We assume that the full state vector is available for feedback, i.e., all aircraft are equipped with sufficient sensors that provide the aircraft positions, linear-velocities, angular velocities and orientation. The design of a formation control scheme follows the steps of the control deign procedure presented in section 4.3.
We propose the following intermediary (virtual) input for each aircraft
Fi = ˙vd−kpiχ(θi)−kdiχ( ˙θi), (5.4) ¨
θi =Fi−ui−v˙d, (5.5) wherekipand kdi are positive scalar gains and the function χis defined in (2.33). The variables θi and ˙θi can be initialized arbitrarily and ui is an auxiliary input to be
determined later. We define the following error variables
ξi =pi−θi, zi =vi−vd−θ˙i := ˙ξi−vd, (5.6) for i∈ N. It should be noted from (5.4)-(5.5) that if the input ui is guaranteed to
be globally bounded and converges asymptotically to zero, the variable θi and its time-derivative are guaranteed to be bounded and converge asymptotically to zero by virtue of Lemma 2.6. Therefore, the formation control objective is attained when the auxiliary input is designed such that zi → 0 and (ξi −ξj)→ δij. To this end, we propose the following input for system (5.5)
ui =−kvizi− n X
j=1
kijξij, (5.7)
where ξij = (ξi−ξj −δij), kvi is a strictly positive scalar gain andkij is the (i, j)th entry of the weighted adjacency matrixKof the communication graph,G = (N,E,K), characterizing the information flow between aircraft. Note that the intermediary control input Fi in (5.4) is guaranteed to be bounded as
kFik ≤δd+σb√3(kip+kid), (5.8) with kv˙dk ≤ δd and σb is defined in property P2 in section 2.6. Therefore, the
extraction condition (4.8) can be satisfied with a natural restriction on the desired linear-velocity and an appropriate choice of the gains kip and kdi. As a result, the necessary thrust input and the desired attitude for each aircraft can be extracted according to Lemma 4.1.
(4.7), as a time-varying reference attitude for the ith aircraft. Using the expression of the intermediary control input in (5.4), explicit expressions for the desired angular velocity and its time-derivative of each aircraft can be obtained as
ωd
i = Ξ(Fi) ˙Fi, (5.9)
˙
ωd
i = ¯Ξ(Fi,F˙i) ˙Fi+ Ξ(Fi) ¨Fi, (5.10)
where ¯Ξ(Fi,F˙i) is the time-derivative of Ξ(Fi) given in (4.10), and
˙ Fi =¨vd−kiph(θi) ˙θi−kdih( ˙θi)¨θi, (5.11) ¨ Fi =vd(3)−kiph˙(θi) ˙θi− kiph(θi) +kidh˙( ˙θi)θ¨i−kdih( ˙θi)( ˙Fi−v¨d−u˙i), (5.12)
where the diagonal matrixh(·) is defined in (2.34) and ˙h(·) is its time-derivative. We propose the following input torque for each aircraft
Γi =Hi(ωi,ωdi,ω˙di,Q˜i) +Jfiβ˙i−kqiq˜i−kiΩ( ˜ωi−βi), (5.13) βi = −kβiq˜i+ 2Ti
kiqmi
S(¯qi)TR(Qi)zi, (5.14)
where kiq, kΩi and kβi are positive scalar gains, ˜qi is the vector part of the unit- quaternion ˜Qi describing the attitude tracking error defined in (2.13), ˜ωi is the an-
gular velocity tracking error defined in (2.15),
Hi(·) = S(ωi)Jfiωi−JfiS( ˜ωi)R( ˜Qi)ωd
i +JfiR( ˜Qi) ˙ωdi, (5.15)
and ωd
i, ˙ωdi are derived in (5.9)-(5.12) with
˙ ui =−kvi ui− 2Ti miR(Qi) TS(¯q i)˜qi − n X j=1 kij(zi−zj). (5.16)
Note that to implement the above control scheme, communicating aircraft need to transmit their variables ξi and zi. Our result is stated in the following theorem.
Theorem 5.1. Consider the VTOL-UAVs modeled as in (5.1)-(5.2) and let the de- sired velocityvd and the controller gains kip and kdi satisfy Assumption 4.1. For each aircraft, let the thrust input Ti and the desired attitude Qdi be extracted according to
Lemma 4.1, and are given by (4.6) and (4.7) respectively, withFi given in (5.4) with
(5.5) and (5.7). Let the input torque for each aircraft be given as in (5.13)-(5.14)and let the communication graph G be connected. Then, starting from any initial condi- tions, the signals vi, (pi −pj) and ω˜i are bounded and vi → vd, (pi−pj) → δij,
˜
Sketch of Proof: First, we can verify that if the desired trajectory and control gains are selected according to Assumption 4.1, the extraction condition (4.8) is always satisfied, and the results of Lemma 4.1 can be used to extract the necessary thrust and desired attitude, from (4.6) and (4.7) respectively, for each VTOL vehicle. The translational error dynamics can be obtained from (5.1) and (5.5)-(5.7) as
˙ zi =−kvizi− n X j=1 kijξij − 2Ti miR(Qi) TS(¯q i)˜qi. (5.17)
In addition, the attitude tracking error dynamics are derived from (5.2) with (2.15) and (5.13) as
JfiΩ˙i =−kqiq˜i−kΩi Ωi, (5.18) with Ωi = ( ˜ωi−βi). The proof of Theorem 5.1 follows from Lyapunov arguments
using the positive definite Lyapunov function
V =1 2 n X i=1 zTi zi+ 1 2 n X j=1 kijξTijξij +ΩTi JfiΩi+ 4(1−η˜i) , (5.19)
leading to the negative semi-definite time-derivative
˙ V = n X i=1 −kvizTi zi−kiΩΩTi Ωi−kiqkβiq˜Ti ˜qi. (5.20) Invoking Barb˘alat Lemma, we can show that zi → 0, Ωi → 0 and ˜qi → 0, which
leads us to conclude that ˜ωi → 0. Also, invoking the extended Barb˘alat Lemma (Lemma 2.3), we show that ˙zi →0, and the translational dynamics (5.17) reduces to
n X
j=1
kijξij = 0, (5.21)
for i ∈ N. Exploiting the properties of the undirected communication graph, we show that this last set of equations leads to (ξi−ξj)→ δij, for all i, j ∈ N. Next, the result of Lemma 2.6 is used to show that θi and ˙θi are bounded and converge asymptotically to zero, which leads to the results of the theorem. A detailed proof of Theorem 5.1 is given in Appendix A.4.1.
Remark 5.1. It is important to mention that the variable βi is used in the input torque (5.13) to compensate for the perturbation term in the translational dynamics
using the linear-velocity tracking error. Also, the time-derivative of this variable is required in the input torque and is given by
˙ βi=−k β i 2 (˜ηiI3+S(˜qi)) ˜ωi+ 2Ti kiqmi d dt(S(¯qi) TR(Q i))zi+S(¯qi)TR(Qi) ˙zi + 2mi kqiTi(gˆe3−Fi) TF˙ i S(¯qi)TR(Qi)zi (5.22)
with dtd(S(¯qi)TR(Qi)) = S( ˙¯qi)TR(Qi)−S(¯qi)TS(ωi)R(Qi). In Abdessameud and
Tayebi (2009c), a different design for this variable is considered, where it has been shown that the effects of the perturbation term in the translational dynamics can be dominated with the choice ofβi =−kiqq˜i under some conditions on the control gains.
It should be noted that the formation control scheme in Theorem 5.1 is based on the introduction of an auxiliary system whose input is designed such that vi and
(pi−pj−δij) converge first to (vd−θ˙i) and (θi−θj) respectively. Then, the variables θi and ˙θi are driven to zero asymptotically, leading to our original objective. A different design of the intermediary control input is possible using classical formation control methods. In fact, we can show that the formation control objective is achieved using the input
Fi = ˙vd−kivχ(vi−vd)−
n X
j=1
kijχ(pij), (5.23)
with pij = (pi −pj −δij) and the control gains being defined as in Theorem 5.1, together with the torque input (5.13) with
βi =−kiβq˜i+ 2Ti
kiqmi
S(¯qi)TR(Qi)(vi−vd), (5.24) and the desired angular velocity and its time-derivative are derived from (5.9)-(5.10) using the the first and second time-derivatives of the intermediary input (5.23).
It can be verified thatFi in (5.23) is a prioribounded as
kFik ≤δd+σb√3 kvi + n X j=1 kij , (5.25)
which depends on the number of neighbors of each aircraft. Therefore, if the commu- nication topology between aircraft is known in advance, we can satisfy condition (4.8) and use the thrust and attitude extraction algorithm in Lemma 4.1. However, when the number of neighbors of each aircraft is large, it is generally difficult to satisfy the extraction condition and achieve a good/acceptable system response. Moreover, the
first and second time-derivatives of (5.23) will be respectively function of the linear- velocities and linear-accelerations of neighboring aircraft. Hence, to implement the above control scheme, communicating aircraft need to transmit their positions, linear- velocities and linear-accelerations. Of course, aircraft linear-accelerations can be com- puted on-line and then transmitted through the communication channels, which will increase the communication requirement between aircraft.
The proposed control scheme in Theorem 5.1 presents several advantages over the above classical design. This can be seen from the proposed intermediary control input that does not depend explicitly on the systems states (linear-velocity tracking error vectors and relative positions). As a result, the upper bound of the interme- diary control input, given in (5.8), does not depend on the number of neighbors of each aircraft. This is important since condition (4.8) can be easily satisfied without any consideration on the communication topology between aircraft. In addition, the extracted input thrust of each aircraft, given in (4.6), is guaranteed to be a priori
bounded as Ti≤mi g+δd+ √ 3σb(kpi +kdi) :=Tib (5.26)
with Tib being a positive constant. In view of (5.26), the designer can set the max- imum allowed input thrust for each aircraft without any a priori knowledge on the information flow between members of the team. Furthermore, the desired angular velocity and its time-derivative can be obtained using only available signals and com- municating aircraft need to transmit only the variablesξiandzi. The implementation
of the control scheme in Theorem 5.1 is shown in Fig.5.1.