Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 27-44, Warsaw 2010 MODEL OF GASODYNAMIC CONTROL SYSTEM FOR GUIDED BOMBS Robert Głębocki Marcin Żugaj WarsawUniversity of Technology, Institute of Aeronautics andAppliedMechanics,Warsaw, Poland e-mail: rgleb@meil.pw.edu.pl, zugaj@meil.pw.edu.pl In this paper, a new concept of the control system for guided bombs is described. Authors propose a gasodynamic method for the air bomb control. In the presented method, the bomb control is realised by a set of single-use impulse control engines. The gasodynamic controlled bomb dynamic and aerodynamicmodels are described. Some results of nume- rical simulations are presented as well. Key words: guided bombs, flight control system Notationa A – inertia matrix B – gyroscopic matrix CX,CY ,CZ – coefficients of aerodynamic force CR,CM,CN – coefficients of aerodynamic moments fa,fA – vector of aerodynamics force and loads, respectively fg,fG – vector of gravity force and loads, respectively fsi – vector of i-th impulse engine force fS – vector of control loads g – gravity acceleration Ix,Iy,Iz – moments of inertia of bomb kS – control signal activates the impulse engine l,m – bomb length andmass, respectively ma,mg,msi – vector of aerodynamic, gravity and i-th impulse engine moment, respectively nS – active engine number P,Q,R – angular velocities, components of state vector x PSi – i-th engine thrust value 28 R. Głębocki, M. Żugaj rc – centre of gravity position vector S – maximum area of the bomb body cross-section in 0yz plane Sx – bomb static mass moments TV ,TΩ – velocity and angle transformation vector, respectively U,V,W – linear velocities, components of state vector x v – vector of linear velocity x – state vector xc – centre of gravity coordinates in x axis, component of cen- tre of gravity vector position rc x1,y1,z1 – bomb position coordinates, components of position and attitude vector y y – position and attitude vector φ,θ,ψ – bomb euler angles, components of the position and atti- tude vector y γSi – angle of i-th engine position ρ – air density Ω – velocities and rates matrix 1. Introduction The contemporary development of air-launched weapons is mostly oriented on design of precision-guided munitions. The percentage of guided weapons in all instruments of war used from air is greater in each subsequent military conflict. For example, at the operation Desert Storm 18 percent of used by US Army bombs were controlled, at the operation Iraq Freedom the same factor was about 66 percent. In the last ten years this developmentwas closely connected with Global Positioning System and Inertial Navigation System. Especially, since GPS has reached full availability, many navigation systems of guided missiles and bombs are based on INS/GPS. Well known is Joint Direct Attack Monition, which was widely used during operations in Kosovo, Afghanistan and Iraq. There are also other constructions like AASM carried out by SAGEMand SPICE carried out byRAFAEL (Fig.1). All these bombs are aerodynamically controlled. Oneof thepossibilities of development of guidedbombs is taking advantage of new concepts for control of the object. In the present control systems, the bomb is controlled with small single-use rocket engines with thrust directed normally to the main axis of symmetry of the object. Model of gasodynamic control system... 29 Fig. 1. Bombs AASM and SPICE 2. Problem description In this paper, a new concept of the control system for guided bombs is descri- bed. The gasodynamic steering kit is proposed instead of aerodynamic one. The bomb is controlled by a set of impulse correction engines. The engines are mounted around the bomb (Fig.4). The correcting impulses from rocket engines are perpendicular to themain symmetry axis of the flying object and influence directly the centre of gravity of the guided bomb.The impulse rocket engines, usedonly one time each, correct the trajectory.Thepresented solution of the control system with impulse correction engines needs slow spin of the bomb. The bomb aft section is fitted with fins to give the bomb aerodynamic stability and spin.The fins are immediately unfolded after the bombdrop and their fixed cant angle gives the object a slow spin (about 20-30rad/s). The rotation velocity of the bomb depends on the velocity of flight. Themuch less range than in the case of aerodynamic control objects require that bombshave to be accurately launched over the target operating area. The control process starts when the pitch angle is higher than 45%. The earlier control system is ineffective. In the next stage of flight, the object is automatically guided to the target. The system is based on a set of single-use impulse engines. It can correct the flight trajectory only about 700m from the uncontrolled one. But the control system hardware is very simple. There are notmovable devices on the bombboard. Itmakes themthe potential to be cheaper andmore reliable than systems with aerodynamic control. A similar gasodynamic control system is successfully used in guided mortar missiles like STRIX carried out by SAAB and BOFORS. The bomb can be dropped from altitude of about four to ten thousand meters using this control system, and the whole fall takes about 30 R. Głębocki, M. Żugaj twenty to thirty seconds. In this concept, guided bombs do not have as long range as JDAM (Joint Direct Attack Munition) but are potentially cheaper and have less complicated hardware. They can be used for precision bombing at the battlefield. 2.1. Dynamics of impulse control of flying objects Classic methods of control of a flying object make assumptions that: • steering forces initially change the moment acting on the object, then this moment rotates the object around its gravity centre; • supporting surfaces get necessary angles of attack and produce steering forces. This way, the object is turned at first around the mass centre, then this movement effects on the mass centre velocity vector. This solution is charac- terized by inertia and a ”long” time gap between control systemdecisions and execution of its commands. This effect delays the control. This fault can be limited by the direct action onmotion of the gravity centre. In the presented method, the control of themissile is performed by the set of correction rocket engines. These engines act on the gravity centre of the object (Fig.2). Fig. 2. Block scheme of the object dynamics In this method of the flying object control we make assumptions that: • steering forces influence first the object gravity centre; • rotation around the gravity centre is an effect of gravity centre transla- tion and aerodynamic interaction. Model of gasodynamic control system... 31 Solution of this kind gives more effective influence on the speed vector. A block scheme of the object dynamics is shown in Fig.2. 2.2. Missile guidance system In the spinning object, one channel is used to control the object in both horizontal and vertical planes. This can be realised by a gasodynamic impulse acting on the object gravity centre. Themethodwas described inmore details in papers by Głębocki and Vogt (2007) and Głębocki and Żugaj (2009). This solution can precisely guide the object to the attacked target. It also makes the operation of the servo-control system easier. Complicated structure of the aerodynamic servo is not needed, either. Also the board power demand for the gasodynamic system is much less than in the aerodynamic one. Electric energy supplies only electronic devices, not control surfaces. Itmakes the equ- ipment on the missile board smaller and easier to made, but it complicates the guidance logic and dynamics of the object controlled flight. As was said earlier, in the presented concept the control is realised by correction engines located around the flying object centre of gravity. In our simulations we tested different numbers of correction engines from 12 to 20. The tracking techniquemade 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. The task of the rocket engines is to correct the course of the bomb in the second stage of flight, when the pitch angle is over 45◦. The control systemho- ming it to the target, enables the object to achieve a direct hit. The correction rocket engines are located in a cylindrical unit, arranged radially around the periphery. Each correction rocket engine can be fired individually only once in a selected radial direction. The correction engine set is placed close to the centre of gravity of the projectile.When the rocket engine is fired, the course of themissile is changed instantaneously. By successive firing of several rocket engines, the object 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 32 R. Głębocki, M. Żugaj direction, position of the correction engine, roll angle and angular velocity ωx along the axis x. The time of correction enginework tk should be as short as possible. Tests have shown that this time should not be longer than 1/4 time of the bomb turn.During this time, the impulse of the correction engine changes the bomb course, which leads the object main symmetry axis. A single-channel direct discontinuous impulse control method imposes re- quirements on control quality for the optimal correction engines firing algori- thmand good dynamic stability of the bomb (Iglesian andUrban, 2000). This control method, in contrast to the aerodynamic control, does not require any compromisebetween stability and controllability, because the stability value of the bomb does not have the upper limit. Stability has to bemuch higher than in the case of aerodynamic control. However, thismethodmakes algorithms of correction engines firing more complicated. The sequence of firing should be such that the unbalance of the bomb is minimal. This algorithm should give the value of themean effect of control proportional to the control signal value. 3. Nonlinear simulation model of guided bomb 3.1. Dynamics model In the simulation, the bomb ismodelled as a rigid bodywith six degrees of freedom. In this case, the control force is produced by a set of impulse engines placed around the bomb centre of gravity. The bomb rotates around its axis of symmetry during the fall. Each engine is activated separately in appropriate angle of the bomb turn andworks in a short period of time. The vector of the engine force is perpendicular to the axis of bomb symmetry and displaces the bomb centre of gravity. Equations of motion of the bomb are derived in the co-ordinate system 0xyz, Fig.3 (Maryniak, 1978) fixed to the bomb body. The centre 0 of the system is placed at an arbitrary point in the bomb axis of symmetry. The 0x axis lies in the axis of bomb symmetry and is directed forward. The 0y axis is perpendicular to the axis of bomb symmetry and points right, the 0z axis points ”down”. Thebombtranslations andattitude angles are calculated in the inertial co- ordinate system 01x1y1z1; the centre of this system 01 is placed at anarbitrary point on the earth surface. The 01z1 axis is placed along the vector of gravity acceleration and points down. The 01x1z1 plane is horizontal, tangent to the earth surface, the 01x1 axis points to the North, and 01y1 axis to the East. Model of gasodynamic control system... 33 Fig. 3. Co-ordinate systems A relationship between the bomb state vector x= [U,V,W,P,Q,R]⊤ and the vector describing position and attitude y= [x1,y1,z1,φ,θ,ψ]⊤ is given by ẏ=Tx (3.1) Thematrix T has form T= [ TV 0 0 TΩ ] (3.2) where the velocity transformation matrix TV is TV =    cosθcosψ sinθ sinφcosψ− cosφsinψ cosφsinθcosψ+sinφsinψ cosθsinψ sinθ sinφsinψ+cosφcosψ cosφsinθsinψ− sinφcosψ −sinθ sinφcosθ cosφcosθ    (3.3) and the transformation matrix for the angles TΩ is TΩ =    1 sinφtanθ cosφtanθ 0 cosφ −sinφ 0 sinφsecθ cosφsecθ    (3.4) The roll angle φ, the pitch angle θ and the azimuth angle ψ describe the attitude of thebomb(Fig.4) and the components of the vector r1 = [x1,y1,z1] describe the bomb position in the 0x1y1z1 system of co-ordinates. 34 R. Głębocki, M. Żugaj Thebombequations areobtainedby summingupthe inertia (left hand side of the equation), gravity fG, aerodynamic fA and control fS loads (forces andmoments) acting on the bomb (Maryniak, 1978) Aẋ+B(x)x=fA(x,y)+fG(y)+fS(y,kS,nS) (3.5) where kS is the control signal activating the impulse engine and nS is the number of the active engine. The left-hand side of equation (3.5) describes the inertia loads in the bomb frame of reference. The inertia matrix A has form A=          m 0 0 0 0 0 0 m 0 0 0 Sx 0 0 m 0 −Sx 0 0 0 0 Ix 0 0 0 0 −Sx 0 Iy 0 0 Sx 0 0 0 Iz          (3.6) where: m is the bombmass, Sx is the bomb static massmoments and Ix, Iy, Iz are the bombmoments of inertia. The gyroscopic matrix B(x) is calculated as B(x)=Ω(x)A (3.7) where the matrix of velocities and rates Ω(x) has form Ω(x)=          0 −R Q 0 0 0 R 0 −P 0 0 0 −Q P 0 0 0 0 0 −W V 0 −R Q W 0 −U R 0 −P −V U 0 −Q P 0          (3.8) The vector of gravity force acting on the body is calculated as fg(y)= mg    −sinθ cosθ sinφ cosθcosφ    (3.9) where g is the gravity acceleration. The point 0 is placed at the bomb centre of gravity, the vector of moment from gravity forces is mg(y)= rc×fg(y) (3.10) Model of gasodynamic control system... 35 Fig. 4. Set of impulse engines where rc = [xc,0,0]⊤ is the vector of centre of gravity position in the bomb system of coordinates (Fig.4). Combining (3.9) and (3.10), the vector of gravity loads acting on the bomb is calculated as fG(y)= [ fg(y) mg(y) ] (3.11) The bomb has a set of impulse engines placed at the bomb body around the centre of gravity (Fig.4). The vector of i-th impulse engine force has form fSi(y,kS,nS)= PSikS    0 −cosγSi sinγSi    (3.12) where PSi is the value of engine thrust, γSi is the angle of engine position (Fig.5). Fig. 5. Position of the impulse engine 36 R. Głębocki, M. Żugaj The number of engine nS gives information about the thrust and angle position of the given engine. The control signal kS is used to activate the engine and is calculated using the control error and actual bomb attitude. It can have value 0 or 1. The vector of moment from the impulse engine forces of each engine is equal mSi(y)= rc×fSi(y,kS,nS) (3.13) The vector of impulse engines forces acting on the bomb is calculated from (3.12) and (3.13) as fS(y,kS,nS)= [ fSi(y,kS,nS) mSi(y) ] (3.14) The bomb aerodynamic loads are calculated using coefficients describing the loads acting on thewhole bomb.The force andmoment vectors are calculated as fa(x,y)= 1 2 ρ(z1)|v| 2S    CX(x) CY (x) CZ(x)    ma(x,y)= 1 2 ρ(z1)|v| 2Sl    CR(x) CM(x) CN(x)    (3.15) where: S is the maximum area of the bomb body cross-section in 0yz plane (Fig.4), l – bomb length, ρ(z1) – air density, v – vector of linear velocity and CX, CY , CZ, CR, CM, CN are force andmoment coefficients. Fig. 6. Aerodynamic parameters of the bomb The aerodynamic loads in the equations of motion are calculated as fA(x,y)= [ fa(x,y) ma(x,y) ] (3.16) Thebomb stabilizers generate the aerodynamicmoment along the x axis. The moment value depends on the angle of incidence, area, shape and position of stabilizers, bomb air speed, angle of attack and angle of sideslip. Model of gasodynamic control system... 37 The equations of bombmotion are combinedwith themodel of the control system.The control system calculates the control signal and selects the proper impulse engine. 3.2. Aerodynamical model TheComputational FluidDynamics (CFD)methodwasused to obtain the aerodynamic coefficients. The bomb has a cylindrical shape and aerodynamic phenomena in the 0xy plane of symmetry are similar as in the 0xy plane of symmetry. So, it was assumed that the aerodynamic load can be calculated as a sum of lateral (0xy) and longitudinal (0xz) loads. The non-rotating bomb was considered in this calculation. The coefficients were derived in the aerodynamical coordinate system 0AxAyAzA (Fig.7) fixed to the bomb tip. The 0AxA axis lies in the axis of bomb symmetry and is directed backwards. The 0AyA axis is perpendicular to the axis of bomb symmetry and points right, and the 0AzA axis points up. Fig. 7. Aerodynamical co-ordinate system of the bomb The lift CL, drag CD and aerodynamic moment Cm coefficients were calculated for theMach number 0.56 and 1.16. Next, the aerodynamic centre was obtained andaerodynamic coefficientswere transformed to this point.The aerodynamic moment at aerodynamic centre is denoted as Cmac (Fig.8). Fig. 8. Aerodynamical coefficients of the bomb 38 R. Głębocki, M. Żugaj The aerodynamic centre is placed at 26.67% and 30% of the bomb length for Ma=0.56 and 1.16, respectively. The coefficients as function of the angle of attack are shown in Figs.9 to 11. Fig. 9. Lift coefficient Fig. 10. Drag coefficient The pitch dynamic derivative dCM , which is given in the body fixed co- ordinate system 0xyz, was obtained aswell. Thederivative changes versus the angle of attack as shown in Fig.12. The force and moment coefficients CX, CY , CZ, CR, C′M, C ′ N are obta- ined by transformation of the CL, CD and Cmac coefficients to the body fixed coordinate system 0xyz for lateral (0xy) and longitudinal (0xz)motion separately. The whole pitch and yaw moments are calculated using equations CM = C ′ M +dCM ·Q CN = C ′ N +dCM ·R (3.17) Model of gasodynamic control system... 39 Fig. 11.Moment coefficient Fig. 12. Pitch derivative coefficient 4. Results Simulation experiments tested dynamic properties of gasodynamic-controlled bombs. The tests were based on the mathematical model described in Sec- tion 4. For simulation experiments, a Matlab/Simulink model was used. Si- mulations were made for bombs with the following parameters: mass 100kg, length 1.5m, and diameter 0.18m. The bombs were tested with three diffe- rent rotation coefficients kω =5.8, 12.3, 37 and were dropped from altitudes H =1000, 2000, 4000mwith the initial speed v =222m/s. Figures 13 and 14 show the us influence of the bombs rotation velocity on their flight. As a result of bomb rotation, we can observe changes in flight trajectories. The gyroscopic effect gets out the trajectory. Figure 13 shows this gyroscopic effect on flight 40 R. Głębocki, M. Żugaj Fig. 13. Gyroscopic effect influencing flight trajectories for different coefficients k ω Fig. 14. Model of gasodynamic control system... 41 trajectories for different coefficients kω. In the case when kω is equal 5.8, the change of the flight trajectory is above 15m. Except for the changes at the flight trajectories, the rotation of the bomb makes small disturbances to the yaw angle (Fig.14a). The pitch angle distur- bances appear in both cases of rotating and non-rotating bombs (Fig.14b). They come from the low stability margin of the bomb. Figure 14c describes changes of the spin velocity of bombs during flight. Figure 14ddepicts relations between thenumberof revolutions andflight times (depending on the drop altitude) and coefficients kω. These characteristics were used to design the bomb control system. In Fig.15, the flight trajectory of a bombwithmass 100kg dropped from 4000m with the initial speed v =180m/s is shown. The bomb has 20 rocket control engines with thrust P = 10kN and the work time tk = 0.05s. The spin velocity ωx is about 30rad/s. Figure 15 gives comparison trajectories for guided and ballistic flight. The bomb can reach the target with error less than 10m within the range of about 700m from the uncontrolled fall point. Figure 16 presents time changes of the pitch angle θ during the control flight. We can see that the control process started since about 15 second. It means that first 1000m in the zg axis is the ballistic stage of the flight. Fig. 15. Controlled and ballistic flight trajectory. The control is realised by 20 correction engines Figure 17 shows the flight trajectory for a bomb with 12 rocket control engines. The other flight condition were similar as in the case in Fig.15: mass 100kg, drop from 4000m, initial speed V = 180m/s, engine thrust P = 10kN, enginework time tk =0.05s, spinvelocity ωx about 30rad/s.Figure 17 reveals comparison trajectories for the guided and ballistic flight. The bomb can reach the targetwith error less than 10mwithin the range of about 400m from the uncontrolled fall point. Figure 18 presents time changes of the pitch 42 R. Głębocki, M. Żugaj Fig. 16. Pitch angle during the controlled flight. The control is realised by 20 correction engines Fig. 17. Controlled and ballistic flight trajectory. The control is realised by 12 correction engines Fig. 18. Pitch angle during the controlled flight. The control is realised by 12 correction engines Model of gasodynamic control system... 43 angle θ during the controlled flight. Similarly as in Fig.16, the control process started since about 15 seconds. It means that first 1000m in the zg axis was the ballistic stage of the flight. 5. Conclusion Numerical experiments have shown large possibilities of the objects control by influencingmotion of their gravity centre. It is possible to use impulse correc- tion rockets to control falling objects like bombs.Thismethod of control leads to more complicated control algorithms but makes the servo-control easier to operate. The servo has only a correction rocket engines set and the electrical system of initiation. The presented model and its properties are based on design of a gasody- namic bomb control systemwith GPS/INS navigation. Acknowledgment Thisworkhasbeen supportedbyPolishMinistry of Science andHigherEducation (No. N514-O/0028/32). References 1. ChenW.H., 2003,Nonlineardisturbanceobserver-enhanceddynamic inversion control ofmissiles, Journal of Guidance, Control, andDynamics,26, 1, 161-166 2. Etkin B., Reid D., 1996,Dynamics of Flight Stability and Control, 3rd Ed., Wiley, NewYork 3. Głębocki R., VogtR., 2007,Guidance system of smartmortarmissile,The Archive of Mechanical Engineering, LIV, 1, 47-63 4. Głębocki R., Żugaj M., 2009, Gasodynamic control system for INS guided air bombs,Aerospace Sciences Conference and Exhibit AIAA, Orlando 5. Iglesian P.A., Urban T.J., 2000, Loop shaping design formissile autopilot, Journal of Guidance, Control, and Dynamics, 23, 3, 516-525 6. Klotz H., Derbak C., 1998, GPS-aided navigation and unaided navigation on the Joint Direct AttackMunition, IEEE 7. Maryniak J., 1978, Dynamiczna teoria obiektów ruchomych, Prace Nauko- we PolitechnikiWarszawskiej, Mechanika, 32,WydawnictwoPolitechnikiWar- szawskiej,Warszawa 44 R. Głębocki, M. Żugaj Model gazodynamicznego systemu naprowadzania lotniczych bomb sterowanych Streszczenie Wpublikacji autorzy przedstawili nowatorski systemnaprowadzania sterowanych bomb lotniczych. W prezentowanej metodzie układ wykonawczy sterowania oparty jest na zestawie jednorazowych impulsowych silników korekcyjnych oddziałujących bezpośrednio na środek ciężkości sterowanego obiektu. W artykule autorzy zapre- zentowali modele dynamiki oraz aerodynamiki bomby sterowanej gazodynamicznie. Oprócz opisu modelu zawarto również wyniki obliczeń aerodynamicznych. Przedsta- wiono opis systemu sterowania oraz wyniki przeprowadzonych badań symulacyjnych. Manuscript received March 18, 2009; accepted for print April 15, 2009