A Variational Approach to the Fuel Optimal Control Problem for UAV Formations

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Introduction
The pivotal role of unmanned aerial vehicles (UAVs) in modern aircraft technology is evidenced by the large number of civil and military applications they are employed in. For example, UAVs successfully serve as platforms carrying payloads aimed at land monitoring (Ramage et al., 2009), wildfire detection and management (Ambrosia & Hinkley, 2008), law enforcement (Haddal & Gertler, 2010), pollution monitoring (Oyekan & Huosheng, 2009), and communication broadcast relay (Majewski, 1999), to name just a few.
A formation of UAVs, defined by a set of vehicles whose states are coupled through a common control law (Scharf et al., 2003b), is often more valuable than a single aircraft because it can accomplish several tasks concurrently. In particular, UAV formations can guarantee higher flexibility and redundancy, as well as increased capability of distributed payloads (Scharf et al., 2003a). For example, an aircraft formation can successfully intercept a vehicle which is faster than its chasers (Jang & Tomlin, 2005). Alternatively, a UAV formation equipped with interferometic synthetic aperture radar (In-SAR) antennas can pursue both along-track and cross-track interferometry, which allow harvesting information that a single radar cannot detect otherwise (Lillesand et al., 2007).
Path planning is one of the main problems when designing missions involving multiple vehicles; a UAV formation typically needs to accomplish diverse tasks while meeting some assigned constraints. For example, a UAV formation may need to intercept given targets while its members maintain an assigned relative attitude. Trajectories should also be optimized with respect to some performance measure capturing minimum time or minimum fuel expenditure. In particular, trajectory optimization is critical for mini and micro UAVs (µUAVs) because they often operate independently from remote human controllers for extended periods of time (Shanmugavel et al., 2010) and also because of limited amount of available energy sources (Plnes & Bohorquez, 2006).
The scope of the present paper is to provide a rigorous and sufficiently broad formulation of the optimal path planning problem for UAV formations, modeled as a system of n 6-degrees of freedom (DoF) rigid bodies subject to a constant gravitational acceleration and aerodynamic forces and moments. Specifically, system trajectories are optimized in terms of control effort, that is, we design a control law that minimizes the forces and moments needed to operate a UAV formation, while meeting all the mission objectives. Minimizing the control effort is equivalent to minimizing the formation's fuel consumption in the case of vehicles equipped 2 Will-be-set-by- IN-TECH with conventional fuel-based propulsion systems (Schouwenaars et al., 2006) and is a suitable indicator of the energy consumption for vehicles powered by batteries or other power sources.
In this paper, we derive an optimal control law which is independent of the size of the formation, the system constraints, and the environmental model adopted, and hence, our framework applies to aircraft, spacecraft, autonomous marine vehicles, and robot formations. The direction and magnitude of the optimal control forces and moments is a function of the dynamics of two vectors, namely the translational and rotational primer vectors. In general, finding the dynamics of these two vectors over a given time interval is a demanding task that does not allow for an analytical closed-form solution, and hence, a numerical approach is required. Our main result involves necessary conditions for optimality of the formations' trajectories.
The contents of this paper are as follows. In Section 2, we present notation and definitions of the physical variables needed to formulate the fuel optimization problem. Section 3 gives a problem statement of the UAV path planning optimization problem, whereas Section 4 provides the necessary mathematical background for this problem. Next, in Section 5, we survey the relevant literature and highlight the advantages related to the proposed approach. Section 6 discusses results achieved by applying the theoretical framework developed in Section 4. In Section 7, we present an illustrative numerical example that highlights the efficacy of the proposed approach. Finally, in Section 8, we draw conclusions and highlight future research directions.

Notation and definitions
The notation used in this paper is fairly standard. When a word is defined in the text, the concept defined is italicized and it should be understood as an "if and only if" statement. Mathematical definitions are introduced by the symbol " ." The symbol N denotes the set of positive integers, R denotes the set of real numbers, R + denotes the set of nonnegative real numbers, R n denotes the set of n × 1 column vectors on the field of real numbers, and R n×m denotes the set of real n × m matrices. Both natural and real numbers are denoted by lower case letters, e.g., j ∈ N and a ∈ R, vectors are denoted by bold lower case letters, e.g., x ∈ R n , and matrices are denoted by bold upper case letters, e.g., A ∈ R n×m . Subsets of R n and R n×m are denoted by italicized upper case letters, e.g., A ⊆ R n and B ⊆ R n×m . The interior of the set A is denoted by int(A). The zero vector in R n is denoted by 0 n , the zero matrix in R n×m is denoted by 0 n×m , and the identity matrix in R n×n is denoted by I n .
For x ∈ R n we write x ≥≥ 0 n (respectively, x >> 0 n ) to indicate that every component of x is nonnegative (respectively, positive). We write ||·|| p for the p-norm of a vector and its corresponding equi-induced matrix norm, e.g., ||x|| p and ||A|| p . The transpose of a vector or of a matrix is denoted by the superscript (·) T , e.g., x T and A T . The cross product between two vectors a and b is denoted by a ∧ b. Given x ∈ R 3 such that x [x 1 ,x 2 ,x 3 ] T , we define

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Throughout the paper we use two types of mathematical statements, namely, existential and universal statements. An existential statement has the form: "there exist x ∈ A such that condition Φ is satisfied." A universal statement has the form: "condition Φ is satisfied for all x ∈ A." For universal statements we often omit the words "for all" and write: "condition Φ holds, x ∈ A." Time is the only independent variable used in this paper and is denoted by t. In this paper, t ∈ [t 1 , t 2 ], where [t 1 , t 2 ] ⊂ R is a fixed time interval and is a priori assigned. A generic member of a formation of n ∈ N UAVs is identified by the subscript i and, hence, i = 1, ..., n. We define r i : [t 1 , t 2 ] → R 3 as the position vector of the center of mass of the i-th vehicle in a given inertial reference frame, σ i : [t 1 , t 2 ] → R 3 as the attitude vector of the i-th vehicle in modified rodrigues parameters (MRPs) (Shuster, 1993), and x i [r T i , σ T i ] T as the state vector of the i-th vehicle. The system's configuration at time t is defined by x T 1 (t), ..., x T n (t) T .
The vector v i : [t 1 , t 2 ] → R 3 denotes the velocity of the center of mass of the i-th vehicle, ω i : [t 1 , t 2 ] → R 3 denotes the angular velocity of the i-th vehicle in a principal body reference frame, and (Neimark & Fufaev, 1972;Shuster, 1993). We assume We define u i,tran : [t 1 , t 2 ] → Γ i,tran (respectively, u i,rot : [t 1 , t 2 ] → Γ i,rot )a st h etranslational acceleration (respectively, the rotational acceleration) provided by the control system of the i-th vehicle in the formation, e.g., u i,tran is the acceleration provided by the propulsion system and u i,rot is the acceleration provided by the ailerons. The vector u i,tran (respectively, u i,rot ) is also referred to as the i-th translational control vector (respectively, the i-th rotational control vector). For a given set of real constants ρ i,1 , ρ i,2 , ρ i,3 , and ρ i,4 such that 0 ≤ ρ i,1 < ρ i,2 and 0 ≤ ρ i,3 < ρ i,4 , Γ i,tran and Γ i,rot are defined as is continuous for all t ∈ (t 1 , t 2 ) with the exception of a finite number of times t at which u(·) may have discontinuities of the first kind, and iii) u(τ)=lim t→τ − u(t), where τ ∈ [t 1 , t 2 ] is a point of discontinuity of first kind for u(t) (Pontryagin et al., 1962). We assume that u i,tran (respectively, u i,rot ) is an admissible control in Γ i,tran (respectively, Γ i,rot ) for each i ∈{1, . . . , n}.

Fuel consumption performance functional
A measure of the effort needed to control the i-th formation vehicle is given by the performance functional T and c is a real constant with units of distance. Without loss of generality we assume that |c| = 1. The performance functional t 2 t 1 ||u i,tran (t)|| 2 dt represents a measure of the fuel consumed over the time interval [t 1 , t 2 ] (Schouwenaars et al., 2006). Path planning for UAV formations is sometimes addressed by minimizing the more conservative performance functional t 2 t 1 ||u i,tran (t)|| 1 dt (Blackmore, 2008). It is important to note that ||u i,rot (t)|| 2 is much smaller than ||u i,tran (t)|| 2 for conventional aircraft and, hence, its contribution to the performance functional (1) is negligible. However, this assumption does not hold for the case of µUAVs (Bataillé et al., 2009).
The control effort for the entire formation can be captured by the performance measure whereũ(t) [u T 1 (t), ..., u T n (t)] T and µ i ∈ [0, 1], with ∑ n i=1 µ i = 1, which represents the relative importance of minimizing the control effort of the i-th vehicle with respect to the others.

Aircraft dynamic equations
Aircraft are subject to external forces and moments from the environment. Specifically, an aerial vehicle is subject to gravitational forces, aerodynamic forces, and aerodynamic moments. Accelerations induced by external forces and external moments acting on a formation vehicle are denoted by a : R 12 → R 3 and m : The unconstrained dynamic equations for the i-th vehicle are given by (Greenwood, 2003) d dt x where r min and r max are real constants such that 0 < r min < r max . Equation (12) ensures that r min ≤| | r 1 (t) − r 2 (t)|| 2 2 ≤ r max and (13) ensures that both vehicles always have the same attitude: Introducing the slack variables s 1 : [t 1 , t 2 ] → R and s 2 : [t 1 , t 2 ] → R, (12) becomes f ineq (s(t), x 1 (t), x 2 (t)) = ||r 1 (t) − r 2 (t)|| 2 2 − r max + 1 2 s 2 1 (t) r min −||r 1 (t) − r 2 (t)|| 2 2 + 1 2 s 2 2 (t) As noted in Section 3.3, the equality constraint (13) can be embedded into (12) to give where s j : [t 1 , t 2 ] → R 3 ,j= 3, 4. Note that in this case, the dimension of f ineq is increased since six additional slack variables have been introduced, which increases computational complexity.

Constrained formation dynamic equations
The formation's kinetic energy is given by König's theorem (Pars, 1965) and for our problem takes the form where m i is the mass of the i-th vehicle, which is assumed to be constant. The dynamic equations of the constrained formation can be written in terms of Lagrange coordinates by applying the Boltzmann-Hammel equation (Greenwood, 2003) to give d dt Equations (10) and (11) are the boundary conditions for (22). It is important to note that the dynamic equation (22) is written in terms of Lagrange coordinates, and hence, accounts for (5) and (6).
Analytical optimization techniques such as Pontryagin's minimum principle, Bellman's theorem, and calculus of variations require the dynamic equations to be written as a first-order ordinary differential equation in explicit form. Therefore, using the hypothesis on q dot , the second-order ordinary differential equation (22) needs to be written in a first-order forṁ Equation (22) or, equivalently, (23) gives a set of 6n − n 4 equations in 2(6n − n 4 ) unknowns, which are q and q dot . Thus, (22) needs to be solved together with (9) (Greenwood, 2003) to give q dot (t)

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A Variational Approach to the Fuel Optimal Control Problem for UAV Formations www.intechopen.com From (21) it follows that the formation's kinetic energy k is not an explicit function of s, and hence, if q is chosen as an explicit function of p components of s(t) ∈ R n 3 , then p of the 6n − n 4 equations in (22) cannot be straightforwardly recast in the explicit form given by (23). In this case, assume, without loss of generality, that q explicitly depends on the first p components of s and substitute the corresponding p equations in (24) with which is obtained by differentiating (7). In this case, the boundary conditions are given by (25) can be replaced by where s ∈C 3 (t 1 , t 2 ). In general, (7) must be differentiated so thats j (t), or one of its higher-order derivatives, explicitly appears and is multiplied by a term that is non-zero for all t ∈ [t 1 , t 2 ]. In this case, the differentiability assumptions on s and f ineq must be modified accordingly.

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By comparing this problem statement to the problem statement given in Section 3.4, it is clear that (5) and (6) are not explicitly accounted for in the above reformulation of the optimization problem. Hence, the constrained optimization problem has been reduced to an unconstrained optimization problem by the introduction of slack variables and Lagrange coordinates.

Transversality condition
Let S : D 1 → D 2 , where D 1 ⊂ R p and D 2 ⊂ R m , be a a continuously differentiable manifold and let the manifold tangent to S at y 0 be given by Every vector v ∈ R p that is normal to the manifold tangent to S at y 0 , that is, v T y = 0 for all y ∈ R p such that (40) holds, is said to verify the transversality condition for S at y 0 .

Pontryagin's minimum principle
Assume that a set of Lagrange coordinates has been found and that the formation's dynamic equations can be written in the form given by (24). Define the costate vectors λ dot : [t 1 , t 2 ] → R 6n−n 4 and λ dyn : [t 1 , t 2 ] → R 6n−n 4 so that the costate equation holds. The boundary conditions for (41) are given in Theorem 4.2 below. Given λ 0 ∈ R, define the Hamiltonian function Finally, define (43) The following theorem is known as the Pontryagin minimum principle. For details on this theorem and its numerous applications to optimal control, see Pontryagin et al. (1962).
Pontryagin minimum principle is a necessary condition for optimality, and hence, it provides candidate optimal control vectors. Sufficient conditions for optimality that are currently available in the literature do not apply to the optimization problem discussed herein.
It is worth noting that, instead of introducing the Lagrange coordinates, the equality constraints (7) and (5) can be accounted for by introducing Lagrange multipliers. This approach requires modifying the assigned performance measure and introducing additional costate vectors (Giaquinta & Hildebrandt, 1996;Lee & Markus, 1968). The dynamics of the costate vectors are characterized by ordinary differential equations known as costate equations, which need to be integrated numerically together with the dynamic equations of the state vector. Therefore, the computational complexity of finding optimal trajectories for large formations increases drastically when Lagrange multipliers are employed (L'Afflitto & Sultan, 2010). Alternatively, finding a suitable set of Lagrange coordinates can be a demanding task and in some cases the Lagrange coordinates may not have physical meaning (Pars, 1965); however, this reduces the dimension of the costate equation and consequently reduces the computational complexity.
Finally, we say the optimization problem is normal if λ 0 = 0, otherwise the optimization problem is abnormal. Normality can be shown by using the Euler necessary condition ). In particular, assume, ad absurdum, that λ 0 = 0. Now, if (41) and (44) imply that λ dot (t)= 0 6n−n 4 and λ dyn (t)=0 6n−n 4 for some t ∈ [t 1 , t 2 ], then assertion i) of Theorem 4.2 is contradicted. Therefore, λ 0 = 0, and hence, the optimization problem is normal. In this case, we assume without loss of generality that λ 0 = 1.

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Analytical and numerical approaches to the optimal path planning problem
Finding minimizers to (2) subject to the constraints (3) -(6) can be formulated as a Lagrange optimization problem (Ewing, 1969), which has been extensively studied both analytically and numerically in the literature. Analytical methods rely on either Lagrange's variational approach using calculus of variations or on the direct approach. In the classical variational approach, candidate minimizers for a given performance functional can be found by applying the Euler necessary condition. In order to find the minimizers, candidate optimal solutions need to be further tested by applying the Clebsh necessary condition, Jacobi necessary condition, Weierstrass necessary condition, as well as the associated sufficient conditions (Ewing, 1969;Giaquinta & Hildebrandt, 1996).
This classical analytical approach is not practical since applying the Euler necessary condition involves solving a differential-algebraic boundary value problem, whose analytical solutions are impossible to find for many practical problems of interest. Moreover, numerical solutions to this boundary value problem are affected by a strong sensitivity to the boundary conditions (Bryson, 1975). Furthermore, verifying the Jacobi necessary condition or the Weierstrass necessary condition can be a dauting task (L'Afflitto & Sultan, 2010).
A variational approach to the optimal path planning problem for a single vehicle, known as primer vector theory, was addressed by Lawden (1963). Lawden's problem was formulated using the assumptions that the acceleration vector a induced by external forces due to the environment is function of only the position vector, the vehicle is a 3 DoF point mass, and the state and control are only subject to equality constraints (Lawden, 1963). Primer vector theory is successfully employed in spacecraft trajectory optimization (Jamison & Coverstone, 2010), orbit transfers (Petropoulos & Russell, 2008), and optimal rendezvous problems (Zaitri et al., 2010), however, vehicles are often assumed to be point masses subject to only gravitational acceleration. Among the few studies on primer vector theory applied to vehicle formations, it is worth noting the work of Mailhe & Guzman (2004), where the formation initialization problem is addressed. Applications of primer vector theory to 6 DoF single vehicles have been employed to optimize the descent on Mars (Topcu et al., 2007). These studies, however, assume that the spacecraft is subject to a constant gravity acceleration, the control variables are the translational acceleration and the angular rates, and the translational acceleration can be pointed in any direction by rotating the vehicle.
Pontryagin's minimum principle is a variational method that is equivalent to the Weierstrass necessary condition with the advantage of addressing constraints on the control more effectively than applying the classical variational approach. State constraints need to be addressed by applying an optimal switching condition on the costate equation (Pontryagin et al., 1962), which generally increases the complexity of the problem. In the present formulation, the constraints on the formation are addressed by employing Lagrange coordinates, which does not introduce further conditions on the costate vector dynamics.
The direct approach in the calculus of variations, which is more recent than the variational approach, is based on defining a minimizing sequence of control functions u n (t) in some set Γ such that lim n→+∞ u n (t)=u(t) is a minimizer of the performance measure J[u(·)]. To this end, the following conditions should be met. i) Compactness of Γ, so that a minimizing sequence contains a convergent subsequence, ii) closedness of Γ, so that the limit of such a subsequence is contained in Γ, and iii) lower semicontinuity of the sequence 234 Recent Advances in Aircraft Technology www.intechopen.com {u n (·)} ∞ n=0 , that is, if lim n→+∞ u n (t)=u(t), then J[u(t)] ≤ lim inf n→+∞ J[u n (t)], u n ∈ Γ. Finally, it is also worth noting that approximate analytical methods can be used to solve the optimal path planning problem such as shape-based approximation methods (Petropoulos & Longuski, 2004), which are generally less effective due to the arbitrary parameterization of the minimizers (Wall, 2008).
Most of the results on the fuel consumption optimization employ numerical methods (Betts, 1998), which can be categorized as indirect or direct. Indirect numerical methods, which mimic the variational approach, suffer from high computational complexity since adjoint variables must be introduced. Alternatively, direct numerical methods are computationally more efficient, however, they require casting the given problem into a parameter optimization problem (Herman & Conway, 1987). Among the numerical methods commonly in use, it is worth mentioning genetic algorithms (Seereram et al., 2000) and particle swarm optimizers (Hassan et al., 2005).
One of the contributions of the present paper is that it extends Lawden's results on primer vector theory to formations of vehicles modeled as 6 DoF rigid bodies subject to generic environmental forces and moments by applying Pontryagin's minimum principle. As in all classical variational methods, Pontryagin's minimum principle is not suitable for numerically computing the optimal trajectory of a formation. However, Pontryagin's minimum principle allows us to draw analytical conclusions since it provides a generalization of the necessary conditions used by Lawden (1963), allows us to formally implement bounded integrable functions as admissible controls, and allows us to account for control constraints. Prussing (2010) and Marec (1979) have used Pontryagin's minimum principle to address primer vector theory using the same assumptions as Lawden (1963). In contrast, the present work provides additional analytical results for generic mission scenarios and complex environmental conditions for which numerical results can be verified. Furthermore, this paper exploits some properties of the costate space and consequently provides further insight into the formation system dynamics problem.

Necessary conditions for optimality of UAV formation trajectories
The following propositions are needed to develop the necessary conditions for optimality of the UAV formation problem. Proposition 6.1. Consider the performance measure J formation [ũ(·)] given by (2). Then, there exists at least oneũ * such that Proof. Since the integrand of the performance measure (1) is a continuous function defined on the compact set Γ i,tran × Γ i,rot , it follows from Weierstrass' theorem that (1) has a global minimizer on Furthemore, note that (44) implies that . Now, assume, ad absurdum, that λ * 0 = 0 and note that Since Ψ (q) is diffeomorphic and Theorem 4.1 holds, it follows that λ * dyn (t)=0 6n−n 4 . In this case, (41) can be explicitly written as d dt and hence, λ * dot (t)=0 6n−n 4 , which contradicts i) of Theorem 4.2.
If follows from Proposition 6.2 that the path planning optimization problem for a constrained formation is abnormal. Example 6.1 below, however, shows that this problem is normal for unconstrained 3 DoF vehicles, which is a well known result in the literature (Lawden, 1963).
Theorem 6.1. Consider the path planning optimization problem. If and −λ * dyn (t) are parallel, then the performance measure (2) is minimized. Moreover, for all i = 1, . . . , n, the following conditions hold.

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Recent Advances in Aircraft Technology is parallel to u * i,rot (t). Thus, using the triangular inequality, it follows that ), then Pontryagin's minimum principle does not provide any information about the optimal control, and hence, v) and vi) hold.
Analogous to Lawden's (Lawden, 1963) primer vector theory, λ * dyn (t) determine the magnitude and the direction of the control forces, and hence, we denote them as the translational primer vector and the rotational primer vector, respectively. Moreover, the trajectory given by each of the cases in Theorem 6.1 are called arcs. For each i = 1, . . . , n, the arcs corresponding to i) (respectively, ii)) are called maximum translational (respectively, rotational) thrust arcs. Similarly, arcs corresponding to iii) (respectively, iv)) are called null translational (respectively, rotational) thrust arcs. Finally, arcs corresponding to v) (respectively, vi)) are called singular translational (respectively, rotational) thrust arcs. The optimal translational and rotational control vectors for v) and vi) in Theorem

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Theorem 6.2. Consider the path planning optimization problem. Then, there exists c * ∈ R such that Proof. It follows from the Weierstrass -Erdmann condition (Giaquinta & Hildebrandt, 1996) that on an optimal trajectory, holds for all t ∈ (t 1 , t 2 ). Now, since h does not explicitly depend on t, it follows that there exists c * ∈ R such that h q * (t), q * dot (t), 0 6n , λ * dyn (t), λ * dot (t) = c * , which proves (48).
Proposition 6.3. Consider the costate dynamics given by (46). Then, the dynamics of λ * dyn (t) are decoupled from the dynamics of λ * dot (t).
Proof. The result is immediate from the form of (46).
It follows from Proposition 6.3 that the translational primer vector and the rotational primer vector dynamics are independent of the choice of q dot . Moreover, in solving for λ * dyn (t) we need not integrate a system of 2(6n − n 4 ) ordinary differential equations as in (41), but rather a system of (6n − n 4 ) ordinary differential equations, which is very advantageous for large formations.
Proposition 6.4. The translational primer vector and the rotational primer vector are continuously differentiable functions.
In order to elucidate the translational primer vector and rotational primer vector dynamics for a vehicle formation problem, we focus on specific formation configurations and on a specific environmental model. Hence, in the reminder of the paper we concentrate on the case where n v components of v i and n ω components of ω i are also components of q dot . A justification for this model is as follows. Assume that the i-th formation vehicle behaves as unconstrained, e.g., the first vehicle in Examples 4.1 and 4.2, or the dynamics of the i-th vehicle can be addressed as partly unconstrained, e.g., the second formation vehicle in the aforementioned examples. In either of these cases, it is natural to choose the unconstrained components of v i and ω i as some of the components of q dot . This model includes the classical formation configuration known 238 Recent Advances in Aircraft Technology www.intechopen.com as the leader-follower model, whose trajectories are computed as a function of the leader's path (Wang, 1991).
To simplify the environmental model assume that For notational convenience, we will refer to (49) and (50) as a(v i (t)) and ω i (v i (t), ω i (t)), respectively. This assumption on the accelerations induced by external forces and external moments is justified by a common environmental model given by (Anderson, 2001) where g is the constant gravitational acceleration is the unit vector in the direction of the aerodynamic lift (respectively, in the direction opposite to the aerodynamic side force), is the unit vector in the direction of roll (respectively, pitch and yaw), and k i,D , k i,L , k i,S , k i,R , k i,P , and k i,Y , are the drag, lift, side force, roll, pitch, and yaw coefficients, respectively.

Conclusion and recommendations for future research
In this paper, we addressed the problem of minimizing the control effort needed to operate a formation of n UAVs. Specifically, a candidate optimal control law as well as necessary conditions for optimality that characterize the resulting optimal trajectories are derived and discussed assuming that the formation vehicles are 6 DoF rigid bodies flying in generic environmental conditions and subject to equality and inequality constraints. The results presented extend Lawden's seminal work (Lawden, 1963) and several papers predicated on his work.
An illustrative numerical example involving a formation of two vehicles is provided to illustrate the mathematical path planning optimization framework presented in the paper. Furthermore, we show that our framework is not restricted to UAV formations and can be applied to formations of robots, spacecraft, and underwater vehicles.
The results of the present paper can be further extended in several directions. Specifically, an analytical study of the translational primer vector and the rotational primer vector can be useful in identifying numerous properties of the formation's optimal path. In particular, the translational primer vector and the rotational primer vector can be used to measure the sensitivity of the candidate optimal control law to uncertainties in the dynamical model. In this paper, we provide a generic formulation to the optimal path planning problem in order to address a large number of formation problems. However, specializing our results to a particular formation and a particular environmental model can lead to analytical tools that can be amenable to efficient numerical methods. Additionally, nonholonomic constraints have not been accounted in our framework and can be addressed by modifying Theorem 4.1. Finally, in this paper, we penalize vehicle control effort by tuning the constants µ 1 ,...,µ n in (2). In many practical applications, however, it is preferable to trade-off the control effort in a formation of vehicles by optimizing over the free parameters µ 1 ,...,µ n .

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