r After applying I/O linearization, we obtain u i = B vi 1 (b r υ r λz ei ) (9) where λ enotes positive-constant iagonal matrix. The solution of this controlle system, is z ei = e λt z ei () (1) which itself tells that the solution exponentially converges in an internal shape x e an y e. Moreover, solution is wellefine because parameter l is esigne not to be zero, an hence, B vi 1 is non-singular. Curve in Fig. 3 illustrates this property. In aition, zero ynamics given in term of θ i must be prove to be stable in orer to guarantee the use of control law in (9). The ifferential equation for θ i is given by θ i =( λ 1x ei υ r cos θ r l or can be written in the form ) sin θ i +( λ 2y ei υ r sin θ r ) cos θ l i (11) θ i = k 1 sin (θ i + k 2 ) (12) Corollary 1. Property 1 ensures that applying (9) to follower robot will automatically avoi collision with reference robot if the conition below is satisfie, ρ ri (t ) > 2 ( + D) (15) where ρ ri enotes istance between reference an follower robot i an t is initial time. Proof. Recall that (1) ensures exponential convergence of ρ ri. Also, the preetermine r l clearance implies that ρ ri r 2 + l 2. Obviously, if robots o not with collision, i.e. (15) is satisfie, the motions of follower robots are free of collision with reference robot. Remark. Corollary 1 only guarantees collision-free motion between reference an one of follower robot. Hence, there is no guarantee of the motion between follower robots themselves. 3.3 l-l control The aim of this control is to maintain the esire lengths, l 13 an l 23 of consiere robot (Robot 3 in Fig. 4) from its two leaers. Referring to the main algorithm, at any collision occurrence, the lowest priority robot will be switche from VR tracking control to this control. l-l control is originally presente in Desai [5]; however, the convergence of states is not establishe within finite time an is aime for maintaining formation. Here, we utilize l-l control as collision avoiance controller, which requires that formation must be establishe within finite time. The kinematic equations for the Robot 3 is given as follows, where k 1 = 1 l (λ 1 x e υ r cos θ r ) 2 +( λ 2 y e + υ r sin θ r ) 2 k 2 = arctan λ 2 y e + υ r sin θ r λ 1 x e υ r cos θ r The trajectory, θ e w i = k 2 arcsin r is shown to be k 1 exponentially stable by linearization [1] of (12) as follows. θ i = k 1 cos (θ i + k 2 ) θ i = θ i e θ i (13) = k 1 2 w r 2 θ i (14) 1 a 2 an enotes a evia- Note that cos(arcsin (a)) = tion operator. Fig. 4. Notation for l-l control moel.

5 269 J. Jongusuk an T. Mita: Tracking Control of Multiple Mobile Robots: A Case Stuy of Inter-Robot an i 13 i 23 = cos γ 1 D sin γ 1 cos γ 2 D sin γ 2 υ 3 ω 3 υ 1 cos ψ 13 υ 2 cos ψ 23 = B ll u 3 υ ll (16) Again, zero ynamics in term of θ 3 can be prove stable, clearly when two leaers follows parallel straight lines with the same velocities or υ 1 = υ 2 = υ >, ω 1 = ω 2 =, θ 1 () = θ 2 () = θ. From (16), (19), we have θ 3 = υ cos ψ 13 cos (ψ 23 + θ θ 3 ) D sin (ψ 13 ψ 23 ) θ 3 = ω 3 where γ i = θ i + ψ i3 θ 3, (i = 1, 2). Remark. In (16), B ll is singular only if γ 2 γ 1 = nπ, (n =, 1,, 1, ) which etermines the three robots lie on the same line connecting them. We will refer it to singular line later in the context. For the purpose of collision avoiance, a property below is realize by I/O linearization. Property 2. l-l control gives solution that monotonically converges in an internal shape l 13 an l 23 within finite time. Proof. Require that the controlle variables l 13 an l 23 monotonically converge to l 13 an l 23, respectively, within finite time T r, we set i 13 = α 1 (l 13 i 23 α 2 (l 23 l 13 ) p q l 23 ) p q (17) = α l e (18) which is ientical to a terminal attractor moel where (p, q) can be, for example, set to (2, 3). See Appenix B for etails. By arbitrary setting reaching time T r, we obtain a formula for fining α 1, α 2 as follows. α 1 = sign (l 13 l 13 ()) l 1 13 l 13 () 3 /( T r 3 ) α 2 = sign (l 23 l 23 ()) l 1 23 l 23 () 3 /( T r 3 ) The sign( ) an are use to avoi complex solutions. We can generate a feeback control law as follows. u 3 = B ll 1 (υ ll + α l e ), t T r (19) The solution of this controlle system is l e = (l i3 l i3 ()) 1 3 α 3 i 3 t, i =1,2, (2) which proves the properties. Note that the term in parenthesis is linear. υ cos ψ 23cos (ψ 13 + θ θ 3 ) D sin (ψ 13 ψ 23 ) = υ D sin (θ θ 3 ) (21) Lyapunov caniate, V = 1 cos(θ θ 3 ) guarantees the stability of (21). In case of more complicate motions of two leaers, the stability is guarantee by observing that ρ 12 exponentially ecreases accoring to Property 1. This means l 13, l 23 are always etermine or the only singularity γ 1 γ 2 = is not passe. Remark. Note that (2) also gives an interesting convergent motion in an internal shape l 13 an l 23. It matches with our require motion when collision is etecte, i.e., when Robot 3 gets close to the two leaers, with the control (19), it tens to monotonically separate from the leaers avoiing collision. 3.4 Combination of VR an l-l control We can formulate a control of the whole system using the combination of three techniques as state in Section The flow of control process is formerly presente in this section; however, conitions in which the control process can be performe properly are not analyze. We summarize all limitations occurs from the combination as follows. 1) r 2 + l 2 > 4(D + ) 2 (22) 2) l i3 >2(D + ), i =1,2 (23) 3) ρ 12 < l 13 + l 23 (24) Conition (22) is extracte from the first requirement in item (1) of the problem statement an the esign of collision etection in Section 3.1. Conition (23) comes from the restriction of the robots not to get closer than safety regions when l-l control is performe. Again, in l- l control phase, l 13 an l 23 must be esigne in such a way that three robots can form a triangle formation at the en of control, which yiels (24). As state in Section 3.3, when three robots lie on the same line connecting them, the control law u 3 in (19) is not etermine. In aition, the requirement in item (3) of the problem states that communication range is limite, namely,

6 Asian Journal of Control, Vol. 4, No. 3, September R as maximum range. These two facts enable us to efine the accessible area, shae ones in Fig. 5 an Fig. 6, where the low priority robot can move on uring collision avoiance. Accessible area comes from the intersection between the circle R an the area evie by singular line in which the lowest priority robot exists. Hence, in l-l control phase, the esign of l 13 an l 23 (equivalent to the esign of P x ) is simply one by setting the values within accessible area. Physically, P x is the point where follower is suppose to be move to at the en of l-l control. In fact, the esign of P x can be classifie into two cases: esire target is outsie accessible area, Fig. 5, an insie the area, Fig. 6. The two cases can be istinguishe by the conition below. (For the sake of simplicity, we assume that reference robot moves along the positive y-irection.) if θ a < θ b Target is outsie accessible area else Target is insie accessible area where θ a, θ b [, 2π) are measure in clockwise fashion. θ a is an angle measure from follower 2 to follower 1 centere at reference robot while θ b is measure from follower 2 to its formation target TG 2. Remark. In case like Fig. 5, it is strongly recommene to efine P x behin follower 1 to ensure that collision will harly or not repeat after switching to VR tracking control. 3.5 Main issue Finally, we summarize all results in this paper as a main issue in theorem below. Theorem 1. A system of three mobile robots whose moels are efine by (1), operating in an environment escribe by assumptions (i)-(v) an controlle by VR tracking control law: u i in (9) as a formation controller, an l-l control law: u 3 in (19) as a collision controller, gives a motion of the form, Reference Follower 2 ρ ri R, i =1,2 (25) while avoiing inter-robot collision an maintaining a preetermine formation in the steay state. R is positive scalar corresponing to maximum communication length of reference robot. TG 2 θ b TG 1 θ a P x Proof. It can be simply checke that the istribution of the three robots is guarantee to be insie a circle of raius R centere at reference robot by Property 1, Property 2 an esign process in Section 3.4 along the motion if the two follower robots are initially in this circle. Follower 1 R Fig. 5. Target TG 2 is outsie accessible area : θ a < θ b. Reference Follower 1 IV. SIMULATION RESULTS The aim of these simulations is to valiate our tracking control metho for a multiple mobile robot (threerobot case). Our goal is to make the three robots form a leaer-wing motion. Two cases when target is initially locate outsie an insie accessible area are shown. We from the former case where initial parameters are set as follows. θ b θ a Common: Tr =1.5s, D = 2.25, = 1, R = 25 Reference: q r () = [,, π 2 ]T, u r = [5, ] T TG 2 TG 1 P x Follower 1: q 1 () = [6, 6, π 2 ]T, (r, l) = (6, 6) Follower 2: q 2 () = [1, 2, 3π 4 ]T, (r, l) = ( 6, 6) Follower 2 Fig.6. Target TG 2 is insie accessible area: θ a > θ b. R Assume also that reference robot has the highest priority an follower robot 2 has the lowest one. From Fig. 8, we see that motion of follower robot 2 is kept within the maximum communication range R with

7 271 J. Jongusuk an T. Mita: Tracking Control of Multiple Mobile Robots: A Case Stuy of Inter-Robot a esire formation establishe in the steay state. See snapshot in Fig. 7 for an entire motion of system. Accoring to Table 1, four of collision occurrences between the two followers are foun. The first two occurences fall to the case when target is outsie accessible area, while the successive ones fall to the case when target is insie the area. The number of collision can be ecrease by moifying l 13, l 23 an T r in each occurrence, an here we simply use constant values. On the other han, the case when target is initially locate insie accessible area is one by setting, Follower 1: q1() = [8, 3, 5π 4 ]T 8 Reference Follower 2: q2() = [8, 9, 5π 8 ]T Follower 2 Follower 1 where other parameters are ientical to the last example. Again, the motion of follower robot 2 (Figs. 9, 1) is guarantee to be insie communication range. The esign of l 13, l 23 for this case gives only one collision, referring to Table 2. y Reference 1 6 Follower 2 Follower x Fig. 7. Snapshot of motion (target is outsie accessible area) 3 y Relative istance r-l clearance Max. Range R x Fig. 9. Snapshot of motion (target is insie accessible area) r-irection l-irection time Fig. 8. Distance from reference to follower 2 in r l irection when target is outsie. Table 1. Collision information (Target outsie). Relative istance r-l clearance Max. Range R 6 No. Info. t l 13 l 23 case outsie outsie insie insie -2 r-irection l-irection time Fig. 1. Distance from reference to follower 2 in r l irection when target is insie.

8 Asian Journal of Control, Vol. 4, No. 3, September Table 2. Collision information (Target insie). No. Info. t l 13 l 23 case insie V. CONCLUSION In this paper, we have stuie strategies for controlling multiple mobile robots (three-robot case) focuse on a collision-free movement. The follower robots track the gait of reference robot using virtual robot tracking control. The extene l-l control technique use to avoi the collision of the three-tuple robots is applie uring finite time interval in such a way that the lowest priority robot can move away from the others. After the collision-free conition is establishe, the controller is switche back to the virtual robot tracking control an maintain the esire formation an so on. The convergences an stabilities of both controllers are throughly stuie, leaing to the main issue in Theorem 1. The problem arises from the nonrobustness of the terminal attractor function use in l-l control, an hence before we move to real experiment, the function shoul be substitute by the more robust one. We are currently working with this issue. We also plan to enlarge our framework to the problems of obstacle avoiance an formation changes, of which we believe the successful strategies are strongly base on the geometric calculation as state in Latombe [4]. Accoring to Theorem 1, the extension of our work to the system of more than three robots is possible particularly when we have a number of groups of three-tuple robots to be controlle. These topics are the subject of our future work. APPENDIX A. Proof of VR kinematic (7) Proof. Using real robot kinematic equation (1) an time erivative of a right-han moel (6), we obtain x υ = x r cos θθ l sin θθ = υ cos θ + ω( r cos θ lsinθ) (A1) y υ = y r sin θθ l cos θθ θ υ = θ = υ sin θ + ω( r sin θ + lcosθ) (A2) = ω (A3) Rewrite the above equations in riftless matrix form yiels (7). B. Terminal attractor To establish the convergence of z ei in arbitrary finite time, T r, consier the following lemma. Lemma 1. The origin of a ifferential equation p z = λz q, λ > is a terminal attractor with a finite reaching time p q (A4) T r = z()1 λ(1 p, p q ) q (, 1) (A5) where p, q I +, an I + = {positive integer} Proof. Direct integration of (A4) yiels z(t) 1 pq = z() 1 pq (1 p q )λt Requiring z(t r ) = implies formula (A5). (A6) Note that T r can be chosen arbitrary from the choice of λ. We shoul take care of the choice p, q, which is suggeste in proposition below. Proposition 1. The choice of p an q in (A4) shoul be selecte such that (p, q) Ω. Ω ={(p, q) p < q, q is o } (A7) Accoring to (A4) an (A5), p q an 1 p q must be esigne to be o-root in orer to avoi 2i-root of negative values. In aition with the assumption that p q <1, Ω is establishe. ACKNOWLEDGEMENTS This work is part of Grant-in-Ai for COE Research Project supporte by the Ministry of Eucation, Science, Sports an Culture, Japan. REFERENCE 1. Cao, Y.U., A.S. Fukunaga an A.B. Kahng, Cooperative Mobile Robotics, Autonomous Robots, Vol. 4, pp (1997). 2. Balch, T. an R. Arkin, Communication in Reactive Multiagent Robotic Systems, Autonomous Robots, Vol. 1, pp (1994). 3. LaValle, S. an S. Hutchinson, Optimal Motion Planning For Multiple Robots Having Inepenent

9 273 J. Jongusuk an T. Mita: Tracking Control of Multiple Mobile Robots: A Case Stuy of Inter-Robot Goals, IEEE Trans. Rob. Autom., Vol. 14, No. 6, pp (1998). 4. Latombe, J.C., Motion Planning: A Journey of Robots, Molecules, Digital Actors, an Other Artifacts, Int. J. Rob. Res., Vol. 18, No. 11, pp (1999). 5. Desai, J.P., J. Ostrowski an V. Kumar, Controlling Formations of Multiple Mobile Robots, IEEE Proc. Int. Conf. Rob. Autom., pp (1998). 6. Desai, J.P., J. Ostrowski an V. Kumar, Control of Changes In Formations For A Team Of Mobile Robots, IEEE Proc. Int. Conf. Rob. Autom., pp (1999). 7. Kanayama, Y., Y. Kimura, F. Miyazaki an T. Noguchi, A Stable Tracking Control Metho for an Autonomous Mobile Robot, IEEE Proc. Int. Conf. Rob. Autom., pp (199). 8. Jiang, Z.P. an H. Nijmeijer, Tracking Control of Mobile Robots: A Case Stuy in Backstepping, Automatica, Vol. 33, No. 7, pp (1997). 9. Slotine, J.E. an W. Li, Applie Nonlinear Control, Prentice-Hall, NJ (1991). 1. Kogo, H. an T. Mita, Introuction to Control System Theory 2 n E., (in Japanese), Jikkyo Press, Japan (1997). 11. Canuas e Wit, C., B. Siciliano an G. Bastin, Theory of Robot Control, Springer Press, Lonon (1997). Jurachart Jongusuk receive B.S. egrees in control engineering at King Mongkut s Institute of Technology, Lakrabang, Thailan in The M.S. egree in the same fiel at Tokyo Institute of Technology (TIT), Japan in 2. He is currently a Ph.D. stuent in Mita laboratory, TIT. His research interests inclue nonlinear control, multitple UAVs formation control. Tsutomu Mita receive the Dr. Eng. egree in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in From 1975 to 1995, he was with the Chiba University, Chiba, Japan, where he was initially appointe as a Lecturer an became an Associate Professor an then a Professor in 1979 an 199, respectively. From December 1991 to September 1992, he was a Visiting Fellow at the Australian National University, Canberra. Since 1995, he has been a Professor in the Department of Control an Systems Engineering, Tokyo Institute of Technology. His research an eucational interests are control theory an its applications to inustrial esign, an robotics. He is the author of several textbooks, incluing An Introuction to H Control (Tokyo, Japan: Shoko-o, 1994).