Motion equations with distance mismatches

agents, or translation with each agent moving at the same velocity1. Two examples of such planar collective motions have also been studied in [Sepulchre et al., 2008] (for a different problem) in which agents are assumed to have constant unit speed. The collective motion in the 3-D space has an additional degree of freedom, and thus a helical motion becomes possible.

A similar collective helical motion (with parallel motion or circular motion as special cases) for a group of unit-speed agents was discussed in some previous pa- pers; see e.g. [Justh and Krishnaprasad, 2005; Scardovi et al., 2008]. However the problem formulation and motion generation mechanism discussed in [Justh and Kr- ishnaprasad, 2005; Scardovi et al., 2008] are very different to that arising in rigid formation control to be discussed here. These differences include (i) that in contrast to the system model used in [Justh and Krishnaprasad, 2005; Scardovi et al., 2008], we do not assume constant unit speed in agents’ kinematics; (ii) that the collective helical motion discussed here must be consistent with the existence of a rigid formation shape; and (iii) that the rigid motion discussed in this chapter is caused by distance mismatches.

Also, the results in this chapter indicate one interesting mechanism on how to generate rigid motions with specified rigid formation shapes, which may have po- tential applications for controlling and generating rigid motions for undirected rigid formations with inter-agent distance constraints. We also note that in the literature, helical and spiral motions have been considered as useful motions with particular applications, e.g. for gliding robotic fish [Zhang et al., 2014] and [Zhang et al., 2016] (although the mechanism for generating helical motions discussed in [Zhang et al., 2014] and [Zhang et al., 2016] is different to the result in this chapter).

5.1.2 Chapter organization

The remaining parts of this chapter are organized as follows. Section 5.2 presents the problem description and then sets up some key equations of agent motions. In Section 5.3 we focus on the property and convergence of formation shapes, and a relative equilibrium analysis for the additional rigid motion. Section 5.4 shows motion formulas to describe the formation movements in terms of distance mismatches and their applications on steering and controlling rigid formation motions. Section 5.5 concludes this chapter.

5.2 Motion equations with distance mismatches

We recall some notations from Chapter 2. Let dkij denote the desired distance of edgekwhich links agentiand j. The control goal in formation shape stabilization is to drive all the agents to reach a configuration such that a certain set of inter-agent distances can be achieved. We assume that from agenti’s perspective, the specified 1_{The research group led by Prof. Ming Cao at the University of Groningen performed several exper-}

iments using a group of ground robots to verify such rigid motions in rigid formation control. Videos are available athttps://www.youtube.com/user/noeth3rperformed by Héctor Garcia de Marina.

ķ ĺ ĸ 1

d

Figure 5.1: An undirected rigid tetrahedron formation.

target distance between agent iand neighbor j isdij where dij is a positive number which is approximately equal to dkij. Following the gradient descent control law in Section 2.5 of Chapter 2, we consider the following formation control system in which the control law for agentiis described as2

xi =ui =

∑

j∈Ni

(xj−xi)(kxj−xik2−d2ij) (5.1) Note that the above gradient control is distributed in the sense that its implementa- tion only requires measurements of relative positionsof neighboring agents, denoted byxj−xi.

Example: We show an example of a 3-D tetrahedron formation to illustrate the

derivation of the system equations described above. Consider a tetrahedron formation in the 3-D space, which consists of four agents labeled by 1, 2, 3, 4. For the purpose of writing an oriented incidence matrix, suppose that the edges are oriented from ito j just when i < j. Then we can number the edges in the following order: 12, 23, 34, 13, 24, 14; see Figure 5.1. Thus, the following oriented incidence matrix for the undirected graph in Figure 5.1 can be obtained

H=          −1 1 0 0 0 −1 1 0 0 0 −1 1 −1 0 1 0 0 −1 0 1 −1 0 0 1          (5.2)

The relative position vector z is then defined according to (2.1) (in Chapter 2). As an example, one has z1 = x2−x1, i.e., the vector z1 at edge 1 is defined by the relative position between agent 2 and agent 1. Also, from (5.1) one can further obtain 2_{Note that in this chapter, we use}_x_{instead of}_p_{to denote agents’ positions. This is for the consis-}

§5.2 Motion equations with distance mismatches 59

the dynamical system for each agent in the tetrahedron formation control. Again, as an example, the dynamical system for agent 1 can be written as

x1 =u1 =

∑

j∈N1

(xj−x1)(kxj−x1k2−d21j), j=2, 3, 4. (5.3) and the equations for other agents can be obtained similarly.

Unlike the problem settings in [Krick et al., 2009; Dimarogonas and Johansson, 2010; Cortés, 2009; Dörfler and Francis, 2010; Oh and Ahn, 2014a], we assume in this chapter that the perceived distances dij and dji for neighboring agents iand j, respectively, are not necessarily equal. The following formulation follows similarly from [Mou et al., 2016]. The distance inconsistency is assumed to satisfy|dij−dji| ≤ βji where βji is a small nonnegative number bounding the discrepancy from the two agents’ understanding of what the desired distance between them should be. Furthermore, the misbehavior actually stems from the mismatch (the difference, or discrepancy) betweendij anddji rather than the assumption that both dij anddji are only approximately equal todkij. In other words, only the difference between mutual distances in each edge matters in the modelling of distance mismatch. Without loss of generality and to simplify the equations in the sequel, we will henceforth assume that dij exactly equalsdkij for all adjacent vertex pairs(i,j)for whichiis the head of edge kij. Next, denoteµkij = d

ij−d2ji as the constant distance mismatch corresponding to edgek_ij; clearly, one has

d2_ij =d2_kij,d2_ji=d2_kij−µkij (5.4) Letekijdenote the distance error of thek-th edge:

ekij(z) =kzkijk 2₋_d2

kij We denote by N+

i the set of all j∈ Ni for which vertexiis the head of the oriented edge kij, and denote by N_i− the complement of N_i+ in Ni. Then the equation for agent i’s motion in the presence of distance mismatch can be written as (see also [Sun et al., 2013; Mou et al., 2016])

˙ xi =−

∑

j∈Ni (xi−xj)ekij(z) =−

∑

j∈N+ i z_kije_kij(z) +

∑

j∈N− i

z_kij(e_kij(z) +µ_kij) (5.5) wherezkijrefers to thekth block entry of the relative position vectorzfor the edgekij. As noted earlier, for ease of notation we will occasionally usez_kij andz_kinterchange- ably; this will apply todkijanddk,µkijandµk,ekijandekin the following context when the dropping out of the dummy double subscriptij in each vector causes no confu- sion. The error vector, distance vector and mismatched value vector are constructed as e = [e1,e2,· · · ,em]>, d = [d1,d2,· · · ,dm]> andµ = [µ1,µ2,· · · ,µm]>, respectively. In the following, we will use similar techniques as in [Sun et al., 2013] to obtain some

compact forms of the system equations. First note that the rigidity matrix is given as R(z) =Z>H¯, whereZ = diag{z1, z2,· · · , zm}(for the derivation, see Section 2.3 in Chapter 2). DefineJand ¯Jto be the matrices obtained from−Hand−_H¯ _{by replacing} all −1 entries by zeros, which also means that ¯J = J⊗I3. With the definition of ¯J, we can define a m×3n matrixS(z)byS(z) = Z>J¯. By doing this, we are led to the following compact equation:

x=−R(z)>e+S>(z)µ (5.6) which, together with (2.1), implies

z= −_HR¯ >_{(z)e(z) +}_HS_¯ >_(z)

µ (5.7)

Note that ˙e=2Rx˙. In conjunction with (5.6), one obtains ˙

e= −2R(z)R>(z)e+2R(z)S>(z)µ (5.8) In the sequel, we shall refer to (5.6) asthe overall system, (5.7) asthe z system, and (5.8) asthe error system.

Example continued: Following the example of a 3-D tetrahedron formation and

the discussions above, we now derive the motion equation for the tetrahedron formation case in the presence of distance mismatches. As an example, the dynamical system for agent 1 in (5.3) with mismatched distances in edges 1, 4 and 6 can be modified as ˙ x1=u1=

∑

j∈N1 (xj−x1)(kxj−x1k2−d21j+µk), j=2, 3, 4;k =1, 4, 6. (5.9)

where the edge indexk=1, 4, 6 is associated with adjacent agent pairs(1, 2),(1, 3),(1, 4), respectively; see Figure 5.1. The matrix J in this example can be obtained by replacing all−1 entries of −H in (5.2) by zeros, and the rigidity matrix R and the matrix S(z)can be written as R(z) =          −z1 z1 0 0 0 −z2 z2 0 0 0 −z3 z3 −z4 0 z4 0 0 −z5 0 z5 −z6 0 0 z6          ,S(z) =          z1 0 0 0 0 z2 0 0 0 0 z3 0 z4 0 0 0 0 z5 0 0 z6 0 0 0         

By doing this, one can obtain compact equations of system dynamics in the compact

In document Cooperative Coordination and Formation Control for Multi agent Systems (Page 83-87)