Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 2, pp. 363-381, Warsaw 2009 THE CONTROL LAWS HAVING A FORM OF KINEMATIC RELATIONS BETWEEN DEVIATIONS IN THE AUTOMATIC CONTROL OF A FLYING OBJECT Edyta Ładyżyńska-Kozdraś Warsaw University of Technology, Faculty of Mechatronics, Warsaw, Poland e-mail: e.ladyzynska@mech.pw.edu.pl The paper presents some practical applications of control laws in dy- namics of flying objects. The control laws considered have the form of kinematics and geometrical deviation relations between the actual para- meters and those specified that result from guidance of the considered object. The specified parameters were introduced into the control laws as: parameters determined bymotion of the target in the guidedmissile or motion parameters of the beam tracking the target. A general mo- del of dynamical behaviour of the guided object was employed in the considerations. Key words: automatic control of missiles, homing and beam-riding guidance, numerical simulations 1. Introduction Before starting the design process of anti-aircraft missile one should solve the problem of control. A proper control should include the flight control systems that ensure the optimal missile guidance onto a target. The paper aims at presentation of a general model of a flying object with non-holonomic constraints imposed.These constraints take the formof control laws assumed for the examined objectmotion in terms of kinematical relations between deviations from the preset to current values, respectively, of selected parameters. Samplemissiles of different types guidedontomaneuvering targets served to illustrate the approach. It is well known that the crucial element of each control system consists in the guidance algorithm it performs. The algorithm imposes constraints on the object motion; therefore, a proper choice of the control method is essen- tial. When dealing with missile guidance, one can apply two or three-point 364 E. Ładyżyńska-Kozdraś methods, respectively (Ben-Asher and Yaesh, 1998; Blakelock, 1991; Dziopa, 2006;Menon et al., 2003; Zarchan, 2001). In the two-pointmethod, constraints are imposed on themissile-target motion, while in the three-pointmethod the guidance station should be considered as well. To implement automatic control and navigation systems of advanced ob- jects, especially those computer-aided, one should assumeproper physicalmo- dels and develop the mathematical ones capable of representing dynamical properties of the object. Then, proper control laws and kinematical relations of guidance and navigation should be formulated. The kinematical and dyna- mical behaviour of the executing system should be assumed as well and the signaling way of current position parameters. Motion of the object and the program of predetermined trajectory with specific limits imposed should be accepted as well. When developing a model, a kind of compromise should be agreed on the influence of different aspects of the problem; i.e., accuracy of theoretical analysis of the physical problem, complexity of mathematical equ- ations, availability of technicalmeans andpersonal knowledge – skills intuition and experience should be properly balanced. A new, relatively uncomplicated approach to the problem of automatic control of an object has been presented in the paper, basing on a general mathematical model of a flying object under control with the control laws introduced. Themethod efficiency can be improved when the control laws are considered as non-holonomic constraints imposed upon the object motion. 2. General form of control laws The paper presents sample applications of control laws to stabilisation, con- trol, guidance and navigation, respectively, of flying objects under automatic control. The preset flight parameters (bearing the index z) may appear in the control laws in different ways; i.e., as parameters of a steady flight, as parameters resulting from the accepted guidance method, flight program or way of reaching the preset target as well as those from the tracking of terrain obstacles, and finally as parameters ensuring the required flight state. The control laws for flying objects given below have the form of kinema- tical equations of deviations between the preset and current values of flight parameters observed in the roll, pitch, yaw and velocity channels, respectively. Thedifferences appearingbetween the currentparameters and thepreset ones, respectively, determine the deflections.As a result, the forces acting on control surfaces change so that the object returns onto its predetermined trajectory. The control laws having a form of kinematic relations... 365 The accepted control laws have the form (Ładyżyńska-Kozdraś, 2006; Ładyżyńska-Kozdraś andMaryniak, 2003; Maryniak, 1987): — in the pitch channel TH3 δ̇H +T H 2 δH = K H x1 (x1−x1z)+K H z1 (z1−z1z)+K H U (U −Uz)+ (2.1) +KHW(W −Wz)+K H Q (Q−Qz)+K H θ (θ−θz)+δH0 — in the yaw channel TV3 δ̇v +T V 2 δV = K V y11 (y1−y1z)+K V V (V −Vz)+K V P (P −Pz)+ (2.2) +KVφ (φ−φz)+K V R(R−Rz)+K V ψ (ψ−ψz)+ δV0 — in the roll channel TL3 δ̇L +T L 2 δL = K L V (V −V1z)+K L W(W −Wz)+K L P(P −Pz)+ (2.3) +KLφ(φ−φz)+K V Q(Q−Qz)+K V R(R−Rz)+ δL0 — in the velocity channel TT3 δ̇T +T T 2 δT = K T x11 (x1−x1z)+K T z11 (z1−z1z)+K T U(U −Uz)+ +KTW(W −Wz)+K T θ (θ−θz)+K H Q (Q−Qz)+K T φ (φ−φz)+ (2.4) +KTψ(ψ−ψz)+ δT0 where Tij – time constants Kij – amplification coefficients δH,δV ,δL,δT – deflections of: elevator δH, rudder δV , ailerons δL and throttle lever δT , respectively φ,θ,ψ – theangles of roll φ, pitch θ andyaw ψ, respectively (Fig.1) x1,y1,z1 – components of the position vector of the object re- lative to the fixed gravitational frame of reference (Fig.1) U,V,W – components of the velocity vector (Fig.1) P,Q,R – angular velocities of: roll P, pitch Q and yaw R, respectively (Fig.1). Since they are non-intergrable and impose limitations on the system mo- tion, these control laws define two equations of non-holonomic constraints. 366 E. Ładyżyńska-Kozdraś The control laws together with the equations of motion determine the object trajectory and its behaviour along it. The square coefficient of control quality was applied to the control process quality assessment (Ładyżyńska-Kozdraś et al., 2005),withall control channels considered (n =4), i.e. J = 4 ∑ i=1 tk ∫ 0 [yi(t)−yzi(t)] 2 dt (2.5) where yzi(t) – denotes the predetermined course of the variable yi(t) – stands for the actual course of the variable. Since the coefficient given in Eq. (2.5) has not been normalised, it cannot be applied to the analysis of transient processes in which the quantities under control reveal different orders of magnitude. One should, therefore, normalise it using e.g. the formulae for relative deviations (Ładyżyńska-Kozdraś, 2006; Ładyżyńska-Kozdraś andMaryniak, 2003) J = 4 ∑ i=1 tk ∫ 0 [yi(t)−yzi(t) yimax ]2 dt (2.6) where yimax is the maximum preset range of the ith state variable or the preset value yzi of the ith state variable if it takes a non-zero value. It should be noted, however, that depending on the task to be executed and the type of flying object the control laws can be reduced and adapted adequately. This will be shown in simulation examples in the next part of this paper. The kinematical relations representing the objectmotion are the following functions of its linear and angular velocities ṙ= [ẋ1, ẏ1, ż1, φ̇, θ̇, ψ̇] −1 =F(U,V,W,P,Q,R,φ,θ,ψ) (2.7) The kinematical equations of the flying object (bearing the index R) gu- idance onto the target (bearing the index C) can be written as follows ṙRP =f1(VR,VC,εRC,νRC,φC,θC,ψC,φR,θR,ψR) (2.8) ṙ1R =f2(VR,VC,φC,ψC,θC,φR,θR,ψR) In the case when the flying object moves on the predetermined trajectory, the program constraints should be imposed r1 =f3(x1,y1,z1,φz,θz,ψz) (2.9) The control laws having a form of kinematic relations... 367 To ensure the proper course of automatic control of the flying object, one should introduce into the control laws some selected flight parameters that result from thepredeterminedflight trajectory together with flight parameters of the enemy or parameters of the laser terrain penetration. One should also consider dynamical behaviour of the control execu- ting system represented by the following equations given in a general form (Ładyżyńska-Kozdraś andMaryniak, 2003; Maryniak, 2007): — kinematical equation of control in the pitch channel TH1 δ̇H +T H 2 δH =−(M H z0 +K αH z αH +K δH z δH +K H z δ̇H) (2.10) — kinematical equation of control in the yaw channel TV1 δ̇V +T V 2 δV =−(M V z0+K βV z βV +K δV z δV +K V z δ̇V ) (2.11) — kinematical equation of control in the roll channel TL1 δ̇L +T L 2 δL =−(M L z0+K αL z αL +K δL z δL +K L z δ̇L) (2.12) where the values of particular coefficients depend on the design and type of the control system applied (mechanical, electrical or electronic) as well as on its performance quality and MHz0,M V z0,M L z0 – moments of forces necessary for automatic control of: elevator, rudder and aileron displacements, re- spectively KαHz ,K βV z ,K αL z – coefficients of the stiffness forces due to changes in: angles of attack of the elevator unit, sideslip angle and aileron displacement, respectively KδHz ,K δV z ,K δL z – drag coefficients in the control due to its stiffness, dependingus on the angles of control surface deflec- tions KHz ,K V z ,K L z – drag coefficients in the control system due to the velocity of control surface deflections. In the case of beam-riding missile guidance, the dynamical equations of missile motion were combined with kinematical equations of constraints thro- ugh the Maggi equations for non-holonomic systems (Ładyżyńska-Kozdraś et al., 2005;Menon et al., 2003). In the case of amissile homing onto amaneuve- ring target, the Bolztmann-Hamel equations for non-holonomic systems were used (Ładyżyńska-Kozdraś, 2008; Menon et al., 2003). As a result, a system of automatic stabilisations in the roll, pitch and yaw channels, respectively, is obtained. 368 E. Ładyżyńska-Kozdraś The approach to the problem of flying object control presented abovemay be used for a broad range of flying objects; i.e., missile, torpedo, plane or helicopter. Advantages of the approach are particularly visible in the case of a flying object with non-holonomic constraints imposed. 3. Real flight parameters for a sample missile The current parameters represent the real way the flying object behaves on its trajectory during the guidance process. Over the whole flight, the parameters are automatically registered and read out by the control system and depend only on the real behaviour of the object on its trajectory. In the three-point methods (e.g. beam-riding guidance) the missile flight is tracked from the earth, therefore, in this case the earth-fixed frame of re- ference O1x1y1z1 (Fig.1) is considered as the main one, relative to which all kinematical relations true for the currentphaseofmissile flight aredetermined. Fig. 1. Real parameters of the missile in the course of guidance When one employs the two-pointmethod (homing onto a target), themis- sile is homing basing on the information gathered without a delay from its own on-board equipment. In that case the main frame of reference ORxyz is fixed to the moving missile. The control laws having a form of kinematic relations... 369 It can be seen fromFig.1 that the position ofmissile body-fixed coordinate system ORxyz relative to the gravitational missile-fixed system ORxgygzg is determined unambiguously by the angles of roll φR, pitch θR and yaw ψR of themissile, respectively. At the same time the position ofmoving gravitational system relative to the earth-fixed one O1x1y1z1 is determined by the vector of actual missile position rR. Thereal linearvelocity of themissile in the system O1x1y1z1 canbewritten as follows (Fig.1) V RO = U1Ri1+V1Rj1+W1Rk1 (3.1) where U1R = ẋ1R V1R = ẏ1R W1R = ż1R (3.2) while in the missile body-fixed system ORxyz (Fig.1 and Fig.4) the velocity has the following components V RO = URi+VRj+WRk    UR VR WR    =ΛR    ẋ1R ẏ1R ż1R    (3.3) where the transformation matrix has the following form ΛR =    cosψRcosθR sinψRcosθR −sinθR l21 l22 sinφRcosθR l31 l32 cosφRcosθR    (3.4) where l21 =sinφRcosψR sinθR − sinψR sinφR l22 =sinφR sinψR sinθR +cosψRcosφR l31 =cosφRcosψR sinθR +sinψR sinφR l32 =cosφR sinψR sinθR − cosψR sinφR The angular velocity vector can be written as follows (Fig.1) ΩRO =PRi+QRj+RRk (3.5) where PR, QR, RR are angular velocities of roll, pitch and yaw, respectively. The components PR, QR, RR of instantaneous angular velocity are linear functions of the generalised velocities φ̇R, θ̇R, ψ̇R with the coefficients depen- ding on the generalised coordinates φR, θR, ψR    PR QR RR    =ΛΩR    φ̇R θ̇R ψ̇R    (3.6) 370 E. Ładyżyńska-Kozdraś with the transformation matrix of the following form ΛΩR =    1 0 −sinθR 0 cosφR sinφRcosθR 0 −sinφR cosφRcosθR    (3.7) In the case of homing, the angle of attack can be written as follows αR =arctan WR UR (3.8) and the sideslip angle is represented by βR =arcsin VR VRO (3.9) In the case of beam-riding guidance, the angle of attack equals αR =arctan W1R U1R −θR (3.10) while the sideslip angle reads βR =arcsin V1R VRO −ψR (3.11) 4. The preset parameters – kinematical missile-beam-target relations When applying the three-point method (e.g. beam-riding guidance of a mis- sile), the missile reaches the target provided that it is always illuminated by the missile-emitted beam, which means that the radar station (point O1), missile (point OR) and target (point OC) should be situated on the line of si- ght (Fig.2) (Dziopa, 2006; Etkin and Reid, 1996; Ładyżyńska-Kozdraś, 2006; Ładyżyńska-Kozdraś and Maryniak, 2003; Ładyżyńska-Kozdraś et al., 2005; Menon et al., 2003). The condition for target reaching (Fig.2) is VR > VC cosγCw cosγRw (4.1) where γCw, γRw stand for the angles defining the positions of target and missile, respectively, relative to the beam. The control laws having a form of kinematic relations... 371 Fig. 2. Missile and target trajectories within the guiding beam In the course ofmissile guidance itsmotion relative to the origin of guiding beam can be determined by the angular velocity equal to the beam angular velocity. Therefore, the angular velocities of pitch and yaw, respectively, of the beam determine the required flight corrections of the missile under control ε̇w = VC rC sinγCw cosηCw cosθw θ̇w = VC rC sinγCw sinηCw (4.2) The equations of beam position can be easily determined from the trigo- nometric relations (Fig.2), depending on the instantaneous target position εw =arctan y1C x1C θw =arcsin −z1C rC (4.3) The values of parameters preset in the control laws (Eqs. (2.1)-(2.4)) result in this case from the kinematical behaviour of the guidingbeam,which rotates about a fixed point depending on the target maneuvers relative to the earth- fixed frame of reference O1x1y1z1. Thus: • the vector of the preset missile position within the beam relative to the earth-fixed system O1x1y1z1 (Fig.2) can be written as follows rRz = √ x2 1Rz +y2 1Rz +z2 1Rz (4.4) 372 E. Ładyżyńska-Kozdraś where x1Rz = rRcosεw cosθw y1Rz =−rR sinεw cosθw (4.5) z1Rz =−rR sinθw • the vector of the presetmissile linear velocity within the beamunder the ideal guidance reads V Rz = ∂rRz ∂t + ∣ ∣ ∣ ∣ ∣ ∣ ∣ i1 j1 k1 θ̇w sinεw θ̇w cosεw −ε̇w rRcosεw cosθw −rR sinεw cosθw −rR sinθw ∣ ∣ ∣ ∣ ∣ ∣ ∣ = (4.6) = U1Rzi1+V1Rzj1+W1Rzk1 where U1Rz = ṙRcosεw cosθw −2rRε̇w sinεw cosθw −2rRθ̇w cosεw sinθw V1Rz =−ṙR sinεw cosθw −2rRε̇w cosεw cosθw +2rRθ̇w sinεw sinθw W1Rz =−ṙR sinθw −2rRθ̇w cosθw (4.7) • the vector of the presetmissile angular velocity can bewritten as follows ΩRz =ΛRz    θ̇w sinεw θ̇w cosεw −ε̇w    =    PRz QRz RRz    (4.8) where ΛRz is the transformationmatrix one can arrive at after replacing the index R with Rz in ΛR (Eq.(3.4)). The formulae for the preset angles of attack and sideslip during the beam guidance can be derived on the equilibrium condition for the forces acting upon themissile in the horizontal and vertical planes (Fig.3). Equation of equilibrium along the axis ORzA Pzz +TR sinαRz = mgcos(θw +γRw1) Equation of equilibrium along the axis ORyA Pyz = TR sinβRz While (Fig.2) γRw1 =arcsin rRθ̇w VR γRw2 =arcsin rRε̇w cosθw VR (4.9) The control laws having a form of kinematic relations... 373 Fig. 3. Missile trajectories in the vertical and horizontal planes – the preset parameters Then: — angle of attack αRz =arcsin mgcos ( θw +arcsin rRθ̇w VR ) − 1 2 ρSRV 2 RCz TR (4.10) — angle of sideslip βRz =arcsin 1 2 ρSRV 2 RCy TR (4.11) — pitch angle (Fig.3) θRz = θw +γRw1+αRz = (4.12) = θw +arcsin rRθ̇w VR +arcsin mgcos ( θw +arcsin rRθ̇w VR ) − 1 2 ρSRV 2 RCz TR —yaw angle (Fig.3) ψRz = εw+γRw2+βRz = εw+arcsin rRε̇w cosθw VR +arcsin 1 2 ρSRV 2 RCy TR (4.13) where ρ is the air density at a given altitude H =−z1R, (0¬ H ¬ 11000m) ρ = ρ0 ( 1− H 44300 )4.256 ρ0 – air density at the sea level SR – missile reference surface (maximum cross-section of its body) 374 E. Ładyżyńska-Kozdraś TR – missile engine thrust Pyz – resisting force Pzz – aerodynamic lift. 5. The preset parameters – kinematical relations between the missile and target when homing In the two-point missile guidance (homing onto a target), the constraints are directly imposed on the missile-target motion. Let us assume that we de- al with passive homing performed along a ”curve of pursuit” (Dziopa, 2006; Ładyżyńska-Kozdraś, 2006, 2008; Ładyżyńska-Kozdraś and Maryniak, 2002, 2003). In this method, the arrow of missile velocity is always directed at the target position (Fig.4). Fig. 4. Flight parameters of the homingmissile The changes in themissile-target distance rRC and the sight angle ν = θR can be written as functions of parameters of the missile-target motion in the following way The control laws having a form of kinematic relations... 375 ṙRC = VC cos(θR −θC)−VR ν̇ = 1 rRC VR sin(θR −θC) (5.1) VC – target velocity θC – target pitch angle ψC – target yaw angle. When themissile follows the target using the curve-of-pursuit homingme- thod, the preset parameters are those of the target, thus: • the preset angles of roll, pitch and yaw for themissile are equal to those for the target φRz = φC θRz = θC ψRz = ψC (5.2) • the components of thepresetmissile position relative to thegravitational system ORxgygzg are determined by the target position x1z = rRC cosψC cosθC y1z =−rRC sinψC cosθC (5.3) z1z =−rRC sinθC • the components of the preset linear velocity of the missile relative to its body-fixed system read, where ΛC is the transformationmatrix, can be found after replacing the index R with C in ΛR (Eq. (3.4))    URz VRz WRz    =ΛC    ẋ1C ẏ1C ż1C    (5.4) • the components of the preset angular velocity of the missile relative to its body-fixed system can be written as follows, where ΛΩC is the transformation matrix, can be determined by replacing the index R with C in ΛΩR (Eq. (3.7))    PRz QRz RRz    =ΛΩC    φ̇C θ̇C ψ̇C    (5.5) 6. Beam-riding guidance of an earth-to-air missile A simplified sample case of the beam-riding guidance of a Roland-class earth- to-air missile has been analysed (Ładyżyńska-Kozdraś et al., 2005; Menon 376 E. Ładyżyńska-Kozdraś et al., 2003; Zarchan, 2001). The preset parameters of the control laws are determined by the beam kinematical behaviour, the rotation of which about a fixed point depends on target maneuvers. During themissile flight, the current parameters of its flight are registered and compared to those preset, which have been determined by the beam trac- king the target. Therefore, the constraints are imposed by means of combing the motion of the line passing through the control point and the missile with motion of the guide beam. Since the missile control is preformed in the ψ yaw and θ pitch channels in terms of the control surface deflections δH and δV , the control laws given by Eqs. (2.1) and (2.2) should be transformed to assume the following form (assuming a prompt deflection of the control surfaces – no delay in the control system): — in the pitch channel δH = K H z (z1R −z1Rz)+K H U (ẋ1R − ẋ1Rz)+K H W(ż1R − ż1Rz)+ (6.1) +KHQ (QR −QRz)+K H θ (θR −θRz) — in the yaw channel δV = K V y (y1R −y1Rz)+K V V (ẏ1R − ẏ1Rz)+K V P (PR −PRz)+ (6.2) +KVR(RR −RRz)+K V ψ (ψR −ψRz) In the roll channel φ, the missile is automatically stabilised through aile- rons, while there is no control in the velocity channel since the control laws (Eqs. (2.3) and (2.4)) are neglected. Kinematical and geometrical parameters appearing in the control laws (Eqs. (6.1) and (6.2)) are shown in Fig.1 and described by Eqs. (4.1)-(4.8). Numerical simulation of a missile guidance onto to flying plane was per- formed. The equations of missile motions were derived from the Maggi equ- ations for non-holonomic systems (Ben-Asher and Yaesh, 1998; Etkin and Reid, 1996; Greenwood, 2003; Ładyżyńska-Kozdraś et al., 2005; Nizioł and Maryniak, 2005). The coefficients of amplification resulting from the integral criterion employed before (Eqs (2.5) and (2.6)) took the following values KHz =−0.00029 K H U =0.0007 K H W =0.00011 KHQ =−1.36 K H θ =−4.3 K V y =0.00007 KVV =−0.00054 K V P =0.0231 K V R =1.1 KVψ =−0.074 The control laws having a form of kinematic relations... 377 Sample simulation results shown in Fig.5, Fig.6 prove the efficiency of the missile guidance procedure based on the three-point-guidance method. Fig. 5. Flight path – the actual and preset ones, respectively Fig. 6. Histories of the missile elevator and rudder deflections 7. Curve-of-pursuit missile homing onto a manoeuvring target Aflightwas analysed of an air-to-air Sidewinder-classmissile under the curve- of-pursuit homingonto amaneuvering target (Ładyżyńska-Kozdraś, 2008;Me- non et al., 2003; Zarchan, 2001). In this case, the missile control is preformed in the ψ yaw and θ pitch channels in terms of the control surface deflections δH and δV , assuming a prompt deflection of the control surfaces - no delay in the control system. 378 E. Ładyżyńska-Kozdraś After some adaptation, the control laws (Eqs (2.1) and (2.2)) assumed the following form: — in the pitch channel δH = K H z (HR −Hz)+K H W(WR −Wz)+K H Q (QR −Qz)+ (7.1) +KHθ (θR −θz)+ δH0 — in the yaw channel δV = K V y (y1R −y1z)+K V W(W −Wz)+K V R(RR −Rz)+ (7.2) +KVψ (ψR −ψz)+ δV0 The kinematical and geometrical parameters appearing in the control laws (Eqs (7.1) and (7.2)) are shown in Fig.4 and represented by Eqs (5.1)-(5.5). The control laws (Eqs (7.1) and (7.2)) were considered as non-holonomic constraints imposeduponthemotionofmissile under control.Theequations of motionwere derivedusing theBoltzmann-Hamell equations for non-holonomic systems (Ben-Asher andYaesh, 1998; Etkin andReid, 1996;Greenwood, 2003; Ładyżyńska-Kozdraś, 2008; Nizioł andMaryniak, 2005). A Sidewinder-class missile was the case-study. Aerodynamical characteri- stics were determined and verified in terms of a non-controllablemissile. Upon application of the square control quality criterion (Eqs. (2.7) and (2.8)1) the coefficients of amplification appearing in the control laws (Eqs. (7.1) and (7.2)) took the following values KHθ =−0.84 K H W =−0.00005 KHz =0.00032 K H Q =0.0 KVψ =0.24 K V W =−0.0002 KVy =0.00014 K V R =0.0 Sample simulation results are shown inFig.7 andFig.8 which also present the trajectories of both the plane and themissile homing onto it. Themissile finally reaches the maneuvering target. 8. Conclusions The paper proves the efficiency of the applied general model of a flying object under control. The control laws assume form of kinematical relations between The control laws having a form of kinematic relations... 379 Fig. 7. Guiding performance of the missile onto a maneuvering target Fig. 8. Histories of the missile elevator and rudder deflections deviations, i.e. differences between thepreset and currentvalues of selected pa- rameters. The control laws formulated in thatwaymaybe successfully applied to investigations ofmotion of different types of flying objects; bothunmanned; like missile or torpedos and those with crew; like aircraft or helicopters. Depending on the problem to be solved and the type of flying object, the control laws may be reduced and adapted adequately. The advantages of the presented approach are particularly visible when dealing with systems with non-holonomic constraints. High efficiency of the method is revealed when applied to different case studies, which should be emphasised as well. Acknowledgement Thework has been part ofGrant sponsored by the State Committee for Scientific Research in the years 2008-2010. 380 E. Ładyżyńska-Kozdraś References 1. Ben-AsherJ.Z.,Yaesh I., 1998,Advances inMissileGuidanceTheory,AIAA series:Progress in Astronautics and Aeronautics, Reston, VA 2. Blakelock J.H., 1991,Automatic Control of Aircraft andMissiles, JohnWi- ley and Sons Inc., NewYork 3. DziopaZ., 2006,Wybranemetody sterowania rakietamiprzeciwlotniczymibli- skiego zasięgu,NIT – Nauka Innowacje Technika, Oficyna wydawnicza ”MH”, 12, 1 4. Etkin B., Reid L., 1996, Dynamics of Flight. Stability and Control, John Wiley and Sons Inc., NewYork 5. Greenwood D.T., 2003, Advanced Dynamics, Cambridge University Press, Cambridge, UK 6. Ładyżyńska-Kozdraś E., 2006, Prawa sterowania obiektów w ruchu prze- strzennym jakouchybymiędzy parametrami realizowanymi i zadanymi– proste i skuteczne zastosowania przy naprowadzaniu rakiet,Naukowe Aspekty Bezpi- lotowych Aparatów Latających, Zeszyty Naukowe Politechniki Świętokrzyskiej, Kielce 7. Ładyżyńska-Kozdraś E., 2008, Analiza dynamiki przestrzennego ruchu ra- kiety sterowanej automatycznie,Mechanika wLotnictwieML-XIII 2008, J.Ma- ryniak (red.), PTMTS,Warszawa 8. Ładyżyńska-KozdraśE.,Maryniak J., 2002,Dobór zadanychparametrów sterowania w ostatniej fazie lotu rakiety – samonaprowadzania się na manew- rujący cel, IV Międzynarodowa Konferencja Uzbrojeniowa ”Naukowe Aspekty Techniki Uzbrojenia”, Waplewo 9. Ładyżyńska-Kozdraś E., Maryniak J., 2003, Mathematical modeling of anti-aircraftguidancetomovingtargets,ConferenceProceedingTheFifth Inter- national Scientific And technical Conference, Cz.Niżankowski (Edit.), Tarnów- Zakopane 10. Ładyżyńska-Kozdraś E., Wolski K., Maryniak J., Sibilski K., 2005, Modelingofmotionofanautomatically controlledbeam-ridingguidedmissile in terms of theMaggi equations,AIAAAtmospheric FlightMechanics Conference and Exhibit, San Francisco, California 11. Maryniak J., 1987, Prawa sterowania jako więzy nieholonomiczne automa- tycznego sterowania śmigłowca,Mechanika Teoretyczna i Stosowana, 25, 1/2 12. Nizioł J., Maryniak J., red., 2005,Mechanika techniczna, Tom II – Dyna- mika układówmechanicznych, częśćV–Dynamika lotu, 363-472,Wyd.Komitet Mechaniki PAN, IPPTPAN,Warszawa The control laws having a form of kinematic relations... 381 13. Menon P.K., Sweriduk G.D., Ohlmeyer E.J., 2003, Optima fixe-interval integratedguidance-control laws forHitto-Kilmissiles,Proceedings of theAIAA Guidance, Navigation and Control Conference, AIAA 2003-5579CP 14. ZarchanP., 2001,Tactical and StrategicMissile Guidance, AIAA series:Pro- gress in Astronautics and Aeronautics, Reston, VA Prawa sterowania traktowane jako kinematyczne związki uchybów w automatycznym sterowaniu obiektów latających Streszczenie W pracy przedstawiono zastosowania praktyczne praw sterowania w dynamice obiektów latających. Rozpatrywane prawa sterowania stanowią kinematyczne i geo- metrycznezwiązkiuchybówparametrówrealizowanychi zadanychwynikającychz sys- temu naprowadzania badanego obiektu. Zadane parametry lotuwprowadzone zostały do praw sterowania jako parametrywynikające z lotu celu przy sterowaniu rakiet sa- monaprowadzających się, albo jakoparametry ruchuwiązki śledzącej cel. Rozważania przeprowadzono dla ogólnegomodelu dynamiki obiektu sterowanego. Manuscript received August 25, 2008; accepted for print December 30, 2008