Three Different Attitude Measurements of Spinning Projectile Only Using Magnetic Sensors

Three Different Attitude Measurements of Spinning Projectile Only Using Magnetic Sensors

Abstract—current micro-inertial sensors could not provide long-time stability attitude information for the high-spinning projectile because of the drift error. Meanwhile, the method of the navigation and the attitude measurement with respect to the earth's magnetic field is still auxiliary, and it can’t get the attitude angle information only by measuring the three-axis components of the geomagnetic field. In view of the flying characteristics of high-spinning projectile, three different attitude measurements only using magnetic sensors are researched. Through the comparative analysis, the calculating principle, the system composition, the applicable condition and the error range of these methods are explained. Besides, the semi-physical experiments are made to prove the effectiveness of the three attitude measurements. The experiment results indicate that only scalar arithmetic operations are required for the angular measurements and they have the day/night and all weather capability. The three different measurements have the same angle error range which within ±1 degree, but different attitude update rate.

At present, the attitude measurement of a moving body is involved in several fields, for example, in aerial and marine vehicles [1-3], robots, human pose tracking [4], etc. Especially for the military application, it is important to test the projectile flight attitude accurately, which can accelerate the research course and improve the performance of weapon. Accurate measurement of angular motions of spinning projectiles with on-board sensors has long been recognized as a daunting task. The fundamental requirements for such measurement are lightweight, small-size, low power consumption, and easy implementation. The current micro-inertial sensors have relatively low accuracy and the drift could cause remarkable attitude error [5]. Successful attitude measurement with inertial sensors requires the expensive gyroscopes and accelerometers with exceptionally accurate and complex filtering algorithm [6-9].

Geomagnetic field is a vector field as the earth's natural resources. It provides the natural coordinate system for navigation with its rich element such as strength, inclination, declination and gradient, etc. Recent advances in magnetic sensor technologies have resulted in devices small enough, rugged enough, and sensitive enough to be useful in systems capable of making high-speed, high-resolution measurements of attitude relative to magnetic fields when installed on free-flying bodies [10-11]. Because of its high reliability and anti-interference ability, attitude measurements with geomagnetic have became a hot spot in research of flying parameter measurement. In general, most of known systems giving orientations with respect to a magnetic field are based on the known geomagnetic field and require prior knowledge of the field. Because the three-component magnetic sensor cannot provide three independent equations, other methods are combined to calculate one of the three angles of yaw, pitch and roll to obtain another two. The problems make the magnetic sensor is still auxiliary in attitude measuring systems [12-17].

Thomas Harkins and David Hepner have designed an attitude measuring system for spinning bodies, called "MAGSONDE" only with magnetic sensors [18]. The "Zero Crossings Method" for the "MAGSONDE" has been provided in their report. On this basis, an attitude angle measurement based on the ratio of extremum of two orthogonal magnetic sensors is introduced in this paper and this "Extremum Ratio Method" is extended to "Three Orthogonal Ratio Method". Through the theory research and Semi-physical experiments, the three different attitude measuring methods that only using magnetic sensors were compared in detail. In all the three cases, we do not need to know the magnetic field strength, only scalar calculation is required. Meanwhile, the latter two methods have some advantages than Thomas Harkins’ method. All of the three methods satisfy the requirement of high-spinning projectile's attitude determination. The results have an important signification no matter for projectile design, projectile aerodynamics or fuze design. Potential application for these methods includes determination of angular motion histories of experimental, developmental and tactical projectiles.

Three kinds of different attitude measuring methods that are introduced in this paper all based on components of geomagnetic measuring by MEMS magnetic sensors. Before the analysis of measuring principle, the installation of magnetic sensors and the mathematical relationship between attitude angles and sensor's output should be introduced first. Assuming that the gravity center of the spinning projectile is at the origin of the o-xyz coordinate system which fixed in body frame, its axis of rotation on the x axis and its nose pointed in the + x direction.

Installation of magnetic sensors in coordinate o-xyz
The magnetic sensors Msx, Msy and Msz locate respectively along the x axis, y axis and z axis. The sensor Ms1 locate in the o-xy plane and orient at a non-zero angle λ from the spin axis x. According to coordinate system rotation matrix rules, when the attitude angles change, the field strength along the sensitive axes of two sensors are given by
In (1), the vector is the strength and direction of geomagnetic field. The is designated as the angle between and its projection of the projectile axis in o-xy plane. The is the sum of the declination and real yaw and the projectile roll angle is described by the . The true yaw and pitch can be get by calculated , and the local magnetic elements. There are three unknown parameters in (1), but no three independent equations. Therefore, one or more attitude angles must be known from other ways in order to calculation the rest of the attitude angles.

According to the flight characteristic of the high spinning projectile, some basic hypothesis is as follows:
1)The roll rate approximates constant in a spin cycle.
2)Velocity vector is in the firing plane all the time that is is invariable.
3)changes slowly with time relative to the roll rate.
With the above three hypothesis, the "Zero Crossings Method" was anew explained. Based on it, the other two new methods were introduced in this section.

3.1 Zero Crossings Method
The normalized field strength along the sensitive axis for two non-orthogonal sensors Msy and Ms1 throughout several roll cycles is plotted in Fig. 2 with , , . Denoting the two pairs of roll angles at the zero crossings for these sensors as and . By (1), with fixed and known and , the rotation angles between the above two pairs of roll angles at the zero crossings depend on .

Normalized Magnetic Field Strength Along Msy and Ms1
When the body has high roll rate and the changes slowly with time, the ratio as with are considered a constant in a spin cycle. Equation (2), (3) can be deduced from (1) like this,
In (2) and (3), . Equation (2), (3) show that: With fixed and known and , mean of Msy has no change but curve of Ms1 move up and down with different , so that it makes ratio and into corresponding relation. Besides , different make the output curves of Msy and Ms1 translation left and right with the same phase, it has no influence on relationship between and . It follows that the value of ratio only depends on with fixed and known and . (

Ratio versus when and
The ambiguity arising from the symmetry of this ratio about is easily resolved by checking the parity of the field along Ms1 when Msy=0. Sensors do not give roll angles at zero crossings but times at which these crossings occurred, the crossing times can also be used to directly compute instead of roll angles [19]. Thus, the combination of the ratio calibration curve and a parity check completely specifies the angle between the projectile axis and the magnetic field. During the body’s flight, the real-time could be calculated from the calibration curve and the real-time ratio .

3.2 Extremum Ratio Method
Not only the ratios of zero crossing, but also the ratios of maximums and minimums of the two magnetic sensors have corresponding relationship with and it is proved as follows: and change slowly with time compared with , so when Msy and Ms1 reach the maximum or the minimum, that is d Msy /dt=0 and d Ms1 /dt=0, we always have

It is seen from (4) that Msy and Ms1 reach the maximums and minimums at the same time, respectively. When Msy, or Ms1 reach the max-min values, the attitude angles must satisfy (4), so combining (1) and (4), the relationships between the ratios of the maximums and minimums and are
The flight projectile allows changing and with respect to each . That is, for each yaw angle and the corresponding range of , the curve for the ratio of peak and valley value of the two magnetic sensors can be made out by making the projectile rotate one circle. Fig.4 shows the curves of the ratios of maximums and minimums values versus when and Ratio of Extremum versus when and
During the flight, each attitude angle changes in every time and the sensors output actual values. The ratio of extremum could be got from the outputs of the sensors, then the in this occasion could be got from the curve of calibration according to the ratios. Due to the zero crossing exists between the maximum and the minimum, the at the zero crossing time can be obtained using the interpolation method. When Msy=0, we have
Substituting known and calculated into (7), then in this time could be solved. In projectile’s flight, is considered as invariable, but there are errors caused by this approximation method. With above calculated and (1), we can corrected yaw angle by the under Equation

3.3 Three Orthogonal Ratio Method
The “Extremum Ratio Method” makes use of the ratio of the extremum of the two non-orthogonal magnetic sensors Ms1 and Msy to get the attitude information. The thought can be extended to the ratio calculation of three pairwise-orthogonal magnetic sensors.
The output of the magnetic sensor Msx is a constant in a spin cycle under the assumptions of the flight characteristic of the high spinning projectile. When Msz reaches the maximum or the minimum, that is dMsz /dt=0 and, we have

From (4) and (9), we know that Msy and Msz respectively reach the maximums and minimums at different time. The time interval of each extremum is just right a quarter of a roll cycle. Fig.5 shows the ratio relationship of the Msx / Msz and Msx / Msz in A, B, C and D points respectively when , .

Figure 5. Sensor output when ,
Combining (4), (9) and (1), we can get the ratio of three pairwise-orthogonal magnetic sensors at the extreme value point of Msy and Msz

It is seen from (10) that there is the one-to-one correspondence between the ratio of three pairwise-orthogonal magnetic sensors at the extreme value point of Msy and Msz and for each fixed . So, we can also calculate the by the curve of calibration according to Rxymax/min and Rxzmax/min. When Msy=0, we have
The could be solved by substituting known and calculated into (7) and (11).The corrected calculation of the has the same steps as the "Extremum Ratio Method".

All of these three different attitude measuring methods are based on the mathematics corresponding relations between and the certain ratio of these magnetic sensors. Each of the three methods has its own characteristics in some ways with a link in principle. Their differences should be discussed through some comparative analysis.

4.1System Composition and Attitude Update Rate
Both “Zero Crossing Method” and “Extremum Ratio Method” must use four magnetic sensors in order to realize all the three attitude angles measurement. The four magnetic sensors contain a non-orthogonal installation magnetic sensor, which makes some difficulties in the installation of the sensors and the system building. In contrast, the “Three Orthogonal Ratio Method” only needs three pairwise-orthogonal magnetic sensors to achieve the same effect.
In a proper rotation cycle, the “Zero Crossing Method” has only one ratio, so it can work out only a group of attitude angles. The “Extremum Ratio Method” can get a pair of extreme value ratio, accordingly, it can get two groups of attitude angles. There are four characteristic ratios of the “Three Orthogonal Ratio Method” in a spin cycle. Its attitude update rate is four times of the former and twice of the latter.

4.2 Launch Window Range
The necessity of each of the magnetic sensors being orthogonal to the field during a roll cycle defines the range of the magnetic aspect angles within which the particular sensor configuration is able to operate. This applicability region is called the “Magsonde window” [20]. Both of “zero crossing method” and “extremum ratio method” require the Msy and Ms1 have the zero point. Meanwhile, the “Three Orthogonal Ratio Method” requires the Msy and Msz have the zero point.
When or , (1) is simplified into
For sensors Msy, whenever the sensor axis is orthogonal to the field Msy=0. Two possibilities exist, either or . In the first case, or , the axis of rotation is parallel to the magnetic field. In the latter case, the variation of field strength along the sensor axis is sinusoidal, and Msy=0 when or . Similarly, Msz=0 when or and or . For sensor Ms1, Solving (12) for the roll angles at which Ms1=0 yields.
The existence criterion of leads to the requirement that for the occurrence of an orthogonality condition
When or , solving (1) for the roll angles at which Msy=0 and Msz=0 yields, we get (7) and (11). For sensor Ms1, solving (3) for the roll angles at which Ms1=0.
The existence criterion for of leads to the requirement that . In generally, . If , . If , .
With the above analysis, we find that it is no matter what value of , the sensors Msy and Msz always have the zero crossing point. So, the “Three Orthogonal Ratio Method” does not be limited by “Magsonde window”. At the same time, the other two methods only suit for a particular launch widow which depends on the range of possible during the flight.

4.3 Measurement Blind Area
When the axis of rotation is parallel to the local geomagnetic vector ( or ) during the flight, the outputs of Msy and Msz stay at the zero value. At this situation, the true pitch angle is the negative or supplementary angle of the local inclination. When the axis of rotation is in a very small angle area around the local geomagnetic vector, the outputs of the Msy and Msz are such small that the useful signals basically are submerged in measurement noise. All of the three attitude measuring methods cannot have effective role in such an area called “Measurement Blind Area”. The inclination has different value in different latitude, so that the “Measurement Blind Area” changes with the definite location. Because of the A very short time in which the projectile stay in the “Measurement Blind Area”, the influence on the whole of measurement can be ignored. In practical projects, the problem can be solved by the forecast and filter algorithm.

4.4 Error Analysis
The reasons for the errors of these above attitude measurements are analyzed as follows:
1) Suppose that is invariable in the flight for the first estimation, which result in the errors of , to be big. This kind of error exists in all the three attitude measuring methods. The second correct method for eliminating the influence of the hypothesis in these calculation methods could be carried out by repeating calculation steps with the corrected instead of the hypothetical one.
2) In the three methods, the interpolation method is adopted in the calculation process of , i.e. using interpolation to estimate the value of in the pre-calibrated curve. This produces the error of and then leads to the error of when use (7) and (11) to estimate the value of . Meanwhile, the “Zero Crossing Method” in a proper rotation cycle only can get one . So, the faster changes of the in the flight leads the bigger error of this method. The attitude update rate of the “Three Orthogonal Ratio Method” is higher than the other two methods, accordingly, its error of the is the smallest one.
3) In the curve of the sensors output , the precision of interpretation of zero point is lower than the accuracy of extreme value estimate. The results of the “Zero Crossing Method” are two symmetrical values, the parity check needs in its calculation process. The two problems make the calculating precision of this method relatively low.
Through the above the error analysis, we have a conclusion like this: In the same conditions, the attitude error of the “Zero Crossing Method” is the biggest among the three different methods, the precision of “Three Orthogonal Ratio Method” is the highest one.

Based on the research of the theory algorithms, the experiments were performed to verify and compare these three attitude measurements. A semi-physical device was designed for the simulation experiments. The Hardware components diagram of the device is shown in Fig. 6. Fig. 7 illustrates this homemade prototype.

Hardware components diagram of the semi-physical device
The prototype is mainly comprised of a pair of Honeywell’s HMC1021/1043 magnetometers (single-axis/three-axis magnetic sensors with ±6 Gauss measurement range and about 8 μ Gauss noise level), a Texas Instruments’ ADS8365(16-Bit, 250ksps, 6-Channel, Simultaneous Sampling analog-to-digital converter), a CYGNAL’s C8051F320(Full Speed USB, 16kisp flash MCU), and an ATMEL’s AT45DB642D(64Mb serial-interface flash memory with SPI interface).The signals of the magnetic sensors was acquainted and stored in the flash memory. Then, through the USB port, these data was transferred to computer for attitude calculation. The Fig.8 shows the system block diagram of the experimental setup.

In experiments, the hardware prototype was fixed in a three axis turntable (See Fig. 9), it moved follow the turntable. SO, the yaw, pitch and roll motions of the device is the same as the turntable. Firstly, the calibration curves of the each method were made when . Then, semi-physical simulation experiments was processed under the condition that the device rotates with a constant roll rate and was stay at with . These above three attitude measuring methods were used to estimate all the three attitude angles. Fig.10 shows the real output signals of the four magnetic sensors after filter. The calculating errors of the for each method are showed in Fig.11. Table Ⅰ shows the detail results of the semi-physical simulation experiments. Note that the reference values of the pitch and the yaw during the measurement were 80 degree and 30 degree. The output roll angle values of the turntable were considered as the reference of the roll.
The hardware prototype was fixed in three axis turntable

Method Attitude Measuring Error
Attitude updata rate
Zero crossing ±1° ±1° ±1° 1 point per cycle
Extremum ratio ±0.5° ±1° ±0.8° 2 point per cycle
Three orthogonal ratio ±0.2° ±0.6° ±0.8° 4 point per cycle
It is seen from the results of experiment that the three different attitude measuring methods discussed in this paper all can keep the attitude errors less than 1 degree. The error of the “Zero Crossing Method” is relatively large. The Attitude updata rate of the “Three Orthogonal Ratio Method” is the highest.

In the paper, the theories of three different attitude measurements of spinning projectile only using magnetic sensors are discussed and models for the output of each sensor are presented. All of the three methods which only need scalar arithmetic operations have a series of merits and would not be affected by the weather or light. Through the comparison, the differences between the three methods in several sides are analyzed in detail. The semi-physical simulation experiment is done under the given condition and the results show that the measurement system is valid and the errors of the attitude angles are in the range of permission of the facts. These indicate that the measurement system is effective for high-g and high spinning projectile. The “Three Orthogonal Ratio Method” has some advantages in precision and attitude updata rate compared with the other two methods. The resulting angle data from these methods can be used with diagnostic tools for projectile aero ballistic characterization. The research in the paper provides a new theoretical basis for the attitude measurement and navigation with the geomagnetic.

The work is supported by Graduate research and innovation plan project of Jiangsu province of China (Project number: CX10B-119Z).
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