# Mathematical formulation, notations, flight dynamics and instrumentation

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## The concept of smart artillery shells

Gun-fired ammunition are still a prominent part of military arsenals. They are significantly cheaper than missiles, and can easily and promptly be de-ployed on various battlegrounds. So far, their main limitations is their lack of guidance capabilities, as they cannot be controlled once fired. This major limitation is being pushed back as some recently developed actuation tech-nologies have emerged over the last decades such as single mass ejection (or pyrotechnical thrusters [41] as pictured in Figure 1.2.1 and Figure 1.2.2) or the deployment of canards and have proven to be valuable means to deflect and to optimize the projectile trajectory [12]. A prime example is the fold-ing glide canards found on the M982 Excalibur, a 155 mm extended range guided artillery shell developed during a collaborative eﬀort between the US Army Research Laboratory (ARL) and the United States Army Armament Research, Development and Engineering Center (ARDEC). The fins are uti-lized to glide from the top of a ballistic arc towards the target. The same concept is being explored in the ISLs guided long range projectile concept pictured in Figure 1.2.3.
The projectiles considered here are rigid bodies with one central symme-try axis. Usually, they have a reference diameter , called the caliber. Most of the parameters can be deduced from it for similarly shaped projectiles, through a homothetic transformation.
Most shells have an ogive-shaped nose, a cylindrical central part, and possibly a tapering base (boat-tail). The length of gyro-stabilized shells commonly ranges between 4 and 5 calibers, the length of the nose can
vary between 2 5 and 4 calibers and its front can be round-shaped or flat.
Finally, the boat-tail typically has a length 0 8 and a diameter between 0 8 and 0 9 . A typical boat-tail shell with a flat nose is pictured in Figure 1.2.4.
Shells are fired by a cannon which provides them with an initial velocity, and most of the time a significant spin rate for stabilization purposes. After gun-fire, they follow a trajectory solely governed by the external forces and moments acting on them during their flight.
Depending on their shape (which in the scope of the thesis is always assumed to be rotationally symmetric), most shells are gyro-stabilized, i.e. submitted to a high spin rate, so that they are stable under normal flight conditions (see the classical gyroscopic stability criterion [72, Chapter 10]). Very often, the value of the spin-rate required for the stability overwhelms the rate gyro range of operation. To circumvent this, decoupled two-section fuse concepts have been developed, having a slowly (almost despun) rotating part containing sensors and possibly actuators. In this thesis, we will not consider these (relatively costly) solutions, and, instead, consider that the rate gyro is not present.

### Attitude estimation for smart shells applica-tions

In the context of navigation of smart artillery shells, the knowledge of the attitude is particularly useful as it makes it possible, in addition with 3-axis Accelerometer, to estimate the position of the shell in-flight, i.e. solving a navigation problem4. The attitude is also required as an input for control
4 for short time horizons, the aerodynamic forces measured by the 3-axis Accelerometer can be converted into forces in the inertial frame of reference and added to gravity to laws (guidance or terminal guidance controllers), and telemetry applications (e.g. antennas orientation to minimize data loss). It is often required in late parts of the flight where trajectory correction have to be made. A typical ballistic flight for smart shell is depicted in Figure 1.3.1. The steered gliding phase taking place after the apogee is a prime example where attitude information is useful.
Figure 1.3.1: Typical ballistic flight phases for smart artillery shells. (ISL)

#### Outline of the proposed solution

As explained earlier, classic attitude estimation methods can not work as-is onboard a smart shell. We will not use any rate gyro, but when needed, an estimate of the angular velocity will be developed (this estimation will be referred to as a « virtual gyro »). The 3-axis Magnetometer will be used as a body-frame measurement of the Earth magnetic field, whose coordinates 0 in the local frame are known Besides, an additional input will compensate for the missing direction measurement usually given by the 3-axis Accelerometer. The attitude will be represented under the form of a rotation matrix ˆ. A pictorial view of the estimation method is given in Figure 1.4.1.
The « virtual gyro » can be a simple estimation of the dominant roll rate, which will be shown to be easily determined using the large oscillations observed in both transverse accelerometers and transverse magnetometers signals5.
To compensate for the missing direction, one attitude angle will be di-rectly estimated. As will be explained, measuring only one direction makes one able to compute the attitude, up to a rotation by an unknown angle around the single known direction. If an additional « well-chosen » attitude angle is available, then the attitude estimation has only two isolated solu-tion, that can be discriminated easily. The angle under consideration is the pitch angle. It is obtained from the estimate of the velocity w.r.t. the air, as pictured in Figure 1.4.3, which gives an approximation of it under the form of the slope angle. The estimation method will be exposed in Chapter 4. The pitch angle serves as « additional input » for the attitude observer of Figure 1.4.1 as pictured in Figure 1.4.4. This will be treated in Chapter 5.
To compensate for the missing direction, one attitude angle will be di-rectly estimated. As will be explained, measuring only one direction makes one able to compute the attitude, up to a rotation by an unknown angle around the single known direction. If an additional « well-chosen » attitude angle is available, then the attitude estimation has only two isolated solu-tion, that can be discriminated easily. The angle under consideration is the pitch angle. It is obtained from the estimate of the velocity w.r.t. the air, as pictured in Figure 1.4.3, which gives an approximation of it under the form of the slope angle. The estimation method will be exposed in Chapter 4. The pitch angle serves as « additional input » for the attitude observer of Figure 1.4.1 as pictured in Figure 1.4.4. This will be treated in Chapter 5.
The manuscript is organized as follows.
Chapter 2 presents the mathematical notations employed to describe the flight dynamics of the shell under the form of a 6-degrees of freedom model of a rigid body subjected to aerodynamic forces and moments, and gravity. The two types of rotationally symmetric shells (155 mm and Basic Finner) under consideration in the thesis are described. The main equation governing the pitching and yawing motion of the shell is presented. The combined oscillations define an epicyclic motion. The set of strapdown sensors is described. Some data obtained during experimental tests serve to illustrate typical measurements observed in-flight.
In Chapter 3, the signals generated by the epicyclic motion of the shell are processed by frequency detection techniques. The frequency is related to the norm of the velocity w.r.t. the air of the shell. This relation is a key ingredient for the velocity estimator that is developed to account for observability issues near Mach 1.0.
In Chapter 4, the previously developed velocity estimation serves to es-tablish an estimation of the pitch angle of the shell. A simple linear time varying (LTV) formulation serves to establish the convergence of a Luen-berger observer, which can be replaced for sake of improved performance with an extended Kalman filter.
Chapter 5 develops an extension of the attitude complementary filter dealing with a single vector measurement and the knowledge of one angle. The convergence analysis is established.
Finally, Chapter 6 is devoted to the application of all the methods pre-sented above to real flight data, using solely on-board measurements. Com-parisons with high-fidelity measurements from a ground based position radar are provided.

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1 Context and problem statement
1.1 Introduction
1.2 The concept of smart artillery shells
1.3 Attitude estimation for smart shells applications
1.4 Outline of the proposed solution
1.5 Organization of the thesis
2 Mathematical formulation, notations, flight dynamics and instrumentation
2.1 Reference frames and Six-Degrees-of-Freedom description
2.2 Environment model
2.3 Projectile model: dimensional parameters and aerodynamic coefficients
2.4 Flight dynamics
2.4.1 Translational dynamics
2.4.2 Rotational dynamics
2.5 Onboard Sensors
2.5.1 Description of the embedded system
2.5.2 Detrimental effects and mitigation means Eddy currents
Misalignment
Fictitious forces
2.6 Preliminary estimation of the angular velocity around main axis
2.7 Shooting range and external instrumentation
2.8 Testcases considered in the thesis
3 Frequency analysis of the epicyclic rotational dynamics
3.1 Problem statement
3.2 Frequency content of the embedded inertial measurements
3.3 Instantaneous frequency detection: measuring varying frequencies
3.3.1 Definition of the frequency of interest
3.3.2 Envelope filter and FFT
3.3.3 Detecting peaks in the autocorrelation function
3.3.4 Frequency detection using super-resolution
3.3.5 Filtering the estimates
3.4 Design of an observer for the velocity from frequency measurements
3.4.1 System dynamics and output map
3.4.2 Observer design
3.4.3 Convergence analysis
3.5 Illustrative results
3.5.1 Reference velocity
3.5.2 Results
3.6 Conclusion
4 Slope estimation through an analysis of the velocity dynamics
4.1 Slope angle observer
4.1.1 Observer design
4.1.2 Simulation results
4.1.3 Experimental results
4.2 From the slope angle to the pitch angle
4.3 Conclusion
5 An attitude observer from 3-axis Magnetometer and pitch angle
5.1 A quaternion representation of the problem
5.2 Single-direction attitude complementary filter
5.2.1 Recalls on attitude complementary filter
5.2.2 Partial convergence using a single direction
5.3 Complementarity of pitch angle information and magnetic vector measurement
5.3.1 Reduction of the convergence set
5.3.2 A continuity property
5.4 Proposed observer
5.5 Assumptions on the flight
5.6 Main result
5.7 Proof of convergence
5.7.1 Asymptotic behavior
5.7.2 First case: Assumption 3 holds
5.7.3 Second case: Assumption 3 does not hold
5.7.4 Conclusion of the proof
5.8 Practical use of the main result
5.8.1 Observed asymptotic behavior
5.8.2 Interpretation of the solution q#
5.8.3 Practical initialization to ensure convergence towards actual attitude
5.9 Estimation results
5.9.1 Simulation results
On 155 mm shell
On Basic Finner
5.9.2 Experimental results
6 Experimental results of the proposed attitude observer using only on-board sensors
6.1 Multirate Kalman filtering of frequency estimators
6.2 Debiasing of the frequency estimate
6.3 Smoothing under convexity constraints
6.4 Slope and pitch estimation experimental results
6.5 Attitude observer results
7 Conclusion and perspectives
Appendix A Supplementary material
A.1 Transition matrices
A.2 Alternative angles
A.3 Approximation on the shell velocity
A.4 Material for Chapter 3
A.5 Calculations leading to q# expression in Chapter 5
A.6 An equivalence property for Chapter 5
Appendix B Various useful estimation methods
B.1 Complex argument method for single-axis rotation rate estimation
B.2 Linear constrained estimation
B.3 Background on MUSIC frequency estimation algorithm

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