Collision Avoidance for an Aircraft in Abort Landing - Semantic Scholar

J Optim Theory Appl (2010) 146: 233–254 DOI 10.1007/s10957-010-9669-2

Collision Avoidance for an Aircraft in Abort Landing: Trajectory Optimization and Guidance A. Miele · T. Wang · J.A. Mathwig · M. Ciarcià

Published online: 18 February 2010 © Springer Science+Business Media, LLC 2010

Abstract For a host aircraft in the abort landing mode under emergency conditions, the best strategy for collision avoidance is to maximize wrt to the controls the timewise minimum distance between the host aircraft and an intruder aircraft. This leads to a maximin problem or Chebyshev problem of optimal control. At the maximin point of the encounter, the distance between the two aircraft has a minimum wrt the time; its time derivative vanishes and this occurs when the relative position vector is orthogonal to the relative velocity vector. By using the zero derivative condition as an inner boundary condition, the one-subarc Chebyshev problem can be converted into a two-subarc Bolza-Pontryagin problem, which in turn can be solved via the multiple-subarc sequential gradient-restoration algorithm. Optimal Trajectory. In the avoidance phase, maximum angle of attack is used by the host aircraft until the minimum distance point is reached. In the recovery phase, the host aircraft completes the transition of the angle of attack from the maximum value to that required for quasisteady ascending flight. Guidance Trajectory. Because the optimal trajectory is not suitable for real-time implementation, a guidance scheme is developed such that it approximates the optimal trajectory results in real time. In the avoidance phase, the guidance scheme employs the same control history (maximum angle of attack) as that of the optimal trajectory so as to achieve the goal of maximizing wrt the control the timewise minimum distance. In the recovery phase, the guidance scheme employs a time-explicit This paper is based on Refs. [1–3]. A. Miele () · T. Wang Aero-Astronautics Group, Rice University, Houston, TX, USA e-mail: [email protected] J.A. Mathwig PROS Revenue Management, Houston, TX, USA M. Ciarcià Università di Palermo, Palermo, Italy

234

J Optim Theory Appl (2010) 146: 233–254

cubic control law so as to achieve the goal of recovering the quasisteady ascending flight state at the final time. Numerical results for both the optimal trajectory and the guidance trajectory complete the paper. Keywords Collision avoidance · Abort landing · Aerospace engineering · Optimal control · Guidance 1 Introduction The danger of collision between aircraft is particularly severe in areas of dense air traffic, such as the vicinity of an airport. For recent research on trajectory optimization related to collision avoidance, see Refs. [4–11]. In spite of the progresses, the possibility of collision still exists due to a variety of factors such as pilot error, aircraft malfunction, air traffic control system breakdown, bad weather, difficult environmental conditions, and so on. This paper studies both the optimal trajectory and the associated guidance trajectory for an aircraft in the abort landing mode under emergency conditions. Clearly, the best strategy is to maximize wrt to the controls the timewise minimum distance between the two aircraft: a host aircraft and an intruder aircraft. This problem is a maximin problem or Chebyshev problem of optimal control. For the optimal trajectory, the conversion of the Chebyshev problem into a BolzaPontryagin problem is based on a simple observation: at the maximin point of the encounter between the two aircraft, the distance between the two aircraft has a minimum wrt the time; its time derivative vanishes and this occurs when the relative position vector is orthogonal to the relative velocity vector. In turn, the resulting geometric/kinematic condition is an inner boundary condition allowing the separation of the collision avoidance maneuver into two phases (avoidance and recovery) connected via the maximin point. In this way, the one-subarc Chebyshev problem can be converted into a two-subarc Bolza-Pontryagin problem, to be solved via the multiplesubarc sequential gradient-restoration algorithm (SGRA). For the multiple-subarc sequential gradient-restoration algorithm and its applications, see Refs. [12–16]. Because the optimal trajectory is not suitable for real-time execution, a guidance scheme, capable of real-time implementation, is developed so as to approximate the optimal trajectory results. Indeed, in the development of the guidance scheme, the analysis and utilization of the properties of the optimal trajectory is essential (Refs. [17–20]). The results on the optimal trajectory for aircraft collision avoidance in abort landing (see Refs. [1–3]) show that the optimal trajectory can be partitioned into two phases: avoidance and recovery. (i) In the avoidance phase of the optimal trajectory, saturated control (maximum angle of attack) is used until the minimum distance point is reached, provided enough time is available for the recovery phase. (ii) In the recovery phase of the optimal trajectory, the angle of attack gradually changes from the maximin value to that required for quasisteady ascending flight. Based on these properties, a guidance scheme is developed approximating the optimal trajectory in real time. (i) In the avoidance phase of the guidance trajectory,


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maximum angle of attack is used, so as to achieve the goal of maximizing wrt the control the timewise minimum distance. (ii) In the recovery phase of the guidance trajectory, a time-explicit cubic control law is used so as to achieve the goal of recovering quasisteady ascending flight. The paper is organized as follows. Section 2 contains the system description; Sect. 3 formulates the collision avoidance problem as an optimal control problem; Sect. 4 develops the guidance scheme based on the properties of the optimal trajectory; Sect. 5 provides the data used in the examples; Sect. 6 presents numerical results for both the optimal trajectory and the guidance trajectory; Sect. 7 contains the conclusions. For completeness, Appendix A contains details concerning the boundary conditions. Appendix B discusses a cooperative differential game approach to the solution of the collision avoidance problem.

2 System Description This section deals with the kinematic and dynamic equations of motion for the host aircraft. We make the following assumptions: (i) the aircraft is a particle of constant mass; (ii) the flight takes place in a vertical plane; (iii) Newton’s law is valid in an Earth-fixed system. As a consequence, the equations of motion can be written as x = V cos γ ,

(1a)

h = V sin γ ,

(1b)

V = (T /m) cos(α + δ) − D/m − g sin γ ,

(1c)

γ = (T /mV ) sin(α + δ) + L/mV − (g/V ) cos γ ,

(1d)

where the prime denotes derivative with respect to the actual time t. The forces acting on the aircraft include the thrust T , drag D, lift L, and weight W . These forces are represented by the functional relations T = T (h, V , β) ∼ = βTmax (h, V ),

(2a)

D = D(h, V , α),

(2b)

L = L(h, V , α),

(2c)

W = mg.

(2d)

The quantities appearing in (1)–(2) are the longitudinal distance x, altitude h, velocity V , path inclination γ , mass m, acceleration of gravity g, thrust inclination angle δ, angle of attack α, and thrust setting β. 2.1 Inequality Constraints In general, the thrust setting and the angle of attack are the main controls available to the pilot in order to maneuver an aircraft in a vertical plane.

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The thrust setting β and its time rate β must satisfy the inequality constraints 0 ≤ β ≤ 1, 0≤β

(3a)

≤ βmax ,

(3b)

where β = 0 denotes engine shut off, β = 1 denotes engine operating at maximum denotes the max permissible thrust setting time rate. thrust, and βmax At the beginning of the collision avoidance maneuver, the thrust setting must be switched from its initial value β0 (consistent with certain quasisteady descending flight conditions) to the maximum value at the maximum permissible time rate = 0.2/s). Afterward, the thrust setting must be held at the maximum value until (βmax the end of the abort landing maneuver. Hence, the time history of the thrust setting is represented by the relations t, β(t) = β0 + βmax

β(t) = 1,

0 ≤ t ≤ tsβ ,

tsβ ≤ t ≤ θ,

(4a) (4b)

where θ is the final time and tsβ = (1 − β0 )/βmax

(4c)

denotes the switch time at which max thrust setting is reached. With the thrust setting time history described by (4), the only remaining control is the angle of attack. In turn, the angle of attack α(t) and its time rate α (t) are subject to the two-sided inequality constraints αmin ≤ α ≤ αmax ,

(5a)

, −αmax ≤ α ≤ αmax

(5b)

where αmax , αmin , αmax are prescribed constants. The above inequalities can be converted into equalities via the following nonsingular transformation:

α = (1/2)(αmax + αmin ) + (1/2)(αmax − αmin ) sin η,

(6a)

/(αmax − αmin )] sin w, η = [2αmax

(6b)

in which η(t) denotes an auxiliary state variable and w(t) denotes an auxiliary control variable. Because (6) imply α = αmax cos η sin w,

(7)

it is clear that any pair of functions η(t), w(t) consistent with (6) satisfies automatically both inequalities (5). The transformation (6) has two advantages: (i) it is nonsingular; (ii) the angle of attack boundary is reached tangentially, regardless of the value of the auxiliary control. These advantages are obtained at a price: as the angle of attack moves toward its upper or lower boundary, the available α -range shrinks proportionally to cos η, vanishing at the upper or lower boundary.


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2.2 Transformed System In light of (6), the system equations of the host aircraft (1) can be rewritten as x = V cos γ ,

(8a)

h = V sin γ ,

(8b)

V = (βTmax /m) cos(α + δ) − D/m − g sin γ ,

(8c)

γ = (βTmax /mV ) sin(α + δ) + L/mV − (g/V ) cos γ ,

(8d)

/(αmax − αmin )] sin w, η = [2αmax

(8e)

with α = (1/2)(αmax + αmin ) + (1/2)(αmax − αmin ) sin η.

(8f)

In the transformed system (8), the state variables are x(t), h(t), V (t), γ (t), η(t); the new control variable is w(t). Once η(t) is known, the original control α(t) can be recovered via (8f). 2.3 Potential Collision Now, let us assume that both the host aircraft and the intruder aircraft move with constant velocity and direction. Under this scenario, the motion of the host aircraft is described by the linear relations x(t) = x0 + (V cos γ )0 t,

(9a)

h(t) = h0 + (V sin γ )0 t,

(9b)

in which the subscript 0 denotes a quantity evaluated at the initial time t = 0. Under the same scenario, the motion of the intruder aircraft (subscript asterisk) is described by the linear relations x∗ (t) = x0 + (d cos ξ )0 + (V∗ cos χ∗ )0 t,

(10a)

h∗ (t) = h0 + (d sin ξ )0 + (V∗ sin χ∗ )0 t,

(10b)

in which d is the distance between the two aircraft, ξ is the angle between the relative position vector and the x-axis, V∗ is the velocity modulus of the intruder aircraft, and χ∗ (t) is the angle between the velocity vector of the intruder aircraft and the x-axis. Given the initial values V0 , γ0 for the host aircraft, the initial values (V∗ )0 , (χ∗ )0 for the intruder aircraft, and the initial values d0 , ξ0 for the relative position vector, a collision occurs if the following relations are satisfied at some forward time tc > 0: x(tc ) = x∗ (tc ),

(11a)

h(tc ) = h∗ (tc ),

(11b)

(V cos γ )0 tc = (d cos ξ )0 + (V∗ cos χ∗ )0 tc ,

(12a)

with the implication that

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(V sin γ )0 tc = (d sin ξ )0 + (V∗ sin χ∗ )0 tc .

(12b)

Elimination of the collision time tc from the above relations yields the following determinantal form of the collision condition: (V cos γ − V∗ cos χ∗ )0 (d cos ξ )0 (12c) = 0. (V sin γ − V∗ sin χ∗ )0 (d sin ξ )0 The meaning of (12) is that, if (i) the relative velocity vector is collinear with the relative position vector at the initial time t = 0 and if (ii) both the host and intruder aircraft move with constant velocity and direction, then a collision will occur at a subsequent time tc , which is consistent with (12a) and (12b) in light of (12c).

3 Optimization Problem For collision avoidance under emergency conditions, the best strategy is to maximize wrt the controls the timewise minimum distance between the host aircraft and the intruder aircraft. At the maximin point of the encounter, the distance between the two aircraft has a minimum wrt the time, which occurs when the relative position vector is orthogonal to the relative velocity vector. In this way, we obtain an inner boundary condition to be satisfied at the maximin point separating the two main branches of the maneuver: the avoidance branch and the recovery branch. As a consequence, a onesubarc Chebyshev problem can be transformed into a two-subarc Bolza-Pontryagin problem solvable via the multiple-subarc sequential gradient-restoration algorithm (SGRA). The preconditions for the application of the multiple-subarc SGRA to a BolzaPontryagin type optimization problem of collision avoidance are (i) the determination of the inner boundary condition and (ii) the transformation of the problem from the actual time domain to the normalized time domain. 3.1 Inner Boundary Condition The distance between the host aircraft and the intruder aircraft can be written as (13a) d = [(x − x∗ )2 + (h − h∗ )2 ] and its time derivative (prime superscript) is d = (1/d)[(x − x∗ )(x − x∗ ) + (h − h∗ )(h − h∗ ) ].

(13b)

For the convenience of computation, let D = (1/2)d 2 = (1/2)[(x − x∗ )2 + (h − h∗ )2 ]

(14a)

denote the squared distance function. Its time derivative is D = (x − x∗ )(x − x∗ ) + (h − h∗ )(h − h∗ ) .

(14b)


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Inspection of (13b) and (14b) shows that the conditions d = 0 and D = 0

(15a)

are reached simultaneously when (x − x∗ )(x − x∗ ) + (h − h∗ )(h − h∗ ) = 0,

(15b)

that is, when (see the equations of motion) (x − x∗ )(V cos γ − V∗ cos χ∗ ) + (h − h∗ )(V sin γ − V∗ sin χ∗ ) = 0,

(15c)

namely, when the relative position vector is orthogonal to the relative velocity vector. Now, let t = 0 denote the initial time, let t = σ denote the time at which the orthogonality condition (15c) is satisfied, and let t = θ denote the assigned total maneuver time. These critical times allow us to subdivide the extremal arc, covering the interval 0 ≤ t ≤ θ , into two subarcs: the avoidance subarc 0 ≤ t ≤ σ and the recovery subarc σ ≤ t ≤ θ . Hence, we can write the relations θ1 = σ1 ,

θ2 = θ − σ,

θ = θ1 + θ2

(16a)

for the time length of the avoidance subarc, the time length of the recovery subarc, and the total maneuver time. 3.2 Time Normalization Having established that the time length of the first subarc is σ , the time length of the second subarc is θ − σ , with the total maneuver time θ given, we introduce now a time transformation defined in such a way that the normalized time length of each subarc is equal to 1. More precisely, the time transformation is subarc 1,

τ = t/σ,

0 ≤ τ ≤ 1,

subarc 2,

τ = (t − σ )/(θ − σ ),

0 ≤ t ≤ σ, 0 ≤ τ ≤ 1,

(16b) σ ≤ t ≤ θ.

(16c)

3.3 Performance Index Let F denote a generic function of the time. Let F (t) be its one-index representation in the actual time domain. Then, F (τ, i) is its two-index representation in the normalized time domain, with i = 1 for the first subarc and i = 2 for the second subarc. With the above understanding, we replace the one-subarc Chebyshev problem with a two-subarc Bolza-Pontryagin problem. Now, the optimization problem consists in maximizing [wrt the new control w(t) and parameter σ ] the distance function at the point where the inner boundary condition is satisfied. In the actual time domain, the problem is max D(σ )

w(t),σ

→

min [−D(σ )].

w(t),σ

(17a)

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In the normalized time domain, the problem becomes max D(1, 1)

w(τ ),σ

→

min [−D(1, 1)].

w(τ ),σ

(17b)

In light of the continuity of the distance function at the interface between the avoidance and recovery subarcs, (17b) can be rewritten as max D(0, 2)

w(τ ),σ

→

min [−D(0, 2)].

w(τ ),σ

(17c)

In (17), D(1, 1) means D(τ, i) evaluated at the end time τ = 1 of the first subarc i = 1.D(0, 2) means D(τ, i) evaluated at the start time τ = 0 of the second subarc i = 2. 3.4 Penalized Performance Index In the collision avoidance problem for an aircraft in abort landing, it is necessary to prevent the undershooting of the initial altitude h0 . This can be achieved by replacing the performance index (17b) with the following penalized performance index: min[−D(1, 1) + kP ],

(18a)

where k > 0 is a suitable penalty constant and P is the penalty functional P=

2

1

E 2 (τ, i)dτ.

(18b)

i=1 0

In the above relation E(τ, i) measures the violation of the altitude threshold and is defined as follows: E(τ, i) = 0,

if h ≥ h0 ,

E(τ, i) = h0 − h(τ, i),

(18c) if h ≤ h0 .

(18d)

3.5 Differential Constraints Let a dot superscript denote a derivative wrt the normalized time τ . For both subarcs, the differential constraints can be rewritten as x˙ = K[V cos γ ],

(19a)

h˙ = K[V sin γ ],

(19b)

V˙ = K[(Tmax /m) cos(α + δ) − D/m − g sin γ ],

(19c)

γ˙ = K[(Tmax /mV ) sin(α + δ) + L/mV − (g/V ) cos γ ],

(19d)

/(αmax η˙ = K[2αmax

− αmin )] sin w,

(19e)

α = (1/2)(αmax + αmin ) + (1/2)(αmax − αmin ) sin η

(19f)

with


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and subarc 1,

K = σ,

0 ≤ τ ≤ 1,

subarc 2,

K = θ − σ,

0 ≤ τ ≤ 1,

0 ≤ t ≤ σ, σ ≤ t ≤ θ.

(19g) (19h)

3.6 Boundary Conditions The boundary conditions include the initial conditions, the continuity condition at the interface of the subarcs, the inner boundary condition at the interface of the subarcs, and the final conditions. The initial conditions are x(0, 1) = 0,

(20a)

h(0, 1) = h0 ,

(20b)

V (0, 1) = V0 ,

(20c)

γ (0, 1) = γ0 , η(0, 1) = η0 = sin

(20d) −1

{[2α0 − αmax − αmin ]/(αmax − αmin )},

(20e)

with h0 , V0 , γ0 , η0 , hence α0 specified. For details, see Appendix A.1 dealing with quasisteady descending flight. The continuity conditions at the interface of the first and the second subarcs are x(1, 1) = x(0, 2),

(21a)

h(1, 1) = h(0, 2),

(21b)

V (1, 1) = V (0, 2),

(21c)

γ (1, 1) = γ (0, 2),

(21d)

η(1, 1) = η(0, 2).

(21e)

The inner boundary condition at the interface of the first and the second subarcs is ˙ D(1, 1) = 0.

(22)

The final conditions are V (1, 2) = Vθ ,

(23a)

γ (1, 2) = γθ ,

(23b)

η(1, 2) = ηθ = sin−1 {[2αθ − αmax − αmin ]/(αmax − αmin )},

(23c)

with Vθ , γθ , ηθ , hence αθ specified. For details, see Appendix A.2 dealing with quasisteady ascending flight.

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3.7 Bolza-Pontryagin Problem In sum, the Bolza-Pontryagin problem of aircraft collision avoidance can be formulated as that of minimizing the penalized performance index (18), subject to the differential constraints (19), the initial conditions (20), the continuity conditions (21), the inner boundary condition (22), and the final conditions (23). With the actual final time θ given, the unknowns are the state and control variables of the first subarc x(t, 1), h(t, 1), V (t, 1), γ (t, 1), η(t, 1), w(t, 1), the state and control variables of the second subarc x(t, 2), h(t, 2), V (t, 2), γ (t, 2), η(t, 2), w(t, 2), plus the time parameter σ . With these functions known, the angle of attack α(t, 1) for the first subarc and the angle of attack α(t, 2) for the second subarc can be recovered via (19f).

4 Guidance Scheme While the optimal trajectory provides the best maneuver for aircraft collision avoidance, it is not suitable for real-time execution. This is why a guidance scheme must be developed approximating the optimal trajectory and suitable for real-time implementation. To develop such guidance scheme, we recall now the properties of the optimal trajectory, in particular, its relationship with the corresponding control. 4.1 Optimal Trajectory Properties The numerical results of Ref. [1] show that the optimal trajectory can be partitioned into two phases: avoidance and recovery. The objective of the avoidance phase is to maximize wrt the controls the timewise minimum distance between the host aircraft and the intruder aircraft; the objective of the recovery phase is to bring the host aircraft back to quasisteady ascending state. Ideally, the switch from the avoidance phase to the recovery phase occurs at the time when the minimum distance between the two aircraft is reached. The numerical results of Ref. [1] indicate that, to maximize wrt the control the timewise minimum distance, the angle of attack should be brought to its maximum value and held there until the minimum distance point is past, providing the time available for the recovery phase is not too strict. Note that long-time application of maximum angle of attack to raise the altitude implies a velocity loss; should the velocity loss become excessive, this in turn would cause a subsequent altitude loss (see Ref. [1]). This is why prolonged application of maximum angle of attack is to be discouraged. 4.2 Avoidance Phase The thrust setting and the angle of attack are the two controls available for the host aircraft maneuver. For abort landing, the power setting is specified as follows [see (4)]: β(t) = β0 + (1 − β0 )t/tsβ ,

0 ≤ t ≤ tsβ ,

(24a)


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β(t) = 1,

tsβ ≤ t ≤ σ,

(24b)

where σ is the end time of the avoidance phase and tsβ = (1 − β0 )/βmax

(24c)

is the switch time at which max thrust setting β = 1 is reached. Concerning the angle of attack, based on the analysis of the optimal trajectory results, a simple rule can be applied: after detecting the potential collision under emergency conditions, the pilot must bring the angle of attack from the initial value α0 to the maximum value and hold it there until the minimum distance point is past. Therefore, the angle of attack α(t) is specified as follows: α(t) = α0 + (αmax − α0 )t/tsα , α(t) = αmax ,

0 ≤ t ≤ tsα ,

tsα ≤ t ≤ σ,

(25a) (25b)

where σ is the end time of the avoidance phase and tsα = (αmax − α0 )/αmax

(25c)

is the switch time at which max angle of attack is reached. In turn, the above linear law can be smoothed via the following trigonometric transformation: α(t) = α0 + (αmax − α0 ) sin[(π/2)t/tsα ], α(t) = αmax ,

0 ≤ t ≤ tsα ,

tsα ≤ t ≤ σ,

(26a) (26b)

where σ is the end time of the avoidance phase and tsα = (π/2)(αmax − α0 )/αmax

(26c)

is the switch time at which max angle of attack is reached. Note that the switch time predicted via (26c) is always larger than the switch time predicted via (25c). 4.3 Recovery Phase, First Scheme For the recovery phase, maximum power setting is assumed, β(t) = 1,

σ ≤ t ≤ θ.

(27)

Because the main objective of maximizing the minimum distance between the two aircraft has been achieved already in the avoidance phase, it is not necessary for the guidance scheme of the recovery phase to mirror exactly the optimal trajectory results, as long as the guidance scheme can achieve approximately the goal of quasisteady ascending state at the end of the recovery phase. With the above understanding, the following time-explicit cubic guidance scheme is considered in normalized form: α = C1 τ + C2 (1 − τ ) + C3 τ (1 − τ ) + C4 τ 2 (1 − τ ),

(28a)

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with t −σ , θ −σ Hence, in nonnormalized form, τ=

1−τ =

θ −t . θ −σ

t −σ 2 θ −t t −σ θ −t t −σ θ −t + C2 + C3 + C4 , α = C1 θ −σ θ −σ θ −σ θ −σ θ −σ θ −σ

(28b)

σ ≤ t ≤ θ. (28c)

The coefficients C1 , C2 have the values C1 = α(θ ) = αθ ,

C2 = α(σ ) = ασ .

(28d)

The coefficients C3 , C4 must be determined using a quasilinearization process in such a way that, upon integration of the differential system with the guidance law (28), the following final conditions are satisfied: γ (θ ) = γθ .

V (θ ) = Vθ ,

(28e)

We note that computations done with the guidance law (28) are acceptable providing the angle of attack satisfies the inequality αmin ≤ α ≤ αmax .

(29)

If a violation occurs, a modification of (28) is in order. 4.4 Recovery Phase, Second Scheme Next, we introduce the auxiliary control ψ defined by α = (1/2)(αmax + αmin ) + (1/2)(αmax − αmin ) sin ψ,

(30)

which automatically satisfies the inequality (29). Then, in normalized form, we replace the time-explicit cubic guidance scheme (28a) with the following time-explicit cubic guidance scheme: ψ = C1 τ + C2 (1 − τ ) + C3 τ (1 − τ ) + C4 τ 2 (1 − τ ),

(31a)

with t −σ , θ −σ Hence, in nonnormalized form, τ=

ψ = C1

1−τ =

θ −t . θ −σ

t −σ 2 θ −t t −σ θ −t t −σ θ −t + C2 + C3 + C4 , θ −σ θ −σ θ −σ θ −σ θ −σ θ −σ

(31b)

σ ≤ t ≤ θ. (31c)

Now, the coefficients C1 , C2 have the values C1 = ψ(θ ) = ψθ ,

C2 = ψ(σ ) = ψσ

(31d)


245

with 2αθ − αmax − αmin ψθ = sin , αmax − αmin −1 2ασ − αmax − αmin . ψσ = sin αmax − αmin −1

(31e) (31f)

The coefficients C3 , C4 must be determined using a quasilinearization process in such a way that, upon integration of the differential system with the guidance law (30)–(31), the following final conditions are satisfied: V (θ ) = Vθ ,

γ (θ ) = γθ .

(31g)

5 Data for the Examples In this section, we present the data used in the numerical experiments. The host aircraft under consideration is a Boeing B-727 aircraft powered by three Pratt & Whitney JT8D-17 turbofan engines. It is assumed that the host aircraft (attempting to land) is initially at the altitude of 1000 ft on a glide trajectory with path inclination of −3 deg. After detecting a potential collision with the intruder aircraft, the host aircraft switches to the abort landing mode. The intruder aircraft is assumed to fly horizontally at an altitude (h∗ = 900 ft) below that of the host aircraft and to have the same velocity (V∗ = 240 ft/s) as the initial velocity of the host aircraft. The intruder aircraft is uncooperative and the collision avoidance maneuver is performed only by the host aircraft. The landing weight of the host aircraft is W = 150000 lb and its reference surface (wing) is S = 1560 ft2 . 5.1 Aerodynamic Forces The aerodynamic forces include lift and drag, which can be controlled via the angle of attack; the maximum angle of attack is αmax = 17.2 deg, the minimum angle of = 3 deg/s; attack is αmin = 0 deg, and the maximum angle attack time rate is αmax the flap setting is 30 deg. The lift and drag are represented by the relations L = (1/2)C L ρSV 2 ,

(32a)

D = (1/2)C D ρSV 2 ,

(32b)

with ρ the air density, S the reference surface, V the velocity, C L the lift coefficient, and C D the drag coefficient. By a least-square fit of manufacturer-supplied data for flap setting 30 deg, the aerodynamic coefficients can be written as C L = 0.7145 + 6.065α,

0 ≤ α ≤ 0.3 rad,

C D = 0.1394 − 0.08167α + 2.569α , 2

0 ≤ α ≤ 0.3 rad.

(32c) (32d)

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5.2 Thrust It is assumed that the thrust has the form T = βTmax (h, V ),

(33a)

where β is the thrust setting and Tmax is the maximum thrust. The thrust setting is = 0.2/s and tsβ = 3.57 s. The governed by (4) with β0 = 0.286, βmax = 1, βmax assumed ambient temperature is 100 deg Fahrenheit. The dependence of the thrust on the altitude is disregarded and the maximum thrust is assumed to depend on only the velocity. With a least-square fit of manufacturer-supplied data in the velocity range 0–422 ft/s, the thrust can be represented via the relation Tmax /TR = 44.56 − 2.398(V /VR ) + 0.1442(V /VR )2 ,

(33b)

where V [ft/s] is the velocity, VR = 100 ft/s is a reference velocity, and TR = 1000 lb is a reference thrust. The inclination of the thrust with respect to the aircraft reference line is assumed to be δ = 2 deg. 5.3 Boundary Conditions for the Host Aircraft At the beginning of the abort landing maneuver, the host aircraft is in quasisteady descending flight. Specifically, the initial conditions are assumed to be (see the Appendix A.1) x0 = 0,

(34a)

h0 = 1000 ft,

(34b)

V0 = 240 ft/s,

(34c)

γ0 = −3 deg,

(34d) (34e)

with controls β0 = 0.286,

(34f)

α0 = 7.41 deg.

(34g)

At the end of the abort landing maneuver, the host aircraft recovers quasisteady ascending flight. Specifically, the final conditions are assumed to be (see the Appendix A.2) xθ = free,

(35a)

hθ = free,

(35b)

Vθ = 240 ft/s,

(35c)

γθ = 7.96 deg,

(35d)


247

with controls βθ = 1.00,

(35e)

αθ = 6.88 deg.

(35f)

The initial distance between the host aircraft and intruder aircraft is assumed to be d0 = 4000 ft. The final time is set at θ = 40 sec. 5.4 Intruder Aircraft The intruder aircraft is assumed to be flying at a lower altitude (h∗ = 900 ft) horizontally with the same velocity as the initial velocity of the host aircraft (V∗ = 240 ft/s) but in the opposite direction. The intruder aircraft is assumed to be uncooperative so that only the host aircraft performs the collision avoidance maneuver.

6 Numerical Results and Analysis The numerical computation of the optimal trajectory for collision avoidance under emergency conditions was done using the multiple-subarc sequential gradientrestoration algorithm (SGRA, Ref. [12]). The optimal trajectory maximizes wrt the control [the angle of attack α(t), since the thrust setting β(t) is prespecified] the timewise minimum distance between two aircraft: the host aircraft in the abort landing mode and the intruder aircraft flying uncooperatively at constant altitude below that of the host aircraft. The guidance trajectory approximates the properties of the optimal trajectory and can be implemented in the real time sense. Mirroring the optimal trajectory, the guidance trajectory has two phases: avoidance and recovery. In the avoidance phase, the guidance scheme switches the angle of attack to its maximum value and holds it there until the minimum distance point is past. In the recovery phase, the guidance scheme employs the angle of attack (30) in conjunction with the time-explicit cubic guidance law (31) for the auxiliary control ψ so as to achieve the goal of recovering the quasisteady ascending flight state at the final time. For d0 = 4000 ft and θ = 40 sec, the numerical results are shown in Tables 1–2 and Figs. 1–7. Table 1 provides the optimal trajectory data at three critical times (initial time, maximin time, final time); Table 2 provides the guidance trajectory data at the same critical times (initial time, maximin time, final time). Comparing Table 1 and Table 2, Table 1 Optimal trajectory data at critical times (θ = 40 s, d0 = 4000 ft) Time

t

x

h

V

γ

α

d

[sec]

[ft]

[ft]

[ft/s]

[deg]

[deg]

[ft]

t =0

0.00

0

1000

240.0

−3.00

7.41

t =σ

8.57

1917

1348

199.4

28.78

17.20

454

t =θ

40.00

7747

1929

240.0

7.96

6.88

13389

4000

248


Table 2 Guidance trajectory data at critical times (θ = 40 s, d0 = 4000 ft) Time

t

x

h

V

γ

α

d

[sec]

[ft]

[ft]

[ft/s]

[deg]

[deg]

[ft]

t =0

0.00

0

1000

240.0

−3.00

7.41

t =σ

8.57

1919

1346

199.8

28.76

17.20

452

t =θ

40.00

7743

1931

240.1

7.94

6.88

13384

4000

Fig. 1 Trajectory geometry of host aircraft and intruder aircraft. For the host aircraft, the solid line refers to the OT and the dashed line refers to the GT

Fig. 2 Host aircraft. Time history of longitudinal distance: optimal trajectory (solid line) and guidance trajectory (dashed line)

we see that: (i) at the initial time t = 0, both the optimal trajectory and the guidance trajectory have the same state and control; (ii) at the maximin time t = σ , the angle of attack for both the optimal trajectory and the guidance trajectory has the same maximum value α = 17.2 deg; (iii) as expected, the timewise minimum distance of the optimal trajectory (d = 454 ft) is slightly larger than the timewise minimum distance of the guidance trajectory (d = 452 ft); (iv) at the final time t = θ , quasisteady


249

Fig. 3 Host aircraft. Time history of altitude: optimal trajectory (solid line) and guidance trajectory (dashed line)

Fig. 4 Host aircraft. Time history of velocity: optimal trajectory (solid line) and guidance trajectory (dashed line)

state is achieved exactly by the optimal trajectory and approximately by the guidance trajectory. Figures 1–7 further show how the guidance scheme achieves the goals of the optimal trajectory, namely, (i) maximizing wrt the controls the timewise minimum distance in the avoidance phase and (ii) approaching the quasisteady state at the final time of the recovery phase. With reference to the (x, h) domain, Fig. 1 shows the geometry of the optimal trajectory and the guidance trajectory. In this figure, the letters A, B, C denote the aircraft positions at corresponding points. In particular, A denotes the initial point, B denotes the maximin point, and C denotes the final point. Figure 1 shows how the potential collision is avoided by raising the altitude of the host aircraft, with the angle of attack being switched to its maximum value and being held there until the minimum distance point is past. For the host aircraft, comparison of the optimal trajectory and the guidance trajectory shows that, in the avoidance phase (left of point B), the trajectory profiles are almost identical due to the similarity of the control profiles; in the recovery phase (right of point B), the optimal trajectory

250


Fig. 5 Host aircraft. Time history of path inclination: optimal trajectory (solid line) and guidance trajectory (dashed line)

Fig. 6 Host aircraft. Time history of angle of attack: optimal trajectory (solid line) and guidance trajectory (dashed line)

profile and the guidance trajectory profile have some differences due to dissimilar control distribution. With reference to the host aircraft, Fig. 2–7 show the time histories of the state variables for both the optimal trajectory (solid line) and the guidance trajectory (dashed line), specifically: horizontal distance (Fig. 2), altitude (Fig. 3), velocity (Fig. 4), path inclination (Fig. 5), angle of attack (Fig. 6), and distance (Fig. 7). Clearly, the angle of attack profiles of the optimal trajectory and the guidance trajectory are the same for the avoidance phase. While the angle of attack profiles are different for the recovery phase, nevertheless the guidance trajectory achieves the same quasisteady flight condition of the optimal trajectory at the final point C.

7 Conclusions This paper covers both the optimal trajectory and the associated guidance trajectory for a host aircraft in the abort landing mode under emergency conditions. The intruder


251

Fig. 7 Host aircraft. Time history of distance between host aircraft and intruder aircraft: optimal trajectory (solid line) and guidance trajectory (dashed line)

aircraft is assumed to be uncooperative so that the collision avoidance maneuver is performed only by the host aircraft. Major conclusions are as follows. (i) The best strategy for collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between the two aircraft. This results in a maximin or Chebyshev problem of optimal control. Key to this paper is the transformation of the one-subarc Chebyshev problem into a twosubarc Bolza-Pontryagin problem with an inner boundary condition, the orthogonality between the relative position vector and the relative velocity vector. This problem is then solved via the multiple-subarc sequential gradient-restoration algorithm. (ii) The optimal trajectory can be partitioned into two phases, avoidance and recovery, separated by the maximin point. In the avoidance phase, maximum angle of attack is to be applied until the timewise minimum distance point is past, providing the recovery time is not too strict. In the recovery phase, the host aircraft completes the transition of the angle of attack from the maximin value to the value corresponding to quasisteady ascending flight. (iii) A guidance trajectory mirroring the properties of the optimal trajectory is developed. In the avoidance phase, maximum angle of attack is used so as to maximize the timewise minimum distance between the two aircraft. In the recovery phase, the guidance scheme employs a time-explicit cubic control law so as to achieve the goal of recovering at the final time the same quasisteady ascending flight state as that of the optimal trajectory, while respecting the angle of attack constraint αmin ≤ α ≤ αmax . (iv) Even though the collision avoidance scenario for abort landing is quite different from its counterpart for takeoff, the basic concepts concerning the needed control action are very much the same (Refs. [22, 23]).

252


Appendix A: Quasisteady Flight If we neglect the acceleration terms in the dynamic equations (1c)–(1d), we obtain in light of (2) and for β = 1, the following quasisteady flight equations on the tangent and normal to flight path: [Tmax (h, V )/m] cos(α + δ) − D(h, V , α)/m − g sin γ = 0,

(36a)

[Tmax (h, V )/mV ] sin(α + δ) + L(h, V , α)/mV − (g/V ) cos γ = 0. (36b) For any given altitude h, (36) involve the following variables: velocity V , path inclination γ , angle of attack α, and thrust setting β. As a consequence, there exist a double infinity of solutions to (36). To obtain a unique solution, we need to specify two quantities in the quadruplet (V , γ , α, β). A.1 Initial Conditions At the initial time t = 0, the host aircraft is in quasisteady descending flight with velocity V0 = 240 ft/s and path inclination γ0 = −3 deg. With these quantities frozen, (36) can be solved for the angle of attack α and the thrust setting β, yielding the following values [see (34)]: α0 = 7.41 deg,

β0 = 0.286 deg.

(37)

A.2 Final Conditions At the final time t = θ , the host aircraft is in quasisteady ascending flight with velocity Vθ = 240 ft/s and maximum thrust setting βθ = 1. With these quantities frozen, (36) can be solved for the path inclination γ and the angle of attack α, yielding the following values [see (35)]: γθ = 7.96 deg,

αθ = 6.88 deg.

(38)

Appendix B: Differential Game Approach The multiple-subarc sequential gradient-restoration algorithm can also be used to solve the collision avoidance problem as a differential game problem (Ref. [21]) if one assumes that the host aircraft and the intruder aircraft maneuver cooperatively. For the intruder aircraft, suppose that the equality h∗ = 900 ft

(39a)

h∗ ≥ 900 ft.

(39b)

is replaced by the inequality

Under this condition, use of the SGRA shows that the maneuvers of the host aircraft and the intruder aircraft remain precisely the same as discussed in Sect. 7.


253

Next suppose that, for the intruder aircraft, the inequality (39b) is replaced by h∗ ≥ 600 ft.

(39c)

Under this condition, use of SGRA shows that the maneuver of the host aircraft remains the same, while the intruder aircraft descends from 900 ft to 600 ft. As result, the maximin distance of the optimal trajectory changes from d = 454 ft to at least d∼ = 754 ft; hence, it increases by at least 300 ft, which is the difference between the thresholds (39b) and (39c).

References 1. Miele, A., Wang, T., Mathwig, J.A.: Optimal collision avoidance trajectory for an aircraft in abort landing. Rice University, Aero-Astronautics Report No. 353 (2005) 2. Miele, A., Wang, T., Mathwig, J.A.: Guidance scheme approximating the optimal collision avoidance trajectory for an aircraft in abort landing. Rice University, Aero-Astronautics Report No. 354 (2005) 3. Miele, A., Ciarcià, M.: Time-explicit cubic guidance scheme for an aircraft in abort landing. Rice University, Aero-Astronautics Report No. 366 (2009) 4. Raghunathan, A.U., Gopal, V., Subramanian, D., Biegler, L.T., Samad, T.: Dynamic optimization strategies for three-dimensional conflict resolution of multiple aircraft. J. Guid. Control Dyn. 27(4), 586–594 (2004) 5. Hu, J., Prandini, M., Sastry, S.: Optimal coordinated maneuvers for three-dimensional aircraft conflict resolution. J. Guid. Control Dyn. 25(5), 888–900 (2002) 6. Frazzoli, E., Mao, Z.H., Oh, J.H., Feron, E.: Resolution of conflicts involving many aircraft via semidefinite programming. J. Guid. Control Dyn. 24(1), 79–86 (2001) 7. Menon, P.K., Sweriduk, G.D., Sridhar, B.: Optimal strategies for free-flight air traffic conflict resolution. J. Guid. Control Dyn. 22(2), 202–211 (1999) 8. Clements, J.C.: The optimal control of collision-avoidance trajectories in air-traffic management. Transp. Res. B 33(4), 265–280 (1999) 9. Kuo, V.H., Zhao, Y.J.: Required ranges for conflict resolutions in air traffic management. J. Guid. Control Dyn. 24(2), 237–245 (2001) 10. Tornapolskaya, T., Fulton, N.: Optimal cooperative collision avoidance strategies for coplanar encounter: Merz solution revisited. J. Optim. Theory Appl. 140(2), 355–375 (2009) 11. Tornapolskaya, T., Fulton, N.: Synthesis of optimal control for cooperative collision avoidance for aircraft (ships) with unequal turn capabilities. J. Optim. Theory Appl. 144(2) (2010) 12. Miele, A., Wang, T.: Multiple-subarc sequential gradient-restoration algorithm, Part 1: Algorithm structure. J. Optim. Theory Appl. 116(1), 1–17 (2003) 13. Miele, A., Wang, T.: Multiple-subarc sequential gradient-restoration algorithm, Part 2: Application to a multistage launch vehicle design. J. Optim. Theory Appl. 116(1), 19–39 (2003) 14. Miele, A., Wang, T.: Maximin approach to the ship collision avoidance problem via multiple-subarc sequential gradient-restoration algorithm. J. Optim. Theory Appl. 124(1), 29–53 (2005) 15. Miele, A., Wang, T.: Fundamental properties of optimal orbital transfer. Paper IAC-03-A.7.02, 54th International Astronautical Congress, Bremen, Germany (2003) 16. Miele, A., Wang, T., Williams, P.N.: On feasibility of launch vehicle designs. Appl. Math. Comput. 164(2), 295–312 (2005) 17. Miele, A., Wang, T., Melvin, W.W.: Optimization and acceleration guidance of flight trajectories in a windshear. J. Guid. Control Dyn. 10(4), 368–377 (1987) 18. Miele, A., Wang, T., Melvin, W.W., Bowles, R.L.: Gamma guidance schemes for flight in a windshear. J. Guid. Control Dyn. 11(4), 320–327 (1988) 19. Miele, A., Wang, T., Melvin, W.W.: Penetration landing guidance trajectories in the presence of windshear. J. Guid. Control Dyn. 12(6), 806–814 (1989) 20. Miele, A., Wang, T., Melvin, W.W., Bowles, R.L.: Acceleration, gamma, and theta guidance for abort landing in a windshear. J. Guid. Control Dyn. 12(6), 815–821 (1989)

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21. Pontani, M.: Differential games treated by a gradient-restoration approach. In: Buttazzo, G., Frediani, A. (eds.) Variational Analysis and Aerospace Engineering, pp. 379–396. Springer, New York (2009). Chap. 21 22. Miele, A., Wang, T., Ciarcià, M.: Optimal collision avoidance trajectory for an aircraft in takeoff. Aero-Astronautics Report 351, Rice University (2005) 23. Miele, A., Wang, T., Ciarcià, M.: Guidance scheme approximating the optimal collision avoidance trajectory for an aircraft in takeoff. Aero-Astronautics Report 352, Rice University (2005)