Chinese Journal of Aeronautics, 2013,26(2): 449–455 Chinese Society of Aeronautics and Astronautics & Beihang University Chinese Journal of Aeronautics [email protected] www.sciencedirect.com Interlaced optimal-REQUEST and unscented Kalman filtering for attitude determination Quan Wei *, Xu Liang, Zhang Huijuan, Fang Jiancheng Science and Technology on Inertial Laboratory, Key Laboratory of Fundamental Science for National Defense-Novel Inertial Instrument & Navigation System Technology, Beihang University, Beijing 100191, China Received 25 November 2011; revised 13 February 2012; accepted 22 May 2012 Available online 7 March 2013 KEYWORDS Abstract Aimed at low accuracy of attitude determination because of using low-cost components Attitude determination; which may result in non-linearity in integrated attitude determination systems, a novel attitude Hybrid simulation; determination algorithm using vector observations and gyro measurements is presented. The Interlaced filtering; various features of the unscented Kalman filter (UKF) and optimal-REQUEST (quaternion estima- Optimal-REQUEST; tor) algorithms are introduced for attitude determination. An interlaced filtering method is pre- Unscented Kalman filter sented for the attitude determination of nano-spacecraft by setting the quaternion as the attitude representation, using the UKF and optimal-REQUEST to estimate the gyro drifts and the quaternion, respectively. The optimal-REQUEST and UKF are not isolated from each other. When the optimal-REQUEST algorithm estimates the attitude quaternion, the gyro drifts are estimated by the UKF algorithm synchronously by using the estimated attitude quaternion. Furthermore, the speed of attitude determination is improved by setting the state dimension to three. Experimen- tal results show that the presented method has higher performance in attitude determination compared to the UKF algorithm and the traditional interlaced filtering method and can estimate the gyro drifts quickly. ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. 1. Introduction including accuracy and speed, not only depends on the capa- bility of inboard sensors, but also relies on the performance High-accuracy attitude determination is required to achieve of attitude determination algorithms. Besides the accuracy, high-accuracy attitude control and attitude stability for the better the observation ability is, the quicker the speed of nano-spacecraft. As such, attitude determination is important attitude determination is, especially, for the new-generation to any spacecraft. The performance of attitude determination, smart small satellites. A typical system for attitude determina- tion of nano-spacecraft comprises several gyroscopes and a vector-sensor. The development of new sensor technology * Corresponding author. Tel.: +86 10 82339013. promotes the integration of micro-electro-mechanical system E-mail address: [email protected] (W. Quan). (MEMS) gyroscopes and a complementary metal oxide Peer review under responsibility of Editorial Committe of CJA. semiconductor (CMOS) transistor active pixel sensor (APS) as a star sensor, and the quick development of a high performance strapdown stellar-inertial integrated attitude Production and hosting by Elsevier determination system,1 which is especially suitable for the high 1000-9361 ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.org/10.1016/j.cja.2013.02.023 450 W. Quan et al. performance attitude determination of nano-spacecraft.2,3 The UKF and optimal-REQUEST, an interlaced filtering method attitude determination algorithms based on state estimation is presented by setting the quaternion as the attitude represen- and vector observation are most effective because of their high tation, using the UKF and optimal-REQUEST to estimate the accuracy and efficiency.4,5 gyro drifts and the quaternion respectively. When the optimal- At present, there are many real-time attitude estimation REQUEST algorithm is used to estimate the attitude quater- algorithms for spacecraft based on state estimation. The ex- nion, the gyro drifts are estimated by the UKF algorithm only tended Kalman filter (EKF) method is widely used in nonlin- using the three-dimensional state variable which is much ear systems.6–10 However, it needs to calculate the Jacobian quicker than in the case of six dimensions.18 Meanwhile, the matrix for the linearization of nonlinear systems, which results attitude quaternion derived from the optimal-REQUEST, is in low accuracy and output frequency. The unscented Kalman adopted when the UKF estimates the gyro drifts. Further- filter (UKF) method was presented by Juliear and Uhlman in more, the state dimension is only set to three to improve the 1995, in which they used special sample points to fit the non- speed of attitude determination. This method not only has linear state and let the mean and variance of sampling points higher performance of attitude determination, but also quickly equal to those of the states at the sampling moment. Based estimates gyro drifts. on such sampling points in the nonlinear system, the corre- sponding transformation sampling points were obtained, dur- 2. System model ing which the corresponding mean and variance were calculated. For non-linearized systems, this method can 2.1. Gyro measurement model achieve higher accuracy of filtering than the EKF method.11,12 Among attitude determination methods based on vector Supposing that the three sensitive axes of three MEMS gyro- observation, the quaternion estimator (QUEST) method13 is scopes are respectively parallel to the three body-axes of a a very popular algorithm for single frame. However, it shows spacecraft, a general gyroscope model is given by two disadvantages. One is that at least two simultaneous mea- x~ k ¼ xk þ bk þ gv;k ð1Þ surements are needed to estimate attitude completely. The other is that the information of previous measurements is lost bkþ1 ¼ bk þ gu;k ð2Þ entirely. To solve the above problems, the recursive quaternion where x~ is the gyroscope measurement, x the attitude angu- estimator (REQUEST) method was presented by Bar-Itzhack k k lar rate of the gyro input axis, b the gyro drift vector; g and in 1996,14 which was actually a recursive QUEST method. k v;k gu;k are independent zero-mean Gaussian white-noise pro- The K-matrix is sequentially propagated and updated as state 2 2 cesses, and the variance of gv;k and gu;k are rv 1 and ru1 respec- variables. The REQUEST method is suboptimal since the T tively, 1 ¼½111 . propagation noise is suppressed using a fading memory factor. In 2004, the optimal-REQUEST method was presented by 2.2. Star sensor measurement model Choukroun et al.15, in which the REQUEST algorithm was embedded in the framework of a Kalman filter. For that pur- As a high-accuracy vector sensor, the measurement model of pose, the K-matrix elements are modeled as state variables and the star sensor is defined by model equations for the process and the measurement of the K-matrix are developed. The attitude quaternion is then iso- bk ¼ AðqkÞrk þ dbk ð3Þ lated from the estimated K-matrix. The optimal-REQUEST where rk is the stellar reference vector in inertial coordinates, method has the advantage of sequentially processing the mea- which can be obtained from the astronomical ephemeris; bk sured information. It can estimate the attitude even when the measurement vector containing noise in the star sensor acquiring a single vector measurement at each observation coordinates, which are supposed to be the same as the body time, but under the condition of non-collinearity between at coordinates of the spacecraft; dbk the noise of the measurement least two observed lines-of-sight. This method cannot estimate and satisfied by parameters other than attitude, and even then it cannot esti- EðdbÞ¼0 mate and compensate for gyroscope drifts, which results in ð4Þ Eðdb Á dbTÞ¼r2ðI À bbTÞ the divergence of the optimal-REQUEST. 3 2 The EKF and UKF methods can estimate not only attitude where I3 is the third-order identity matrix and r the variance parameters, but also gyro drifts. However, these methods re- of the measurement noise. quire measurements from the star sensor that are disposed AðqkÞ is the attitude matrix from the inertial coordinates to sequentially. Therefore, a traditional interlaced filtering meth- the spacecraft body coordinates, and can be described as a T T od based on the EKF, QUEST, and optimal-REQUEST for function of the quaternion qk ¼½qk q4k : 16 micro-spacecraft was presented by Yu and Fang in 2005 2 T T AðqkÞ¼ððq4kÞ À qk qkÞI3 þ 2qkqk À 2q4k½qk ð5Þ and an adaptive interlaced filtering method for attitude deter- T mination of nano-spacecraft was presented by Quan et al. in where qk ¼½q1k q2k q3k and q4k are the vector and scale 17 2008. However, the optimal-REQUEST algorithm is only part of the quaternion, respectively; ½qk denotes the cross- an approximate algorithm and contains errors beyond those product matrix associated with the vector qk . of the EKF and UKF for attitude information. There is six- 2.3. Spacecraft attitude kinematical model dimensional state variable for filtering models in Refs. 16,17, which leads to low speed of attitude determination. However, using the EKF and UKF directly with the Spacecraft attitude is described using the quaternion qk in the 14 quaternion attitude parameterization yields a non-unit norm body coordinates relative to the inertial coordinates , and the quaternion estimate. According to the various features