Efficiency Analysis of Canards-Based Course CorrectionFuze for a 155-mm Spin-Stabilized Projectile

Eric Gagnon1 and Alexandre Vachon2

Abstract: There are many course correction fuze concepts for improving the precision of a spin-stabilized projectile. Some of them consist ina despun fuze equipped with canards. Canards provide continuous and, possibly, modulable maneuvering capabilities in crossrange anddownrange. This paper analyzes the efficiency of this type of course correction fuze and determines the best configuration for the canards.To do so, four concepts of canards-based course correction fuze are proposed and tested. To properly operate the fuzes, a guidance algorithm,based on point-of-impact prediction, and two autopilots, a poles/zeros cancellation controller and a proportional integrator controller, aredeveloped. The fuzes efficiency is studied with their control authority footprint and achieved performances during Monte-Carlo simulations.All the tests are done with a pseudo-seven-degrees-of-freedom simulator including the developed algorithms. Those tests demonstrate that thefour concepts significantly improve the precision of a spin-stabilized projectile and that, with the proposed algorithms, the best precision isobtained when the canards directly handle the projectile longitudinal acceleration. DOI: 10.1061/(ASCE)AS.1943-5525.0000634. © 2016American Society of Civil Engineers.

Introduction

In recent years, the mission of artillery has evolved from fire sup-port, where relatively large dispersion is allowed, to precise fire.Specially developed precision-guided munitions succeed quite wellat this new task. However, these specific designs represent high-cost options. An alternative strategy is to develop course correctionfuzes (CCF) for existing conventional shells to give maneuveringcapabilities to otherwise unguided projectiles.

The major dispersion of unguided projectiles being in thedownrange direction, some concepts of CCF are oriented to maketrajectory corrections along the longitudinal axis. These concepts(Hollis and Brandon 1999; Kautzsch and Reusch 2003) use dragbrakes, deployable surfaces which decrease projectile velocityand range. Crossrange corrections can be achieved by spinbrakes, devices that decrease the projectile spin rate (Hillstromand Osborne 2005). However, these spin brake concepts yieldonly small capabilities (Grignon et al. 2007). Also, actual dragand spin brakes are single shot deployable control surfaces.Therefore, with these devices, it is impossible to increase neitherthe range nor the drift and to compensate for perturbations occur-ring late in the flight.

An alternative existing CCF concept, providing downrangeand crossrange corrections, consists in a roll-decoupled finsring equipped with canards. Even if it has been demonstrated(Ollerenshaw and Costello 2008) that, for a spin-stabilized projec-tile, the ideal location for control surfaces is the projectile tail, theconcept of canards-based course correction fuze (CCCF) is still

widely studied. It appeared in the 1970s (Reagan and Smith 1975)and, with the evolution and miniaturization of relevant technolo-gies, has regained interest lately [J. A. Clancy et al., “Fixed canard2-d guidance of artillery projectiles,” U.S. Patent No. 6,981,672 B2(2006); Wernert et al. 2008; Bybee 2010]. It has been demonstrated(Gagnon and Lauzon 2008) that, even if it is less efficient than con-tinuous feedback control concept combining drag brake and spinbrake, roll-decoupled CCCF is an efficient way to increase projec-tile precision. Also, in opposition to drag and spin brakes, CCCFprovides continuous control, allowing accounting for disturbancesoccurring late in the flight. Furthermore, the canards can be actuatedin order to provide scalable forces when the predicted miss does notrequire large corrections. However, a force perpendicular to the spinaxis of a spin-stabillized projectile and applied in front of its centerof mass, like canards generate, results in out-of-plane swerve mo-tion (Ollerenshaw and Costello 2008). Worst, the projectile can bedestabilized if a large force is applied near its nose (Lloyd andBrown 1979). This paper specifically studies four different conceptsof CCCF in order to identify, under the proposed guidance andcontrol algorithms, the best choice. To properly operate the fuzes,guidance, navigation, and control (GNC) algorithms are required.

Projectile guidance algorithms usually fit in one of the threefollowing categories: trajectory shaping, trajectory tracking, or pre-dictive guidance. Trajectory shaping algorithms (Park et al. 2011)compute a new nonballistic trajectory. This method generally re-quires a high level of maneuverability, which is difficult to achievewith canards control on a spin-stabilized projectile. Trajectorytracking (Rogers and Costello 2010; Robinson and Strömbäck2013) consists in following the expected ballistic trajectory. Thistype of guidance is easy to implement, but tends to require highercontrol effort through the whole flight than predictive guidance(Teofilatto and De Pasquale 1998). Predictive guidance (Calise andEl-Shirbiny 2001; Fresconi 2011) uses a ballistic model of the pro-jectile to estimate its point of impact (PoI) and compute correctionsaccordingly. The precision of this method is, therefore, highly af-fected by the model (Fresconi et al. 2011). In order to include thedrifting motion of the spin-stabilized projectile in the trajectory pre-diction, the modified point mass model (Lieske and Reiter 1966) isused in the proposed predictive guidance algorithm.

1Defence Scientist, Weapons Systems, Defence Research and Develop-ment Canada, 2459 de la Bravoure Rd., Quebec City, QC, Canada G3J 1X5(corresponding author). E-mail: Eric.Gagnon@drdc-rddc.gc.ca

2Research Engineer, Numerica Technologies Inc., 3420 rue Lacoste,Quebec City, QC, Canada G2E 4P8. E-mail: Alexandre.Vachon.NUMERICA@drdc-rddc.gc.ca

Note. This manuscript was submitted on July 3, 2015; approved onMarch 9, 2016; published online on July 11, 2016. Discussion period openuntil December 11, 2016; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Aerospace Engineer-ing, © ASCE, ISSN 0893-1321.

© ASCE 04016055-1 J. Aerosp. Eng.

The first works studying spin-stabilized projectile control algo-rithms are based on decoupled lateral and vertical axis controllers,similar to missiles autopilot design, and use simple linear controltechniques: integrator on acceleration (Calise and El-Shirbiny2001) or loop-shaping on pitch/yaw rates (Gagnon and Lauzon2008). Recently, more-advanced control techniques, multivariatefiltered proportional-integral-derivative controller (Theodoulis et al.2013b) and H∞ synthesis (Theodoulis et al. 2015) on load factorsand pitch/yaw rates, and linear-quadratic synthesis (Fresconi et al.2015) on accelerations and pitch/yaw rates, are applied on a modelderived from the complete seven-degrees-of-freedom (DoF) dy-namic of the projectile. However, as the main objective of this workis not the development of a control algorithm, a simple and versatileautopilot is designed by poles/zeros cancellation on, similar to earlyworks, a linearized six-DoF model with decoupling of the lateraland vertical axes. Furthermore, as the concepts are not equippedwith inertial measurement devices, the control function is designedto manage only the accelerations. For the CCCF concept requiringprojectile drag control, a proportional integrator scheme is alsodeveloped.

The navigation function, which uses the sensor measurements toestimate the position, velocity, acceleration, and spin rate of theprojectile, does not differ from navigation algorithm of other flyingvehicles where various types of Kalman filter are used. Kalmanfilters are the subject of extensive research in the area of autono-mous and assisted navigation (Welch and Bishop 2006), and arealso used for guided projectile applications (Fairfax and Fresconi2012). This work does not focus on the navigation aspects and,therefore, employs only a linear Kalman filter to perform thenavigation task.

The objective of this paper being a relative comparison betweenthe CCCF concepts, the proposed GNC algorithms were chosen fortheir versatility and simplicity. They are not necessarily to the levelof the latest algorithms in their respective fields, but they are suit-able for the comparison. The obtained results are therefore nuancedwith respect to the algorithms. Those results are obtained by sim-ulations of a pseudo-seven-degrees-of-freedom (7DoF) simulatorof a spin-stabilized projectile equipped with the four studied CCCFconcepts. A pseudo-7DoF simulator is a 7DoF model (Costelloand Peterson 2000) in which one DoF, the fuze spin rate, is variedwithout any dynamics. Therefore, no control loop, as developed byTheodoulis et al. (2013a), is required for this channel.

Concepts Descriptions

The four concepts of course correction fuzes have a similar design.The fuzes are equipped with four identical canards mounted on adespun ring (Fig. 1).

All the canards have the same trapezium-shaped planar area anddiamond airfoil (Fig. 2). The fuze dimensions and weight reparti-tion are also identical for the four types of CCCF.

The ring is free to rotate, and its spin rate is controlled by modi-fying the counter-electromotive torque of an alternator. Since thealternator is a unidirectional actuator, an external torque is requiredto move the fins ring in the opposite direction. This external torqueis generated by the canards differential deflection. Before the start-ing point of the guided flight, the spin rate is held constant. There-fore, over a complete cycle, the canards do not generate a residuallifting force. During the guided portion of the flight, the spin rate isforced at 0 and, as a pseudo-7DoF model is used, the fins ring isimmediately positioned at the desired roll angle.

The fuzes are equipped with a global positioning system (GPS)receiver and an algorithm, which have the capability to determine

the spin rate and roll angle of the projectile. Combined with a rotaryencoder, the spin rate and roll angle of the fins ring can be esti-mated. As the GPS receiver measures the projectile position andvelocity only, a Kalman filter is required to estimate its accelera-tion. Furthermore, as no inertial measurements are available, thenavigation function cannot estimate the projectile pitch, yaw, andtheir rates.

The fuzes are installed on a typical 155-mm spin-stabilized shellto form the guided projectile. The aerodynamic coefficients of theshell and of the fins ring are computed independently. The shellcoefficients, obtained with PRODAS version 3.5.3, are stored inthree-dimensional tables, decomposed in terms of the Mach num-ber, total angle of attack, and aerodynamic roll angle. On their part,the fins ring coefficients form seven-dimensional tables obtainedfrom the subtraction of two sets of Missile Datcom (Rosema et al.2011) predictions, one set representing the projectile equipped witha CCCF and the other representing the projectile equipped with aconventional fuze. Therefore, the aerodynamic interactions be-tween the fins ring and the projectile body are encapsulated in thefins ring coefficients. The fins ring tables are function of the Machnumber, total angle of attack, aerodynamic roll angle, and fourcanards deflection angle. At very high deflection angles, requiredfor Concept 4, the aerodynamics exhibit a nonlinear behavior. Asthe simulator estimates the coefficients with linear interpolationbetween the flight condition points, the gap between consecutivedeflection angles has been tightened in order to capture, as muchas possible, this nonlinear behavior.

Changeable Roll Angle Ring Equipped with Four FixedCanards (Concept 1)

This concept is composed of four canards fixed at the same deflec-tion angle. In order to produce a torque around the projectile spin

Fig. 1. Front view of canards-based course correction fuze, showingpositive deflection angle

Fig. 2. Canards shape schematic

© ASCE 04016055-2 J. Aerosp. Eng.

axis which despins the fins ring, two opposite canards, Fins 2and 4, are deflected positively (Fig. 1). The two other canardsare deflected in opposite directions: Fin 1 is positively deflectedwhile Fin 3 deflection is negative. They produce a lifting aero-dynamic control force that is oriented by changing the fins ringroll angle, controlled by the counter-electromotive torque of analternator.

Changeable Roll Angle Ring Equipped with Two FixedCanards and Two Actuated Canards (Concept 2)

The only difference between this concept and the previous one isin the maneuvering canards, Fins 1 and 3. In this concept, theirdeflection angle is modulated by an actuator. In order to be coherentin the comparison, their maximal deflection angle is identical to thedeflection angle of Concept 1. Fins 2 and 4 remain fixed and the rollangle of the fins ring is still controlled by the counter-electromotivetorque of an alternator.

Regulated Roll Angle Ring Equipped with FourActuated Canards Having Similar Deflection Limits(Concept 3)

This concept is based on complete decoupling of vertical and lateralmaneuvers. One pair of actuated canards, Fins 1 and 3, generatesthe lateral maneuvers, while the other pair generates the verticalones. The canards of each pair are deflected in opposite directionsto produce an aerodynamic control force perpendicular to the pro-jectile spin axis. In order to be coherent in the comparison, themaximal deflection angle is similar to the deflection angle of Con-cept 1. The despun torque is obtained by a bias of 0.5°, in positivedirection, on the deflection angle of each canard. The counter-electromotive torque of an alternator is used to maintain the finsring roll angle at 0° once the fuze is activated.

Regulated Roll Angle Ring Equipped with FourActuated Canards Having Different DeflectionLimits (Concept 4)

This concept modulates the projectile drag by deflecting one pair ofcanards, Fins 2 and 4, to high deflection angles. By controlling thedrag, these canards provide maneuverability along the longitudinalaxis. Their maximal deflection angle is set at 44.5°, generating arange similar to those of the other concepts. The lateral maneuversare obtained by the other pair of canards, Fins 1 and 3, which aredeflected in opposite directions in order to generate an aerodynamiccontrol force perpendicular to the projectile spin axis. In order to beconsistent, their maximal deflection angle is identical to the deflec-tion angle of Concept 1. As for Concept 3, the despun torque isobtained by a bias of 0.5°, in a positive direction, on the deflectionangle of each canard. Once the fuze is activated, the fins ring ismaintained at 0° of roll angle by controlling the counter-electromotivetorque of an alternator.

Summary of Concepts Characteristics

Table 1 presents the canards deflection ranges and required actua-tors, canards deflection actuator (CDA), and counter-electromotiveactuator (CEA), for each concept. For the actuated concepts, thecanards of a pair (Fins 1 and 3 or Fins 2 and 4) are connectedon the same actuator and therefore move simultaneously. This workdoes not study how the required actuators can be included in thefuze, but assumes that it is feasible.

Autopilot Design

Concepts 1–3 use the same autopilot duplicated on two controlaxes. The first one manages the acceleration along the vertical axis,while the second autopilot works along the lateral axis. This auto-pilot is developed in order to manage the acceleration with a pair ofcanards generating a force perpendicular to the projectile spin axis.On its part, as it is equipped with a pair of canards configured inorder to generate a force parallel to the projectiles spin axis, thevertical autopilot of Concept 4 is replaced by an autopilot managingthe axial acceleration.

Lateral and Vertical Axes Autopilots

The first step in developing an autopilot is to obtain a mathematicalrepresentation of the projectile. As the autopilot has to work withthe projectile rotational and translation motions, a complete 6DoFnonspinning model (Wernert et al. 2010) is required. The proposedmodel includes the normal force (CNα

), Magnus force (CNpa),

pitching/yawing moment (Cmα), Magnus moment (Cmpa

), andpitch/yaw damping moment (Cmq). Assuming a symmetrical pro-jectile with small angle of attack and sideslip angle, the state spacerepresentation of the equations of motion is (Calise and El-Shirbiny2001)

266664α

β

q

r

377775¼

2666666664

Zα Zm 1 0

−Zm Zα 0 −1Mα Mm Mq −pIx

Iy

Mm −MαpIxIy

Mq

3777777775

266664α

β

q

r

377775þ

266664

0 Zc

Zc 0

0 Mc

−Mc 0

377775�fy

fz

�

ð1Þwhere

Zα ¼ − qsrCNα

mjvj ð2Þ

Zm ¼qsrdrpCNpa

mjvj2 ð3Þ

Mα ¼qsrdrCmα

Iyð4Þ

Mm ¼qsrd2rpCmpa

Iyjvjð5Þ

Mq ¼qsrd2rCmq

Iyjvjð6Þ

Zc ¼1

mjvj ð7Þ

Table 1. Actuators and Canards Deflection Ranges for Each Concept

Concept Actuators

Canards deflection ranges

Fin 1 [°] Fin 2 [°] Fin 3 [°] Fin 4 [°]

1 1 CEA 8 8 −8 82 1 CEA, 1 CDA ½−8, 8� 8 ½−8, 8� 83 1 CEA, 2 CDAs ½−7.5; 8.5� ½−7.5; 8.5� ½−7.5; 8.5� ½−7.5; 8.5�4 1 CEA, 2 CDAs ½−8, 8� [0.5, 44.5] ½−8, 8� [0.5, 44.5]

© ASCE 04016055-3 J. Aerosp. Eng.

Mc ¼lfIy

ð8Þ

and sr and dr are the referential area and referential diameter of theprojectile, respectively; m its mass; v its velocity vector; p its spinrate; Ix its inertia around its spin axis; Iy its inertia around its pitch/yaw axis; q is the dynamic pressure; lf is the distance, along theprojectile spin axis, between the fins center of pressure and projec-tile center of gravity; and inputs fy and fz are the magnitudes of thelateral and vertical aerodynamic control forces.

The controlled variables being the lateral and vertical acceler-ations, the output equation of the state space representation is

�ay

az

�¼�−Zmjvj Zαjvj 0 0

Zαjvj Zmjvj 0 0

�26664α

β

q

r

37775þ

�Zcjvj 0

0 Zcjvj

��fy

fz

�

ð9Þ

An analysis of the Relative Gain Array supports the pairing ofthe vertical force with the vertical acceleration, and the lateral forcewith the lateral acceleration. The cross axis relative gains beingsmaller than 0.33 for all the domain of operation, a decentralizedcontroller using direct axis is then designed. A multivariable con-troller, as in Theodoulis et al. (2015), or the implementation ofdecouplers (Gagnon et al. 1998) might improve the autopilot per-formance, but they are not mandatory. Furthermore, the transferfunctions of the dominant pairings are identical; an autopilot canthen be designed for one axis and duplicated on the other.

The aerodynamic coefficients and, hence the Z andM variables,vary as a function of the projectile airspeed, spin rate, and altitude.Therefore, the autopilots must be scheduled to match the modelvariations generated by these three parameters.

The vertical axis transfer function, extracted from the state spacerepresentation, can be represented by the following transferfunction:

GpzðsÞ ¼ az

fz¼ðs2 þ 2ζ3ω3sþ ω2

3Þð sω4þ 1Þð− s

ω5þ 1Þ

ðs2 þ 2ζ1ω1sþ ω21Þðs2 þ 2ζ2ω2sþ ω2

2Þð10Þ

Here, the location of the poles and zeros is interesting. The com-plex zeros, defined by their natural frequency, ω3, damping factor,ζ3, and a pair of complex poles (ω1 and ζ1) have a high naturalfrequency and are located near each other. The other complex poleshave a much smaller natural frequency (ω2).

The proposed autopilot is obtained by poles/zeros cancellationenhanced by an integrator to ensure no static error. As the high-frequency poles and zeros are much higher in frequency than theachievable closed-loop dynamics and are close enough to naturallycancel each other, they are omitted in the cancellation process. Theunstable zero cannot be canceled, but a stable pole is added at thesame frequency to reduce high-frequency noise. The resulting auto-pilot is therefore

GczðsÞ ¼fz

acz − az¼ kcðs2 þ 2ζ2ω2sþ ω2

2Þ�sω4þ 1

��sω5þ 1

�s

ð11Þ

The autopilot gain (kc) is chosen to obtain a phase margin of60°. Gain margin is not enforced during autopilot design, but ananalysis of the controller shows that, for all the design points,the margin is always greater than 6 dB. The combination of bothmargins ensures some robustness properties.

The projectile being considered symmetrical, and the lateral axisautopilot is identical

GcyðsÞ ¼fy

acy − ay¼ kcðs2 þ 2ζ2ω2sþ ω2

2Þ�sω4þ 1

��sω5þ 1

�s

ð12Þ

The autopilots are scheduled to match the variations of the Z andM variables due to the projectile airspeed, spin rate, and altitudeevolution. Those variations are visible in the transfer function gain,and in the natural frequency and damping factor of the poles andzeros. Autopilot sequencing is done by linear interpolation of thosevariables (kc, ω2, ω4, ω5, and ζ2) between the selected designpoints. Ten points, evenly spaced between the minimal and maxi-mal values of the conventional projectile flight envelope, are se-lected for each of the three sequencing parameters. With thisapproach, all the parameters combinations must be computed, evenif some are not expected to happen.

The robustness properties of the controllers are validated by aμ-analysis, for robust performance, on the multivariable system.Uncertainties of 40% on the Magnus, 10% on the pitch/yaw damp-ing and control forces, and 5% on the other aerodynamic coeffi-cients (Theodoulis et al. 2015) are considered in the analysis, andthe desired performances are those of the nominal closed-loop re-sponses for all flight conditions. The μ-analysis is conducted at dif-ferent flight times of the guided portion of the high quadrantelevation (QE) referential trajectory. The resulting contour plotof the structured singular value upper bound for robust performanceis presented in Fig. 3. As the bound is below the unity for all flighttimes and frequencies, robust performance over the entire trajectoryis ensured.

Concept 1 Control Function OutputFor this concept, only the fins ring roll angle is controlled. To becoherent with the orientation defined in Fig. 1, the fins ring rollangle is obtained by mixing the lateral and vertical autopilotsoutput

ϕc ¼ π þ arctanfzfy

ð13Þ

Concept 2 Control Function OutputsThe fins ring roll angle of this concept is also generated withEq. (13), and the magnitude of the aerodynamic control force is

Fig. 3. (Color) μ-plot of robust performance along the high QEreferential trajectory

© ASCE 04016055-4 J. Aerosp. Eng.

modulated by the deflection angle of the maneuvering canards. Thelift generated by one fin can be approximated by

fl ¼ qsrfCNδδ ð14Þ

Based on this approximation, the deflection angle of the maneu-vering fins (Fins 1 and 3) is computed as

−δ1 ¼ δ3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2y þ f2z

qqð2srf ÞCNδ

ð15Þ

where srf is the aerodynamic referential area of a fin. The sign in-version between Fins 1 and 3 comes from the sign convention ofFig. 1.

Concept 3 Control Function OutputsIn this concept, the autopilots output are not mixed. They are ratherdirectly used to determine the required deflection angles for bothpairs of canards. Considering the approximation of Eq. (14) for thelift force, the deflection angle of the lateral axis fins (Fins 1 and 3)and of the vertical axis fins (Fins 2 and 4) are computed as

δ1 ¼ − fyqð2srf ÞCNδ

þ 0.5 δ3 ¼fy

qð2srf ÞCNδ

þ 0.5 ð16Þ

δ2 ¼ − fzqð2srf ÞCNδ

þ 0.5 δ4 ¼fz

qð2srf ÞCNδ

þ 0.5 ð17Þ

The sign inversions between Fins 1 and 3 and between Fins 2and 4 come from the sign convention of Fig. 1, while the offset of0.5° on each canard is applied to despin the fins ring.

Longitudinal Autopilot

As the projectile rotational motion has only a small influence on itsaxial airspeed, Newton’s second law can be used to describe thelongitudinal transfer function, which is therefore a single gain cor-responding to the inverse of the projectile mass

GpxðsÞ ¼ ax

fx¼ 1

mð18Þ

As a result, a nonscheduled, unitary gain, proportional integratorautopilot is sufficient to control the projectile axial motion

GcxðsÞ ¼fx

acx − ax¼ τsþ 1

τsð19Þ

The integrator time constant, τ , is set in order to obtain a settlingtime of approximately 1 s.

Concept 4 Control Function OutputsFor the lateral axis, the autopilot of Eq. (12) is implemented, andEq. (16), without the 0.5° offset, is used to compute the canardsdeflection angle from the autopilot output.

Assuming that the drag force of a fin can be computed by

fd ¼ qsrf ðCDδδ2 þ CD0

Þ ð20Þ

Then, the deflection angle of Fins 2 and 4 is

δd ¼ δ2 ¼ δ4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifx − qð2srf ÞCD0

qð2srf ÞCDδ

sð21Þ

Guidance Algorithm

The guidance algorithm is responsible for determining the actionsrequired to reach the target. The proposed algorithm is based on thecomputation of zero-effort misses (ZEMs) from predicted impactlocations. These ZEMs are converted into acceleration setpointsfor the autopilots.

The guidance algorithm is based only on the translationalmotion of the projectile, which can be modeled by a point massaffected by external forces. These forces are the gravity, drag, lift,and Magnus.

The drag force acts in the opposite direction of the projectilevelocity

f D ¼ − qsrCD

mvjvj ð22Þ

The lift force is parallel to the yaw of repose vector (αr)

f L ¼qsrCLα

mαr ð23Þ

where the latter can be modeled as (McCoy 2012)

αr ¼Ixp

qsrdrCmα

g ×

vjvj

ð24Þ

The Magnus force is perpendicular to the plane formed by thevelocity and yaw of repose vectors

fM ¼qsrCNpa

mpdrjvj

vjvj × αr

ð25Þ

In the previous equations, CD, CLα, CNpa

, and Cmαare the aero-

dynamic coefficients, namely drag coefficient, lift coefficient due tothe angle of attack, Magnus force coefficient, and pitching momentcoefficient due to the angle of attack. The gravity vector (g)is assumed to be constant at 9.81 m=s2 and is always pointingdownwards.

In order to consider the drifting motion inherent to a spin-stabilized projectile, the modified point mass equation (Lieskeand Reiter 1966) is used in a line-of-sight (LoS) coordinate system.This system has its x-axis and y-axis in the horizontal plane. Itsx-axis is parallel to the cannon muzzle azimuth and points down-range, while its y-axis points to the right when looking downrange.Its z-axis completes the right-handed system and points down-wards. The system origin is the launch point. The modified pointmass equation is

mv ¼ f D þmgþ f L þ fM ð26Þ

Impact Location Prediction

The guidance algorithm begins by estimating the time to go, i.e., thetime required to reach the target altitude. To obtain it, the verticalvelocity differential equation, the third component of Eq. (26), isused. Along the LoS frame’s z-axis, the lift and Magnus forces aremuch smaller than the drag and gravity forces. Therefore, they canbe neglected and the velocity derivative becomes

vz ¼ − qsrCD

mjvj vz þ g ð27Þ

Considering the drag coefficient and projectile airspeed constantfor the rest of the flight, qsrCD=mjvj becomes a constant denotedby d in the following. Eq. (27) is then a time linear equation andcan be integrated from the current time (t ¼ 0) until the time to go(t ¼ tgo)

© ASCE 04016055-5 J. Aerosp. Eng.

vzðtgoÞ − vzð0Þ ¼ −dðzT − zð0ÞÞ þ gtgo ð28Þ

where zT is the target altitude, which is considered null in the fol-lowing. By considering Eq. (27) as a time-invariant linear differ-ential equation, it can be solved by an exponential. Specifically,at t ¼ tgo, it becomes

vzðtgoÞ ¼ vzð0Þe−dtgo þgð1 − e−dtgoÞ

dð29Þ

Eq. (28) is introduced in Eq. (29), and the resulting equation isreorganized to form a transcendental equation in tgo

tgo ¼ln�dzð0Þþgtgoþvzð0Þ−g

dvzð0Þ−g

d

�−d ð30Þ

This equation can be solved iteratively using a Newton-Raphsonscheme. With a good initial guess, it takes fewer than five iterationsto solve it at a precision of 0.01 s. At the first call of the guidancelaw, the expected time to go of the referential trajectory can be usedfor this initial guess. In the following calls, the time to go of theprevious solution is used as actual initial guess.

This estimated time to go is used to compute the impact locationin the horizontal plane. The projectile motion along the LoS framex-axis is not significantly affected by the Magnus, gravity, and liftforces. As a result, the first component of Eq. (26) can be written as

vx ¼ −dvx ð31ÞConsidering Eq. (31) as a time-invariant linear differential equa-

tion, its solution at t ¼ tgo is an exponential

vxðtgoÞ ¼ vxð0Þe−dtgo ð32ÞEq. (31) can also be integrated from the current time (t ¼ 0) up

to the time to go (t ¼ tgo)

vxðtgoÞ − vxð0Þ ¼ −dðxðtgoÞ − xð0ÞÞ ð33ÞCombining Eqs. (32) and (33), the estimated longitudinal

impact position is

xðtgoÞ ¼ xð0Þ − vxð0Þe−dtgo − vxð0Þd

ð34Þ

For the motion along the lateral axis, the Magnus and lift forcesmust be considered. The second component of Eq. (26) is then

vy ¼ −dvy þ CLαIxpg

Cmαdrmjvj2

vx −CNpa

p2Ixg

Cmαmjvj4 vyvz ð35Þ

The velocity along the y-axis is coupled with those of the x-axisand z-axis. With this coupling, an analytic solution of Eq. (35) isdifficult to establish. However, the estimated lateral position atimpact can be approximated by a time second-order polynomial

yðtÞ ¼ vyð0Þt2 þ vyð0Þtþ yð0Þ ð36Þwhere vy is obtained from Eq. (35). Then, the estimated lateralposition at impact becomes

yðtgoÞ ¼−dvyð0Þ þ CLα

Ixpg

Cmαdrmjvj2

vxð0Þ

− CNpap2Ixg

Cmαmjvj4 vyð0Þvzð0Þ

t2go þ vyð0Þtgo þ yð0Þ ð37Þ

From the current projectile state, defined by its position [xð0Þ]and velocity [vð0Þ] vectors, the impact location is then obtained by

estimating the time to go with Eq. (30), which is introduced inEqs. (34) and (37) to obtain the estimated longitudinal and lateralpositions at impact.

Dual Regime Algorithm

The PoI predictions are based on the hypothesis that the aerody-namic coefficients are constant for the whole flight, which is ob-viously not the case. Projectile aerodynamics vary significantlywith the Mach number and exhibit two distinct behaviors, depend-ing on whether the projectile is subsonic or supersonic. Also, oncethe projectile gets to a subsonic regime, it should not return tosupersonic speed later in its flight. Therefore, to improve the pre-dictions, two sets of constant aerodynamic coefficients are used.The projectile state at the end of the supersonic flight is predictedwith one set and, from there, the PoI is predicted with the other set.From the referential trajectory, the time at which the projectile be-comes subsonic (tm1) can be estimated and used to compute thesupersonic time to go

ts ¼ tm1 − t0 ð38ÞThe supersonic time to go is used in Eqs. (29), (32), (34),

and (37) to obtain, respectively, the vertical velocity, axial velocity,axial position, and lateral position of the projectile when it fallsbelow the sonic barrier. To be coherent with the time second-orderpolynomial approximation of the crossrange position, the cross-range velocity is approximated by a time first-order polynomial

vyðtm1Þ ¼−dvyð0Þ þ CLα

Ixpg

Cmαdrmjvj2

vxð0Þ

− CNpap2Ixg

Cmαmjvj4 vyð0Þvzð0Þ

ts þ vyðt0Þ ð39Þ

On its part, the vertical position is obtained by combiningEqs. (28) and (29)

zðtm1Þ ¼ zð0Þ − vzð0Þðe−dts − 1Þ þ gð1−e−dtsd − tsÞd

ð40Þ

The six components of the position and velocity vectors at theend of the supersonic regime are then known and used as initialstate of the algorithm to perform the impact location prediction.When the projectile is subsonic at the initial time, the predictionalgorithm is used directly on the measured projectile positionand velocity, eliminating the supersonic prediction step.

Zero-Effort Misses Estimation

The zero-effort misses are obtained by subtracting the targetlongitudinal (xT) and lateral (yT ) positions from the predictedlongitudinal and lateral positions of the PoI

ZEMy ¼ yðtgoÞ − yT ð41Þ

ZEMx ¼ xðtgoÞ − xT ð42Þ

The impact location predictions based on the modified pointmass model significantly differ from those based on the complete6DoF model (Fresconi et al. 2011). To mitigate this issue, the targetposition is predicted at the same rate and with the same predictionmodel as the PoI, but with its current state obtained from the refer-ential trajectory.

The resulting ZEMs are converted into autopilot accelerationsetpoints. For Concept 4, the conversion is

© ASCE 04016055-6 J. Aerosp. Eng.

ago ¼ kg½−ZEMx −ZEMy 0 �T ð43Þ

This conversion gives a setpoint for the longitudinal autopilot,and another for the lateral one.

The three other concepts require a setpoint along the verticalaxis rather than along the longitudinal one. As a force along thevertical axis modifies the longitudinal impact location, the longi-tudinal ZEM is converted in a vertical acceleration setpoint. Inthe ascending phase of high QE launch, a negative force in the ver-tical axis decreases the projectile impact location, the accelerationsetpoint must then be inversed. For high QE descending portionand for low QE launch, a negative force in the vertical axis in-creases the projectile longitudinal impact location, a sign inversionis therefore not required in the setpoint

ago ¼(kg½ 0 −ZEMy −ZEMx �T if γl > 45° and γ > 0

kg½ 0 −ZEMy ZEMx �T otherwise

ð44Þ

where γl is the launch QE, in degrees, and γ is the flight path angle.The guidance gain, kg, which appears in both conversions, is the

tuning parameter of the guidance algorithm. An optimization routinebased on the bisection method has been created in order to findthe value of the gain that minimizes the circular error probability(CEP), evaluated fromMonte-Carlo runs of 100 simulations. A differ-ent optimization is run for each concept and for each firing condition.

Eqs. (43) and (44) represent the accelerations required toeliminate the ZEMs. However, there is also expected accelerationcoming from the projectile aerodynamics and gravity. These twoacceleration components are estimated with the same variables asthe impact location estimation

aeo ¼qsr

−CD

vojvj þ CLα

αro þ drpCNpa

jvj�vojvj × αro

�m

þ go ð45Þ

The expected acceleration is added to the acceleration setpoints[Eqs. (43) or (44)] in order to reduce autopilot workload.

Results

The previous algorithms are implemented into a pseudo-7DoFsimulator of a typical 155-mm shell equipped with a CCCF, whichcan be configured to represent any of the four proposed concepts.The simulator is implemented in a nonspinning reference frame(Wernert et al. 2010), with constant gravity and U.S. Standard1976 atmospheric model.

Different factors may contribute to projectile dispersion. Thefactors simulated are the winds, muzzle velocity, and gun aiming.Table 2 offers the standard deviation applied on each of them. Thewinds are constant and their magnitudes are relative to the mostrecent meteorological measurements, which are compensated bythe muzzle alignment.

The first series of tests studies the footprints of the CCCF con-cepts to see which one better matches the conventional projectiledispersion. The second series of tests studies guided projectilesdispersion and range, through Monte-Carlo simulations realizedwith the perturbation standard deviations given in Table 2. Eachseries of tests is done at two different quadrant elevations, oneunder 800 mils, considered as the low QE case, and the other oneover 800 mils, considered the high QE case. Prior to these twoseries of tests, an analysis of the time history of the projectile andalgorithms parameters for typical guided flights was conducted.This analysis did not show any problematic behavior or tendenciesto bring the projectile close to instability.

The fuze activation time is dictated by the GPS receiver lock timeand the flight-path inclination. The GPS receiver must be locked andthe projectile must have reached a given flight-path angle beforestarting the guided flight. This flight-path angle is chosen in orderto maximize fuze efficiency. During the unguided flight and forthe referential trajectory generation, the canards are fixed in orderto minimize the drag. Therefore, the PoI of the referential trajectory,which represents the target location, differs for each concept.

Footprint Analysis

The footprint of a projectile equipped with a CCF corresponds tothe displacement achievable with this type of fuze. It is obtained byfixing the control variables at their extrema for the whole durationof the guided flight. If a projectile equipped with a conventionalfuze lands inside the guided projectile footprint, the CCF shouldhave enough control authority to compensate for the disturbancesand directly hit the target. However, because of sensor imperfec-tions, guidance and control limitations, and the lack of ability tofully compensate for perturbations occurring late in the flight, aperfect hit rarely occurs. In theory, the optimal CCF design is ob-tained when its footprint area perfectly fits the conventional fuzedispersion pattern.

The footprint is composed of the results of unperturbed simu-lations. One simulation per control variable combination is run, andthe points of impact (PoIs) of all combinations create the footprint.For Concepts 1 and 2, the footprint is driven only by the fins ringroll angle, which is incremented by a step of 2° between each sim-ulation. For Concepts 3 and 4, the footprint is affected by the ca-nards deflection angle. One set of canards is deflected to itsmaximal/minimal values, and the deflection angle of the otherset is incremented by step of 0.5° between each simulation. Thesame process is then repeated with the second set of canards beingkept at its extrema, while the first set is incremented by step of 0.5°.Figs. 4 and 5 present the obtained footprints for the four types ofCCCF. They also include 1,000 PoIs obtained with a conventionalfuze (yellow dots).

The graphics indicate a better potential for Concept 4 than forthe three others. Its elongated footprint shape better fits the down-range dispersion of the conventional fuze. Based on the footprintanalysis only, Concept 3 does not seem to provide a significant im-provement over Concepts 1 or 2. Only a small area when both pairsof canards are close to their extrema is covered by Concept 3 andnot by the two others. Obviously, actuating the canards does notchange the footprint area; with the latter representing the maximalachievable displacement, the footprints of Concepts 1 and 2 arethen strictly identical.

Monte-Carlo Simulations Results

The second series of tests simulates 4,000 perturbed guided rounds,1,000 per fuze, for each firing condition. The conventional fuzesimulations of the footprints analysis are also used in this section.

Table 2. Standard Deviations of Considered Perturbations

Perturbation Standard deviation

Crossrange wind velocity 3 m=sDownrange wind velocity 3 m=sMuzzle velocity 2 m=sFiring azimuth 1 milsQuadrant elevation angle 1.5 mils

© ASCE 04016055-7 J. Aerosp. Eng.

From these simulations, the range and precision of the fuzes areanalyzed. Fig. 6 presents the range diminution of each CCCF, rel-ative to the conventional fuze range, for both QEs.

The range being inversely proportional to the control surfacesdeflection angles, Concept 4 should reach a shorter range than theother concepts. However, because all of its canards are set atsmaller deflection angles during the unguided portion of the flight,the difference is not very significant. Similarly, due to the canardsdeflection angles during the unguided portion of the flight, Concept3 reaches the longest range of the four proposed concepts, whileConcept 2’s range is slightly longer than that of Concept 1.

The resulting trajectories of both QEs are presented in Fig. 7.In this figure, the firing locations are located at 0 of downrangeand distributed on the crossrange axis; each type of fuze is firedfrom a different crossrange.

For both QEs, the CCCFs trajectories are less dispersed than theconventional fuze ones, which is an indicator of the guidance andcontrol algorithms efficiency. However, Fig. 7 does not give mean-ingful information about the precision at target. The analysis of theprecision is initially done with the miss distance cumulative distri-bution. As guided projectiles are expected to have a concentrationof misses close to the target, the usual CEP hides useful informationrequired to draw conclusions about their performances. The missdistance cumulative distribution is a graphical representation of thepercentage of rounds included in a circle. The graphic abscissa isthe radius of the circle and its ordinate is the ratio of rounds that

have landed in the corresponding circle’s radius over the total num-ber of rounds. The CEP is then the abscissa value for the 0.5 ratio.The graphics at low and high QE are, respectively, presented inFigs. 8 and 9. As expected, the four CCCF concepts are able todrastically increase the precision of a spin-stabilized projectile, butConcept 4 provides the highest precision level. For nearly all missdistances, its cumulative distribution is higher than the three otherCCCF concepts. Concept 4’s best precision is in line with that inGagnon and Lauzon (2008), who states that continuous feedbackcontrol concept combining drag brake and spin brake, like Concept4, is more precise than roll-decoupled CCF.

At low QE, Concept 3 is nearly as good as Concept 4, with thedifferences generated by the conventional PoIs very large misses,i.e., those impacting outside the Concept 3 footprint. Also, the per-formance gap between Concept 3 and Concepts 1 and 2 is largerthan the footprints (Fig. 4) suggest. This is due to the guidancealgorithm, which, because of the conventional PoIs dispersion, hasthe tendency to favor the longitudinal axis over the lateral one.At this elevation, the downrange ZEMs are larger than the lateralones. As the guidance gain is the same for both axes [Eq. (44)], theguidance algorithm asks for larger corrections along the downrangeaxis. As the fins ring angle in Concepts 1 and 2 are based on theratio of the two axes, the guidance algorithm then produces smallercorrections in the crossrange direction (Fig. 10). Furthermore, as

Fig. 4. (Color) CCCFs footprint and conventional PoIs at low QE

Fig. 5. (Color) CCCFs footprint and conventional PoIs at high QE

Fig. 6. (Color) Relative range diminution of the CCCFs

Fig. 7. (Color) Trajectories of the CCCFs and conventional fuze forboth QEs

© ASCE 04016055-8 J. Aerosp. Eng.

Concept 1’s control force is not modulated, this concept tends tooscillate around the referential trajectory and to overcorrect theclose misses, resulting in slightly lower precision than Concept 2.

At high QE, Concept 4 remains the best CCCF concept, and theperformance gap with the other concepts is much more significant,which is coherent with the footprint patterns. Also, at this QE, thedownrange ZEMs are marginally larger than the crossrange ones.The prioritization effect of the guidance law is, therefore, less sig-nificant. The axes decoupling of Concept 3 does not provide a sig-nificant improvement of the precision, in comparison to Concepts 1and 2. As for the low QE tests, because of the inability to modulateits control force, Concept 1 has slightly larger misses than the otherCCCF concepts.

A second way to analyze the precision of the CCCF conceptsis by comparing the PoI patterns on a map centered on the target.Figs. 10 and 11 present those maps for both QE, revealing patternsthat are coherent with the footprints and guidance law behavior.

At low QE, the PoIs of Concepts 3 and 4 are concentrated nearthe target. Concept 3 has a few residual misses corresponding to theconventional fuze misses impacting close or outside its footprintboundary. For their part, Concepts 1 and 2 have miss patterns thatare largely due to the guidance algorithm which, as explained ear-lier, favors downrange corrections at this QE.

At high QE, Concept 4 has some residual misses in crossrange,which are related to the conventional fuze PoIs that are not coveredby its footprint. For their part, Concepts 1, 2, and 3 exhibit largedispersions in downrange. These dispersions give a misleading im-pression of the achieved performances because the PoI map doesnot display the PoI concentrations. Analysis of the PoI map in par-allel with the cumulative distribution graphic (Fig. 9) is thereforerequired. The cumulative distributions show that more than 60% ofthe misses are close to the target. Therefore, only the less than 40%remaining misses contribute to the large dispersions observed inFig. 11. With an analysis of the PoIs and their displacement be-tween the unguided and guided concepts, the large misses of theguided fuzes can be traced back to conventional PoIs located out-side the guided fuze footprints. Also, because of the prioritizationeffect of the guidance law, Concepts 1 and 2 still have some re-sidual crossrange misses.

Conclusion

This paper studied four canards-based course correction fuzes.Simulations of these concepts with the developed guidance andcontrol algorithms demonstrated that the four concepts are ableto drastically improve the precision of a spin-stabilized projectile.Furthermore, the observed miss distributions are coherent with thefootprints and guidance law behavior. Under the proposed guidanceand control algorithms, a concept able to control the longitudinal

Fig. 8. (Color) Cumulative distribution of the CCCFs and conventionalfuze at low QE

Fig. 9. (Color) Cumulative distribution of the CCCFs and conventionalfuze at high QE

Fig. 10. (Color) PoIs of the CCCFs and conventional fuze at low QE

Fig. 11. (Color) PoIs of the CCCFs and conventional fuze at high QE

© ASCE 04016055-9 J. Aerosp. Eng.

acceleration, like the proposed regulated roll angle ring equippedwith four actuated canards having different deflection limits, pro-vides precision significantly better than that offered by conceptsusing only the lift force as an aerodynamic control force. For thelatter type of concepts, the axes decoupling provides an improve-ment. The miss distribution pattern is similar for all tested quadrantelevations and the precision is slightly better. Even if, among theproposed concepts and with the proposed algorithms, the conceptwith fixed canards provides the worst precision, the improvement,in comparison with an unguided fuze, is significant enough to iden-tify it as a mechanically simple and efficient solution.

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