Description

Formation Maneuver Control for High-Order Multiagent Systems
Soowhan Yi
Mechanical Engineering
syi94@uw.edu

I. INTRODUCTION
To solve this formation maneuver control problem, the paper [1] utilizes stress matrices among other solutions that realizes those manuevers at the same time, such as berycentric coordinates-based approach and complex Laplacians-based approach, because the berycentric coordinate-based approach requires relative rotation matrix and complex Laplacians-based approach is only applicable in some special dimension, and therefore stress matrix-based approach is more flexible and realizable. In “Affine Formation Maneuver Control of Multiagent Systems” [2], the formation maneuver control problem is solved with stress matrix-based approach and achieves translational, rotational, scaling, and shearing maneuver concurrently. However, it does not consider the case where the leaders of the formation have time varying acceleration. The followers following leaders with time varying acceleration would have coupled inputs as those followers have to use global information of acceleration of the leaders to calculate control input.
Therefore the paper suggests two layered leader-follower control strategy with stress matrix to fully distribute the control strategy for followers in affine formation maneuver. Also, for the leaders, it achieves autonomous maneuver control algorithm with information of their desired trajectory and finite time derivatives.
II. MATHEMATICAL FORMULATION
In order to properly formulate this multiagent formation manuever control problem, the paper utilizes two layered leader follower strategy and stress matrix in undirected graph. First, in graph theory, this stress vector ω = [ω1,ω2,··· ,ωm] ∈ Rm is said to be in equilibrium, when it
satisfies,
X
ωij(pj − pi) = 0,
j∈Ni where this pi and pj are the position of i and j agents, and this vector ωij(pj − pi) represents the tension force between the two agents. In a matrix form, above equation can be expressed as
(Ω ⊗ Id)p = 0
where , and Ω ∈ Rn×n satisfying
 0, i ̸= j,(i,j) ∈/ ε

[Ω]ij = −ωij, i ̸= j,(i,j) ∈ ε Pj∈Ni ωik, i = j.
By letting these tension force in each agents with its neighbors to sumed to be 0, then their formations would be structurally rigid. This structural rigidity of the formation is proved to be universally rigid if and only if there exists a stress matrix Ω such that Ω is positive semidefinite and rank(Ω) = n − d − 1. [3] Also in [2] it shows the desired trajectories of each leaders can be expressed as

where ri is the nominal configuration of each agents. With this time varying desired trajectory information of each agents, trajectory tracking control algorithm is designed fo the leaders’ formation. It also introduces the leader-follower strategy that utilizes above stress matrix and leaders positions to calculate the desired trajectories of followers. It shows that the number of leaders selected in a dimension d in order to be affinely localizable should be d+1. Also it shows that limt→∞ pf = where
,
where Ωff has to be non-singular. Therefore it suggests the number of leaders in dimension d and contraint on stress matrix in order to find convergable route for followers. Then it utilizes this desired positions of the leaders and followers to build the control objectives for leaders and followers, respectively.
lim pl − p∗l = 0
t→∞ lim pf − p∗f = pf − Ω¯−1Ω¯
t→∞ ff flpl = 0
The formulation of this problem heavily builds upon the “Affine Formation Maneuver Control of Multiagent Systems” [2] paper. However, this paper [1] is novel in a way that it builds upon the original paper with higher order so that it can render fully distributed control strategy for the followers and achieve autonomous manuever for group of leaders.
III. ANALYSIS
A. Tracking Control Algorithm for First Leader
Assuming that the trajectory and its finite time derivative information is already given, the paper designes tracking control algorithm for leaders. First it uses a finite time backstepping approach to achieve for the first leader.
Using the existing control objectives as auxiliary variable and a virtual variable that is to be determined later, auxiliary variable can be designed as such,
Z11 = p1 − p∗1
Z1i = pi1−1 − α1(i−1)(t).
Then, if we can determine α1(i−1)(t) and know all the time derivative information of itself, then we would be able to design the tracking control algorithm for the first leader. By using above equations, the paper achieves the relation between auxiliary variables in higher order.
Z˙11 = p˙1 − p˙∗1 = Z12 − α11 − p˙∗1 α11(t) = p˙∗1 − k11sigβ(Z11) − k12Z11, where
sigr(x) = [sigr(x1),sigr(x2),···sigr(xn)]
sigr(xi) = sgn(xi)|xi|r

−1

and sgn(xi) = 0
1 xi < 0
xi = 0. Also k11and k12 are positive xi > 0
control gains, and 0 ≤ β ≤ 1. Therefore, substituting above α11 equation to the Z˙11 equation, yields Z˙11 = Z12 − k11sigβ(Z11) − k12Z11.
Repeating the same procedure for Z12 and continuing on Z1i,
(1)
Z12 = p2 − α11(t)
Z˙12 = p(2)2 − α˙11 = Z13 + α12 − α˙11 α12 = α˙11Z˙11 + p¨∗1 − k11sigβ(Z12) − k12Z12 − Z11 Z1i = pi1−1 − α1(i−1)(t)
Z˙1i = Z1(i+1) − k11sigβ(Z1i) − k12Z1i − Z1(i−1) α1i = α˙1iZ˙1i − k11sigβ(Z1i) − k12Z1i − Z1(i−1)
Z˙1n = −k11sigβ(Z1n) − k12Z1n − Z1(n−1)ui = αin.
Finally, it utilizes the nth order intergrator α1n for the control inputs and its stability is verified with sum of quadratic form of Lyapunov functions.

Then,

Therefore

Here we know that 0 ≤ β ≤ 1, and k11 and V1n are positive.
Therefore V˙1n is negative definite and
B. Tracking Control Algorithm for Second Leader Group
Before designing control algorithm for second leader group, it needs to have desired trajectory and time derivative information of those second leader group. To estimate those information in a distributed way, the second leader group utilizes the desired trajectory and its nth order derivatives of the first leader to generate desired trajectory and the time derivatives of second leader group. Since we know the desired trajectory,b(t), and time derivatives of the first leader,b(j)(t), where j = 0,1,··· ,n, desired trajectory and time derivatives of second leader group can be estimated with
M
ˆb˙i(j) = −ρjsgn[Xsgn(ωik)(ˆb(ij) −ˆb(kj))], k=1
where i ∈ 2,··· ,M and ρj is positive constant. Also the time derivative information of A(t) can be estimated in a distributed way.
M aˆ˙i(j) = −ρjsgn[Xsgn(ωik)(aˆi(j) − aˆ(kj))], k=1
where is agent i’s estimation of a(t), and a(t) is column vector form of A(t).
Then combining those two estimates generates desired trajectory and time derivative information of each agents.
.
With this generated desired trajectory of each agents, again, finite time backstepping appoarch is used to generate the control inputs of each agents.
Zi1 = pj − p∗j = Zi2 − αi1 − p˙∗j αi1(t) = p˙∗j − ki1sigβ(Zi1) − kj2Zi1 Zik = pk1−1 − α1(k−1)(t) αik = α˙ikZ˙ik − k11sigβ(Zik) − k12Zik − Z1(i−1) ui = αin
C. PDm Control Algorithm for Followers with Constant Gains
The author was inspired from “Containment Control of Multiagent Systems With Dynamic Leaders Based on a PInType Approach” [4] and proposed PDm control algorithm for followers. Using all the desired trajectory information and their time derivative information of neighbors of itself, the agent is able to generate adaptive gains using offline computation. The author proposes control algorithm for followers
u˙i = −ηi X ωij[k0(pi − pj) + k1(p˙i − p˙j)
j∈Vl SVf
.
Using the dynamics of leaders and followers and their compact form,
x˙i = (B ⊗ Id)xi,i ∈ Vl x˙i = (B ⊗ Id)xi + (C ⊗ Id)ui,i ∈ Vf x˙l = (IM ⊗ B ⊗ Id)xl
x˙f = (Inf ⊗ B ⊗ Id)xf + (ΛΩff ⊗ C ⊗ Id)xf
− (ΛΩfl ⊗ CK ⊗ Id)xl
where , and

the tracking error can be defined as
.
From this tracking error equation, ηi will be designed to let this tracking error to converge to zero. By taking the derivative of it,
X˙f(t) = x˙f + [Ω−ff1Ωfl ⊗ I(m+1)d]x˙l
= (Inf ⊗ B ⊗ Id)xf + (ΛΩff ⊗ C ⊗ Id)xf
− (ΛΩfl ⊗ CK ⊗ Id)xl
+ [Ω−ff1Ωfl ⊗ I(m+1)d](IM ⊗ B ⊗ Id)xl
= (Inf ⊗ B ⊗ Id − ΛΩff ⊗ CK ⊗ Id)Xf(t)
With this time derivative information, construct the Lyapunov function and take the derivative.
V = XfT(Ωff ⊗ P ⊗ Id)Xf
V˙ = X˙fT(Ωff ⊗ P ⊗ Id)Xf + XfT(Ωff ⊗ P ⊗ Id)X˙f
Using the above X˙f(t) equation and Algebraic Riccati Equation,
BTP + PB − PCCTP + Im+1 = 0
X˙f(t) =(Inf ⊗ B ⊗ Id − ΛΩff ⊗ CK ⊗ Id)Xf(t)
V˙ = XfT(Ωff ⊗ (PB + BTP) ⊗ Id)Xf
− 2XfT(ΩffΛΩff ⊗ (PCCTP) ⊗ Id)Xf
= −XfT[Ωff ⊗ Im+1 ⊗ Id]Xf + XfT[(Ωff
− 2ΩffΛΩff) ⊗ PCCTP ⊗ Id]Xf
To prove stability of this Lyapunov function, (Ωff −
2ΩffΛΩff) ≤ 0, since Ωff ⊗Im+1⊗Id ≥ 0 and PCCTP ≥
0. If we choose ηi as
,
then
,
for any y ∈ Rnf.
D. PDm Control Algorithm for Followers with Adaptive Gains
With constant control gain larger than , the trajectory would be depending on the formation of the graph. So the author suggests using adaptive gains with Algebraic Riccati Equation.
u˙i = −ηi X ωij[k0(eij) + k1(eij(1) ) + ··· + km(e(ijm))] j∈Vl SVf
ηˆ˙i = γisTi (PCCTP ⊗ Id)si
si = X ωij(xi − xj).
j∈Vl SVf
Considering the previous Lyapunov function and building upon it,
V1 = XfT(Ωff ⊗ P ⊗ Id)Xf
N
V2 = V1 + X (ηˆi(t) − η¯)2/γi i=M+1
V˙1 = −XfT[Ωff ⊗ Im+1 ⊗ Id]Xf + XfT[(Ωff
− 2ΩffΛΩff) ⊗ PCCTP ⊗ Id]Xf
= −XfT[Ωff ⊗ Im+1 ⊗ Id]Xf
+ XfT[Ωff ⊗ PCCTP ⊗ Id]Xf
− 2[Ωff ⊗ I(m+1)dXf]T[Λ(ˆ t) ⊗ PCCTP ⊗ Id]
× [(Ωff ⊗ I(m+1)d)Xf]
N
V˙2 = V˙1 + X 2(ηˆi(t) − η¯)[siT (PCCTP ⊗ Id)si].
i=M+1
Substituting V˙1 into V˙2 equation achieves,
∴ V˙2 = −XfT[Ωff ⊗ Im+1 ⊗ Id]Xf
+ XfT[Ωff ⊗ PCCTP ⊗ Id]Xf
N
− 2¯η X sTi (PCCTP ⊗ Id)si
i=M+1
= −XfT[Ωff ⊗ Im+1 ⊗ Id]Xf
+ XfT[Ωff ⊗ PCCTP ⊗ Id]Xf
− 2¯ηXfT[ΩffΩff(PCCTP ⊗ Id)]Xf.
Applying the same logic that was applied in previous section, to obtain the stabilityV˙2 ≤ 0,
.
IV. SIMULATION
A. Stress matrix Design
The paper designs the stress matrix according to the methods presented in [5] and [2], and the nominal configuration follows,
r1 = [0;0] r5 = [−2;1] r2 = [−1;1] r6 = [−2;−1] r3 = [−1;0] r7 = [−3;1] r4 = [−1;−1]
,
r8 = [−3;−1]
and figure 1 shows the paper’s the nominal configuration and the corresponding weights.

Fig. 1. nominal formation and stresses
B. Manevuer Design
Before simulating, the desired trajectory has to be generated according to the given desired position and velocity value at time tk and tk+1. With the previous assumption that desired trajectory,pi = A(t)ri + b(t), is known and nth order differentiable, the paper assumes 3rd order time differentiable equation,
b(t) = c3(t − tk)3 + c2(t − tk)2 + c1(t − tk) + c0,
where ci ∈ R2 are constant vectors at t ∈ [tk,tk+1] for all the leader positions. With known desired position and velocity profile at two different given time, then the desired trajectory can be found with

where Stk,vtk,Stk+1,vtk+1 are the desired position and velocity profile at tk and tk+1. Also the scaling maneuver is designed with,
A(t) = ϕ(t)I2 ϕ(t) = ς3(t − tk)3 + ς2(t − tk)2 + ς1(t − tk) + ς0,
where ς3,ς2,ς1,ς0 are known desired scaling factor at tk and tk+1. With the same matrix equation used for translational maneuver, scaling maneuver constants,ςi ∈ R,i = 0,1,···3, can be also be inferred. Also the rotational and shearing design is described in a same manner, where

are rotational and shearing matrix for each design, and θ(t) ∈ [0,2π) and ψ(t) ∈ (0,1) are rotation angle with respect to the nominal configuration and shearing factors respectively.
C. Simulation and parameters
The paper also utilizes MATLAB to calculate the unique solution of the Algebraic Riccati Equation to get values of K. K = [46.72143.817197.9471143.817]
Therefore ηi is chosen to be 56 bcause . Other constant parameters are chosen abstract
ρi = 12,β = 0.5,ki1 = 1,ki2 = 16,i = 1,···4
From figure 2 we can see that the paper successfully simulates the designed affine manuever.

Fig. 2. Formation maneuver trajectories
Here is my attempt, figure 3, to redo the work. It seems like the tracking control algorithm for leaders group is working but it is probably details that I might have missed. But here is my link for the code and future simulation demonstration and effort. https://github.com/SoowhanYi94/ME597

Fig. 3. Formation maneuver trajectories
V. CONCLUSION
Overall the paper heavily depends on the previous works
[2], [4] and [3]. However the contribution of the paper lies on the fully distributing the followers control strategy for affine formation maneuver and making the leaders maneuver autonomous with given desired trajectory and its finite time derivative information. Also, the PIn type control algorithm does require higher order information than the agents’ dynamics, and therefore can not realize the adaptive gain because simply we do not have those information in this setting. Therefore its novelty also lies on usage of different distributed control method for the followers and autonomous tracking control method for leaders.
The assumptions here is that the desired trajectory and its finite time derivative information for each agents are accessible and differentiable at nth order. This means that the discontinuous manuever is not available for the leaders and they would be vulnerable sudden change in enviornment. Also the number of leaders are selected at least more than 1 + d in which d is the dimension of the dynamics, that affinely spans Rd, which are realizable with 3 4 agents being in the leader group in 2 or 3 dimensions.
However, another assumption here is that connection between leaders group and the first leader should always exist. Since the continuous maneuver of the first leader and the connection among leaders group are vulnerable to the disturbances, the leaders group should render relatively slow speed for the accurate tracking control algorithm.
Therefore those assumptions are applicable to the situation where agents are maneuvering through unknown map as it needs slowly maneuver to accurately map the enviornment and avoid any obstacle. Also it has to be applied where the sudden change in enviornment is not expected. For this reason, I suggest, in the future, the research can be improved with conducting performance measure using variance of the disturbance response on this leader-follower affine formation maneuver. Could it be further researched for disturbance rejection control strategy using [6]?
REFERENCES
[3] A.Y. Alfakih. On the dual rigidity matrix. Linear Algebra and its Applications, 428(4):962–972, 2008.