A Distributed Two-Layer Framework for Teleoperated Platooning of Fixed-Wing UAVs via Decomposition and Backstepping

We propose a novel distributed control framework for teleoperated platooning of multiple three-dimensional (3D) fixed-wing unmanned aerial vehicles (UAVs), consisting of the following two layers: 1) virtual frame layer, which generates the target 3D nonholonomic motion of the virtual nonholonomic frames (VNFs) using the nonholonomic decomposition (D. J. Lee, 2010) and backstepping in such a way that the VNFs are to maintain the platoon formation in a distributed manner while respecting the directionality of the fixed-wing UAV; and 2) local control layer, which drives each fixed-wing UAV to track their respective VNF with their under-actuation and aerodynamic disturbance effects fully taken into account by using the backstepping and Lyapunov-based design techniques. Convergence and stability of salient aspects of each layer and their combination are theoretically established. Simulations with 25 fixed-wing UAVs and a haptic device are also performed to validate the theory with their multimedia provided at https://youtu.be/Z3Mo66KIsns.


I. INTRODUCTION
F ORMATION control of multiple unmanned vehicles is drawing an increasing interest due to its promise to materialize numerous powerful commercial and military applications, with their implementation becoming ever closer to reality with the recent rapid advancement of hardware and software technologies. Formation control of unmanned aerial vehicles (UAVs) can further extend these applications into the three-dimensional (3D) air environment. Teleoperated platooning (i.e., the forefront leader UAV operated by a remote (or onboard) operator (e.g., via first-person view (FPV) camera), while the follower UAVs flying with certain relative spatial relation maintained among them all through the leader) is very a promising direction for Manuscript  this formation control, at least for the near future [2], as it allows us to incorporate human intelligence to cope with unexpected situations (often not fully resolved by the state-of-the-art autonomy) in a familiar setting (e.g., car driving), while minimizing the human-associated costs (e.g., fatigue, casualty, labor, etc.). In this letter, we consider this problem of teleoperated platooning of the fixed-wing UAVs.
The two major categories of the UAVs are the rotor-based UAVs (e.g., quadrotor drones [3]) and the fixed-wing UAVs [4]. As compared to the rotor-based UAVs, the fixed-wing UAVs are known to possess longer flying time/distance and better robustness to wind disturbance. They are however also known to be subject to the directionality constraint, that is, in a typical flying condition, their Cartesian velocity should be predominantly along the forward direction (i.e., along the main vehicle axis) to produce enough lift force for the vehicle. To address this directionality of the fixed-wing UAVs, many prior works have adopted the two-dimensional (2D) nonholonomic unicycle model (with its permissible velocity modeling the UAV's forward velocity) and proposed distributed formation control schemes for that (e.g., [5]- [9]). Yet, these 2D results (e.g., [5]- [9]) are not directly applicable to the 3D teleoperated platooning of the fixed-wing UAVs (e.g., simply augment those 2D control with some other vertical motion control), since, with the directionality, their velocity vector is rotating together with the body, thus, their motion can only be properly captured/analyzed by incorporating their full kinematics in SE (3).
In this letter, we propose a novel distributed two-layer control framework for the teleoperated platooning of multiple fixedwing UAVs evolving in SE (3), which can accommodate arbitrary/unpredictable (remote or onboard) operator command or that from automatic mission planner. Our framework consists of the following two layers: 1) virtual frame layer, which generates the target 3D nonholonomic motion of virtual nonholonomic frames (VNFs) using the nonholonomic passive decomposition [1] and backstepping in such a way that the VNFs are guaranteed to maintain the platoon formation among them in a distributed manner (i.e., requiring only peer-to-peer (P2P) communication); and 2) local control layer, which drives each physical fixedwing UAV to track the target 3D nonholonomic motion of their respective VNF with the effects of their control under-actuation and the aerodynamic disturbance fully taken into account by using the backstepping and the Lyapunov-based control design techniques. See Fig. 1 for an illustration of the teleoperated platooning. Our framework renders the multiple fixed-wing UAVs to behave as a deformable string or structure "pulled by" one of its corners [10] -see Section IV. Although our framework can accommodate other formation shapes, in this letter, we also focus on the line-topology platooning as shown in Fig. 1, since it allows the multiple UAVs to navigate through obstacles or pass through a narrow passage one by one via their collective serpentine motion, particularly relevant to urban air mobility (UAM) scenarios.
In contrast to many results for the 2D nonholonomic agents (e.g., [5]- [9]), those for the distributed formation control of 3D nonholonomic agents are very rare and, to our knowledge, the recent works of [11], [12] are the only results for this. The results of [11], [12], however, are not fully distributed, as they assume each agent have access to the leader's attitude; and do not fully consider the underlying dynamics of the fixed-wing UAV either, which is not nonholonomically constrained but under-actuated. On the other hand, backstepping [13] has been widely-used for the control of a single under-actuated system (e.g., [3]), however, not has been so for the distributed control of multiple agents in SE(3). One of rare results for this is [14], where a distributed formation control was presented for the rotor-based UAVs in SE(3) on a balanced graph. This rotor-based UAV is however omni-directional (i.e., can assume any Cartesian velocity w.r.t. the body), thus, the nonholonomic constraint adopted in this letter to respect the directionality of the fixed-wing UAVs is not necessary therein. To our knowledge, our proposed framework is the very first result for the distributed control of the fixedwing UAVs, which can fully incorporate their directionality and kinematics in SE(3) by utilizing the nonholonomic decomposition [1] and backstepping, while also ensuring the robustness against the aerodynamic disturbance and well-behaving of the attitude motion even under their under-actuation by utilizing the Lyapunov-based control design and backstepping. This adoption of backstepping renders our framework inherently robust following the well-known robustness property of the backstepping [13]. This current letter also extends our prior result of [15] for the 2D kinematic nonholonomic unicycles to the 3D under-actuated dynamic fixed-wing UAVs.
The rest of the letter is organized as follows. Section II introduce the virtual frame layer and its distributed VNF platooning strategy based on the nonholonomic decomposition and backstepping. The local control layer for each fixed-wing UAV is detailed in Section III based on the backstepping and Lyapunov-based design. Simulations of teleoperated platooning with 25 fixed-wing UAVs are performed in Section IV. Section V concludes the letter.

A. Nonholonomic Decomposition of VNFs
Let us consider the VNF {N j } for the j-th UAV evolving in SE(3), whose Cartesian position is denoted by p N j ∈ R 3 and its rotation matrix by R N j ∈ SO(3). The position and the rotation are expressed w.r.t. the north-east-down (NED) inertial frame {O} with its vectors being e 1 , e 2 , e 3 for the N, E, and D directions. The kinematics of {N j } is then given by: where v N j ∈ R 3 is the velocity and ω N j ∈ R 3 is the angular rate with hat map· : R 3 → so (3), which is a skew-symmetric operator satisfyingxy = x × y for all x, y ∈ R 3 . The motion of the VNF {N j } can then be described by where T j is the transformation matrix in the special Euclidean group, ξ j is the spatial velocity or so-called the (body) twist in screw theory, and 0 a×b ∈ R a×b is a zero matrix. In order to capture the directionality of the fixed-wing UAV, that is, the majority of its velocity is along its forward direction while its sideways velocities are much smaller, we set the following nonholonomic constraint: Aξ j = 0 2×1 (2) with A := (0 2×1 , I 2 , 0 2×3 ) ∈ R 2×6 and I a ∈ R a×a is a identity matrix. Note that this constraint (2) ensures the sideways velocities of the VNF are zero (i.e., e T 2 v N j = e T 3 v N j = 0). To attain the platooning behavior in a distributed manner, we design the virtual frame layer action for each pair of a preceding j-th VNF and its following (j + 1)-th VNF. For this, we define the pairwise twist as with the pairwise nonholonomic constraint: is a block diagonal matrix operator. From the constraint (3), we define the unconstrained distribution [1] as D := diag(e 1 , I 3 , e 1 , I 3 ) ∈ R 12×8 (4) which characterizes the null-space of diag(A, A) in (3). Every possible twist of the VNF should satisfy ξ j,j+1 ∈ col(D ), where col(·) is a column space operator.
The platooning behavior can then be attained by enforcing the following pairwise virtual constraint: where h j,j+1 ∈ R 3 is called a formation map [1] and l j ∈ R 3 is the relative position vector expressed in {N j }. This constraint (5) ensures that the relative position of the preceding VNF as observed in the following VNF is equal to l j , i.e., with the trailing condition l (1) j := e T 1 l j > 0 also assumed. This also means that the preceding VNF is within the limited field-of-view of the following VNF, pointing the direction of l j , regardless of the formation shape and its curvature [15]. Note also that h j,j+1 can be understood as a formation error in the inertial frame. See Fig. 1 for an illustration of the platooning under (5). We now state the control objective of the virtual frame layer to achieve the platooning among the VNFs by enforcing (5).
From the constraint (5), we define the normal distribution: The orthogonal complement of this normal distribution is named as tangential distribution and obtained by where R N j,j+1 := R T N j R N j+1 is the relative rotational matrix. This Δ j,j+1 characterizes the twist tangential to the level set: We decompose the unconstrained distribution (4) under the virtual constraint (5) by the locked and quotient distributions [1] as follows. The locked distribution is the intersection between the unconstrained and tangential distributions s.t. D l j,j+1 := D ∩ Δ j,j+1 , which stands for the twist space enforcing h j,j+1 fixed while satisfying the nonholonomic constraint (3). There exists a subspace of col(D l j,j+1 ) which does not affect the preceding twist ξ j . It does not appear in the 2D results of [15] as it determines the roll rate of the VNF. For this, we define a new distribution D j+1 := [0 6 ; I 6 ] ∈ R 12×6 which characterize only the follower's motion. Then, we can decompose the locked distribution according to the dependency on ξ j into 1) the j-independent locked distribution: and 2) the j-dependent locked distribution: . One the other hand, the quotient distribution is a relative complement of the locked distribution w.r.t. the unconstrained distribution s.t.
The quotient distribution contains both the normal and tangential components to the level set H h j,j+1 [15]. Complete normal-tangential decoupling is possible only if the strong decomposability is granted [1], which is not the case here.
From the decomposition, the pairwise twist can be written by where S j,j+1 is the decomposition matrix [15] and u j,j+1 are the inputs for the virtual frame layer, which are to be distributed in Section II-B. From now on, let us omit the subscript {j, j + 1} to shorten notations, for instance, h := h j,j+1 , R N := R N j,j+1 , Δ := Δ j,j+1 , D := D j,j+1 , and u := u j,j+1 . Then, the distribution inputs to be designed are u LI ∈ R, u LD ∈ R 4 , and u C ∈ R 3 .

B. Distributed Virtual Layer Input Design
As in (9), the distribution inputs u directly specify the twist. If we directly assign the inputs, however, the VNF acceleration may not be smooth, which is incompatible with the local tracking control in Section III, as it requires the jerk of position. Thus, in this Section II-B, we design update laws ofω N j+1 ,v N j+1 for the VNFs to maintain the platooning while being compatible with the local control in Section III.
Let us define the reference inputs u ref instead of directly defining their real inputs u . Writing the first six rows of (9) with the reference distribution inputs, we get from (6), (7), and (8). Suppose that the preceding twist ξ j and its derivatives are already given. Choosing the condition of the reference inputs as ξ ref j = ξ j , the following equality ; ω N j ) must be satisfied. Substituting this equality to (9) and extracting the last six rows, we can compute the reference twist of the following frame by One of the common roll motion of the fixed-wing UAVs is to use a desired bank angle (i.e., roll angle of fixed-wing aircraft) as in [16], where the desired bank angle can be obtained by its steering acceleration and the gravitational acceleration g > 0.
Thus, we define the sum of the accelerations as (10). Then our bank angle error is ϕ j+1 := arctan2(a (2) j+1.g , −a (3) j+1.g ) ∈ R where a (2) j+1.g = e T 2 a j+1.g ∈ R and a (3) j+1.g = e T 3 a j+1.g ∈ R. Using the error, we design the update law of the roll rate: where b ω , b ϑ > 0 are stabilizing gains, k ω , k ϑ > 0 are driving gains, ϑ j+1 ∈ R is the "filtered" bank angle error approaching to the bank angle error ϕ j+1 . The new variable ϑ j+1 is introduced here to avoid a direct impact of ϕ j+1 on u ref LI . In (12), we usė ϑ j+1 + u ref LI , since ϑ j+1 evolves on the VNF, whose roll rate is u ref LI . The hyperbolic tangent function discounts the effect of the bank angle error when a (2) j+1.g , a (3) j+1.g are both small. Considering the matrix form of (11), (12), we can find exponentially stable control gains b ω , k ω , b ϑ . In the steady state, we also have and it is the same as the conventional bank angle control, which determines the roll rate with the bank angle error. Geometric relations of this roll rate control are visualized in Fig. 2.
2) Virtual Constraint and Backstepping: Let us consider the virtual constraint (5) to determine the reference quotient distribution input u ref C with the first Lyapunov function: Fig. 2. The filtered bank angle error ϑ j+1 approaches to the bank angle error ϕ j+1 from (12), and the roll motion of the VNF follows ϑ j+1 from (11).
which is positive definite. The time derivative of (13) yieldṡ while it can be reconstructed by (0 6×1 ; e ξ j+1 ) = B T ν 1 . To establish the stability, we set the reference quotient distribution input by We then haveV (4,1,1) is a positive definite matrix. Note that (14) is a twice differentiable equation.
We get u ref LI from (11), (12) and u ref C from (14). Substituting these reference distribution inputs to (10), we can compute the reference twist ξ ref j+1 and its derivatives. Now the backstepping approach is employed to obtain the update laws of the twist. For this, we augment a quadratic potential of ν 1 to (13) by with a constant γ 1 > 0, which leads us to an update law: with k 1 > 0 and C := (1, 0 1×3 ) ∈ R 1×4 , where ν 2 := C(ν 1 + k 1 ν 1 + γ 1 BΔ ⊥ h) ∈ R is the remained error of the update law on the velocity aspect. From the update law, the derivative of the angular rate is designated aṡ where e ω N j+1 := ω N j+1 − ω ref N j+1 is the angular rate error. Similarly, we augment a quadratic potential of ν 2 to (15) Fig. 3. Heading-hold control aims to regulate the symmetric plane alignment error r j and the heading vector error s j as much as permissible by the higher priorities in the null-space hierarchical control (27). by with a constant γ 2 > 0. Then it suggests an update law: is the velocity error. The following Prop. 1 summarizes the main properties of our virtual frame layer control and the attainment of the Control objective 1 in Section II-A. Note also that, from the twist updates of the virtual frame layer (16) and (18), we can calculate the acceleration of (j + 1)-th VNF byp N j+1 = R N j+1 (v N j+1 +ω N j+1 v N j+1 ) and the jerk by ...
, which are to be used for the local control layer in Section III.

III. LOCAL CONTROL LAYER
This Section III-A presents the local control layer, which aims to drive each fixed-wing UAV to track their respective VNF {N j } robustly even with their under-actuation. For this, let us attach a frame {B j } to the j-th UAV, whose e 1 , e 2 , e 3 represent the heading, main-wing, and down direction as in Fig. 3. We then model the fixed-wing UAV as a combination of the translation dynamics and the attitude kinematics of {B j } evolving in SE (3) with the assumption of availability of low-level attitude control: where m j ∈ R is the mass, p B j ∈ R 3 is the position vector, R B j ∈ SO(3) is the rotation matrix, ω B j ∈ R 3 is the body angular rate, and f j ∈ R 3 is a force on the body-fixed frame written by is the aerodynamic force of the fixed-wing UAV as modeled in [17], ρ ∈ R is the temperature/altitude-dependent air density, v j : is the relative wind velocity, and v w ∈ R 3 is the wind velocity in {O} measurable as stated in [18]. In this letter, we consider uncertainty only in the aerodynamic force f v j (ρ, v j ), as this is typically much larger as compared to other uncertainties (e.g., inertia parameters, control/communication imperfectness, etc.).
Control objective 2: Consider the fixed-wing UAV (19), (20) evolving in SE (3). We then want the position p B j of the UAV to track that of its VNF (i.e., p N j ), and its attitude R B j matches that of the VNF (i.e., R N j ) as much as permissible (i.e., within the null-space of the (p B j , p N j )-tracking objective -see (27)), while ensuring robustness against the aerodynamic uncertainty and boundedness of all the states even with their under-actuation.

A. Position Tracking Control Design
We first consider the position tracking problem between the fixed-wing UAV and its corresponding VNF. For this, we extend the framework of [3] to the case of fixed-wing UAVs. Define a desired acceleration for the position tracking: where e j := p B j − p N j ∈ R 3 is the position error between the UAV and its VNF, and ν a ∈ R 3 is the acceleration error due to the under-actuation. To enforce e j ,ė j → 0 3×1 , we define ζ j := (ė j ; e j ) ∈ R 6 and the following Lyapunov function: with 6 × 6 matrices: where ⊗ is the Kronecker product. This P, Q are both positive definite if we choose a small s.t. 0 < < b a . For backstepping, we augment W 1 and obtain: where γ > 0 is a constant. This then suggests us the following update law:ν To attain the control law, we differentiate (19) by The air density ρ is typically slowly varying, thus, we assumeρ ≈ 0. We rearrange the above equation by G 1 u B j = d 1 with for the UAV control input u B j := (ḟ t j ; ω B j ) ∈ R 4 . Here, notice the redundancy from (24), that is, the size of the control input R 4 is larger than that of the position tracking objective R 3 . Thus, we can introduce another control objective, heading-hold objective as detailed in the next Section III-B, using the null-space hierarchical control.

B. Heading-Hold Control Design
Recall from Section II that the VNF does not have sideways velocities and follows the conventional roll attitude control. This means that conditions for heading-hold can be achieved by matching the plane of symmetry of the fixed-wing UAV (a plane perpendicular to its main-wing) with that of the VNF. Thus the objective function can be written byṙ j = −k r r j , where r j := (R B j − R N j )e 2 is the symmetric plane error as shown in Fig. 3 and k r > 0 is a positive gain. This objective function can then be organized into G 2 u B j = d 2 with Since G 2 in (25), together with G 1 in (24), is still lacking in spanning R 4 (i.e., rank([G 1 ; G 2 ]) < 4 for some rare cases), we additionally define the third objective function asṡ j = −k s s j where s j := (R B j − R N j )e 1 is the heading vector error as shown in Fig. 3 and k s > 0 is a positive gain. Then, similarly to (25), we can obtain G 3 u B j = d 3 with This last objective (26) turns out to be very rarely activated, since, in almost all cases, rank([G 1 ; G 2 ]) = 4 and the lowest priority is to be assigned to (26). More precisely, we combine (24), (25) and (26) into a form of the null-space hierarchical control s.t.
In order to deal with the aerodynamic uncertainty, the aerodynamic force is set to is the additive uncertainty. If the body accelerationp B j can be measured with some sensors (e.g., accelerometer), we can obtain ν a ,ȧ des j , and f v j but cannot compute their derivatives so that the controller implements ∂f v j /∂v j instead of ∂f v j /∂v j in (24). In the case that the body accelerationp B j cannot be measured due to some practical reasons, whose proof is provided in this Prop. 2, the controller also estimates the acceleration byp B j − R B jf v j /m j and estimateṡ a des j , u ν a using the estimated acceleration. Then the tracking objective implementation (24) is replaced byḠ 1 u B j =d 1 withḠ For further development, we define H := G 1Ḡ , which are both bounded (i.e., σ H : v j → |λ max (H)| and σ F : v j → |λ max (F )| are bounded for all v j ) due to assumptions. Then the evolution of the translation dynamics can be written byν δ is also bounded due to assumptions. The time derivative of the Lyapunov function is now bounded bẏ where κ H := max(b a + k, |b 2 a − γ|, |b a k a − γ |) ∈ R and κ F := max (1, b a , k a ) ∈ R with ν a 2 ≤ (ζ j ; ν a ) 2 . The following inequality sandwiches the Lyapunov function by so that the error (ζ j , ν a ) converges to the bounded ball of Note that we can always find a set of control gains satisfying the convergence condition if the magnitude of the uncertainty and its derivative meets σ H (v j ) < 1.
On the other hand, with the assumed boundedness of ) and the convergence of (ζ j , ν a ), we can show thatṗ B j , v j , a des j ,p B j ,ȧ des j are bounded and so are d 1 , d 2 , d 3 as well. This then implies On top of Prop. 1 and Prop. 2 for the control objective achievement and the well-behaving of the relevant states in the virtual frame layer and the local control layer, respectively, we also provide the following Thm. 1 to further specify the key properties of the combined teleoperated platooning behavior of the multiple under-actuated fixed-wing UAVs.
Theorem 1: Consider the teleoperated platoon consists of n-VNFs (1) and n-fixed-wing UAVs (19), (20) with the virtual frame layer (16), (18) and the local control layer (27). Suppose that: a) the wind velocity v w (t) and aerodynamic disturbance f v j (ρ, v j ) in (19) are Lipschitz continuous and bounded; and b) the twist ξ 1 and its derivativesξ 1 ,v N 1 of the forefront (commanding) leader VNF are bounded. Then, the formation objective and the tracking objective exponentially converge, i.e., h j,j+1 → 0 3×1 (with ξ j+1 → ξ ref j+1 ) for ∀j = 1, 2, · · · n and ζ j → 0 6×1 for j = 1, 2, · · · n (or ζ j ultimately bounded with the aerodynamic uncertainty as stated in Prop. 2). Further, the control inputs of the VNFs (ω N j+1 ,v N j+1 ) and the UAVs (ḟ t j , ω B j ) are also remained to be bounded. Proof: Note that this Thm. 1 is essentially a combination of Prop. 1 and Prop. 2 using the integrated Lyapunov function V := are also bounded. Again with bounded ν 2 → 0, the twist ratė ξ j+1 is bounded andḧ,ü ref Thus, bounded ξ 1 ,ξ 1 ,v N 1 ensures the boundedness of the control input (ω N j ,v N j ) of all the VNFs and that of the UAV control input (ḟ t j , ω B j ) while nullifying the necessity of the assumption a) of Prop. 2.

IV. SIMULATION
In order to validate our proposed framework, we perform real-time simulations with a human operator in the loop. A commercial Omega.3 1 is used as a haptic device and its 3DOF Cartesian position u user ∈ R 3 is used to control the forefront leader VNF {N 1 }. For this, we set the reference acceleration for {N 1 } to be e T 1v ref  (11), (12). The twist update law of the forefront VNF is written bẏ reusing the gains of (16), (18) whereω N 1 ,v N 1 are computed by usingu user which is provided by the device and smooth enough during the simulations. We also introduce some haptic feedback to assist the user: 1) preventing a crash to the ground; 2) leveling the flight path; and 3) ensuring sufficient forward velocities. Note that other types of control (e.g., mission planner) can be applied to the leader UAV.
The aerodynamic coefficients C j are given by C Z j := C 0 Z + C α Z sin α j , C Y j := C β Y sin β j , C X j := C 0 X + C A X (C Z j ) 2 where α j = sin −1 (e T 3 v j / v j ) is the angle of attack and β j = sin −1 (e T 2 v j / v j ) is the side-slip angle. Aerodynamic uncertainties are modeled by estimated aerodynamic coef-ficientsC 0 Z = 0.15,C α Z = 4.5,C β Y = 0.7,C 0 X = 0.015,C A X = 0.15 and included in the simulations. The body acceleration is assumed to be measurable (e.g., accelerometer).
The real-time simulation environment (i.e., visual display, physics simulation, haptic interface, etc.) is custom built using The formation objective (i.e., h j,j+1 ) and the tracking objective (i.e., e j ) converge to zero, implying that the UAVs attain the desired platoon formation. The side velocity v (2) B j is also regulated due to the heading control, and converges to zero at level flight. Adequate angle of attack (or downward velocity v (3) B j ) is formed to generate the lift force during the simulation.

V. CONCLUSION
We proposed a novel distributed control framework for a teleoperated platooning of 3D fixed-wing UAVs consists of: 1) virtual frame layer, which renders a platooning of 3D virtual nonholonomic frames (VNFs) with only peer-to-peer communication required; and 2) local control layer, which drives each fixed-wing UAV to track corresponding VNF with the underactuated property and aerodynamic uncertainty considered. Real-time human-in-the-loop simulations are also performed to show the theory. Some possible future research directions include: 1) incorporation of collision avoidance and formation feedback [20]; 2) extension to other communication topologies both for cooperative perception or control; and 3) experimentation with real fixed-wing UAVs.