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Shipboard is not an absolute rigid body. Many factors could cause deformations which lead to large errors of mounted devices, especially for the navigation systems. Such errors should be estimated and compensated effectively, or they will severely reduce the navigation accuracy of the ship. In order to estimate the deformation, an unscented particle filter method for estimation of shipboard deformation based on an inertial measurement unit is presented. In this method, a nonlinear shipboard deformation model is built. Simulations demonstrated the accuracy reduction due to deformation. Then an attitude plus angular rate match mode is proposed as a frame to estimate the shipboard deformation using inertial measurement units. In this frame, for the nonlinearity of the system model, an unscented particle filter method is proposed to estimate and compensate the deformation angles. Simulations show that the proposed method gives accurate and rapid deformation estimations, which can increase navigation accuracy after compensation of deformation.

Inertial measurement units (IMUs) are widely used in ships, aircraft and land vehicles to provide navigation information such as attitude, velocity and position [

Shipboard is not an absolute rigid body. Many factors could cause deformations which lead to larger misalignment angles between the coordinates of the main INS and slave INSs [

When deformation occurs, the attitude information provided by the main INS contains not only the attitude angle between navigation frame and the local coordinate frames of every equipment, but also the deformation angle between the installation spot of the main and slave INS [

For nonlinear Gaussian systems, the Extended Kalman Filter (EKF) is a widely used method [

In recent years, many improved PF methods were proposed, especially for important density function and resampling methods [

In this work, we consider the shipboard deformation as a nonlinear Markov process and build an attitude plus angular rate match mode of aided alignment considering deformation angles. The deformation angles are estimated by an UPF-based method. Then, simulations of the proposed estimation method and aided alignment method are given. The main contribution of this paper is proposing an aided alignment method with deformation estimated by an UPF method.

The rest of the paper is organized as follows: the nonlinear shipboard deformation model is presented in Section 2. In Section 3, an attitude plus angular rate match mode is proposed as an estimation frame. In Section 4, we propose a UPF-based method to estimate the deformation angles. In Section 5, simulations are used to verify the deformation estimation and compensation method. The conclusions are given in Section 6.

Suppose the main INS is installed on point

For INS, the propagation of attitude error can be presented by a

The attitude error vector of the main and slave INS is _{m}

For _{m}

In aided alignment, the output of the main INS is treated as a standard for the slave INSs. We ignore the positioning error and constant drift of gyro of the main INS and obtain:

The navigation equation of the slave INS is:

As described before, the computational error of main INS can be ignored. Therefore, the navigation equation is:

The specific force can be presented as:
_{s}

We can obtain the following equation:

The error of the computed Earth rotation angular rate caused by the position error in the system noise is considered. Because the position error is very small, the gravity can be regarded as the same. Therefore, we simplify

And then, we get the following equations:

Defining
_{m}

In practice, it can be rewritten as:

The differential equation of the static deformation angle is:

Assuming the dynamic shipboard deformation angle is:

Dynamic deformation occurs when the ship is rotated by waves and winds. During navigation computation, we can use a second order Markov process driven by white noise to describe such a motion and assume the deformation processes of the three axes of the ship are independent. Let
_{η}_{i}_{i}_{i}_{w}

During sailing, a lot of environmental disturbance will bring different levels of ship deformation and influence the aided alignment accuracy of the shipboard INS. In order to test the influence of deformation on aided alignment, we set the initial conditions of the INS as follows: the constant drift of the gyro is [0.01 0.01 0.01]^{T}^{T}^{T}^{T}

Simulation results demonstrate that the deformation angles have a certain influence on aided alignment, increasing with deformation. The installation error angles reflect the shipboard deformation estimated by the IMU. From comparison of

This method compares the outputs of gyro and computed attitude angles between the main INS and slave INS to estimate the gyro drift and deformation. The system equation is as follows: first, we choose the state variable
_{2}= −_{1}, _{3}, is a unit matrix, _{4}= −_{1}, _{5}, and _{6} are unit matrices,
_{8}=_{x}_{y}_{z}_{1}(

We choose the measurement

In the PF method, we choose a set of discrete random sample points (particles) with weights to approximate the

Initialization:

For

For

Importance sampling procedure:

For

Using UKF to update new particles:

Compute sigma points:

Forward propagation particles (time update):

Acquire new measurement (measurement update):

Sampling

Assuming

Sampling

Assuming

For

For

Choosing procedure:

We use high/low importance weight

Output: As the regular PF:

From the above procedure, we embed the UKF into the PF frame, and can easily introduce new measurements into the state estimation. Although the

In order to verify the proposed method, we choose the shipboard deformation nonlinear model from

It is demonstrated in

Under some severe conditions, large amplitude swings caused by wave and wind disturbances lead to large dynamic deformations. In order to test the effectiveness of UPF in dynamic deformation, we increase the pitch, roll, and azimuth to 5°, 4° and 2°. Choosing 200 particles, we have the estimation results of UPF in

From the results we can conclude that under the same conditions, deformation increases will cause longer convergence times. This is because of the large variation of deformation with the same particle number. Increasing the particle number will reduce the filter convergence time, but the computation burden will increase according to the theoretical analysis. For small deformations, UPF will achieve higher accuracy; for large deformations, UPF will also converge in a shorter time. The accuracy of both situations is less than 1′.

The constant drift of the gyro will be obtained by initial alignment or calibration methods at the beginning when the ship departs from the dock. Moreover, such biases are not important in shipboard deformation estimation. Therefore, even though the parameter of gyro's constant drift is in the state equation of the proposed method, we only discuss the simulation results of the shipboard deformation parameters.

In order to verify the efficiency of the UPF method, we choose the traditional EKF method for comparison. Simulations of static and dynamic deformation are shown in

In order to verify the efficiency of the proposed method, we present a simulation of the attitude plus angular rate match mode in aided alignment. The initial conditions are the same as Section 2.2. Assuming three compensated shipboard deformation angles are [0.5′ 9.2′ 3.4′]. Estimations of attitude error angles and installation error angles are simulated in

Compared to the simulations in

An unscented particle filter method for estimation of shipboard deformation based on an inertial measurement unit is presented. In this method, we first build the nonlinear shipboard deformation model, and produce a simulation of accuracy reduction due to deformation. An attitude plus angular rate match mode is proposed to estimate the shipboard deformation during aided alignment. For the nonlinearity of the system model, a UPF method is proposed to obtain the deformation angles and compensate to aid alignment.

We use the UPF method to estimate the nonlinear model of large shipboard deformation. For different deformation angles, estimations can rapidly converge with high accuracy. The amplitude of deformation determines the convergence speed of UPF. Simulations demonstrate the estimation comparison of UPF and EKF. For static and dynamic deformation angles, UPF, which is a more effective estimation method of deformation, has higher estimation accuracy and converges more rapidly than EKF. After shipboard deformation compensation, simulation shows the accuracy of aided alignment is also increased. In future work, multiple IMUs should be considered for the ship deformation measurement. Multiple spots will provide full and effective ship deformation measurements.

This work was supported by the grant from National Natural Science Foundation of China (No. 61104189, 61104192 and 61127004), Excellent Young Scholar Research Fund of BIT (No. 2012YG0605), and NCET-11-0784. The authors would like to thank all the editors and anonymous reviewers for improving this article.

The authors declare no conflict of interest.

Dynamic deformation of a ship.

Attitude error estimations of the first set.

Installation error estimations of the first set.

Attitude error estimations of the second set.

Installation error estimations of the second set.

Flow chart of the proposed method.

UPF estimation error of static deformation (200 particles).

UPF estimation error of static deformation (500 particles).

UPF estimation error of dynamic deformation (200 particles).

UPF estimation error of dynamic deformation (500 particles).

Static deformation estimation comparison of UPF and EKF.

Dynamic deformation estimation comparison of UPF and EKF.

Estimation error of attitude error angles.

Estimation error of installation error angles.