Applications of Guidance

Transcription

1 Chapter 1 Applications of Guidance Module 11: Lecture 34 PN Based Impact Angle Constrained Guidance Keywords. Impact Angle Control, Orientation Guidance 1.1 PN Based Impact Angle Constrained Guidance In many advanced guidance applications (see the list of references at the end of the chapter) it is required to intercept the target from a particular direction, that is, achieve a certain impact angle. In this section we will cover an application of this nature based on the paper by Ratnoo and Ghose (008). Lu et al. (006) have solved the problem of guiding a hypersonic gliding vehicle in the terminal phase to a stationary target using adaptive guidance. In their law, the missile applies maximum lateral acceleration in a sense opposite to the sense of rotation of line of sight and orients itself in a feasible geometry for the PNG to achieve the desired impact angle. In surface-to-surface engagements with high heading errors such an approach is not feasible as applying maximum lateral acceleration in the initial phase of the missile flight will cause immense induced drag. Secondly, due to rotation of the missile velocity vector in a sense opposite to the rotation of line-of-sight will drive the missile away from the collision course and also increase the time of flight. 14

2 Guidance of Missiles/NPTEL/01/D.Ghose 15 In this section, a proportional navigation based guidance law is proposed for capturing all possible impact angles in a surface to surface planar engagement against a stationary target. The achievable set of impact angles is derived for pure proportional navigation guidance law with N. To achieve the remaining impact angles an orientation guidance scheme is proposed for the initial phase of the missile trajectory. The orientation guidance law is also proportional navigation with the navigation constant being N<and is a function of the initial engagement geometry and the desired impact angle. After following the orientation trajectory, the missile can switch over to a navigation constant N to achieve the desired impact angle. It is to be noted that varying the value of the navigation constant the proportional navigation guidance law gives a set of impact angles against stationary targets. However, studies on classical proportional navigation guidance (See Shneydor(1998)) reveal that, for N<, the missile lateral acceleration shoots to infinity as the missile-target range goes to zero. But by using N near interception in the proposed guidance law, we avoid the lateral acceleration command to shoot up to infinity and still achieve the desired impact angle. 1. Impact Angles Against a Stationary Target with PN Consider the planar engagement scenario as shown in Fig. 1.1 (a). The target is stationary and the guidance objective is to intercept the target along a desired impact angle denoted as α mf. Here α m and θ are the missile heading and the line of sight angle, respectively. Proportional navigation guidance law is defined as, m = N θ (1.1) Integrating (1.1), and knowing that for successful interception of a stationary target the final missile heading should be toward the target, i.e., α mf = θ f, we get α mf α m0 θ f θ 0 = N (1.) For the successful interception of a stationary target the final missile heading should point towards the target, i.e,

3 xx xx xx 16 Guidance of Missiles/NPTEL/01/D.Ghose Z u (am) vm Missile αm θ Missile O Launch Point Target αmf X αm0 xx xx xx xx Target αm0 xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx (a) (b) 3 αm=0, θ= π/ 1 Missile αm0, θ0=0 xxxxxxxxxxxxxxxxxxxxxxxxxxxxx 3 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxx θ π/ 0 θ. (c) (d) Figure 1.1: (a) Engagement geometry (b) PPN impact angle zone (c) Orientation trajectory (d) Line of sight rate variation on orientation trajectory

4 Guidance of Missiles/NPTEL/01/D.Ghose 17 Using (1.) and (1.3) θ f = α mf (1.3) N = α mf α m0 (1.4) α mf θ 0 Solving for the impact angle (α mf ) α mf = θ 0 (1 1 N ) α m0 (N 1) (1.5) Eqn. (1.5) shows that the final impact angle is a function of the navigation constant N, other parameters being constant for a given initial engagement geometry. The limiting impact angles using PPN are { θ0 α α mf = m0 if N = if N θ 0 (1.6) that is., α mf [θ 0 α m0, θ 0 ], N (1.7) The achievable impact angles using PN guidance for a surface to surface engagement (with θ 0 =0) lie in the shaded region as shown in Fig. 1.1 (b). Impact angles with N<satisfying (1.4) can not be achieved by PN guidance since the lateral acceleration demand goes to infinity near the interception (See Shneydor (1998)). 1.3 Orientation Guidance In a surface to surface engagement the desired set of impact angles should contain all angles from 0 to π. As shown above, classical PPN (N ) guidance will not cover the desired range of impact angles completely. For all impact angles outside the range given by (1.7) we propose an orientation guidance for initial phase of the missile flight. The missile follows the orientation trajectory as shown in Fig. 1.1(c) until the value of N satisfying the following relation becomes equal to. α mf α m θ f θ = N (1.8)

5 18 Guidance of Missiles/NPTEL/01/D.Ghose After which the missile follows PPN guidance with N =. As shown in Fig. 1.1(c) the achievable impact angle band using PPN guidance at the time of firing the missile is the shaded region 1. As the missile reaches point on the orientation trajectory, the achievable band shifts to the shaded region. The purpose of the orientation guidance is to eventually take the missile to the point 3 in Fig. 1.1(c). At point 3, θ = π and α m =0and thus using (1.7) we find that the impact band covers α mf = π to α mf = π as shown by the shaded region 3. The union of all shaded impact angle regions formed by tracing the orientation trajectory is α mf [0, π]. The properties of orientation guidance are obtained analytically in the next subsection Orientation guidance command For orientation guidance we propose the guidance law a m = Nv m θ (1.9) To execute the orientation maneuver, i.e., to take the missile from θ =0and α m = α m0 to θ = π and α m =0, we choose the navigation constant as Note that N = α m0 0 0 ( π ) = π α m0 (1.10) N (0, ) α m0 (0,π) (1.11) From (1.9) and (1.10), the orientation guidance command is given by a m = π α m0v m θ (1.1) 1.3. Properties of the orientation trajectory Using (1.1) we have, on the orientation trajectory m = a m = v m π α θ m0 (1.13) Integrating with respect to time, α m = π α m0θ + α m0 (1.14) Eqn. (1.14) relates the missile heading and the line of sight angle on the orientation trajectory.

6 Guidance of Missiles/NPTEL/01/D.Ghose 19 Proposition 1 On the orientation trajectory the line of sight rate θ <0 and the missile velocity vector rotation rate α m < 0 Proof. For a stationary target θ = v m R sin (α m θ) (1.15) Using (1.56) in (1.15) [ θ = v m R sin α m0 + ( ) ] π α m0 1 θ (1.16) θ <0, for all θ [0 π/], α m0 (0,π) (1.17) With θ = { v mr sin α m0 if θ =0 vm R if θ = π (1.18) The line of sight rate variation with respect to line of sight angle is shown with a solid line in Fig. 1.1(d). Also, using (1.13) From (1.19) and (1.17) m = π α θ m0 (1.19) m < 0, θ [0 π/], α m0 (0,π) (1.0) Proposition On the orientation trajectory, θ [ π/ 0] [θ α m θ] =[ π 0] Proof. Let q 1 = θ α m (1.1) q = θ (1.) Differentiating q 1 with respect to time 1 = θ ( ) m θ (1.3)

7 0 Guidance of Missiles/NPTEL/01/D.Ghose Using (1.13) in (1.3) 1 = θ ( ) π α m0 (1.4) Since θ <0 (Proposition 1) and α mo (0,π) 1 < 0 (1.5) Similarly for q, using (1.) and Proposition 1 = θ <0 (1.6) Let q 1i and q i be the values of q 1 and q at the initial point (see point 1 on the missile trajectory in Fig. 1.6 (c)) on the orientation trajectory. At the initial point θ =0and α m = α m0. Using (1.6) and (1.), we have q 1i = α m0,q i = 0 (1.7) Also, let q 1t and q t be the values of q 1 and q at the terminal point (see point 3 on the missile trajectory in Fig. 1.6 (c)) on the orientation trajectory. At the terminal point θ = π/ and α m =0. Using (1.6) and (1.), we have q 1t = π, q t = π/ (1.8) Thus θ [ π/ 0] [θ α m θ] = θ [ π/ 0] [q 1 q ] = [q 1i q i ] [q t q i ] [q 1t q 1i ]( 1 < 0, < 0) = [ α m0 0] [ π/ 0] [ π α m0 ]=[ π 0](1.9)