Abstract
This chapter proposes a new optimal guidance law that directly utilizes, instead of compensating, the gravity for accelerating missiles. The desired collision triangle that considers both gravity and vehicle’s axial acceleration is analytically derived based on geometric conditions. The concept of instantaneous zero-effort-miss is introduced to allow for analytical guidance command derivation. The proposed optimal guidance law is derived by using the optimal error dynamics proposed in Chap. 2. The relationships of the proposed formulation with conventional PNG and guidance-to-collision (G2C) are analyzed and the results show that the proposed guidance law encompasses previously suggested approaches. The significant contribution of the proposed guidance law lies in that it ensures zero final guidance command and enables energy saving with the aid of utilizing gravity turn. Nonlinear numerical simulations clearly demonstrate the effectiveness of the proposed approach.
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References
Gazit R, Gutman S (1991) Development of guidance laws for a variable-speed missile. Dyn Control 1(2):177–198
Gazit R (1993) Guidance to collision of a variable-speed missile. In: Proceedings of the first IEEE regional conference on aerospace control systems. IEEE, Westlake Village, CA, pp 734–737
Shima T, Golan OM (2012) Exo-atmospheric guidance of an accelerating interceptor missile. J Franklin Inst 349(2):622–637
Reisner Daniel, Shima Tal (2013) Optimal guidance-to-collision law for an accelerating exoatmospheric interceptor missile. J Guid Control Dyn 36(6):1695–1708
Conn AR, Gould NIM, Toint PL (2000) Trust region methods, vol 1. SIAM, Philadelphia, PA
Kanzow Christian, Yamashita Nobuo, Fukushima Masao (2004) Levenberg-marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J Comput Appl Math 172(2):375–397
Zarchan P (2012) Tactical and strategic missile guidance. American Institute of Aeronautics and Astronautics, Reston, VA
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Appendix. Closed-Form Solution of Eqs. (6.16)–(6.18)
Appendix. Closed-Form Solution of Eqs. (6.16)–(6.18)
Noting from Eqs. (6.16)–(6.18) that the solution requires the integration of
where \(h\left( \gamma _{M} \right) \) is a function of secant and tangent functions. For an arbitrary function \(\beta \left( \gamma _{M} \right) \), we have
which reveals that the general solution of \(\int h\left( \gamma _{M} \right) \left| \sec \gamma _{M} +\tan \gamma _{M} \right| ^{n} d\gamma _{M} \) can be obtained by equalizing \(h\left( \gamma _{M} \right) \) with \(\frac{d\beta \left( \gamma _{M} \right) }{d\gamma _{M} } +n\beta \left( \gamma _{M} \right) \sec \gamma _{M} \).
Based on the properties of secant and tangent functions, the function \(\beta \left( \gamma _{M} \right) \) in solving Eq. (6.16)–(6.18) can be formulated as a general form as
where \(a_{i} \), \(i=1,2,3,4,5\), are unknown constant coefficients to be determined.
Differentiating Eq. (6.62) with respect to \(\gamma _{M} \) gives
Substituting Eqs. (6.62) and (6.63) into Eq. (6.61) yields
For Eq. (6.16), we have \(h\left( \gamma _{M} \right) =\sec ^{2} \gamma _{M} \), \(n=\kappa \). Comparing the coefficients results in
Solving Eq. (6.65) gives the unknown coefficients as
Then, the closed-form solution of Eq. (6.16) is given by
For Eq. (6.17), we have \(h\left( \gamma _{M} \right) =\sec ^{2} \gamma _{M} \), \(n=2\kappa \). Replacing \(\kappa \) with \(2\kappa \) in Eq. (6.65) leads to the closed-form solution of Eq. (6.17) as
For Eq. (6.18), we have \(h\left( \gamma _{M} \right) =\sec ^{2} \gamma _{M} \tan \gamma _{M} \), \(n=2\kappa \). Comparing the coefficients gives the following coupled equations
Solving Eq. (6.69) yields
which gives the closed-form solution of Eq. (6.18) as
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He, S., Lee, CH., Shin, HS., Tsourdos, A. (2020). Gravity-Turn-Assisted Optimal Guidance Law. In: Optimal Guidance and Its Applications in Missiles and UAVs. Springer Aerospace Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-47348-8_6
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DOI: https://doi.org/10.1007/978-3-030-47348-8_6
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