A Class of Coning Algorithms Based on a Half-Compressed Structure

Aiming to advance the coning algorithm performance of strapdown inertial navigation systems, a new half-compressed coning correction structure is presented. The half-compressed algorithm structure is analytically proven to be equivalent to the traditional compressed structure under coning environments. The half-compressed algorithm coefficients allow direct configuration from traditional compressed algorithm coefficients. A type of algorithm error model is defined for coning algorithm performance evaluation under maneuver environment conditions. Like previous uncompressed algorithms, the half-compressed algorithm has improved maneuver accuracy and retained coning accuracy compared with its corresponding compressed algorithm. Compared with prior uncompressed algorithms, the formula for the new algorithm coefficients is simpler.


Introduction
In the recent decades since Jordan [1] and Bortz [2] introduced the two-stage attitude updating algorithm for strapdown inertial navigation systems (SINS), the design of efficient coning algorithms which include designing an efficient coning correction structure and achieving the optimized structure OPEN ACCESS coefficients for coning correction accounting for a portion of the rotation vector has always been an attractive topic.
Jordan [1] first presented the two-sample algorithm structure for non-commutativity error compensation. Miller [3] first presented the three-sample algorithm structure and the concept of designing the coning correction algorithm for optimum performance in a pure coning environment by using a truncated coning frequency Taylor-series expansion formulation for updating the rotation vector errors corresponding to updating the quaternion error. On the basis of Miller's idea, Ignagni [4] summarized the generally uncompressed algorithm structure for coning correction and proposed several coning correction algorithms. Lee [5] applied Miller's idea and concluded that there exists redundancy in the uncompressed coning correction structure under coning motion conditions. Based on Lee's conclusion, Ignagni [6] proved that the cross product of both integral angular rate samples is independent of absolute time and a function of merely the relative time interval between sampling points under coning motion conditions, and derived the first compressed coning correction structure. Different from the previous coning algorithms for gyro error-free outputs, Mark [7] disclosed a method of tuning high-order coning algorithms to match the frequency response characteristics of gyros with filtered outputs. Based on the compressed coning correction structure, Savage [8] further expanded Miller's idea, and raised an idea of using the least square method to design the coning correction algorithm for balanced coning performance in a given discrete coning environment. Song [9] concentrated on the improvement of maneuver accuracy of coning algorithms, and developed an approach for recovering maneuver accuracy in previous coning algorithms based on the uncompressed structure by combining the earliest time Taylor-series method and the latest frequency methods. This paper proposes a new half-compressed coning correction structure which is analytically proven to be equivalent to the traditional compressed coning correction structure under coning motion conditions. On the basis of the equivalency of the two types of structures, the half-compressed algorithm coefficients can be derived directly from the past compressed coning algorithm coefficients, rather than being specifically designed for coning or maneuver environments. The building of a reasonable structure makes for a simpler formula for the coefficients, retained coning accuracy and improved maneuver performance for the new half-compressed algorithm.

Attitude Algorithm Structure
The classical attitude updating computation formula [1,2,8] in modern strapdown inertial navigation systems is given by: where n is a navigation coordinate frame, b is a body coordinate frame, n where t is a time, l α is the integral of the gyro sensed angular rate ω from time 1 − l t to time l t , and l δφ denotes the coning correction.

Coning Correction Structure
In recent decades, strapdown attitude algorithm design activity has centered on developing routines for computing the coning correction using various approximations to the updating rotation vector l φ in Equation (2). The traditional numerical computation algorithm formula for updating rotation vector l φ has the form of the integrated angular rate l α and the coning correction l φ δˆ [4,6,8,9]: where each α Δ is an angular increment sample over a fixed time interval k T , the α Δ s are adjacent and spaced sequentially forward in time, Based on the pure coning motion properties, the compressed algorithm form [6,8] equivalent to the uncompressed form for the coning correction l φ δˆ in Equation (3) is given by: where s K is the coning correction coefficient equivalent to the sum of (3), and other signs are defined as those in Equation (3).
Both traditional coning correction forms defined by Equations (3) and (4) are equivalent under coning motion conditions, but not equivalent under maneuver conditions. Song [9] indicated that the algorithms based on the uncompressed form of Equation (3) would give much higher maneuver accuracies, but have a much heavier computation load than those based on the compressed form of Equation (4) after intensive design under maneuver conditions. This paper proposes a half-compressed algorithm form different from the former forms for coning correction l φ δˆ given by: where s J is the coning correction coefficient depending on the half-compressed structure, s θ which can be directly achieved from the process of computing l α in Equation (3) is an angular increment beginning at time k l NT t − and ending at time

Coning Correction Structure Equivalency and Algorithm Design
Assume that the body is undergoing the pure coning motion defined by the angular rate vector [4,6,8]: where t is a time, ( ) t ω is an angular rate vector in the body frame at time t , a and b are amplitudes of the angular oscillations in two orthogonal axes of the body, Ω is frequency associated with the angular oscillations, and I , J are unit vectors along the two orthogonal axes of the body.
Under the coning motion defined by Equation (6), Ignagni [4] had derived the cross product where K is an unit vector orthogonal to the unit vectors I and J , and β is a coning frequency parameter relevant to k T . Simplification of the coning algorithm form of Equation (3) in the form of Equation (4) in [6], utilizing the coning property expressed by Equation (7) also allows the coning algorithm form of Equation (5) to be simplified as the form of Equation (4) with the relationship of coefficients s J s and s K s: matrix whose upper triangular components are ones, and others are zeros concealed in Equation (8).
It is easily proved that is a non-singular matrix. Thus A and B are linear representation. That means, N-sample coning algorithms designed by using the same optimum method and based on the correction structures of Equation (4) and Equation (5) have the same coning correction value under pure a coning motion. Therefore, Equation (5) and Equation (4) which gives the optimized coefficients s J s applicable to the form of Equation (5) from already designed s K s of Equation (4).

Algorithm Performance Evaluation
Above we have verified that the half-compressed structure of Equation (5) and the compressed structure of Equation (4) are equivalent under pure coning motion. The uncompressed algorithm based on Equation (3) presented by Song [9] also has the same accuracy as the compressed algorithm based on Equation (4) under a coning motion. Thus, the algorithms designed by using the same optimum method and based on Equations (3)-(5) have the same coning correction accuracy under a pure coning environment.
Below is an error model used for evaluating the coning algorithm accuracy under maneuver environments. Assume that the body is undergoing a maneuver angular motion characterized by the angular rate vector: where j g is a coefficient vector based on the form of Equation (10), M is the coefficient vector number, t is a time. To evaluate the algorithm accuracy in maneuver environments, Equation (10) can be rewritten as another equivalent form: where i g is a coefficient vector based on the form of Equation (11). Through investigating Equation (10) and Equation (11), we can get the relationship of i g s and j g s: is a M by M square matrix whose the j th row and i th column component is ( ) 1 ji l t γ − , G is a M by one column matrix whose the i th row component is i g , and G is a M by one column matrix whose the j th row component is j g .
According to the maneuver error analysis method given by [9], the error of a coning algorithm based on uncompressed correction structure of Equation (3) under the maneuver motion expressed by Equation (11) with 5 M ≥ can be built as: l l e t z g g t t z g g t t z g g z g g t t z g g z g g t t z g g z g g z g g t t o t t δφ δφ According to Equations (3)-(5), Equation (13) can be used for analyzing the maneuver errors of coning algorithms based on the compressed correction structure of Equation (4) and the half-compressed correction structure of Equation (5), when the coefficients K s in Equation (4) and the coefficients J s in Equation (5) are respectively expanded into the coefficients ς s in Equation (3) with the following relationships:

Algorithm Examples and Simulation
To illustrate the properties of coning algorithms, algorithm errors computed using the optimized coning correction coefficients designed by using the frequency Taylor Tables 1 and 2 respectively give the 3-to-5-sample FTSc and LMSc algorithm coefficients s K s [5,6,8,9]. Using Equation (9), we can obtain the 3-to-5-sample FTShc and LMShc algorithm coefficients s J s given in Tables 3 and 4 from coefficients s K s in Tables 1 and 2. According to Equation (15), expanding the coefficients s K s in Tables 1 and 2 give the expanded coefficients ς s in Tables 5 and 6 for maneuver accuracy evaluation. According to Equation (16), expanding the coefficients s J s in Tables 3 and 4 give the expanded coefficients ς s in Tables 7 and 8 for maneuver accuracy evaluation.    Table 5. Coefficients expanded from the FTSc coefficients in Table 1.  Table 6. Coefficients expanded from the LMSc coefficients in Table 2.  Table 7. Coefficients expanded from the FTShc coefficients in Table 3.  Table 8. Coefficients expanded from the LMShc coefficients in Table 4.  Tables 9 and 10 give the 3-to-5-sample FTSuc and LMSuc algorithm coefficients ς s designed by Song [9] from the coefficients s K s in Tables 1 and 2, respectively.    The maneuver environment set for algorithm accuracy evaluation is the extreme 2 s angular rate profile pictured in Figure 2   Accordingly, maximum maneuver errors of several concerned algorithms over 2 s maneuver are listed in Table 17. Comparing the data in Tables 11 to 16, it is indicated that the absolute values of z 3 for FTSc, FTShc and FTSuc algorithms are zeros, LMSc, LMShc and LMSuc algorithms have the same non-zero z 3 , the absolute values of z 4 for 3 to 5 sample FTShc (or LMShc) algorithms are respectively about one third, one sixth and one tenth those of z 4 for 3 to 5 sample FTSc (or LMSc) algorithms, while the absolute values of z 4 for 3 to 5 sample FTSuc (or LMSuc) algorithms are much smaller than those for 3 to 5 sample FTShc (or LMShc) algorithms. If the low order term with z in Equation (13) is the main supply of maneuver error, it would be concluded that the maneuver accuracy of FTShc algorithm is higher than that of FTSc algorithm if the FTSuc algorithm is compared with the FTShc algorithm, and the maneuver accuracy of LMShc algorithm is higher than that of LMSc algorithm if the LMSuc algorithm is compared with LMShc algorithm ignoring the error term with z 3 . The simulation results in Table 17 are basically consistent with the analytical conclusion above, whereas the maximum maneuver error of the LMSuc3 algorithm is bigger than that of the LMSc3 and LMShc3 algorithms owing to the coupling of the first two error terms with z 3 and z 4 under this particular 2 s maneuver condition.
According to the 21.3 μrad error contribution from sensors in Table 2 of [8] during the similar maneuver environment, all concerned algorithms with the maximum errors in Table 17 contributing less than 1% of 10 to 20 μrad are compatible with the overall INS accuracy requirement of 0.01 deg/h. Like the uncompressed algorithm (reference [9]), the half-compressed algorithm has significantly more accuracy than the past compressed algorithm, and the new algorithm is more than adequate for modern INS applications. The formula for the algorithm coefficients for the new half-compressed algorithm is simpler compared to the uncompressed algorithm.

Conclusions
The new half-compressed coning algorithm can be directly derived from the past compressed coning algorithm. The new algorithms are highly efficient overall in coning and maneuver environments. Compared with the past uncompressed algorithm, the formula for the new algorithm coefficients is simpler. Like the past uncompressed algorithm, the half-compressed algorithm and its corresponding compressed algorithm have the same coning accuracy, while the maneuver accuracy of the half-compressed algorithm is significantly higher than the past compressed algorithm, and more than adequate for modern INS applications.