DUAL QUATERNION CURVE INTERPOLATION ALGORITHM FOR FORMATION SATELLITES

Li Jing,Wang Huinan,Liu Haiying

(1.College of Automation Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing,210016,P.R.China;2.College of Astronautics,Nanjing University of Aeronautics and Astronautics,Nanjing,210016,P.R.China;3.Academy of Frontier Science,Nanjing University of Aeronautics and Astronautics,Nanjing,210016,P.R.China)

INTRODUCTION

Compared with traditional satellites,the micro-satellite gives much better performance with lower weight and cost,shorter developing and manufacturing cycles[1].But it also poses many shortcomings such as bad performancein diversity and precision.Thus formation flying is introduced to satellites maneuver for the purpose of overcoming above mentioned shortcomings[2].Such applications can be seen in distributed radar,electronic reconnaissance,3-Dimaging and spaceinterferometer as well.However the bunching problem causes a sharp rise of complexity in flight maneuver.The difficulty expands with the increase of satellite number.Position and attitude information becomes very vital in solving the above mentioned problem.

The most straightforward way to obtain relative position and attitude information of formation flying satellite is using differential technique of the absolute position and attitude information.The relative position and attitude information gathered by such technique is low precision.The recent researches incline utilizing GPS or GPS-like methods[3-5],or are based on vision meterage technique[6-7].And all separate satellite trajectory and attitude in forma tion into translation and rotation.Using vector and direction-cosine matrix or quaternion respectively to solve translation and rotation.This isolated technique increases the possibility of coupling error of translation and rotation,which certainly adds system complexity.

The trajectories of moving objects can be found by interpolation of given positions(points and orientations). Under such assumption,a method utilizing dual quaternion curve is used to interpolate the trajectories of formation flying satellites.At first,the mathematics model of dual quaternion curveis constructed.The trajectory of principal satellite can be found by interpolating method such as translation-rotation(TR)or translation-rotation-translation(TRT). Then,the relative position and attitude information of deputy satellite can be found by working out the dual quaternion kinematics equations.

1 TRADITIONAL SATELLITE POSITION AND ATTITUDE ALGORITHMS

1.1 Orbital parameter method

Orbital parameter method utilizes the number of principal and deputy satellite orbits to get the relative position and velocity information.Suppose that orbital parameters a,e,i,K,k,f,E,and M denote the long semi-axis,the eccentricity,the orbit inclination angle,the longitude ascending node,the perigee angular,the true anomaly,the eccentric anomaly,and the mean anomaly respectively,u=k+f is the latitude angular.Therefore,under principal satellite frame,the relative position and velocity of deputy satellite are

where subscript M and m denote the principal and the deputy satellite systems respectively,n denotes theaverageorbital angular velocity.Orbital parameter method can describe relativeposition of formation flying satellites.

1.2 Quaternion method

Quaternion method is brought in as a compensation of the orbital method to depict the relative attitude information.A quaternion is made up of one scalar part and three vector parts,shown as

where q0is the scalar part and q thevector part.The four parameters satisfy the following obligation equation

2 MATHEMATICAL MODEL OF DUAL QUATERNION CURVE

Dual quaternion curveis based on the quaternion and dual algebra theory.And the curve is powerful tool for rigid body kinematics,since it can easily describe the combination of both translation and rotation[8],shown as

3 DUAL QUATERNION CURVE INTERPOLATION ALGORITHM OF PRINCIPAL SATELLITE TRAJECTORY

Because of the influence caused by disturbing force etc,the satellitetrajectory no longer follows the ideal elliptic routine.In order to achieve precise control and meterage,it is crucial to obtain the trajectory information at anytime.

As mentioned before,a movement is made up of several continuous displacements.Take the(m+ 1)th displacement for example[9]

The(m+ 1)th interpolated displacement satisfies the following condition

The real factorλi≠ 0 is arbitrary,the rotation part can be written as followsis assumed to be obtained after a normalization on rot(Pi),that is

The signs ofλi satisfy

The sign of thefirst position R(0)i is arbitrary,the remaining signs are determined by Eq.(10).Theinterpolating condition is obtained by solving Eq.(8),that is

3.1 TR method

The interpolating function expression formula is

where coefficient bj can be obtained by

Unify the translation and rotation parts into one dual quaternion q(t)[10],that is

where X is the real factor,coefficient pjunknown.In order to avoid the polar point,the scalar part is normalized to 2.Then the interpolating condition is

The algorithm is as follows:

(1) Normalize the rotation part using Eq.(9),then assign the sign using Eq.(10).

(2)Work out Eq.(11)to obtain Cj.

(3)Work out Eq.(15)to obtain p j.

(4)Combine translation and rotation parts using Eq.(14),then work out Eq.(13)to obtain bj.

3.2 TRT method

The TR method is influenced by the selection of origin.The dual quaternion curve jumps when the interpolating position is close to the origin.A sharp change of curvature is apparently seen.In order to avert such occasion,a refined TRT method is introduced[11]. The rotational motion qrot(t)of degree k is composed of two translational Q-motionsof degree l 1 and l 2 respectively.

Unify the translation and rotation parts into rotation part,shown as

The interpolating condition is

The refined TRT method can effectively avoid the strong vibration around the origin.However,for object like satellites whose size is far smaller than its trajectories,the error is quite big.So the TRT method is used only when the deputy satellite is close to the principal satellite trajectory.Fig.1 shows the trajecties obtained by interpolating with both TR and TRT methods.The curvature is smooth and has no jump.

Fig.1 Principal satellite trajectories obtained by interpolating

4 DUAL QUATERNION CURVE MODELING OF DEPUTY SATELLITE POSITION AND ATTITUDE

4.1 Transform of deputy to principal satellites

Dual quaternion is a powerful tool in rigid body kinematics and coordinate transform.As can been seen in Fig.2,coordinate system O revolves an angle ofθaround special vector,and makes a translation of d alongat the same time,then becomes a new coordinate system N.

Fig.2 Geometrical display of screw

The above mentioned rigid body movement can easily be expressed with dual quaternion curve,shown as

where r denotes the position vector between two origins.

Therefore,the kinematics equation can be expressed as

The relative movement between deputy and principal satellites is depicted by the dual quaternion,and each centroid is the origin.They can be rewritten as

4.2 Dual quaternion curve updating

This paper chooses spiral vector method to calculate the renovated dual quaternion curve between deputy and principal satellites[14],that is wheredenotes the spiral vector along spiral axis,that is

The fourth order trigonometric series approximation is used

And the kinematics equation for is

From aboveequations,we can obtain the updating formula

After updating,position and attitude dual quaternion of deputy to principal satellites is

where q Mmtk+1is the attitude quaternion,and relative position vector of deputy to principal satellites is

Judging from Eq.(29),if the initial position and attitudeinformation of two satellites is given,and the relativevelocity and angular ratearemeasured,it is realistic to obtain the real-time relative position and attitude information of deputy to principal satellites.

5 DATA SIMULATION

Orbital parameters of two satellites are as follows.

The relative attitude angular velocities to the inertial system are k M=[0 0 1],k m=[0 0 1],respectively.The initial attitude angles areθM0=[0.05°0.05°0.05°],θm0=[0.05°0 0].

The simulating interval for position and attitude error isΔt=0.1 s,and simulation lasts for t=6 000 s.

The real value of relative position and attitude is denoted by r~,and the calculated value is,the simulation error isδr=r~-Comparing the ideal position and attitude data calculated by orbital parameter and quaternion methods with the data obtained by the algorithm in this paper,the position and attitude error is obtained.Figs.3-6 illustrate the position and the attitude angle errors obtained using orbital parameter and dual quaternion methods respectively.The attitude angleincludes rolling,pitch and yaw angles.

In Fig.3,the errors along axes x,y and z are 0— 3.9,0— 2.1,0— 3.7 cm respectively.In Fig.4,the corresponding errors are 0— 1.1,0—0.9, 0—1.8 cm respectively. Comparison of Figs.3,4 shows an obvious improvement in error using the dual quaternion method.The position error curve shows a periodical trend,which can satisfy long time functioning formation flying requirement.

Comparing Figs.5,6,the estimation precision of the roll and pitch angleis basically equivalent.However in the estimation of yaw angle,the precision using dual quaternion method is much better,which confines the attitude angle error to within 0.03°.Results show that using dual quaternion method,the convergence rate is much faster and stability of the estimation error is much better.

Fig.3 Position errors using orbital parameters method

Fig.4 Position errors using dual quaternion method

Fig.5 Attitude angle errors using quaternion method

Fig.6 Attitude angle errors using dual quaternion method

6 CONCLUSION

This paper constructs a position and attitude unified model for satellite formation flying based on the dual quaternion method.First TR and TRT interpolation methods are used to calculate the principal satellite trajectory during formation flying.Then using dual quaternion modeling,the position and attitude information integration of deputy satellite is accomplished.The simulation results show that the proposed algorithm can obviously improve both position and attitude precision.

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