Collision Avoidance Trajectory for an Ekranoplan

M. Ciarcià1 and C. Grillo, 2Università degli Studi di Palermo, Palermo, Italy, 90128

The risk of collision is one of the crucial aspects for the applications of Ekranoplans in civil transportation. In fact, the extremely low flight altitude of these aircraft increases dramatically the chances of interference between their flight path and the multitude of obstacles populating the surrounding area. In this work we consider the optimal collision avoidance problem between a cruising Ekranoplan and a steady obstacle located on its flight path. First we solve the optimal control problem imposing that the collision avoidance maneuver lies on the longitudinal plane identified by the initial cruising conditions. In the second part of this work we state the three-dimensional version of the same collision avoidance problem.

I. Introduction he risk of collision is one of the crucial aspects for the applications of Ekranoplans in civil transportation. In fact, the extremely low flight altitude of these aircrafts increases dramatically the chances of interference

between their flight path and the multitude of obstacles populating the surrounding area. Examples of those potential obstacles are: ships, small boats, and stumbling blocks.

T In the first part of this work we investigate the bi-dimensional optimal collision avoidance problem between a

cruising Ekranoplan and a steady obstacle located on the ground imposing that the collision avoidance maneuver lies on the longitudinal plane. The following assumptions are employed: (i) initially the Ekranoplan is moving in a quasisteady levelled cruise trajectory; (ii) after the avoidance manoeuvre a recovery manoeuvre is executed by the aircraft to return to the initial cruise path; (iii) both the avoidance manoeuvre and the recovery manoeuvre lie on the same vertical plane identified by the initial cruise trajectory. The investigation of the collision avoidance performances on the longitudinal flight is encouraged by the relatively large quasi-flat turn radius of the Ekranoplans. The second part of this work is dedicated to state the equivalent three-dimensional case of the collision avoidance problem. In detail we collect the complete set of differential constraints that describe the lateral motion of the aircraft considering the ground effect.

The approach to the problem considered, is to maximize wrt the controls the timewise minimum distance between the aircraft and the obstacle. This yields to a maximin problem or Chebyshev problem of optimal control, which is not solvable in a direct way. Hence, a technique is performed to transform the Chebyshev problem into a Bolza problem.1 Once reduced in this form the optimization problem can be solved numerically applying the multiple-subarc sequential gradient-restoration algorithm.2,3

The main target of this research is to determine the relationship between the optimal avoidance maneuver and the control to execute it. In turn, this relationship is basilar to the development of a guidance scheme capable of approximating the optimal trajectory in real time.

It is worth to notice that the peculiar aerodynamic characteristics of the Ekranoplans joined to their relatively weak manoeuvrability make this application of optimal control techniques particularly challenging. We believe that the results of the present approach would lead the future investigations toward a suitable collision avoidance strategy. Key Words: Collision avoidance, ekranoplan, optimal control, Chebyshev problems, Bolza problems, multiple-subarc sequential gradient-restoration algorithm.

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1 Postdoctoral Research Fellow, Dipartimento di Ingeneria dei Trasporti, Università degli Studi di Palermo, Viale delle Scienze Ed. 8, Palermo, Italy,90128 , AIAA Member. 2 Associate Professor, Dipartimento di Ingeneria dei Trasporti, Università degli Studi di Palermo, Viale delle Scienze Ed. 8, Palermo, Italy,90128 , AIAA Member.

AIAA Atmospheric Flight Mechanics Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-5931

Copyright © 2009 by Marco Ciarcià. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

II. Algorithm In this research, the multiple-subarc sequential gradient-restoration algorithm (SGRA) is applied to solve the

collision avoidance problem involving a host aircraft and an obstacle. The multiple-subarc SGRA is an important extension of the single-subarc SGRA. The single-subarc SGRA was

developed by Miele et al during the period 1968 to 1986. It has proven to be a powerful tool for solving optimal trajectory problems of atmospheric and space flight. Applications and extensions of this algorithm have been reported in the US, Japan, Germany, Spain, and other countries around the world; in particular, a version of this algorithm is used at NASA-JSC under the code name SEGRAM, developed by McDonnell Douglas Technical Service Company.4

While the single-subarc SGRA deals with the optimization of a single system with initial and final boundary conditions, the multiple-subarc SGRA deals with the optimization of multiple systems with initial, final, and inner boundary conditions. In the multiple-subarc SGRA,2,3 a large and complicated overall system is decomposed into several subsystems along the time domain: each subsystem, having relatively simple properties, corresponds to a subarc; the connection between consecutive subsystems takes place via inner boundary conditions.

In the application of the multiple-subarc SGRA, the actual time is denoted by t, the total number of subarcs is denoted by s, and the actual time length at each subarc is denoted by θi , i = 1, 2,…, s. Then, the end times of all the subarcs are given by the recurrence relation

ti = ti-1 + θi , i = 1, 2,…, s, (1a)

in which we set

t0 = 0. (1b)

We denote by τ the dimensionless virtual time, which is such that the time length of each subarc is normalized to 1. For a specified subarc i, the direct transformation from actual time to virtual time can be written as

τ = (t - ti-1) / θi , i = 1, 2,…, s ; (2a)

the corresponding inverse transformation from virtual time to actual time can be written as

t = ti-1 + θi τ , i = 1, 2,…, s . (2b)

Then, with reference to the virtual time domain, the Bolza-Pontryagin optimization problem is formulated as follows: minimize the functional

I = + g(y, π), (3a) 1

10( , ,π τ, ) d τ

s

if x u i

=,∑∫

with respect to the n-dimensional state vectors x(τ, i), the m-dimensional control vectors u(τ,i), and the p-dimensional parameter vector π which satisfy the n-dimensional differential constraints

x = φ(x,u,π,τ,i), 0 ≤ τ ≤ 1, i = 1, 2,…, s , (3b)

plus a general c-dimensional boundary condition

ψ( ,π) 0y = , c ≤ 2ns, (3c)

incorporating the initial conditions, final conditions, and inner boundary conditions. In the above relations, s is total number of subarcs, i is the index identifying a particular subarc, and the 2ns-vector y is given by

y = [xT(0,i), xT(1,i),…, i = 1,2,…,s]T. (3d)

Clearly, the vector y includes the initial and final state vectors of all the subarcs. The time scaling factors θi , characterizing the mapping from the virtual time to the actual time, can be either fixed or free, depending on the optimization problem to be solved. In the latter case, the time scaling factors θi can be treated as elements of the unknown parameter vector π.

We note that, for stiff systems, it might be convenient to convert a single-subarc problem into a multiple-subarc problem with continuous inner boundary conditions. Indeed, proper increase of the number of subarcs can enhance considerably the robustness of the solution.5

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III. 2D System Description The motion of the aircraft is described considering the following assumptions: (i) the flight path lies in a vertical

plane; (ii) the aircraft is a particle of constant mass; (iii) the Earth-fixed coordinate system is inertial. Under these conditions, the equations of motion can be written as

x′ = V cosγ, (4a)

h′ = V sin γ, (4b)

V′ = (T/m)cos(α + αthrust) – D/m – g sinγ, (4c)

γ′ = (T/mV)sin(α + αthrust) + L/mV – (g/V) cosγ. (4d)

In Eqs. (4) appear the following quantities: the longitudinal distance x, altitude h, velocity V, path inclination γ, mass m, acceleration of gravity g, thrust inclination angle αthrust, and angle of attack α. Also, the prime ′ denotes derivative with respect to the actual time t. The thrust T, the drag D, the lift L, and the weight W are forces acting on the aircraft. These forces can be represented by the functional relations

T = T(h, V, δthrottle) ≅ δthrottle Tmax(h, V), (5a)

D = D(h, V, α), (5b)

L = L(h, V, α), (5c)

W = mg, (5d)

where δthrottle is the thrust setting. A. Inequality Constraints

It assumed that the collision avoidance maneuver is performed with full thrust, which means we can impose

T = Tmax(h, V) (6)

for the entire duration of the maneuver. As regard the angle of attack α(t), it range between a minimum value and a maximum value

αmin ≤ α ≤ αmax , (7a)

analogously its time rate α′(t), is subject to the similar constraint

−α′max ≤ α′ ≤ α′max , (7b)

where αmax, αmin, α′max are prescribed positive 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 , (8a)

η′ = [2α′max/(αmax − αmin)] sin ξ, (8b)

in which η(t) denotes an auxiliary state variable and ξ(t) denotes an auxiliary control variable. Eqs. (8) imply

α′= α′max cos η sin ξ, (9)

which implies that any pair of functions η(t), ξ(t) consistent with Eqs. (8) satisfies automatically the inequalities (7). The transformation (8) has two advantages: (i) it is nonsingular; (ii) the angle of attack boundary is reached

tangentially, regardless of the value of the auxiliary control. It is important to observe that these advantages are obtained at a price. In fact 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 boundary or lower boundary.

Eqs. (8), allow us to rewrite the system of equations for the ekranoplan as

x′ = V cosγ, (10a)

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h′ = V sin γ, (10b)

V′ = (Tmax/m)cos(α + αthrust) – D/m – g sinγ, (10c)

γ′ = (Tmax/mV)sin(α + αthrust) + L/mV – (g/V) cosγ, (10d)

η′ = [2α′max/(αmax − αmin)] sin ξ, (10e)

where

α = (1/2) (αmax + αmin) + (1/2) (αmax − αmin) sin η . (10f)

In the transformed system (10), the state variables are x(t), h(t), V(t), γ(t), η(t); the new control variable is ξ(t). Once η(t) is determined, the original control α(t) can be recovered using Eq. (10f). B. Potential Collision

If the aircraft moves along a leveled trajectory with constant speed, its motion is described by the relation

x(t) = x0 + V0 t , (11a)

h(t) = h0 , (11b)

where the subscript 0 denotes the quantities at the initial time t = 0. We assume that the obstacle is steady and located at the same initial altitude of the aircraft and at an initial distance d0 from it

x*(t) = x0 + d0 , (12a)

h*(t) = h0 . (12b)

Given the initial velocity of the aircraft V0, and the initial distance of the obstacle d0, the collision occurs if the following relations are satisfied at some forward time tc > 0:

x(tc) = x*(tc) , (13a)

h(tc) = h*(tc) , (13b)

with the implication that

V0 t = d0 . (14)

IV. 2D Optimal Control Problem Statement For collision avoidance under emergency conditions, the best strategy is to maximize wrt the controls the

timewise minimum distance between the aircraft and the obstacle. At the maximin point of the encounter, the distance between the ekranoplan and the obstacle 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 one-subarc Chebyshev problem can be transformed into a two-subarc Bolza problem solvable via the multiple-subarc sequential gradient-restoration algorithm (SGRA).6,7

With the purpose to determine the inner boundary condition mention above, let the distance between the host aircraft and the obstacle be written as

d = √ [(x – x*)2 + (h – h*)2] (15a)

and its time derivative is

d′ = (1/2) [(x – x*) (x – x*)′ + (h – h*) (h – h*)′]. (15b)

Let also

D = (1/2) d 2 = (1/2) [(x – x*)2 + (h – h*)2] (16a)

denote the squared distance function whose time derivative is

D′ = (x – x*) (x – x*)′ + (h – h*) (h – h*)′. (16b)

From (12b) and (13b) result that the conditions

d′ = 0 and D ′ = 0 (17a)

are reached simultaneously when

(x – x*) (x – x*)′ + (h – h*) (h – h*)′ = 0, (17b)

that is, when (see the equations of motion)

(x – x*) V cos γ + (h – h*) V sin γ = 0, (17c)

namely, when the relative position vector is orthogonal to the relative velocity vector. In the actual time domain, let θ1 denote the time length of the avoidance subarc; let θ2 = θ – θ1 denote the time

length of the recovery subarc; let θ = θ1 + θ2 denote the assigned time length of the entire maneuver. The two subarcs are connected to one another at the maximin point where the inner boundary condition (17c) is satisfied.

In order to bring the equations of the relative motion into the format required by the multiple-subar SGRA, it is necessary to normalize the interval of integration to unity. Having established that the time length of the first subarc is θ1 and the time length of the second subarc is θ2 , we introduce now a transformation from the actual time t to the virtual time τ rendering the virtual time length of each subarc equal to 1. More precisely, the time transformation is

subarc i = 1, τ = t / θ1, 0 ≤ τ ≤ 1, 0 ≤ t ≤ θ1, (18a)

subarc i = 2, τ = 1+(t – θ1)/ θ2, 1 ≤ τ ≤ 2, θ1 ≤ t ≤ θ. (18b)

A. Performance Index For the final time θ given, the optimization problem consists in maximizing wrt the new control ξ(t) and

parameters θ1 , the distance function at the point where the inner boundary condition is satisfied.

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1DIn the actual time domain, the problem is

111

( ), ( ), max ( ) min - ( )

w tw tD

θθθ → θ

D

D

. (19a)

Let F(t) be the one-index representation, in the actual time domain, of a generic function of the time F, and let F(τ, i) be its two-index representation in the virtual time domain, with i = 1 for the first subarc and i = 2 for the second subarc. In the virtual time domain, the problem becomes

(19b) 11 ( , ), ( , ),

max (1,1) min - (1,1) ,w iw i

Dτ θτ θ

→

with D(1, 1) meaning D(τ, i) evaluated at the end (τ = 1) of the first subarc (i = 1). With the introduction of this new two-index representation the one-subarc Chebyshev problem has been replaced with a two-subarc Bolza problem. Alternatively, in light of the continuity of the distance function, (19b) can be rewritten as

(19c) ( , ), ( , ), 11

max (0, 2) min - (0, 2) ,w iw i

Dτ θτ θ

→

with D(0, 2) meaning D(τ, i) evaluated at the start (τ = 0) of the second subarc (i = 2). B. Penalized Performance Index

In order to prevent the undershooting of the initial altitude h0 the performance index (19b) is replaced with the following penalized performance index:

min [−D(1, 1) + kP], (20a)

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

. (20b) 1 1

2 21 2

0 0 = (τ, 1)dτ (τ, 2)dτP E Eθ + θ∫ ∫

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

E(τ, i) = h0 - h(τ, i) , if h ≤ h0 , (20c)

E(τ, i) = 0 , if h ≥ h0 . (20d)

C. Differential Constraints Due to the normalization of the interval of integration, the equation of motion, for the first subarc, can be

rewritten as x = θ1 [V cosγ], (21a)

= θh 1 [V sin γ], (21b)

= θV 1 [ (Tmax/m)cos(α + αthrust) – D/m – g sinγ], (21c)

= θγ 1 [ (Tmax/mV)sin(α+αthrust) + L/mV – (g/V) cosγ], (21d)

= θη 1 [2α′max /(αmax – αmin) sinξ], (21e)

where the dot superscript denote a derivative wrt the virtual time τ. (21f) For the second subarc, the differential constraints can be rewritten as x = θ2 [V cosγ], (22a)

= θh 2 [V sin γ], (22b)

= θV 2 [ (Tmax/m)cos(α + δ) – D/m – g sinγ], (22c)

= θγ 2 [ (Tmax/mV)sin(α+ δ) + L/mV – (g/V) cosγ], (22d)

= θη 2 [2α′max /(αmax – αmin) sinξ]. (22e)

D. Boundary Conditions The initial conditions are

x(0,1) = x0 = 0, (23a)

h(0,1) = h0, (23b)

V(0,1) = V0, (23c)

γ(0,1) = γ0 = 0, (23d)

η(0,1) = η0 = sin-1{[2α0 – (αmax + αmin)] / (αmax – αmin)}. (23e)

The continuity conditions at the interface of the first and the second subarcs are

x(1,1) = x(0,2), (24a)

h(1,1) = h(0,2), (24b)

V(1,1) = V(0,2), (24c)

γ(1,1) = γ(0,2), (24d)

η(1,1) = η(0,2). (24e)

Also, at the interface, the inner boundary condition is

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(1,1) = 0. (25) D

The final conditions are

h(1,2) = hθ , (26a)

γ(1,2) = γθ = 0. (26b)

E. Bolza Problem In conclusion, the Bolza problem for the aircraft collision avoidance maneuver can be formulated as that of

minimizing the performance index (20), subject to the differential constraints (22)-(23), the initial conditions (23), the continuity conditions (24), the inner boundary condition (25), and the final conditions (26).

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), ξ(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), ξ(t, 2), plus the time parameter θ1. With these functions known, the angle of attack α(t, 1) for the first subarc can be recovered via (21f) and the angle of attack α (t, 2) for the second subarc can be recovered via (22f).

V. Data for Examples The ekranoplan considered is an Alekseev A-90 Orlyonok powered, in cruise conditions, by a Kuznetsov NK-

12MK turboprop engine with contra-rotating propeller. It is assumed the aircraft is flying along a leveled trajectory. The cruising weight is W = 125000 kg and its wing surface is S = 306.4 m2 . 8-10

The aerodynamic forces involved in the longitudinal dynamic are lift and drag, which are controlled via the angle of attack α. The angle of attack is bounded between the values αmax = 10 deg and αmin = -10 deg. To model the rotational inertia of the aircraft, the time rate of the angle attack is limited imposing α′max = 3 deg/s. The lift and drag can be expressed by the relations

L = (1/2)ρV 2 S CL, (27a)

D = (1/2)ρV 2 S CD, (27b)

with ρ the air density, S the wing reference surface, V the velocity, CL the lift coefficient, and CD the drag coefficient, were

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0 L L L C C Cα= α + , (28a)

, (28b) 2D D0C C K C= + L

were

D0 0.021C = . (28e)

From previous work of our research group dedicated on modeling the aerodynamic force in ground effect conditions,8 results that in ground effect flight, the aerodynamic other aerodynamic coefficients CLα , CL0 , and K are strongly dependent on the distance h between the ekranoplan and the water surface

, (28c) ( ) ( )1.26 1.11L 4.463 0.1938 / 0.0279 /C h c −α

⎡= + +⎣ h c − ⎤⎦

+

c

, (28d) ( ) 0.7749L0 0.02636 / 0.399C h c −=

, (28f) ( ) 0.6060.0951 0.0275 /K h −= −

where c is the average chord length of the wing. The thrust expression considered for the turboprop engine is

maxmax Pη , 50 m/s 148 m/sT

VΠ

= ≤ V ≤ , (29)

where V [m/s] is the velocity, ηP = 0.85 is the propeller efficiency, and Πmax = 15000 [hp] is the maximum engine power. It is imposed that the avoidance maneuver and recovery maneuver are both operated at maximum thrust setting, δthrottle = 1. The inclination of the thrust with respect to the aircraft reference line is assumed to be αthrust = 0 deg. Note that because of the small variation of the flight altitude, the density ρ is assumed constant.

At the beginning of the collision avoidance maneuver, the ekranoplan is in quasisteady leveled flight. Specifically, the initial conditions are

x0 = 0, (30a)

h0 = 2.5 m, (30b)

V0 = 105 m/s, (30c)

γ0 = 0, (30d)

η0 = sin-1{[2α0 – (αmax + αmin)] / (αmax – αmin)}, (30e)

with

α0 = 1.17 deg. (30f)

After a transient period, at the end of the collision avoidance maneuver, the host aircraft recovers the quasisteady leveled flight. Specifically, the final conditions are

xθ = free, (31a)

hθ = 2.5 m, (31b)

Vθ = free, (31c)

γθ = 0, (31d)

ηθ = free. (31e)

The obstacle initial distance d0 is assumed ranging from 250 m to 1250 m. Namely we consider the following obstacle coordinates

250 m ≤ x* ≤ 1250 m, (32a)

h* = 2.5 m. (32b)

VI. Numerica Results of the 2D Avoidance Maneuver The numerical computation of the optimal trajectories for collision avoidance was done using the multiple-

subarc sequential gradient-restoration algorithm (SGRA). Since the thrust setting control is set at the maximum value δthrottle = 1, the only remaining control is the angle of attack α(t) or, equivalently, the auxiliary control ξ. The task of the multiple-subarc SGRA is to maximize wrt the control α(t) and the time parameter θ1, the timewise minimum distance between the ekranoplan and the obstacle: the vehicle is flying along a cruise leveled trajectory and the obstacle is steady on the sea surface. The numerical results are shown in Table 1 and Figures 1-3.

Table 1 shows the effect of the initial distance d0 on the maximin distance d(θ1). As the initial distance varies from 250 m to 1250 m, the maximin distance increases from 8.5 m to 370.7 m. In other words, an earlier detection of the obstacle will provide a larger safety margin for the ekranoplan.

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In Figs. 1-4 are represented the results of the collision avoidance for an initial distance d0 = 1250 m. For this case the optimal trajectory is shown in Fig. 1. It appears that the collision avoidance maneuver is symmetrical, bell shaped, having its top over the obstacle. As consequence, the maximin point occurs in correspondence of the maximum height reached by the ekranoplan.

In Fig. 2 is shown the time history of the angle of attack. It increases rapidly at the beginning of the maneuver to produce the acceleration upward and, after a reduction in correspondence of the maximin point, it increases again before the end of the maneuver to null the downward velocity.

Figure 3 represent the time history of the velocity. As we can see, the maximum reduction of speed doesn’t exceed 5% of the

d0 [m] d(θ1) [m] θ1 [s] θ [s] 250 8.5 2.39 10 500 56.1 4.93 20 750 156.3 7.94 25 1000 297.0 11.91 30 1250 370.7 16.54 30

Table 1. 2D maneuver. Effect of the initial distance on the maximin distance.

Figure 1. 2D maneuver, d0 = 1250 m. Optimal Trajectory.

Figure 3. 2D maneuver, d0 = 1250 m. Time history of the velocity.

Figure 2. 2D maneuver, d0 = 1250 m. Time history of the angle of attack.

Figure 4. 2D maneuver, d0 = 1250 m. Time history of the lift coefficient.

cruise velocity. This small value is the direct consequence that, after the jump, the cruise condition must be recovered in a short amount of time.

Figure 4 show the time histories of the lift coefficient. While the initial growth of the lift coefficient is a direct consequence of the increment of the angle of attack, its consequent reduction is an effect of the speed drop and,

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more important, it is the result the dramatic reduction of the ground effect. The rising of the lift coefficient in the final part of the maneuver, in spite of a constant angle of attack and a slight drop of the speed, is a consequence of the rising ground effect in proximity of the sea surface.

VII. 3D Collision Avoidance Problem Statement This section describe the correspondent 3D extension of the 2D problem so far considered. In this case the

dynamic equations of the aircraft are described in body axes (principal axes). Under the same hypotheses considered previously: (i) the aircraft is a rigid body; (ii) the Earth is flat; (iii) the Earth-fixed coordinate system is inertial; the equations of motion, become

u′ = – qw + rv – g sinθ + Tcos(αT)/m + X/m, (33a)

v′ = – ru + pw + g cosθ sinφ + Y/m, (33b)

w′ = qu – pv + g cosθ cosφ + Tsin(αT)/m + Z/m, (33c)

p′ = qr (Iy – Iz)/Ix + Lroll/Ix, (33d)

q′ = rp (Iz – Ix)/Iy + M/Iy, (33e)

r′ = pq (Ix – Iy)/Iz + N/Iz, (33f)

ϕ′ = p + q sinϕ tanθ + r cosϕ tanθ, (33g)

θ′ = q cosφ – r sinφ, (33h)

ψ′ = (q sinφ – r cosφ) secθ, (33i)

x′ = u cosθ cosψ + v (-cosφ sinψ + sinφ sinθ cosψ) + w (-sinφ sinψ + cosφ sinθ cosψ), (33l)

y′ = u cosθ sinψ + v (cosφ cosψ + sinφ sinθ sinψ) + w (-sinφ cosψ + cosφ sinθ sinψ), (33m)

h′ = u sinθ – v sinφ cosθ – w cosφ cosθ. (33n)

In Eqs. (33) appear the following quantities: the navigation coordinates x, y, and h; the velocity components along the body axes u, v, and w; the angular velocity components along the body axes p, q, and r; the orientation angles φ, θ, and ψ. Also, the prime ′ denotes derivative with respect to the actual time t. The constants Ix, Iy, and Iz, are the inertia moments of the vehicle, and m is its mass. T is the thrust magnitude already defined in chapter V. The components X, Y, and Z are the projections along the body axes of the aerodynamic force F, the components Lroll, M, and N are the projections along the body axes of the aerodynamic torque C. These components are given by the following relations

X = (1/2)ρV 2 S CX, (34a)

Y = (1/2)ρV 2 S CY, (34b)

Z = (1/2)ρV 2 S CZ, (34c)

Lroll = (1/2)ρV 2 Sb CLroll, (34d)

M = (1/2)ρV 2 Sc CM, (34e)

N = (1/2)ρV 2 Sb CN, (34f)

where b is the wing span. The aerodynamic coefficients involved in the lateral dynamic CX, CY, CZ, CLroll, CM, CN have the following expression

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LC X Dcosαcosβ sinαC C= + , (35a)

, (35b) rY Yβ Yp Yr Yδ rβC C C p C r C= + + + δ

LC Z Dsin αcosβ cosαC C= + , (35c)

( )roll left right r aL Lβ Lwing Lwing Lp Lr Lδ r Lδ a1β δ6

C C C C C p C r C C= + − + + + + δ , (35d)

eM M0 Mα Mq Mα Mδ eα q α

2 2c cC C C C C CV V

= + + + + δ , (35e)

rright left

2 2N Nβ right left Np Nr Nδ r Nδ aLwing Lwing

1β δ6 2 2

b bC C C K C K C p C r C CV V

⎛ ⎞= + − + + + +⎜ ⎟⎝ ⎠ a

δ . (35f)

In Eqs. (35d), (35f) appear the terms which are the average lift coefficients

respectively of the left wing and the right wing assuming a trapezoidal distribution of the lift between the tip and the symmetry plane of the aircraft. Note that this difference is caused by the effects, if in proximity of the sea surface, of the ground effect. In detail, their expressions are:

left rightLwing Lwing, and C C

left L G L GLwing

1 ( ) ( sin2 2

bC C h h C h h⎛= = + = +⎜⎝ ⎠

)⎞φ ⎟ , (36a)

right L G L GLwing

1 ( ) ( sin2 2

bC C h h C h h⎛= = + = −⎜⎝ ⎠

)⎞φ ⎟ , (36b)

The induced drag coefficients Kdown and Kup are determined with (28f) in correspondence of the two wing tips

0.606

left 0.0951 0.0275 sin /2GbK h

−⎡ ⎤⎛ ⎞= − + φ⎜ ⎟⎢⎝ ⎠⎣ ⎦

c⎥ , (37a)

0.606

right 0.0951 0.0275 sin /2GbK h

−⎡ ⎤⎛ ⎞= − − φ⎜ ⎟⎢⎝ ⎠⎣ ⎦

c⎥ . (37b)

The velocity magnitude V, the angle of attack α, and the sideslip angle β, can be expressed as function of the velocities components

2 2 2V u v w= + + , (38a)

warctgu

α = , (38b)

2 2 2

arcsin arcsinv vV u v w

β = =+ +

. (38c)

The stability derivates are Yβ Yp, Yr, Lβ, Lp, Lr, Mα, Mq, Mα, Nβ, Np, Nr,, C C C C C C C C C C C C 8

, (39a) Yβ 1.076C = −

, (39b) Yp 0.473C = −

Yr 1.280C = , (39c)

, (39d) Lβ 0.330C = −

1.313

Yp 1.331 0.2765 hCc

−⎡ ⎤⎛ ⎞= ⎢− − ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

, (39e)

0.7061 1.313

Lr 0.4249 0.0079 0.002373hCc c

− −⎡ ⎤⎛ ⎞ ⎛ ⎞= ⎢ + + ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

h , (39f)

1.26 1.1

Mα 0.6695 0.02907 0.0004185hCc c

h− −⎛ ⎞ ⎛ ⎞= − − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, (39g)

, (39h) Mq 12.2778C = −

1.1

Mα 2.8 0.258 hCc

−⎛ ⎞= − ⎜ ⎟⎝ ⎠

, (39i)

, (39l) Nβ 0.5210C =

0.7061 1.3135

Np

0.5988 1.313

0.27192 0.68 10 0.5668 0.02949 0.00878

0.0951 0.02791 48607.2 2035.8

h hCc c

h hc c

− −−

− −

⎡ ⎤⎛ ⎞ ⎛ ⎞= + ⋅ ⎢ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎥ ∗⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ − ⎥ ⎢ + ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

, (39m)

20.7061 1.313

Nr 1.0281 0.01474 0.004394h hCc c

− −⎡ ⎤⎛ ⎞ ⎛ ⎞= ⎢− − − ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

, (39n)

the coefficient CM0 and the control derivative CMδe have values 9

M0 0.111C = − , (40a)

. (40b) eMδ 1.95C = −

The remaining control derivates depend strongly on the ground effect and, because of the complexity of their analytical expressions, the correspondent polynomial approximation will be considered instead

r aNδ Nδ, ,C Cr rYδ Lδ Lδ, , and C C C

a

8

r

6 5 4 3 2

Yδ 0.0091 0.1568 +1.0642 3.6057 6.3479 5.4266 +2.4255h h h h h hCc c c c c c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, (41a)

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r

6 5 4 3 2

Lδ 0.0019 0.0333 +0.2259 0.7655 1.3476 1.1521 +0.5149h h h h h hCc c c c c c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, (41b)

a

6 5 4 3 2

Lδ 0.0001 +0.0025 -0.0174 0.0616 0.1172 0.1144 -0.1357h h h h h hCc c c c c c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, (41c)

r

6 5 4 3 2

Nδ 0.0052 +0.0901 -0.6115 2.0718 3.6475 3.1181 1.3937h h h h h hCc c c c c c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, (41d)

a

5 4 3 2

Nδ 0.0002 +0.0012 0.0043 0.0081 0.0079 +0.0091h h h h hCc c c c c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

. (41e)

A. Inequality Constraints Also in this case we assume that the collision avoidance maneuver is performed with full thrust which means we can impose

T = Tmax (42)

for the entire duration of the maneuver. As we have shown in the 2D case, to bound the variations of the angle of attack α(t) and its time rate α′(t) to their correspondent ranges, we have introduced a state equation, with an associated state variable η(t), and an auxiliary control variable ξ(t):

α = (1/2) (αmax + αmin) + (1/2) (αmax − αmin) sin η , (43a)

η′ = [2α′max/(αmax − αmin)] sin ξ. (43b)

Analogously, to bound the variations of the sideslip angle β(t) in the range

−βmax ≤ β ≤ βmax , (44a)

and its time rate β′ in the range

−β′max ≤ β′ ≤ β′max , (44b)

we introduce a new state variable ζ(t) and an auxiliary control variable χ(t) by the following non-singular transformation:

β = βmax sin ζ . (45a)

ζ′ = β′max/βmax sin χ, (45b)

where β′max is the maximum time rate. Note that both the transformations (43) and (45) are both nonsingular regardless of the values of the auxiliary controls ξ(t) and χ(t).

As concern the controls δe , δr , and δ a , to bound their variations in the ranges

, (46a) min maxe e eδ δ≤ δ ≤

, (46b) max maxr r r-δ ≤ δ ≤ δ

, (46c) min maxa a aδ δ≤ δ ≤

we introduce the following non singular transformations

, (47a) max min max mine e e e e(1/ 2)(δ δ ) (1/ 2)(δ -δ ) sinδ = + + ε

maxr rδ sinδ = κ , (47b)

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, (47c) max min max mina a a a a(1/ 2)(δ δ ) (1/ 2)(δ -δ ) sinδ = + + ω

where ε(t), κ(t) and ω(t) are the correspondent auxiliary control variables. As result, the dynamic system that describes the motion of the Ekranoplan is given by the composition of Eqs.

(33), Eq. (43b) and Eq.(45b). Such system is characterized by fourteen state variables (x, y, h, u, v, w, p, q, r, φ, θ, ψ, η, ζ) and five control variables (ξ, χ, ε, κ, ω). B. Potential Collision If the aircraft moves along a levelled trajectory with constant speed, its motion is described by the relation

x(t) = x0 + V0 t , (48a)

y(t) = 0 , (48b)

h(t) = h0 , (48c)

where the subscript 0 denotes the quantities at the initial time t = 0. We assume that the obstacle is steady and located at the same initial altitude of the aircraft and at an initial distance d0 from it

x*(t) = x0 + d0 , (49a)

y*(t) = 0 , (49b)

h*(t) = h0 . (49c)

Given the initial velocity of the aircraft V0, and the initial distance of the obstacle d0, the collision occurs if the following relations are satisfied at some forward time tc > 0:

x(tc) = x*(tc) , (50a)

y(tc) = y*(tc) , (50b)

h(tc) = h*(tc) , (50c)

with the implication that

V0 t = d0 . (51)

C. Optimization Problem Also for the 3D case, the application of the multiple-subarc SGRA to a Bolza-type optimization problem of

collision avoidance is subsequent to the determination of the inner boundary condition. With this purpose, let the distance between the host aircraft and the obstacle be written as

d = √ [(x – x*)2 + (y – y*)2 + (h – h*)2] (52a)

and its time derivative is

d′ = (1/2) [(x – x*) (x – x*)′ + (y – y*) (y – y*)′ + (h – h*) (h – h*)′]. (52b)

Let also

D = (1/2) d 2 = (1/2) [(x – x*)2 + (y – y*)2 + (h – h*)2] (53a)

denote the squared distance function whose time derivative is

D′ = (x – x*) (x – x*)′ + (y – y*) (y – y*)′ + (h – h*) (h – h*)′. (53b)

From (11b) and (12b) result that the conditions

d′ = 0 and D′ = 0 (54)

are reached simultaneously when

(x – x*) (x – x*)′ + (y – y*) (y – y*)′ + (h – h*) (h – h*)′ = 0, (55)

that is, when the relative position vector is orthogonal to the relative velocity vector.

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Also in this case, let θ1 denote the time length of the avoidance subarc; let θ2 = θ – θ1 denote the time length of the recovery subarc; let θ = θ1 + θ2 denote the assigned time length of the entire maneuver. The two subarcs are connected to one another at the maximin point where the inner boundary condition (53b) is satisfied. In order to bring the equations of the relative motion into the format required by the multiple-subarc SGRA, we introduce here the same time normalization for the 2D case.

In the actual time domain, the problem is to determine the time history of the controls ξ, χ, ε, κ, ω and the values of the time length of the first subarcs θ1 such that

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1D 1( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), 11

max ( ) min - ( )t t t t tt t t t t

Dξ χ ε κ ω θξ χ ε κ ω θ

θ → θ

D

. (56a)

In the virtual time domain, the problem becomes

(56b) ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), 11

max (1,1) min - (1,1) ,t t t t tt t t t t

Dξ χ ε κ ω θξ χ ε κ ω θ

→

with D(1, 1) meaning D(τ, i) evaluated at the end (τ = 1) of the first subarc (i = 1). With the introduction of this new two-index representation the one-subarc Chebyshev problem has been replaced with a two-subarc Bolza problem. D. Penalized Performance Index In order to prevent the undershooting by the wing tip of the initial altitude h0 the performance index (16b) is replaced with the following penalized performance index:

min [−D(1, 1) + kP], (57)

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

. (58) 1 1

2 21 2

0 0

= (τ, 1)dτ (τ, 2)dτP E Eθ + θ∫ ∫

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

E(τ, i) = h0 – h(τ, i) – (b/2) sin |φ(τ, i)| , if h – (b/2) sin |φ| ≤ h0 , (59a)

E(τ, i) = 0 , if h ≥ h0 . (59b)

E. Bolza Problem As result, we have determined the elements of the Bolza problem for the 3D collision avoidance maneuver. For sake of brevity we have omitted to state the boundary conditions. These can be determined using the same criteria of the 2D collision avoidance problem.

VIII. Conclusion In the first part of this work we investigate the bi-dimensional optimal collision avoidance problem between a

cruising Ekranoplan and a steady obstacle located on the ground imposing that the collision avoidance maneuver lies on the longitudinal plane. The following assumptions are employed: (i) initially the Ekranoplan is moving in a quasisteady levelled cruise trajectory; (ii) after the avoidance manoeuvre a recovery manoeuvre is executed by the aircraft to return to the initial cruise path; (iii) both the avoidance manoeuvre and the recovery manoeuvre lie on the same vertical plane identified by the initial cruise trajectory. The investigation of the collision avoidance performances on the longitudinal flight is encouraged by the relatively large quasi-flat turn radius of the Ekranoplans.

The second part of this work is dedicated to state the equivalent three-dimensional case of the collision avoidance problem. In detail we collect the complete set of differential constraints that describe the lateral motion of the aircraft considering the ground effect.

The approach to the problem considered, is to maximize wrt the controls the timewise minimum distance between the aircraft and the obstacle. This yields to a maximin problem or Chebyshev problem of optimal control, which is not solvable in a direct way. Hence, a technique is performed to transform the Chebyshev problem into a Bolza problem. Once reduced in this form the optimization problem can be solved numerically applying the multiple-subarc sequential gradient-restoration algorithm.

The main target of this research is to determine the relationship between the optimal avoidance maneuver and the control to execute it. In turn, this relationship is basilar to the development of a guidance scheme capable of approximating the optimal trajectory in real time.

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The results show that the angle of attack changes with the maximum speed. Eventually, for the cases with longer initial distances from the obstacle, the positive angle of attack bound is reached, meaning the ekranoplan tends to reach the stall conditions. The characteristics of the control that we have found make it particularly suitable for guidance purposes.

References 1 Miele, A., and Wang, T., “Optimal Collision Avoidance in Aerospace under Emergency Conditions”, Paper IAC-05-

C1.5.01, 55th International Astronautical Congress, Fukuoka, Japan, 2005. 2Miele, A., and Wang, T., “Multiple-Subarc Sequential Gradient-Restoration Algorithm, Part 1: Algorithm Structure”,

Journal of Optimization Theory and Applications, Vol. 116, No. 1, pp. 1-17, 2003. 3Miele, A., and Wang, T., “Multiple-Subarc Sequential Gradient-Restoration Algorithm, Part 2: Application to a Multistage

Launch Vehicle Design”, Journal of Optimization Theory and Applications, Vol. 116, No. 1, pp. 19-39, 2003. 4Rishikof, B. H., McCormick, B. R., Pritchard, R. E., and Sponaugle, S. J., “SEGRAM: A Practical and Versatile Tool for

Spacecraft Trajectory Optimization”, Acta Astronautica, Vol. 26, Nos. 8-10, pp. 599-609, 1992. 5Nocedal, J., and Wright, S.J, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, New York,

NY, 1999. 6Miele, A., and Wang, T., “Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential

Gradient-Restoration Algorithm”, Journal of Optimization Theory and Applications, Vol. 124, No. 1, pp. 29-53, 2005. 7Miele, A., and Wang, T., “Optimal Trajectories and Guidance Schemes for Ship Collision Avoidance”, Journal of

Optimization Theory and Applications, Vol. 129, No. 1, 2006. 8 Grillo, C., and Gatto, C. “Dynamic Stability of Wing in Ground Effect Vehicles: a general model”. FAST 2005, St.

Petersburg, Russia. 9 Hall, I.A. An Investigation into the Flight Dynamics of Wing in Ground Effect Aircraft Operating in Aerodynamic Flight.

MSc thesis, Cranfield University, 1994. 10Frazzoli, E., Mao, Z. H., Oh, J. H., and Feron, E., “Resolution of Conflicts Involving Many Aircrafts via Semidefinite

Programming”, Journal of Guidance, Control, and Dynamics, Vol. 24. No. 1, pp. 79-86, 2001. 11Menon, P. K., Sweriduk, G. D., and Sridhar, B., “Optimal Strategies for Free-Flight Air Traffic Confliction Resolution”,

Journal of Guidance, Control, and Dynamics, Vol. 22. No. 2, pp. 202-211, 1999. 12Clements, J. C., “The Optimal Control of Collision-Avoidance Trajectories in Air-Traffic Management”, Transportation

Research, Vol. 33B, No. 4, pp. 265-280, 1999.