JPH0587585A - Atitude detector by strapdown system - Google Patents

Abstract

PURPOSE:To simplify the constitution and maintain detection accuracy by constituting of two gyros, two accelerometers and a magnet azimuth detector. CONSTITUTION:From a sensor block part 1' to an operator 2', X, Y acceleration signals AX, AY from X, Y accelectrometers 11A, 12A, X, Y gyro signals GX, GY from X, Y gyros 11G, 12G and azimuth signal H from a magnet azimuth detector 13Y' are sent. In the operator 2', by using the signal H in a coordinates corrector 30 and a Z gyro calculator 31, an equivalent Z gyro signal GZ and corrected azimuth signal HC are obtained. Also by using signals AX, AY in a Z acceleration operator 40, an equivalent Z acceleration signal Az is obtained so that a posture angle transmission role angle signal ROL, a pitch angle signal PIT and an azimuth signal YAW are calculated. X, Y gyro bias signals Bx, By included in the signals GX, GY are estimated by operation in a gyro bias operator 50 and is subtracted from the signals GX, GY in a gyro bias correction part 51.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a strapdown type attitude detection device using a gyro, an accelerometer and a magnetic direction detector.

[0002]

2. Description of the Related Art As a conventional strap-down type posture detecting device, one shown in FIGS. 5 and 6 has been proposed. FIG. 5 is a block diagram showing an entire example of a conventional strapdown-type posture detection device. This strapdown-type posture detection device includes a sensor block unit 1 and a calculation unit 2. The sensor block unit 1 of FIG. 5 has a mounting shaft 10 that matches the right-handed orthogonal three-axis coordinate system (in FIG. 5, the Z-axis direction is the downward vertical direction, and the arrow in the figure indicates the detection amount of each sensor (+)). Input direction of X), the X accelerometers 11A and X for detecting acceleration in the X axis direction are provided on the X axis.
An X gyro 11G that detects an angular velocity around the axis is attached, and a Y accelerometer 12A that detects an acceleration in the Y axis direction and a Y gyro 12G that detects an angular velocity about the Y axis are attached to the Y axis. A Z accelerometer 13A for detecting acceleration in the Z axis direction, a Z gyro 13G for detecting angular velocity around the Z axis, and a magnetic azimuth detector 13Y for detecting azimuth angle around the vertical axis in the direction of gravity acceleration are attached to the axis. Has been. The three-axis accelerometers 11A, 12A, 13A and the three-axis gyro 11G of the sensor block unit 1,
Output signals from the 12G and 13G and the magnetic azimuth detector 13Y are input to the arithmetic unit 2. The calculation unit 2 calculates using this input signal and outputs the roll angle signal ROL, the pitch angle signal PIT, and the azimuth angle signal YAW of the mounting shaft 10 of the sensor block unit 1 as the attitude transmission angle.

FIG. 6 is a block diagram showing the configuration of an example of the arithmetic unit 2 of FIG. The output signals of the three-axis gyros 11G, 12G, and 13G of the sensor block unit 1 are X gyro signal GX, Y gyro signal GY, and Z gyro signal G.
It is input to the CTM arithmetic unit 20 of the arithmetic unit 2 as Z (hereinafter, these three gyro signals may be referred to as Gi). In the CTM calculation unit 20, the input gyro signal Gi is used to change the coordinate system of the mounting shaft 10 from the local coordinate system (the X-axis direction is the north direction and the Z-axis direction is the gravity direction). A 3 × 3 matrix to be converted into a coordinate system) is calculated.

Furthermore, the attitude angle with respect to the local coordinate system is calculated by the elements of this 3 × 3 matrix, and the roll angle signal R with respect to the X axis is used.
The pitch angle signal PIT for the OL and Y axes and the azimuth angle signal YAW for the Z axis are calculated and output as the output of the calculation unit 2. However, 3 × 3 calculated by the CTM calculator 20
The matrix has an error in the posture angle due to the difference between the initial value of the 3 × 3 matrix used in the calculation and the initial value of the mounting shaft 10 of the sensor block unit 1. Regarding the horizontal component of the attitude angle error generated in the CTM calculation unit 20, the horizontal calculation unit 21
And the upright calculation unit 22 are used for correction.

A 3 × 3 matrix signal C from the CTM calculator 20
TMS and 3-axis accelerometer 1 of this sensor block 1
The output signals of 1A, 12A, and 13A are X acceleration signals A
As the X, Y acceleration signals AY, Z acceleration signals AZ (hereinafter, the three acceleration signals may be referred to as Ai), the calculation unit 2
Is input to the horizontal calculation unit 21. In the horizontal calculation unit 21, the three input acceleration signals Ai and matrix signals C
Using TMS, the X-axis direction attitude angle error ERRX and the Y-axis direction attitude angle error ERRY of the attitude angle error with respect to this local coordinate system are calculated and output. This attitude angle error ERR
X and ERRY are input to the standing up calculation unit 22. In the upright operation unit 22, the X-axis correction signal CX of the CTM operation unit 20
And the Y-axis correction signal CY
It is calculated using X and ERRY and output.

Further, the vertical component of the error of the attitude angle generated in the CTM calculating section 20 is corrected by using the azimuth calculating section 23. In the azimuth calculation unit 23, the output signal of the azimuth detector 13Y attached to the Z-axis of the sensor block unit 1 is input as the azimuth signal H via the calculation unit 2 and is calculated in the CTM calculation unit 20. The output azimuth signal YAW is also input, and the Z-axis correction signal CZ of the CTM calculator 20 is calculated and output using the input azimuth signal H and azimuth signal YAW. This X-axis correction signal C
X, Y axis correction signal CY and Z axis correction signal CZ are CTM
A loop that is input to the arithmetic unit 20 and corrects an error in the attitude angle is configured.

More specifically, when there is a difference between the initial value of the mounting shaft 10 of the sensor block unit 1 and the initial value of the CTM calculation unit 20 of the calculation unit 2, the reason why the error included in the attitude angle transmission finally disappears is explained. State. Mounting shaft 10 at this time
Is rotated γ _O around the Z axis, then β _O around the Y axis, and finally α _O around the X axis with respect to the local coordinate system. Is assumed to match the local coordinate system. The acceleration signal Ai at this time is approximately represented by the following equation.

[0008]

[Equation 1]

[0009] Here, g is the gravitational acceleration, T _O is 3 × 3 matrix, is as shown by the number 2.

[0010]

[Equation 2]

Here, α _O , β _O , and γ _O are the mounting shaft 10
It is a small angle with respect to the local coordinate system of. The initial value of the matrix signal CTMS of the CTM calculator 20 is given by the following equation.

[0012]

[Equation 3]

In the horizontal calculation section 21, in order to calculate the error angle in the X and Y axis directions of the local coordinate system, the input matrix signal CTMS and acceleration signal Ai are multiplied by the following equation.

[0014]

[Equation 4]

Here, ERRi is X, Y. This is the error angle of the Z axis. In this equation 4, the X-axis error angle is detected as −β _O g and the Y-axis error angle is detected as α _O g. Error angles ERRX and ERRY of horizontal components detected by the horizontal calculation unit 21
Is input to the stand-up calculator 22. In the stand-up calculation unit 22, using the input error angles ERRX and ERRY,
After the calculation of the following equation, the X-axis correction signal CX and the Y-axis correction signal CY are output.

[0016]

[Equation 5]

[0017]

[Equation 6]

Here, μ is a proportional gain. Magnetic azimuth detector 13Y attached to the Z-axis of the attachment shaft 10
The azimuth signal H output from the CTM calculator 20 and the azimuth angle signal YAW transmitted from the CTM calculator 20 are input to the azimuth calculator 23. Since the magnetic azimuth detector 13Y attached to the Z axis of the attachment shaft 10 detects the azimuth around the vertical axis of the gravitational acceleration direction, this azimuth is the same as the azimuth signal YAW for the local coordinate system. It has a coordinate system.

In the azimuth calculation unit 23, the azimuth signal H and the azimuth angle signal YAW are calculated by the following equation to obtain the Z-axis correction signal CZ.

[0020]

[Equation 7]

[0021]

[Equation 8]

[0022]

[Equation 9] Here, τ is an azimuth time constant.

The X-axis correction signal CX is obtained from Eq. 5, the Y-axis correction signal CY is obtained from Eq. 6, and the Z-axis correction signal is obtained from Eq. CTM calculator 20
, The input correction signals CX, CY and C of the three axes
Z is used to create a modified matrix CTMC, and the modified matrix CTM
By using the C and the CTM signal CTMS, the calculation of the following equation can be performed to correct the attitude angle error.

[0024]

[Equation 10]

Here, t represents the present time, and Δt
Represents the time update interval, (t + Δt) represents the next time (t + Δt) from the current time t, and (t, Δt)
Represents the amount of change between time t and time t + Δt.

[0026]

[Equation 11]

Since the equation 11 is described by a discrete matrix, if the equation is newly rewritten in the form of the differential equation, the error of the attitude angle is as follows.

[0028]

[Equation 12]

Therefore, if the equation 12 is solved,

[0030]

[Equation 13]

Here, the subscript O represents an initial value. Therefore, when the CTM calculation unit 20 corrects the formula 11, the posture angle error included in the matrix signal CTMS is represented by the formula 13, and its steady value becomes zero, and the posture angle (roll Angle ROL, pitch angle PIT and azimuth YA
The component due to the error due to the difference between the initial value of the mounting axis included in W) and the initial value of the CTM calculator 20 finally disappears, and the attitude angle becomes a value with respect to the local coordinate system of the mounting axis 10. Next, a method of calculating a 3 × 3 coordinate conversion matrix for converting the coordinate system of the mounting axis 10 into the local coordinate system by using the input 3-axis gyro signal Gi in the CTM calculator 20 will be described. .. The 3-axis gyro signal Gi is an angle obtained by integrating the angular velocity over time (t, t + Δt).
Expressed as a formula,

[0032]

[Equation 14]

Here, ωi is the angular velocity detected by the X-axis, Y-axis and Z-axis gyros. 3-axis gyro signal Gi
Is the change amount at the interval Δt, and this change amount is calculated as the interval Δt.
The coordinate transformation matrix is obtained by accumulating each time as in the following equation. That is,

[0034]

[Equation 15]

Here, CTMS is a 3 × 3 coordinate transformation matrix Γ (t, Δt), which is an angle matrix changed in time [t, t + Δt],

[0036]

[Equation 16]

When t = 0, CTMS (0) is an initial value, and from Equation 3,

[0038]

[Equation 17] Becomes

In practice, the calculation of Γ (t, Δt) of equation 15 and CTM (t, Δt) of equation 10 described above is executed in the CTM arithmetic unit 20. Newly, the operation is described as follows.

[0040]

[Equation 18]

Further, the roll angle signal ROL, the pitch angle signal PIT, and the azimuth angle signal Y transmitted from the CTM calculator 20.
The method of calculating the AW is as follows. First, the coordinate axes of the coordinate transformation matrix are made to coincide with the local coordinate axes, the azimuth angle φ is rotated about the Z axis, and then the pitch angle θ is rotated about the Y axis.
The 3 × 3 coordinate transformation matrix CTMS when rotated and finally rotated by the roll angle φ is as follows.

[0042]

[Formula 19]

Therefore, the posture angle transmission signals are given by the following equations from the equation (19).

[0044]

[Equation 20]

[0045]

However, in the above-mentioned conventional device, since the strap-down system is constituted by using three gyros and three accelerometers, the mounting mechanism portion of the sensor block portion 1 is There is a problem in that it becomes complicated, and there are restrictions in weight reduction and size reduction. More importantly, since the price of the gyro and the accelerometer supports most of the price of the device itself, there is an inconvenience that the conventional device cannot easily reduce the price. Further, if the gyro used does not have a very small drift change due to temperature change, vibration, etc., there is a disadvantage that the posture change transmission includes an error due to the drift change. In view of this point, the present invention is to provide a novel strap-down type posture detection device that has improved the above-mentioned inconvenience.

[0046]

As shown in FIGS. 1 and 2, for example, as shown in FIGS. 1 and 2, a strap-down type posture detecting device according to the present invention has a right-handed orthogonal triaxial coordinate 0-XYZ in a mounting shaft,
An X accelerometer 11A for detecting acceleration in the X axis direction and an X gyro 1 for detecting angular velocity around the X axis are provided on the X axis.
1G is attached, a Y accelerometer 12A for detecting acceleration in the Y-axis direction and a Y gyro 12G for detecting angular velocity around the Y-axis are attached to the Y-axis, and a Z-axis is attached to the Y-axis. Magnetic azimuth detector 13 for detecting the azimuth angle of
A sensor block portion 1'on which Y'is attached, and X and Y accelerometers 11A and 12A of this sensor block portion 1 '.
The X and Y acceleration signals output from the X and Y gyros 11G and 12G, and the azimuth signal output from the magnetic azimuth detector 13Y 'are input, and the attitude angle and roll are input. A calculation unit 2'which calculates and transmits an angle signal, a pitch angle signal and an azimuth signal.
It is composed of

Further, the strap-down type posture detecting device of the present invention is, for example, as described above with reference to FIGS.
The function of calculating the azimuth angle about the vertical axis in the direction of gravity acceleration, the function of calculating the angular velocity of the mounting axis about the Z axis, the function of calculating the acceleration in the Z axis direction, the X and Y gyro signals. Is provided with a function of estimating and calculating the X and Y gyro bias components and a function of removing the X and Y gyro bias components.
And 12A, X and Y gyros 11G and 12G, respectively, so that the strap-down method can be used.

[0048]

In accordance with the present invention, X and Y accelerometers 11A and 1
Attitude angle, roll angle signal, pitch based on X and Y acceleration signals output from 2A, X and Y gyro signals output from X and Y gyros 11G and 12G, and azimuth signals output from magnetic azimuth detector 13Y '. Since the angle signal and the azimuth signal are obtained, the two gyros 11
With the configuration of G, 12G, the two accelerometers 11A and 12A, and the magnetic azimuth detector 13Y ', it is possible to obtain the attitude detecting device with the same accuracy as the conventional one.

Further, according to the present invention, since it has a function of estimating and removing the bias component of the gyro to be used, a relatively small and inexpensive gyro such as a vibration gyro or a gas rate gyro can be used.

[0050]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT An embodiment of the strap-down type posture detecting device of the present invention will be described below with reference to FIGS.
In FIGS. 1 and 2, parts corresponding to those in FIGS. 5 and 6 are designated by the same reference numerals, and description thereof will be omitted. 1 is different from the conventional example of FIG. 5 in that a Z accelerometer 13A and a Z gyro 13G, which are attached to the Z axis of the attachment shaft 10 of the conventional sensor block section 1, are included in the sensor block section 1'of this example. A magnetic azimuth detector 13Y, which is removed from the Z axis of the mounting shaft 10 and detects the azimuth angle of the mounting shaft 10 around the Z axis, instead of the magnetic azimuth detector 13Y that detects the azimuth angle around the vertical axis of the gravitational acceleration direction. 5 is different from the conventional example of FIG. 5 in that in the calculation unit 2 ′, the coordinate correction unit 30, the Z gyro calculation unit 31, and the Z acceleration calculation unit 4 are included.
0, a gyro bias calculation unit 50, and a gyro bias correction unit 51 are newly provided.

In FIG. 1, an X accelerometer 11A for detecting an acceleration in the X axis direction and an X for detecting an angular velocity around the X axis are provided on the X axis of the mounting axis 10 which coincides with the right-handed orthogonal three-axis coordinate system. A gyro 11G is attached, and a Y accelerometer 12A and a Y accelerometer 12A for detecting acceleration in the Y axis direction are attached to the Y axis.
A Y gyro that detects the angular velocity around the axis is attached,
The Z-axis is a sensor block unit 1'where a magnetic azimuth detector 13Y 'for detecting the azimuth angle around the Z-axis is attached.
From the X acceleration signal AX from the X accelerometer 11A and the Y acceleration signal AY, X from the Y accelerometer 12A.
The X gyro signal GX from the gyro 11G, the Y gyro signal GY from the Y gyro 12G, and the azimuth signal H'from the magnetic azimuth detector 13Y 'are supplied. The X acceleration signal AX and the Y acceleration signal A supplied to the calculation unit 2 '
The Y, X gyro signal GX, the Y gyro signal GY, and the azimuth signal H'are used for calculation, and the posture angle of the mounting shaft 10 of the sensor block unit 1 ', the roll angle signal ROL, the pitch angle signal PIT, and the azimuth signal YAW are calculated. Is output as the attitude transmission angle. A more detailed operation of this example will be described with reference to FIGS. 2, 3, and 4.

FIG. 2 is a block diagram of the arithmetic unit 2'of this example. In the calculation unit 2 ', the operations of the CTM calculation unit 20, the horizontal calculation unit 21, and the stand-up calculation unit 22 are the same as those in FIG. 6, and therefore the description thereof will be omitted, and the newly provided coordinate correction unit 30, Z gyro calculation unit 31, and Z acceleration. The operations of the calculation unit 40, the gyro bias calculation unit 50, and the gyro bias correction unit 51 will be described. The coordinate correction unit 30 uses the input azimuth angle signal H ′ and the roll angle signal ROL and pitch angle signal PIT input from the CTM calculation unit, and the corrected azimuth signal HC of the azimuth angle of the local coordinate system, An azimuth equivalent signal Hz of the coordinate system of the mounting shaft 10 is calculated and output. The calculation of the coordinate correction unit 30 is the following formula.

[0053]

[Equation 21]

[0054]

[Equation 22]

Here, f ₁ and f ₂ are functions depending on the variables in the parentheses. The correction signal HC is output to the azimuth calculation unit 23, and the azimuth equivalent signal Hz is output to the Z gyro calculation unit 31. In the Z gyro operation unit 31,
The equivalent Z gyro signal Gz ′ is calculated using the input azimuth equivalent signal Hz and output to the CTM calculator 20. This equivalent Z gyro signal Gz 'is obtained by obtaining the difference between the previous azimuth equivalent signal Hz (t-Δt) and the current azimuth equivalent signal Hz (t). Therefore, this equivalent Z gyro signal Gz 'is expressed by the following equation.

[0056]

[Equation 23]

Here, Δt is a calculation time interval.
Since the equivalent Z acceleration signal Az ′ is obtained by the equation 23, the arithmetic operation of the CTM arithmetic unit 20 using the triaxial gyro signals becomes possible. In the Z acceleration calculation unit 40,
Using the input X acceleration signal AX and Y acceleration signal AY, the equivalent Z acceleration signal Az 'is calculated to calculate the horizontal calculation unit 2
Output to 1. First, the mounting shaft 1 of the sensor block 1 '
When the coordinate system of 0 is stationary, the acceleration that acts is only the gravitational acceleration, so referring to FIG.

[0058]

[Equation 24]

Here, g is the acceleration of gravity. Therefore,
From Eq. 22, the equivalent Z acceleration signal Az 'is obtained by the following equation.

[0060]

[Equation 25]

Or, if approximated,

[0062]

[Equation 26]

[0063] The operations of the gyro bias calculation unit 50 and the gyro bias correction unit 51 will be specifically described. First, the error correction model of the attitude detection device shown in FIG. 1 is as follows.

[0064]

[Equation 27]

[0065]

[0066]

[Equation 28]

[0067]

[0068]

[Equation 29]

[0069]

[0070]

[Equation 30]

[0071]

[0072]

[Equation 31]

[0073]

[0074]

[Equation 32]

It is

[0076]

[Expression 33]

At this time, the influence of the X, Y gyro bias signals B _X , B _Y disappears on the right side of the equation 24.

[0078]

The operations of the gyro bias calculation unit 50 and the gyro bias correction unit 51 will be described below.

[0080]

[Equation 34]

Also, the mounting shaft 1 in the sensor block 1 '
When the coordinate system of 0 is moving with relatively large acceleration,
The acting acceleration is gravitational acceleration and motion acceleration.
Therefore, the equivalent Z acceleration signal Az ′ obtained by the calculation of the equation 25 or the equation 26 in the Z acceleration calculator 40 using the detected X and Y acceleration signals AX and AY is the correct value of the gravitational acceleration of the Z axis. do not become. Therefore, X of the mounting shaft 10,
An X speed meter and a Y speed meter are provided on the Y axis.
Speed signals VX and VY for the X and Y axes from the speedometer
The equivalent Z acceleration signal Az ′ that is not affected by the motion acceleration is obtained by performing the calculation of the following equation using That is, in the Z acceleration calculation unit 40, the equation 25 is modified to

[0082]

[Equation 35]

Then, by modifying equation 26,

[0084]

[Equation 36]

It is assumed that Due to the above, the mounting shaft 10
X, Y gyro 11G, 12G, X, Y accelerometer 1
The 1A and 12A and the magnetic azimuth detector 13Y 'can constitute the same attitude detecting device as the conventional one. Note that the present invention is not limited to the above-described embodiments, and needless to say, various other configurations can be adopted without departing from the gist of the present invention.

[0086]

According to the present invention, the same accuracy as that of the conventional attitude detecting device can be realized with the structure of two gyros, two accelerometers and the direction sensor, and there is an advantage that the structure is simple. In the conventional attitude detection device, the total price of the three gyros, the three accelerometers, and the azimuth sensor occupies most of the price, so that one gyro and one accelerometer according to the present invention are used. The reduction has the benefit of directly reducing the price.

Further, according to the present invention, one gyro and one gyro
The reduction in the number of individual accelerometers simplifies the structure of the mechanical section, which can be expected to reduce the number of electronic circuits, and the reduction in the number of parts enables flexible mounting arrangements, which also facilitates weight reduction. There is a profit to be.

Furthermore, according to the present invention, since it has a function of estimating and removing the gyro bias component to be used, it is possible to use a relatively small and inexpensive gyro such as a vibration gyro and a gas rate gyro. Even if the magnetic azimuth detector used in accordance with the present invention detects the azimuth angle around the vertical axis in the direction of gravitational acceleration, the Z-axis of the mounting axis 10 can be provided. Even in the type that detects the azimuth angle, there is an advantage that it can be dealt with by making a slight correction. Further, when the posture detection device of the present invention makes a relatively large acceleration motion, it is mounted on the X and Y axes of the mounting shaft 10. By using the X and Y velocities of the X and Y velocimeters, it is possible to easily perform posture angle transmission that is not affected by acceleration.

[Brief description of drawings]

FIG. 1 is a configuration diagram showing an embodiment of a strap-down type posture detection device of the present invention.

FIG. 2 is a configuration diagram showing an example of a calculation unit 2'of the present invention.

FIG. 3 is a diagram for explaining the present invention.

FIG. 4 is a diagram for explaining a gyro bias calculation unit 50 and a gyro bias correction unit 51 of FIG.

FIG. 5 is a configuration diagram showing an example of a conventional attitude detection device.

FIG. 6 is a configuration diagram showing an example of a conventional calculation unit 2.

[Explanation of symbols]

 1'Sensor block section 2'Calculation section 11A X accelerometer 11G X gyro 12A Y accelerometer 12G Y gyro 13Y 'Magnetic direction detector 20 CTM calculation section 21 Horizontal calculation section 22 Standing calculation section 23 Direction calculation section 30 Coordinate correction section 31 Z gyro calculation unit 40 Z acceleration calculation unit 50 Gyro bias calculation unit 51 Gyro bias correction unit

Front Page Continuation (72) Inventor Shinichi Nakamura 2-16-46 Minami-Kamata, Ota-ku, Tokyo Within Tokimetsu Co., Ltd.

Claims (2)

[Claims] 1. An X-accelerometer for detecting acceleration in the X-axis direction and an X-gyro for detecting angular velocity around the X-axis are attached to the X-axis of the mounting axis that coincides with the right-handed orthogonal triaxial coordinate O-XYZ. A Y accelerometer that detects acceleration in the Y axis direction and a Y gyro that detects angular velocity around the Y axis are attached to the Y axis, and an azimuth angle around the Z axis is detected on the Z axis. A sensor block unit to which a magnetic azimuth detector is attached, X and Y acceleration signals output from the X and Y accelerometers of the sensor block unit, X and Y gyro signals output from the X and Y gyros, Then, using the azimuth signal output from the magnetic azimuth detector as an input, the attitude angle, roll angle signal, pitch angle signal and azimuth angle signal are calculated,
A strapdown-type posture detection device, characterized in that it is configured with a calculation unit for transmitting. 2. The strapdown-type posture detection device according to claim 1, wherein the computing unit has a function of computing an azimuth angle about a vertical axis in the direction of gravity acceleration, and an angular velocity of the mounting shaft about the Z axis. A function for calculating the acceleration in the Z-axis direction, a function for estimating and calculating the X and Y gyro bias components included in the X and Y gyro signals, and a function for removing the X and Y gyro bias components. A strapdown-type attitude detection device, wherein a strapdown-type calculation using two of each of the X and Y accelerometers and two of the X and Y gyros is possible.