Due to the fact that attitude error of vehicles has an intense trend of divergence when vehicles undergo worsening coning environment, in this paper, the model of dynamic coning environment is derived firstly. Then, through investigation of the effect on Euler attitude algorithm for the equivalency of traditional attitude algorithm, it is found that attitude error is actually the roll angle error including drifting error and oscillating error, which is induced directly by dynamic coning environment and further affects the pitch angle and yaw angle through transferring. Based on definition of the cone frame and cone attitude, a cone algorithm is proposed by rotation relationship to calculate cone attitude, and the relationship between cone attitude and Euler attitude of spinning vehicle is established. Through numerical simulations with different conditions of dynamic coning environment, it is shown that the induced error of Euler attitude fluctuates by the variation of precession and nutation, especially by that of nutation, and the oscillating frequency of roll angle error is twice that of pitch angle error and yaw angle error. In addition, the rotation angle is more competent to describe the spinning process of vehicles under coning environment than Euler angle gamma, and the real pitch angle and yaw angle are calculated finally.
Attitude algorithm is a key part of navigation technology and guarantees directly control system accuracy and navigation accuracy of aircrafts, ships, and vehicles. Through decades of persistent efforts by researchers, to expand applicability of attitude algorithm, an outstanding twostage structure of attitude algorithm is refined and outlined [
Synthesizing and analyzing research achievements above, we find that the recognized coning error and the pseudoconing error are generated directly while calculating rotation vector increment from gyro in rotation vector update cycle, and the optimization of attitude algorithm is mainly achieved by using the total model and the simplified model of the classical coning environment to restrain drifting component of coning error. However, the oscillating error of attitude exists when the optimization of the twostage structure of attitude algorithm is used for vehicles under classical coning environment. Although attitude error is not large enough for nonspinning vehicles to attract more attentions, it has an intense trend of divergence over time when coning environment is becoming worse. However, up to now, there is no literature to exploit this issue, and the indepth study of attitude algorithm of vehicles under high dynamic environment is hardly carried on. Therefore, it is necessary for us to investigate the real attitude of vehicles and disclose the hidden relationship between vehicles’ attitude and dynamic coning environment.
The scheme of this paper is as follows. In Section
Before studying the effect of dynamic coning environment on attitude algorithm, the derivation of modelling dynamic coning environment is made in this section.
Generally, the classical coning environment is defined as a condition where two orthogonal axes in a vehicle simultaneously experience sinusoidal oscillations that are mutually phaseshifted by 90 degrees [
Referring to [
Substituting (
From (
Since the common optimization algorithm is developed under static coning environment, it is reasonable to consider how attitude algorithm is affected by dynamic coning environment. In this section, we will investigate this question.
To describe the movement of vehicles with respect to the reference frame, the common frames are introduced firstly.
The rotation order of
As is shown in Figure
Rotational relationship of
The optimization of attitude algorithm is developed by using twostage structure of attitude algorithm under the classical coning environment, and the twostage structure algorithm in modern strapdown inertial navigation systems is given by [
According to the common optimizing methods of attitude algorithm by using angular increment, some coefficients are designed to restrain the direct component of
One stage is updating the attitude in an attitude updating cycle, and it is referred to in (
By analyzing the optimizing process, the socalled coning error is independent directly to vehicles’ movement due to (
It is convenient to use the twostage structure algorithm to design coefficients and restrain drifting error of attitude under static coning environment, but it is difficult to investigate the effect of dynamic coning environment clearly by using twostage structure algorithm because it is quite difficult to derive simple and clear descriptive equations for oscillating error and drifting error. Nevertheless, twostage structure algorithm and the Euler attitude algorithm both use Euler angles to define the relationship between the body frame and the earth frame [
According to the rotation relationship in Figure
In (
Substituting (
Under dynamic coning environment, there is no rolling motivation for nonspinning vehicles, so the real roll angle should be zero. However, we note that the roll angle located on the right of the third element of (
In (
From the derivation above by using Euler attitude algorithm, we confirm that attitude of vehicles is affected to generate drifting error and oscillating error under dynamic coning environment.
From the first and second elements of (
With the small angle approximation, the pitch angle error and yaw angle error can be obtained by comparing (
In fact, the pitch angle error and yaw angle error are all induced by the transferred roll angle error under dynamic coning environment, so attitude error is actually a sort of induced error by dynamic coning environment. This is the reason why so many researchers pay more attentions to attitude error by coning environment, in, specially, the roll angle error.
From the analysis above, attitude algorithm is affected by dynamic coning environment badly. To calculate attitude accurately, in this section, we propose a coning algorithm of spinning vehicles based on a cone frame and cone attitude and build the relationship between cone attitude and Euler attitude.
Compared with common angular movement, coning motion of spinning vehicles is a sort of angular movement of longitudinal axis of vehicles under coning environment. Actually, this special movement directly shows precession and nutation of longitudinal axis, and selfrotation is accompanying. Since attitude error is induced by coning motion of periodicity, a reasonable method is describing coning motion accurately and calculating attitude without error. Therefore, cone frame and cone angles should be defined specially to describe angular movement.
Referring to gyrodynamics [
The rotation order of
In the rotation order,
As is shown in Figure
Rotational relationship of
According to the rotation relationship in Figure
From (
Equation (
By transforming (
We noted that (
Because the dynamic coning model is derived by using the cone frame, cone attitude is not affected by coning environment. In other words, there is no error in the third axis although the other two axes are oscillating periodically:
The coning environment can be described in the earth frame or the cone frame, so the relationship of Euler attitude and cone attitude is built according to the geometrical relationship in Figure
Geometrical relationship of Euler attitude and cone attitude.
Therefore, the pitch angle and yaw angle can be calculated by
According to Figure
Symbol of




+ 

− 
Symbol of




+ 

− 
To disclose the effect of coning environment on traditional attitude algorithm and verify the validity of the cone attitude algorithm, numerical simulations are designed on purpose in this section.
We choose a nonspinning vehicle for simulations because it is convenient to check whether the roll angle or the rotation angle is zero. We assume that the initial Euler attitude of the nonspinning vehicle is
Varying
Simulation  1  2  3  4 


360  720  1080  1440 
Varying
Simulation  1  2  3  4 

Λ (°/s)  0.011  0.023  0.034  0.046 
Attitude error by Table
Attitude error by varying
The whole process of attitude error
Amplification of attitude error
Attitude error by varying Λ.
The whole process of attitude error
Amplification of attitude error
Combining Tables
Triaxial angular velocity.
Cone attitude angles
Pitch angle and yaw angle of the vehicle.
The whole process of pitch angle and yaw angle
Amplification of pitch angle and yaw angle
Attitude error of vehicles has an intense trend of divergence when vehicles undergo worsening coning environment. In this paper, we firstly derive the model of dynamic coning environment. Then through investigation of the effect on Euler attitude algorithm for the equivalency of traditional attitude algorithm, we find that attitude error is actually the roll angle error including drifting error and oscillating error, which is induced directly by dynamic coning environment and further affects the pitch angle and yaw angle through transferring. Based on the definition of the cone frame and cone attitude, we propose a cone algorithm by rotation relationship to calculate cone attitude and establish the relationship between cone attitude and Euler attitude of spinning vehicle. Finally, numerical simulations are made with variant precession frequency and changing rate of cone halfangle to verify the effect of dynamic coning environment on attitude algorithm and validity of cone algorithm. The results show that the induced error of Euler attitude fluctuates by the variation of precession and nutation, especially by that of nutation, and the oscillating frequency of roll angle error is twice that of pitch angle error and yaw angle error. In addition, the rotation angle is more competent to describe the spinning process of vehicles under coning environment than Euler angle gamma, and the real pitch angle and yaw angle are calculated finally.
This paper is beneficial for calculating real attitude of spinning bodies, understanding mechanism of attitude error by coning environment, and developing attitude algorithm, and it also provides a new view on the effect of coning motion. Our future work will focus on the optimization of attitude algorithm under dynamic coning environment and realtime attitude algorithm of spinning vehicles under high dynamic movement.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study is financially supported by the National Natural Science Foundation of China (nos. 61471046, 61473039, and 11202023).