An adaptive proportional navigation guidance law for guided mortar projectiles

Transcription

1 JDMS Applications An adaptive proportional navigation guidance law for guided mortar projectiles Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 2016, Vol. 13(4) Ó The Author(s) 2016 DOI: / dms.sagepub.com Yongwei Zhang, Min Gao, Suochang Yang and Dan Fang Abstract Mortar projectiles have large launch angles and curved trajectories, which bring new problems to the application of proportional navigation guidance (PNG). In this article, the ballistic characteristics of mortar projectiles and their impact on the application of PNG are analyzed, and an adaptive proportional navigation guidance (APNG) scheme with a variable coefficient is proposed. The guidance performance of two guidance laws is compared through simulation. Results of the simulation show that the path command angle with PNG is different from the mortar path angle, which leads to poor guidance. On the other hand, the path command angle with APNG is similar to the mortar path angle and easy to track for the guided mortar projectiles, so the application of APNG can significantly improve the firing precision of guided mortar projectiles. The result of an example trajectory and Monte Carlo simulations testifies to the effectiveness of the APNG scheme. Keywords Guided mortar projectile, adaptive proportional navigation guidance, firing precision 1. Introduction Guidance law has an important impact on the firing precision of guided munitions. Proportional navigation guidance (PNG) is widely used in various types of guided munitions due to its computational simplicity, robustness, and implementability. 1 Many scholars have proposed a variety of modified forms of proportional navigation law for different needs or constraints, especially for the need against maneuvering targets and the impact angle constraint. Kim et al. 2 proposed a modified PNG with a timevarying bias using a nonlinear planar engagement and Lyapunov-like function. Kim et al. 3 proposed the biasshaping method for biased PNG with terminal-angle constraints. A two-phase guidance scheme for impact angle control on the basis of the conventional PNG was proposed by Erer and Özgören 4 and Ratnoo and Ghose. 5,6 Analysis of pure PNG against maneuvering targets was presented by Ghawghawe and Ghose 7 for a varying bounded target maneuver and by Guelman 8,9 for constant target maneuver. Ghosh et al. 10 proposed a variation of the pure PNG law, to account for target maneuvers, in a realistic nonlinear engagement geometry, and presented its capturability analysis. The guided mortar projectile in this paper is designed against nonmaneuvering targets, and the guided mortar projectile has large impact angles, so conventional PNG is taken as the guidance scheme. However, the ballistic characteristics of guided mortar projectiles bring new problems to the application of PNG. In this paper, the impact of mortar ballistic characteristics on the application of PNG is analyzed, and an adaptive proportional navigation Electronic Engineering Department, Shijiazhuang Mechanical Engineering College, Hebei, China Corresponding author: Yongwei Zhang, Electronic Engineering Department, Shijiazhuang Mechanical Engineering College, No.97 Heping West Road, Shijiazhuang, Hebei, , China. yongwei1112@sina.com

2 468 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 13(4) Figure 1. Schematic of the guided mortar projectile. guidance (APNG) scheme with a variable coefficient is put forward. The outline of the paper is as follows: Section 2 presents the guidance scheme of the guided mortar projectile. Section 3 presents the problems in the application of PNG. Section 4 proposes the APNG scheme. Section 5 describes the results of simulations, and a conclusion is provided in Section Guidance scheme The schematic of the guided mortar is shown in Figure 1. The projectile weight, diameter, mass center location from the nose tip, roll inertia, and pitch inertia are 15.0 kg, 120 mm, m, kg m 2, and 0.70 kg m 2, respectively. PNG is introduced for guidance in the longitudinal plane. The motion relationship model of the projectile and target is established in the longitudinal plane, as shown in Figure 2, where T is the target position, x T,y T are position vector components of the target, M is the projectile position, x m,y m are position vector components of the projectile, q is the line-of-sight (LOS) angle, θ is the path angle, v m is the projectile velocity, and r is the distance between the target and the projectile. The LOS rate is expressed as follows: ð _q ¼ x T x m Þv m sinðþ θ ðy T y m Þv m cosðþ θ r 2 ð1þ where _q is the LOS rate. The change rate of the path angle should be proportional to the LOS rate according to the definition of PNG, then: _θ ¼ k _q ð2þ where _θ is the change rate of the path angle and k is the navigation gain. Integrating Equation (2), we obtain the following: θ cx ¼ θ 0 þ kq ð q 0 Þ ð3þ where θ cx is the path command angle, θ 0 is the path angle when the guidance starts, and q 0 is the LOS angle when the guidance starts. The trajectory apex time is usually set as the guidance start time. The required overload in longitudinal plane can be expressed as follows: U θ ¼ Aðθ θ cx Þ ð4þ where A is the gain factor. It can be shown from Equation (4) that the objective of the guidance system is making the actual path angle θ equal to the path command angle θ cx. Equation (5) shows the guidance law used in the horizontal plane: U σ ¼ k z z m þ k vz v zm ð5þ where U σ is the required overload in the horizontal plane, z m is the horizontal position of the projectile, and v zm is the horizontal velocity of the projectile; k z and k vz are userdefined gains, and they are set as 1 in this paper. The control method of the canards used in the guided mortar projectile is letting the deflection angle of canards follow a sinusoidal signal of the projectile roll angle, 11 as shown in Equation (6): 8 < δ t ¼ δp 0 sinðγ þ π=2 φþ δ 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uθ 2 þ U σ 2 ð6þ : φ ¼ a tanðu σ =U θ Þ where δ t is the canards deflection angle, δ 0 is the amplitude of the canards deflection angle, u is the canards control phase, and γ is the roll angle. 3. Description of the problem The guided mortar projectile has large launching elevation angles; the range of the launching elevation angles of the guided mortar projectile is deg. The trajectory with the maximum launching elevation angle (80 deg) and the trajectory with the minimum launching elevation angle (45 deg) are shown in Figure 3. It can be shown from Figure 3 that the trajectory with the maximum launching elevation angle is more sharply curved than the trajectory with the minimum launching elevation angle. Figure 4 shows the change rate of the path angle of the trajectory with the maximum launching elevation angle and the trajectory

3 Zhang et al. 469 Figure 2. Relation schematic between the mortar projectile and the target. Figure 4. Path angle change rate. variation range of the change rate of the path angle has an important impact on the application of PNG. Suppose that the change rate of the path angle and the LOS rate of the actual trajectory have the following relationship: Figure 3. Ballistics of the mortar projectile. with the minimum launching elevation angle. The path angle change rate in Figure 4 starts from the trajectory apex time. It can be shown from Figure 4 that the change rate of the path angle of the trajectory with the minimum launching elevation angle has a small variation range: the change rate of the path angle at the trajectory apex time is 3.4 deg/s, and the change rate of the path angle before impact is 1.6 deg/s. However, the change rate of the path angle of the trajectory with the maximum launching elevation angle has a large variation range: the change rate of the path angle at the trajectory apex time is 14.3 deg/s, and the change rate of the path angle before impact is 0.3 deg/s. The large _θ ¼ k _q ð7þ where k# is the ratio between the change rate of the path angle and the LOS rate of the actual trajectory. The actual trajectory is the trajectory subject to initial value perturbations and atmospheric disturbances. Through integrating Equation (7), the actual path angle can be expressed as follows: θ ¼ θ 0 þ k ðq q 0 Þ Substituting Equations (3) and (8) into Equation (4), we obtain the following: U θ ¼ Aðk k Þðq q 0 Þ ð9þ It can be shown from Equation (9) that the difference between the navigation gain k and the ratio between the change rate of the path angle and the LOS rate of the actual trajectory k# has an important impact on the application of PNG; the navigation gain k should match k# to reduce the required overload. Two nominal trajectories, one with the minimum launching elevation angle and the other with the maximum launching elevation angle, are taken as examples to show the change trends of k# in flight, as shown in Figure 5. The k# in Figure 5 starts from the trajectory apex time. The nominal trajectory is the uncontrolled trajectory subject to no initial value perturbations and atmospheric disturbances.

4 470 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 13(4) Figure 5. Ratio of path angle change rate and line-of-sight rate. Figure 5 shows the following. 1) For the trajectory with the minimum launching elevation angle, the variation range of k# is from 2.4 at the trajectory apex time to 2.2 before impact. The small variation range of k# is advantageous for the application of PNG. For example, the navigation gain could be set as 2.3, then the required overload will maintain within a small range according to Equation (9). 2) For the trajectory with the maximum launching elevation angle, k# has a large variation range. k# decreases from 21.3 at the trajectory apex time to 1.2 before impact. The large variation range is disadvantageous for the application of PNG. The required overload will be large no matter what value the navigation gain is set at according to Equation (9), and this is the problem brought by the ballistic characteristic to the application of PNG in guided mortar projectiles. 4. Adaptive proportional navigation guidance An APNG law is designed to solve the problem described above. APNG is designed to let the navigation gain of guided mortar projectiles change according to their position instead of using the constant navigation gain, so that the navigation gain can match k# all along in flight to reduce the required overload and improve firing accuracy. The key idea of APNG is making the navigation gain equal to k# of the nominal trajectory all along in flight, that is: k ¼ k 0 n ð10þ Figure 6. Application process of adaptive proportional navigation guidance. where the subscript n indicates the nominal trajectory and kn 0 is the ratio between the change rate of path angle and the LOS rate of the nominal trajectory. APNG possesses several advantages compared with PNG. Firstly, the required overload is 0 for the projectile without disturbances according to Equations (9) and (10), so APNG is the most efficient for a projectile without disturbances. Secondly, the k# value of the actual trajectory with disturbances such as launch perturbations and atmospheric wind usually has little deviation with the nominal trajectory, so the required overload of the actual trajectory with APNG will be maintained within a small range according to Equation (9). The application progress of the APNG is shown in Figure 6. The application progress of APNG is as follows. 1) The ground computer calculates the nominal trajectory and obtains the ballistic position and corresponding k#. Determine a set of projectile positions X and the corresponding k# s K with equal time intervals. 2) Load the two-dimensional array [X K] to the onboard computer. 3) The guidance system gets real-time navigation gain k through the two-dimensional interpolation with the real-time position and the twodimensional array [X K]. It can be shown from the application progress of APNG that the computational burden of the onboard computer will not increase significantly compared to PNG, because only a two-dimensional interpolation algorithm is added.

5 Zhang et al. 471 Table 1. Initial conditions for example simulation. State Unit Unperturbed value Perturbed value Launching elevation angle deg Launching azimuth angle deg Initial velocity m/s Wind m/s 0 3 Wind direction deg Table 2. The two-dimensional array for adaptive proportional navigation guidance. Parameter Value X (m) K Figure 7. Altitude versus range graphs. PNG: proportional navigation guidance; APNG: adaptive proportional navigation guidance. 5. Results and discussion Some simulations have been done to verify the effectiveness of APNG. The performance of APNG will be compared with PNG in these simulations. 5.1 Example trajectory An example trajectory is simulated to demonstrate guidance performance. The horizontal plane is set as uncontrolled to stand out from the trajectory performance of APNG and PNG in the longitudinal plane. The nominal trajectory is simulated using the unperturbed value of initial conditions shown in Table 1. The uncontrolled trajectory and controlled trajectory are simulated using the perturbed value of initial conditions shown in Table 1. The two-dimensional array for APNG is shown in Table 2; the navigation gain decreases from 9.30 to 2.26 with the change of projectile position. The navigation gain for conventional PNG is set as 5.8, because a series of Monte Carlo simulations with different navigation gains indicate that PNG obtains the best result when the navigation gain is 5.8. The canards deflection angle is limited to 15 deg in the simulations. Figure 7 compares the trajectory response obtained using the different guidance schemes. The figure shows a nominal trajectory, an uncontrolled trajectory, a PNG controlled trajectory, and an APNG controlled trajectory. The miss distance is 86.5 m in the uncontrolled case. PNG and APNG significantly reduce miss distance from 86.5 m in

6 472 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 13(4) Figure 8. Comparison of path angle with proportional navigation guidance. Figure 10. The canards deflection histories with proportional navigation guidance. Figure 9. Comparison of the path angle with adaptive proportional navigation guidance. the uncontrolled case to 23.3 m and 0.8 m, respectively. Note that APNG has a better trajectory correction performance. Figure 8 shows the change of path angle with PNG. It can be shown from Figure 8 that the change trend of the path command angle with PNG is greatly different to the change trend of the uncontrolled path angle, where the path command angle with PNG decreases proportionally from the start of guidance; however, the controlled path angle cannot decrease proportionally due to the constraint of the trajectory correction capability of canards, so the controlled path angle cannot follow the change of the path command angle. Figure 11. The canards deflection histories with adaptive proportional navigation guidance. Figure 9 shows the change of path angle with APNG. It can be shown from Figure 9 that the change trend of the path command angle with APNG is similar to the change trend of the uncontrolled path angle, so the controlled path angle can follow the change of the path command angle easily. The difference between the controlled path angle and the path command angle is maintained at a small range. The canards deflection histories of the controlled trajectory with PNG are shown in Figure 10. The amplitude of the canards deflection angle maintains at 15 deg most of the time, because the controlled path angle is greatly different from the path command angle all along in flight. The control phase maintains at 180 deg during s, and maintains at 0 deg after s. The

7 Zhang et al. 473 Figure 12. Impact point dispersion. difference of the control phase before s and after s indicates the disadvantage of PNG, which will lead to the loss of trajectory correction capability of canards. Figure 11 plots the canards deflection histories of the controlled trajectory with APNG, where the amplitude of the canards deflection angle maintains at 15 deg during s, and reduces gradually after s. The control phase maintains at 0 deg most of the time. In contrast, APNG is more effective in making use of the trajectory correction capability of canards compared with PNG. 5.2 Dispersion simulations A Monte Carlo analysis was conducted to determine the closed-loop performance of APNG across a full spectrum of uncertainty in initial conditions. The guided mortar projectiles are launched at an altitude of 0 m toward a target on the ground with altitude and cross-range equal to zero at a range of 3000 m. APNG and PNG are included for the guidance in the longitudinal plane, and the guidance scheme expressed in Equation (5) is included for the guidance in the horizontal plane in the simulations. All initial conditions are taken to be independent normally distributed random variables, with mean values and standard deviations as given in Table 3. Wind direction is a uniform random variable between 0 and 2p. The signal noise and bias standard deviations are listed in Table 4. Figure 12 shows the distribution of impact points in two dimensions for a statistical sample of 200 trajectories. Figure 12(a) shows the impact point distribution of the uncontrolled case, and the Circular Error Probability (CEP) is 90.2 m. The CEP is defined as the least radius of

8 474 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 13(4) Table 3. Initial conditions and disturbances. Parameter Unit Mean value Std. deviation Launching elevation angle deg Launching azimuth angle deg Initial velocity m/s Wind m/s 0 2 Table 4. Signal noise and bias standard deviations. Signal Unit Noise std. deviation Bias std. deviation x m 8 4 y m 8 4 z m 8 4 v x m/s v y m/s v z m/s γ deg 5 3 the circle, the center of which is at the hit point of the reference trajectory, which contains 50% of all hit points. 12 The CEP of the controlled projectiles with PNG is reduced to be 21.8 m, as shown in Figure 12(b), while the CEP of the controlled projectiles with APNG is reduced to be 11.0 m, as shown in Figure 12(c). The greater reduction of CEP of the controlled projectiles with APNG indicates its effectiveness. 6. Conclusions Mortar projectiles have large launch angles and curved trajectories, which brings new problems to the application of PNG. APNG is put forward to solve the problem, which is designed to let the navigation gain of guided mortar projectiles change according to their position instead of using constant navigation gain to reduce the required overload and improve firing accuracy. The simulation of an example trajectory and Monte Carlo simulations were done to verify the effectiveness of APNG; the result of the example trajectory shows that APNG a has better trajectory correction performance, and the result of Monte Carlo simulations shows that the uncontrolled mortar projectiles have a CEP of 90.2 m; the CEP was reduced to 21.8 m with PNG and, as a contrast, the CEP was reduced to 11.0 m with APNG. The result of the example trajectory and Monte Carlo simulations testifies to the effectiveness of APNG. The proposed APNG is effective against a stationary target, but the proposed method is not feasible if each mortar projectile is to be fired toward a differently located or a moving target. Declaration of conflicting interest The authors declare that there is no conflict of interest. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. References 1. Murtaugh SA and Criel HE. Fundamentals of proportional navigation. IEEE Spectr 1966; A41: Kim BS, Lee JG and Han HS. Biased PNG law for impact angular constraint. IEEE Trans Aerosp Electron Syst 1998; 34: Kim TH, Park BG and Tahk MJ. Bias-shaping method for biased proportional navigation with terminal-angle constraint. J Guid Contr Dynam 2013; 36: Erer KS and Özgören MK. Indirect impact-angle-control against stationary targets using biased pure proportional navigation. J Guid Contr Dynam 2012; 35: Ratnoo A and Ghose D. Impact angle constrained interception of stationary targets. J Guid Contr Dynam 2008; 31: Ratnoo A and Ghose D. Impact angle constrained guidance against nonstationary nonmaneuvering targets. J Guid Contr Dynam 2010; 32: Ghawghawe SN and Ghose D. Pure proportional navigation against time-varying target maneuvers. IEEE Trans Aerosp Electron Syst 1996; 32: Guelman M. Proportional navigation with a maneuvering target. IEEE Trans Aerosp Electron Syst 1972; 8: Guelman M. Missile acceleration in proportional navigation. IEEE Trans Aerosp Electron Syst 1973; 9: Ghosh S, Ghose D and Raha S. Capturability of augmented pure proportional navigation guidance against time-varying target maneuvers. J Guid Contr Dynam 2014; 37: Zhang Y, Gao M, Yang S, et al. Optimization of trajectory correction scheme for guided mortar projectiles. Int J Aerosp Eng 2015; 2015: Article ID Pavković B and Pavić M. Frequency-modulated pulse-jet control of an artillery rocket. J Spacecraft Rockets 2012; 49: Author biographies Yongwei Zhang is a doctor in the Electronic Engineering Department at Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, China. Min Gao is a full professor in the Electronic Engineering Department at Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, China ( gaomin1103@yeah.net).

9 Zhang et al. 475 Suochang Yang is a full professor in the Electronic Engineering Department at Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, China ( yangsuochang_jx@sina.com). Dan Fang is a lecturer in the Electronic Engineering Department at Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, China ( fangdan1979_jx@sina.com).