Synthesis of Distributed Control of Coordinated ... - Semantic Scholar

1170

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 5, MAY 2011

[12] P. K. Sahoo and T. Riedel, Mean Value Theorms and Functional Equations, 1st ed. Singapore: World Scientific, 1999. [13] K. Eriksson, D. Estep, and C. Johnson, Applied Mathematics: Body and Soul. Berlin, Germany: Springer, 2010. [14] A. Zemouche, M. BoutayebG, and G. I. Bara, “Observer design for nonlinear systems: An approach based on the differential mean value theorem,” in Proc. 44th IEEE Conf. Decision Control, Eur. Control Conf., Seville, Spain, 2005, pp. 6353–6358. [15] “RT3000 Inertial and Measurement System User Manual,” Oxford Technical Systems, 2004. [16] R. Rajamani, D. Piyabongkarn, J. Y. Lew, K. Yi, and G. Phanomchoeng, “Tire road friction coefficient estimation—Real-time estimation methods for active automotive safety applications,” IEEE Control Syst. Mag., vol. 30, no. 4, pp. 54–69, Aug. 2010.

Synthesis of Distributed Control of Coordinated Path Following Based on Hybrid Approach Ying Lan, Gangfeng Yan, Member, IEEE, and Zhiyun Lin, Senior Member, IEEE

Abstract—The coordinated path following problem, steering a fleet of unicycles along a path while achieving a desired inter-vehicle formation, is investigated. The path is a paved road in the world or a marked curve on the floor of work area, for which each vehicle can measure the distance to the path and the heading error as well as the curvature of the path segment in its sensing range. Moreover, the vehicles can measure the arc distances of their neighbors that lie in their sensing range. A hybrid control approach is proposed to solve the problem. Depending on the posture with respect to the partly observed path, each vehicle either interacts with one of the others or works at the single-agent level. It is shown that the path following error of each vehicle is eventually reduced to zero and vehicles of every weakly connected component asymptotically converge to formations with equal arc distances. Index Terms—Coordinated path following, distributed control, hybrid approach, multi-robot systems.

I. INTRODUCTION The problem of steering a group of autonomous vehicles along given spatial paths while achieving desired inter-vehicle formation patterns is referred to as the coordinated path following problem [7]. It has been a topic of interest over the last few years. Motivated by military applications, previous work has been mainly on marine robots and spacecrafts [4], [11], [12], [15]. The common strategy includes two components: path following and multi-vehicle coordination. In [17], a group of vehicles modeled as Newtonian particles are steered to generate patterns on closed smooth curves, which has been applied to ocean sampling. Vehicle kinematics are considered in [6]–[9], [15], where the paths are Manuscript received March 17, 2009; revised November 03, 2009, April 13, 2010, September 09, 2010, and January 04, 2011; accepted January 08, 2011. Date of publication January 24, 2011; date of current version May 11, 2011. The work was supported in part by the National Natural Science Foundation of China (60875074) and Qianjiang Talents Program (2009R10027). Recommended by Associate Editor Z. Qu. The authors are with the Asus Intelligent Systems Lab, College of Electrical Engineering Zhejiang University, Hangzhou 310027, China (e-mail: linz@zju. edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2107070

parameterized according to the desired formation and each vehicle executes path following along its assigned path with dynamical synchronization. Recently, underactuated vehicles, which are independently controlled in surge, pitch and yaw, are studied in [5], where a decentralized controller is proposed such that the vehicles converge to follow desired straight paths with desired speed and formation. For more practical consideration and control efficiency, limited communication between vehicles and communication failures are taken into account in [1], [2], [6], [7], [16]. In contrast to existing work on coordinated path following where the path is defined in a global coordinate system and known to all the vehicles, we assume in the note that no global coordinate system exists and each vehicle can only sense a part of the path locally as well as its neighbors. The path is a paved road in the world or a marked curve on the floor of work area. The vehicles can then use a sensor to follow this path. For instance, by a perspective camera, the path can be identified and measured based on texture analysis or boundary detection techniques. Moreover, the sensor on each vehicle has a limited field of view. Consequently, the methods above may not be applicable. In the note, a hybrid control approach is proposed to solve the coordinated path following problem under a local setup. The vehicles are assumed to be positioned in a region close to the path. Depending on the posture, each vehicle either runs a coordination algorithm or works at the single-agent level. It is then shown by Nagumo’s theorem that if a vehicle initially has its state in a set in which the coordination algorithm is run, its state remains in the set no matter where the other vehicles are. On the other hand, if a vehicle does not have its state in this set initially, its trajectory enters the set in finite time. As a result, the control scheme results in a finite number of switches, and all the vehicles do get into the set in finite time and run the coordination algorithm thereafter. Thus, the distance from each vehicle to the path and the heading difference of each vehicle with respect to the path asymptotically converge to zero. Moreover, the arc distance from each vehicle to its neighbor approaches a desired value and so vehicles belonging to a weakly connected component are evenly spaced on the path. In our control strategy, though each vehicle may sense several other vehicles in its sensing range, only the local measurement of one neighbor called pre-neighbor is used for the coordinated control purpose. The idea of interacting with one neighbor to make the information measurement at the minimum is similar to the cyclic pursuit strategy studied in [13], but ring-like topology is considered in [13] to reach a circular motion. Also, different from [13] and other approaches in [8], [9], [15], which require the vehicles to pre-specify their neighbors according to the labels, the vehicles in the note are indistinguishable and the total number of vehicles in the group participating in coordination is scalable. Moreover, comparing with [6] and [7] where the absolute position and speed need to be known, our approach relies only on local information of the path and employs minimum links. II. PRELIMINARIES AND PROBLEM STATEMENT In the note, we consider the problem of tracking a spatial path for a group of n unicycles as well as attaining a desired inter-vehicle formation. For n unicycles labeled 1 through n, the posture of unicycle i is described by qi = (xi ; yi ; i )T 2 2 2 [0; ), called its state, where (xi ; yi ) denotes the position of its representing point defined in an inertia coordinate frame W , and i is its orientation with respect to the x-axis of W . The kinematic model for unicycle i with pure rolling and non-slipping is given as

0018-9286/$26.00 © 2011 IEEE

q_i =

x_ i y_ i _i

=

vi cos i vi sin i !i

:

(1)


Fig. 1. Path 0 with a direction, path following errors ( ; l .

), and arc distance Fig. 2. Coordinated path following.

The control inputs vi and !i stand for its linear speed and angular speed, respectively. The path that the vehicles are required to follow is a paved road in the world or a marked curve on the floor of work area. For instance, by a perspective camera, the path can be identified and measured based on texture analysis or boundary detection techniques. For the path denoted by 0, we give the following assumptions. Assumption A1: The path 0 is spatially smooth with its curvature at any point p less than a constant 0 , i.e., j(p)j < 0 . Assumption A2: Any point whose distance to the path is less than R0 := 1=0 has a unique closest point on 0. Assumption A3: The path 0 is not parameterized and is not available globally. With these assumptions, if vehicle i is positioned at a location with the distance to the path 0 less than R0 , the nearest point on 0 to vehicle i is unique and denoted by pi . We consider a direction on the path 0 and define i = (i ; i ) the path following error to 0 (see Fig. 1), where i 2 called the location difference is the distance from vehicle i to the nearest point pi on 0 but has a sign (that is, i > 0 when vehicle i is on the left side of 0 in the direction of the path and i < 0 when it is on the other side), and i 2 [0; ) is the orientation difference defined as the difference between the heading of vehicle i and the tangent direction of 0 at pi . Denote the curvature at a point p 2 0 by (p). With respect to the path 0, the dynamics thus become

_ i

= vi sin i ; p )v cos _ i = !i + (1+ (p )

;

1171

i = 1; . . . ; n

Fig. 3. Partition of S where S is the rectangle excluding the boundary of .

=

Fig. 4. An example of not weakly connected interaction graph. (The wedges in the figure not only indicate the positions of unicycles but also their orientations (the head direction of the wedges), so the orientation differences j j and j j of vehicle 3 and 4 are greater than =2 and thus they are not in S but S ).

(2) A. Interaction Topology and Hybrid Control Law

which describe the evolution of the path following error. The derivation of (2) is referred to [14]. Suppose each vehicle i carries an onboard sensor, with which it is capable of measuring its current location difference i and orientation difference i as well as the curvature (pi ) when it locates within R0 distance away from the path. Moreover, it can sense its neighboring vehicle j in the sense that lij can be measured where lij is the arc length between pi and pj , the nearest points on the path of vehicle i and j , respectively (see Fig. 1). It is assumed in the paper that the sensor range Rs of each vehicle is much larger than 2R0 . In the note, we focus on the situation that vehicles are initially positioned within R0 to the path and investigate distributed control laws using only local available information such that the vehicles eventually converge to follow the path while achieving an inter-vehicle formation. Fig. 2 is a schematic illustration for the problem of coordinated path following.

III. SYNTHESIS OF HYBRID CONTROL In this section, we synthesize control laws for the coordinated path following problem and present the convergence analysis.

Note that when a vehicle i is positioned within R0 units to the path 0 as shown in Fig. 2, equivalently i = (i ; i ) is in S := f(; )j 2 [0R0 ; R0 ]; 2 [0; )g which is illustrated in Fig. 3. For the coordinated path following problem, it is expected that the path following error i converges to 0 for all i and the arc distance lij between any vehicle i and the vehicle ahead converges to L (a desired value less than (Rs 0 2R0 )). Taking into account the kinematic constraints of the vehicles, we partition S into two parts S1 and S2 as in Fig. 3, where S1 = f(; ) 2 S jjj R0 ; j j a; ja + R0 j aR0 g and S2 = S n S1 . The parameter a satisfies 0 < a < minf(=2); R0 g. When vehicles are in S1 , they are heading the same direction as the path. For this situation, the vehicles run a coordination algorithm with only forward motions. When vehicles are in S2 , they run another algorithm to get themselves into S1 first and then switch to the motion coordination algorithm. Before presenting our control law, we introduce the interaction topology first. For the vehicles in S2 , they do not interact with others and work at the single-agent level. For the vehicles in S1 , they interact with one of their neighbors in their sensing range called pre-neighbor according to the following definition. For a vehicle i in S1 , another vehicle j is called vehicle i’s pre-neighbor if 1) vehicle j lies within R0 units to the path 0;

1172


2) vehicle j ’s nearest point pj on 0 is in front of vehicle i’s nearest point pi in the direction of the path without any other vehicle’s nearest point in the middle; 3) the arc distance lij is less than R, where R Rs 0 2R0 and R > L. Simply, we say that vehicle i and vehicle j are neighbors if one of them is the pre-neighbor of the other. Remark 3.1: In terms of the definition above, if vehicle i and j are neighbors, the Euclidean distance between vehicle i and j is less than Rs , which means they are mutually sensible. We now use a directed graph to describe the interaction topology of the vehicles. The nodes in the graph correspond to the vehicles. A directed arc from node i to node j exists if vehicle i is in S1 and it has a pre-neighbor j . We call it the interaction graph and say it weakly connected if there is an undirected path between any pair of nodes. Remark 3.2: The interaction graph in our setup may not always be weakly connected. In particular, when there are some vehicles in S2 or no other vehicles are found in a vehicle’s sensing range, the interaction graph is not weakly connected. For example, in Fig. 4, the interaction graph is not weakly connected as vehicles 3 and 4 are in S2 and they do not interact with others. Moreover, the interaction graph may dynamically change over time as the pre-neighbor of each vehicle may change depending on the states of the vehicles. For notation simplicity, we use i to denote the arc distance from vehicle i to its pre-neighbor. That is, if vehicle j is vehicle i’s preneighbor, then i = lij . In the special case when a vehicle has no pre-neighbor, we artificially let i = L. We propose the following control law for each vehicle i:

if i 2 S1 ; 0 if i 2 S2 ;

(p ) = [c + s(i 0 L)] 1+cos 1 ( + k !i = vi 0 k i i + 2 sin i )

vi

0

vi !i

= 0k1 Sat (sin ) p )v cos = 0k2 Sat ( i ) 0 (1+ (p )

(3)

) cos 1+(p )

(p

(4)

1

1, k (R0 =a), and s( ) is the sign function with where in (3), c s(0) = 0, and in (4), > 0 is a small constant, = sin(), 0 < k1 < k2 , and Sata ( ) is the saturation function defined as

1

Sata (x)

=

x; a;

0a;

jxj a, 0 x < a, 0a < x < 0.

Remark 3.3: The formula of (3) is considered such that it results in the following closed-loop dynamics: _ i

= vi sin i ; _ i = 0 vi (i + k i + 2 sin i ) k

for which 1) the state remains in S1 after it gets into S1 ; 2) it can be verified by Lyapunov-based techniques that the path following error converges to zero. In addition, the sign function s(i 0 L) in vi is used to adjust the inter-vehicle space. Remark 3.4: The control law (4) ensures that 1) the state of a vehicle enters S1 in finite time when it is originally in S2 ; 2) the state trajectories do not leave S . The control law of form (4) is obtained via the feedback linearization technique resulting in a desired linear closed-loop system with its equilibrium at the origin, for which both 1) and 2) are assured. The saturation function in (4) is used in order to avoid singularity.

B. Convergence Analysis In this subsection, we show that the proposed control law (3) and (4) solves the coordinated path following problem under certain technical conditions. Firstly, we show that if a vehicle initially has its state in S1 , then its state remains in S1 no matter where the other vehicles are. Theorem 3.1: For a group of vehicles under the control law (3) and (4), if a vehicle i satisfies i (t0 ) 2 S1 at time t0 , then i (t) 2 S1 for all t t0 . Proof: Substituting (3) into (2) generates _ i

vi sin

= f (i ) :=

i

0 vk (i + k i + 2 sin i )

:

Describe the boundary of S1 (see Fig. 3) as follows:

= f(; = f(; 3 = f(; 4 = f(; 5 = f(; 6 = f(; 1

2

)j = 0R0 ; 2 [0; a]g ; )j 2 [0R0 ; 0]; = ag ; )ja + R0 = aR0 ; > 0g ; )j = R0 ; 2 [0a; 0]g ; )j 2 [0; R0 ]; = 0ag ; )ja + R0 = 0aR0 ; > 0g :

Let nk , k = 1; . . . ; 6, be the normal vectors of k pointing outside of S1 . Then it can be inferred from Nagumo’s Theorem ([3], page 26) that, if all the inner products of nk and the vector fields f (i ) at the corresponding boundary are non-positive, then no trajectory leaves S1 . Now we check the condition of Nagumo’s Theorem for the system. Notice that, for all the states on the boundaries of S1 , we can conclude vi > 0 from the formula of vi in (3). For the boundary 1 , n1 = [01; 0]T and n1 1 f (i ) = 0vi sin i < 0 holds for all states in 1 . For the boundary 2 , n2 = [0; 1]T and the inner product is n2 1 f (i ) = 0(vi =k )(i + k i + 2 sin i ). For states in 2 , i + k i 0R0 + ka 0 due to k (R0 =a). TTherefore, n2 1 f (i ) 0. For the boundary 3 , n3 = [a; R0 ] and n3 1 f (i ) = (vi =k)[(ka 0 2R0 ) sin i 0R0 i 0kR0 i ]. Recall that vi > 0 and k > 0. Moreover, notice that for states on 3 , i 0, i 0, and 0 < sin i < i . Thus, one obtains that (ka 0 2R0 ) sin i < ka i kR0 i and therefore n3 1 f (i ) < 0 for all i on 3 . Similarly, the condition of Nagumo’s Theorem can be verified for the boundaries 4 ; 5 ; 6 . Hence, the set S1 is positively invariant for the dynamics of i , which means that i (t0 ) 2 S1 implies i (t) 2 S1 for all t t0 . Secondly, we show that if a vehicle initially has its state in S2 , then its trajectory enters S1 in finite time without leaving S . Theorem 3.2: For a group of vehicles under the control law (3) and (4), if a vehicle i satisfies i (t0 ) 2 S2 at time t0 , then there exists a time t1 > t0 such that i (t1 ) 2 S1 . Proof: For states in S2 , the control law (4) is used. Consequently, if i 2 [0 + ; 0] [ [; 0 ], the resulting closed-loop system with the control law (4) is

= 0k1 i _ i = 0k2 i . It is a linear system with an

_ i

equilibrium at the origin. Since k1 < k2 , its phase portrait looks like Fig. 5. Hence, it can be easily obtained that all trajectories enter S1 in finite time and meanwhile the trajectries remain in S . If i 2 ( 0 ; ), then the resulting closed-loop system has the following nonlinear form due to the saturation function:

= 0k1 sin _ i = 0k2 i :

_ i

;


1173

2)

Fig. 5. Phase portrait.

Notice that for states on the boundary = , we have _ = 0k2 < 0, on the boundary = 0R0 , we have _ = k1 (R0 sin = ) > 0, and on the boundary = R0 , we have _ = 0k1 (R0 sin = ) < 0. So no trajectory leaves S for the above nonlinear dynamics. Moreover, we are able to find a vector = [0; 01] such that 1 _ = k2 is 2 ( 0 ; ), which greater than a small positive number for all i

i

i

i

i

i

i

i

means that the trajectories flow along the direction of to the region with linear dynamics and then enter into S1 . If i 2 [0; ), then the resulting closed-loop system is

_i = 0k1 sin ; _ i = 0k2 : By the same argument, we can show that no trajectory leaves S . Moreover, we have 1 _ i = k2 greater than a small positive number for all i 2 [0; ), which means that the trajectories flow along the direction of to S1 . If i is initially in [0; 0 + ) or (0; 0), using the same argument above, it can be obtained that the trajectories remain in S and enter into S1 in finite time. Thirdly, we show that if all the vehicles have their states in S1 and the arc distance of two neighboring vehicles is positive, then the arc distance between them remains positive, which guarantees that no collision occurs in the process of coordinated path following. Moreover, we show that for this case, the connectivity of the interaction graph is preserved by our control law. That is, if the arc distance of two neighboring vehicles is less than R, then it is less than R all the time. Theorem 3.3: For a group of vehicles under the control law (3) and (4), suppose i (t0 ) 2 S1 for all i = 1; . . . ; n. Then the following holds: 1) If i (t0 ) > 0, then i (t) > 0 for all t t0 . 2) If i (t0 ) < R, then i (t) < R for all t t0 . Proof: 1) By our assumption that i (t0 ) 2 S1 for all i, it then follows from Theorem 3.1 that i (t) 2 S1 for all t t0 . Thus, for t t0 , vir := (cos i =(1 + (pi )i ))vi = c + s(i 0 L). Let vehicle j be the pre-neighbor of vehicle i at time t0 . Then the dynamics of the arc distance between vehicle i and its preneighbor j can be written as

_i = vjr 0 vir = [c + s(j 0 L)] 0 [c + s(i 0 L)] = s(j 0 L) 0 s(i 0 L):

(5)

By our condition we know (t0 ) > 0. Next we show that (t) > 0 for all t t0 . Suppose by contradiction that becomes 0 at T and (t) > 0 for t 2 [t0 ; T ). By continuity of (t), there exists a small > 0 such that (t) < L for t 2 [T 0 ; T ]. Hence, in the time interval [T 0 ; T ], s( (t) 0 L) = 01 and therefore i

i

i

i

i

i

i

_i (t) = s(j (t) 0 L) 0 s(i (t) 0 L) = s(j (t) 0 L) + 1 0. T From this, we can get i (T ) = i (T 0 ) + T 0 _i ( )d i (T 0 ) > 0, a contradiction to the assumption that i (T ) = 0. Hence, i (t) > 0 for all t. Note that R > L by our assumption. Hence, when i (t) enters into the range (L; R), we have s(i (t) 0 L) = 1 and _i (t) = s(j (t) 0 L) 0 s(i (t) 0 L) = s(j (t) 0 L) 0 1 0. By the similar argument as above, we obtain that if i (t0 ) < R, then i (t) < R for all t t0 .

Remark 3.5: From Theorem 3.1 and Theorem 3.2, it is seen that all the vehicles in S2 will eventually enter S1 and then remain in S1 . Moreover, it can be inferred from Theorem 3.3 that the interaction graph eventually becomes fixed since more arcs might be established but no arc is dropped after all the vehicles enter S1 . Therefore, there is only a finite number of switches after which the system becomes continuous. Fourthly, we show that if all the vehicles have their states in S1 and the interaction graph is weakly connected, then the group of vehicles achieve an inter-vehicle formation evenly spaced with the desired arc distance while following the path. Theorem 3.4: Consider a group of vehicles under the control law (3), (4) and suppose i (t0 ) 2 S1 for all i. Then 1) limt!1 i (t) = 0 for all i; 2) if 0 < i (t0 ) < R for all i, then limt!1 i (t) = L for all i. Proof: (i) 1) Consider a positive definite function V1 = (1=2)[(i + k i )2 + 2 (i ) ], where i = 1; 2; . . . ; n. Taking the derivative along the solution of (2) with the control law (3), one obtains

V_ 1(i) = (i + k i )(_ i + k _ i ) + i _i = 0 vi (i + k i )2 0 kvi i sin i : By assumption that i (t0 ) 2 S1 , then it follows from Theorem 3.1 that i (t) 2 S1 for all t t0 . Thus, j i (t)j a < (=2) and i sin i 0 for t t0 . Note that i sin i = 0 if and only if i = 0 and recall that vi is always positive, so we obtain that V_ 1(i) is negative definite with respect to the origin. Hence, for any i = 1; . . . ; n, i (t) ! 0 as t ! 1. 2) Since 0 < i (t0 ) < R holds for all i, it follows from Theorem 3.3 that the interaction graph is always weakly connected and the pre-neighbor of every vehicle does not change. So without loss of generality, we assume that vehicle 1 is the pre-neighbor of vehicle 2, vehicle 2 is the pre-neighbor of vehicle 3, and so on. In the group, only vehicle 1 does not have a pre-neighbor. Now we (i 0 L)2 . The function V2 consider a function V2 = (1=2) n i=1 is positive when the arc distances i s are not all equal to L, and is 0 when they are all equal to L. Taking the derivative leads to V_ 2 = ni=1 (i 0 L)_i . Recall that since vehicle 1 has no pre-neighbor, in the control law, it is artificially set 1 = L all the time and thus _1 = 0. For any i 2, if i 0 L > 0, then s(i 0 L) = 1 and _i = s(i01 0 L)01 0. If i 0 L < 0, then s(i 0 L) = 01 and _i = s(i01 0 L)+1 0. Therefore, (i 0 L)_i 0, i = 1; . . . ; n, which gives V_ 2 = ni=1 (i 0 L)_i 0. Furthermore, V_ 2 = 0 implies i 0 L = 0 for all i. (This can be seen by contradiction. Suppose that there is an i3 such that i 0 L 6= 0. Then in order to make V_ 2 = 0, _i has to be zero, which means i 01 0 L 6= 0. Repeating this argument eventually leads to 1 0 L 6= 0, a contradiction to the assumption 1 = L). Hence, by Lyapunov stability theory, we obtain limt!1 1 (t) = 1 1 1 = limt!1 n (t) = L. Finally, we show that for initial states in S , the group of vehicles converge to follow the path and achieve an inter-vehicle formation when the interaction graph is eventually weakly connected.

1174


Fig. 6. Trajectory of seven unicycles in the plane for t

2 [0; 5].

Theorem 3.5: Consider a group of vehicles under the control law i (t) = 0 for (3), (4) and suppose i (t0 ) 2 S for all i. Then, limt all i. Proof: If i (t0 ) 2 S , it means that i (t0 ) is either in S1 or in S2 . For the first case, by Theorem 3.1 the state remains in S1 thereafter. For the second case, by Theorem 3.2, there exists a time t1 t0 such that i (t1 ) 2 S1 . Consequently, there exists a time t so that all the vehicles have their states in S1 at t . Thus, it follows from Theorem 3.4 that limt i (t) = 0 for all i. Remark 3.6: From the above proof, we see that all the vehicles starting in S will eventually enter S1 . From Theorem 3.4, it can be concluded that if vehicle j becomes the pre-neighbor of vehicle i at any time after all the vehicles enter S1 , then the distance between them (namely, lij (t)) asymptotically converges to L and the pre-neighbor relationship is maintained. However, the pre-neighbor relationship established before all the vehicles enter S1 may not be preserved. This is the main drawback of the approach because the vehicles in S2 do not work in a cooperative fashion. But if the sensing radius Rs is large enough and the vehicles start very close, it is more possible to have the interaction graph weakly connected eventually. Remark 3.7: We have discussed how to solve the coordinated path following problem when the vehicles are positioned within R0 units to the path. Typically, it is the case for path following. For the more rare case that there are some vehicles far away from the path, we could use some tracking control laws for these vehicles to track the vehicles or landmarks around the path. Thus they can be attracted into the region within the distance R0 around the path in finite time and our control law can then be applied.


2 [0; 10].


2 [0; 20].

!1

!1

3

3

IV. SIMULATION RESULTS In this section, we present several simulations to illustrate our control strategy for coordinated path following. First, we consider seven unicycles (labeled 1 through 7) with the initial states (5; 4; 00:2 ), (4; 04; 0:7 ), (2; 02; 0:55 ), (05; 1; 0:25 ), (010; 010; 0:75 ), (010; 5; 00:15 ), and (015; 15; 0:65 ), respectively. In the simulation, R0 = 5, Rs = 15:5, L = 3:5, and R = 5:5. The parameter a used to partition S in Fig. 3 is selected to be 1.2. The control parameters in (3) and (4) are given as follows: c = 1:5, k = 5, = 0:5, k1 = 3, and k2 = 5. For the vehicles having distance to the path larger than R0 , the tracking control law in [10] is used to steer them into the region within R0 units to the path. The simulated trajectories of seven unicycles under the hybrid control are shown in Figs. 6–8, from which it can be seen that our proposed control law solves the coordinated path following problem. Seven vehicles are eventually evenly spaced with the desired arc distance between the neighboring vehicles while moving along the path.

Fig. 9. Following a set a parallel paths while achieving a desired “in-line” formation pattern.

In the figures, the dashed curve is the desired path to be followed and two real lines are the boundaries of the R0 -band (within R0 units to the path). From the simulation, we can also see that the interaction graph dynamically changes over time. At t = 0, it can be checked that vehicles 1, 4, and 5 have their states in S1 , vehicle 2 and 3 have their states in S2 , and vehicles 6 and 7 have their states outside of S (not in the R0 -band) initially. Moreover, it can be checked that at t = 0, only vehicle 4 has a pre-neighbor which is vehicle 3 and vehicle 5 has a pre-neighbor which is vehicle 4, so the interaction graph is not weakly connected initially. At t = 5, it can be observed from Fig. 6 that vehicles 1, 2, 3, 4, 5, and 6 all get into S1 , but vehicle 7 is still outside of the R0 -band. At this moment, the interaction graph becomes: vehicle 5’s pre-neighbor changes


to vehicle 6, vehicle 6 has vehicle 4 as its pre-neighbor, vehicle 4’s pre-neighbor changes to 2, vehicle 2 has vehicle 3 as its pre-neighbor, and vehicle 3’s pre-neighbor is vehicle 1. At t = 10 after vehicle 7 enters S1 , it gets a pre-neighbor, which is vehicle 5, while the others have the same pre-neighbor as at t = 5. From t = 10 to 20, it is seen from Fig. 7 and Fig. 8 that the interaction graph is ensured to be invariant as all the vehicles have their states in S1 at t = 10, which verifies our analytical result. Next, with a little bit modifications, we show that our proposed approach can be used to control a group of vehicles to move on a set of parallel paths and achieve a desired “in-line” formation pattern. Each vehicle in the group is able to identify its own target path and obtain the local measurements of its location difference, orientation difference, and the curvature of its nearest point on its own target path. Moreover, the pre-neighbor of each vehicle and the arc distance to its pre-neighbor are all defined with respect to its own target path. For the purpose of achieving an “in-line” formation pattern, the desired arc distance between any two neighbor vehicles is L = 0. This is different from the case of following a single path where L = 0 is not expected since L = 0 on a single path implies all the vehicles come to stick together. However, the results for both situations are almost similar. That is, if the interaction graph is weakly connected at any time, the vehicles eventually achieve an “in-line” formation pattern (namely, the arc distance between any pair of neighbors is 0). The simulated trajectories of five vehicles following a set of parallel paths while achieving a desired “in-line” formation are given in Fig. 9.

V. CONCLUSION In the note, the coordinated path following problem, steering a fleet of unicycles along a given path while achieving a desired inter-vehicle formation pattern, is investigated. A hybrid control approach is proposed to solve the problem where the path is a paved road in the world or a marked curve on the floor of work area, which can be sensed locally by onboard sensors. Initially, the vehicles are assumed to be positioned within R0 units to the path, which is typically the case in practical applications. Depending on the states with respect to the path, the vehicles either run a motion coordination algorithm or work at the single-agent level so that eventually the path following error is reduced to zero and the arc distance used to describe the inter-vehicle formation converges to a desired constant. It is shown in the note that as long as the interaction graph is weakly connected at some time and the arc distance of any two neighboring vehicles is nonzero initially, the connectivity of the graph is preserved under the proposed control law and no collision could occur. However, before the interaction graph becomes weakly connected, there is no such guarantee. The interaction graph dynamically changes over time as the system evolves. There is also no guarantee for the interaction graph to be weakly connected after some time when initially it is not weakly connected. These are the deficiencies of the approach.

REFERENCES [1] A. P. Aguiar and A. Pascoal, “Coordinated path-following control for nonlinear systems with logic-based communication,” in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, 2007, pp. 1473–1479. [2] J. Almeida, C. Silvestre, and A. Pascoal, “Coordinated control of multiple vehicles with discrete-time periodic communications,” in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, 2007, pp. 2888–2893. [3] J. P. Aubin, Viability Theory. Boston, MA: Birkhauser, 1991.

1175

[4] R. Beard, L. Lawton, and F. Hadaegh, “A coordination architecture for spacecraft formation control,” IEEE Trans. Control Syst. Technol., vol. 9, no. 6, pp. 777–790, Jun. 2001. [5] E. Borhaug, A. Pavlov, and K. Y. Pettersen, “Straight line path following for formations of underactuated underwater vehicles,” in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, 2007, pp. 2905–2912. [6] R. Ghabcheloo, A. P. Aguiar, A. Pascoal, C. Silvestre, I. Kaminer, and J. Hespanha, “Coordinated path-following in the presence of communication losses and time delays,” SIAM J. Control Optim., vol. 48, no. 1, pp. 234–265, 2009. [7] R. Ghabcheloo, A. Pascoal, C. Silvestre, and I. Kaminer, “Nonlinear coordinated path-following control of multiple wheeled robots with bi-directional communication constraints,” Int. J. Adaptive Control Signal Processing, vol. 21, pp. 133–157, 2007. [8] I. F. Ihle, M. Arcak, and T. I. Fossen, “Passivity-based designs for synchronized path following,” Automatica, vol. 43, no. 9, pp. 1508–1518, 2007. [9] I. Kaminer, O. Yakimenko, A. Pascoal, and R. Ghabcheloo, “Path generation, path following and coordinated control for time-critical missions of multiple UAVs,” in Proc. Amer. Control Conf., Minneapolis, MN, 2006, pp. 4906–4913. [10] Y. Lan, G. Yan, and Z. Lin, “A hybrid control approach to multi-robot coordinated path following,” in Proc. 48th IEEE Conf. Decision Control, Shanghai, China, 2009, pp. 3032–3037. [11] L. Lapierre, D. Soetanto, and A. Pascoal, “Coordinated motion control of marine robots,” in Proc. 6th IFAC Conf. Manoeuvering Control Marine Craft (MCMC), Girona, Spain, 2003, pp. 250–255. [12] N. E. Leonard, D. Paley, F. Lekien, R. Sepulchre, D. Frantantoni, and R. Davis, “Collective motion, sensor networks, and ocean sampling,” Proc. IEEE, vol. 95, no. 1, pp. 48–74, Jan. 2007. [13] J. Marshal, Z. Lin, M. Maggiore, and B. Francis, “Pursuit strategies for autonomous agents,” Lecture Notes Control Inform. Sci., vol. 309, pp. 137–151, 2004. [14] A. Micaelli and C. Samson, Trajectory Tracking for Unicycle-Type and Two-Steering-Wheels Mobile Robots Institut National de Recherche en Informatique et en Automatique(INRIA), France, Tech. Rep., 1993, Tech. Rep.. [15] R. Skjetne, I. Ihle, and T. I. Fossen, “Formation control by synchronizing multiple maneuvering systems,” in Proc. 6th IFAC Conf. Manoeuvering Control Marine Craft, Girona, Spain, 2003, pp. 280–285. [16] F. Vanni, A. P. Aguiar, and A. Pascoal, “Cooperative path-following of underactuated autonomous marine vehicles with logic-based communication,” in Proc. NGCUV’08-IFAC Workshop Navig., Guid. Control Underwater Veh., Killaloe, Ireland, 2008, pp. 1–6. [17] F. Zhang, D. M. Fratantoni, D. Paley, J. Lund, and N. E. Leonard, “Control of coordinated patterns for ocean sampling,” Int. J. Control, vol. 80, no. 7, pp. 1186–1199, 2007.