Gradient Descent Optimization-Based Self-Alignment Algorithm for Stationary SINS
Jingchun Li, Wei Gao, Ya Zhang, Zicheng Wang, Harbin Institute of Technology, China
Location: Big Sur
The main task of stationary SINS self-alignment is to determine the initial value of the attitude with the gravity vector, angular rate of the earth's rotation and local latitude information. Euler angle, direction-cosine matrix (DCM), quaternion and rotating vector are the most used methods to represent the attitude. Particularly, DCM-based method has been widely employed for SINS stationary alignment. However, due to the non-orthogonal problem, DCM-based method may bring in computational errors for the calculation of attitude. Besides, it requires at least nine constraint equations to determine the nine elements in the DCM matrix. Compared with DCM-based method, there are four elements in quaternion, which means it needs less constraints to determine a quaternion vector. Additionally, there is also no non-orthogonal problem in the quaternion-based method. Therefore, we employ the quaternion-based method to represent the attitude in this paper.
Besides, the attitude determination problem using multiple measurements is formulated as a Wahba problem. Our next work to determine the four elements in quaternion vector turns to solve the least square estimation of the Wahba problem. Therefore, a gradient descent optimization-based initial alignment algorithm (GD1) for stationary SINS is proposed to determine the initial attitude in this paper. For stationary base alignment, the theoretical gravity vector and angular rate of the earth's rotation in navigation frame are definite constants once we have the local latitude information. And then we can construct an objective function, which is a least-squares cost function, to calculate the attitude quaternion vector with these vectors in navigation frame and their practical measurements from IMUs in body frame. Hence, the gradient descent optimization method is employed to compute the minimum of the nonlinear objective function with repeated iterations along the negative gradient direction.
Moreover, most existing SINS alignment methods, to the best of our knowledge, all require the local latitude information to be known in advanced. The local latitude information, however, may be inaccurate or difficult to obtain in some situations. To deal with the stationary SINS alignment problem with no latitude information, furthermore, we propose an improvement on our gradient descent optimization algorithm (GD2), which can complete the initial alignment process without using the latitude information.
Firstly, utilizing the tri-axis accelerometers outputs alone, we can determine an orthogonal coordinate frame, of which the horizontal plane is aligned with that of the navigation frame. And then we can obtain the components of the angular rate of earth rotation in the new orthogonal coordinate frame with the measurements from tri-axis gyroscopes. Besides, it is noted that there are only y-axis and z-axis components of the angular rate of earth rotation in the navigation frame. Thus, based on the geometry constraints, we can represent the angular rate vector in the navigation frame with its components in the new orthogonal coordinate frame, without using the local latitude information, and then we obtain a new objective function without the latitude parameter. Furthermore, we can determine the attitude quaternion vector by using the GD1 method to solve the new objective function.
Moreover, we also find that the practical measurements from IMUs contain high frequency noises, which highly degrade the performance of the proposed algorithms. Therefore, we design an IIR low-pass filter to pre-process the outputs from IMUs before employing the proposed alignment algorithm, which can resist these high frequency disturbances.
To verify the performances of the proposed SINS alignment algorithms, simulations and experiments for SINS initial alignment are employed, and the results demonstrate the rapidness and accuracy.