Cooperative Control of Multi-Tilt Tricopter Drones Applying a ‘Mixed’ Negative Imaginary and Strict Passivity Technique

This paper develops a cooperative controller for a fleet of multi-tilt tricopter drones modelled as Negative Imaginary (NI) systems via closed-loop linearisation exploiting Sliding-Mode Control (SMC) technique. The multi-tilt tricopter model is derived from the first principles, and the SMC method is invoked to obtain a closed-loop linear model of the tricopter plant with six outputs and six inputs. A subspacebased system identification algorithm is developed to identify the linearised drone model enforcing the NI property. The primary control objective is to design a distributed ‘mixed’ NI and strictly Passive control protocol so that the tricopter agents asymptotically reach the desired formation and keep tracking the target. Instead of the Lyapunov approach, the characteristic loci method is used to prove the asymptotic convergence of the trajectory tracking error. Finally, an exhaustive simulation case study is carried out on a fleet of six multi-tilt tricopter drones to demonstrate the feasibility of the cooperative controller presented in this work.


I. INTRODUCTION
Cooperative control of UAVs (e.g. quadcopters, tricopters) has drawn the importance of academia and industry due to their widespread applications in reconnaissance [1], search and rescue [2], inspection and mapping [3], target localisation and tracking, precision agriculture, etc. Commercial drones have become more efficient, reliable, compact in size, durable, and offer longer flight time and higher payload capacity with the advent of modern gyroscopes and accelerometers, high-speed embedded processors and highpower batteries. Different from the classic quadcopter, where the three dimensional position tracking is indirectly accomplished by controlling the attitude, multi-tilt tricopters can attain attitude and position control independently and simultaneously (complete 6-DOF control) due to their airframe. This means that, multi-tilt tricopters (or simply tricopters) overcome the limitations of the classic quadcopter in this regard, and translation in the x and y directions can be achieved without modifying the attitude (roll, pitch and yaw angles). This paper aims to design a simple yet effective cooperative controller for a group of networked tricopter drones utilising the NI theory, characteristic loci technique and SMC-based linearisation principles. Negative Imaginary (NI) systems theory has attracted the interest of control theorists and practising engineers due to its potential to solve various real-life engineering challenges, like vibration control of lightly-damped systems [4], cantilever beams [5], [6]; nano-positioning applications [7]; cooperative train platooning [8]; control of dissipative systems [9]- [11]; etc. Due to a simple robust stability criterion depending only on the loop gain at zero frequency (known as the DC loop gain condition [4]), NI theory has gained a lot of traction. In a SISO setting, an NI transfer function's imaginary part remains negative for all ω ≥ 0. Among the strict subclasses within the NI class, Strictly NI (SNI [4]) and Output Strictly NI (OSNI [9], [12]- [14]) appear frequently in the literature. The primary motivation for applying NI theory in this work is that a certain class of tricopter drones may be converted into (or modelled as) an NI system in closed loop via feedback linearisation or SMC-based linearisation and hence, a network of such tricopters can then be controlled by a distributed SNI controller. This idea deploys the cooperative control scheme as the outer-loop controller while the SMCbased linearising control law works in the inner loop. The article [15] has proceeded along this direction and offered significant contributions on designing a cooperative control scheme for quadrotor UAVs. In this paper, we have used an SMC-based linearisation technique in contrast to the feedback linearisation technique (used in [16] for the same purpose) since the latter lacks in preserving robustness properties due to relying on the inverse-based nonlinearity cancellation. To describe the linearised tricopter closedloop system as an NI system (the inner-loop), a subspacebased closed-loop identification algorithm is developed to guarantee that the resulting model is NI. A distributed SNI controller based on output-feedback is then proposed to guarantee formation tracking for a group of tricopter drones. To prove stability and show that the tracking error (which includes both formation and cooperative tracking) converges, we utilize a new approach that uses the features of the characteristic loci from a network of SNI and NI plants rather than commonly used Lyapunov methods.

A. Coordinate rotations
Using the Euler z, y, x rotation sequence [17], a body may be transformed from the earth frame (which is inertial) to body-fixed frame using the following rotation matrix where η = [φ θ ψ] , c φ = cos φ and s φ = sin φ. In similar fashion, the Euler angle rates may be transformed from the body to inertial frame using [17] (2)

B. Passive and NI properties
This subsection includes the frequency-domain definitions of Passive and NI systems. We will first go through the definitions of Passive and Strictly Passive systems. Note that, for LTI systems, passive and Positive-real (PR) properties are equivalent. In fact, LTI passive systems are characterised by PR transfer functions.
Definition 1: (Passive System) [18] A system Σ(s) ∈ R m×m with no RHP poles is called Passive if Σ(jω) + Σ(jω) * ≥ 0 ∀ω ∈ R except those ω 0 ∈ ω where s = jω 0 is a pole of Σ(s). The multiplicity of the pole s = jω 0 cannot be more than one and the residue matrix ∆| s=jω0 lim We will now cover NI and SNI definitions.

C. Characteristic loci theory
The concept of characteristic loci and its application in determining closed-loop asymptotic stability of LTI MIMO systems were introduced during 1969-1973 by MacFarlane and Belletrutti [20]. This concept is analogous to a multi-loop Nyquist criterion that offers a pretty convenient graphical stability analysis tool for MIMO systems. The characteristic lociλ i (s), where i ∈ {1, 2, . . . , m}, of any square LTI system Σ(s) ∈ R m×m is a conformal transform of the complex function det[Σ(s)] into another complex (argand) plane when s follows the standard s-plane D-contour in a CW direction (see Fig. 4a).
Theorem 1: [20] The closed-loop interconnection of two LTI systems Σ(s) and Σ c (s), connected via negative feedback, is asymptotically stable if and only if the total number of the CCW encirclements about the (−1 + j0) point by the characteristic lociλ i (jω) of L(s) Σ(s)Σ c (s) ∀i ∈ {1, 2, . . . , n} is equal to the number of Right-half plane poles of L(s). When L(s) ∈ RH m×m ∞ , none of the characteristic lociλ i (jω) should encircle the (−1 + j0) point.

D. Multi-agent NI and SNI systems
This paper has exploited the multi-agent NI theory for modelling and controlling a multi-tricopter system whose linearised (via an inner-loop sliding-mode controller) dynamics can be well fitted into a six-channel SNI system model. The interactions or communications among the agents are modelled by an algebraic graph G , which is characterised by its Laplacian matrix L .
Property 1: The communication topology G among the agents is undirected and there exists at least one root node (target or virtual leader) that has direct/indirect connection to all the agents in the network. As a consequence of Property 1, the graph Laplacian matrix L = L ∈ R N ×N and the pinning-gain matrix P G = diag{w 0,1 , w 0,2 , · · · , w 0,N } ∈ R N ×N hold the relationship L + P G > 0. The first subscript '0' indicates the root node, which stands for the target or virtual leader or simply the reference generator. It is an exo-system that may have a dynamics independent of the dynamics of the agents. For the heterogeneous cases, the graph observes the property We will now review some fundamental properties of multiagent NI systems. [21] first established that a homogeneous network of NI (or SNI) agents, given byΣ(s) = (L + P G ) ⊗ Σ(s), retains the NI (or SNI) property andΣ(0) > 0 (or < 0) ⇔ Σ(0) > 0 (or < 0). In recent times, [22] and [23] have derived a crucial property of a multi-agent NI system exploiting the results of [24]. It asserts that all the characteristic lociλ i (jω) of a homogeneous multi-agent NI (or SNI) system lie in the union of the fourth and third complex plane quadrants (also known as the characteristic loci plane, discussed in Subsection II-C).

III. TRICOPTER MODELLING, LINEARISATION AND SYSTEM IDENTIFICATION
This section describes the modelling and linearisation of the tricopter drone. The model identification of the linearised tricopter while enforcing the NI constraints is also presented. The modelling part is inspired by [16] and [25].

A. Forces and moments
From Fig. 1, the total rotor force (equivalent force from the three rotors) in body-fixed coordinates [25] is given by From Fig. 1, let G oi ∀i ∈ {1, 2, 3} be the 3D distance of rotor i respectively from the tricopter's centre of mass, and O i ∀i ∈ {1, 2, 3} be the application points of f i respectively, with l 2 = 1 2 l 0 , l 1 = √ 3 2 l 0 and l 0 being the arm length. Then, the total rotor torque in body-fixed coordinates [25] is given The drag torque produced as a result of each spinning propellers may be expressed in inertial coordinates as The sum total of torques produced by the tricopter is therefore obtained as and actuator commands Ω are related by the following expression Remark 1: It can be shown that M is always full rank and invertible if k t , l 0 > 0, as utilized in Fig.2.

B. Rigid-body model
From Newton-Euler modelling techniques [17], the tricopter dynamics is given as and from (2) where ξ e , ω b , denote the 3D linear position and angular velocity respectively, I 3 ∈ R 3 is the identity matrix and I = diag{I x , I y , I z } is the inertia matrix.

C. Linearisation using SMC
Assuming small angles for φ and θ, Γ ≈ I 3 via (2). This impliesη =ω b in (6) which when expanded yields Also, let us assume that for motion in z-axis (vertical motion), Σf z = 0, Σf x = Σf y = 0; for motion in x-axis, Σf y = 0, Σf x = 0, Σf z = 0; and for motion in y-axis, Σf y = 0, Σf x = 0, Σf z = 0. These assumptions are sensible since thrust is essential for motion in all axes (to maintain altitude), but f x is mainly for motion in the x-axis while f y is mainly for motion in the y-axis. Applying these assumptions to the translational dynamics from (5) results in Let us now define a sliding surfaces φ =ė φ + λ φ e φ for the roll dynamics (7) following [26], where λ φ is positive and the roll angle error is e φ = φ − φ d . If a reaching lawṡ = −k φ sgn(s φ ), k φ > 0 is selected and a Lyapunov candidate V φ = 1 2s 2 φ > 0 is considered for stability analysis, theṅ which implies and by rearranging It is evident that an equilibrium point exists ats φ = 0 and is asymptotically stable via [26] because V φ is positive definite and the surfaceṡ φ satisfiesV φ ≤ 0. Hence, the linearising input τ φ will stabilize the roll angle. By using similar derivations in (8)- (12),

D. Tricopter model identification imposing NI constraints
A fairly accurate NI model of the linearised tricopter discussed in Section III-C, having six input-output channels with respect to the input variables x d , y d , z d , φ d , θ d and ψ d , needs to be identified in closed-loop. We collect data by passing a square wave as input to each channel and recording the output signal. The identification algorithm is described in [27] except that in our case, the estimation of the B matrix is done as a constrained optimization problem involving the NI property. The NI property has been embedded through the relationship AY + Y A ≤ 0 and B + AY C = 0 (referred to as the NI Lemma [4]). We have estimated the A and C matrices relying on the procedure in [27] and have obtained the following six transfer functions of the SMC-linearised tricopter model, which are stable NI. We will now dive into another key result of this paper. Here we will develop a simple but easy-to-implement coop-erative control scheme (shown in Fig. 3) for a fleet of SMClinearised tricopter drones exploiting the idea of a distributed 'mixed' SNI + strictly Passive controller.  Fig. 3: A formation tracking scheme for a multi-tricopter system Σ(s) which exhibit SNI property using a distributed 'mixed' SNI + strictly Passive controller Σ c (s).
Theorem 2: Consider a homogeneous multi-tricopter system whose SMC-linearised model Σ(s), as derived in (13)- (18), has a (6 × 6) SNI transfer function matrix. Let the communication topology G satisfy Property 1. Let r = r 1 r 2 , . . . , r N denote the reference vector and h = h 1 h 2 , . . . , h N represent the desired formation configuration vector. If a 'mixed' SNI + strictly Passive controller Σ c (s) ∈ RH 6×6 ∞ with Σ c (0) > 0 is used as in Fig. 3. Then, there always exists a finite β ∈ (0, β ] such that the proposed scheme in Fig. 3 drives all agents to converge to the state trajectory of the leader agent and the agents asymptotically reach the desired formation, under the influence of the following 'mixed' SNI + strictly Passive distributed controller Proof. The proof has been divided into two major parts. Part 1 derives the closed-loop asymptotic stability of the proposed cooperative control scheme shown in Fig. 3 and Part 2 shows that the consensus (or formation) error of the agents decays to zero. Part 1: Letλ i (s) be the characteristic loci of the transfer function mapping [(L +P G )⊗Σ c (s)Σ(s)] of the networked loop shown in Fig. 3. To proceed with the proof exploiting the characteristic loci technique, we define two sets of the Laplace variable s, Ψ ±j and Ψ ∞ , as marked in the s-plane D-contour shown in Fig. 4a. From Theorem 1, the closedloop interconnection (in negative feedback) of [I n ⊗ Σ(s)] and [(L + P G ) ⊗ βΣ c (s)], like in Fig. 3, is asymptotically stable as far as none of theλ i (jω) encircles the (− 1 β + j0) point for any β ∈ (0, β ]. This condition is graphically illustrated through Fig. 4b, that is, allλ i (jω) completely remain inside the Cyan-coloured region. In the following proof, the index i ∈ {1, 2, . . . , nm} will be uniformly used. Fig. 4: (a) The s-plane Nyquist D-contour; and (b) All λ i (jω) of (L + P G ) ⊗ Σ c (s)Σ(s) remain inside the Cyancoloured region when Σ(s) is SNI and Σ c (s) is 'mixed' SNI + strictly Passive with Σ(0) > 0 and Σ c (0) > 0.
Depending on Step 1 and Step 2, we can conclude that all λ i (s) remain inside the Cyan-coloured region (in Fig. 4b) which guarantees that even the worst-case critical point (− 1 β + j0) will remain unencircled by all the characteristic loci. This means that the proposed cooperative control scheme remains asymptotically stable for a finite range of β > 0.
Part 2: Since the networked closed-loop system shown in Fig. 3 is asymptotically stable (as derived in Part 1), the formation tracking (or consensus reaching) error will asymptotically decay to zero. In a notational expression, lim t→∞ ξ i (t) = 0 or lim t→∞ [r i (t) + h i (t) − y i (t)] = 0 ∀i. This part naturally results from [28,Theorem 1]. Consequently, we can infer that the linearised multi-tricopter system will attain the desired formation (or consensus) specified by r and h.

V. CASE STUDY AND SIMULATION RESULTS
We consider a fleet of six identical tricopter drones and the control objective is to attain a predefined timevarying formation and keep tracking the target (or the leader). According to the proposed cooperative control scheme, we choose a (6 × 6) 'mixed' SNI + strictly Passive controller transfer function matrix, in which i) n 1 (s) = During the first phase (for t < 10 seconds), the tricopter agents form a four-sided shape similar to a triangle around the target and in the next phase (for t ≥ 10seconds), they form a double echelon-like formation surrounding the target, and noting the change in the formation reference from r i1 to r i2 at t = 10 seconds. It can be noted from Fig. 5 that the six tricopters achieve leader-following formation tracking (implying consensus is also achieved) as t tends to infinity (t → ∞). It is worth mentioning that despite the changes in r i and h i , the stability of the overall network is maintained and the tricopters achieved the new formation successfully.

VI. CONCLUSIONS
This paper considers the cooperative control problem of a group of tricopter drones connected via a network. A continuous-time system identification algorithm is developed to identify a custom-made multi-tilt tricopter model in a closed loop that enforces the Negative Imaginary (NI) property. Sliding-mode Control (SMC) strategy has been used to linearise the highly-coupled and nonlinear dynamics of the tricopters. We then proposed a simple yet effective and easy-to-implement cooperative control scheme for a class of multi-tricopter systems that exhibits NI property upon applying an SMC-based linearisation technique in the inner loop. The theoretical proof of this scheme relies on the characteristic loci theory in contrast to the typical Lyapunovbased approaches found in the cooperative control literature. A detailed Matlab simulation case study is also provided to validate the effectiveness of the proposed scheme.