A guidance and control strategy for a class of 2D trajectory correction fuze with fixed canards is developed in this paper. Firstly, correction control mechanism is researched through studying the deviation motion, the key point of which is the dynamic equilibrium angle. Phase lag of swerve response is the dominating factor for correction control, and formula is deduced with the Mach number as argument. Secondly, impact point deviation prediction based on perturbation theory is proposed, and the numerical solution and application method are introduced. Finally, guidance and control strategy is developed, and simulations to validate the strategy are conducted.
An important objective for future artillery projectiles is high delivery accuracy, in other words, improved aim to reduce round expenditure. In last few decades, several interesting concepts of guided projectiles were proposed, and they could be divided into three categories in terms of the control mechanisms employed: aerodynamic surfaces (nose-mounted canards, tail fins, and wedge-shaped paddle) [
Swerve response of the spin-stabilized is a key point of the dynamic characteristic to be studied, when control input is added. Ollerenshaw and Costello gave the close-form formulas for the magnitude and phase angle of a projectile excited by a control force in terms of fundamental and pointed out that spin-stabilized projectiles respond out of phase to control-force inputs forward of the center of pressure [
The trajectory correction fuze incorporates all the necessary electromechanical devices for guidance, navigation, and control system of the airframe. Due to the limit of the volume of the fuze, all the electromechanical devices must have high integration degree, which requires the guidance algorithm to have small amount of calculation but high accuracy.
The five main types of guidance most considered in guided ammunition contexts are trajectory shaping, model predictive guidance, path following, impact point prediction control, and proportional navigation. When only small correction to a trajectory is possible, only the later four are applicable. In model predictive guidance a model of the projectile and the environment is used in each update instant to compute a sequence of control actions, given the current state [
In this paper, guidance and control algorithm for a class of spin-stabilized projectile with trajectory correction fuze is proposed. Section
Trajectory correction fuze is attached to the projectile by screw thread, and by the bears in the fuze the guided projectile is divided into two parts: the front part called correction part and the aft part called body part. On the correction part, two sets of fixed canards are mounted; one has the same cant angle called steering canards to provide control force, and the other called rotation canards has the opposite cant angle to rotate the correction part by the rotational moment, as is shown in Figure
Sketch map of correction part.
The definition for the roll angle of the correction part is also given in Figure
Two reference coordinate systems are needed in the present analysis: ground launch system and quasi body system. The ground launch frame is an approximation of the earth inertial reference frame. It is a fixed, nonrotating frame, and it neglects the earth’s curvature and rotation. Its
The quasi body reference frame
Coordinate transformation relation.
During the flight, the two parts of the projectile spin in different directions due to the effort of the aerodynamic forces. In order to describe the motion of the projectile, three translational and four rotational rigid body degrees of freedom are introduced. The translational degrees of freedom are the three components of the mass center position vector. The rotational degrees of freedom are the Euler yaw and pitch angles as well as the correction part roll and body part roll angles.
Equations (
In this section, control mechanism strategy is researched. Firstly, formula for dynamic equilibrium angle caused by the trajectory correction is deduced based on the equations of the angle motion. Then, swerve response is studied by researching the deviation motion, and the formulas for the magnitude and the phase angle are acquired. Finally, formula for the phase angle of a spin-stability projectile with Mach number as argument is got.
When deducing the equation of angle motion, a few assumptions are invoked.
(1) The argument is changed from time
(2) The attack and sideslip angles are small, so
(3) Compared to
(4) The projectile is mass balanced, such that the centers of gravity of both the correction part and the body part lie on the rotational axis of symmetry:
Define
The point mass trajectory model is the first order approximation of the real trajectory, which only takes the gravity as well as the zero yaw axial force in consideration. In the flight, other aerodynamic forces and moments act on the projectile and make the flight contrail diverge from the point mass trajectory. The motion which is perpendicular to the point mass trajectory is called the deviation motion [
Define the components of the deviation motion as
The steering canards mounted on the fuze have fixed cant angle, and trajectory correction is realized by adjusting the roll angle of the correction part. So the key point of the control for this class of spin-stabilized projectile is the phase angle between command angle and the direction response.
Take a certain class of spin-stabilized projectile with fixed canards as example, as is shown in Figure
A class of spin-stabilized projectile with fixed canards.
The projectile was fired at elevation 45 deg, and standard meteorological condition was taken. In simulation correction control began with 10 s, and the command angle was set at 0 deg, 90 deg, 180 deg, and −90 deg, respectively. The results gotten are shown in Figure
Curves of phase angle.
Define
Phase shift.
So an assumption is applied, that phase shift for all the command angles is the same one if the state variables of the projectile are given. Since magnitude of
Polynomial fitting for phase shift.
In this section theory of impact point deviation prediction based on perturbation theory is presented firstly. Then the numerical method and application are introduced. At the same time, for projectiles using GPS receiver as the trajectory measurement tool, a second-order fading memory filter is designed. Finally, prediction accuracy is studied by simulation.
During flight disturbances always exit, and the projectile deflects from nominal trajectory, which causes the impact point deviation. Define the impact point of flight path influenced by disturbances to be
When using this method, note the following. The impact point Predictive accuracy should meet requirements.
In order to predict the deviation in real time, the guidance system needs to acquire the information of velocity and position; at the same time it should get the partial derivatives and disturbance magnitudes. In application, partial derivatives and nominal trajectory are loaded on the onboard computer prior to flight. The information of velocity and position can be acquired by global position system (GPS) receiver, or inertial measurement unit (IMU). The projectile mentioned above takes GPS receiver.
The unguided trajectory whose elevation is 45 deg is used as nominal trajectory, and the trajectory whose elevation is in the neighborhood of 45 deg is used as the disturbance trajectory. The advantage is that the impact point deviation is the same value along the trajectory.
Finite difference method is invoked here. Take
Here velocity step and position step are defined, which are velocity and position disturbance. In this part, how to select the velocity step, position step, and order of the Taylor expansion is presented. Select the trajectory whose elevation is 45.5 deg as disturbance trajectory.
The predictive results of downrange deviation using first and second-order partial derivatives are shown in Figure
Predictive results with different order partial derivatives.
Figure
Predictive results with different velocity and position step.
Table
Partial derivatives.
Range |
|
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|---|
5146.29 | 26.41 | 26.48 | −4.4 | 1.18 | −0.0013 | −0.31 | 1.25 | 47.98 | 0.069 | 0.99658 |
6707.75 | 27.53 | 25.95 | −4.45 | 1.04 | −0.0017 | −0.57 | 1.26 | 48.6 | 0.0689 | 0.99688 |
8144.93 | 27.8 | 25.87 | −4.06 | 0.86 | −0.0021 | −0.84 | 1.26 | 48.38 | 0.062 | 0.99718 |
9487 | 27.7 | 25.74 | −3.59 | 0.71 | −0.0024 | −1.05 | 1.22 | 47.5 | 0.055 | 0.99747 |
10753.4 | 28.7 | 26.12 | −3.33 | 0.66 | −0.0028 | −1.09 | 1.13 | 46.2 | 0.055 | 0.99775 |
11962.3 | 30.35 | 26.06 | −2.99 | 0.62 | −0.003 | −0.91 | 1.01 | 44.2 | 0.056 | 0.99801 |
13128.3 | 31.0 | 24.94 | −2.96 | 0.604 | −0.0035 | −0.71 | 0.86 | 41.7 | 0.0564 | 0.99825 |
14260.2 | 30.7 | 23.19 | −2.19 | 0.58 | −0.0039 | −0.46 | 0.75 | 39.2 | 0.057 | 0.99848 |
15363.2 | 29.74 | 21.79 | −1.6 | 0.57 | −0.0042 | −0.15 | 0.73 | 36.4 | 0.0565 | 0.99870 |
16440.2 | 28.25 | 20.2 | −1.4 | 0.55 | −0.0045 | −0.62 | 0.81 | 33.4 | 0.053 | 0.99892 |
17492.6 | 26.19 | 18.31 | −0.87 | 0.54 | −0.0048 | −1.03 | 0.88 | 30.5 | 0.0503 | 0.99913 |
18519.1 | 23.73 | 16.57 | −0.29 | 0.53 | −0.005 | −1.01 | 0.93 | 27.5 | 0.0495 | 0.99930 |
19518.4 | 21.6 | 14.4 | −0.33 | 0.52 | −0.005 | −0.82 | 0.84 | 24.5 | 0.0495 | 0.99944 |
20487.7 | 19.5 | 12.42 | −0.16 | 0.51 | −0.0058 | −0.66 | 0.76 | 21.5 | 0.049 | 0.999574 |
21422.9 | 17.04 | 10.39 | −0.07 | 0.5 | −0.006 | −0.54 | 0.66 | 18.6 | 0.0486 | 0.99968 |
22319.2 | 14.62 | 8.83 | −0.15 | 0.49 | −0.006 | −0.42 | 0.59 | 15.6 | 0.0479 | 0.999768 |
23171.8 | 12.0 | 6.82 | 0.028 | 0.48 | −0.006 | −0.33 | 0.45 | 12.7 | 0.0471 | 0.999838 |
23975.6 | 9.27 | 4.92 | −0.05 | 0.47 | −0.007 | −0.24 | 0.32 | 9.64 | 0.0463 | 0.999891 |
24726.5 | 6.32 | 3.42 | 0.17 | 0.46 | −0.007 | −0.16 | 0.23 | 6.51 | 0.0455 | 0.999926 |
25421.5 | 3.03 | 1.43 | −0.004 | 0.46 | −0.007 | −0.08 | 0.086 | 3.13 | 0.0449 | 0.999943 |
Nominal trajectory loaded on the onboard computer is discrete values with range
Figure
Predictive results with different rows of nominal trajectory.
Downrange predictive results
Cross range predictive results
GPS receiver has measurement error and will cause predictive error. So a filter is necessary to work in the loop to reduce error. Here second-order fading memory filter is taken, and the algorithm is as follows:
The filter results of the predictive deviation are shown in Figure
Results of second-order fading memory filter.
Downrange filter results
Cross range filter results
As is mentioned above, the unguided trajectory whose elevation is 45 deg is used as nominal trajectory, and the trajectory whose elevation is in the neighborhood of 45 deg is used as the disturbance trajectory. Here trajectory with elevation being 45.5 deg is the disturbance trajectory, and its impact point deviation is a constant, as is shown in Figure
Results of second-order fading memory filter when fire elevation is 45.5 deg.
Downrange filter results
Cross range filter results
Impact point deviation prediction based on perturbation theory for spin-stabilized projectiles has high prediction accuracy and has small calculation amounts. It is suitable for application on the trajectory correction fuze.
For the spin-stabilized projectile mentioned above, the onboard GPS receiver gets the velocity and position information of the real trajectory every 0.1 s and then onboard computer samples the information, predicts impact point deviation, generates command signal, and controls the actuator to make the correction part steady at a setting angle to correct the real trajectory, which is also a disturbance trajectory, as is shown in Figure
Process of guidance and control.
The objective of correction control is changing the state variables to make the predictive derivation 0 to realize impacting the target precisely, by adjusting the roll angle of steering canards mounted on the correction part. As correction control mechanism is studied in Section
As is shown in Table
Sketch map of predictive derivation.
As expected, Figure
Correction authority.
As is mentioned in Section
In ascending segment
In descending segment
In order to validate the effectiveness of the control strategy simulation of control trajectory was conducted with the elevation being 44 deg. From Figures
Range versus altitude.
Range versus cross range.
Figure
Predictive results.
Sideslip angle.
When an unguided projectile mentioned above is fired, range and cross range are all smaller than the two of nominal trajectory, and the downrange deviation takes the dominating part. So the phase angle of swerve response needs to be set in the neighborhood of 0 deg to correct the trajectory. Since phase lag exits, command angle should be set at the position ahead the phase angle of swerve response, as is shown in Figure
Curve of command angle.
Adding all kinds of disturbances, as is shown in Table
Disturbance parameters.
Parameter | Unit | Deviation value |
---|---|---|
Mass | kg | ±0.5 |
Length | mm | ±5 |
Moment of inertia | % | ±4 |
Muzzle velocity | m/s | ±5 |
Launching elevation angle | deg | ±0.06 |
Launching azimuth angle | deg | ±0.02 |
Wind velocity | m/s | ±2 |
Wind direction | deg | ±15 |
Results of Monte Carlo targeting.
Guidance and control strategy for a class of spin-stabilized projectile with 2D trajectory correction fuze is presented in this paper. Based on correction control mechanism study, control strategy using impact point derivation prediction based on perturbation theory is proposed. Simulation results show that the strategy designed is suitable for this class of projectiles, and the miss distance is reduced effectively.
The authors declare that there is no conflict of interests regarding the publication of this paper.