Rendezvous in space with minimal sensing and

Rendezvous in space with minimal sensing and coarse actuation Soumya R. Sahoo ∗ Ravi N. Banavar ∗∗ Arpita Sinha ∗∗∗ ∗

Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai, India (e-mail: [email protected]) ∗∗ Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai, India (e-mail: [email protected]) ∗∗∗ Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai, India (e-mail: [email protected]) Abstract: In this paper we propose a control law for achieving rendezvous of autonomous vehicles capable of moving in three-dimensional (3D) space by using minimal data and quantized control. A preassigned graph uniquely assigns the pursuer-target pair in a cyclic manner. The measurement required for the proposed control law is the position from which the target vehicle moves out of the field-of-view of the pursuing vehicle. A Lyapunov function is chosen to find a range for the field-of-view which would guarantee rendezvous under the proposed control law. Keywords: consensus; rendezvous; minimal sensing; quantized control; UAVs 1. INTRODUCTION Autonomous vehicle systems have found potential applications in military operations, search and rescue, environment monitoring, commercial cleaning, material handling, and homeland security. While single vehicles performing solo missions have yielded some benefits, greater benefits will come from the cooperation of a team of vehicles. A multi-agent system is robust to failure than a single agent. It is more effcient than individual agents in certain cases. It is also possible to reduce the size of the agents and operational cost, increase system reliability in case of a multi-agent system. This has aroused interest of the control community in cooperative control and consensus algorithms. Ren et al. [2005] mention various consensus algorithms in multi-agent coordination. Control strategies to achieve consensus like formation control or rendezvous under cyclic pursuit have been developed; see, e.g., Bruckstein et al. [July 1991], Lin et al. [2003], Marshall et al. [June 2004], Marshall et al. [November 2004], Sinha and Ghose [November 2006], Pavone and Frazzoli [2007], Ramirez et al. [2009]. Sometimes tasks have to be performed with minimal data available. Minimal data availability may be due to lower bandwidth available, security reasons, compact design of agents or availability of less sensors due to failure of other sensors of the agents. Focusing on minimal data helps increase the robustness of the system, results in simpler and compact design of agents, and lowers the production cost. Both quantised control and reduced sensing focus on minimal data. In Yu et al. [December 2008] it has been shown that using minimal data, rendezvous of agents on a plane can be achieved. The agent just detects whether its target(agent assigned to it) moves out of its left or right side of the windshield and the turns accordingly to bring the latter into its view(or windshield). However, autonomous agents like UAVs, autonomous underwater vehicles (AUVs) always move in a 3D space. Further, these agents may not be on a plane initially. So the control laws in

a planar case will not hold and these need to be modified for agents capable of moving in 3D space. In this paper we present the rendezvous of agents in 3D space. The agent in our case is considered to be an aerial vehicle. The paper is organized as follows: section 2 talks about the vehicle model, the sensors and the control law used to achieve rendezvous. Section 3 deals with some graph theory, the formation of a Lyapunov function and a condition on the angle of the windshield. In section 4 simulation results for the problem have been presented and discussed. 2. PROBLEM FORMULATION We assume n agents in the system. Each agent is an UAV. The schematic of the agent is shown in Fig. 1. The agent can yaw and pitch with constant angular speed, and move straight with constant speed. The agent, however, cannot roll or slip laterally. Each agent has a conical field-of-view with infinite range within which it trys to maintain its target. The target for ith agent is (i + 1)mod n. The control is applied when the target moves out of the windshield. The vehicle model, sensors and control have been discussed in the following subsections. 2.1 Vehicle Model Let pi = (xi , yi , zi ) ∈ R3 be the position of the ith agent in an earth-fixed frame and (αi , βi , γi ) be the Euler angles corresponding to the Z-Y-Z Euler angle convention for transformation from the body-fixed frame to the earth-fixed frame. See Fig. 1. • The forward (linear) velocity of the vehicle is only along its body Xb axis and its magnitude (vi ) remains constant. • The vehicle can rotate about its body Yb -axis (pitch) and about its body Zb -axis (yaw). See Figure 2. The pitch and yaw rates assume values from the discrete set, ωyib ∈ {−ωi , 0, +ωi } and ωzib ∈ {−ωi , 0, +ωi }, ωi is

Yb pitch Xb yaw Zb Z Orientation(αi , βi , γi ) Y

  " # α˙i cos αi cos βi sin αi cos βi − sin βi 1  β˙i  = − sin αi sin βi − cos αi sin βi 0 sin βi − cos αi − sin αi 0 γ˙i   0 × ωyib  ωzib (2) Equations (1) and (2) describe the kinematic model of the ith vehicle. 2.2 Sensors

X Fig. 1. Schematic of vehicle and transformation from bodyfixed frame to earth-fixed frame

Z ib ωzib ωyib

The sensor of each agent has a conical view with the half angle of the cone being φ ∈ (0, π) as shown in Figure 3. This fieldof-view is termed as the windshield. The range of view within this angle is assumed infinite. It is also assumed that another vehicle cannot occlude the view of the agent if it appears within the windshield. The field-of-view, when seen from inside the agent, appears like a disc. This disc can be divided into four quadrants as shown in Figure 4. The sensors do not give the actual distance between agents. They only give a discrete output based on the quadrant from which the assigned agent(the target) moves out.

Yib φ

vi

Xib

Fig. 2. Velocity of agent and controls applied w.r.t. body axes constant and ωi > 0. The agents are considered identical and hence vi = vj and ωi = ωj . Since the model involves rigid body motion (translation and rotation) in space, we relate the inertial velocity components to the body velocity components through the Euler angles as " # " # x˙ i vi y˙ i = Rzi3 Ryi2 Rzi1 0 (1) z˙i 0 where (x˙ i , y˙ i , z˙i ) are the velocity components in the inertial frame and (vi , 0, 0) are the velocity components in the body-fixed frame. Rzi3 , Ryi2 , Rzi1 are rotation matrices parametrized by the Euler angles as " # cos αi sin αi 0 Rzi1 = − sin αi cos αi 0 0 0 1 " Ryi2 =

cos βi 0 − sin βi 0 1 0 sin βi 0 cos βi

" Rzi3 =

cos γi sin γi 0 − sin γi cos γi 0 0 0 1

#

#

The Euler angle rates are related to the vehicle angular velocities as

Fig. 3. The conical field-of-view of the agent 2.3 Controls As mentioned earlier, the sensors only give a discrete output based on the quadrant from which the target agent moves out. Let the output set be O and the sensor measurement of the ith vehicle be (oyi , ozi ) ∈ O. (oyi , ozi ) takes the following values according to the escape of the target agent j from view: (1, 1) agent j disappears from the 1st    quadrant (from right and top or     only right)     (−1, 1) agent j disappears from the 2nd     quadrant (from left and top or   only top or only left) (oyi , ozi ) = (−1, −1) agent j disappears from the 3rd     quadrant (from left and bottom or     only bottom)     (1, −1) agent j disappears from the 4th    quadrant (from right and bottom)   (0, 0) agent j is in the view (3) in which agent j is the target agent of i. These values are also indicated in Fig. (4). The output of the sensors actuate the controllers for necessary action. The control law can be defined based on the output as: (ωyib , ωzib ) = (ozi , oyi )ωi (4) As can be seen, this control law involves no history or state estimation. Now we assume a cyclic connection and try to find a condition on the windshield angle φ and the angular speed ωi such that using this control law the agents achieve rendezvous.

Let V : R3n → R be a function which is defined as X V = li,j

z+ (−1, 1)

(1, 1)

y−

y+

(−1, −1)

(1, −1) z−

Fig. 4. Field-of-view is a disc when seen from agent 3. GRAPH THEORY AND LYAPUNOV CANDIDATE 3.1 Concepts from graph theory The multi-agent system can be denoted by a directed graph G = (V, E) with the agents being its nodes ∈ V. The directed edge ei,j ∈ E(G), the edge set, if agent i is pursuing or communicating with or can sense agent j. In our case, if agent j is assigned to agent i then the edge ei,j exists. Assignment of agent i means the target agent which i is supposed to pursue. A graph defining the assignments of the agents in the multiagent system is called an assignment graph. For simplicity and without any loss of generality we assume that (mod(i, n) + 1) is assigned to i and so on. This type of assignment results in the formation of a cyclic graph. A pursuit defined by a cyclic graph is called a cyclic pursuit. Let the distance between agents i and i + 1 be denoted by li,i+1 . Initially there are n agents and hence G has n vertices. As agent i catches up with agent i + 1, i and i + 1 move as one entity i + 1. This is called merging. The merging operation is triggered when the distance between the pursued and the pursuer reduces to the merging radius ρ or less. The merging radius is the distance between the pursued and the pursuer after which they merge and move as one entity. Merging occurs if ei,i+1 ∈ E(G), li,i+1 ≤ ρ After merging, agent i − 1 which was pursuing i starts pursuing i + 1. The node i is deleted and the edges ei−1,i and ei,i+1 are deleted from E(G) and a new edge ei−1,i+1 comes into effect. The number of nodes also is reduced. The graph G is said to be live if it has at least one edge(Yu et al. [December 2008]). A graph with a single vertex is also a live-graph. If G is not live then there exists more than one vertex with no edge. Lemma 1. (Lemma 1, Yu et al. [December 2008]) The assignment graph G is live for all t ≥ 0, under arbitrary evolution of the agent position if and only if it is connected at time t = 0. 3.2 A Lyapunov candidate We construct a function which is the sum of the distances between the agents i and i + 1. It is positive definite because the distances will always be positive and will only go to zero when all the agents have rendezvous. We need the function to decrease monotonically so that it reaches zero. So, its timederivative has to be negative definite. To ensure this derivative to be negative definite we later on find a condition on the windshield angle φ.

(5)

ei,j ∈E(G)

Since V is the sum of distances, it will be always positive and will only go to zero when edges do not exist. So V is a suitable candidate for being the Lyapunov function of the system. V is also called a graph compatible Lyapunov function as it is based on the digraph G (Yu et al. [December 2008]). The timederivative of V is given by X V˙ = l˙i,j (6) ei,j ∈E(G)

Lemma 2. (Lemma 2, Yu et al. [December 2008]) A graphcompatible Lyapunov function V is rendezvous positive definite if and only if its assignment graph G is connected. Here we are considering the rendezvous problem in a cyclic case. It is assumed that agent i is assigned to pursue agent (mod(i, n) + 1). The digraph G is also cyclic. It has n nodes with n edges. 3.3 Condition on the windshield angle pi+1 ψi+1 vi

li,i+1

θi+1

φi

v i+1 φi+1 li+1,i+2

pi pi+2 Fig. 5. Two consecutive agents in cyclic pursuit As shown in Fig. 5, which is in 3D space, • pi , pi+1 , pi+2 are the positions of agents i, i + 1 and i + 2 respectively in the earth-fixed frame. • li,i+1 , li+1,i+2 are the lines-of-sight of agent i and i + 1 respectively. • φi is the angle between the velocity vector v i and li,i+1 . • φi+1 is the angle between velocity v i+1 of the agent i + 1 and li+1,i+2 . • The angle between li,i+1 and li+1,i+2 is θi+1 . • ψi+1 is the angle between v i+1 and li,i+1 . • The rate at which the distance between the ith and (i + 1)th agent reduces is dependent on the resultant of the component of the respective velocity vectors along the line joining the two agents i.e. li,i+1 Thus l˙i,i+1 can be expressed as l˙i,i+1 = −vi+1 cos(ψi+1 ) − vi cos(φi )

(7)

Summing (7) over all i we get n X ˙ V = −vi (cos(φi ) + cos(ψi ))

(8)

i=1

Assuming all agents to have unit speed, (8) becomes n X V˙ = − (cos(φi ) + cos(ψi )) i=1

(9)

A necessary condition for rendezvous to occur is that V˙ is strictly less than zero. Theorem 3. Unit speed cyclic pursuit of n agents with kinematics given by (1) and (2) will rendezvous if the agents maintain their targets within the windshield and the windshield angle φ satisfies π/2 n=2 0 0. Hence, trajectories n with V˙ > 0 also exist when φ ∈ ( nπ , π]. Existence of V˙ ≥ 0 is not desirable. So we rule out the domain φ ∈ [ nπ , π]. Now we have to find a condition on φ such that it is guaranteed that V˙ is always less than 0 for any initial condition. We now state another result on a closed polygonal line that we employ in the proof of our main result. If βi is the angle formed by the polygonal line at the i-th vertex such that 0 < βi < π and αi be the supplementary angle of βi then the following lemma holds. Lemma 6. (Lemma 3, Bruckstein et al. [July 1991]) In any 3 closed Pn polygonal line with Pnn segments in R , the inequalities β ≤ (n − 2)π and α ≥ 2π, hold. Equality occurs i=1 i i=1 i iff the polygonal line is a planar convex polygon. Now we find a condition on φ such that V˙ < 0. When n = 2, ψ1 = φ2 and ψ2 = φ1 . So from (9), 2 X (cos (φi ) + cos (ψi )) V˙ = − i=1

For V˙ < 0, we should have P2 − i=1 (cos (φi ) + cos (ψi )) < 0 ⇒ −2(cos (φ1 ) + cos (φ2 )) < 0 ⇒ cos (φ1 ) + cos (φ2 ) > 0

(13)

(13) holds if φ < π/2 imposing cos (φ1 ) > 0 and cos (φ2 ) > 0. Thus, for n = 2, when φ < π/2, V˙ < 0. We now check for n ≥ 3. From Lemma 5 and the fact that φi < φ we have, 0 < φi < φ < π/n (14) For any closed polygonal line we can always consider the smaller angle as interior angle. So 0 < θi < π (15) From (14) and (15) we have 0 < θi + φi < π + π/n (16) For 0 ≤ ψi ≤ π and 0 < θi + φi ≤ π, from (12) we have, ψi < θ i + φ i ⇒ cos (ψi ) > cos (θi + φi ) (17) ⇒ − cos (ψi ) < − cos (θi + φi ) For 0 ≤ ψi ≤ π and π < θi + φi < π + π/n, from (12) we have, ψi < 2π − (θi + φi ) ⇒ cos (ψi ) > cos (2π − (θi + φi )) (18) ⇒ cos (ψi ) > cos (θi + φi ) ⇒ − cos (ψi ) < − cos (θi + φi ) Hence, from (17) and (18) we have − cos ψi < − cos (θi + φi ) (19) From (9) and (19) we have

V˙ ≤ −

n X (cos φi + cos (θi + φi )) =: V˙ 1

(20)

i=1

V˙ 1 can now be written as the sum of two functions, f and g as follows n X f := − cos (θi + φi ) (21)

Note that by converting the inequality constraint to an equality constraint we have introduced an additional variable β into the problem. For a stationary point we take the partial derivatives with respect to θi ’s i = 1 . . . n, β and λ of (28) and equate it to zero to give sin (θi + φi ) = λ, ∀ i 2λβ = 0 H=0

i=1

g := −

n X

cos φi

(22)

i=1

(29)

From 2λβ = 0 we have the following possibilities for β and λ. (β = λ = 0), (β 6= 0, λ = 0), (β = 0, λ 6= 0)

Note that −n ≤ f, g ≤ n. On the basis of (16) we can have two sets:Θout = {(θ1 , . . . , θn )|∃ i, (θi + φi ) ∈ [0, π/2)} (23) Θin = {(θ1 , . . . , θn )|∀ i, (θi + φi ) ∈ [π/2, π + π/n)} (24) We have two disjoint sets defined as above as behavior of f is different in Θout and Θin . We now analyse the behavior of f in Θout . If (θ1 , . . . , θn ) ∈ Θout then atleast for one i, (θi + φi ) must belong to [0, π/2). Let for i = k, θk + φk ∈ [0, π/2). So

For the first 2 solutions, when λ = 0 we have from (29), ⇒

sin (θi + φi ) = 0, ∀i (θi + φi ) = ±nπ, ∀i, n = 0, ±1, ±2, . . .

Since we are analyzing f in Θin , the values of (θi + φi ) must belong to [π/2, π + π/n), ∀i. So we need to consider the following case only: (θi + φi ) = π And thus

cos (θk + φk ) > 0 ⇒ − cos (θk + φk ) < 0

n X (θi + φi ) = nπ

Now,

(30)

i=1

Pn f = − i=1 cos (θi + φi ) Pn−1 = − cos (θk + φk ) − i=1,i6=k cos (θi + φi ) < 0 + (n − 1) = (n − 1)

From (14) and (26), we have X 0< (θi + φi ) < (n − 1)π, ∀i

Hence, f 0 and hence from (29) we have λ > 0 and therefore, sin (θic + φi ) > 0, ∀i. We are considering the Θin set. So, there is only one value of (θic + φi ) such that sin (θic + φi ) > 0. But this holds for all i. So, (θ1c +φ1 ) = (θ2c + φ2 ) = · · · = (θnc + φn ). Thus, f (Θ, Φ) has a single stationary point in Θin . Lemma 8. If the stationary point is a point of maxima in Θin , then the maximum value obtained by f is (n − 2)π + nφ (33) f ≤ −n cos n Proof. At the single stationary point since sin (θic + φi ) = λ > 0, ∀i, (θ1c + φ1 ) = (θ2c + φ2 )P = . . . = (θnc + φn ). Thus, at the n

stationary point, (θic + φi ) =

i=1

(θic +φi ) . n

∇θ1 ,··· ,θn ,β,λ fmod = (sin (θ1 + φ1 ) − λ, X · · · , sin (θn + φn ) − λ, −2λβ, −( θi + β 2 − (n − 2)π) We now compute the second derivative of fmod to charaterize a maxima or minima.

∇2θ1 ,··· ,θn ,β,λ fmod |(θic +φi , ∀i) =  cos (θ1c + φ1 ) 0 ··· 0 0 0 cos (θ2c + φ2 ) · · · 0 0   .. .. .. .. ..  .  . . . .   0 0 · · · cos (θnc + φn ) 0  0 0 ··· 0 −2λ −1 −1 ··· −1 0

in (10) then rendezvous is guaranteed. In the following section we have presented some simulation results.

 −1 −1 ..   .   −1 0

4. SIMULATION RESULTS FOR RENDEZVOUS IN 3D We have considered a “five-agent” system. The agents start from random points in space with the respective target agent inside the field-of-view. The forward speed for each agent is 10units/second and ωi = 200rad/second ∀ i. The simulation was done for various values of φ.

0

All the principal minors of −(∇2θ1 ,··· ,θn ,β,λ fmod |(θic +φi , ∀i) ) are positive. So −(∇2θ1 ,··· ,θn ,β,λ fmod |(θic +φi , ∀i) ) is positive definite and hence, (∇2θ1 ,··· ,θn ,β,λ fmod |(θic +φi , ∀i) ) is a negative definite matrix. Thus, the stationary point is a point of maxima. Pn f = − i=1 cos (θic + φi ) c = −n cos (θP i + φi ) n

= −n cos

i=1

≤ −n cos ≤ −n cos

(θic +φi ) n

Pn

(n−2)π+ n

(n−2)π+nφ n

i=1

φi

Fig. 8: Five agents with φ = 0.2π/5, ω = 200, v = 10

Lemma 9. For n agents in cyclic pursuit with unit speed, f (Θ, Φ) satisfies (n − 2)π + nφ f (Θ, Φ) ≤ max n − 1, −n cos (34) n Proof. f is a continuous function in Θout ∪ Θin . From (25) and (33) the n maximum value that f can oachieve will be less than max n − 1, −n cos (n−2)π+nφ . n In continuation of the proof of Theorem 3, for V˙ 1 < 0, −g has to bengreater than f . To −g has to be greater than ensure thiso max n − 1, −n cos (n−2)π+nφ . So, n (n − 2)π + nφ max n − 1, −n cos < −g n From (22), we have, n X (n − 2)π + nφ max n − 1, −n cos < cos (φi ) n i=1 (35) From (14) 0 < φi < φ < nπ . So, cos φ < cos φi . So, to satisfy (35) for all φi ’s the following must be true. o n < n cos (φ) max n − 1, −n cos (n−2)π+nφ n n o (n−2)π+nφ n−1 ⇒ max n , − cos < cos (φ) n −1 ⇒ φ < min cos π

Fig. 9: V of five agents with φ = 0.2π/5, ω = 200, v = 10 Now we increase φ.


n−1 n

,

n

Hence, for n ≥ 3, and V˙ 1 to be less than zero, we should have n−1 π −1 φ < min cos , (36) n n As from (20), V˙ < V˙ 1 , so for the same range of φ, V˙ < 0 also. Hence, the condition on φ is proved for n ≥ 3.

Fig. 11: V of five agents with φ = 0.5π/5, ω = 200, v = 10

If the agents maintain their target always within their windshield and have their windshield angle satisfying the condition

It can be seen that the increase in φ increases the rendezvous time.





Further increase in φ increases the time for rendezvous. By intuition as the windshield angle increases the controls become more coarse. The target-agent also stays for a longer time in the field-of-view of the pursuing agent. To check whether rendezvous ocurred if condition (10) was violated, we carried out a few simulations.




Fig. 15: V of five agents with φ = 1.02π/5, ω = 200, v = 10 It was seen that for φ violating the condition (10) the agents still rendezvous. The condition on φ is a sufficient condition. However, for sufficiently larger value of φ the agents diverge. See Fig. 16 and 17. Fig. 20: Five agents with φ = 0.2π/5, ω = 2, v = 10

REFERENCES



Fig. 23: V of five agents with φ = 0.2π/5, ω = 200, v = 100 It can be observed from Fig. 18-23 that as the ωv ratio decreases the agents come much closer to each other because the agents can turn rapidly and bring their target-agent back to their fieldof-view. When ωv > 1 the distance between the agent also increases and the distance between them exhibits a sinusoidal pattern whose amplitude increases with increase in the ratio.

5. SUMMARY AND FUTURE SCOPE OF WORK Using a system of autonomous agents, each of whom has a forward speed, and the yaw and pitch control is available, we have shown that without communication, and with minimal sensing and quantized control we are able to achieve rendezvous. We have obtained a sufficient condition on the windshield angle φ which guarantees rendezvous. The simualtion results match the predicted results. As an extension to the present work some strategies for collison avoidance between agents i and j, (j 6= i + 1) can be investigated. One could also investigate if the agents can rendezvous at a specific location in space.

ACKNOWLEDGEMENTS The authors acknowledge useful discussions with Anup Menon.

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