VISNAV integrated relative navigation and

Journal of Systems Engineering and Electronics Vol. 22, No. 2, April 2011, pp.283–291 Available online at www.jseepub.com

GPS/VISNAV integrated relative navigation and attitude determination system for ultra-close spacecraft formation flying Xiaoliang Wang, Xiaowei Shao∗ , Deren Gong, and Dengping Duan Institute of Aerospace Science & Technology, Shanghai Jiaotong University, Shanghai 200240, P. R. China

Abstract: For the improvement of accuracy and better faulttolerant performance, a global position system (GPS)/vision navigation (VISNAV) integrated relative navigation and attitude determination approach is presented for ultra-close spacecraft formation flying. Onboard GPS and VISNAV system are adopted and a federal Kalman filter architecture is used for the total navigation system design. Simulation results indicate that the integrated system can provide a total improvement of relative navigation and attitude estimation performance in accuracy and fault-tolerance.

Keywords: control and navigation, relative navigation, federal Kalman filter, spacecraft formation flying, global position system (GPS), vision navigation (VISNAV).

DOI: 10.3969/j.issn.1004-4132.2011.02.015

1. Introduction Ultra-close (baseline less than 100 m) spacecrafts formation flying has attracted much attention and has actually been used recently since it can be implemented as interferometry observe missions, or as a test programme for real autonomous rendezvous, docking and proximity operations [1,2]. While relative navigation and attitude determination (RNAD) accuracy directly impacts the effect and success of such missions. References [3,4] demonstrate that fuel usage for active control of formation flying satellites is a strong function of the relative navigation error, and in particular, the relative velocities. The larger the error, the higher the fuel usage will be, and the shorter the mission life. At the same time, precise relative attitude knowledge also provides a key factor to maintain mission requirements. Generally, relative attitude is not specially considered during spacecraft formation flying with long baseline. But for the formation of ultra-close distance with specifically missions, the RNAD problem should be solved properly. The traditional way to solve this is separation; relative navigation is provided by one subsystem as GPS or by Manuscript received August 19, 2009. *Corresponding author.

laser radar (Ladar) measurement. Those subsystems are switched from one to another when the baseline of formation spacecrafts altered, such as from GPS subsystem to Ladar subsystem when in close proximity. Different subsystems correspond to different formation phases. At the meantime, relative attitude is given by the attitude subsystem as star vector sensors and gyro output. This design has no redundant consideration and can be a real disaster when a single sensor fails [5]. For the future application of RNAD in ultra-close formation missions, two requirements are specially desired: first, the RNAD design should be preserved fault-tolerant in case of sensor-malfunction, this requires RNAD information should be provided by at least two RNAD subsystems; second, a better performance of accuracy is preferred, which focuses on the optimal fusion algorithm designed for the proposed RNAD system. Precise RNAD can be achieved through different ways. One kind of candidate to resolve this is using GPS. Corazzini [6] presented a promising RNAD system architecture using carrier-phase difference GPS (CDGPS) technology in 2D spacecraft formation laboratory environment. By using multi-antennas configuration in the simulation spacecrafts model, the attitude information were obtained by intra-vehicle carrier phase single different measurement, and the relative positions between each vehicle were provided by inter-vehicle double differences measurement. Corazzini also demonstrated several nonlinear estimation algorithms for the proposed RNAD architecture. However, the RNAD architecture has to process 69 measurement equations at each time-step and was not computationally efficient. Furthermore, in case the sensor failure was not considered in the total system design, which was crucial for practical implementation. Reference [7] introduces a new optical sensor named vision navigation (VISNAV) which can provide multiple line-of-sight vectors from a spacecraft to another, so, both relative navigation and attitude information can be derived simultaneously. Reference [7] also provides

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a navigation demonstrated of 100 m to zero approach of two vehicles using VISNAV system, and the Gaussian least squares differential correction (GLSDC) algorithm was used for the states estimation. Reference [8] presents a RNAD method for spacecraft formation flying using LOS vectors measurement from the VISNAV sensor coupled with gyro output, and a nonlinear extended Kalman filter (EKF) algorithm is used for the states estimation. Although both Gunnam and Kim’s RNAD system performed well in accuracy and is robust with large initial states error, for practical implementation, it still needs to be integrated with other RNAD system in case sensor fails. Furthermore, the LEDs in those papers used as light sources (called beacons) are considered as ideal points which can not be modeled in actual scenarios. This paper integrates GPS and VISNAV equipment to form a RNAD system for ultra-close spacecrafts formation flying. A federal Kalman filter (FKF) [9] architecture is adopted for the RNAD states estimation and optimal fusion of GPS/VISNAV subsystem. J2 disturbances influence in the relative orbit dynamics and LED emit energy distribution influence in the VISNAV subsystem measurement models are specially considered, which are omitted by all the papers mentioned above. For the states estimation in the presence of J2 effect which can be considered as a model error in relative orbit dynamics, an adaptive fading Kalman filter (AFKF) proposed by [10, 11] is used for the relative navigation of GPS subsystem, which can provide optimal states estimation theoretically. The QUEST algorithm is used for the attitude determination of GPS subsystem. The predictive filter is adopted for the total RNAD solution of VISNAV subsystem. The GPS/VISNAV integrated system proposed in this paper has the following virtue characteristics: first, it can provide optimal relative information on both position and attitude; second, absolute navigation and attitude information can also be derived through GPS subsystem which is not particularly mentioned in this paper; third, the most importantly, the total system preserves fault-tolerant performance in case of any subsystem malfunction. The organization of this paper proceeds as follows. First, relative coordinate systems and positional equations of motion are given and quaternion kinematics for relative attitude propagation are also provided, which consists of the stated equations for the total states estimation system. Then, a system structure model used for the GPS/VISNAV integrated system and GPS/VISNAV measurement models for the RNAD are given, also their estimation methods, respectively. Finally, simulation results and conclusions are presented.

2. Navigation system model 2.1 Relative orbit motion equations The spacecraft about which all other spacecrafts are orbiting is referred to as the chief. The remaining spacecrafts are referred to as the deputies. The relative orbit position vector ρ is expressed in components by ρ = [x y z]T in the direction of radial in-track and cross-track respectively. Detailed derivation of the relative equations of motion for eccentric orbits can be found in [12,13]. If the relative orbit coordinates are small compared with the chief orbit radius, then the equations of motion are given by ⎧ rc r˙c ⎪ 2 ˙ ˙ ⎪ x ¨ − xθ 1 + 2 = wx − 2θ y˙ − y ⎪ ⎪ p rc ⎪ ⎪ ⎪ ⎨ ˙ x˙ − x r˙c − y θ˙2 1 − rc = wy (1) y ¨ + 2 θ ⎪ rc p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z¨ + z θ˙ 2 rc = w ⎩ z p where p is semilatus rectum of the chief, rc is the chief orbit radius and θ is the true anomaly rate of the chief. Also, wx , wy , wz are acceleration disturbances which are modeled as zero-mean Gaussian white-noise processes. The true anomaly acceleration and chief orbit-radius acceleration are given by r˙c θ¨ = −2 θ˙ (2) rc rc 2 ˙ r¨c = rc θ 1 − (3) p 2.2 Relative attitude quaternion kinematics The quaternion is defined by q = [q T q4 ]T , with q = [q1 q2 q3 ]T = eˆ sin(ϑ/2) and q4 = cos(ϑ/2), where eˆ is the axis of rotation and ϑ is the angle of rotation. In this paper, quaternion q is used for the representation of the relative attitude, which is the map of vectors in the chief frame to vectors in the deputy frame, expressed by q = qd ⊗ qc−1

(4)

where qc , qd represent the attitude quaternion of chief and deputy spacecraft, respectively. ⊗ denotes the quaternion nonlinear operator more details which can be found in [14,15]. The relative quaternion kinematics are given as q˙ =

1 Ξ(q)ωdc 2

(5)

where ωdc is the relative angular velocity defined by ωdc = ωd − A(q)ωc and

Ξ(q) =

q4 I3×3 + [q×] −q T

(6) (7a)

Xiaoliang Wang et al.: GPS/VISNAV integrated relative navigation and attitude determination system for ultra-close ...

⎡

⎤

0 −q3 q2 [q×] = ⎣ q3 (7b) 0 −q1 ⎦ −q2 q1 0 where ωc , ωd are the angular velocities of the chief and deputy, respectively. A(q) is the attitude matrix represented by the relative quaternion. The complete close-form solution for the state transition matrix of (5) can be found in [8], which is crucial for the discrete-time propagation of the relative quaternion.

3. Navigation system architecture 3.1 System structure The integrated GPS/VISNAV RNAD system proposed in this paper consists of four major parts, as shown in Fig. 1, which includes reference system, GPS subsystem, VISNAV subsystem and fault isolate/optimal fusion section. The reference system provides precise relative orbit motion in the presence of J2 disturbances and attitude information for the generation of simulated measurement data, and as the evaluation criterion for the proposed integrated RNAD system. GPS and VISNAV subsystems

Fig. 1

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contain corresponding measurement which provides the GPS/VISNAV measurement data Y1 , Y2 and a filter model which gives the corresponding RNAD solution and states ˆ 1 , P1 , X ˆ 2 , P2 . Isolate segestimation covariance matrix X ment execute fail detection and isolate function and optiˆ F , PF based mal fusion give final optimal RNAD results X on the output of the GPS/VISNAV subsystem. For consideration of better RNAD accuracy and faulttolerant performance for the proposed integrated system, no reset feedback architecture [9] is used in this paper as can be easily figured out in Fig. 1, since a fault in any one local filter (LF) which is caused by one subsystem malfunction can not contaminate any other LF solution. The well-functioned LF solutions can still be used when the fault solution has been isolated in time. Furthermore, this design is still more accurate than any of the individual LFs operating stand-alone while not globally optimal. The following sections give detailed introduction to those four parts and provide the mathematical model adopted in this paper.

Navigation system structure

3.2 Reference systems

3.3 GPS subsystem

Reference systems shown in Fig. 1 consist of two parts: the relative motion and the relative attitude. Unlike traditional GPS/INS integrated navigation system design [16], which usually considers INS as the reference system since it can provide completely precise navigation and attitude information in short period, reference relative motion is provided in this paper by propagation of two body models with J2 disturbances for both the chief and the deputy spacecrafts. Then the relative motion can be derived easily, and the reference relative attitude is provided by propagation of relative quaternion kinematics given before. Since the relative motion with J2 disturbances can not be precisely modeled by (1) – (3), the navigation filter can failed to perform convergence if the traditional nonlinear filter is adopted. Detailed filter design will be introduced later.

Carrier-phase difference GPS for the relative orbit and attitude determination for spacecraft formation flying is emerging as a very promising low-cost method. It is assumed to be three antennas fixed in each spacecraft in this paper, which forms two baseline vectors in body frame for the attitude and navigation estimation. 3.3.1 GPS measurement model The carrier-phase single differences measurement can be used for both relative navigation and attitude determination. For the purpose of relative navigation, single differences measurement can be obtained by differencing between the chief antenna and deputy antenna assumed to be located in both mass centers [17]. For the purpose of absolute attitude determination (AAD), single differences measurement can be obtained by differencing between the

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master antenna and slaver antenna located in each spacecraft. The single differences carrier phase measurement model used in this paper is as follows k k + cτi − cτj + ηij ΔΦ kij + λNijk = Dij

(8)

k is the measured carrier phase of antenna i and where ΔΦij j from GPS k. λ is the wave length of GPS carrier phase (19 cm for L1 , 24 cm for L2 ). Nijk is the single differences integer which can be determined by an integer search algorithm that has to be performed at startup before. τi , τj are k is the measurement the clock errors of receiver i and j. ηij k noise. Dij is the projection of the baseline vector onto the kth GPS line-of-sight (LOS) vector which contains the information of the relative position or attitude. The clock error τ can be modeled as two-order Markov process as follows

τ˙ = f + w1 , f˙ = w2 ,

w1 ∼ N (0, Sτ ) w2 ∼ N (0, Sf )

(9)

X = [ρT ρ˙ T rc r˙c θ θ˙ τc fc τd fd ]T

where c, d represent the chief and deputy spacecraft, respectively. The nonlinear state-space model can be easily derived by combination of (1), (2) and (8). In this nonlinear model, the chief radius and true anomaly, clock error of both chief/deputy spacecraft, as well as their derivations respectively, are estimated. An introduction of AFKF and filter design for a nonlinear system can be found in [10,11], which is not the focus of this paper and not explicitly shown here. 3.3.3 Relative attitude determination Traditional absolute attitude determination can be achieved when at least two baselines are given in the body frame which satifies [18] J(A) = min A

(10)

where f is the frequency drift. 3.3.2 Relative position estimation For the GPS subsystem relative navigation LF design, the relative motion equation given by (1)–(3) can not be considered as a perfect model with actual J2 disturbances, and the filter performance will decrease when normal nonlinear Kalman filter (KF) is used. Since then, AFKF introduced by [10,11] is used for the relative navigation estimation of GPS subsystem. AFKF algorithm can adaptively adjust the forgetting factors to the optimality condition compared with traditional KF structure. Then filter divergence can be solved in case of an erroneous model. Based on the description of the GPS measurement model and filter selection, the total states for the relative navigation estimation can be modeled as follows

n

(Φk − BAsk )T W (Φk − BAsk ) (12)

k=1

where n is the number of GPS satellites. Φk = [Φk1 Φk2 . . . Φkm ]T is the carrier phase measurement from T T T GPS. W is the weight matrix. B = [bT 1 b2 . . . bm ] is the matrix of baseline vectors in spacecraft body frame. A is attitude matrix. sk is the unit vector of kth GPS LOS direction in reference frame. Attitude can be determinated using QUEST algorithm, a very famous one proposed by [19]. For the relative attitude determination (RAD), sk should be replaced with the baseline vector of the chief spacecraft as the benchmark for RAD. The carrier phase measurement, Φk , is replaced with a virtual parameter which is materially the dot product of the baseline vector on the deputy spacecraft and that on the chief. If the carrier phase measurement from GPS k for the baseline i, j of chief/deputy k T spacecraft is denoted as Φik = bT i sik , Φj = bj sjk , the virtual measurement can be given as follows

⎡ −1 n n n −1 ⎤ n ⎦ Φij = tr[bi · bj ] = tr ⎣ sik sT sik Φik Φjk sjk sjk sT ik jk k=1

k=1

where tr denotes the matrix trace operation. The relative attitude determination can also be solved using QUEST methods similarly as absolute attitude determination. Detailed description of RAD problem using GPS observation was introduced in [20]. 3.4 VISNAV subsystem The VISNAV system comprises an optical CCD sensor of a new kind located in the deputy spacecraft’s body frame

(11)

k=1

(13)

k=1

combined with specific light sources (beacons) fixed in the chief spacecraft’s body frame, which can be used for close range photogrammetry-type applications [21]. The relationship between the relative position/attitude and the VISNAV observe equations is briefly reviewed in this section. Special error sources caused by LED characteristic and RNAD using the nonlinear predictive filter are also introduced.


3.4.1 VISNAV measurement model The observing equations of a VISNAV system can be written as ˜bi = Ari + wi , wT Ari = 0 i (14)

i = 1, 2, . . . , N

where ˜bi denotes the ith measurement in CCD focal plane coordinate given by ⎡ ⎤ −ςi 1 ˜bi = ⎣ −ξi ⎦ (15) f 2 + ςi2 + ξi2 f ⎤ ⎡ Xi − x 1 ⎣ Yi − y ⎦ ri = (Xi − x)2 + (Yi − y)2 + (Zi − z)2 Z −z i

(16) where N is the total number of observations, (ςi , ξi ) are the image space observations for the ith LOS, (Xi , Yi , Zi ) are the known object space locations of the ith beacon, (x, y, z) are the unknown object space location of the sensor, f is the known focal length, and A is the unknown attitude matrix, associated to the orientation from the object plane (chief) to the image plane (deputy). The object of the VISNAV system is given observations (ςi , ξi ) and object space locations (Xi , Yi , Zi ) for i = 1, 2, . . . , N determine the attitude A and position(x, y, z). The sensor error wi is approximated by Gaussian, which satisfies (17) E{wi } = 0 E{wi wiT } = γσi [I3×3 − (Ari )(Ari )T ]

(18)

where γ is a coefficient determinated by the LED and CCD hardware structure. Generally speaking, 95 % of LED scattering energy distributes in a circle area and is collected by a CCD camera. The distribution of scattering energy in the circle area can be simulated by 2D normal distribution. The closer the distance between LED and CCD camera along axis direction and radial direction, the more energy the CCD camera captures and the less level of measured noises it contains. The detailed description of such relationship is as follows. Let θ (rad) be the half angle shaped by the LED emit taper with 95 % energy inside, as shown in Fig. 2. Then, the cut area in the distance of R (m) is

Fig. 2 LED emit taper

θ Sθ =

0

2πR2 sin αdα = 4πR2 sin2

287

θ 2

(19)

Assume that LED power is P0 (mW), caliber D = f /1.3, caliber area is π(D/2)2 . Considering actual applications, the filter is needed to eliminate light disturbance. If the total loss coefficient k = k1 k2 k3 = 0.2 caused by lens, the filter and the propagator, the energy collected by the CCD camera is 2 2 D π (D/2) (20) = k · P0 Pr = k · P0 Sθ 4R sin(θ/2) It is clear that energy received by CCD camera has a relationship with 1/R2 when half angle of LED taper is fixed. For the noise coefficient γ appeared in (18), the following expressions are adopted f (0, 0) , R 10 m f (x, y) 1 1 f (0, 0) γ = − lg , R > 10 m 2 R2 f (x, y) γ=1

(21) (22)

where f (x, y) is the 2D normal probability density given by 2 2 x x 1 1 exp − + f (x, y) = 2πσ1 σ2 2 σ1 σ2 (23) where σ1 , σ2 are the covariances in two directions in the circle area, which are the function of R. The relationship can be modeled as σ1 = σ2 = σ0 R, where σ0 is the covariance when the distance is equal to 1m. 3.4.2 Attitude/position estimation The nonlinear predictive filter for VISNAV system introduced by [22], since it can provide optimal states estimates in the presence of significant errors in the system model. The states in the filter are given by the quaternion q and the position vector p, so x = [q T pT ]T . The propagation model used here for attitude determination is given by the quaternion kinematics model, and the propagation model for position determination is assumed to be given by a simple first-order process 1 q )dq qˆ˙ = Ξ(ˆ 2

(24)

pˆ˙ = dp

(25)

where the model error vector in this case is given by T dT x = [dT q p ] . The real/estimated observation equation in the predictive filter is given by yi = Ari , yˆi =

288


Aˆ ri (i = 1, 2, . . . , N ). For practical applications, the sample interval should be well below the Nyquist’s limit. Detailed derivation of the predictive filter can be found in [23] which is not shown here. 3.5 Isolate and optimal fusion After the processing of both GPS/VISNAV local filters, the optimal fusion of those relative states can be achieved finally after the fault isolate section. Generally, fault detection can be archived though many ways like residual evaluation, a commonly-used one which is adopted in this paper. A fault in any one local filter can be detected in time and can not contaminate any other local filter solution through this isolate design. For the final fusion section, one thing is for sure, different navigation systems introduce different errors that must be considered in filter states. While only the common states appeared in both GPS and VISNAV systems (the relative navigation and attitude information) are considered in the part on fusion in this paper. After the estimation of common states in each navigation system and the corresponding covariance matrix denoted ˆ 2 , P2 , the optimal fusion state and its coˆ 1 , P1 and X as X variance can be given by a simple fusion procedure as ˆ F = (P −1 + P −1 )(P −1 X ˆ 1 + P −1 X ˆ2) X 1 2 1 2 and

PF = (P1−1 + P2−1 )−1

4. Simulation results Simulation results are presented in this section, which shows the RNAD performance of the proposed GPS/VISNAV integrated navigation system for ultra-close spacecrafts formation flying as pendulum configuration in the distance of about 50 m. Orbit parameters used in this simulation are given in Table 1. Table 1 Formation orbit parameters Elements Radial/km Eccentricity Inclination/(◦ ) Argument of perigee/(◦ ) RAAN/(◦ ) Mean anomaly/(◦ )

Chief 7 178 0 51.7 180 0 0

Deputy 7 178 0 51.700 1 180 0 −0.000 4

Both formation spacecrafts are three axes stabilization. Six beacons are assumed to be existing on the chief’s body frame X1 = [0 1 − 0.05]T X2 = [0 1 0.05]T X3 = [0 − 1 − 0.05]T X4 = [0 1 0.05]T

X5 = [0 X6 = [0

0.5 0]T − 0.5 0]T

with half taper angle 25◦ in the direction of x-axis of the chief’s body frame. The boresight direction of the camera image frame is assumed to be in the direction of x-axis of the deputy’s body frame for sake of simplicity. Also, three antennas are assumed on both chief and deputy spacecrafts with position Xm = [0 0 0]T Xs1 = [0 1 0]T Xs2 = [−1 0 0]T in the body frame. The simulation time for relative motion of the two spacecrafts is 3 000 s with time interval of 10 s for the integrated system. Simulated time step for GPS subsystem is 10 s while VISNAV subsystem is 0.01 s with consideration of the Nyquist’s limit. The multi-time-step problem can be solved through the output of the VISNAV subsystem once every 10 s. These six beacons are assumed to be visible to the CCD on the deputy throughout the entire simulation run. Simulated VISNAV measurements are generated by using (14) with the measurement error standard deviation given by 0.000 5◦ and σ0 = 0.1 m while carrier phase measurements noise for both chief and deputy is set to be 0.5 mm. The initial clock error and frequency drift are set to be 4 × 10−4 s and 3 × 10−11 s/s with Sτ = 10−4 (s)2 and Sf = 10−11 (s/s)2 for the two spacecrafts. GPS data used in this paper come from IGS web site [24]. The spectral densities of the process noise compo√ nents wx , wy , wz in (1) are given by 5 × 10−11 m/s3/2 each. The real relative attitude motion is given by propagating (11) using an initial quaternion given by q(t0 ) = [0 0 0 1]T and angular velocities given − 0.001 0]T rad/s and ωd = [0 by ωc = [0 T −0.001 0] rad/s. The tops of Figs. 3 and 4 show the final fusion relative position and velocity errors in this simulation. Relative position knowledge is within 0.1 meter for each LVLH direction while the relative velocity knowledge is within 0.01 meter per second. The final fusion relative attitude errors during the simulation run are shown in top of Fig. 5, the knowledge of which is within 0.05◦ for the roll, pitch and yaw directions. Still, the filter can also provide estimation of clock error and frequency drift for both the chief and deputy spacecrafts, and estimation of radius, radius rate, true anomaly, true anomaly rate of chief spacecraft orbit which are not shown here.


Estimation comparison can be achieved when one of the two navigation filters completely shuts down, which can be simulated as one navigation system corrupted. Figs. 3– 5 also show the corresponding estimation errors of the relative position, velocity and attitude for the fusion/GPS only/VISNAV only mode. The corresponding root mean square (RMS) analysis is provided in Table 2. It can be seen that the integrated system design used in this simulation can provide an optimal data fusion procedure. The accuracy improvement of final fusion attitude estimation compared with GPS is 0.24% and 31% with VISNAV, 12% and 57% for position estimation accuracy improvement of GPS/VISNAV and 10%, 59% for velocity estimation accuracy improvement. Moreover, the relative distance of Pendulum formation flying used in this simulation can be varied with a little change of Mean anomaly of the deputy spacecraft.

Fig. 5

Comparison of relative attitude error Table 2

Items

289

Direction Pitch Attitude/(◦ ) Roll Yaw Radial Position/m In-track Cross-track Radial Velocity In-track (m/s) Cross-track

RMS analysis

Fusion RMS 0.020 641 0.014 338 0.013 854 0.009 716 0.007 998 7 0.009 927 5 0.001 343 4 0.001 116 3 0.001 374 6

GPS RMS VISNAV RMS 0.026 454 0.025 975 0.022 12 0.024 808 0.016 235 0.021 994 0.011 359 0.020 169 0.008 680 7 0.022 097 0.011 191 0.022 466 0.001 545 7 0.002 992 2 0.001 183 4 0.003 150 9 0.001 534 8 0.003 191 7

At the end of this simulation section, more simulation results are provided with the relative distance from 10 m to 100 m by intervals of 10 m. The mean anomaly used by the deputy spacecraft is given in Table 3. Table 3 Fig. 3

Fig. 4

Comparison of relative position error

Comparison of relative velocity error

Mean anomaly of deputy spacecraft vs relative distance

Mean anomaly of deputy spacecraft/(◦ ) −0.000 08 −0.000 16 −0.000 24 −0.000 32 −0.000 40 −0.000 48 −0.000 56 −0.000 64 −0.000 72 −0.000 80

Relative distance/m 10 20 30 40 50 60 70 80 90 100

Figs. 6–8 show the final fusion results of three axes relative attitude error and relative position, and velocity error varied with relative distance between two spacecrafts described previously. The solid lines denote the fusion RMS of each state while the dash and dot lines denote the GPS only/VISNAV only RMS. It is clear that the proposed integrated system in this paper can provide an optimal and reliable fusion estimation of relative states when the relative distance varies. The accuracy of VISNAV subsystem

290


changes dramatically since it is essentially due to the VISNAV measurement model described in (14) to (18). The states estimation accuracy provided by the VISNAV system not only depends on the accuracy of the CCD sensor and the number of beacons, but also on the “spread” of the beacons as well as the distance from the beacons and CCD [8, 25]. At the meantime, the GPS subsystem provides a reliable and consistent relative states estimation solution despite of the relative distance between spacecrafts,

above, it may draw a conclusion that GPS can provide an ideal sensor for formation flying of spacecrafts while the VISNAV system can be aided as an assistant sensor when “small” distance is satisfied. As a new image navigation technology, VISNAV can be widely used in many aspects, especially in deep space missions, while GPS can provide this capability just near the earth. Although GPSlike technologies are proposed for the application of deep space missions [26], they are also subject to the generic GPS performance-limiting effects, including geometric dilution of precision, multipath errors, clock errors of both transmitter and receiver errors, etc.

5. Conclusions

Fig. 6 Attitude RMS

A GPS/VISNAV integrated RNAD system for ultra-close spacecraft formation flying is proposed in this paper for the purpose of RNAD accuracy improvement and better fault-tolerant performance. FKF architecture is adopted for the system states estimation and optimal fusion for the GPS/VISNAV subsystem. J2 disturbances in the relative orbit motion and LED emit energy distribution influence in the VISNAV subsystem are specially considered. AFKF, QUEST and predictive filter algorithms are used for the LF design in the FKF structure. Simulation results indicate that the proposed GPS/VISNAV integrated system can provide better RNAD accuracy and fault-tolerant performance for ultra-close spacecraft formation flying. The accuracy improvement of final fusion estimation compared with GPS/VISNAV is 0.24%, 31% for attitude, 12%, 57% for position and 10%, 59% for velocity.

References Fig. 7

Position RMS

Fig. 8 Velocity RMS

only if the relative position vector is small enough compared with the chief orbit radius (less than 5 km in practical) for the existence of (1). From the analysis given

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Biographies Xiaoliang Wang was born in 1981. He is a Ph.D. candidate with Institute of Aerospace Science & Technology, Shanghai Jiaotong University. His research interests include relative states measurement and estimation for spacecraft formation flying. E-mail: [email protected]

Xiaowei Shao was born in 1974. He is a Ph.D. and associate professor with Institute of Aerospace Science & Technology, Shanghai Jiaotong University. His research interests include SAR mission in space and space vehicle GNC technology. E-mail: [email protected]

Deren Gong was born in 1982. He is a Ph.D. candidate with Institute of Aerospace Science & Technology, Shanghai Jiaotong University. His research interests include satellite formation flying and tethered satellite formation. E-mail: [email protected]

Dengping Duan was born in 1966. He is a professor with Institute of Aerospace Science & Technology, Shanghai Jiaotong University. His research interests include satellite technology and space vehicle GNC technology. E-mail: [email protected]