Accepted Manuscript Guidance and Control for High Dynamic Rotating Artillery Rockets Raúl de Celis, Luis Cadarso, Jesús Sánchez PII: DOI: Reference: S1270-9638(17)30205-5 http://dx.doi.org/10.1016/j.ast.2017.01.026 AESCTE 3908 To appear in: Aerospace Science and Technology Received date: Revised date: Accepted date: 22 May 2016 29 November 2016 26 January 2017 Please cite this article in press as: R. de Celis et al., Guidance and Control for High Dynamic Rotating Artillery Rockets, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.01.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Guidance and Control for High Dynamic Rotating Artillery Rockets Raúl de Celis 1 and Luis Cadarso. 2 European Institute for Aviation Training and Accreditation, Rey Juan Carlos University, Fuenlabrada, Madrid, Spain, 28943 and Jesús Sánchez 3 National Institute for Aerospace Technology, San Martin de la Vega, Madrid, Spain, 28330 The acceleration autopilot with a rate loop is the most commonly implemented autopilot, which has been extensively applied to high-performance missiles. Nevertheless, for spinning rockets, the design of the guidance and control modules is a challenging task because the rapid spinning of the body creates a heavy coupling between the normal and lateral rocket dynamics. Nonlinear modeling of the rocket dynamics, control design as well as guidance algorithms are performed in this paper. Moreover, discrete-time guidance and control algorithms for the terminal phase, which is based on proportional navigation, are performed. Finally, complete nonlinear simulations based on realistic scenarios are developed to demonstrate the robustness of the proposed solution with respect to uncertainty regarding launch, environment and rocket conditions. The performance of the proposed navigation, guidance and control system for a high-spin rocket leads to significant reductions in impact point dispersion. Keywords: rockets, artillery, flight mechanics, guidance, control. I. Introduction Precision has always been recognized as an important attribute of weapon development. One of the greatest advantages of the precision weapon is the confidence that ‘collateral damage’ is minimized. In addition, using force in some circumstances would be either unacceptable or call into question the viability of continued military action in absence of precision weapons [1]. A precision-guided munition (PGM) is a guided munition intended to precisely hit a specific target, and to minimize collateral damage. Considering that the damage effects of explosive weapons decrease with distance, even modest improvements in accuracy enable a target to be attacked with fewer or smaller bombs. The precision of these weapons is dependent both on the precision of the measurement system used for location determination and the precision in setting the coordinates of the target. The latter critically depends on intelligence information, not all of which is accurate. If the targeting information is accurate, satellite-guided weapons (including inertial navigation in the event of signal loss) are significantly more likely to achieve a successful strike in any given weather conditions than any other type of precision-guided munition. Development of low-cost navigation, guidance and control technologies for unguided rockets is a unique engineering challenge. Over the past several decades, numerous solutions have been proposed, primarily for large artillery projectiles or for slowly rolling airframes. Guided projectiles are divided into three categories in terms of the control mechanisms employed: aerodynamic surfaces [3, 4], jet thrusters [2, 5], and inertial loads [6, 7]. Guided projectile concepts involving aerodynamic control surfaces are divided into two categories: fin-stabilized and spinstabilized. Spin-stabilized guided projectiles are generally equipped with a roll-decoupled trajectory correction fuse, designed to provide trajectory correction, while at the same time the high spin rate of the aft part contributes to airframe stability. However, the high spin rate creates an important coupling between the normal and lateral axes of the body, which makes the dynamic characteristic rather complex. For such projectiles, previous work has proven that flight instabilities occur for spin-stabilized projectiles maneuvering perpendicular to the gravity field when the control effectiveness is sufficiently high [5]. In [8], it is stated that in a guidance system, a suitable guidance law and navigation constant are fundamental elections. Investigation and comparison of the system behavior for guidance laws under different navigation constants are developed. The capture region of the general ideal proportional navigation guidance law is analyzed in [9]. A new longitudinal autopilot to address the finite-time tracking problem is presented in [10]. In [11], it is 1 Assistant Professor, Rey Juan Carlos University, Fuenlabrada, Madrid, Spain. Associate Professor, Rey Juan Carlos University, Fuenlabrada, Madrid, Spain. Corresponding author. 3 Lieutenant Colonel, Chief (NMT), Area of Propulsion and Guidance, National Institute for Aerospace Technology, Madrid, Spain. 2 1 addressed a nonlinear terminal guidance/autopilot controller with pulse-type control inputs. In [12], a threedimensional integrated guidance and control law with impact angle constraint is developed, using the dynamic surface control and extended state observer techniques. A decrement of coupling effects of a tactical missile by designing a robust autopilot for its roll channel is achieved in [13]. A description of an application of a new terminal guidance law for missiles in three-dimensional (3D) environment during the terminal phase is performed in [14]. A guidance and control strategy for a class of 2D trajectory correction fuse with fixed canards for a spinning projectile is developed in [15]. Correction control mechanism is researched through studying the deviation motion and impact point deviation prediction based on perturbation theory. In [16], it is detailed the design and simulation of an attitude guidance and control scheme for a spinning aerospace vehicle. Two single-input/single-output controllers are used, which in turn issue flap deflection commands. In [17], it is noticed that the stable region of the design parameters for the autopilot shrinks significantly under the spinning condition. It is also observed that the stable region for design parameters is further narrowed when an integrator is introduced into the acceleration loop while the steady-state accuracy is dramatically improved. In [18], it is also observed that an unexpected and unstable coning motion occur for spinning missiles after burnout. In order to address this unstable motion, the governing equation of the coning motion, and the dynamics of the fin actuators under the associated hinge moment are derived. The necessary and sufficient conditions of the coning motion stability are then analytically derived and further validated through nonlinear six degrees-of-freedom simulations for a representative scenario. A discrete-time, proportional-derivative navigation guidance law, for the terminal phase of an engagement, with emphasis on the effect of a digital implementation is proposed in [19]. In [20], it is presented a complete design, concerning the guidance and autopilot modules for a class of spin-stabilized fin-controlled projectiles. The proposed concept is composed of two sections: the rapidly spinning aft part contains the charge, whereas the front part, which is roll decoupled from the aft. A. Contributions The main contributions of this paper are the development of a flight model in order to simulate the dynamics of a highly spinning rocket which features a decoupled fuse. This model is employed in the development of a novel 3D guidance law for gyroscopically stabilized artillery rockets (i.e., spin rates in the hundreds of rotations per second), which is derived from proportional navigation. This novel control law allows the development of a simple but effective and robust single-input single-output controller, which is able to handle the heavy coupling between the normal and lateral rocket dynamics. This new approach allows simple but effective control algorithms and, therefore, simplification in guidance fuses’ electronics. Simulations based on a 140-mm rocket validate the effectiveness of the presented approach. The rocket is equipped with a rotating force mechanism and non-controlled thrust which is only available during a limited period of time after launch. The rotating force mechanism consists of a decoupled fuse from the aft part of the rocket. The main advantage of the overall setup is that it maintains the inherent dynamic stability properties of a rapidly spinning body due to the aft part, while at the same time, the front part, is able to be fit to any unguided rocket, hence transforming it into a guided one [20]. Nonlinear simulations based on realistic scenarios are performed to demonstrate the robustness of the proposed solution with respect to uncertain launch, environment and rocket conditions. The paper proceeds first presenting the nonlinear rocket dynamic model. Second, an integrated navigation and guidance approach is exposed, followed by the design of a controller. Example of ballistic and controlled flight simulation results are conferred, which analyze performance. II. Rocket Flight Dynamic Model This section describes the nonlinear flight dynamic model used in this study, including dynamics and aerodynamics, and control actuation. An example rocket concept is described, followed by a description of the mathematical model for each relevant component. B. Rocket The guidance and control formulation proposed in this study applies to a 140-mm axisymmetric rocket with wrap around stabilizing fins. This system features a supersonic launch and a rocket spin rate of approximately 150 Hz. The maneuver mechanism is placed at the front part of the spin stabilized rocket, with a roll-decoupled fuse. It consists of a section composed of fixed canard surfaces in order to generate a rotating control force and its associated moment. This mechanism is composed of a rotary motor linked to the front section (see Figure 3). The 2 most representative data of the rocket is presented in Table 1. The thrust curve depending on flight time is exposed in Figure 1. The propellant mass decreases according to Eq. (1): ௧ ்(ఛ) ݉(݉ = )ݐ െ ଶଷସ଼.ଶ ݀߬ , (1) where m(t) is the rocket mass function of time, m is rocket initial mass and T(t) is thrust modulus, which is also function of time. Thrust curves have been fitted using data from actual shots. Note that once the rocket has been launched there is no control on the thrust force. Finally, all the aerodynamic coefficients for the rocket under study are acquired by numerical computation and tested against actual shots (see Appendix A for a comprehensive description of the notation). Their variation with the Mach number is represented in Figure 2. Table 1 Representative data of the studied rocket Parameter Maximum thrust Burn-out time Initial mass Propellant mass Ix0 Iy0 XCG0 Caliber Value 29160 N 2.70 s 77.40 kg 21.60 kg 0.36 kg m2 35.63 kg m2 1.30 m 0.14 m Fig. 1 Thrust curve depending on flight time. 3 Fig. 2 Aerodynamic coefficients for the rocket under study depending on the Mach number. C. Flight Dynamic Model Three axes systems are defined to express forces and moments: earth axes, body axes, and working axes. Earth axes are defined by sub index e. xe pointing north, ze perpendicular to xe and pointing nadir, and ye forming a clockwise trihedral. Working axes are defined by sub index w. xw pointing to the target, yw perpendicular to xw and pointing zenith, and zw forming a clockwise trihedral. ܼܣ is the angle between xe and xw, in other words, the initial azimuth. Body axes are defined by sub index b. xb pointing forward and contained in the plane of symmetry of the rocket, zb perpendicular to xb pointing down and contained in the plane of symmetry of the rocket, and yb forming a clockwise trihedral. The origin of body axes is located at the center of mass of the rocket and they are severely coupled to the roll-decoupled fuse. Earth, body and working axes systems are illustrated in Figure 3. Fig. 3 The 140-mm axisymmetric rocket with wrap around fins and a roll-decoupled fuse and reference systems. Total forces and moments on the rocket are given by (2) and (3), respectively: ሬሬሬሬሬሬሬԦ ሬԦ ሬԦ ሬሬԦ ሬԦ ሬԦ ሬሬሬԦ Ԧ ܨ ௫௧ = ܦ+ ܮ+ ܯ+ ܲ + ܶ + ܹ + ܥ, (2) ሬሬሬሬሬሬሬሬሬԦ ሬԦ ሬሬሬሬሬԦ ሬሬሬሬሬሬԦ Ԧ ܯ ௫௧ = ܱ + ܲெ + ܯெ + ܵ, (3) 4 ሬԦ is the drag force, L ሬԦ is the lift force, M ሬሬሬԦ is the Magnus force, P ሬԦ is the pitch damping force, T ሬԦ is the thrust where D ሬ ሬሬԦ ሬ Ԧ ሬ Ԧ ሬሬሬሬԦ force, W is the weight force and C is the Coriolis force, O is the overturn moment, P is the pitch damping moment, ሬሬሬሬሬሬԦ ሬԦ M is the Magnus moment and S is the spin damping moment. Rocket forces in working axes include contributions from drag, lift, Magnus, pitch damping, thrust, weight and Coriolis forces, which are described by the following: ሬԦ = െ గ ݀ଶ ߩ ቀܥ + ܥ ߙ ଶቁ ԡሬሬሬሬԦ ݒ௪ ԡሬሬሬሬԦ, ݒ௪ ܦ మ బ ଼ (4) గ ଶ ܮሬԦ = െ ݀ଶ ߩ ቀܥഀ · ߙ + ܥ య ߙ ଶ ቁ (ԡݒ ሬሬሬሬԦ ݔ௪ െ (ݔ ሬሬሬሬሬԦ ሬሬሬሬԦ)ݒ ௪ ԡ ሬሬሬሬሬԦ ௪·ݒ ௪ ሬሬሬሬԦ), ௪ ଼ (5) ሬሬԦ = െ గ ݀ଷ ߩ (ܮ ሬሬሬሬሬԦ ܯ ݔ௪ (ݔ ሬሬሬሬሬԦ ݒ௪ ௪ · ሬሬሬሬሬԦ) ௪ × ሬሬሬሬԦ), (6) ಿ గ ଶ ሬሬሬሬሬԦ ԡݒ ܲሬԦ = ݀ଷ ߩ ሬሬሬሬԦ ݔ௪ ௪ ԡ (ܮ௪ × ሬሬሬሬሬԦ), (7) ሬԦ = ܶ(ݔ)ݐ ܶ ሬሬሬሬሬԦ, ௪ (8) ሬሬሬԦ = ݉݃ ሬሬሬሬሬԦ, ܹ ௪ (9) ሬሬԦ × ݒ ሬሬሬሬԦ, ܥԦ = െ2݉ȳ ௪ (10) ഀ ഀ ଼ ூೣ ଼ ூ where d is the rocket caliber, ɏ is the air density, Cୈబ is the drag force linear coefficient, Cୈ మ is the drag force ಉ square coefficient, Ƚ is the total angle of attack, Cಉ is the lift force linear coefficient, C య is the lift force cubic ಉ ܮ௪ is the rocket angular momentum expressed in working axes, coefficient, C୫ is the Magnus force coefficient, ሬሬሬሬሬԦ I୶ ܽ݊݀ I୷ are the rocket inertia moments in body axes, C୯ is the pitch damping force coefficient, xሬሬሬሬሬԦ ୵ is the rocket ሬ ሬԦ nose pointing vector expressed in working axes, gሬሬሬሬሬԦ is the gravity vector in working axes, ȳ is the earth angular ୵ ሬሬሬሬԦ is the rocket velocity expressed in working axes. Likewise, rocket moments in working axes speed vector, and ݒ ௪ include contributions from overturning, pitch damping, Magnus, and spin damping moments and are described by the following: గ ݒ௪ ԡଶ (ݒ ሬሬሬሬԦ ݔ௪ ܱሬԦ = ଼ ݀ଷ ߩ ቀܥெഀ + ܥெ య ߙ ଶ ቁ ԡሬሬሬሬԦ ௪ × ሬሬሬሬሬԦ), ഀ ഏ య ௗ ఘ ఴ ሬሬሬሬሬԦ ܲெ = ூ ഏ ర ௗ ఘ ఴ ሬሬሬሬሬሬԦ ܯ ெ =െ ூ ೣ ܵԦ = ሬሬሬሬሬԦ ሬሬሬሬሬԦ ݔ௪ ሬሬሬሬሬԦ), ܥெ ԡݒ ሬሬሬሬԦ ௪ ԡ (ܮ௪ െ ൫ܮ௪ · ሬሬሬሬሬԦ൯ݔ ௪ ሬሬሬሬሬԦ ܥ ቀ൫ܮ ݔ௪ ሬሬሬሬԦ ݔ௪ ሬሬሬሬሬԦ൯ ሬሬሬሬԦቁ, ௪ · ሬሬሬሬሬԦ൯൫(ݒ ௪ · ሬሬሬሬሬԦ)ݔ ௪ െݒ ௪ ഏ ర ௗ ఘ ఴ ூೣ ݒ௪ ԡ൫ሬሬሬሬሬԦ ܥ௦ ԡሬሬሬሬԦ ݔ௪ ሬሬሬሬሬԦ, ܮ௪ · ሬሬሬሬሬԦ൯ݔ ௪ (11) (12) (13) (14) where Cಉ is the overturning moment linear coefficient, C య is the overturning moment cubic coefficient, C౧ is ಉ the pitch damping moment coefficient, C୫୫ is the Magnus moment coefficient and Cୱ୮୧୬ is the spin damping moment coefficient. The control forces and moments in body axes are modeled in (15) and (16), respectively: 0 ԡ௩ೢ మ ሬሬሬሬሬԦ = ఘௌೣ ሬሬሬሬሬԦԡ ܿݏ (߶ ܥ (ߙ) ܨܥ െ ߶) ൩ ே ସ െ߶(݊݅ݏ െ ߶) (15) 5 ሬሬሬሬሬሬԦ = ݔ ሬሬሬሬԦ ܥM ሬሬሬሬሬሬሬሬԦ ್ × ܥF (16) ሬሬሬሬԦ is the control force, CM ሬሬሬሬሬሬԦ is the control moment, Sୣ୶୮ is the reference surface for fins, C (Ƚ) is the fin where CF ሬሬሬሬሬሬሬሬԦ normal force coefficient, Ԅୡ is the control force angle, Ԅ is the roll angle and ݔ ್ is the center of gravity position expressed in body axes. In order to solve the motion of the rocket a body reference frame, which is coupled to the fuse, is used. Because, the fuse is uncoupled from the rear part which spins at high rates, it is required to model this spinning motion to account for rear Magnus force and moment, and gyroscopic effects. The spin rate of the rear part of the rocket is modeled as provided in Eq. (17): ഏ ర ௗ ఘ = െ( 200ߜ(ݐ ) െ ఴ ூ ೣ ݒ௪ ԡฮ൫ሬሬሬሬԦ ܥ௦ ԡሬሬሬሬԦ ሬሬሬሬԦ൯ݔ ܮ · ݔ ሬሬሬሬԦฮ)݀ݐ, (17) where ߜ(ݐ ) is a Dirac’s delta, ሬሬሬሬԦ ݒ is the rocket velocity ܮ is the rocket angular momentum expressed in body axes, ሬሬሬሬԦ expressed in body axes and ሬሬሬሬԦ xୠ is the rocket nose pointing vector expressed in body axes. Note that initial spin speed is modeled as an impulse which correlates to experimental data. It is assumed that the fuse mass is negligible, which implies that non-appreciable reactions are involved between fuse and aft part. Taking this into account, the aft effect is expressed as an extra addition of external forces and ௗ௩ ሬሬሬሬሬሬሬሬሬԦ ್ ሬሬሬሬሬԦᇱ × ሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬԦ +߱ ݉ݒ and moments to the Newton-Euler equations expressed in the ܾ reference system: ܨ ௫௧ = ௗ௧ ܫ௫ 0 0 + ሬሬሬሬԦ ᇲ ௗ ್ ᇱ ᇱ ᇱ × ሬሬሬሬԦ ᇱ =ቆ ᇱ are the angular speed and momentum, ሬሬሬሬԦ 0 ܫ௬ 0 ߱ ሬሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬሬԦ ቇ ܯ = + ߱ ܮ , where ߱ = and ܮ ݍ ௫௧ ௗ௧ 0 0 ܫ௬ ݎ respectively, of the joint body, namely rocket and fuse. The aerodynamic (18) -(19) and gyroscopic (20) contributions of the aft part are computed separately and moved to the left part of the Newton-Euler equations. ܫ௫ గ ଷ ሬሬԦ ܯ = െ ଼ ݀ ߩ ூ ቆ ݍቇ 0 ೣ ݎ 0 ഏ ర ௗ ఘ ሬሬԦெ = െ ఴ ܯ ூೣ ܫ௫ ܥ ቌቆ ݍቇ 0 ݎ 0 ܫ௫ ሬሬሬሬԦ ܩ = െ 0 0 0 ܫ௬ 0 0 ܫ௬ 0 0 ܫ௬ 0 0 0 · ݔ ሬሬሬሬԦ ሬሬሬሬԦ × ሬሬሬሬԦ) ݒ (ݔ ܫ௬ 0 0 · ݔ ሬሬሬሬԦ ሬሬሬሬԦ · ݔ ሬሬሬሬԦ)ݔ ሬሬሬሬԦቍ ൫(ݒ ሬሬሬሬԦ൯ െݒ ܫ௬ 0 ᇱ +߱ 0 ௗ ߱ ሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬԦ × ሬሬሬሬԦ ܮᇱ ௗ௧ ܫ௬ (18) (19) (20) ௗ௩ ሬሬሬሬሬሬሬሬሬԦ ್ ሬሬሬሬሬሬሬԦ ሬሬԦ ሬሬሬሬሬԦ × ሬሬሬሬሬሬሬሬԦ ݉ݒ ܨ ௫௧ + ܯ = ௗ௧ + ߱ (21) ሬሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬሬԦ ሬሬሬሬԦ ௗሬሬሬሬԦ್ + ߱ ܯ ሬሬሬሬሬԦ × ሬሬሬሬԦ ܮ ௫௧ + ܯெ + ܩ = (22) ௗ௧ ሬሬԦ୰ is the Magnus force of the rotating part of the rocket, M ሬሬሬሬሬሬሬሬԦ where ሬM ୰ is the Magnus moment of the rotating part of ሬሬሬሬԦ୰ is the gyroscopic moment of the rotating part of the rocket, p, q and r are the angular speed the rocket, G ሬሬሬሬሬԦ. components of the fuse, ߱ Given the force and moment models above, the equations of motion for the rocket are formulated using a Newton–Euler approach. The inertial, flat-Earth coordinate system (denoted by frame e) and the body-fixed coordinate system b DUHUHODWHGE\(XOHUUROOࢥSLWFKșDQG\DZȥDQJOHV The equations of motion given by Eq. (21) and Eq. (22) are integrated forward in time using a fixed time step Runge–Kutta of fourth order to obtain a single flight trajectory. Note that control action is input to this system through specification of the roll DQJOH RI WKH XQFRXSOHG IXVH ࢥc. The result for a non-controlled trajectory, i.e., a ballistic trajectory, calculated by this model is represented in Figure 4 IRUDQLQLWLDOșRIGHJUHHV 6 Fig. 4 $EDOOLVWLFWUDMHFWRU\IRUDQLQLWLDOșRIGHJUHHV Fig. 5 Initial elevation angle and Azimuth Correction (࢘࢘ ) depending on target distance to launch point. III. Navigation and Guidance Law In order to determine the elevation angle of the shot and to correct lateral deviations produced by Coriolis, Magnus and Gyroscopic effects depending on target location, a set of non-controlled simulations were done and an initial Azimuth is added to ܼܣ in order to obtain the initial Azimuth: ܼܣ௧ = ܼܣ + ܣ . Initial elevation angle and ܣ depending on target distance to launch point are illustrated in Figure 5. After this correction, navigation and guidance are provided by a modified proportional law defined in Eq. (23) and Eq. (24). Guidance is activated if and only if the variable ܥܰܩ௧ takes value 1; it takes value 1 if flight time is JUHDWHUWKDQVHFRQGVDQGș-15 degrees. This means that guidance is only activated when the rocket is in the last part of the ballistic flight because only small lateral deviations are pretended to be corrected. Errors to be introduced in the controller are defined by the following equations: െܺଵೢ మೢ ௗ ௗ ቍቑ Ʌୣ୰୰ = െܰ · ቮௗ௧ െܺଶೢ ቮ ௗ௧ ቐatan ቌ మ ା మ ටభೢ యೢ െܺଷೢ యೢ ߰ୣ୰୰ = atan ൬ భೢ ൰ െ atan ቀ ௫ೢయ ௫ೢభ ቁ (23) (24) where, ܺଵೢ ሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬԦ ܺଶೢ = ܺ ்௧ೢ െ ܺ௪ ܺଷೢ (25) 7 being X୮ଵ౭ , X୮ଶ౭ , X୮ଷ౭ the line of sight vector components expressed in working axes, ሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬԦ Xୟ୰ୣ୲౭ the target position ሬሬሬሬሬԦ expressed in working axes, X ୵ the rocket position expressed in working axes, Ʌୣ୰୰ the pitch error, ɗୣ୰୰ the yaw error and N the Proportional Navigation Law Constant. IV. Control System Control is processed by a double loop feedback system, which uses accelerations and angular speeds in body axes. The inner loop is only used as a system of stability augmentation. The control angle for the rotating force is defined in Eq. (26), taking Pitch (Ʌୣ୰୰ ) and Yaw (߰ୣ୰୰ ) errors as inputs. Figure 6 represents the philosophy of the controller. It has three main inputs: the acceleration of the rocket, the pitch error and the yaw error. Roughly speaking, the controller calculates the needed pointing angle of the aerodynamic force calculating the arctangent of the quotient of the pitch and yaw errors. This gives an angle at which the aerodynamic force, in the yb-zb plane, must point to reach the target. Nevertheless, the gyroscopic effect due to the spinning part of the rocket causes the response difficult to govern, i.e., imposing a ߶ = 90 degrees will not make the rocket to respond upwards. Therefore, it is also required to measure the acceleration of the rocket, without accounting for gravity, and adjust zero the difference between the angle that forms the projection of the aerodynamic force in the yb-zb plane with yb and ߶ . By comparing this required and measured accelerations in body axes, referred to the fuse as defined, it is possible to correct gyroscopic effect, as it is implicit in the measured acceleration. ߶ = ܥܰܩ௧ ൜ܭ ܽ ݊ܽݐ൬ ್ ൰ െ ܽ ݊ܽݐቀ ܭ ௗ ௗ௧ ൬ܽ ݊ܽݐ൬ ್ ್ ൰ െ ܽ ݊ܽݐቀ ౨౨ ିభ ·ఏ ౨౨ ିభ ·ఏ మ (ట౨౨ ିభ ట) ್ ቁ൨൰ െ ܽ ݊ܽݐቀ మ (ట౨౨ ିభ ట) ቁ൨ + ܭூ ܽ ݊ܽݐ൬ ౨౨ ିభ ·ఏ మ (ట౨౨ ିభ ట) ್ ್ ൰ െ ܽ ݊ܽݐ൬ ౨౨ ିభ ·ఏ మ(ట౨౨ ିభ ట) ቁൠ (ܥଵ = 0.01, ܥଶ = 100) ൰൨ ݀ ݐ+ (26) acc୶ୠ , acc୷ୠ , accୠ are the rocket accelerations expressed in body axes, Ʌ is the pitch angle, ɗ is the yaw angle, GNCୡ୲ is the activation parameter for GNC, K is the PID proportional constant, K ୍ is the PID Integral constant, and K ୈ is the PID derivative constant. Fig. 6 Controller scheme. In order to develop the discrete-time guidance and control algorithms for the terminal phase the following process is utilized: Model-In-the-Loop (MIL), Software-In-the-loop (SIL), Processor-In-the-Loop (PIL) and Hardware-In-theLoop (HIL). In the MIL stage, all the models in Sections II, III, and IV are implemented and tested; here, continuous time is used for all the components of the system. Once the MIL stage is successfully accomplished, the autopilot, that is, the models in Sections III and IV are enabled for discrete time and tested together with the model of the plant in Section II (i.e., SIL stage). Then, in the PIL stage, the autopilot is loaded in the final hardware (e.g., processor or FPGA) and tested again together with the model of the plant. Finally, a gyroscopic table of three degrees of freedom is employed to perform tests with real hardware such as processor and sensors (i.e., HIL tests). 8 V. Simulation Results Simulation results are presented using the nonlinear flight dynamic model to demonstrate closed-loop performance of the presented navigation, guidance and control novel approach and contrast performance with ballistic flight. We used MATLAB/Simulink R2015a on a laptop with a processor of 2.6 GHz and 8 GB RAM. The rest of this section presents an example trajectory, which it is used to study the stability of the system and its response to step commands in the control force, and Monte Carlo simulations in order to expose the robustness of the proposed algorithm to perturbation in the rocket and environment. A. Example Trajectory A representative flight simulation is performed for the proposed rocket where the target is placed at 21000 m range. Figure 7 illustrates the ballistic and controlled trajectories, and the spinning rate (revolutions per second, rev/s) of the rear part of the rocket. Note that the controlled trajectory impacted less than 2 m from the target, in contrast to a ballistic flight from the same initial conditions, which impacted about 350 m away. More interesting than the trajectory plots for this case is the response of the system to step commands. In Figure 8 it is displayed the delay between pretended and measured control force angle for different step commands in the control variable (߶ ) (angles of 45, 90 and 135 degrees for the aerodynamic force in the rocket). Note that in the first steps of this simulation ߶ is not controlled as GNC laws are not activated. It is very important to adjust PID parameters from controller to obtain a fast response, in order to correct trajectory accurately. Figure 8 presents that an applicable response is reached in an acceptable period of time for trajectory performance. Note that before the control input, the aerodynamic force in the rocket spins freely. For the PID gains, ܭ , ܭூ , and ܭ , values of 1.2, 1.2 and 0.0125 have been chosen, respectively, all of them based on simulation test analysis. Fig. 7 Example Trajectory. Fig. 8 Response of the rocket to different step commands in the control variable (ࣘࢉ ). B. Monte Carlo Simulations Monte Carlo analysis was conducted to determine closed-loop performance across a full spectrum of uncertainty in initial conditions, target coordinates, atmospheric conditions, and thrust properties. For atmospheric conditions, variations in turbulence are studied using the specification MIL-F-8785C and the Dryden Wind turbulence model. A 9 set of 10000 Monte Carlo replications are performed; distribution parameters are listed in Table 2. 5000 of the replications are performed with the control subsystem disenabled, that is, ballistic shots, and the rest of the replications with the control subsystem enabled. Five different targets are studied, which are located at different distances from the launching point, namely, 18500, 19000, 19500, 20000 and 20500 m. Table 2 Monte Carlo initial condition distribution parameters Parameter Initial ߶ Target Coordinates Wind Speed Wind Direction Thrust at each time instant Initial azimuth deviation Mean 0º Nominal Coordinates 10 m/s 0º T(t) 0º Standard deviation 2º 3m 5 m/s 20º 10 N 2º Figure 9 expresses impact point dispersion patterns for each of the ballistic and controlled cases. Also, the circular error probable (CEP) is observed. Filled points represent the impacts of controlled flights while not filled points the impacts of ballistic flights. The controlled flights exhibit tighter impact groupings. Spread in the impact distribution does remain in the guided flights due to the difficulties discussed, such as high normal and lateral dynamics coupling. Figure B.1 displays the same information as Figure 9 but it zooms in, in order to better see the dispersion area of all the controlled flights. The circular error probable (CEP) for each of the targets and for ballistic and controlled flights are exposed in Table 3. The last row in the table presents the obtained CEPs when considering all the flights together. Using the guidance and control algorithms introduced in this paper, the CEP is reduced from the ballistic case by about a 90%. Note that the dispersion of impact points and the CEP are greater for targets at lower distances from the launch point. This is due to the fact that the gains of the controller have been adjusted for the case where the distance to the target is 20500 m. Despite of the constant gains, the reduction of the CEP is significant. Moreover, note that the process of adjusting the gains is straightforward. Figure 10 represents the impact data in a different manner. This histogram exposes for each distance from the impact point to the target the number impact points. Overall, the results presented in this section support the hypothesis that the presented guidance and control algorithms are used to effectively guide these types of rockets and potentially improve system effectiveness. Table 3: The CEP for each of the targets and for ballistic and controlled flights Distance to Target (m) 18500 19000 19500 20000 20500 Total Ballistic Flight (m) 352.84 374.78 371.74 360.15 372.35 367.06 Controlled Flight (m) 29.35 29.93 28.82 19.55 12.61 23.38 10 Fig. 9 Complete dispersion area of all the flights. Fig. 10 Histogram of number of impacts for each distance to the target. VI. Conclusions A novel 3D navigation, guidance and control system was proposed for high dynamic rockets which feature high spinning rates and a non-controlled thrust. The proposed guidance algorithm is based on a modified proportional navigation law: flight data is used to compute pitch and yaw errors to compute control command, where pitch error calculations come from proportional navigation concepts. A computer model approach was used to characterize the input–output response of the rocket, and to develop discrete-time guidance and control algorithms. These algorithms were tested in a gyroscopic table and implemented in a processor in order to perform simulations. Example trajectories and Monte Carlo simulations demonstrated that the proposed approach can improve the CEP by approximately 93% compared with an unguided rocket. 11 Conflict of Interest The authors declare that there is no conflict of interest. Acknowledgments The authors are grateful to “Instituto Tecnológico La Marañosa” of the National Institute for Aerospace Technology (INTA) for solid modeling of the concept and to for discussions regarding modeling weapon delivery accuracy. 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"Robust Discrete-Time Proportional-Derivative Navigation Guidance,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 3, 2012, pp. 1007-1013. doi: 10.2514/1.55783] [20] Theodoulis, S., Gassmann, V., Wernert, P., Dritsas, L., Kitsios, I., and Tzes, A., “Guidance and Control Design for a Class of Spin-Stabilized Fin-Controlled Projectiles,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 2, 2013, pp. 517531. doi: 10.2514/1.56520 13 Appendix A Nomenclature m(t) ՜ Rocket mass function of time m ՜Rocket initial mass T(t) ՜ Thrust modulus function of time I୶ , I୷ ՜Rocket initial inertia moments in body axes X େୋ ՜ Rocket initial position of center of gravity xୣ , yୣ , zୣ ՜ Earth axes definition xୠ , yୠ , zୠ ՜ Body axes definition x୵ , y୵ , z୵ ՜ Working axes definition ሬሬԦ ՜ Drag Force D ሬԦ L ՜ Lift Force ሬሬሬԦ M ՜ Magnus Force ሬԦ ՜ Pitch Damping Force P ሬTԦ ՜ Thrust Force ሬሬሬԦ ՜ Weight Froce W ሬԦ C ՜ Coriolis Force ሬሬሬሬԦ CF ՜ Control Force ሬሬሬԦ M୰ ՜Magnus Force of the rotating part of the Rocket ሬሬԦ ՜ Overturn Moment O ሬሬሬሬԦ P ՜ Pitch Damping Moment ሬሬሬሬሬሬԦ M ՜ Magnus Moment ሬԦS ՜ Spin Damping Moment ሬሬሬሬሬሬԦ CM ՜ Control Moment ሬሬሬሬሬሬሬሬԦ M ୰ ՜ Magnus moment of the rotating part of the Rocket ሬሬሬሬԦ୰ ՜ Gyroscopic Moment of the rotating part of the Rocket G d ՜ Rocket Caliber ɏ ՜ Air Density Sୣ୶୮ ՜ Reference Surface for fins Cୈబ ՜ Drag Force Linear Coefficient Cୈಉమ ՜ Drag Force Square Coefficient Cಉ ՜ Lift Force Linear Coefficient Cಉయ ՜ Lift Force Cubic Coefficient C୫ ՜ Magnus Force Coefficient C୯ ՜ Pitch damping force coefficient ሬሬሬሬሬሬሬሬԦ ߱ ᇱ ՜ angular speed of the joint body (rocket and fuse) ሬሬሬሬሬሬԦ ܮᇱ ՜ momentum of the joint body (rocket and fuse) Cಉ ՜ Overturning moment linear coefficient Cಉయ ՜ Overturning moment cubic coefficient C౧ ՜ Pitch damping moment coefficient C୫୫ ՜ Magnus Moment Coefficient Cୱ୮୧୬ ՜ Spin damping moment Coefficient C (Ƚ) ՜ Fin Normal force Coefficient Ƚ ՜ Total Angle of Attack ሬሬሬԦ ՜Rocket Velocity expressed in earth (j=e), body (j=b) v or working (j=w) axes ሬሬሬԦ x ՜ Rocket nose pointing vector expressed in earth (j=e), body (j=b) or working (j=w) axes ሬሬሬԦ L ՜Rocket Angular Momentum expressed in earth (j=e), body (j=b) or working (j=w) axes ሬሬሬሬሬԦ g ୵ ՜ Gravity Vector in working axes AZ ՜ Initial Azimuth AZ୧୬୧୲୧ୟ୪ ՜ Correction Parameter for initial Azimuth A େ୭୰୰ ՜ Azimuth needed to correct trajectory ሬሬԦ ȳ ՜ Earth angular speed vector p, q, r ՜ Angular speed components of the fuse p୰ ՜ Angular speed of the rotating part of the rocket Ԅୡ ՜ Control Force Angle Ԅ ՜ Roll Angle Ʌ ՜ Pitch Angle ɗ ՜ Yaw Angle GNCୡ୲ ՜ Activation Parameter for GNC Ʌୣ୰୰ ՜ Pitch Error ɗୣ୰୰ ՜ Yaw Error N ՜ Proportional Navigation Law Constant X ୮ଵ౭ , X ୮ଶ౭ , X ୮ଷ౭ ՜Line of Sight Vector Components expressed in working axes ሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬሬԦ X ୟ୰ୣ୲౭ ՜ Target Position expressed in working axes ሬሬሬሬሬԦ X ୵ ՜ Rocket Position expressed in working axes K ՜ PID Proportional Constant K୍ ՜ PID Integral Constant Kୈ ՜ PID Derivative Constant acc୶ୠ , acc୷ୠ , accୠ ՜Rocket Acceleration Expressed in Body Axes 14 Appendix B Figure B.1 represents impact point dispersion patterns for each of the ballistic and controlled cases. Also, the circular error probable (CEP) is observed. Not filled points represent the impacts of controlled flights while filled points the impacts of ballistic flights. Note that controlled flights are biased downrange of target if they are out of CEP. The reason for that is controlled trajectory through Proportional Navigation requires energy in order to keep glide path. For unfavorable random conditions on Montecarlo simulations this required energy is taken from rocket speed, as there is not thrust during guided phase. This is translated into a loss of range for unfavorable launching conditions. Incidentally, most of the controlled shots are inside controlled CEP as it is reasonable. Fig. B.1 Impact point dispersion patterns for ballistic and controlled flights 15
