Strapdown Sculling Velocity Algorithms Using Novel Input Combinations

1 School of Mechanical and Electrical Engineering, Nanjing Forestry University, Nanjing 210037, China 2Jiangsu Province 3D Printing Equipment and Manufacturing Key Lab, Nanjing Normal University, Nanjing 210042, China 3School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China 4School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China


Introduction
Improvements in maneuverability of flight vehicles necessitate SINS capable of efficiency in highly dynamic environments.To improve the precision of SINS, sculling motion is usually employed as a standard input to evaluate the performance of the velocity algorithm in a dynamic environment.Savage provided an analytical description of sculling motion in two forms [1,2].In that study a digital algorithm capable of calculating the sculling integral term was described.An analytical model for the evaluation of error build-up under band-limited random process input for the digital integration algorithm was developed in a later study [3].It was demonstrated that the digital integration process introduced a random walk type error in the output that was directly proportional to the root-mean-square input amplitude, directly proportional to the square-root of the input bandwidth, and inversely proportional to the digital integration update frequency.Ignagni derived a class of optimized sculling algorithms and demonstrated a duality between the derived class of sculling algorithms and a previously derived class of coning algorithms [4].Kelly developed a generic equivalency between coning and sculling integrals and algorithms.The equivalency allows a coning algorithm based on incremental angle input to be converted to its corresponding sculling algorithm using a simple mathematical formula [5].In [6] two alternative approaches were developed for deriving strapdown navigation sculling algorithms.A key point of the two approaches is the use of additional gyro/accelerometer output signals which are the increments of the angular rate/specific force multiple integrals over the iteration interval.However, all these conventional velocity algorithms adopt incremental angle/specific force increments or angular rate/specific force as algorithm inputs.However modern inertial sensors produce different types of output now.For example, certain fiber optic gyros (FOGs) and the microelectromechanical systems (MEMS) gyros have angular rate sampled output, but the laser gyro usually has integrated angular rate output.The quartz accelerometer output is usually specific force, but some other type of accelerometer output carrier's specific force increments.If the SINS use a gyro with an output of angular rate and an accelerometer with an output of specific force increments, these conventional sculling velocity algorithms are no longer very applicable.To solve this problem, a novel sculling algorithm using incremental angle/specific force inputs or angular rate/specific force increments inputs is developed in this paper.The novel algorithm can calculate out the carrier velocity directly without converting the dimension of inertial sensor outputs values.Owing to this, the precision of the algorithm is considerably high.

Conventional Sculling Algorithms
Velocity rate equation is The carrier velocity in navigation coordinates at time   is then obtained as the integral of (1) from time  −1 , evaluated at time   as [1] () where  is the computer cycle index, ΔV  is the integrated transformed specific force increment, and ΔV / is the gravity/Carioles velocity increment.Calculating ΔV / is a relatively simple process and is therefore omitted.ΔV  can be expanded as [5] Δ where In (3) ΔV  is the velocity rotation correction and ΔV  is the sculling correction.ΔV  in (3) is a continuous integral form.The discrete algorithm for ΔV  , which is usually referred to as sculling algorithm, can be obtained from the coning algorithm using a simple duality formula [4,5,7].For example, the 2-interval optimal sculling algorithm using incremental angle/specific force increments inputs is [5,[8][9][10]] Its coning algorithm counterpart, namely, the 2-interval optimal coning algorithm using incremental angle input, is [11] The 2-interval optimal sculling algorithm using angular rate/specific force inputs is [12,13] where  is the attitude and velocity update period  =   − −1 .Its coning algorithm counterpart using angular rate input is [14] where ℎ is the algorithm iteration interval.In a 2-interval algorithm there is H=2h.

Sculling Algorithm Using Incremental Angle/Specific Force Input
If the inertial sensors produce integrated angular rate (incremental angle)/specific force outputs, the conventional sculling algorithms such as ( 5) and ( 7) cannot calculate out the carrier's velocity directly.The velocity algorithm must include a step for conversion of integrated angular rate (incremental angle) into angular rate by digital differentiation in order to use (7) or else convert specific force into specific force increments by digital integration in order to use (5).These steps will increase computational error.To solve this problem we have developed a novel sculling algorithm using incremental angle/specific force input.

Optimal Sculling Algorithm.
A typical sculling motion is defined as [15]  = Ω cos (Ω) ,  =  sin (Ω)  (13) where  is the amplitude of the angular vibration, c is the amplitude of the specific force vibration, J, K are the unit vectors along the two body axes (y, z) about which the oscillations are occurring, and Ω is the frequency associated with the angular and specific force oscillations.
The incremental angle and specific force increments of the sculling motion are Substituting ( 13) and ( 14) into the ΔV  term in (3) yields the truth sculling correction: As is seen in (15), in a sculling environment the true sculling correction ΔV  is a constant value.For a 2interval system, the outputs of inertial sensors over a velocity update period are The generalized form of the sculling correction using incremental angle/specific input can be given as where  is the number of iteration intervals over the velocity update period.For a 2-interval algorithm, N=2.It follows from ( 16) that Obviously, the sculling correction ΔV  is only determined by |j-i|.Therefore (17) can be simplified as For a 2-interval sculling algorithm, N=2.Substituting (18) into (19) gives Sculling correction Δ V should be equal to the true sculling correction Δ  , Δ V = Δ  .Using ( 19) and ( 15), the simultaneous equation for the unknown parameter  can be obtained.Then the equation can be simplified by Taylor expanding "ΩH": The solution for Δ V = Δ  is  1 =1/3.Hence the 2-interval optimal sculling algorithm using incremental angle/specific force input is derived: Mathematical Problems in Engineering The per-unit time algorithm error is In ( 2) and ( 3), the integrated transformed specific force increment ΔV  is composed of ΔV  , ΔV  , and ΔV  .ΔV  and Δ  in (3) can be calculated by digital integration: Substituting ( 24) into (3) gives Substituting ( 22), (24), and (25) into (3) gives the integrated transformed specific force increments: As described previously, ΔV / in (2) can be calculated out easily; indeed it can be calculated approximatively as a constant during one update period and omitted here.  −1 in (2), i.e., the carrier velocity at  −1 , has been calculated out during the last algorithm update period.Thus the carrier velocity at time   can be calculated out by substituting (26) into (2).

Formalized Optimal Sculling Algorithm.
For an N-interval sculling velocity algorithm using incremental angle/spe-cific force input, the sample number of the accelerometer outputs (specific force) is N+1, and the number of the gyro outputs (integrated angular rate) is .The gyro/accelerometer outputs in a sculling environment defined by (13) are As previously mentioned, N-interval sculling velocity algorithm can be formulated as (19).Substituting ( 27) into (19) we obtain Applying (28) with Taylor series expansion for the coefficient terms "ΩH", we obtain Applying (15) with Taylor series expansion for the coefficient terms "ΩH" gives From Δ V = Δ  we can obtain where The solution to (31) is G = A −1 D. Details regarding the optimal coefficients are shown in Table 1.
Comparing the developed sculling algorithm given by ( 22) with the conventional sculling algorithms represented in ( 5) or (7), we can see the advantages of the developed algorithm are that it is able to calculate out the sculling correction and then the velocity at   directly without any requirements to perform the dimension conversion of inertial sensor outputs.

Sculling Algorithm Using Angular
Rate/Specific Force Increments Input where a, b, A, B, and  are the polynomials coefficients.We then obtain the following: Comparing ( 9) and ( 10) with ( 33) and (34) we can see that there is a duality between the sculling algorithm using incremental angle/specific force inputs and the sculling algorithm using angular rate/specific force increments inputs.Eqs. ( 9) and (10) have the identical mathematical forms to (33) and (34), and they equal each other when the terms Δ, f, aH, bH 2 , A, BH, and CH 2 in ( 9) and ( 10) are replaced by ΔV, , AH, BH 2 , a, bH, and cH 2 respectively.The 2-interval generalized sculling algorithm using angular rate/specific force increments inputs can therefore be given as 4.2.Formalized Optimal Sculling Algorithm.For N-interval sculling velocity algorithm using angular rate/specific force increments inputs, the sample number of the accelerometer outputs (specific force increments) is N-1, and the sample number of the gyro outputs (angular rate) is .
The gyro/accelerometer outputs in a sculling environment defined by (13) are Similar to the derivations of ( 15)-( 17), the generalized form for sculling correction algorithm using angular rate/specific force increments inputs is It can be derived from (36) that Obviously, the sculling correction Δ V in (37) is only controlled by |j-i|.Therefore (37) can be simplified as  outputs.Therefore, the sculling correction has been compensated effectively and the velocity determination precision is improved dramatically.
From Figure 2 we can see that the longitude errors of both algorithms are larger than the latitude errors of both algorithms.This is because the eastward velocity error is the main reason for the longitude error.As mentioned in the third paragraph before, the eastward velocity errors of both algorithms are larger about one order than those northern velocity errors.
The increase of the longitude error curve of the traditional algorithm is particularly rapid, because when the inertial sensors output integrated angular rate/specific force and the carrier is in a sculling motion environment, the traditional algorithm cannot compensate the sculling correction well.It will induce a rapid error propagation in eastward velocity and then a rapid increasing longitude error finally.
However, the position errors of the proposed algorithm are reduced by more than one order of magnitude compared with those of the traditional sculling algorithm.This is because the improved precision in velocity determination will also result in improved precision in position determination.

Trial Study
As is shown in Figure 3, a dynamic car test was performed to validate the performance of our proposed sculling algorithm.For this 20-minute test, the constant gyro drift is 0.01 ∘ /h; the accelerometer bias is 5 × 10 −5 g.The update period for the velocity algorithm is 0.01s.A GPS/INS integrated navigation system (SPAN-CPT, NovAtel in Sweden) is also used as the benchmark for attitude, velocity, and position.NovAtel has claimed that the performance of this integrated navigation system is better than 0.06 ∘ for heading and 0.02 ∘ for pitch and roll, 0.015 m/s for horizontal velocity, and 1.2m for horizontal position accuracy.
Trial results for the comparative study between the novel and traditional algorithm are shown in Figures 4 and  5. Obviously, the accuracy of horizontal (eastward/northward) velocity and horizontal position (longitude/latitude) is improved when using the novel algorithm.These trial results are consistent with both theoretical analysis (Section 3.3) and simulation results before (Figures 1 and 2).

Conclusions
To solve the issue that in certain situations the output parameters of inertial sensor do not match the input parameters required by the conventional sculling algorithms, an optimal sculling algorithm using incremental angle/specific force input or angular rate/specific force increments inputs is Mathematical Problems in Engineering   developed in this paper.The developed sculling algorithms can calculate out the carrier velocity directly without the dimension conversion of inertial sensor outputs.Accordingly, the developed algorithms provide higher precision than conventional sculling algorithms in the SINS in which the employed inertial sensors have the integrated angular rate/specific force or angular rate/specific force increments outputs.Our novel sculling velocity algorithm thus has great useful value in such cases.

Nomenclature
: Specific force measured by accelerometers    : Direction cosine matrix : Update period △  : Incremental angle vector over the ith subminor interval : Angular velocity vector.

Figure 1 :
Figure 1: Comparison of the horizontal velocity determination results between two algorithms under sculling environment.

Table 1 :Figure 2 :
Figure 2: Comparison of the horizontal position determination results between two algorithms under sculling environment.

Figure 4 :
Figure 4: Comparison of horizontal velocity determination results of the dynamic car test between two algorithms.

Figure 5 :
Figure 5: Comparison of horizontal position determination results of the dynamic car test between two algorithms.