International Journal of Aerospace Engineering

Volume 2015, Article ID 904913, 11 pages

http://dx.doi.org/10.1155/2015/904913

## Cone Algorithm of Spinning Vehicles under Dynamic Coning Environment

^{1}School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China^{2}Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, Beijing 100081, China^{3}Beijing Key Laboratory of High Dynamic Navigation Technology, Beijing Information Science & Technology University, Beijing 100101, China

Received 20 August 2015; Revised 30 October 2015; Accepted 10 November 2015

Academic Editor: Hikmat Asadov

Copyright © 2015 Shuang-biao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 two-stage structure of attitude algorithm is refined and outlined [1], which consists of attitude matrix update cycle by Jordan [2] and rotation vector update cycle by Bortz [3]. Particularly, the crucial coning correction of noncommutativity, calculated from gyro data in a separate algorithm, is the critical process executed in rotation vector update cycle to improve the attitude accuracy, and it has attracted more attentions from demanding researchers in navigation technology. This is because the coning error induced by coning environment can affect navigation accuracy and control precision of vehicles [4, 5]. Miller used a quaternion algorithm with three intervals from gyro to calculate attitude and approximate coning correction error for the coning correction coefficients design under a pure coning condition [6]. Lee et al. improved the quaternion algorithm with four intervals and compared drift error with other fewer-interval algorithms [7]. Based on the Miller achievement, Ignagni proposed a two-speed structure for coning correction and introduced nine algorithms to realize optimization of coning correction coefficients [8]. Jiang and Lin proposed an improved strapdown coning algorithm and disclosed the essential relationship between rotation vector and quaternion [9]. Li et al. divided minor coning correction into a number of subminor intervals and delivered a generalized coning compensation algorithm that is independent of the number of incremental angles [10]. Tang and Chen proposed a coning correction structure containing cross-product of angular rates, cross-product of angular increments, and cross-product of angular rate and increment, and it can analyze effect on attitude by basing on time Taylor series and frequency Taylor series [11]. Meanwhile, the gyro frequency response is a primary cause for pseudo-coning error that is another form of coning error, and the traditional coning correction algorithms cannot work to high-order accuracy. Mark and Tazartes proposed a tuning high-order coning algorithm to match the frequency response characteristics of the gyro and avoided overcompensation for pseudo-coning correction [12]. Kim and Lee added self-vibration of gyro to the causes of pseudo-coning. They showed that vibration had a form generated by different frequency inputted to two orthogonal axes, and coning motion was just a specific case [13]. Savage thought that stochastic dynamic environment sensed by high precision gyro could also cause pseudo-coning error, so he used explicit frequency shaping by minimum least-squares estimation for stochastic data to design coning correction coefficients [14]. Song et al. combined with Miller’s and Savage’s methods and delivered a supplementary coning error equation to achieve superior maneuver accuracy [15]. Considering measurement error of gyro, Fu et al. developed a two-time scale model by singular perturbation technique to realize coning and pseudo-coning correction [16]. Wang et al. used a new coning motion model that contains pure constant coning environment and spinning motion of vehicles to examine the ignored triple-cross-product term of noncommutativity error in attitude rotation vector and test the performance of previous algorithms [17]. Patera disclosed that no attitude error was propagated by using a slewing frame under oscillating coning environment [18] and extended the slewing frame to improve attitude propagation attitude for cases of time varying coning motion [19]. Unfortunately, the method based on the slewing frame cannot provide the common attitude of spinning vehicles under complex coning environment because the relationship of the slewing frame and the common attitude of vehicles is unclear.

Synthesizing and analyzing research achievements above, we find that the recognized coning error and the pseudo-coning 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 two-stage 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 in-depth 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 2, we derive the model of dynamic coning environment by rotation vector. In Section 3, we provide the traditional two-stage structure of attitude algorithm and investigate the effect of dynamic coning environment on attitude algorithm. In Section 4, based on the definition of the cone frame and cone attitude, we propose a cone algorithm for cone attitude by using the rotation relationship and build the relationship between cone attitude and Euler attitude. In Section 5, the simulations are made to explain the effect of dynamic coning environment on tradition attitude algorithm and verify the validity of the cone algorithm. In Section 6, we conclude our research in this paper and provide our future work.

#### 2. Dynamic Coning Environment

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 phase-shifted by 90 degrees [14]. The third axis, which is orthogonal with the other two oscillating axes, will precess around an axis and generate a conical surface in inertial space. Under the classical coning environment, cone half-angle and oscillating frequency are all constant, and the precession frequency of the third axis is equal to oscillating frequency. Obviously, this process has constant cone half-angle and constant precession frequency, so the classical coning environment is static. In contrast, the dynamic coning environment is a complex process with varying cone half-angle, so rotation vector can be defined aswhere is the changing rate of cone half-angle and is the precession frequency. Assuming that and are constant during small time interval, the derivation of (1) can be obtained:

Referring to [3], the angular velocity can be described as

Substituting (1) and (2) into (3), the angular velocity of dynamic coning environment in vehicles can be obtained:

From (4), we see that if the changing rate of cone half-angle is zero but the cone half-angle is nonzero, (4) can be simplified aswhere is the constant cone half-angle. It is clear that (5) is the model of the classical coning environment, and it is just a special case that has no nutation rate and no rotation [3, 6, 9].

#### 3. Effect of Dynamic Coning Environment on Attitude Algorithm

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.

##### 3.1. Definition of Relative Frames

To describe the movement of vehicles with respect to the reference frame, the common frames are introduced firstly.

*The Body Frame* . The origin is located at the center of mass of a vehicle, axis coincides with longitudinal axis of a vehicle, axis is pointing up and perpendicular to axis in symmetry plane of a vehicle, and axis is obtained by the right-hand rule.

*The Earth Frame* *.* The origin is located at the point of launch site, axis is pointing to a target, axis is pointing up and perpendicular to axis in vertical plane, and axis is obtained by the right-hand rule.

The rotation order of and is as follows:

As is shown in Figure 1, the relationship of and can be described by a transformation matrix:where represents transformation from the body frame to the earth frame, stands for the earth frame, stands for the body frame and Euler attitude angles , , and are the pitch angle, yaw angle, and roll angle, respectively. The range of is , the range of is , and the range of is .