2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009
ThA19.3
A Hybrid Control Approach to Cooperative Target Tracking with Multiple Mobile Robots Ying Lan, Gangfeng Yan, and Zhiyun Lin Abstract— The paper proposes a novel approach based on hybrid control and reachability specification for cooperative target tracking with multiple unicycle-type robots. Robots are steered to get closer to the target when they are far and then switch to coordinate their motion in order to cooperatively capture the target by eventually moving on a circle around the target with equal angular distance between each other.
I. I NTRODUCTION In recent years multi-vehicle systems have attracted much attention, due to their various applications and advantages that they are able to execute a wide variety of missions with increased robustness as compared to their single vehicle counterpart. Interesting research problems include consensus, rendezvous, formation control, motion coordination, and cooperative target tracking [2], [5], [8], [13], [15], [17]. Due to distributed and local nature of multi-vehicle systems, one main challenge is to achieve a collective motion in such a situation that each vehicle takes control actions based only on information from local sensors and limited communication. Recently, it has been addressed using numerous techniques from different aspects. For example, leaderfollower approach is considered in [19] to form a group shape configuration; leaderless coordination with nearest neighbor rules or cyclic pursuits strategy is studied in [7], [14], [15]; and in [9], [10], a steering law for achieving both rectilinear and circular formation is proposed. Cooperative target tracking is one form of motion coordination, where a group of vehicles reach desired relative positions and orientations with respect to the target, obstacles, and other group-mates. Ceccarelli et al. [3] propose a control law to achieve collective circular motion around a virtual fixed/moving target, but only local stability properties are proved. In [11], a collective target tracking controller based on oscillator models is studied. It consists of velocity matching and centroid target tracking and requires full information of the target and other vehicles. Recently, outputfeedback control for group target tracking is considered in [4], where an interlaced observer is designed to estimate the unmeasurable velocity and then each robot broadcasts signals, including its absolute coordination, orientation, and velocity estimation, to other robots in its communication range. Game theoretic approaches are also used to formulate the target-tracking problem in the framework of noncooperative game [6], [18]. The authors are with the Department of Systems Science and Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027 P. R. China (Email:
[email protected] ) The work is supported by National Natural Science Foundation of China under Grant 60875074.
978-1-4244-4524-0/09/$25.00 ©2009 AACC
In this paper, we study cooperative target tracking problems in multiple mobile robots with nonholonomic constraints. A novel approach based on hybrid control and reachability specification is proposed. Firstly, we devise an explicit distributed state-dependent hybrid control to first steer vehicles closer to the target and then coordinate their motion to achieve collective circular motion around the target to “capture” the target. In the equilibrium configuration, the unicycles move in a cooperative fashion with equal angular distance. Secondly, by introducing a coordinate transformation for the group of unicycles, nonholonomic constraints are removed to overcome the difficulty in control synthesis. Thirdly, sensor limitation is also taken into account. Each unicycle uses only local available information such as distance and bearing angle to others as the feedback input and yet a global collective behavior achieving cooperative target tracking occurs. Rigorous analysis is given for trajectory convergence, set invariance, and reachability in the process of cooperative target tracking. Finally, a simulation is provided to verify the hybrid control method. Several proofs are omitted in the paper due to space limitation. Readers are referred to [12]. II. P RELIMINARIES AND P ROBLEM S TATEMENT A. Vehicle Kinematic Model Consider a group of unicycles labelled 1 through n in the plane. For any unicycle i (i = 1, . . . , n), its pose is described (i) (i) (i) (i) (i) (i) by qc = (xc , yc , θc )T ∈ R2 ×[−π, π), where (xc , yc ) denotes its representing point of the unicycle defined in an (i) inertia coordinate frame W, the angle θc is its orientation of the unicycle with respect to the x-axis. Thus, the kinematic model for unicycle i with pure rolling and non-slipping in W is given as: (i) (i) (i) vc cos θc x˙ c (1) q˙c(i) = y˙ c(i) = vc(i) sin θc(i) . (i) (i) θ˙c ωc (i)
(i)
The control inputs vc and ωc stand for the linear speed and angular speed, respectively. In addition, suppose there is a stationary target (beacon) in the plane labelled 0. Its position in W is denoted by z0 .
B. Local Sensing Information For each vehicle i, we construct a moving frame, the Frenet-Serret frame that is fixed on the vehicle with origin at the representing point and x-axis coincident to the orientation of the vehicle. In our setup, vehicles are indistinguishable
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and no vehicle can access the absolute positions of other pose of the virtual vehicle i in the inertia coordinate frame vehicles or its own. When vehicle i senses vehicle j (or the W. Its kinematic model has the same form as (1) with control (i) (i) target 0), vehicle i can only measure the distance dij (di0PSfrag ) inputs vr and ωr . replacements between them and the bearing angle αij (αi0 ) with respect to its own Frenet-Serret frame (see Fig. 1). The bearing angle (i) θe αij is defined in [−π, π). These local information can be (i) measured by local sensors, e.g., onboard cameras. θc (i) yc
(i)
ye j
PSfrag replacements
dij
Σ
θr R
*
x
αij
W
y i
Fig. 1.
(i)
(i)
yr
Fig. 3.
Local measurement (distance dij and bearing angle αij ).
We assume that vehicles within a disk-like region D0 centered at the target with radius 2R (R > 0 is a constant) can sense each other and the target. C. Problem Statement The cooperative target tracking problem consists of finding distributed control law for each vehicle using only local available information such that the group of vehicles eventually converges to uniform circular motion around the target and are evenly spaced (see Fig. 2). The radius of the circle, which is R in the paper, is pre-specified and known by all vehicles.
(i) xc
(i) xr
Illustration for virtual vehicles and coordinate variables.
Next we construct the virtual Frenet-Serret frame Σi that is fixed to the virtual vehicle i and defined in the same way as the Frenet-Serret frame on vehicle i. Then we are able to define the configuration of vehicle i in the virtual FrenetSerret frame Σi as (i) (i) cos θr sin θr 0 qe(i) := − sin θr(i) cos θr(i) 0 (qc(i) − qr(i) ). 0 0 1 (i)
Thus, the dynamics for qe is (i) (i) (i) (i) (i) (i) ωr ye − vr + vc cos θe x˙ e (i) (i) q˙e(i) = y˙ e(i) = −ωr(i) x(i) . e + vc sin θe (i) (i) ˙θe(i) ωc − ω r
By the definitions of the virtual vehicle i and its associated (i) (i) virtual Frenet-Serret frame Σi , one obtains that xe and x˙ e (i) ωr(i) remain equal to zero. Denote χr = − (i) the curvature of
R
vr
(i)
γ. In our setup, χr = 1/R for all i = 1, . . . , n. Then from the first equation of the above formula, it follows that PSfrag replacements
(i)
vr(i) = Fig. 2.
Cooperative target tracking.
(i)
(i)
Rvc cos θe (i)
R + ye
(i) and ωr(i) = −χ(i) r vr = −
(i)
vc cos θe (i)
R + ye
(i)
D. Problem Reformulation We now introduce a coordinate transformation, originally developed for path following [1], [16], and reformulate the cooperative target tracking problem in new state space. Let γ be the clockwise circular orbit (circle with a direction of motion) centered at the stationary target with radius R. For each unicycle i = 1, . . . , n, we define a corresponding (i) (i) virtual vehicle i whose position (xr , yr ) is the orthogonal projection of the moving unicycle i’s position onto γ and (i) whose orientation θr is tangent to γ in the direction of motion (see Fig. 3 for an illustration). In other words, the position of the virtual vehicle i is the nearest point on γ to vehicle i. Note that it is unique when vehicle i is not co(i) (i) (i) (i) located with the target. Denote qr = (xr , yr , θr )T the
under the constraint R + ye 6= 0. Hence, it suffices to look at the reduced system ( (i) (i) (i) y˙ e = vc sin θe (i) i = 1, . . . , n. (2) (i) (i) v cos θe(i) θ˙e = ωc + c (i) R+ye
The dynamics (2) above describe the evolution of the distance and heading difference to γ. (i) Notice that the constraint R + ye 6= 0 is guaranteed if vehicle i is not at the target’s location, which is reasonable in practice. Now define S0 := {(ye , θe )|ye ∈ (−R, R], θe ∈ [−π, π)} . (i)
(i)
Denote φ(i) = (ye , θe )T . It can be easily seen that (i) (i) φ(i) (0) ∈ S0 is equivalent to (xc , yc ) ∈ D0 \ {z0 }. Hence, a necessary step to solve the cooperative target tracking
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(i)
(i)
problem in D0 is to devise distributed control vc and ωc for each vehicle so that for all initial state φ(i) (0) ∈PSfrag S0 , replacements i = 1, . . . , n, the following holds: (i) φ(i) (t) ∈ S0 for all t; (ii) φ(i) (t) → 0 as t → ∞. ψl j The first condition above guarantees that no vehicle leaves (i) ψ l+1 θe D0 and the second one means that they eventually tend to move on the circle. Another additional condition is to assure dij di0 that the vehicles are evenly spaced on the circle eventually. αij αi,0 Let ψ1 , ψ2 , . . . , ψn be the angles counterclockwise defined between two directions originating from the target pointing i towards two neighbor vehicles, which are called central k angles in the paper. The neighbor vehicles are defined in the sense that there is no other vehicles between them by Fig. 5. Local available information in the new coordinate system. rotating a ray (originating from the target) from one to the other counterclockwise. The central angles ψi are defined to belong to [0, 2π). An illustrative example is given in Fig. 4. Thus, if the distributed control are designed to additionally III. H YBRID C ONTROL S YNTHESIS guarantee that In this section, a hybrid control approach is proposed for (iii) limt→∞ ψ1 (t) = · · · = limt→∞ ψn (t), cooperative target tracking in D0 . A group of vehicles with (iv) ψi (t) > 0 for all t ≥ 0 and i = 1, . . . , n, any initial configuration may cross over each other such that then the vehicles are evenly spaced on the circle eventually. the neighbor relationship changes over time. Thus it may not be possible to have one smooth control that not only achieves PSfrag replacements cooperative target tracking but also makes the set S0n (the Cartesian product of n copies of S0 ) positively invariant k j ψ2 (namely, no vehicle leaves D0 ). Hence, we expect to find a subset of states in S0 containing the origin and a distributed ψ3 ψ1 control law for cooperative target tracking so that as long as ψ4 a vehicle enters the set, it remains in it thereafter. And for the vehicles initially located outside of the set but in S0 , we expect to drive them into it in finite time without getting out i l of S0 . In terms of this idea, we partition S0 into a collection of sets S, S1 , . . . , S4 (see Fig. 6), where PSfrag replacements Fig. 4. Illustration for central angles in an example of four vehicles. Finally, the local sensing information is reformulated in the new coordinate system, which will be used for control synthesis in the paper. Based on the assumption of local (i) (i) sensing information we made, the states ye and θe are available for each vehicle (see Fig. 5) in terms of the following relations:
θe π
S4
(i)
ye = di0 − R, (i)
(i)
(i)
θe = θ c − θ r =
−R
−αi0 − π2 , −αi0 + 23 π,
a
−a
S
αi0 ∈ [−π, π/2], αi0 ∈ (π/2, π),
S3
R
ye
−a S2
where di0 and αi0 are the distance and bearing angle of the target to vehicle i, respectively. Moreover, the central angles ψl and ψl+1 that have vehicle i’s location as their endpoint are also available to the vehicle i in terms of trigonometric relationship. As an example, the central angles ψl and ψl+1 in Fig. 5 can be calculated through ψl = g(dij , di0 , αij , αi0 ),
S1
a
−π Fig. 6.
S = {(ye , θe ) ∈ S0 : |ye | ≤ a, |θe | ≤ a, |ye + θe | ≤ a}, S1 = {(ye , θe ) ∈ S0 \ S : a − ≤ θe < π}, S2 = {(ye , θe ) ∈ S0 \ S : −π ≤ θe ≤ −a + }, S3 = {(ye , θe ) ∈ S0 \ S : ye > 0, −a + < θe < a − }, S4 = {(ye , θe ) ∈ S0 \ S : ye < 0, −a + < θe < a − }.
ψl+1 = g(dik , di0 , αik , αi0 ),
where g(·) is a function related to trigonometric functions. Though in D0 , each vehicle is able to sense all the others and the target, but it uses only local sensing information of the target and two neighbors (the one ahead and the one after that are possibly changed over time) in coordination.
A partition of S0 .
The parameter a satisfies 0 < a < min {π/2, R} and > 0 is a small number. Thus, the cooperative target tracking
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problem in the region D0 can be solved by addressing the following two subproblems. Problem 3.1: If φ(i) (0) ∈ S for all i = 1, . . . , n, devise vi and ωi for each vehicle using only local available information such that (i) φ(i) (t) ∈ S for all t ≥ 0, (ii) φ(i) (t) → 0 as t → ∞ , (iii) limt→∞ ψ1 (t) = · · · = limt→∞ ψn (t) > 0, Problem 3.2: If φ(i) (0) ∈ S0 \ S for some i = 1, . . . , n, devise vi and ωi for vehicle i using only local available information such that (i) φ(i) (t) ∈ S0 for all t ≥ 0, (ii) φ(i) (t) enters S in finite time. A. Solving Problem 3.1 To solve Problem 3.1, we devise the following distributed control for each vehicle when its state is in S, i.e., as long as φ(i) ∈ S, i = 1, . . . , n: (i) R+ye(i) vc = [c − Γ (ψi − ψi−1 )] (i) cos θe i (3) h (i) (i) (i) (i) (i) cos θe(i) ωc = vc − k1 (ye + kθe + 2 sin θe ) − (i)
Proof: First, consider a positive definite function i 1 h (i) (i) V1 = (ye + kθe(i) )2 + (ye(i) )2 , i = 1, 2, . . . , n. 2 Take the derivative along the solution of (2) with controller (3). After several steps of calculation, we obtain (i) V˙ 1
(i)
(i)
(i)
(i) (i)
Since φ(i) (0) ∈ S for all i, φ(i) (t) ∈ S for all t by Theorem 3.1, which implies π |θe(i) (t)| ≤ a < , ∀t > 0. 2 (i)
(i)
(i)
(i)
Hence, θe sin θe ≥ 0. Furthermore, θe sin θe = 0 if and (i) (i) only if θe = 0. Recall that vc is always positive. So we (i) know that V˙ 1 is negative definite with respect to the origin. Hence, for any i = 1, . . . , n, φ(i) (t) → 0 as t → ∞.
(4)
Second, we consider a function
R+ye
where c > 0, k ≥ 1 are constants and the function Γ : R → 0, x ≥ 0, R is defined as Γ(x) = −1, x < 0. With the distributed control above, we first show in Theorem 3.1 that if a vehicle initially has its state in S, then its state remains in S, i.e., the set S is positively invariant for the close-loop dynamics of vehicle i. Theorem 3.1: For any vehicle i with control law (3), if φ(i) (0) ∈ S, then φ(i) (t) ∈ S for all t ≥ 0. Next, we show that if all vehicles initially have their states in S and if the central angles of any two neighbor vehicles are nonzero, then the central angles keep nonzero as the system evolves under control law (3). That means, if all vehicles’ states are in the set S and no two vehicles are located in a same ray, then the neighboring relationship does not change. For notation simplicity, we renumber the vehicles by rotating a ray counterclockwise such that ψi has endpoints located at vehicle i and i + 1 for all i = 1, . . . , n. In this paper, we use a circular index. That is, we use the same notation i + 1 for all i = 1, . . . , n − 1, but when i = n, it means that the index returns to 1. Theorem 3.2: Suppose φ(i) (0) ∈ S for all i = 1, . . . , n and consider control law (3) for each vehicle. If ψi (0) > 0 for all i = 1, . . . , n, then ψi (t) > 0 for all i = 1, . . . , n and for all t ≥ 0. Remark 3.1: That ψi (t) > 0 for all i and all t ≥ 0 guarantees that no collision occurs between vehicles. Finally, we show that each vehicle converges to a desired circular orbit (φ(i) (t) → 0 as t → ∞) and they are evenly spaced (limt→∞ ψ1 (t) = · · · = limt→∞ ψn (t) > 0). That is, the distributed control solves Problem 3.1. Theorem 3.3: If initially φ(i) (0) is in S for all i and no two vehicles are on a same ray originated at the target, then the group of unicycles achieve cooperative target tracking using distributed control law (3).
(i)
= (ye + kθe )(y˙ e + k θ˙e ) + ye y˙ e (i) (i) (i) (i) (i) (i) = −vc (ye + kθe )2 − kvc θe sin θe .
n
V2 =
1X 2 (ψi − ψi−1 ) . 2 i=1
Then V2 is positive when the angles are not all equal and is 0 when they are all equal. Taking the derivative leads to V˙ 2 =
n X
(ψi − ψi−1 )(ψ˙ i − ψ˙ i−1 ).
i=1
Denote βi = ψ˙ i − ψ˙ i−1 . Thus, βi = Γ(ψi+1 − ψi ) + Γ(ψi−1 − ψi−2 ) − 2Γ(ψi − ψi−1 ). Recall that Γ(·) is either 0 or -1. Hence, if ψi − ψi−1 ≥ 0, then Γ(ψi − ψi−1 ) = 0 and βi = Γ(ψi+1 − ψi ) + Γ(ψi−1 − ψi−2 ) ≤ 0. If ψi − ψi−1 < 0, then Γ(ψi − ψi−1 ) = −1 and βi = Γ(ψi+1 − ψi ) + Γ(ψi−1 − ψi−2 ) + 2 ≥ 0.
(5)
Thus, (ψi − ψi−1 )βi ≤ 0,
i = 1, . . . , n,
which gives V˙ 2 =
n X
(ψi − ψi−1 )βi ≤ 0.
i=1
Furthermore, V˙ 2 = 0 implies ψi − ψi−1 ≥ 0 for all i. (This can be seen by contradiction. Suppose that there is an i∗ such that ψi∗ − ψi∗ −1 < 0. Then in order to make V˙ 2 = 0, βi∗ has to be zero, which means ψi∗ +1 − ψi∗ < 0 from (5). Repeating this Pnargument, it follows that ψi+1 − ψi < 0 for all P i. Thus, i=1 (ψi − ψi−1 ) < 0, which is a contradiction n to i=1 (ψi − ψi−1 ) = 0.)
When V˙ 2 = 0, we Pnestablish that ψi − ψi−1 ≥ 0 for all i. Taking the fact i=1 (ψi − ψi−1 ) = 0 into account, we
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PSfrag replacements θe
π
obtain that ψi − ψi−1 = 0 for all i. Hence, it follows that lim ψ1 (t) = · · · = lim ψn (t).
t→∞
t→∞
Moreover, Pn we known ψi (t) > 0 all the time by Theorem 3.2 and i=1 ψi = 2π. Hence, the vehicles are eventually spaced on the circle with equal angular distance. In conclusion, the group of unicycles with distributed control law (3) achieves cooperative target tracking. Remark 3.2: In practice, it is reasonable that no two vehicles are on a same ray due to measurement errors and/ro perturbations.
a− −R
−a
a
S
R
ye
−a −π
Fig. 8.
Phase portrait of the resulting linear closed-loop system.
B. Solving Problem 3.2 We devise distributed control in this subsection to solve problem 3.2. Based on the partition in Fig. 6, we construct controllers for each vehicle such that the abstract state PSfrag replacements transition in Fig. 7 happens. S1
S5
S4 Fig. 7.
S2
S
(i)
(i)
If sin θe < δ, (namely, θe ∈ (π − , π)), then the resulting closed system is ( (i) y (i) sin θ y˙ e = −k1 e δ e , (i) (i) θ˙e = −k2 (θe − a + ). (i)
Note that on the boundary θe = π, θ˙e = −k2 (π − a + ) < (i) R sin θe(i) 0; on the boundary ye = −R, y˙ e = k1 > 0; and δ (i)
R sin θ (i)
e on the boundary ye = R, y˙ e = −k1 < 0. Hence δ no trajectory leaves S0 with the above nonlinear dynamics. Moreover, we are able to find a vector ξ = [0, −1]T such that
S3
State transition graph.
In the paper we use the notation X → Y to represent the reachability specification from one state set X to its adjacent state set Y, namely, for all initial state ψ (i) (0) ∈ X there exists T > 0 such that (a) ψ (i) (t) ∈ X for all 0 ≤ t < T ; (b) ψ (i) (t) ∈ Y when t = T . First, we solve the problem S1 → S. In S1 , we use feedback-linearization technique to design a control law so that the resulting closed-loop system is a linear system in the plane with its equilibrium point at (0, a − ) and its phase portrait like Fig. 8. Thus, the control law is given by ye(i) vc(i) = −k1 (i) Satδ (sin θe ) (6) (i) θe(i) ωc(i) = −k2 (θe(i) − a + ) − vc cos(i) R+ye
where k1 > k2 > 0 are constants and Satδ (·) is a saturation function used to avoid the singularity and is defined as |x| ≥ δ x, δ, 0≤x 0 for all θe(i) ∈ (π − , π), which means all trajectories all the way flow down to the linear region and then enter S. Based on symmetrical properties, we have the following control law for the problem S2 → S: ye(i) vc(i) = −k1 (i) , Satδ (sin θe ) (7) (i) θe(i) ωc(i) = −k2 (θe(i) + a − ) − vc cos(i) . R+ye
Corollary 3.1: Under control law (7), S2 → S. Next, we consider states in S3 . We choose the following control law ( (i) (i) vc = −θe , (8) (i) v (i) cos θe(i) ωc = −1 − c . (i) (R+ye )
Theorem 3.5: Under control law (8), S3 → S ∪ S2 . The similar argument applies to S4 so that the control law ( (i) (i) vc = θ e (9) (i) v (i) cos θe(i) ωc = 1 − c (i)
with δ chosen to be sin(). Theorem 3.4: Under control law (6), S1 → S. (i) (i) Proof: If sin θe ≥ δ, (i.e., θe ∈ [a−, π −]), the resulting closed-loop system with control law (6) is ( (i) (i) y˙ e = −k1 ye , (i) (i) θ˙e = −k2 (θe − a + ),
C. Hybrid Controller
whose phase portrait looks like Fig. 8. It can be seen that the trajectories initiated in S1 enter S without crossing any other boundaries.
We now construct a distributed hybrid control for cooperative target tracking in D0 . For each vehicle, define a hybrid control based on the following table (for instance, using
(R+ye )
solves the reachability problem S4 → S ∪ S1 . Corollary 3.2: Under control law (9), S4 → S ∪ S1 .
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control law (6) when vehicle i’s state lies in S1 ): (6) S1
(7) S2
(8) S3
(9) S4
(3) S
R EFERENCES (10)
Then we have the following main result. Theorem 3.6: Hybrid control (10) solves the cooperative target tracking problem for almost all initial states in D0 . Proof: For each unicycle i, the control law switches depending on its state φ(i) (t). By Theorem 3.4–3.5 and Corollary 3.1–3.2, it follows that every vehicle enters S in finite time according to the state transition in Fig. 7. Moreover, no vehicle leaves S when it enters S by Theorem 3.1. Hence, without loss of generality, say at T1 , all vehicles’ states are in S. By Theorem 3.3, the group of vehicles eventually achieves cooperative target tracking if no two vehicles are on a same ray at T1 . Since the set of states for which two vehicles are on a same ray is of low dimension, the conclusion follows. IV. S IMULATION We present a simulation of five unicycles with the stationary target at (0, 0). The radius of desired enclosing circle is R = 3m. the initial postures of unicycles are randomly generated. Control parameters are chosen as c = 0.5, a = 1.5 < π2 , = 0.1, k = 4, k1 = 5, k2 = 3. Fig. 9 shows the trajectories in the plane, where blank wedges represent the initial postures and filled wedges represent the final postures of the simulation. All unicycles come to run on the enclosing circle clockwise and are evenly spaced.
PSfrag replacements
4 3 2 1 0 −1 −2 −3 −4 Fig. 9.
−2
0
2
4
6
Trajectories of four unicycles in the plane.
V. C ONCLUSION The paper deals with the cooperative target tracking problem based on hybrid control and reachability specification. Vehicles are steered to a region near the target with certain orientation condition satisfied and then coordinate their motion to cooperatively capture the target through an enclosing circular motion around the target. The hybrid nature of independent motion towards the target without coupling in the further range and coordinated motion in the closer range of the target meets the requirement of cooperative target tracking in practice.
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