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Gasodynamic Control System for INS Guided Bomb

Robert Głębocki1 and Marcin śugaj2 Warsaw University of Technology, 00-665, Warsaw, Poland

The paper describes the analysis and simulation results of a new guidance, navigation and control concept for guided bombs. The presented control system is simple and inexpensive. It is based on a set of one time used impulse engines and inertial navigation system which is set up before the drop by GPS. The engines are mounted around the bomb. There are not movable devices on the bomb’s board. The correcting impulses from the rocket engines are perpendicular to main symmetry axis of the bomb and influence directly the centre of gravity of the guided munitions. In the paper, the whole control system of the bomb is described. Numerical analysis presents a few cases of bombs’ controlled flights.

Nomenclature A = amplitude of oscillation a = cylinder diameter Cp = pressure coefficient Cx = force coefficient in the x direction Cy = force coefficient in the y direction c = chord dt = time step Fx = X component of the resultant pressure force acting on the vehicle Fy = Y component of the resultant pressure force acting on the vehicle f, g = generic functions h = height i = time index during navigation j = waypoint index K = trailing-edge (TE) nondimensional angular deflection rate A = inertia matrix B = gyroscopic matrix CX, CY, CZ = coefficients of aerodynamic force CR, CM, CN = coefficients of aerodynamic moments fa = vector of aerodynamics force fA = vector of aerodynamic loads fg = vector of gravity force fG = vector of gravity loads fsi = vector of i-th impulse engine force fS = vector of control loads g = gravity acceleration Ix, Iy, Iz = bomb moments of inertia kS = control signal activates the impulse engine l = bomb length m = bomb mass ma = vector of aerodynamics moment mg = vector of gravity moment ______________________________ 1 Associate Professor, Department of Automation and Aeronautical Systems, AIAA Member 2 Associate Professor, Department of Automation and Aeronautical Systems

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-311

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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msi = vector of i-th impulse engine moment nS = active engine number P, Q, R = angular velocities, components of the state vector x PSi = i-th engine thrust value rc = centre of gravity position vector S = maximum area of the bomb body cross-section in 0yz plane Sx = bomb static mass moments TV = velocity transformation vector TΩΩΩΩ = angle transformation vector U, V, W = linear velocities, components of the state vector x v = vector of linear velocity x = state vector xc = centre of gravity coordinates in x axis, component of the centre of gravity vector position rc x1, y1, z1 = bomb position coordinates, components of the position and attitude vector y y = position and attitude vector ϕ, θ, ψ = bomb euler angles, components of the position and attitude vector y γSi = angle of i-th engine position ρ = air density ΩΩΩΩ = velocities and rates matrix

I. Introduction

uring last fifty years one can observe the development of guided bombs and missiles. Last ten years this development is closely connected with Global Positioning System and Inertial Navigation System. Especially

since GPS reach full availability, many navigation systems of guided missiles and bombs base on INS/GPS. Well known one is Joint Direct Attack Monition (figure 1) wide used during operations in Kosovo, Afghanistan and Iraq. We also have another constructions like AASM carried out by SAGEM and SPICE carried out by RAFAEL. All of these constructions are aerodynamic controlled. They are combinations of a “dumb” bomb and a set of add-ons, a low-cost guidance kit that converts free-falling bombs into guided weapons. The kit’s major parts are a tail section, which contains an INS/GPS equipment, and body strakes that provide extra stability and lift.

Figure 1. Joint Direct Attack Monition concept of operations. [Boeing materials]

D

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In this paper a new concept of a control system for guided bombs is described. The gasodynamic steering kit is proposed instead of aerodynamic one. The system based on a set of one time used impulse engines and inertial navigation system calibrated, only before drop, by airplane’s GPS. It can correct the flight trajectory only about 700m from the uncontrolled one. But the control system’s hardware is very simple. There are not movable devices on the bomb’s board. It makes them the potential to be cheaper and more reliable than systems with aerodynamic control. Similar gasodynamic control system is successful used in guided mortar missiles like STRIX carried out by SAAB and BOFORS. The bomb can be dropped from altitude about four to six thousand meters using this control system, and the whole fall takes about twenty to thirty seconds. The INS system is accurate for that short time and distance of flight. At this concept guided bombs will have not as long range as JDAM but will be potential cheaper and have les complicated hardware. They can be use for precision bombing at the battlefield.

II. Problem description As it was mentioned before the bomb is controlled by a set of impulse correction engines. Engines are mounted

around the bomb. The correcting impulses from rocket engines are perpendicular to main symmetry axis of the flying object and influence directly the center of gravity of the guided munitions. Impulse rocket engines, used only one time each, correct the flying trajectory. The presented solution of the control system with impulse correction engines needs slow spin of the bomb. The bomb’s aft section is fitted with fins to give the missile aerodynamic stability and the angular velocity. The fins are immediately unfolded after the drop and their fixed cant angle gives the object a slow spin (about 30rad/s). The rotation velocity of the bomb depends on the velocity of flight. The much les range than in case of aerodynamic control objects require that bombs have to be accurately launched over the targets’ operating area and the control process starts when pitch angle is higher than 45%. Earlier control system is ineffective. At the next phase of flight object is automatically guided to the target. The first and necessary condition of target interception and successful attack is to launch the bombs into such an area.

A. Dynamics of flying objects’ impulse control. Classic methods of control of a flying object make assumptions that:

• steering forces initially change the moment acting on an object, than this moment rotates the object around its gravity center;

• supporting surfaces get necessary angles of attack and produce steering forces. This way, the object is turned at first around the mass center, than this movement effects on the mass center

velocity vector. This solution is characterized by inertia and a “long” time gap between control system’s decisions and its commands execution. This effect delays the control. This is an important fault in a situation when the precision guidance of the object to the target is needed in a short time, or when control process needs a very quick reaction to the information coming to the object. The whole guided phase lasts for about 15 to 20 seconds. This fault can be limited by the direct action on the motion of the gravity center. In the presented method, the control of the missile is performed by the set of the correction rocket engines. These engines are acted on the gravity center of the object (figure 2).

In this method of a flying object control we make assumptions that:

• steering forces first exert an influence on the object gravity center; • the rotation around the gravity center is an effect of a gravity center translation and an aerodynamic

interaction. Solution of this kind gives more effective influence on the speed vector. The block scheme of an object dynamic

is shown in figure 2.

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Gravity centreMovement

around gravity centre

Aerodynamic

Environmentgravitation

aerodynamical coefficients

aerodynamical coefficients

flight speed

movement of gravity centre

parameters of atmosphere

steering and motive force

angle movement

Figure 2. Block scheme of the object dynamic.

B. Missile guidance system At the rotating object, one channel is used to control the object in both horizontal and vertical planes. It can be

realized by a gasodynamic impulse acting on to the object gravity center. Method was described in more details in papers [2] and [5]. This solution can give us a precision object guiding to the attacked target. It also makes the operation of servo control system easier. A complicated mechanics of the aerodynamic servo is not needed, either. Also on board power demand for gasodynamic system is much les than in aerodynamic one. Electric energy supplies only electronic devices, not control surfaces. It makes equipment on the missile board smaller and easier to made, but it complicates the guidance logic and dynamics of the object controlled flight.

C. Navigation unit Inertial navigation systems usually can only provide an accurate solution for a short period of time. The INS

accelerometers will produce an unknown bias signal that appears as a genuine specific force. This is integrated to produce an error in position. Additionally, the INS software must use an estimate of the angular position of the accelerometers when conducting this integration. Typically, the angular position is tracked through an integration of the angular rate from the gyro sensors. These also produce unknown biases that affect the integration to get the position of the unit. The GPS gives an absolute drift-free position value that can be used to reset the INS solution or may be blended with it by use of a mathematical algorithm such as a Kalman filter [1] and [3]. The angular orientation of the unit may be inferred from the series of position updates from the GPS. The change in the error in position relative to the GPS may be used to estimate the unknown angle error. In the presented concept of the bomb’s impulse control system, the whole flight takes about 30 seconds. Based on data from [1] and ourselves tests, it is possible to use only INS information for navigation object. INS has to be calibrated only direct before the bomb drop.

III. Nonlinear simulation model of aircraft

In the simulation a bomb is modeled as a rigid body with six degrees of freedom. In this case the control force is produced by set of impulse engine placed around the bomb center of gravity. The bomb rotates around it’s axis of symmetry during the fall. Each engine is activated separately in appropriate angle of bomb turn and works in a short

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period of time. The vector of the engine force is perpendicular to the axis of bomb symmetry and cause the bomb center of gravity displacement.

The bomb equations of motion are derived in the co-ordinate system 0xyz (figure 3) [2] fixed to the bomb’s body. The center 0 of the system is placed at the arbitrary point in the bomb axis of symmetry. The 0x axis lays in the axis of bomb symmetry and is directed forward. The Oy axis is perpendicular to the axis of bomb symmetry and points right, the Oz axis points “down”.

The bomb translations and attitude angles are calculated in the inertial co-ordinate system 01x1y1z1; the center of this system 01 is placed at an arbitrary point on the earth surface. The 01z1 axis is along the vector of gravity acceleration, points down. The 01x1z1 plane is horizontal, tangent to earth surface, the 01x1 axis points to the North and 01y1 axis to the East.

Figure 3. Co-ordinate systems. A relationship between bomb state vector [ ]TRQPWVU=x and vector describing position and

attitude [ ]Tψθφzyx 111=y is given by:

Txy =& . (1)

The matrix T has the structure:

=

Ω

V

T0

0TT , (2)

where the velocity transformation matrix TV has the form

⋅⋅−⋅−⋅⋅⋅+⋅⋅⋅⋅+⋅⋅⋅−⋅⋅⋅

=θφθφθ

ψφψθφψφψφθψθ

ψφψθφψφψφθψθ

coscoscossinsin

cossinsinsincoscoscossinsinsinsincos

sinsincossincossincoscossinsincoscos

VT . (3)

and the transformation matrix for angles TΩ is

⋅⋅−

⋅⋅=

θφθφ

φφ

tgθφtgθφ

seccossecsin

sincos

cossin

0

0

1

ΩT . (4)

The roll angle φ, the pitch angle θ and the azimuth angle ψ describe attitude of the bomb (figure 4) and the vector r1=[x1, y1, z1] describes the bomb position in the 0x1y1z1 system of co-ordinates.

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The bomb equations are obtained by summing up inertia (left hand side of the equation), gravity Gf ,

aerodynamic Af and control Sf loads (forces and moments) acting on object:

( ) ( ) ( ) ( )SS nk ,y,fyfyx,fxx SGA ++=+ xBA& , (5)

where kS is the control signal activates the impulse engine and nS is the number of active engine. The left hand side of equation (5) describes the inertia loads in bomb frame of reference. The inertia matrix A

has the form:

−

−=

zx

yx

x

x

x

I000S0

0I0S00

00I000

0S0m00

S000m0

00000m

A, (6)

where: m is the aircraft mass, Sx is the object static mass moments and Ix, Iy, Iz are the bomb moments of inertia. The gyroscopic matrix ( )xB is calculated as:

( ) ( )AxΩxB = , (7)

where matrix of velocities and rates ( )xΩ has form:

( )

−−−−

−−−

−−

=

00

00

00

0000

0000

0000

PQUV

PRUW

QRVW

PQ

PR

QR

xΩ. (8)

The vector of gravity force acting on the body is calculated as:

( )

⋅⋅

−⋅=

φθ

φθ

θ

mg

coscos

sincos

sin

yfg, (9)

where: g is gravity acceleration. The point 0 is placed at the bomb center of gravity, the vector of moment from gravity forces is equal:

( ) ( )yfrym gCg ×= , (10)

where [ ]T00Cx=Cr is the vector of center of gravity position in bomb system of coordinates (figure 4).

Combining (9) and (10) the vector of gravity loads acting on the bomb is calculated as: ( ) ( ) ( )[ ]Tymyfyf ggG = . (11)

The bomb has set of impulse engines placed at the bomb body around the center of gravity (Fig. 2). The vector of i-th impulse engine force has form:

( )

−⋅=

Si

SiSSiSS kPnk

γγ

sin

cos,

0

y,fSi, (12)

where PSi is the value of engine thrust, γSi is the angle of engine position (figure 5).

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Figure 4. Set of impulse engine. The number of engine nS gives information about thrust and angle position of specific engine. The control signal

kS is used to activate the engine and is calculated using control error and actual bomb attitude. It can have value 0 or 1.

Figure 5. Position of the impulse engine. The vector of moment from impulse engine forces of each engine is equal:

( ) ( )SS nk ,y,frym SiCSi ×= . (13)

The vector of impulse engines loads acting on the bomb is calculated from (12) and (13) as:

( ) ( ) ( )[ ]Tymy,fy,f SiSiS SSSS nknk ,, = . (14)

The bomb aerodynamics loads are calculated using coefficients describing loads acting on the whole object. The force and moment vectors are calculated as:

( ) ( )( )( )( )

⋅⋅⋅=x

x

x

vyx,fa

Z

Y

X

C

C

C

Szρ2

121 , ( ) ( )

( )( )( )

⋅⋅⋅⋅=x

x

x

vyx,ma

N

M

R

C

C

C

lSzρ2

121 , (15)

where: S is the maximum area of the bomb body cross-section in 0yz plane (figure 6), l – the bomb length, ρ(z1) – air density, v – vector of linear velocity.

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Figure 6. Bomb aerodynamic parameters. The aerodynamic force CX, CY, CZ and moments CR, CM, CN coefficients obtained from CFD (Computational

Fluid Dynamics) calculations. They depend on bomb angle of attack and slip angle. The aerodynamic loads in the equations of motion are calculated as:

( ) ( ) ( )[ ]Tyx,myx,fyx,f aaA = . (16)

The bomb stabilizers generate the aerodynamic moment in x axis. The moment value depends on angle of incidence, area, shape and position of stabilizers, bomb air speed, angle of attack and angle of sideslip.

The equations of bomb motion are combined with model of the control system. The control system calculates the control signal and selects the proper impulse engine.

IV. Description of control devices and realization method

As we said earlier in presented concept of control method, the control is realized by correction engines located around the flying object’s center of gravity (figure 7). In our simulations we tested different number of correction engines from 12 to 20. The tracking technique also makes it possible to introduce several course corrections in a rapid succession. If necessary, all rocket correction engines can be used for the control process in the last few seconds of the flight.

Figure 7. Rocket control engines. The task of the rocket engines set is to correct the course of the bomb in the second phase of the flight, when the

pitch angle is over 45o. Control system homing it to the target, to achieve a direct hit. Correcting rocket engines are located in a cylindrical unit, arranged radial around the periphery. Each one can be fired individually only once in a selected radial direction.

The correction engine set is placed close to the center of gravity of the projectile. When the rocket engine is fired, the course of the missile is changed instantaneously. By successive firing of several rocket engines, the object

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is steered with high precision onto the target. The chosen steering system gives a very fast response to the guidance signals.

The decision when the correcting rocket engine should be fired depends on the value of the control error and its derivative. The frequency of firing of the correcting engines N is defined as the number of rotations of the mortar missile between the correcting engines firing. N increases with the control signal value K. The direction of control forces depends on the time of firing of the control engine. The time of control engines firing depends on the target direction, the position of the correction engine, the roll angle and the angular velocity along the x axis ωx.

The time of the correction engine work tk should be as short as possible (figure 8). Tests have shown that this time shouldn’t be longer than ¼ time of a mortar missile turn. During this time, the impulse of the correction engine changes the mortar missile course, which leads the missile main symmetry axis. The angles of attack and of side-slip increase. These angles generate lateral oscillations. The time of suppression of these oscillations depends on the frequency of the correcting engines firing N (figure 9) and the missile angular velocity along the x axis ωx. The shortest time of oscillations suppression is observed when N is equal 1. The amplitudes of the angle of attack and the angle of side-slip don’t generate significant disturbances of the target detection. However, some combinations of N, ωx, and of the velocity can lead to a resonance. The angle of attack and the angle of side-slip may considerably increase.

A single channel direct discontinuous impulse control method imposes requirements on a control quality for optimal correcting engines firing algorithm and good dynamic stability of the bomb. This control method, in contrast to an aerodynamic control method, doesn’t require any compromise between stability and controllability, because the stability value of the bomb isn’t limited. However, this method makes the algorithms of the correcting engines firing more complicated. The sequence of the correcting engines firing should be such that the unbalance of the bomb is minimal. This algorithm should give the value of the mean effect of control proportional to the control signal value.

Figure 8. Change of value of: a) The control impulse, b) The control force dependence on the correcting engine work time, c) The work time (black color) in proportion to the single rotation time.

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Figure 9. The angle of attack during the control phase of flight. Case a) N=3, case b) N=6.

V. Results The aim of the study was to find dynamic properties and possibilities of the impulse control of the flying object

by the presented methods. The investigations were carried out on a numerical model of dynamics of the control missile. The model was prepared in a Matlab/Simulink environment. It was a system of differential equations. The model was non-linear and discontinuous. It described space motion of the bomb in all phases of the flight, from the drop to the impact to the target. The description of movement is sufficiently general for the investigation (analysis) of the control process with differential guided methods.

At figure 10 flight trajectory for bomb with mass 100kg dropped from 4 000m, with initial speed V=180m/s. Bomb has 20 rocket control engines. Engine thrust P=10kN, engine’s work time tk=0.05s. Spin velocity ωx is about 30 rad/s. Figure 10 presented comparison trajectories for guided and ballistic flight. Bomb can reach target with error les than 10m in range about 700m far from the uncontrolled fall point. Figure 11 presents time changes of the pitch angle Θ during the control flight. We can see that control process started since about 15 second. It means that first 1000m in Zg axis is the ballistic phase of the flight.

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Figure 10. Control and ballistic bomb’s flight trajectory. Control is realized by 20 correction engines.

Figure 11. Pitch angle during bomb’s control flight. Control is realized by 20 correction engines. Figure 12 shows flight trajectory for bomb with 12 rocket control engines. Another flight conditions are similar

like at the case from figures 10 and 11: mass 100kg, bomb is dropped from 4 000m, initial speed V=180m/s, engine thrust P=10kN, engine’s work time tk=0.05s. Spin velocity ωx is about 30 rad/s. Figure 12 presented comparison trajectories for guided and ballistic flight. Bomb can reach target with error les than 10m in range about 400m far from the uncontrolled fall point. Figure 13 presents time changes of the pitch angle Θ during the control flight. Similar to figure 11 that control process started at about 15 second. It means that first 1000m in Zg axis is the ballistic phase of the flight.

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Figure 12. Control and ballistic bomb’s flight trajectory. Control is realized by 12 correction engines.

Figure 13. Pitch angle during bomb’s control flight. Control is realized by 12 correction engines.

VI. Conclusion Numerical experiments have shown large possibilities of the objects’ control by the influence on the motion of

their gravity center. It is possible to use the set of impulse correction rockets to control falling objects like bombs. The accuracy and control quality, attainable, at the phase of a computer simulation, gives good prognostics for the

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possibilities of practical use. Bombs can reach targets with accuracy about 10m. This result does not include error from INS. This method of control leads to more complicated control algorithms but makes the servo control easier to perform. The servo has only the set of correction rocket engine and the electrical system for initiation.

The investigations have shown that for successful control the object needs a proper energy capability in the potential of correction rocket engines. The amount of energy depends on mass of the bomb. The division of energy between rocket engines and the engines’ times of work are up to designer’s decision.

Acknowledgments This work was supported by Polish Ministry of Science and High Education (No. N514-O/0028/32)

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1998. 2 Glebocki R., Vogt R. „Guidance system of smart mortar missile” The archive of mechanical engineering, Vol LIV

No 1, 2007, pp 47-63 3 Chen W. H., “Nonlinear Disturbance Observer-Enhanced Dynamic Inversion Control of Missiles”, Journal of

Guidance, Control, and Dynamics, Vol. 26, No 1,2003,pp. 161-166. 4 Iglesian P.A., Urban T.J. “Loop Shaping Design for Missile Autopilot”, Journal of Guidance, Control, and

Dynamics, Vol. 23, No 3,2000,pp. 516-525. 5 Głębocki R., Vogt R., śugaj M. “Smart mortar missile attitude detection based on the algorithm that takes

advantage of the artificial neural networks” AIAA GNC conference August 2006