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A nonlinear suboptimal guidance law is presented in this paper for successful interception of ground targets by air-launched missiles and guided munitions. The main feature of this guidance law is that it accurately satisfies terminal impact angle constraints in both azimuth as well as elevation simultaneously. In addition, it is capable of hitting...
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... frame). Hence, in this research ^ a z and ^ a y are replaced by a z c and a y c , respectively, while generating the commands, where as they are replaced by a z and a y , respectively, while doing the simulation studies. This is done to validate the results in the presence of autopilot delays in the system. The engagement geometry is shown in Fig. 1 ...
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... without thrust are considered. Figure 5 Figure 10 shows a close-up of the same plot in the terminal engagement. Figure 11 shows guidance commands. ...
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... 5 Figure 10 shows a close-up of the same plot in the terminal engagement. Figure 11 shows guidance commands. Detailed results are given in Table 3. Final errors in terminal impact angles are achieved within the specified convergence tolerance of 0.4 It is to be noted that larger terminal impact angles, such as m f 180 deg :, are indeed capturable, but the algorithm takes considerable time to converge. ...
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... lateral acceleration are held as a z t a y t 0, 8 t > k and V m t V m k, 8 t k, and the miss distance is computed. Figure 12 depicts ZEM plots for a munition without thrust for capturing a target maneuvering sinusoidally with V t 20 m=s and with a lateral acceleration of the form a y t 2 g sin kt. It clearly shows that the APN law spends more energy than the MPSP guidance in terms of the lateral acceleration issued. ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. ...
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... parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. ...
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... errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. ...
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... deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It ...
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... give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history ...
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... 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be ...
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... criterion of comparison is also considered in this section. The integrals of the sum-of-squares of both the lateral acceleration commands, a z and a y , are of interest and are computed for the engagement scenario of Fig. 17 as follows: where t increases monotonically from 0 to t f . Figure 21 shows the plots of the integral of the sum-of-squares of both the lateral acceleration commands against each time instant during the whole flight. The final value at the terminal time can be obtained for the MPSP guidance as s t f 217:93, and that for the ...
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... integrals of the sum-of-squares of both the lateral acceleration commands, a z and a y , are of interest and are computed for the engagement scenario of Fig. 17 as follows: where t increases monotonically from 0 to t f . Figure 21 shows the plots of the integral of the sum-of-squares of both the lateral acceleration commands against each time instant during the whole flight. The final value at the terminal time can be obtained for the MPSP guidance as s t f 217:93, and that for the explicit guidance can be obtained as s t f 443:17. ...
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... be obtained for the MPSP guidance as s t f 217:93, and that for the explicit guidance can be obtained as s t f 443:17. The higher values for the explicit guidance can perhaps be attributed to the fact that the explicit guidance results in higher lateral acceleration in the initial phase of the flight to align the collision geometry (see Fig. 19). In turn, the slope of s t is steeper for explicit guidance in the initial phase of the flight. As compared with this, the nonlinear guidance appears to distribute the guidance effort more evenly due to its iterative nature. This results in a less steep slope of the graph of s t. A few remarks are in order before concluding the ...
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... targets, additional offline program- ming can render the performance of the explicit schemes quite well. In such scenarios, MPSP may not offer an additional advantage. However, although explicit guidance schemes are based on optimal control theory, they seem to spend more energy for highly maneuvering targets when compared with MPSP, as shown by Fig. 21. It should also be noted that APN (and a particular form of explicit guidance laws) does suffer from singularity (or very high value) of the guidance command towards the end of the engagement, a problem successfully attacked by MPSP owing to its iterative nature. Hence, for certain engagement scenarios, it is reasonable to pay the ...
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... subsection gives the simulation results for targets moving in a straight line. Vehicles without thrust are considered. Figure 5
Figure 10 shows a close-up of the same plot in the terminal engagement. Figure 11 shows guidance commands. Detailed results are given in Table 3. Final errors in terminal impact angles are achieved within the specified convergence tolerance of 0.4 It is to be noted that larger terminal impact angles, such as m f 180 deg :, are indeed capturable, but the algorithm takes considerable time to converge. Although the time to converge is shorter than the total time of flight, such results are not included in the results set. A number of observations can be made. MPSP guidance does not go through the sinusoidal pattern as that exhibited by APN. This feature of MPSP shows that it spends less overall control effort. Also, it is easier to track such commands as a z and a y using control surface deflections. MPSP guidance commands are thus less vulnerable to magnitude and rate ...
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... subsection gives the simulation results for targets moving in a straight line. Vehicles without thrust are considered. Figure 5
Figure 10 shows a close-up of the same plot in the terminal engagement. Figure 11 shows guidance commands. Detailed results are given in Table 3. Final errors in terminal impact angles are achieved within the specified convergence tolerance of 0.4 It is to be noted that larger terminal impact angles, such as m f 180 deg :, are indeed capturable, but the algorithm takes considerable time to converge. Although the time to converge is shorter than the total time of flight, such results are not included in the results set. A number of observations can be made. MPSP guidance does not go through the sinusoidal pattern as that exhibited by APN. This feature of MPSP shows that it spends less overall control effort. Also, it is easier to track such commands as a z and a y using control surface deflections. MPSP guidance commands are thus less vulnerable to magnitude and rate ...
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... concept of zero effort miss (ZEM) is a useful parameter when analyzing the performance of any guidance scheme. The definition of ZEM, as given by Zarchan [1], is the predicted final separation between the missile and its target, assuming zero acceleration from the current time. ZEM gives a fair estimate of how much guidance command is needed to align with collision geometry. Zero acceleration is perceived as no corrective action in this definition. In this paper, the same definition of ZEM is adopted with a slight addition in that the target is allowed to maneuver with lateral acceleration, a y t , and velocity, V t . ZEM at time instant, k, is computed as follows. Missile lateral acceleration are held as a z t a y t 0, 8 t > k and V m t V m k, 8 t k, and the miss distance is computed. Figure 12 depicts ZEM plots for a munition without thrust for capturing a target maneuvering sinusoidally with V t 20 m=s and with a lateral acceleration of the form a y t 2 g sin kt. It clearly shows that the APN law spends more energy than the MPSP guidance in terms of the lateral acceleration issued. Moreover, the profile of APN guidance is dependent on target ...
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... criterion of comparison is also considered in this section. The integrals of the sum-of-squares of both the lateral acceleration commands, a z and a y , are of interest and are computed for the engagement scenario of Fig. 17 as follows: where t increases monotonically from 0 to t f . Figure 21 shows the plots of the integral of the sum-of-squares of both the lateral acceleration commands against each time instant during the whole flight. The final value at the terminal time can be obtained for the MPSP guidance as s t f 217:93, and that for the explicit guidance can be obtained as s t f 443:17. The higher values for the explicit guidance can perhaps be attributed to the fact that the explicit guidance results in higher lateral acceleration in the initial phase of the flight to align the collision geometry (see Fig. 19). In turn, the slope of s t is steeper for explicit guidance in the initial phase of the flight. As compared with this, the nonlinear guidance appears to distribute the guidance effort more evenly due to its iterative nature. This results in a less steep slope of the graph of s t. A few remarks are in order before concluding the discussion on comparison. The explicit guidance laws offer a competitive solution, which is computationally simple yet effective, for many engagement scenarios. Also, for stationary targets, additional offline program- ming can render the performance of the explicit schemes quite well. In such scenarios, MPSP may not offer an additional advantage. However, although explicit guidance schemes are based on optimal control theory, they seem to spend more energy for highly maneuvering targets when compared with MPSP, as shown by Fig. 21. It should also be noted that APN (and a particular form of explicit guidance laws) does suffer from singularity (or very high value) of the guidance command towards the end of the engagement, a problem successfully attacked by MPSP owing to its iterative nature. Hence, for certain engagement scenarios, it is reasonable to pay the price of additional complex computations to achieve a good real time impact angle performance with as good of a range of accuracy, if not better, than the existing guidance ...
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... criterion of comparison is also considered in this section. The integrals of the sum-of-squares of both the lateral acceleration commands, a z and a y , are of interest and are computed for the engagement scenario of Fig. 17 as follows: where t increases monotonically from 0 to t f . Figure 21 shows the plots of the integral of the sum-of-squares of both the lateral acceleration commands against each time instant during the whole flight. The final value at the terminal time can be obtained for the MPSP guidance as s t f 217:93, and that for the explicit guidance can be obtained as s t f 443:17. The higher values for the explicit guidance can perhaps be attributed to the fact that the explicit guidance results in higher lateral acceleration in the initial phase of the flight to align the collision geometry (see Fig. 19). In turn, the slope of s t is steeper for explicit guidance in the initial phase of the flight. As compared with this, the nonlinear guidance appears to distribute the guidance effort more evenly due to its iterative nature. This results in a less steep slope of the graph of s t. A few remarks are in order before concluding the discussion on comparison. The explicit guidance laws offer a competitive solution, which is computationally simple yet effective, for many engagement scenarios. Also, for stationary targets, additional offline program- ming can render the performance of the explicit schemes quite well. In such scenarios, MPSP may not offer an additional advantage. However, although explicit guidance schemes are based on optimal control theory, they seem to spend more energy for highly maneuvering targets when compared with MPSP, as shown by Fig. 21. It should also be noted that APN (and a particular form of explicit guidance laws) does suffer from singularity (or very high value) of the guidance command towards the end of the engagement, a problem successfully attacked by MPSP owing to its iterative nature. Hence, for certain engagement scenarios, it is reasonable to pay the price of additional complex computations to achieve a good real time impact angle performance with as good of a range of accuracy, if not better, than the existing guidance ...
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... criterion of comparison is also considered in this section. The integrals of the sum-of-squares of both the lateral acceleration commands, a z and a y , are of interest and are computed for the engagement scenario of Fig. 17 as follows: where t increases monotonically from 0 to t f . Figure 21 shows the plots of the integral of the sum-of-squares of both the lateral acceleration commands against each time instant during the whole flight. The final value at the terminal time can be obtained for the MPSP guidance as s t f 217:93, and that for the explicit guidance can be obtained as s t f 443:17. The higher values for the explicit guidance can perhaps be attributed to the fact that the explicit guidance results in higher lateral acceleration in the initial phase of the flight to align the collision geometry (see Fig. 19). In turn, the slope of s t is steeper for explicit guidance in the initial phase of the flight. As compared with this, the nonlinear guidance appears to distribute the guidance effort more evenly due to its iterative nature. This results in a less steep slope of the graph of s t. A few remarks are in order before concluding the discussion on comparison. The explicit guidance laws offer a competitive solution, which is computationally simple yet effective, for many engagement scenarios. Also, for stationary targets, additional offline program- ming can render the performance of the explicit schemes quite well. In such scenarios, MPSP may not offer an additional advantage. However, although explicit guidance schemes are based on optimal control theory, they seem to spend more energy for highly maneuvering targets when compared with MPSP, as shown by Fig. 21. It should also be noted that APN (and a particular form of explicit guidance laws) does suffer from singularity (or very high value) of the guidance command towards the end of the engagement, a problem successfully attacked by MPSP owing to its iterative nature. Hence, for certain engagement scenarios, it is reasonable to pay the price of additional complex computations to achieve a good real time impact angle performance with as good of a range of accuracy, if not better, than the existing guidance ...
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... criterion of comparison is also considered in this section. The integrals of the sum-of-squares of both the lateral acceleration commands, a z and a y , are of interest and are computed for the engagement scenario of Fig. 17 as follows: where t increases monotonically from 0 to t f . Figure 21 shows the plots of the integral of the sum-of-squares of both the lateral acceleration commands against each time instant during the whole flight. The final value at the terminal time can be obtained for the MPSP guidance as s t f 217:93, and that for the explicit guidance can be obtained as s t f 443:17. The higher values for the explicit guidance can perhaps be attributed to the fact that the explicit guidance results in higher lateral acceleration in the initial phase of the flight to align the collision geometry (see Fig. 19). In turn, the slope of s t is steeper for explicit guidance in the initial phase of the flight. As compared with this, the nonlinear guidance appears to distribute the guidance effort more evenly due to its iterative nature. This results in a less steep slope of the graph of s t. A few remarks are in order before concluding the discussion on comparison. The explicit guidance laws offer a competitive solution, which is computationally simple yet effective, for many engagement scenarios. Also, for stationary targets, additional offline program- ming can render the performance of the explicit schemes quite well. In such scenarios, MPSP may not offer an additional advantage. However, although explicit guidance schemes are based on optimal control theory, they seem to spend more energy for highly maneuvering targets when compared with MPSP, as shown by Fig. 21. It should also be noted that APN (and a particular form of explicit guidance laws) does suffer from singularity (or very high value) of the guidance command towards the end of the engagement, a problem successfully attacked by MPSP owing to its iterative nature. Hence, for certain engagement scenarios, it is reasonable to pay the price of additional complex computations to achieve a good real time impact angle performance with as good of a range of accuracy, if not better, than the existing guidance ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. It is observed that the explicit guidance law with n 2 performs as well as the MPSP law. The terminal range for both of the guidance laws is less than 1 m, and the error in the terminal impact angles is less than 0.05 deg. for both guidance methods. It can be noted that the MPSP scheme converges in five iterations when explicit guidance serves as a guess history. However, MPSP takes seven iterations for the same engagement scenario when the guess history is generated using the PN-based guidance. It can be noted that the need for nonlinear guidance is not justified for this particular case, as explicit guidance laws can achieve good results. It is also of interest to compare the performance in the case of maneuvering targets. It is specified to It is observed that the range error of the explicit guidance law with n 2 is 0.26 m. The errors in the terminal impact angle m is 0:8 deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history when it successfully achieves the terminal conditions R miss 1 m, j m f m t f j 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be clearly observed by comparing Figs. 12 and 20 that explicit guidance is superior than ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. It is observed that the explicit guidance law with n 2 performs as well as the MPSP law. The terminal range for both of the guidance laws is less than 1 m, and the error in the terminal impact angles is less than 0.05 deg. for both guidance methods. It can be noted that the MPSP scheme converges in five iterations when explicit guidance serves as a guess history. However, MPSP takes seven iterations for the same engagement scenario when the guess history is generated using the PN-based guidance. It can be noted that the need for nonlinear guidance is not justified for this particular case, as explicit guidance laws can achieve good results. It is also of interest to compare the performance in the case of maneuvering targets. It is specified to It is observed that the range error of the explicit guidance law with n 2 is 0.26 m. The errors in the terminal impact angle m is 0:8 deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history when it successfully achieves the terminal conditions R miss 1 m, j m f m t f j 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be clearly observed by comparing Figs. 12 and 20 that explicit guidance is superior than ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. It is observed that the explicit guidance law with n 2 performs as well as the MPSP law. The terminal range for both of the guidance laws is less than 1 m, and the error in the terminal impact angles is less than 0.05 deg. for both guidance methods. It can be noted that the MPSP scheme converges in five iterations when explicit guidance serves as a guess history. However, MPSP takes seven iterations for the same engagement scenario when the guess history is generated using the PN-based guidance. It can be noted that the need for nonlinear guidance is not justified for this particular case, as explicit guidance laws can achieve good results. It is also of interest to compare the performance in the case of maneuvering targets. It is specified to It is observed that the range error of the explicit guidance law with n 2 is 0.26 m. The errors in the terminal impact angle m is 0:8 deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history when it successfully achieves the terminal conditions R miss 1 m, j m f m t f j 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be clearly observed by comparing Figs. 12 and 20 that explicit guidance is superior than ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. It is observed that the explicit guidance law with n 2 performs as well as the MPSP law. The terminal range for both of the guidance laws is less than 1 m, and the error in the terminal impact angles is less than 0.05 deg. for both guidance methods. It can be noted that the MPSP scheme converges in five iterations when explicit guidance serves as a guess history. However, MPSP takes seven iterations for the same engagement scenario when the guess history is generated using the PN-based guidance. It can be noted that the need for nonlinear guidance is not justified for this particular case, as explicit guidance laws can achieve good results. It is also of interest to compare the performance in the case of maneuvering targets. It is specified to It is observed that the range error of the explicit guidance law with n 2 is 0.26 m. The errors in the terminal impact angle m is 0:8 deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history when it successfully achieves the terminal conditions R miss 1 m, j m f m t f j 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be clearly observed by comparing Figs. 12 and 20 that explicit guidance is superior than ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. It is observed that the explicit guidance law with n 2 performs as well as the MPSP law. The terminal range for both of the guidance laws is less than 1 m, and the error in the terminal impact angles is less than 0.05 deg. for both guidance methods. It can be noted that the MPSP scheme converges in five iterations when explicit guidance serves as a guess history. However, MPSP takes seven iterations for the same engagement scenario when the guess history is generated using the PN-based guidance. It can be noted that the need for nonlinear guidance is not justified for this particular case, as explicit guidance laws can achieve good results. It is also of interest to compare the performance in the case of maneuvering targets. It is specified to It is observed that the range error of the explicit guidance law with n 2 is 0.26 m. The errors in the terminal impact angle m is 0:8 deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history when it successfully achieves the terminal conditions R miss 1 m, j m f m t f j 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be clearly observed by comparing Figs. 12 and 20 that explicit guidance is superior than ...
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... and sinusoidally maneuvering targets are considered. The parameters and the method of simulation is the same as that outlined in section IV. Figure 13 shows the engagement scenario for an unthrusted missile engaging with a stationary target. The corresponding histories of 3-D angles and lateral acceleration commands are shown in Figures 14 and 15, respectively. It is specified to achieve m f 20 deg : and m f 30 deg : with the initial conditions of m 0 m 0 10, x m 0 0, y m 0 z m 0 4000 m, x t 0 10000, y t 0 1000, and M 1:6. It is observed that the explicit guidance law with n 2 performs as well as the MPSP law. The terminal range for both of the guidance laws is less than 1 m, and the error in the terminal impact angles is less than 0.05 deg. for both guidance methods. It can be noted that the MPSP scheme converges in five iterations when explicit guidance serves as a guess history. However, MPSP takes seven iterations for the same engagement scenario when the guess history is generated using the PN-based guidance. It can be noted that the need for nonlinear guidance is not justified for this particular case, as explicit guidance laws can achieve good results. It is also of interest to compare the performance in the case of maneuvering targets. It is specified to It is observed that the range error of the explicit guidance law with n 2 is 0.26 m. The errors in the terminal impact angle m is 0:8 deg :, and that of m is 3:8 deg :. These errors are corrected by MPSP in nine iterations to give 0.7 m of range error and less than 0.05 of angle error in both the terminal angles. Figure 16 shows this engagement scenario for an unthrusted missile engaging with a sinusoidally maneuvering target. A close-up view in the terminal range is given by Fig. 17. The corresponding 3-D angle histories and lateral acceleration commands are given in Figs. 18 and 19, respectively. A more detailed comparison of the sinusoidal target maneuver can be found in Table 4, which shows that the nonlinear guidance performs better for steeper terminal values of m f for the initial conditions in question. It is noted that this table shows the results in which the explicit guidance serves as a guess history when it successfully achieves the terminal conditions R miss 1 m, j m f m t f j 2:5 deg :, and j m f m t f j 2:5 deg :. Other- wise, the APN is chosen as the guess history to enable MPSP to continue. The results show that the range of terminal impact angles obtained by the explicit law is narrower than those achieved by the MPSP scheme. Furthermore, the sinusoidal pattern can be seen in the time history of the 3-D angles in Fig. 18. As in the preceding sections, a ZEM plot similar to the APN guidance history is also presented in Fig. 20. It can be observed that the ZEM plot of the MPSP technique decreases almost everywhere monotonically, whereas the ZEM plot of the explicit guidance shows more dependence on the nature of the target maneuver. However, it can be clearly observed by comparing Figs. 12 and 20 that explicit guidance is superior than ...
Context 28
... a z c and a y c are the commanded lateral accelerations in the z and y directions, respectively (in the velocity frame). Hence, in this research ^ a z and ^ a y are replaced by a z c and a y c , respectively, while generating the commands, where as they are replaced by a z and a y , respectively, while doing the simulation studies. This is done to validate the results in the presence of autopilot delays in the system. The engagement geometry is shown in Fig. 1
...
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... In addition to the analytic guidance law mentioned above, some numerical optimization algorithms are also used to solve the guidance problem, such as model predictive static programming (MPSP). Dwivedi et al. used the MPSP algorithm to calculate the suboptimal solution of trajectory optimization in midcourse guidance [7,8], reentry ballistic guidance law of reusable space vehicle [9], impact-angle constraint suboptimal guidance [10], and unmanned aerial vehicle (UAV) autonomous landing [11]. The MPSP algorithm is essentially a Newton iterative algorithm [12]. ...
Missile interception problem can be regarded as a two-person zero-sum differential games problem, which depends on the solution of Hamilton-Jacobi-Isaacs (HJI) equation. It has been proved impossible to obtain a closed-form solution due to the nonlinearity of HJI equation, and many iterative algorithms are proposed to solve the HJI equation. Simultaneous policy updating algorithm (SPUA) is an effective algorithm for solving HJI equation, but it is an on-policy integral reinforcement learning (IRL). For online implementation of SPUA, the disturbance signals need to be adjustable, which is unrealistic. In this paper, an off-policy IRL algorithm based on SPUA is proposed without making use of any knowledge of the systems dynamics. Then, a neural-network based online adaptive critic implementation scheme of the off-policy IRL algorithm is presented. Based on the online off-policy IRL method, a computational intelligence interception guidance (CIIG) law is developed for intercepting high-maneuvering target. As a model-free method, intercepting targets can be achieved through measuring system data online. The effectiveness of the CIIG is verified through two missile and target engagement scenarios.
... The existing MPSP-based guidance methods are mainly focused on the angle constraint only. Oza [21] designed an angle-constrained guidance method for an air-to-ground missile and verified the feasibility of MPSP guidance. Maity [22] further introduced the static Lagrange multiplier in MPSP guidance, improving the computational efficiency. ...
... Refs. [21][22][23][24][25][26][27] verified that the MPSP algorithm is feasible for solving guidance problems online. However, during integral prediction, modeling errors will cause the accumulation of estimation errors, affecting the algorithm's performance and stability. ...
... (2) As one of the predictive guidance methods, MPSP-based guidance can avoid the model mismatch and thus can derive a better guidance performance, which has been verified in Refs. [21][22][23][24][25][26][27]. (3) The existing MPSP-based guidance methods are mainly focused on the angle constraint only. ...
This paper proposes a CKF-MPSP guidance method for hitting stationary targets with impact time and angle constraints for missiles in the presence of modeling errors. This innovative guidance scheme is composed of three parts: First, the model predictive static programming (MPSP) algorithm is used to design a nominal guidance method that simultaneously satisfies impact time and angle constraints. Second, the cubature Kalman filter (CKF) is introduced to estimate values of the influence of the inevitable modeling errors. Finally, a one-step compensation scheme is proposed to eliminate the modeling errors’ influence. The proposed method uses a real missile dynamics model, instead of a simplified one with a constant-velocity assumption, and eliminates the effects of modeling errors with the compensation scheme; thus, it is more practical. Simulations in the presence of modeling errors are conducted, and the results illustrate that the CKF-MPSP guidance method can reach the target with a high accuracy of impact time and angles, which demonstrates the high precision and strong robustness of the method.
... Missile has gradually become the main offensive weapon in the modern battlefield owing to its high speed, long range and high accuracy [1][2][3]. Therefore, positively developing interception technology for highly maneuvering targets has great significance to national defense and security [4]. At present, the proportional guidance law is widely used in the field of missile interception because of its simple structure and easy implement in practice [5]. ...
... where , k and are positive constants, function ) ( sat is introduced in [35]. Then, substituting (5), (6) and (7) into (3), and the robust sliding mode guidance law can be expressed as ...
... Theorem 1: Consider the three-dimensional intercepting mathematical model for maneuvering target denoted by (3). The robust guidance law (10), adaptive law (11) and weight update law of RBF neural network (13) guarantee that the line-of-sight angular rates L and L converge to a small neighborhood of zero in finite time. ...
This paper investigates the three-dimensional guidance and control problem of missile intercepting highly maneuvering target, whose acceleration information is difficult to accurately predict. With the three-dimensional guidance model for intercepting single target established by using the principle of zeroing the rate of line-of-sight (LOS), a novel intelligence guidance law has been designed through backstepping sliding mode control method, radial basis function (RBF) neural network and adaptive control technique. Then, a Lyapunov-based stability analysis demonstrates that all the signals are bounded, and the LOS rates ultimately converge to a neighborhood of the origin. Following advantages are highlighted in this paper: (i) the target information is online estimated and compensated by the RBF neural network, which indicates that the proposed guidance law is easily put into practice only relying on the position information of target. (ii) an adaptive gain term is designed in the control system, which greatly reduces the inherent chattering of sliding mode method. At last, simulations are conducted, and results illustrate the effectiveness and superiority of the designed guidance law.
... Therefore, these approaches need to deal with a large number of variables, leading to the increase of optimization scale and computation complexity. Another class of methods formulate the problem with the state error system, including model predictive static programming (MPSP) [21,22] and its variants, mainly the (trigonometric series-based) model predictive convex programming (MPCP) [17,23,24], and the linear pseudospectral model predictive control (LPMPC) [25,26]. These methods establish the sensitivity relation between state increments and control corrections, thus expressing state increments as a function of only control corrections rather than the control and state in state-space based methods. ...
... Note that the initial state increment 1 x can be determined via state estimation. In [21][22][23], 1 x is assumed zero since 1 ...
... CHARACTERISTICS OF COMPARED METHODS[17,18,[21][22][23][24][25][26][27][28][29] ...
This paper presents a pseudospectral convex optimization-based model predictive static programming (PCMPSP) for the constrained guidance problem. First, the sensitivity relation between the state increment and control correction is reformulated using Legendre-Gauss (LG) and Legendre-Gauss-Radau (LGR) pseudospectral transcriptions. Second, the convex optimal control problem associated with the trajectory optimization is defined by introducing the quadratic performance index. Third, modifications to the initial guess solution and reference trajectory update are introduced to enhance the accuracy and robustness of the algorithm. Finally, a model predictive guidance law is designed based on the proposed PCMPSP algorithm for the air-to-surface missile guidance with impact angle constraint. The simulation results show that the PCMPSP has lower sensitivity to the initial guess trajectory, higher accuracy, as well as faster convergence speed than existing convex programming methods. Moreover, the robustness of the proposed guidance law to uncertainties is demonstrated through the Monte Carlo campaign.
... A variety of challenging guidance problems have been solved using this technique as well that include guidance of missiles and re-entry vehicles. For ex. guidance of air-to-ground missiles [18] and re-entry guidance of re-usable launch vehicles [19] etc. ...
... Note that P i j ∈ R q×m . Next, from Eqs. (18) to (20) , after doing the required algebra, the inequality constraints can be represented in terms of control deviations as ...
... Both of these techniques minimize miss distance while also ensuring the desired impact angles at the collision. Model Predictive Static Programming (MPSP [18] ) is also designed for the same purpose. Whereas the philosophy of the BPN is largely inspired by the geometric analysis, the GENEX and MPSP are derived from optimal control theory problem formulations. ...
This paper proposes a high-precision three-dimensional nonlinear optimal computational guidance law in the terminal phase of an interceptor that ensures near-zero miss-distance as well as the desired impact angle. Additionally, it achieves these ambitious objectives while ensuring that the lead angle and lateral acceleration constraints are not violated throughout its trajectory. This ensures (i) the target does not escape the field of view of its seeker at any point in time (a state constraint) and (ii) it does not demand unreasonable lateral acceleration that cannot be generated (a control constraint). The guidance problem is formulated and solved using newly proposed Path-constrained Model Predictive Static Programming (PC-MPSP) framework. All constraints, both equality and inequality, are equivalently represented as linear constraints in terms of the errors in the control history, thereby reducing the complexity and dimensionality of the problem significantly. Coupled with a quadratic cost function in control, the problem is then reduced to a standard quadratic optimization problem with linear constraints, which is then solved using the computationally efficient interior-point method. Results clearly demonstrate the advantage of the proposed guidance scheme over the conventional Biased PN as well as the recently proposed GENeralized EXplicit (GENEX) guidance techniques. Numerical simulations with variation in initial conditions and Monte-Carlo simulations with parameter uncertainty demonstrates its the robustness of PC-MPSP guidance as well.
... where ε is a positive constant. Substituting (22) into (21), we have ...
... It can be found that dt f will reduce the performance index until the optimal solution is reached. Now the modified forms of control and terminal time have been obtained, as shown in (9) and (22), respectively. However, to figure out δu and dt f , the costate variable λ needs to be solved first. ...
... In this section, the guidance problem of air-to-ground missiles, which can be regarded as an OCP with unspecified terminal time, is used to test LCPM. The simulation example used is similar to that in reference [22], but the difference is that the terminal time in [22] is fixed, while the terminal time in our work is free. To verify the adaptability of the method in different situations, three different conditions are selected for simulation. ...
In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application.
... Model predictive static programming (MPSP) is a method for solving nonlinear OCPs proposed by Padhi et al [4]. This method can be used to solve the OCPs with hard terminal constraints and quadratic performance indexes, which has been applied to the guidance of air-to-ground missile [5]. Yang et al. proposed a linear Gauss pseudospectral model predictive control (LGPMPC) method, which can be used to solve the same problems [6]. ...
... The dynamic model of air-to-ground missile can be referred to [5]. ...
This paper aims to propose a method to solve the nonlinear optimal guidance problems with various performance indexes. First, the conjugate gradient method is derived based on the performance indexes of general form and specified form, and the execution steps of the optimal guidance method are obtained. Secondly, the pseudospectral collocation scheme is used to solve some intermediate variables in the algorithm, and the differential equations are transformed into algebraic equations, so as to improve the computational efficiency. Finally, the proposed method is applied to the air-to-ground missile guidance problem. By comparing with the simulation results of GPOPS-II, the optimality of the proposed method is verified. Additionally, the high computational efficiency of the algorithm is also verified.
... In [6], the handover of two interceptors with simultaneous cooperative interception was considered and the locations of the interceptors were designed by maximizing the overall engagement envelopes, but the distributions of the target movement information errors were not taken into account. For the design of cooperative guidance law, the main purpose is to guarantee that all the interceptors encounter the target at the same time or at a certain angle with respect to each other [7][8][9][10][11][12][13][14][15], and the miss distance of each interceptor is zero with respect to the predicted interception point. However, when all the interceptors are aiming at the same predicted interception point at the handover moment, the maximum maneuverable distance may be less than all the miss distances of the interceptors with respect to the target, which means that all the interceptors are unable to hit the target in the terminal guidance. ...
In this paper, a design approach for simultaneous cooperative interception is presented for a scenario where the successful handover cannot be guaranteed by a single interceptor due to the target maneuver and movement information errors at the handover moment. Firstly, the concepts of the reachable interception area and predicted interception area are introduced, a performance index function is constructed, and the probability of a successful handover is described by considering the coverage of the predicted interception area. Taking the probability of successful handover as a constraint, the simultaneous cooperative interception design problem is formulated based on area coverage. Then, an area coverage optimization algorithm is presented to design the spatial distributions of the interceptors. In order to enhance the handover probability, a simultaneous cooperative interception design approach is proposed to obtain the number of interceptors and the corresponding spatial distributions. Finally, simulation experiments are carried out to validate the effectiveness of the proposed approach.
... MPC has a wide area of application in both industrial and research community. For example, MPC was employed in power electronic devices in [Cortes et al. 2008;Kouro et al. 2009], heating systems in [Široký et al. 2011], process control in [Bartee et al. 2009;Mhaskar 2006], wind turbines in [Lio, Rossiter, and Jones 2014], marine surface vessels in [Fahimi 2007;Oh and Sun 2010], unmanned aerial vehicles in [Alexis, Nikolakopoulos, and Tzes 2011;Dalamagkidis, Valavanis, and Piegl 2011;Alexis et al. 2011;Bouffard, Aswani, and Tomlin 2012;Kang and Hedrick 2009;Eklund, Sprinkle, and Sastry 2012;Chemori and Marchand 2008], projectiles and missiles guidance in [Gross and Costello 2014;Oza and Padhi 2012], parallel robots in [Vivas and Poignet 2005], holonomic robots in [Kanjanawanishkul and Zell 2009] and nonholonomic robots with point-stabilization in [Gu and Hu 2005;Kuhne, Lages, and Silva 2005;Worthmann et al. 2016;Worthmann et al. 2015], trajectory tracking in [Mehrez, Mann, and Gosine 2013;Xie and Fierro 2008;Mehrez, Mann, and Gosine 2015;Dixon et al. 2000;Michalek and Kozlowski 2010;Lee et al. 2001] and path following prblems in [Aguiar, Hespanha, and Kokotović 2008;Nielsen, Fulford, and Maggiore 2010]. ...
Autonomous control and navigation of mobile robots received a lot of attention due to the ability of robots to carry out sophisticated tasks in a complex environment with a high level of precision and efficiency. The majority of control problems related to mobile robots involved go-to-goal, object tracking, and path following consist a target with pre-defined behavior. As such, the control design does not take into account the future behavior of the target. In surveillance, interception, pursuit-evasion problems, the future behavior of the target must be taken into consideration. These problems where the agent plays against an adversary are best tackled using game theory which provides the best strategy for winning. However, game-theoretic algorithms required a lot of information on the opponent to take into account the optimal strategy of the opponent, which is the worst-case scenario from the perspective of the player. This information requirement often restricts the application of game theory on mobile robots. Also, the majority of the works found in the literature proposed offline solutions applied to holonomic systems. This PhD thesis proposed three different solutions to non-cooperative game problems based on the opponent's information available to each player. The proposed solutions are online in nature with the ability to incorporate obstacles avoidance. Also, the controllers designed are applied on nonholonomic mobile robots first in a simulation then validated experimentally in a similar environment. In the first part of the work, the point-stabilization problem in a complex environment was handled using Nonlinear Model Predictive Control(NMPC) with Static and dynamic obstacles avoidance which revolves around the target position. Secondly, the problem was modified to involve a moving target that has a conflicting objective to form a pursuit-evasion game. The problem was solved using Nonlinear Model Predictive Control where two stabilizing approaches are compared. The NMPC method works such that only the current states of the opponent are known to each player. Next, a game-theoretic algorithm that requires all the information of the opponent was proposed. Next, another method called Limited information Model Predictive Control was proposed to solve the same problem where only the current position of the opponent is known. The methods are compared in terms of capture time, computation time, ability to incorporate obstacle avoidance, and robustness to noise and disturbance. A new problem that lies at the intersection between the point stabilization and pursuit-evasion problem was formulated and solved using game-theoretic model predictive control. The problem is called the Differential Game of Target Defense(DGTD) which involves interception of a moving object before reaching a static target. Finally, all the proposed controllers are validated experimentally using two mobile robots and the Motion Capture platform of the laboratory.
... Anti-Tank Guided Missiles (ATGM) are an essential and important element of arming most armies around the world, as they are one of the practical and relatively inexpensive means to defeat enemy armored forces in both defensive or offensive operations or even in special forces operations [1] . Anti-armor missiles could be often divided into 3 Generations or more, as the command to line of sight guidance law is the main concept for both first and secondgeneration ATGMs, the 1st and 2nd generation of ATGMs are flawed by the necessity to be a man in the loop during the flight period of the missile until it collides with the target, and the guidance trajectory enables missiles to hit the armor from the front and the sides where the armor is thicker and more resilient to missiles' strikes. ...
