This paper presents an optimum control scheme of firing time and firing phase angle by taking impact point deviation as optimum objective function which takes account of the difference of longitudinal and horizontal correction efficiency, firing delay, roll rate, flight stability, and so forth. Simulations indicate that this control scheme can assure lateral impulse thrusters are activated at time and phase angle when the correction efficiency is higher. Further simulations show that the impact point dispersion is mainly influenced by the total impulse deployed, and the impulse, number, and firing interval need to be optimized to reduce the impact point dispersion of rockets. Live firing experiments with two trajectory correction rockets indicate that the firing control scheme works effectively.

Dispersion characteristics of the trajectory correction projectiles can be dramatically improved by outfitting with a suitable trajectory correction flight control system. The commonly used executive organs are moveable canards [

The work reported here describes the implementation of a flight control system with impulse thrusters on a representative 122 mm artillery rocket. The flight control consists of a finite number of impulse thrusters mounted forward on the rocket body, computes position and velocity errors through comparing the position and velocity measured by GPS with prespecified (reference) trajectory, fires thrusters to change velocity direction, and assists the rocket to follow a prespecified trajectory.

Research and development on the use of impulse thrusters in order to improve the precision of projectiles has been going on for decades. Brandeis and Gill [

The thrusters’ application on rockets has been originally considered by Harkins and Brown [

Each impulse thruster imparts a single, short-duration, large force to the rocket in the plane normal to the rocket axis of symmetry, the control scheme of impulse thrusters mainly involves two aspects: the firing time and the firing phase angle. Many researchers [

In this paper, to reduce the impact point dispersion of the trajectory correction rocket using impulse thrusters, a 6-DOF trajectory model with lateral force is established, and then the control algorithm of firing time and firing phase angle is put forward, taking impact point deviation as optimum objective function.

The numerical simulation is based on a rigid body six degrees of freedom model typically utilized in flight dynamic analysis of rockets. Figure

Schematic of layout and lateral force of thrusters.

Layout of thrusters

Lateral force

The lateral force in quasibody reference frame can be described as

Transforming the lateral force from the quasi body to the aeroballistic reference frame, we have

The translational kinetic differential equations of the rocket in aero-ballistic reference frame are given in

Lateral moment of an impulse thruster in quasibody reference frame is given by

Other motion equations of the rocket not involving lateral forces or lateral moments can be obtained in [

Figure

Schematic of trajectory correction.

Because the aim of trajectory correction is to reduce the impact point deviation, it is foremost to make best use of the energy of impulse thrusters to obtain the minimum impact point deviation. Therefore, the remaining impact point deviation after an impulse thruster was activated, namely,

The optimization variables firing time and firing phase angle are relatively independent and assumed that the firing time of an impulse thruster has been identified; then the firing phase angle is the only optimization variable of an impulse thruster. To analyze the impact of firing phase angle on the trajectory correction performance, the lateral force is seen as a constant because the thrusters are active over a very short duration of time, and

Trajectory correction performance with equal and unequal converting coefficients.

With equal converting coefficients

With unequal converting coefficients

Given that the correction distance of an impulse thruster is proportional to the thruster impulse,

(1) When

(2) When

In fact,

Solution flow of

To reduce the number of iterations and improve the solution accuracy, solution limits can be acquired by (

The solution accuracy can reach 0.1 deg with

The firing phase angle, denoted by

Schematic of firing phase angle.

Converting coefficients versus flight time.

When the phase angle

After the optimization of firing phase angle, the other optimization variable of objective function is the firing time of thrusters. The general firing time control algorithm has two strategies.

(1) Time elapsed from the previous impulse thruster firing must be longer than a specified time duration

(2) If predicted impact point deviation is greater than a specified distance, activate the thruster as soon as possible.

The impact of flight time on trajectory correction performance of thrusters is not considered by strategy (2), which may lead to the thrusters being activated at the time when the correction efficiency is lower. Gao and Zhang [

As shown in Figure

Schematic of velocity increment of a thruster.

It can be known from (

To sum up, the firing time control algorithm can be improved as the following equation:

To investigate the correction performance of impulse thrusters and verify the effectiveness of the firing control scheme, some simulations of a rocket have been done by numerical integration of the equations described above using a fourth-order Runge-Kutta algorithm. The rocket configuration used in the simulation study is a representative 122 mm artillery rocket, 2.99 m long, fin-stabilized, with four pop-out fins on its rear part. The main rocket motor burns for 2.55 s and imparts an impulse of 54247 N-s to the rocket. During the main rocket motor burns, the forward velocity of the rocket is increased from 46.9 m/s to 935.7 m/s. The rocket weight, mass center location from the nose tip, roll inertia, and pitch inertia before and after burn is 66.1/43.0 kg, 1.43/1.21 m, 0.16/0.12 kg-m^{2}, and 48.42/36.36 kg-m^{2}, respectively. The rocket is launched at sea level toward a target on the ground with altitude and cross range equal to zero at a range of 28000 m. The thruster ring is assumed to be located at 0.869 m from the nose tip of the rocket and contains 50 individual thrusters where each individual thruster imparts an impulse of 15 N-s on the rocket body over a time duration of 0.02 s. The minimum firing interval of thrusters

The time-varying data of uncontrolled and controlled trajectories with optimum firing control scheme against a nominal command trajectory for the example rocket are compared in Figure

Correction performance of the optimum firing control scheme.

Total number of thrusters tired versus time

Total attack angle versus time

Horizontal impact point deviation versus time

Longitudinal impact point deviation versus time

Cross range versus range

Attitude versus range

Figure

It can be known from Figures

Figures

Figure

Initial conditions and disturbances.

Parameter | Unit | Mean value | Standard deviation |
---|---|---|---|

Launching elevation angle | deg | 42.51 | 0.2 |

Launching azimuth angle | deg | 0.1 | 0.2 |

Impulse of the main rocket motor | % | 100 | 0.1 |

Wind | m/s | 0 | 2 |

Impact point distribution.

Impact point distribution of uncontrolled rockets

Impact point distribution of rockets with general firing control scheme

Impact point distribution of rockets with the optimum firing control scheme

The result of Monte Carlo simulations is shown in Figure

Figure

The impact of parameters of thrusters on CEP and flight stability of rockets.

CEP versus number of thrusters and individual thruster impulse

CEP versus total impulse of thrusters and individual thruster impulse

Maximal total attack angle versus individual thruster impulse and minimum firing interval

Figure

Figure

Based on the above discussion, some suggestions are put forward to determine the thruster configuration parameters of trajectory correction rockets. Firstly, the total impulse should be determined according to CEP needed, because the impact point dispersion is mainly influenced by the total impulse of lateral thrusters deployed. Secondly, individual thruster impulse should be determined according to the total impulse, limits of layout space on the rocket, flight stability of rockets, cost, and so forth. If the individual thruster impulse is too small, there may be too many thrusters needed to be mounted on the rocket. On the other hand, if the individual thruster impulse is too large, the flight stability may deteriorate. Thirdly, the minimum firing interval should be determined according to the individual thruster impulse, flight stability of rockets, correction efficiency of thrusters, and so forth. If the individual thruster impulse is large, a relatively large minimum firing interval should be set to guarantee the flight stability of rockets. If the individual thruster impulse is small, a relatively small minimum firing interval time should be set to insure thrusters can be activated at the segment of trajectory while the correction efficiency is higher.

Live experiments have been done to verify the effectiveness of the firing control scheme. The characters of the trajectory correction rockets (TCR) have been launched same as the rocket described in simulations above. Two TCRs have been launched, toward a target on the ground with altitude equal to 200 m and cross range equal to zero at a range of 28000 m; the launching elevation angle was set to 43.6 deg, according to the weather of the time. Figure

The trajectory of two TCRs.

Attitude versus range

Cross range versus range

This paper establishes the 6-DOF trajectory model of a rocket with lateral force and presents an optimum control scheme of firing time and firing phase angle by taking impact point deviation as optimum objective function which takes account of the difference of longitudinal and horizontal correction efficiency, firing delay, roll rate, flight stability, and so forth. The result of Monte Carlo simulations shows that the uncontrolled rockets have a CEP of 359 m; the CEP was improved to 38 m with the general firing control scheme; as a contrast, the CEP was improved to 20 m with the optimum firing control scheme. The average thruster consumption and its standard deviation of the rockets with the general firing control scheme are 31.7 and 15.5, while the average thruster consumption and its standard deviation of the rockets with the optimum firing control scheme are 25.1 and 13.3. The decrease of CEP and the reduction of thruster consumption testify the effectiveness of firing control optimization. The variations of rocket impact point dispersion are analyzed with different impulse and number of impulse thrusters. It is shown that the impact point dispersion is mainly influenced by the total impulse of impulse thrusters deployed and steadily decreases as the total impulse is increased. The impulse, number, and firing interval need to be optimized to insure the flight stability of rockets and impulse thrusters activated at time when the correction efficiency is higher. Two trajectory correction rockets have been launched to verify the effectiveness of the firing control scheme; the final impact point deviations of two trajectory correction rockets are, respectively,

The authors declare that there is no conflict of interests regarding the publication of this paper.