SICE Journal of Control, Measurement, and System Integration, Vol. 3, No. 4, pp. 229–236, July 2010
Experimental Validation of Switching Strategy for Tracking Control with Collision Avoidance in Non-Cooperative Situation Using Toy Model Cars Kiminao KOGISO ∗ , Makoto NOGUCHI ∗ , Kazuyoshi HATADA ∗ , Naoki KIDA ∗ , Naofumi HIRADE ∗ , and Kenji SUGIMOTO ∗ Abstract : This paper presents some experimental validation results of an already-proposed switching control method for simultaneous achievement of collision avoidance and tracking control for a vehicle in a non-cooperative situation. To validate the method, an experimental control system is made, in which the vehicle is a toy model car possible to remotely control via infrared ray and a camera is used to measure the vehicle’s state. After presenting the constructed control system, the effectiveness of the method is investigated with the results obtained from the several control experiments. Key Words : experiment, collision avoidance, model predictive control, pursuer-evader game, reachable set.
1. Introduction Autonomous vehicle robots have gradually become widely used in several settings, including industrial manufacturing. The autonomous vehicles necessarily require some appropriate safe actions, for example, to go to a certain destination or to accomplish some tasks without any contacts with an obstacle. For such a technological requirement, much research related to the vehicles has been conducted in fields such as robotics, control engineering, mechanical engineering, and image processing. In particular, in control engineering, various kinds of problems with the vehicles have been considered, for example, trajectory tracking [1],[2], collision avoidance [3]–[5], cooperative control and formation control [6]–[11]. Also, there are studies that examine experimental validation or verification. As remarkable examples, there are application of cooperative control to a quadrotor helicopter in [12] and application of nonlinear model predictive control for aerial pursuer-evader games in [13]. However, there is still a problem with real-time computation. The situation that we consider is an extreme case of the one dealt with in [14],[15], as follows. There are two vehicles on a flat field. One independently works under the tracking control to a given reference trajectory. The other becomes abnormal and is coming closer to the normal one because of a sensor trouble or an unexpected reason. In such a non-cooperative and dangerous case, the normal vehicle needs a certain strategy to avoid a collision with the abnormal one, in order to keep tracking to the reference safely. If the strategy could decide a certain action that considers the tracking control and the collision avoidance at the same time, it would be quite ideal. Such an ideal strategy is, however, hard to achieve because the simultaneous consideration is a conflicting request in general. Therefore, it seems natural to consider that the vehicle tries to track to the reference trajectory as long as the abnor∗
Department of Information Systems, Nara Institute of Science and Technology, Ikoma, Nara 630-0192, Japan E-mail:
[email protected] (Received November 5, 2009) (Revised January 29, 2010)
mal vehicles are far away. If the abnormal one comes closer, the normal one should take an evasive action until it succeeds in dodging the abnormal one without any collision. After that, the normal vehicle returns to the tracking mode. Such a strategic algorithm is reasonable because it can be separated into three parts: a collision avoidance method, a trajectory tracking control method, and a switching strategy to connect the two methods. This separation makes it possible to concentrate on constructing the respective methods and helps online computation cost reduced to be reasonable, comparing with the nonlinear model predictive control [13]. This is the idea behind the switching strategy proposed in [14],[15]. In this paper, then, we experimentally validate the switching control method [14],[15] for the simultaneous achievement in the non-cooperative situation. To do that, first, the authors construct an experimental control system that consists of a camera, a personal computer, a toy model car, and some other parts. The camera is used to measure the position of the toy model car on the field, which is called visual sensing. The computer takes multiple roles of a controller that generates a control signal based on the switching control method and of an image processing for acquisition of the car’s position, orientation, and velocity. The car is four-wheeled with front steer and rear drive wheels and can be remotely controlled via infrared ray, which is a product of TOMY Corporation and reduces the amount of time and money required to prepare a controlled object. After an identification of the car, the switching control algorithm that requires a solver of the model predictive control [16] for the tracking control and the reachable set for the prediction of the collision [17], are implemented into the computer. Some experimental results obtained from the control system are illustrated to confirm the effectiveness of the previously proposed method. The structure of this paper is as follows. In section 2 the proposed method is briefly introduced. Section 3 gives an account of the experimental control system, and the investigation results are presented in section 4. Section 5 concludes this paper.
2. Introduction of Control Method Validated This section briefly introduces the algorithm to control the
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vehicle for the simultaneous achievement of the collision avoidance and the tracking to the reference. For more details, please see [14],[15]. 2.1 Model of Car For the tracking control problem, let us consider the dynamics of the (evader) vehicle ⎤ ⎡ ⎢⎢⎢ ve cos xe3 (t) ⎥⎥⎥ ⎥ ⎢ x˙e (t) = fe (xe (t), ue (t)) = ⎢⎢⎢⎢ ve sin xe3 (t) ⎥⎥⎥⎥ (1) ⎦ ⎣ ue (t) in absolute coordination, where xe = [ xe1 xe2 xe3 ]T is state variable and xe1 ∈ , xe2 ∈ , and xe3 ∈ , are a horizon, vertical positions, and an angle, as figured in Fig. 1 (a). As for the collision avoidance, letting the evader placed on a zero, the two-vehicle dynamics in relative coordination is considered by x˙(t) = f (x(t), ue (t), u p (t)) ⎤ ⎡ ⎢⎢⎢ −ve + v p cos x3 (t) + ue (t)x2 (t) ⎥⎥⎥ ⎥ ⎢⎢⎢ v p sin x3 (t) − ue (t)x1 (t) ⎥⎥⎥⎥ , = ⎢⎢ ⎦ ⎣ ue (t) − u p (t)
(2)
where x = [ x1 x2 x3 ]T is a state variable in the relative coordination, and x1 ∈ , x2 ∈ , and x3 ∈ , denote a relative horizon, vertical positions, and angle, respectively, as figured in Fig. 1 (b). ue ∈ and u p ∈ mean a yaw-rate input for the evader and the pursuer, and they are allowed to take a value between the bounded range, i.e., −ub ≤ ue (t), u p (t) ≤ ub , ∀ t ∈ R+ . The assumption regarding the strategy is made in this paper that the pursuer knows no a priori information on the reference trajectory. If the evader’s action of being closer to the trajectory is blocked by a smart pursuer’s strategy, such as intermediately cutting in between the trajectory and the evader, then the reachability does not always exist. Such a case of the smart pursuer is the most difficult to realize the tracking control without any collisions. In addition, ve ∈ and v p ∈ are a constant forward velocity under the assumption of v p ≤ ve . The collision avoidance problem considered follows the formulation of the pursuerevader game because the result about the reachable set in the game theory is quite powerful and useful. However, if one wishes the formulation is broadened, which means that the relation of v p > ve is allowed, it comes to be easier for the pursuer to catch up with the evader. In this case, the collision avoidance may become quite complicated and difficult to solve. Thereby, suppose that v p is equal and less than ve . 2.2 Tracking Control The model predictive control (MPC) method is employed to achieve the trajectory tracking. The method in discrete time is constructed with a zero-order-holder, introducing step k ∈ Z+ . With a measured state xk|k at instance kT s , the model predictive controller minimizes a N-horizon objective function, i.e., J(xk|k , ukN ), J ∗ (xk|k ) = min N uk ∈N
s. t. discrete-time dynamics,
(3)
constraint conditions,
where xk|k denotes x(kT s ) with sampling period T s > 0 or simply x(k), and the controller finds the minimizer ukN∗ that gives
(a) The absolute coordination for the tracking control mode.
(b) The relative coordination for the collision avoidance mode.
Fig. 1 The graphical interpretation of how to take the coordinates.
the optimal value J ∗ (xk|k ). And then, the control signal is generated in a MPC fashion, i.e., u(k) = ukN∗ (1). For the realization of the MPC strategy, first, the linearization is performed using minimal change (Δxe , Δue ) around an operating point (x0 , u0 ) based on Jacobian. The following linearized model is obtained: x˙e (t) = AΔxe (t)+ BΔue (t)+C, where ∂ fe ∂ fe A = ∂x , B = ∂u , C = fe (x0 , u0 ). By approximations of e e Δxe (t) ≈ xe (kT s )− xe ((k −1)T s ), Δue (t) ≈ ue (kT s )−ue ((k −1)T s ) and x˙e (t) ≈ (xe ((k + 1)T s ) − xe (kT s ))/T s with t = kT s , the discretized model can be written below, ˆ e (k) + Bu ˆ e (k) + C, ˆ xe (k + 1) = Ax
(4)
where T s is omitted for simplification and each matrix is Aˆ = I + T s A, Bˆ = T s B, Cˆ = T s (C − Axe (k − 1) − Bue (k − 1)). Secondly, the objective function of the strategy is formulated with the approximating form (4). Setting N-horizon to evaluate N ∈ 3N can be written, the objective, a future trajectory xe,k
N xe,k
⎡ ⎤ ⎢⎢⎢ xe,k+1|k ⎥⎥⎥ ⎢⎢⎢ x ⎥ ⎢⎢⎢ e,k+2|k ⎥⎥⎥⎥⎥ N ⎢ ⎥⎥⎥ = Axe,k|k + Bue,k = ⎢⎢ + C, .. ⎢⎢⎢ ⎥⎥⎥ . ⎢⎣ ⎥⎦ xe,k+N−1|k
(5)
where A, B, and C are appropriate matrices, xe,k|k = xe (k), and N := [ue (k), ue (k + 1), · · · , ue (k + N − 1)]T . As the objective is ue,k made the tracking performance, it is given by N N 2 J(xe,k|k ) := ||xe,k − xre f,k ||Q N T N N = (ue,k ) Hue,k + 2 f (xe,k|k )T ue,k + const. N N + C − xre where H = BT QB, f (xe,k|k )T = (Axe,k|k + Bue,k f,k )QB, N 3N ∈ means the given N-horizon reference traand xre f,k jectory and Q is a weight matrix taking the following form, Q = diag[Q1 , Q2 , · · · , QN ], Qi = diag[q1i , q2i , q3i ], where i ∈ {1, 2, · · · , N}. And then, the constraint condition in (3) is formulated. The vehicle model has the pointwise-in-time constraint on the yaw rate. In this case, the constraint corresponding to N step horiN ≤ ub 1 zons can be summarized into a single inequality: Φue,k with the vector 1, whose components are all one, and the matrix Φ ∈ 2N×N of the form Φ = diag{[1, −1]T , · · · }. Consequently, with the objective J(xe,k|k ), the linearized prediction (5), and the inequality constraint, the following optimization can be obtained as the MPC strategy (3),
min
N ue,k ∈N
N T N N N (ue,k ) Hue,k + 2 f (xe,k|k )ue,k s.t. Φue,k ≤ um 1,(6)
and it follows the MPC fashion.
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2.3 Collision Avoidance Our approach to the collision avoidance uses a notion of a backward reachable set, appeared in a pursuit-evasion game theory [17],[18]. The backward reachable set has an interesting property, i.e., stepping inside the reachable set means that even if the evader employed any allowable input signal, the pursuer can take a certain signal to necessarily lead the collision. The collision and the backward reachable set are defined below. Definition 1 We say that the collision occurs if the state of the dynamics (2) goes inside a collision set G0 , i.e., x(t) ∈ G0 holds, where the set G0 is given by G0 = { x ∈ | with a positive value β given as a radius. 3
x12
+
x22
≤ β}
Definition 2 Given the dynamics (2) with the initial state x0 and the collision set G0 , we call the following G(τ) for τ ∈ [0, T (x0 )] as a backward reachable set.
where T (x0 ) = inf{t ∈ R+ | x(t; x0 ) ∈ G0 }. Remark 1 The reachable set G(τ) can be obtained by an approximated calculation method proposed in [17]. The literature discusses the existence and the boundedness of the set as well. From the definitions, if the state of (2) exists in the backward reachable sets, then the collision occurs as long as the pursuer uses the appropriate input. Therefore, we see that the best way is for the state not to step inside the reachable set. To do that, we have to know the optimal inputs for the evasive action. It is known that they can be derived from point of view of the game theory. Let us consider the two-person non-cooperative zero-sum game, then a cost function is common and is defined by g(x(T (x0 ))) := x12 (T (x0 )) + x22 (T (x0 )) − β. Here, a value function V is expressed as V(x0 ) = minue maxu p g(x(T (x0 ))), and Hamiltonian is defined as H(x, ue , u p ) := ψT f (x(t), ue (t), u p (t)), where ψ = [ψ1 ψ2 ψ3 ]T = ∂V 3 ∂x ∈ . Based on (2), in this case, the Hamiltonian is obtained,
+ ue (ψ1 x2 − ψ2 x1 + ψ3 ) − u p ψ3 .
(7)
Now we know that the following inequality in optimality condition [18] holds, H(x∗ , ue , u∗p ) ≤ H(x∗ , u∗e , u∗p ) ≤ H(x∗ , u∗e , u p )
(8)
where x∗ is an optimal trajectory, u∗e and u∗p are the optimal inputs. Therefore, from (7) and (8) the optimal inputs can be gotten, which are expressed in the bang-bang control fashion. u∗e = sign(ψ1 x2 − ψ2 x1 + ψ3 ), u∗p = sign(ψ3 ).
(9)
Accordingly, the optimal input of the evader is +1 or −1 in this collision avoidance problem. In [14],[15] the example of this case is illustrated, but it is omitted here.
(b) The flow chart of the hysteresis-based switching algorithm.
Fig. 2 State-dependent switching plane and the switching algorithm with the switching plane.
2.4
G(τ) := {x0 ∈ 3 | ∀ue ∃s ∈ [0, τ] x(s; x0 ) ∈ G0 }
H(x, ue , u p ) = − ve ψ1 + v p ψ1 cos x3 − v p ψ2 sin x3
(a) The illustration of the switching surface with S (α1 ) and S (α2 ) surrounding the reachable and the collision sets.
Switching Algorithm
This part shows how to switch between the game-based collision avoidance and the MPC-based trajectory tracking in order to achieve our goal. Introducing a set S (α) ⊂ 3 gives timings to switch. The set covers the backward reachable set and the collision set, i.e., ∀α > 0 and x0 S (α) := {x ∈ 3 | ρ(x, G0 ∪ G(τ)) ≤ α ∀ τ ∈ [0, T (x0 )]} where ρ is a distance, defined by ρ(x, Y) = inf y∈Y ||y − x||, and ρ(x, Y) = 0 if x ∈ Y. The reason of introducing it is to avoid the collisions and to detect being close to the reachable set before the pursuer comes in G0 or G(τ). The switching strategy algorithm with the set S (α) was proposed in order to achieve the collision avoidance and the trajectory tracking. The essence of the switching strategy is as follows. Since the state (2) in the backward reachable set necessarily leads to the collision in the finite time, the evader decides to take an evasive action, i.e., if x ∈ S (α), then the optimal control (9) is inputted to the evader. And if the evasive action is not needed, i.e., x S (α), then the evader tries to track to the given trajectory using the model predictive controller. This flow gives a very simple switching algorithm. Furthermore, following [19], in order to prevent a phenomenon of chattering, the strategy includes a hysteresis that is realized by S (α1 ) and S (α2 ) with 0 < α1 < α2 , Thereby, the above hysteresis-based switching algorithm can be described by the following procedure, and in Fig. 2 (b) a flow chart of it is illustrated. [Hysteresis-based Switching Algorithm] Step 0: Init hysteresis threshold α = α1 . Step 1: Measure x(kT s ) of (9) at sampling time t = kT s . Step 2: If x(kT s ) ∈ S (α), then the evader inputs u∗e (kT s ) of (9), and set α = α2 . Step 3: If x(kT s ) S (α), then the evader inputs ukN∗ (1) of (9), and set α = α1 . Step 4: Goto Step 1 with k = k + 1. This algorithm can achieve our goal that if x(0) S (α1 ) holds, then the collisions do not occur and the evader can track to the reference. In the rest of this paper, we will validate the effectiveness of the algorithm using an experimental control system with the toy model car.
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3. Experimental Control System A practical system for controlling the vehicles was constructed to perform the experimental validation. This section gives succinct accounts of an overview of the system and key elements of the apparatus. 3.1 System Overview The schematic overview of the experimental control system is presented in Fig. 3. The essential components of the system are the vehicle as a control object, the camera to get information on the vehicle’s positions, the personal computer to do the image processing for the measurement of the vehicle’s positions and to calculate adequate control input, and the i/o module to send the calculated control to the vehicle with the infrared ray. The model number and specifications of them are shown in Table 1. Additionally, all of the programs for measurement and calculation are written in C# running on Windows Vista. 3.2 Vehicle The vehicles are toy model cars called “Q-steer,” which come with an associated infrared controller kit. To distinguish between the evader and the pursuer, we equipped the evader vehi-
cle with a red LED, the pursuer with a blue one, and respective batteries as well, as shown in Fig. 4. They can be remotely operated by receiving the command signals from the infrared controller. This experiment restricts their operations to three actions: turning to the right, left and going forward at a constant speed, due to the specifications of the toy car. Therefore, the control signals become on-off and discrete in this system, and so, we decided to make rough approximations of the continuous signal obtained from the model predictive controller, in order to get the corresponding three (actions) values rounded by certain thresholds. We give a few more details about the i/o module here. The infrared controller has three on-off switches for the respective actions and discrete value signals to send to the evader come out of the computer via the USB cable. Then, we made a short circuit that enables the discrete signals to be altered to on-off signals for the individual switches, using “Gainer” and some relays. Figure 5 is a photo of the constructed circuit connected to the controllers for automatic controls of the evader. 3.3
Sensing
This control system employs visual sensing to measure the positions and velocities of the vehicles. The camera utilized is fixed on a rigid frame and can capture a gray-scale image of size 640×480 every 1/90 [ms], where camera calibration was performed by the well-known technique of [20]. The images are fed to the personal computer through the IEEE 1394b cable and the PCI-Ex video capture board. Since the vehicles are equipped with the different color LEDs, their positions can be obtained by image processing, where their velocities are esti-
(a) Schematic illustration of whole system.
Fig. 4
(b) Annotated picture of whole system. Fig. 3 Overview of practical vehicle control system.
Toy model cars equipped with the LED and batteries.
Fig. 5 i/o module and the infrared controller for automatic controls of the evader.
Table 1 Apparatuses and software. category model car camera personal computer I/O module video capture board software & libraries
product name (Company) Q-steer FIAT 500 (TOMY) FireDragon CSFV90CC3 (Toshiba Teli Corp.) Windows Vista (Microsoft) Gainer (RT Corp.) + relay (OMRON) Zenkuman PFW-87 (Techno Scope) Visual C# , Zenkuman ZCL-2 (Techno Scope)
spec. forward, right, left, 47 × 28 [mm]. 90 fps, VGA, IEEE1394b Intel Xeon 3 GHz, 4 GB mem. – PCI-Ex, IEEE1394b –
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Fig. 6 Identification of parameters ve and v p .
Fig. 7 Tracking control by MPC with the start point [0.538, -0.462]T .
mated with the two latest measured positions. 3.4 Controller and Sampling Period The rest of the main processes in the computer includes decisions of the control approach, by checking if the state is inside or outside the set S . The collision avoidance mode gives a control signal that corresponds to the right or left turn with the maximum yaw rate. The model predictive control mode performs the linearization, solving the convex quadratic programming optimization (with the C# library made by customizing the solver “QuadProg++” for C++, which is provided free of charge), and altering the optimal value to be discrete. To accomplish all of the processes in real time, therefore, the sampling period is set to 80 [ms]. 3.5 Identification of Vehicle Dynamics With the constructed control system, the identification was done to get a model of the vehicle, assuming that the nonlinear model (1) and (2) can almost approximate practical behaviors of the toy car. The comparison of two trajectories between experiment and the nonlinear model when taking a right, is illustrated in Fig. 6. The figure shows that the simulated trajectory with the speed 0.42 [m/s], denoted by plots of the thick circle, seems almost the same behavior as the actual one of the square-marked plots. Then, adjusting the yaw rate of the model in turning we estimated the real yaw rate as 2.6 [rad/s]. Naturally, −2.6 [rad/s] will be used when taking a left.
4. Experimental Validation This section shows some obtained experimental results by using the control system described in the previous section. The design procedure follows the order of the subsections, 4.1, 4.2, and 4.3, that is, (1) to set the weights and the horizon of the MPC (6), (2) to prepare the feedback rule of (9), and then, (3) to determine the parameters α1 and α2 of the set S . 4.1 Tracking Control Before applying the model predictive control, the information on the reference is to be set a priori. Throughout this experiment the reference is supposed to be oval as illustrated with the thick circle in Fig. 7. After setting N = 2, q1i = 50, q2i = 50, and q3i = 1, we performed the tracking control experiment of the single vehicle. The result of the obtained trajectory is shown in Fig. 7, plotted by the square, where the vehicle was placed on the start point [0.538, −0.462]T . Although this result
Fig. 8
Backward reachable set.
is not extremely good regarding tracking performance, it seems to be the best we can achieve using this control system. Further comments on the reason for this and how to improve the performance will be stated in section 4.4. Here, we would like to continue on with what the practical control system can present. Regarding how to determine the weights and the horizon, some experiential comments are added here. The model parameters obtained through right identification and comparison, as shown in Fig. 6, are used to perform sufficient numerical evaluation with several parameter values, and then, an adequate one was implemented in the practical system and tested by experiment. This procedure is repeated until the satisfactory result is obtained. Furthermore, as for the horizon, it was determined over the small numbers. This is because the offered method employs the linearization of the nonlinear plant to predict the plant’s behavior; in other words, the prediction will become increasingly inaccurate as the steps increase, which has been already checked numerically. 4.2
Collision Avoidance
First, let the collision set be defined by a circle with a radius of 0.05 [m], taking into account the size of the vehicle. Secondly, the backward reachable set is calculated, as shown in Fig. 8 in this case. From the figure, we see that the reachable set includes a pipe of the collision sets along the x3 axis and it looks like the rest of it is a helically spreading shape centering on the pipe. Then, following the collision avoidance algorithm with the obtained reachable set, the experiment only on the collision avoidance was done, where α1 = 0.01 [m] set by experimental trial and error with the concern of economical
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avoidance. The experimental result in taking evasive action is illustrated in Fig. 9, using some figures in time sequence. Each figure corresponds to a different time span and the proper data in each figure is highlighted, where the gray plots mean past data. From the figures, the algorithm based on the reachable set works correctly. 4.3 Tracking Control Considering Collision Avoidance This section shows the main experimental result. The control approach consists of the model predictive control, collision avoidance, and the switching strategy for the two modes with α2 = 0.08 [m] that is also determined by experimental trial and error with the concern of less chattering for good tracking. The evader is automatically controlled and the pursuer is manually
operated by one of the authors. The partial trajectories of the obtained experimental result, which shows the evasive and tracking actions, are illustrated in Fig. 10. Figures 10 (a)-(d) show the trajectory from 7.20 to 13.6 [s], where from 7.20 to about 9.00 [s] the collision avoidance mode operates and after that the mode switches to the tracking control mode. Similarly, Figs. 10 (e)-(h) show from 15.40 to 22.00 [s], where until 18.40 [s] it is in the avoidance mode and then the mode switches. Therefore, we can see that the control approach proposed in [14],[15] is a method possible to be implemented into a practical system and the effectiveness of the method is shown by the experimental validation result of Fig. 10. 4.4
(a) From 0 to 1.60 second.
(b) From 1.68 to 2.40 second.
Discussion
Through this experiment, we can state that the already proposed method is effective, but there is the matter that the tracking performance is not so good, as pointed out in the previous section. The main reason for this seems to be the long sampling period. Some numerical simulation results show that as the sampling period becomes shorter, the trajectory comes to fit the reference more closely. This fact implies that if the operating system could be replaced by a real-time one with decent CPU performance, the tracking performance error should decrease. Therefore, it is ideally possible to improve the system to be more flexible and appropriate for the verification, but it will require much money and time. We would like to build such an improved system given the opportunity.
5. Conclusion
(c) From 2.48 to 4.08 second.
(d) From 4.00 to 4.88 second.
Fig. 9 Both trajectories for about 5 seconds, where collision is avoided.
This paper has shown the effectiveness of the switching method for tracking control taking into account the collision avoidance, proposed in [14],[15], by using a laboratory-size control system that the authors made for control of toy model cars. The tasks include system improvement, including replacement of the operating system with a more appropriate one, and several utilizations of this system for verification and validation
(a) From 7.20 to 8.08 seconds.
(b) From 8.16 to 8.64 seconds.
(c) From 8.72 to 9.92 seconds.
(d) From 10.0 to 13.6 seconds.
(e) From 15.44 to 16.80 seconds.
(f) From 16.88 to 18.40 seconds.
(g) From 18.48 to 20.40 seconds.
(h) From 20.48 to 22.00 seconds.
Fig. 10
The trajectories of the experimental result: the tracking control with the collision avoidance.
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of control methods.
6. Acknowledgements This research was partially supported by Creative and International Competitiveness Project 2008 ‘Catch Me If You Can —Tom and Jerry—’ of Educational Program to Promote Creativity and International Competitiveness in Information Science, and the Support Program for Improving Graduate School Education, at the Graduate School of Information Science, Nara Institute of Science and Technology. The authors gratefully appreciate the opportunity to perform this experimental validation. References [1] M. Egerstedt and X. Hu: A hybrid control approach to action coordination for mobile robots, Automatica, Vol. 38, No. 1, pp. 125–130, 2002. [2] H. Fang, R. Fan, B. Thuilot, and P. Martinet: Trajectory tracking control of farm vehicles in presence of sliding, Robotics and Autonomous Systems, Vol. 54, No. 10, pp. 828–839, 2006. [3] Y. Yoon, J. Shin, H.J. Kim, Y. Park, and S. Sastry: Modelpredictive active steering and obstacle avoidance for autonomous ground vehicles, Control Engineering Practice, Vol. 17, No. 7, pp. 741–750, 2009. [4] E. Lalish, K. A. Morgansen, and T. Tsukamaki: Decentralized reactive collision avoidance for multiple unicycle-type vehicles, Proc. 2008 American Control Conference, pp. 5055– 5061, 2008. [5] J.L. Fernandez, R. Sanz, J.A. Benayas, and A.R. Dieguez: Improving collision avoidance for mobile robots in partially known environments: The beam curvature method, Robotics and Autonomous Systems, Vol. 46, No. 4, pp. 205–219, 2004. [6] Q. Li and Z.-P. Jiang: Formation tracking control of unicycle teams with collision avoidance, Proc. 47th IEEE Conference on Decision and Control, pp. 496–501, 2008. [7] K.N. Krishnanand and D. Ghose: Formations of minimalist mobile robots using local-templates and spatially distributed interactions, Robotics and Autonomous Systems, Vol. 53, No. 34, pp. 194–213, 2005. [8] B.D.O. Anderson, C. Yu, and J.M. Hendrickx: Use of metaformations for cooperative control, Proc. 17th International Symposium on Mathematical Theory of Networks and Systems, pp. 2381–2387, 2006. [9] Y. Li and X. Chen: Stability on multi-robot formation with dynamic interaction topologies, Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1325–1330, 2005. [10] X. Jing: Behavior dynamics based motion planning of mobile robots in uncertain dynamic environments, Robotics and Autonomous Systems, Vol. 53, No. 2, pp. 99–123, 2005. [11] H. Fukushima, K. Kon, and F. Matsuno: Distributed model predictive control for multi-vehicle formation with collision avoidance constraints, Proc. 44th IEEE Conference on Decision and Control, pp. 5480–5485, 2005. [12] G.M. Hoffmann: Autonomy for sensor-rich vehicles: Interaction between sensing and control actions, PhD thesis, Department of Aeronautics and Astronautics, Stanford University, 2008. [13] J.M. Eklund, J. Sprinkle, and S. Sastry: Implementing and testing a nonlinear model predictive tracking controller for aerial pursuit/evasion games on a fixed wing aircraft, Proc. 2005 American Control Conference, pp. 1509–1514, 2005. [14] N. Isoda, K. Kogiso, and T. Asai: Switching strategies of collision avoidance and tracking control for vehicles based on noncooperative game and model predictive control, Proc. 2007 IEEE Multi-conference on Systems and Control, pp. 178–183,
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Kiminao KOGISO (Member) He was born in Sapporo, Japan, in 1975. He received the B.S., M.S., and Ph.D. degrees in Mechanical Engineering from Osaka University, Japan, in 1999, 2001, and 2004, respectively. He had worked as a postdoctoral researcher of the 21st Century COE Program in Nara Institute of Science and Technology, from 2004 to 2005. Since 2005, he has been serving as an Assistant Professor in the Department of Information Systems, Nara Institute of Science and Technology. His current research interests include constrained control, networked control, and implementation. He is a member of ISCIE, JSME and IEEE.
Makoto NOGUCHI (Student Member) He finished the Advanced Course, Yonago National College of Technology, in 2006. In 2006, he received B.S. degree from National Institution for Academic Degrees and University Evaluation, Japan. In 2008, he received the M.S. degree from Nara Institute of Science and Technology, Japan. Currently, he is a Ph.D. student. His research interests include control system design and robot control, and so on.
Kazuyoshi HATADA (Student Member) He received his B.S. degree from Ritsumeikan University, Japan, in 2008, and is currently a Master Course student in Nara Institute of Science and Technology, Japan. He is a member of ISCIE.
Naoki KIDA He received his B.S. degrees from Kansai University, Japan, in 2008, and is currently a Master Course student in Nara Institute of Science and Technology, Japan.
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Naofumi HIRADE He received his B.S. degrees from Kansai University, Japan, in 2008, and is currently a Master Course student in Nara Institute of Science and Technology, Japan.
Kenji SUGIMOTO (Member) He received the Master’s degree and Doctor’s degree from Kyoto University in 1982 and 1989, respectively. After working with Mitsubishi Electric Corporation, he became an Assistant Professor at Kyoto University in 1985. He was an Associate Professor at Okayama University and Nagoya University. Since 1999, he has been a Professor at Nara Institute of Science and Technology. His current research interests include control theory and system science. He is a member of IEEE and ISCIE.
