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In the initial alignment process of strapdown inertial navigation system (SINS), large initial misalignment angles always bring nonlinear problem, which causes alignment failure when the classical linear error model and standard Kalman filter are used. In this paper, the problem of large misalignment angles in SINS initial alignment is investigated, and the key reason for alignment failure is given as the state covariance from Kalman filter cannot represent the true one during the steady filtering process. According to the analysis, an alignment method for SINS based on multiresetting the state covariance matrix of Kalman filter is designed to deal with large initial misalignment angles, in which classical linear error model and standard Kalman filter are used, but the state covariance matrix should be multireset before the steady process until large misalignment angles are decreased to small ones. The performance of the proposed method is evaluated by simulation and car test, and the results indicate that the proposed method can fulfill initial alignment with large misalignment angles effectively and the alignment accuracy of the proposed method is as precise as that of alignment with small misalignment angles.

For its merits of complete independence, strong anti-interference ability, comprehensive, and high update frequency of navigation message, strapdown inertial navigation system (SINS) has been widely used for military application [

The compassing alignment method based on compass effect and the optimal estimation methods based on Kalman filter (e.g., transfer alignment, zero-velocity alignment) are the mostly used techniques to deal with alignment problems in SINS [

Error propagation model and filter method are two key issues to address when Kalman filter or methods derived from Kalman filter are used to fulfill SINS alignment [

Recently, SINS error propagation models and nonlinear estimation methods for large misalignment angles are widely reported in the literatures. Error propagation models can be divided into two types: one is nonlinear model for large misalignment angles, among which the large azimuth misalignment error model is the most representative [

For the purpose of simplifying error model without adding computational load, an initial alignment method for large initial misalignment angles is proposed based on classical linear error propagation model and standard Kalman filter, in which the state covariance matrix of Kalman filter is reset several times before the steady filtering process is achieved. With the parameters resetting, the state covariance from Kalman filter can truly represent the true one and the high utilization of measurement vector can be ensured. Coupled with the help of closed loop correction mode, large misalignment angles can be converted into small ones, so that alignment can be finished effectively.

The rest of this paper is organized as follows. The classical linear error propagation model and standard Kalman filter used in zero-velocity alignment are described in Section

Choose velocity errors and misalignment angles of SINS as the state variables, and then the state vector of the system can be constructed as

The system state equation can be constructed as [

When the vehicle is static, the zero-velocity information in navigation frame is used as external reference and the measurement vector can be constructed as

The system measurement equation can be constructed as [

The system and measurement equations given in Sections

The “system” in (

Kalman filter with closed loop correction mode.

Inertial measurement unit (IMU) is composed of medium or low precision sensors. The gyro constant bias is set as 0.1°/h and random bias is white Gaussian noise with zero mean and standard variance 0.1°/h. The accelerometer constant error is 500 ug and random error is also white Gaussian noise with zero mean and standard variance 500 ug. With the above assumption about sensor precision, the limit alignment accuracy can be obtained by zero-velocity alignment method as follows: pitch—−1.7189′ (minute of degree); roll—1.7189′; and yaw—−27.0248′.

The vehicle is without linear movement, but with swinging motion. The swinging amplitudes of pitch, roll, and yaw are set as 1.8°, 2.4°, and 2.8°, respectively, and the corresponding oscillation cycle is 6, 8, and 10 s. Based on the above ideal motion values, the ideal sensor output can be gotten by back-stepping of navigation solution. When sensor errors are added into ideal sensor output, the true sensor data can be obtained, with which navigation solution and alignment operation can be run. Meanwhile, the ideal motion values can be used as references to judge the alignment results with different initial misalignment angles.

The initial parameters for Kalman filter are set as

There is no unified definition about the magnitude of large misalignment angles in SINS until now. To show how the initial misalignment angles with different magnitudes will affect alignment results, the following five sets of different misalignment angles are used in simulation: Condition 1

In simulation, the update cycle of sensor data, navigation parameter, and alignment filtering are all set as 10 ms, and the statistical period is set as 1 s.

The simulation lasts for 900 s, and the misalignment angle curves of different conditions are shown in Figure

Alignment results with different initial misalignment angles.

Pitch

Roll

Yaw

The simulation results in Section

Nevertheless, the simulation results in Figure

Comparison of state covariance.

Yaw in Condition 1

Yaw in Condition 5

In order to evaluate the effect of

Prediction for state vector

Update for state vector

Calculation for gain

Prediction for state covariance

Update for state covariance

In the above equations, the subscript

Furthermore, by analyzing the relationship between (

The above analysis indicates that Kalman filter can use measurement vector with a higher utilization degree during the whole alignment process under the condition of small misalignment angles, while under the condition of large misalignment angles, Kalman filter cannot because the

Based on the above analysis, it can be concluded that, under large misalignment condition, the change of state covariance from Kalman filter cannot truly represent that of state vector, and Kalman filter cannot utilize external reference navigation information effectively. It should be one of the key reasons which lead to the alignment failure when the classical linear error propagation model and standard Kalman filter are used for zero-velocity alignment with large misalignment angles.

In alignment process, in order to get a higher convergence speed and make full use of external reference information, initial

Based on the understanding about the effect of

Initial alignment process with the state covariance resetting.

During the alignment process based on parameter resetting, at the moment

As mentioned in Section

But during alignment process, the resetting operation should be executed several times, and accordingly the stable process of filtering should be done at the same times passively. Thus, a disadvantage about this method is the prolonged alignment time.

The aim of this paper is to find an alignment method for SINS with large initial misalignment angles without adding model complexity and adding computational load; thus no general or adaptive rules about the selection of resetting cycle and times are studied. In the rest of this paper, fixed resetting cycle and times are selected.

In engineering, a certain safety factor should be added to the above parameters to ensure a successful alignment; thus the alignment time will further be prolonged.

The simulation is conducted under the same conditions as those in Section

The simulation lasts for 900 s, and the

Alignment results with state covariance resetting.

Pitch

Roll

Yaw

Comparison of state covariance.

The curves in Figure

The curves in Figure

From the above analysis, it can be concluded that, with the resetting to the state covariance of Kalman filter, initial alignment for SINS with large initial misalignment angles can be fulfilled with classical linear error propagation model and standard Kalman filter.

Considering the small angular motion without linear movement caused by engine vibration under shutdown condition, the car test is executed to verify the validity of the proposed alignment method. The fiber-optic gyro SINS as shown in Figure

Instrument precision of SINS.

Gyro | Accelerometer | ||
---|---|---|---|

Zero bias | <0.03°/h | Zeros bias | ±5 × 10^{−5} g |

Random walk coefficient | <0.005°/h | Zero bias stability | <4 × 10^{−4} g |

Scale factor error | <20 ppm | Scale factor error | <100 ppm |

Accuracy of PHINS.

Azimuth | Level attitude |
---|---|

0.05° secant latitude (without GPS aid) | Less than 0.01° |

SINS and PHINS in car.

Three different alignment schemes as follows are compared: Scheme 1: standard Kalman filter with small initial misalignment angles; Scheme 2: standard Kalman filter with large initial misalignment angles; Scheme 3: standard Kalman filter with large initial misalignment angles but with resetting operation. For the three schemes, the filtering frequencies are all set as 100 Hz and the classical linear error propagation model is used for the calculation.

The test is run as offline semiphysical simulation based on the saved data. Before fine alignment, analytical coarse alignment is executed. The results of coarse alignment can be considered as small misalignment angles because the sensors used in this test are high and the vibration caused by engine is small. To simulate large misalignment angles, the angles

During the alignment process in Scheme 3, the

The curves of alignment results from different schemes are shown in Figure

Alignment results with resetting in car test.

Pitch

Roll

Yaw

Table

Statistic results of Scheme 1, Scheme 3, and PHINS.

Pitch | Roll | Yaw | ||||
---|---|---|---|---|---|---|

Mean | Std. | Mean | Std. | Mean | Std. | |

PHINS (°) | −0.8292 | 0.0011 | −0.2441 | 0.0046 | 85.5903 | 0.0024 |

Scheme 1 (°) | −0.8200 | 0.0092 | −0.2433 | 0.0056 | 84.6823 | 0.3092 |

Scheme 3 (°) | −0.8202 | 0.0098 | −0.2455 | 0.0061 | 85.0917 | 0.1906 |

From Figure

In Section

The problem of large misalignment angles in SINS initial alignment is investigated in this paper, and simulation results verified that nonlinear problem brought by large initial misalignment angles will cause alignment failure when the classical linear error model and standard Kalman filter are used. The reasons for alignment failure are discussed in detail, and the analysis indicates that the state covariance of Kalman filter is smaller than the true one and the measurement vector cannot be effectively utilized during the steady filtering process. At the same time, the simulation also shows that there is a convergence trend for misalignment angles at the beginning of alignment process when the state covariance is much larger than the true one.

Based on the analysis of state covariance, an initial alignment method for SINS is designed, in which the classical linear error propagation model and standard Kalman filter with multiresetting are used. The state covariance of Kalman filter is reset several times before the steady filtering process and ensures the state covariance larger than the true one, which improves the utilization degree of measurement vector. This proposed method does not change the classical error propagation model and filter structure and owns the merits of simple model and less calculation.

Simulation and car test results both indicate that the proposed method can fulfill initial alignment with large initial misalignment angles effectively and the alignment accuracy can be as precise as that with small ones.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation (61273056) and the Chinese university research and operation expenses (104.205.2.5).