Guidance and Control Design for a Class of Spin-Stabilized Projectiles with a Two-Dimensional Trajectory Correction Fuze

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.


Introduction
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 (nosemounted canards, tail fins, and wedge-shaped paddle) [1][2][3][4][5][6][7][8][9], jet thrusters (gas or explosive thrusters) [10][11][12][13], and inertial loads (translating or rotating internal masses) [14,15].The first category of control mechanism presents some advantages over the other two: on one hand, the aerodynamic surfaces provide continuous time correction easier than jet thrusters; on the other hand, they potentially give the designer more control authority and bear more resemblance to missile trajectory control system.Furthermore, guided projectile concepts involving aerodynamic control devices can be divided into two categories: fin-stabilized (such as the US M712 Copperhead and the M982 Excalibur) and spin-stabilized (such as the US XM 1156 Precision Guidance Kit).Spin-stabilized guided projectiles are generally equipped with a roll-decoupled trajectory correction fuze, designed to provide range and lateral trajectory correction, while at the same time they rely on the high spin rate of the aft part for airframe stability.But the high spin rate creates an important coupling between the normal and lateral axes of the body, which makes the dynamic characteristic rather complex.
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 closeform 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 [16].Fresconi and Plostins studied various control mechanism strategies for spin-stability projectiles and gave the correction authority under different conditions [17].But this paper just introduced the strategies, and the conclusion could not be used in engineering application directly.
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.

International Journal of Aerospace Engineering
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 [18][19][20].The accuracy of the guidance is decided by the model of the projectile.If we want high accuracy, a complex model is needed, and the more state variables are acquired, the higher accuracy is got.The disadvantage of this method is that the amount of calculation is too large.In path following guidance, the current position of the projectile is compared with the nominal trajectory and a control is applied to bring the projectile to the nominal trajectory in some near future [5].A drawback with path following is that it only uses the position to correct the trajectory but it neglects the velocity, which means that the projectile will fly through the nominal trajectory and further corrections are needed.Trajectory integration is one of the impact points prediction, which makes use of numerical calculation of the trajectory to get the impact point.The predictive accuracy depends on the trajectory model, and the higher accuracy of the model, the higher precision of the predictive.The disadvantage of this method is that for long range shoots of the spin-stability the trajectory is complex, which will cause a large amount of calculation [21,22].Proportional navigation is a useful guidance method.But to our best knowledge, only projectiles with jet thrusters as actuators have taken this method [12,13], and if the method can be used for other two concepts is needed to be researched.Of course, some new methods are proposed for guidance [23,24].Perturbation based guidance for a generic 2D trajectory correction fuze is proposed by Robinson, which uses perturbation theory to predict state variables of the projectile [25].The advantage is that the onboard computer can compute the control signal quickly by the parameters loaded prior to flight, but there is still a drawback that the guidance algorithm only takes drag force in consideration.
In this paper, guidance and control algorithm for a class of spin-stabilized projectile with trajectory correction fuze is proposed.Section 2 presents the trajectory model of this projectile.Section 3 studies the response of trajectory correction.Section 4 introduces a new impact point prediction methods and its application.Section 5 describes the process of guidance and control, and conclusion is provided in Section 6.

Trajectory Model
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 1.
The definition for the roll angle of the correction part is also given in Figure 1.Define the roll angle to be 0 deg, when the steering canards are in horizontal plane and provide upward force, whereas when providing downward force, the roll angle is at the position of 180 deg.And the positive direction of the correction part is left, when looking from nose to the base.

Reference Coordinate System.
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  and  axes are in the horizontal plane and its -axis is normal to the x-z plane, pointing upward.The original is set at the end of the gun tube, and the axis points ahead along the launching direction, while the -axis points to the right when looking forward from the gun.
The quasi body reference frame Ox 4 y 4 z 4 is fixed to the projectile with its original at the projectile center of mass.Its  4 -axis aligns with the centerline of the projectile and points forward out of the nose, its positive  4 -axis to the right, and its  4 -axis upward.Note that  4 - 4 plane is fixed in the vertical plane, and it is more convenient to describe the motion of the projectile.The sequence of rotation from the ground launch reference is pitch , yaw , as is shown in Figure 2.

Trajectory Model.
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 ( 1)-( 4) represent the translational and rotational kinematic and dynamic equations of motion for a dual spin projectile, and all these equations are expressed in the quasi body reference frame: 4 ,  4 ,  4 are components of the total force expressed in the quasi body reference frame, V 4 , V 4 , V 4 are velocity vector components of the composite center of mass expressed in the quasi body frame,  4 ,  4 are components of the angular velocity vector expressed on the and -axis of the quasi body reference frame,  is the mass of the projectile, and  is the yaw angle: , ,  are position vector components of the composite center of mass expressed in the inertial frame and  is the pitch angle ,   are moments of both the front and the aft parts expressed on the -axis of the quasi body reference frame,   ,   are moments of the projectile expressed on the and -axis of the quasi body reference frame,   ,  4 are components of the angular velocity vector of both the front and the aft parts expressed on the  4 -axis of the quasi body reference frame,  4 ,  4 are components of the angular velocity vector expressed on the and -axis of the quasi body reference frame,   ,   are components of the total moment of both the front and the aft parts expressed on the -axis of the quasi body reference frame,   ,   are components of the total moment expressed on the and -axis of the quasi body reference frame, and   ,   are roll angles of the correction and body part: Loads on the composite projectile body are due to weight and aerodynamic forces.All the aerodynamic coefficients are acquired by numerical computing, and the forces as well as moments present as follows: International Journal of Aerospace Engineering is axial force coefficient,    is normal force derivative coefficient,    ,    are Magnus force coefficient of the correction and body part,  ℎ is roll moment coefficient of the front part,    is overturn moment coefficient,   ,   are roll damping moment coefficient of the correction and body part,   ,   are yaw and pitch rate damping moment coefficient,   ,   are Magnus moment aerodynamic coefficient of the correction and body part, ,  are attack and sideslip angles, ,  are reference area and length,  is diameter of the projectile, and V is the total velocity.The attack and sideslip angles are computed as follows: When the projectile is controlled, the correction part will be set at a given command angle relative to the ground launch frame, and the roll velocity of the correction part will be 0.So the control equations can be added as follows: where   and   are the roll angle and roll angle velocity components on the  4 -axis of the quasi body frame and   is the command angle.

Control Mechanism Strategy
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.

Dynamic Equilibrium Angle Caused by Correction.
When deducing the equation of angle motion, a few assumptions are invoked.
(1) The argument is changed from time  to arc length , and the arc length is defined as follows: (2) The attack and sideslip angles are small, so (3) Compared to V 4 ,  4 , and  ℎ , the quantities V 4 , V 4 ,  4 , and  4 are small, so that the total velocity Products of small quantities and their derivatives are negligible.
(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: Neglecting the products of small quantities, the derivation of ( 10) is that Let Δ =  − ,  =  4 +  4 ,   =   = ; substituting (5) into (13), after a series of deducing, the equation of angle motion is acquired: International Journal of Aerospace Engineering 5 G and  are the gravity term and control force term, respectively, and the meaning of the symbols is as follows: Coefficients frozen method is taken here, which means that all the aerodynamic coefficients will not change in the near future.So ( 14) is a linear differential equation, and its analytic solution can be obtained.But here the gravity term will not be considered, because this paper just studies the swerve response of correction control, so the following equation is got: All the following research is based on the assumption that the projectile is stable.From ( 16), we know that the angle motion caused by correction control can be divided into fast and slow modes of oscillation, and the oscillation will decay as the projectile flies downrange, because the real parts of the eigenvalues are negative.So the particular solution is only considered, which is called dynamic equilibrium angle caused by correction control.Define Δ 0 as a particular solution for ( 16), and 1 and  2 are roots for homogeneous equation, so Let   1   1  +   2   2  = 0 and differentiate Δ 0 .After deducing, we get Let  * =   − (  γ  +   γ  )/V.The quantity of  * is small and can be seen as a constant in a small time period.Integrating the   1,2 , we have Substituting ( 17) into ( 16), it gives

Swerve Response of Spin-Stability Projectiles.
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 International Journal of Aerospace Engineering mass trajectory.The motion which is perpendicular to the point mass trajectory is called the deviation motion [26].Define the components of the deviation motion as   and   , so we have Taking the arc length  as argument and taking γ  =   = 0, it gives Neglecting the third term which is small and substituting ( 21) into (23), it gives V  , V  and   ,   are the velocity and position components of the deviation motion, respectively.  is the command angle.Equation ( 24) can be changed into The assumption is invoked that velocity and aerodynamic coefficients of the projectile do not change in the small period.  is the component of the flying path on the x-axis of the ground frame, and  is the pitch angle, which is also assuming not change: Let Con = −(  +   − (  γ  /V))(((  −     ) +   (  γ  )/(V))/( + )) +   , and the real and imaginary parts are as follows: Figure 3: A class of spin-stabilized projectile with fixed canards.
Equation ( 24) can be changed into ( So we can get magnitudes and phase angle of the velocity and position of the swerve response as follows: = arctan ( Con real sin   + Con imag cos   Con real cos   − Con imag sin   ) . ( and   are magnitudes of the velocity and position of the swerve response, whereas   is the phase angle.

Phase Angle Analysis.
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 3. Muzzle velocity is 897 m-s −1 , and the mass is 45.5 kg.A simulation experiment is conducted for the phase angle analysis.
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 4.The four curves have the same changing trend.
Define   to be the phase shift, which is the angle between the command angle and the direction of the response, and the curves of   are shown in Figure 5.The four curves have small difference, and when the time  is 42.6 s, the biggest magnitude difference is got, which is about 3 deg.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   is only relative to velocity and command angle, the phase shift   is a function with Mach number as the argument.Take the method of polynomial fitting (Figure 6), and a formula is got: where Ma is on behalf of the Mach number.

Impact Point Deviation Prediction Based on Perturbation Theory
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.

Impact Point Prediction Based on Perturbation Theory.
0 ( 0 ,  0 ,  0 ) and   (  ,   ,   ) are the positions of the gun and target, respectively.Given the muzzle velocity, the trajectory impacting the target can be acquired definitely.For unguided projectiles the trajectory is the only one, because the range of elevation of the gun is restricted, and the trajectory is called nominal trajectory.Define   = Φ(, , , V  , V  , V  ) to be function of nominal trajectory, with  as argument and V  , V  , V  , ,  as state variables.
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  Im ( Im ,  Im ,  Im ).If  Im is close to target   , perturbation theory can be used to predict the impact point.(, , ) is any point on nominal trajectory, and  Im can be derived using Taylor expansion: ( DL and DH are the downrange and lateral deviation of the impact point.When using this method, note the following. (1) The impact point  Im should be close to the target   , and the disturbance magnitudes should not be big ones.

Numerical Method and Application
Taking analytic method to study the predictive accuracy is very hard, while numerical method is a useful way.Numerical method is taken in this paper, and application of partial derivative, nominal trajectory is researched.
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.

Numerical Computing of Partial Derivatives.
Finite difference method is invoked here.Take Φ  /V  and Φ  /V  as example.Add disturbance ΔV  on the state variable V  of any point on the nominal trajectory, and the impact point   (  ,   ,   ) is got.It gives The second and third order partial derivatives are also computed like this.
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 7, when velocity step and position step are 2 m-s −1 and 5 m, and rows of partial derivatives and nominal trajectory are 20 and 75, respectively.Compared with result of first order, result of second-order has big amplitude, and the convergence is slow.So first order partial derivatives is preferred.
Figure 8 shows the predictive results with different velocity steps and position steps, when rows of partial derivatives and nominal trajectory are 20 and 75, respectively.When position step is 5 m, results are almost the same.But when  velocity step is given as 2 m-s −1 , results have big difference after range reaches 22 km.That is because the impact point is far away from the target, since magnitude of position step is too big.Here we select that velocity step and position step are 2 m-s −1 and 5 m.
Table 1 shows twenty rows of partial derivatives of nominal trajectory, and values of each line are almost liner.Simulation results show that selecting 20 rows is suitable, since adding rows will not improve the accuracy of prediction.with range  as argument.In application standard value of state variables is acquired by interpolation to the nominal trajectory, and the error between standard value and real value will cause predictive error.Adding rows of nominal trajectory to reduce the range interval is an effective way to improve the predictive accuracy.Figure 9 shows the results of prediction with different rows of nominal trajectory, when using nominal trajectory as disturbance trajectory.The downrange predictive results change as half sinusoid, and the period is just the sampling period, which means that at the sampling point the predictive error is 0. The bigger the sampling period is, the bigger the predictive error is.When the rows reach 75, the downrange error is less than 0.8 m, and the lateral error is less than 0.1 m, whereas, it just occupies no more than 5 k of RAM.

Fading Memory
Filter.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: L and  L−1 are estimated values of the variable at the current and last instant, Δ *  is the measurement value,  ̂L  and  ̂L −1 are estimated values of first-and second-order derivatives,   is sampling instant,  Δ is gain factor, and  Δ and  Δ are gain coefficients.
The filter results of the predictive deviation are shown in Figure 10, and the results meet the requirements.

Prediction Accuracy.
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 11.Downrange predictive result reflects the real deviation before adding GPS errors, and the predictive error is not more than 2 m.After adding GPS errors, the downrange predictive results also can meet the accuracy requirement.For lateral prediction, till 40 s the convergence of prediction is got, and the results are aligned with the lateral deviation.Figure 11 presents that the lateral prediction results can describe the deviation; just the errors are a little big, but still not more than 15 m.
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.

Process of Guidance and Control
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 12.The key points are the command signal generation and strategy of guidance and control.

Command Signal Generation.
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 3, the pivotal problem to be researched is the relationship between changes of state variables and impact point derivation.
As is shown in Table 1, factors influencing predictive derivation are velocity and position deviations between the real trajectory and nominal trajectory, which are described in the gound launch frame.The dominating factors of downrange derivation are velocity components deviations on the -axis and -axis, and position components deviation on the -axis, whereas the lateral dominating factors are velocity components deviations on the -axis and -axis, and position components deviation on the -axis.But the changes caused by correction control, called correction values, are presented in the quasi body frame, and they are velocity and position components on the  4 -axis and  4 -axis, which are defined as ΔV 4 , ΔV 4 , Δ 4 , and Δ 4 .They need to be transformed and presented in the ground launch frame: ΔV  , ΔV  , ΔV  , Δ, Δ, and Δ are correction values presented in ground launch frame.Δ will be neglected, since it is small.Correction control is to make the real trajectory approach nominal trajectory to reduce the predictive derivation by correction values.DL and DH are the predictive impact point deviations of downrange and lateral direction.As is shown in Figure 13, the direction of   -axis and   -axis is downrang and lateral direction, respectively.Vector  im is the predictive derivation, and angle between the vector and   -axis is The direction of swerve response for correction control is the opposite direction of   ; that is Since phase lag   exits, the command signal can be computed as follows:  till the projectile impacts the ground.If correction control begins with 30 s, the largest downrange correction authority from far to target is about 260 m, and from near to target is about 300 m, whereas the lateral correction authority is about 260 m.As is mentioned in Section 5, the predictive derivations have high accuracy after 40 s, and the lateral derivation has predictive errors before 40 s, but it can still reflect the sign of the lateral derivation, which means that from the predictive derivation we can know the predictive impact point is on left or right of the target.Here the control strategy is divided into two segments: in ascending segment only lateral correction is invoked, while comprehensive correction is conducted in descending segment.The command signals in the two segments are shown as follows: In ascending segment  Figure 17 presents the result of predictive impact point deviation, and it shows that the two curves are close in upon the target in general trend.There is no gyroscope installed on the guidance and control system, so there is no angular feedback.The control force makes the sideslip angle dither, as is shown in Figure 18, and the control trajectory shows a small wobble, which is reflected in Figure 17.The magnitude of sideslip angle is about 10 deg, but after that it declines, which indicates in flight the projectile is kept in stable condition.
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 19.

Results of Monte Carlo
Targeting.Adding all kinds of disturbances, as is shown in Table 2, such as winds, muzzle velocity perturbation, and errors of launching elevation angle, miss distances without guidance and control are shown in Figure 20, which indicates that downrange miss distance is bigger than the lateral one.When guidance and control are taken, the miss distance is much smaller, and CEP is not more than 30 m. Simulation results show that control strategy based on impact point derivation prediction proposed in this

Figure 1 :
Figure 1: Sketch map of correction part.

Figure 7 :
Figure 7: Predictive results with different order partial derivatives.

Figure 8 :
Figure 8: Predictive results with different velocity and position step.

Figure 11 :Figure 12 :
Figure 11: Results of second-order fading memory filter when fire elevation is 45.5 deg.