Lever Arm Compensation of Autonomous Underwater Vehicle for Fast Transfer Alignment

Transfer alignment is used to initialize SINS (Strapdown Inertial Navigation System) in motion. Lever-arm effect compensation is studied existing in an AUV (Autonomous Underwater Vehicle) before launched from the mother ship. The AUV is equipped with SINS, Doppler Velocity Log, depth sensor and other navigation sensors. The lever arm will cause large error on the transfer alignment between master inertial navigation system and slave inertial navigation system, especially in big ship situations. This paper presents a novel method that can effectively estimate and compensate the flexural lever arm between the main inertial navigation system mounted on the mother ship and the slave inertial navigation system equipped on the AUV. The nonlinear measurement equation of angular rate is derived based on three successive rotations of the body frame of the master inertial navigation system. Nonlinear filter is utilized as the nonlinear estimator for its capability of non-linear approximation. Observability analysis was conducted on the SINS state vector based on singular value decomposition method. State equation of SINS was adopted as the system state equation. Simulation experiments were conducted and results showed that the proposed method can estimate the flexural lever arm more accurately, the precision of transfer alignment was improved and alignment time was shortened accordingly.


Introduction
Transfer alignment is the process of initializing the position, velocity and attitude of a slave INS, using the data supplied by another INS known as the master inertial navigation system [John and Leondes (1972)]. As the initial attitude errors cause the navigation errors to increase much rapidly than the initial velocity and position errors [Cheng, Wang and Liu (2014)], thus the relative attitude error of the slave INS with respect to the master INS is a major error source to result in the position error growth after launching the inertial guided weapons [Zhu and Cheng (2013)]. A digital filter method was applied in compensating the lever arm effect [Xu and Wan (1994) O X Y Z stands for the body frame.
b O is the swaying center of the ship, also is known as the gravitational center of the body. The position of the center of gravity is calculated according to the general design of the load distribution, assuming that the gravity center is fixed and the master INS installation position is in coincidence with b O , the slave INS accelerometer is fixed point P mounted on the carrier coordinates, position vector coordinate origin, P point position relative to the inertial coordinate vector at P, relative to the position vector vector coordinate origin. Obviously, they have the following relationship: The position of the gravity center is usually calculated according to the load distribution of design, which is always fixed and coincides with the master inertial navigation system equipping position. The accelerometers of slave inertial navigation system are equipped at the fixed point p in the body frame. The position vector of the origin of the carrier coordinate system is 0 R , the position vector of p to inertial frame origin is p R , the position vector of p to the carrier frame origin is p r , abiding the following rules: The differential of the upper formulae to time can be obtained: The differential of the upper formulae to time can be obtained: According to the relative micro quotient principle of vector differential, it can be obtained: presents linear acceleration of p to carrier frame.
The same reason can be obtained： The linear acceleration of p to inertial frame can be expressed as: The carrier structure is rigid in the study of lever arm effect. Other methods are needed to compensate the error caused by the flexible deformation. P is fixed to the carrier frame here.
The installation point should be in the swing center of the carrier in the ideal situation, 0 p r = , the lever arm effect did not exist in the situation. The master inertial navigation is equipped at the carrier swing center, while the slave inertial navigation system cannot satisfy the needs in transfer alignment. Lever arm effect cannot be neglected in real application. The latter two components are caused by the lever arm effect, sensed by the slave inertial navigation system, not sensed by the master inertial navigation system. f δ stands for the lever arm acceleration, the standard equation of the lever arm effect error is, Suppose the vehicle did not move linearly, that is , the above formulae can be simplified as The lever arm effect acceleration can be expressed in the navigation frame as follows And for 0 cos 3 Observability analysis Before AUV entering the water, transfer alignment is conducted before integrated navigation period. Transfer alignment uses the navigation information of position, velocity and attitude from master inertial navigation system equipped on the big ship. Observability analysis was conducted on the SINS state vector based on SVD method. The measurement information is the vehicle velocity (north velocity and east velocity) and velocity from the master inertial navigation system. The motion process of initial alignment on moving base can be designed as follows, the gyro drift is estimated under three-axis swaying and linear acceleration. The accelerometer bias is estimated under linear acceleration and turning, the angle error can be estimated under linear constant navigation. The observability degree of the state variables under three-axis swaying with velocity, heading angle error and position matching is presented in Tab. 1. From Tab. 1, eastern velocity error, northern velocity error, up heading angle error, longitude error, latitude error, eastern gyro drift, northern gyro drift, up gyro drift are high in observability degree and the filter effects are good. Eastern accelerometer bias and northern accelerometer bias are low in observability degree and the filter effects are not good. From the analysis above, the velocity error can be well estimated with velocity matching while heading angle and position error cannot be well estimated. The heading angle error can be better estimated with velocity and heading matching, while the position error cannot be well estimated. The heading angle error, velocity error, position error can be well estimated with velocity, heading and position matching.

Revised lever-arm effect compensation method
When the ship is under mooring condition, the linear acceleration and the linear velocity is zero, the error equation of the inertial system can be reduced as follows, ' n is the navigation frame, b is the body frame, V δ is the body velocity error, φ is the vehicle attitude error, ' n ie ω is the earth rotation rate on the navigation frame, C is the strapdown matrix.
In the strapdown inertial navigation system, Kalman filter is applied in the estimation of lever arm length. The accelerometer error and the gyro drift gyro are expanded as the states in the Kalman filter. The system state equation is as follows,

Simulation and results
In the simulation experiments, the performance of each transfer alignment algorithm is evaluated. The ship speed is about 30knots. The angular motions of the ship are generated as follows Suppose the initial longitude is 118°, the initial latitude is 32°, the initial height is 0. The initial longitude error is 5'', the initial latitude error is 5'', and the initial height error is 5''. The initial eastern velocity is 10m/s, the initial northern velocity is 10m/s, the initial eastern velocity error is 0.1m/s, the initial northern velocity is 0.1m/s, the initial heading 0 H is 45°, the initial pitching angle is 0°, the initial rolling angle is 0°. Initial heading, pitching and rolling angle error is 10 ,10 , 30 ′′ ′′ ′′ respectively. Accelerometer constant drift and random drift is 100ug. The gyro constant drift and random drift is 0.1°/h. The swing amplitude of heading, pitching and rolling is 14°, 9°and 12° respectively. The swing period of heading, pitching and rolling is 6 s, 8 s, and 10 s respectively. The ship swing model which is suffering the wind and tide under the mooring condition is as follows, 0 sin(2 / ) sin(2 / ) The lever arm length is [ ]

Conclusions
This paper presents an efficient transfer alignment approach for swaying base big ship navigation system. A novel algorithm associated with the transfer alignment is employed to obtain the mathematical platform for the navigation system. Considering the environmental disturbances and the sensor drift as the main error source, a nonlinear filter approach is applied to the system to reduce the lever arm effect on the acceleration measurement. Observability analysis is conducted to different vehicle motion. Simulation experiments were conducted and the results showed that the novel method is able to improve the rapidity and precision of transfer alignment, overcoming the lever arm effect and disturbances existing in the host inertial navigation system and the slave inertial navigation system in the application of big ship navigation initial alignment.