A New Analytic Alignment Method for a SINS

Analytic alignment is a type of self-alignment for a Strapdown inertial navigation system (SINS) that is based solely on two non-collinear vectors, which are the gravity and rotational velocity vectors of the Earth at a stationary base on the ground. The attitude of the SINS with respect to the Earth can be obtained directly using the TRIAD algorithm given two vector measurements. For a traditional analytic coarse alignment, all six outputs from the inertial measurement unit (IMU) are used to compute the attitude. In this study, a novel analytic alignment method called selective alignment is presented. This method uses only three outputs of the IMU and a few properties from the remaining outputs such as the sign and the approximate value to calculate the attitude. Simulations and experimental results demonstrate the validity of this method, and the precision of yaw is improved using the selective alignment method compared to the traditional analytic coarse alignment method in the vehicle experiment. The selective alignment principle provides an accurate relationship between the outputs and the attitude of the SINS relative to the Earth for a stationary base, and it is an extension of the TRIAD algorithm. The selective alignment approach has potential uses in applications such as self-alignment, fault detection, and self-calibration.


Introduction
If the Strapdown inertial navigation system (SINS) remains static on the ground with limited vibrations, the attitude of the SINS with respect to the Earth can be obtained directly from the gravity and rotational velocity vectors of the Earth if they are not collinear [1][2][3][4]. This approach is called analytic alignment (AA). The problem is about finding the transformation matrix from one coordinate system to the other when the components of two abstract vectors are given in two different coordinate systems. TRIAD [5,6] is an algorithm that does just that. A minimum of two non-coplanar vector pairs are required for a solution using TRIAD algorithm. The TRIAD algorithm is applied to the AA for a SINS, and then it is called the analytic coarse alignment method [1][2][3][4]. If the vibrations are so severe that the outputs of the gyroscopes are much larger than the rotational velocity of the Earth, AA cannot be performed. There are many approaches to the alignment problem including the inertial frame alignment method [7,8], Kalman fine alignment [9][10][11], and compass alignment [12,13]. All of these methods require part or all of the inertial navigation process, where the velocity and attitude of the SINS update with the time. Because the average velocity of the SINS is at zero for the base in the presence of vibrations, the attitude could then be computed. These alignment methods for the base in the presence of vibrations can be classified as inertial navigation computational alignment (INCA). However, INCA requires more alignment time and involves massive calculation compared to the AA method. Self-alignment methods can be divided into AA and INCA based on the principles used.
Because vibration is common in actual systems, INCA has been widely studied for several decades [7][8][9][10]12]. There have been few studies on AA, mainly because it has difficulty achieving alignment in the presence of vibrations. Reference [1] presents a basic AA method, analytic coarse alignment. One additional vector can be generated with the vector cross product between the gravity and rotational velocity vectors of the Earth. Then, a 3 × 3 matrix can be formed given the three vectors. In a SINS, the attitude information is stored either as a coordinate transformation matrix or as attitude angles (roll, pitch and yaw). The coordinate transformation matrix can be obtained directly through a one-time matrix calculation. Jiang [2] noted that there are three other vectors that can be generated from the cross-product of the gravity and rotational velocity vectors. Among these five vectors, four lie in the plane of the meridian and the other is perpendicular to this plane. The analytic coarse alignment problem is then to form a 3 × 3 matrix that consists of three linearly independent vectors. There are six possible sets that can be used to build a 3 × 3 matrix for the analytic coarse alignment. Two simple and significant sets are analyzed in [2] in detail. Although both methods are derived from the same measurements of the local gravity and rotational velocity vectors, their error formulations are not completely identical. The six analytic coarse alignment methods and a direct method for a SINS on stationary base were compared using error analyses in [3]. The error characteristics of the direct method can be evaluated because the direct method is equivalent to one of the six analytic coarse alignment methods.
It has been the standard practice for many years to use all six outputs from the IMU for self-alignment. A new AA method called selective alignment in ground platforms is proposed which requires only three outputs from the IMU and a few properties from the remaining outputs such as the sign and the approximate value. The selective alignment method has potential uses in problems such as optimal AA, fault detection, and self-calibration.
The selective alignment method shows that two non-coplanar vector pairs are redundant for determining the transformation matrix, however, a minimum of two non-coplanar vector pairs are required for a solution using TRIAD algorithm. The selective alignment may be applied in other attitude determination problems such as spacecraft attitude estimation where TRIAD algorithm is a basic method [6,14]. The formulas for the selective alignment are given through geometrical analysis in this paper. The principle could also be obtained through algebra analysis and we give a brief description about this in a recent conference paper [15]. After all, the geometrical analysis for the selective alignment shows us a much more intuitive and figurative relationship between the IMU outputs and its attitude relative to the Earth on the stationary base.
The remainder of the paper is organized as follows: We review the analytic coarse alignment method in Section 2. The basic principle of selective alignment is presented in Section 3. To prevent a zero value in the denominator or a negative value in the radicand at certain extreme attitudes, certain adjustments are required, the details of which are presented in Section 4. The results of simulations and experiments to test the validity and the use of the selective alignment method are presented in Section 5. Section 6 provides concluding remarks.

Analytic Coarse Alignment
The analytic coarse alignment is an application of the TRIAD algorithm to the AA for a SINS [2,6]. The process of finding the matrix using TRIAD is as follows [2,6,16]. Let 1 u and 2 u denote the column vectors whose elements are, respectively, the components of the two abstract vectors when resolved in one coordinate system (typically a body frame), and let 1 v and 2 v denote the column vectors whose elements are, respectively, the components of the abstract vectors when resolved in the other coordinate system (typically a reference frame). The reference-to-body coordinate transformation matrix A satisfies: The algorithm calls for the computation of the following column vectors in the body frame: u u u u , 4 3 1 = × r r r , 5 3 2 = × r r r (2) and the following corresponding column vectors in the reference frame: of which four vectors are at the same plane except 3 s which is perpendicular to the plane. The coordinate matrix A can be obtained by 1 3 3 , , , , where , 1, 2, 4,5 i j = , and i j < . There are six different sets to yield A . When the SINS is static with respect to the Earth, there are only two inputs for the IMU: the gravity and rotational velocity vectors of the Earth, which are denoted by g and ie w , respectively, and are shown in Figure 1. A local ENU (East-North-Up) frame is used as the reference frame, or navigation frame, denoted by Neglecting sensor errors and base vibrations, the following measurements are available: where b f contains the outputs of a three-axis accelerometer with outputs of a three-axis gyroscope with and g is the gravitational with ie w and L denoting the rotation rate of the Earth and the latitude, respectively, and b n C is the navigation-to-body coordinate transformation matrix. From Equations (4) and (5), the following two equations can be obtained, and these are typically used in analytic coarse alignment methods [2,3]. 1 , , , ,

Selective Alignment
If one of the IMU outputs, y f for example, is known, the angle between the b y -axis and g can be computed. If y w is also given, the angle between the b y -axis and ie w can be determined. As shown in O x y z − frame can be obtained if one more axis is computed. For example, if z w is also known, the b z -axis will be on the conical surface with its center at ie w . In addition, because the b z -axis is normal to the b y -axis, the b z -axis must be in the plane that is normal to the b y -axis. There are usually two lines of intersection of a plane and a conical surface that represent the b z -axis, as shown in Figure   y -axis will be formed by the intersections of the two conical surfaces; (b) Two lines that represent the b z -axis will be formed by the intersection of a plane and a conical surface.

Derivation
Now let us derive mathematical expressions for the four solutions of be coordinates in the navigation frame, and let be coordinates in the body frame. After ascertaining can be obtained by Assume three outputs of the IMU are given, which are y f , y w , and z w . If y w is given, as shown in Referring to Figure 3b, in the navigation frame the following equation holds: If y f is given, the angle between OB  and the z -axis is known, which implies Combining Equations (10) and (11) Solving Equation (12) Similarly, if z w is given, we have cos sin As shown in Figure 3a, because OC  is the unit vector of the positive b z -axis and OC OB ⊥   , the following equations hold Solving Equation (15) yields two sets of solutions 2 tan cos Then, OA   can be obtained as shown in Equation (17) OA Hence, from Equation (7), b n C can be obtained. There are four sets of solutions for b n C given y f , y w and z w . Next, we consider the case where y f , y w and x f are given. Likewise, given x f rather than z w , we have Because OA   is the unit vector of the positive b x -axis and OA OB ⊥    , the following equations hold Therefore, OC  is given by Hence, b n C can be obtained from Equation (7). There are four sets of solutions for b n C given y f , y w , The coordinate transformation matrix b n C has been derived in two typical cases. For both cases, one output of the three-axis accelerator and one output of the three-axis gyroscope, both in the same axis, should be selected. The case where the remaining measurement is another output of the three-axis gyroscope is denoted by FWW, e.g., y f , y w , and z w , and the case where the remaining measurement is another output of the three-axis accelerator is denoted by FWF, e.g., y f , y w , and x f .
There are six possible sets for FWW and six possible sets for FWF, as shown in Table 1.  Output 1 x For FWW, the three selected outputs are denoted as f α , w α and w β , where the subscripts α , β and γ represent x , y or z and should be different. The body frame is given by where a, b and c in Equation (23) and N in Equation (24) denotes For FWF, the three selected outputs are denoted as f α , w α and f β . The subscripts α , β and γ denote x , y or z and should be different. The body frame is given by Equations (22), (24), and (25): where a , b , and c in Equation (25)  2 a x y f b x z g f c y z y g

Choice of Solutions
After ascertaining n C can be obtained from Equation (7 (5), which are shown in Table 2. It is easy to discern that the attitude for Set 1 in Table 2 is the correct solution by comparing the calculated and measured values of the outputs using one of the following sets of criteria: The values compared will depend on the outputs selected. Through the selective alignment method, the attitude can be determined from any three outputs of the IMU in Table 1 in addition to at least one force component and one angular velocity component in the same axis. Some limited information from the remaining three outputs such as the signs and approximate value is also essential. Furthermore, if, for example, x f , y f , and z w are known, z f can be calculated from Equation (26). Then, the attitude by the selective alignment method can be obtained.

Exception in Selective Alignment
In practice, there are errors in the IMU outputs such as bias, and vibrations in the base will also affect the IMU outputs and thus the alignment. So, it is possible that x α in Equation (22) will have an imaginary value, and thus there is no solution. Furthermore, there are singularities in Equations (23) and (25) when x α and y α equal zero, respectively. Thus, the problems are divided into two classes, those in which the denominator should be nonzero and those in which the radicand should be nonnegative. We consider the denominator problem first.

Nonzero Denominator
For FWW, if x α equals zero, from Equation (15) In Equation (28) α -axis coincides with ie w . Thus, the β -axis should be in the plane that is perpendicular to ie w . Because w β was also selected and must be zero, the β -axis must also be in the plane that is perpendicular to ie w . Two coincident planes have an infinite number of lines of intersection, so only the b α -axis can be obtained in this particular case.
For FWF, if y α is equal to zero, from Equation (19)  = , ,− , which indicates that the α -axis coincides with g . Thus, the β -axis should be in the plane that is perpendicular to g . Because f β was selected and must be zero, the β -axis must also be in the plane that is perpendicular to g . Two coincident planes have an infinite number of lines of intersection, so only the b a -axis can be obtained in this particular case.

Nonnegative Radicand
To ensure that the value of x α in Equation (22) is not imaginary, ( ) 2 2 y z α α + should be less than 1.
However, it is possible that f α can be slightly greater than g because of sensor errors or base vibrations. And, errors in w α may also result in a negative radicand. Constructing algebraic rules to prevent a negative value of the radicand is complicated and abstract. Graphically, a negative value in the radicand occurs when the two conical surfaces in Figure 2a do not intersect or the plane and the conical surface in Figure 2b do not intersect. Therefore, negative values in the radicand can be avoided if these surfaces intersect. If the angle between the b α -axis and g is η , then cos f g α η = . Given w α and denoting the angle between the b α -axis and ie w as θ , we have cos ie w w α θ = . Assuming η is accurate, θ must lie in a certain range that depends on η , or the two conical surfaces shown in Figure 2a  The range of θ based on η is given by   In this case, g and ie w can be considered constants. Given η and θ , we can plot the range of w α as a function of f α . It follows that , must be within the ellipse shown in Figure 5b to ensure that the two conical surfaces in Figure 2a intersect and the radicand in Equation (22) will not be negative. After ascertaining the b α -axis, given another w β or f β , the b β -axis can be obtained. As shown in Figure 2b, the b β -axis is formed by the intersection of the conical surface and the plane. As shown in Figure 6, MN is a plane normal to the b α -axis. The conical surfaces AOA′ and COC′ do not intersect the plane MN , but the conical surface BOB′ does. It follows that only if B is in the area between OM and ON ′ will the conical surface BOB′ intersect the MN plane. In other words, the conical surface BOB′ is defined by w β or f β , which should satisfy If Equation (32) holds, the conical surface resulting from f β or w β intersects the plane normal to the b α -axis and the radicand in Equations (23) and (25) will not be negative.

Exception Handling in the Algorithm
To ensure that the selective alignment method performs robustly, various exceptions must be addressed. In the first stage, where the α -axis is determined, if f α is slightly greater than g because of bias in the accelerometer or base vibrations when the α -axis is nearly parallel to g , f α should be if not, the algorithm substitutes for f β or w β so that the equality holds.
In practice, the algebraic operations may not result in values that are identically zero because of the limits of floating-point operations on the computer, so values within a certain tolerance around zero should be considered zero in the algorithm. We use 12 12 10 10 x − − − < < to judge if x is equal to zero. Usually, four results can be obtained from the three selective outputs based on the selective alignment. There might be only two results or one result when there is only one intersection in Figure 2. If no intersection occurs because of bias errors or vibration, the value of the relevant selected output is changed to ensure an intersection. Because all six original outputs are given, the correct value of the attitude is picked out by minimizing the following cost function for ease of computer programming, although two signs or one approximate value from the remaining outputs could be enough to obtain the right result, as explained in Section 3.2.
' ' is the assignment symbol.
where b w and b f are the original outputs of the IMU, b n C is computed from Equation (7), which may have four different values, and ⋅ is the norm operator.
To handle exceptions, a binary number with five bits, 5  ; otherwise, α -axis → g (The symbol " → " denotes "approximate to") The exceptions rarely happen unless the SINS is level (one of the body axes is along the direction of the gravity) or one of the body axes is parallel to the axis of rotation of the Earth. If 5 1 b = , the attitude cannot be obtained. The other four exceptions arise because of measurement errors or base vibrations, which do not significantly affect the selective alignment after corresponding exception handling. Figure 7 shows a flowchart for the selective alignment algorithm.

Simulations and Experiments
Simulations and experiments were conducted to validate the performance of the proposed selective alignment method.

Static Alignment Example
Similar to the coordinate transformation matrix b n C , the Euler angles, i.e., roll, pitch, and yaw, can also be used to express the attitude [4].
Simulations were conducted to test the selective alignment method. Because the alignment is usually performed in a short period of time, only the bias and noise in the IMU were considered. With the SINS remaining static on the ground, the AA methods including the selective alignment method and the analytic coarse alignment method were used to obtain the attitude angles. The values of the various parameters were chosen as follows: Simulation conditions for the static alignment: 32 L =°, × , Table 3. The attitude angles obtained using analytic alignment (AA) methods in the simulation.   Table 4. Table 4. The attitude angles obtained using AA methods in the simulation.   The IMU outputs were sampled for ten seconds. The average values were computed, and these are denoted as b w and b f . The difference between the norm of b w and ie w was 0 556 /h .°, and the difference between the norm of b f and g was 4 1 0121 10 g − .

×
. It is obvious that several of the outputs of the three-axis gyroscope are faulty. The selective alignment method can be further used to identify

Alignment in the Vehicle with Vibration
Because vibration is common in actual systems, the alignment in the base with vibration is studied. The vehicle remained static for ten minutes, then fire the engine and let the engine idle. The outputs of the three-axis gyroscope were greatly affected because of vibration caused by the idle engine. And the yaw angle obtained through AA could be greatly influenced. The outputs from the IMU for two minutes during the time when the engine was idling were used for the application of the AA. And the alignment results using fine alignment method given the outputs of the whole time were considered as the true attitude.
Since the six outputs from the IMU have a redundancy for the AA. The accuracy of the alignment could be increased compared to other AA methods by using two or more sets of selections for the selective alignment and grouping the alignment results in a certain way. The roll and pitch angles can be determined due to the outputs of the three-axis accelerometer, while outputs from the three-axis gyroscope are required to obtain the yaw angle. The outputs of the three-axis gyroscope are much more easily influenced than the three-axis accelerometer in the base with vibration. The following AA method based on the selective alignment is presented. The yaw angle obtained through the selective alignment with the selection of ( ) , , x y x f f w is denoted by The yaw angle obtained through the selective alignment with the selection of ( ) , , x   The experiment was conducted for fifty times. Assume the yaw angle using the i -th method shown in Figure 11 for the j -th experiment is ˆi j ψ , and the corresponding true value using fine alignment method is j ψ . i δ and i σ are used to test the average alignment error and the stability, respectively, for the i -th method shown in Figure 11. They are given by are redundant for determining the transformation matrix, which could be studied further in other attitude determination problems where TRIAD algorithm is applied such as spacecraft attitude estimation.