Research Article Relative Status Determination for Spacecraft Relative Motion Based on Dual Quaternion

Mathematical Prolems in Engineering Volume 214, Article ID 62724, 7 pages http://dx.doi.org/1.1155/214/62724 Research Article Relative Status Determination for Spacecraft Relative Motion Based on Dual Quaternion Jun Sun, 1,2 Shijie Zhang, 1 Xiande Wu, 3 Fengzhi Guo, 3 and Yaen Xie 3 1 Research Center of Satellite Technology, Harin Institute of Technology, Harin 158, China 2 Shanghai Institute of Space Flight Control Technology, Shanghai 23, China 3 College of Aerospace and Civil Engineering, Harin Engineering University, Harin 151, China Correspondence should e addressed to Xiande Wu; xiande wu@163.com Received 26 Feruary 214; Accepted 7 April 214; Pulished 8 May 214 Academic Editor: Weichao Sun Copyright 214 Jun Sun et al. This is an open access article distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. For the two-satellite formation, the relative motion and attitude determination algorithm is a key component that affects the flight quality and mission efficiency. The relative status determination algorithm is proposed sed on the Extended Kalman Filter (EKF) and the system state optimal estimate linearization. Aiming at the relative motion of the spacecraft formation navigation prolem, the spacecraft relative kinematics and dynamics model are derived from the dual quaternion in the algorithm. Then taking advantage of EKF technique, comining with the dual quaternion integrated dynamic models, considering the navigation algorithm using the fusion measurement y the gyroscope and star sensors, the relative status determination algorithm is designed. At last the simulation is done to verify the feasiility of the algorithm. The simulation results show that the EKF algorithm has faster convergence speed and higher accuracy. 1. Introduction There are higher demands for the satellite navigation technologyandcontrolaccuracyinspacemissionsthanefore. The accuracy of satellite attitude determination is the key factor for the performance of the attitude control. Hence, how to effectively improve the accuracy of satellite attitude determination has received increased attention during the last years. With rapid multitarget acquisition and pointing and tracking capailities, agile spacecraft has great practical value and wide prospect for application. Moreover, agile spacecraft has many advantages such as lighter mass, smaller size, cheaper price, and shorter development cycle, which makes great differences to space technology compared with classical spacecraft. However,itisdifficulttoestalishrelativekinematics and dynamics model for doule agile satellites collaorated earth oserving mission ecause of the rapid attitude maneuveraility of agile satellites. Classical modeling approaches separately address the orit and attitude prolems, which complicates models and the spacecraft attitude control. To descrie spiral motion, dual quaternion was proved to e themostcompactandefficienttoolamongotherapproaches. Considering the advantages of dual quaternion in descriing spiral and relative motion, uilding doule agile satellites integrated orit and attitude dynamics equations not only can enhance the computing efficiency ut also may avoid the effect of orit and attitude decoupling. The Kalman Filter which has many advantages such as fast calculation, small memory resource, and real-time is a recursive algorithm sed on minimum mean-square error [1, 2]. The sic Kalman Filter is limited to a linear assumption. The spiral motion of spacecraft in practical applications, however, can e nonlinear. As a result, spacecraft kinematics and dynamics equations are nonlinear [3 5]. In view of the effect of nonlinearity, this paper, using EKF (Extended Kalman Filter) technology, comined with the dual quaternion integrated dynamics modeling, designs a navigation algorithm of gyroscope/star sensor comined

2 Mathematical Prolems in Engineering X Z Y The orit of 2 r ra x z Spacecraft 2 y a z a r x a The orit of 1 y Spacecraft 1 Figure 1: The orit of the relative motion. measurement and carries on the simulation to verify the feasiility of the algorithm. The paper is organized as follows. Section 2 descries the relative kinematic and dynamic equations sed on dual quaternion. Section 3 designs the measurement models of gyroscope/star sensor, ecause satellite attitude determination mostly adopts gyroscope/star sensor cooperative determination. Based on the models designed, the state and measurement equations which are linearized in Section 4 are derived. Section 5 shows simulation results. Finally, Section 6 contains the main conclusions. 2. The Relative Dynamic Equations Based on Dual Quaternion 2.1. Kinematic Equation Based on Dual Quaternion. Figure 1 shows two rigid-ody spacecraft oriting the earth: one is spacecraft 1 and the other is spacecraft 2. Their ody fixed frames, respectively, are O a -x a y a z a and O -x y z.we canseetherelativemotionofthetwospacecraftasthe spiral motion of the O -x y z relative to the O a -x a y a z a. The relative motion is descried y dual quaternion; the expression is q = q +ε 1 2 ra q = q +ε 1 2 q r, (1) where q is attitude quaternion of the spacecraft 1 relative to spacecraft 2 and r = r r a is relative position vector of two-spacecraft center mass. Thekinematicequationfortherelativemotionoftwo spacecraft expressed y dual quaternion is governed y where 2 q = ω a q = q ω, (2) ω a = ωa +εka = ωa +ε( ra + ra ωa ), (3) ω = ω +εk = ω +ε( r + ω r ). ω a, ω are attitude velocity of the spacecraft 2 relative to spacecraft 1 expressed in O a -x a y a z a and O -x y z. r a, r are relative position vectors of the center mass of the spacecraft expressed in O a -x a y a z a and O -x y z. ω is relative velocity motor of two spacecraft expressed in O -x y z, it can e expressed as ω = ω q ωa a q, (4) where ω, ωa a are velocity motors of two spacecraft expressed in their own ody fixed frames, q is spiral motion of O -x y z with respect to O a -x a y a z a,and q is conjugation of q. 2.2. Dynamic Equation Based on Dual Quaternion. For rigidody spacecraft, the dual inertia operator is defined as M = me + Î =m d dε E +εi m d dε +εi xx εi xy εi xz = εi yx m d [ dε +εi yy εi yz, [ εi zx εi zy m d ] dε +εi zz] where m is the mass of spacecraft, I is the inertia matrix, and E is a 3 3 identity matrix: (5) M = d dε I +ε 1 E. (6) m Based on single spacecraft dynamic equation F c = (d/dt)ĥ c = M(d/dt) ω c + ω c M ω c = M ω c c + ω c M ω c, differentiating (4), and comining with the single spacecraft kinematic equation expressed y dual quaternion q = (1/2) ω i q = (1/2) q ω and q = (1/2) ω q, assume that [ ω + ] = (1/2)([ ω ] [ ω ]),theequationcanewritten as ω = ω q q ωa a q ω a a q q ωa a q = ω q ω a a q q ωa a 1 2 q ω + 1 2 ω q ωa a q

Mathematical Prolems in Engineering 3 = ω q ω a a q + 1 2 ([ + ω ] [ ω ]) q ωa a q. The following equation is derived from the single spacecraft dynamic equation: ω Sustituting (8)into(7)yields ω (7) = M ( ω M ω )+ M F. (8) = M ( ω M ω )+ M F q ω a a q +[ ω ] q ωa a q. Equation (8) isrelativedynamicequationsedondual quaternion. ω is otained through (4) and expressed as (9) ω = ω + q ωa a q. (1) ω is the derivative of ω in O -x y z, M is dual inertia operator of spacecraft 2, and F isdualforceactingonthe spacecraft 2 at the center of mass. According to the demands, we do simplified assumptions: assume that the orit of spacecraft 1 is circular orit, and spacecraft 1 without maneuver; thus ω a a = [ ω]t + ε[ωr a ] T. The following equation is otained: ω a a =, (11) where ω is orit angular velocity of spacecraft 1 and r a is orit radius of spacecraft 1. Relative dynamic equation can e simplified as ω = M ( ω M ω ) + M F + [ ω ] q ωa a q. 3. Gyroscope and Star Sensor Measurement Model (12) 3.1. Gyroscope Measurement Model. Gyroscope is one of attitude measurement sensors which are the most widely used in modern navigation control areas, generally output data in the form of angular velocity or angular velocity increment with respect to inertial space. Comined with other sensors, such as accelerometers, it can navigate and posit the carrier. Generally, directing output of the gyroscope is projected in the gyroscope coordinate system of angular velocity in inertial system. For simplicity, assuming gyroscope measurement coordinate system coincides with the ody coordinate system of spacecraft. Gyroscope mathematical model can e represented y this: ω (t) =ω(t) +β(t) +η V (t), β (t) =η u (t). (13) ω is continuous measurement output value of the gyroscope, β(t) is gyroscope drift, and η V (t) and η u (t) are Gaussian white noise process with uncorrelated zero mean, and they conformed: E{η V (t) η T V (t)} =I 3 3σ 2 Vδ (t τ), (14) E{η u (t) η T u (t)} =I 3 3σ 2 u δ (t τ). 3.2. Star Sensor Measurement Model. Star sensor is currently of the highest accuracy, the most widely used attitude sensor; stellar is its reference source of attitude measurement, and it can output vector direction of stars in star tracker coordinates. It provides high-accuracy measurements of spacecraft attitude control and navigation. The output of star sensor can e a reference axis vector and also e Euler angles or quaternions. Euler angles otained through the star sensor data processing unit and the star sensor output is: ψ m =ψ+v ψ, θ m =θ+v θ, φ m =φ+v φ. (15) ψ m, θ m, φ m are measurements of star sensor, ψ, θ, andφ are true attitude angles, and V s =[V ψ V θ V φ ] T is star sensor measurement noise. 4. Relative Attitude Determination Algorithm Based on Extended Kalman Filter (EKF) 4.1. State Equation and Linearization. State variale X(t) = [ q ω ] T. Nonlinear continuity equation is expressed as where X (t) =f(x (t),t) +w(t), (16) f (X (t),t) =[ q ω ] T = [ [ M [ ( ω M ω )+M 1 2 q ω F q ( ωa a ) q +[ ω ] q ( ωa a ) q ]. (17) ]

4 Mathematical Prolems in Engineering 3.3 F x (N) 2 1 1 2 3 4 5 6 7 8 T x (N m).2.1.1 1 2 3 4 5 6 7 8 5 1 F y (N) T y (N m).5 5 1 2 3 4 5 6 7 8.5 1 2 3 4 5 6 7 8 F z (N) 5 5 1 2 3 4 5 6 7 8 Figure 2: Control force. T z (N m).1.5.5.1 1 2 3 4 5 6 7 8 Figure 3: Control moment. Make the formula (16) to e linearization and discretization; we can get X k =Φ k,k X k,k +W k, (18) where Φ k,k =I+C k T, T is sampling time: C k = f X T X=X k = [ [ q q T ω q T q ω T ω. (19) ] ω T ] 4.2. Measurement Equation and Linearization. The measurement equation is as follows: Z (t) =h(x (t),t) + V (t), (2) where v(t) is measurement noise. This is a nonlinear equation of variale X. Making it to e linearization and discretization, so 4.3. EKF Filtering Algorithm Process State single-step forecast: Z k =H k X k +V k. (21) x k,k = x k +f( x k )T. (22) Single-step forecast error of mean square: P k,k =Φ k,k P k Φ T k,k +Q k, (23) where Φ k,k is single-step transfer matrix of system. Filtration transmission gain: K k =P k,k H T k [H kp k k H T k +R k]. (24) H k is measurement matrix and R k is measurement noise matrix. Status estimated value: x k = x k,k +K k [z k h( x k,k )]. (25) Filtering error of mean square: P k =[I K k H k ]P k,k. (26) 5. Simulation and Result Analysis Under the ckground of the final collaorative period of two-spacecraft earth oserving (relative distance is 1- meters), we carry on the simulation sed on kinematics and dynamics equations. Simulation verifies the two spacecraft s kinematics and dynamics equations, exerts perturtive force and moment to spacecraft 2, and verifies that the relative velocity and angular velocity of the spacecraft change along with the variation of force and moment. Integrating the two spacecraft s kinematics and dynamics equation, integration step is.1 seconds. Suppose that the spacecraft 1 moves in circular orit whose height is 5 kilometers. Because spacecraft 1 is directional to earth, its ody coordinate system coincides with orital coordinate system.

Mathematical Prolems in Engineering 5 q 1 q 2 q 3 q 4 1.5 1.5.5.5 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8.5.5 1 2 3 4 5 6 7 8.5.5 1 2 3 4 5 6 7 8 Dynamics Filtering Star sensor Figure 4: Attitude quaternion. The three axes of the spacecraft 2 s ody coordinate system coincide with its inertia principal axis, its mass is 1 kg, and its moment of inertia is 722 J=[ 82 ] kg m 2. (27) [ 876] The attitude and position of spacecraft 2 relative to the spacecraft 1 at initial time are q 5 q 6 q 7 q 8 y ( ) x ( ) 5 5 5 1 2 3 4 5 6 7 8 1 5 5 1 2 3 4 5 6 7 8 5 5 1 2 3 4 5 6 7 8 2 2 4 1 2 3 4 5 6 7 8 Figure 5: Quaternion. 1 5 5 1 2 3 4 5 6 7 8 1 2 1 2 3 4 5 6 7 8 1 5 q () =[.3772.4329.6645.4783] T, r () =[ 2 1]T m, ω () =[.96.14.15]T rad/s, r () =[.25.382.74]T m/s. (28) z ( ) 5 1 2 3 4 5 6 7 8 Dynamics Star sensor Filtering Figure 6: Attitude angular.

6 Mathematical Prolems in Engineering w x (rad/s) w y (rad/s).2.1.1 1 2 3 4 5 6 7 8.4.2.2 1 2 3 4 5 6 7 8 1 3 5 r x (m) r y (m) 1 2 1 2 3 4 5 6 7 8 5 5 1 2 3 4 5 6 7 8 w z (rad/s) 5 1 2 3 4 5 6 7 8 r z (m) 15 1 5 Dynamics Star sensor Filtering 5 1 2 3 4 5 6 7 8 Figure 7: Attitude angular velocity. Figure 8: Relative position. The attitude and position of the goal are q (t) =[1 ] T, r (t) =[ ]T m, ω (t) =[ ]T rad/s, r (t) =[ ]T m/s. (29) v x (m/s) 6 4 2 2 1 2 3 4 5 6 7 8 The measurement noise mean square error of star sensor σ s = 1/36,initialgyroconstantdrift =[5 5 5] T deg/h, mean square error of measurement noise σ g1 =.5 deg/h, andmeansquareerrorofslopewhitenoiseσ g1 =.1 deg/h. Relative motion of two spacecraft changes in coordinate system O -x y z is as shown. Figures 2 3, respectively, are perturtive force and moment to spacecraft 2 Figures 4, 5, 6, 7, 8, and9, respectively,areattitude quaternion,attitudeangular,attitudeangularvelocity,relative position, and relative velocity in the case of dynamics, star sensor, and filtering. They are parameters estimation of EKF algorithm. 6. Conclusion Based on Extended Kalman Filter, comined with the spacecraft relative dynamics equation which is descried y dual v y (m/s) v z (m/s) 6 4 2 2 1 2 3 4 5 6 7 8 3 2 1 1 2 3 4 5 6 7 8 Figure 9: Relative velocity.

Mathematical Prolems in Engineering 7 quaternion, this paper designs the relative attitude algorithm of gyroscope/star sensor comined measurement. The simulation results show that the EKF algorithm has faster convergence speed and higher accuracy. Conflict of Interests The authors declare that there is no conflict of interests regarding the pulication of this paper. Acknowledgments This work is partially funded y the Fundamental Research Funds for the Central Universities (no. HEUCF21318), the Natural Science Foundation of Heilongjiang Province (no. A21312), the National Natural Science Funds (no. 113728), the Harin Science and Technology Innovation Talent Youth Fund (no. RC213QN17), and the Key Laoratory of Datase and Parallel Computing, Heilongjiang Province, the National High Technology Research and Development Program of China (no. 213AA12294). References [1] M. Bohner and N. Wintz, The Kalman filter for linear systems on time scales, Mathematical Analysis and Applications,vol.46,no.2,pp.419 436,213. [2] A.J.Barragan,B.M.Al-Hadithi,A.Jimenez,andJ.M.Andujar, A general methodology for online TS fuzzy modeling y the extended Kalman filter, Applied Soft Computing, vol. 18, no. 5, pp. 277 289, 214. [3] C. Hajiyev and H. E. Soken, Roust estimation of UAV dynamics in the presence of measurement faults, Aerospace Engineering,vol.25,no.1,pp.8 89,212. [4] K. Xiong, T. Liang, and L. Yongjun, Multiple model Kalman filter for attitude determination of precision pointing spacecraft, Acta Astronautica,vol.68,no.7-8,pp.843 852,211. [5] H. E. Soken and C. Hajiyev, Pico satellite attitude estimation via Roust Unscented Kalman Filter in the presence of measurement faults, ISA Transactions, vol.49,no.3,pp.249 256, 21.

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