Relative State Transition Matrix Using Geometric Approach for Onboard Implementation

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1 doi:.528/jatm.v9i3.762 Relative State ransition Matrix Using Geometric Approach for Onboard Implementation Mankali P Ramachandran ABSRAC: In the present study relative state transition matrix was obtained. It matches the relative motion of 2 satellites while including the oblate perturbation. he used formulation applies the geometric approach and is in Cartesian frame. he relative state transition matrix uses absolute state transition matrix of individual satellites. hereon, simplification in computing methods and on-board implementation at controls are explored in a leader/follower coordination method. Numerical experiments illustrate the accuracy when the baseline separation is equal to 2 km and eccentricity is.5 and.5 for all the inclinations. KEYWORDS: State transition matrix, Formation flying, Orbit, Oblate. INRODUCION Formation flying is about harnessing and controlling the relative orbital dynamics. When using multiple observations obtained from a formation of the low Earth orbiting satellites, the combined data is enhanced. he Clohessy-Wiltshire (CW) equation that describes a relative motion is valid for a small relative distance between the satellites compared to the radius and it assumes that the reference orbit is circular besides using a spherically uniform gravitational field at the centre (Alfriend et al. 2). In practice this is not true for a low Earth orbiting satellite. Maintaining a formation with respect to the circular reference orbit requires prohibitive fuel consumption. It needs frequent orbit corrections to reduce the deviations caused by perturbations. he first significant perturbation is due to the effect of oblate Earth denoted as J 2. When the reference orbit is initially nominally circular, a small eccentricity usually occurs due to the J 2 effect, which also causes a regression in the orbital plane. It may be noted that an advanced scheduling of the satellite payload data collection on the Earth is usually based on a ground system. his mapping system uses an orbit model that includes mean oblate effect. hus, in general, it is more important to include the J 2 effect in the on-board reference orbit of the satellite. State transition matrix (SM) plays an important role in the orbit control. It occurs in the system dynamics equation that is used to control the satellite. he SM that is implemented on-board a satellite needs to be simpler, computationally efficient, and yet scaled close to match the accompanying reference on-board orbit model. Here, the reference orbit on.indian Space Research Organisation Satellite Centre Flight Dynamics Group Bangalore/Karnataka India. Author for correspondence: Mankali Padmanabha Ramachandran Indian Space Research Organisation Satellite Centre Flight Dynamics Group Old Airport Road 567 Bangalore/Karnataka India mprama@isac.gov.in Received: Aug. 2, 26 Accepted: Jan. 24, 27

2 Relative State ransition Matrix Using Geometric Approach for Onboard Implementation 329 each of the satellites is a first-order motion that includes the J 2 effect (Roy 982). In this paper, it was obtained an approximate computationally simpler relative SM in Cartesian frame to match this relative motion. here are various formulations of relative SM and each assumes certain orbital characteristics (Alfriend et al. 2). Here a rectangular Cartesian frame is used since it helps in the direct use of GPS navigation systems data. Incrementing the maneuver velocity is easier to understand with respect to the orbit frame of reference. Cartesian frame has been found to perform satisfactorily under propagation for formation types that are not largely separated along-track (de Brujin et al. 2). he first SM is that of the circular CW equation. he assumptions in the CW model, as mentioned earlier, are the limitations in its use. he SM is derived in Cartesian frame from the analytical solution of the CW differential equation (Alfriend et al. 2). When the eccentricity of the reference orbit is small, a SM can be derived in Cartesian frame, as carried out by Melton (2). his involves a lengthy expression as a function of time and is a series with terms which are powers of the eccentricity (Melton 2). When the orbit is more elliptical then the Keplerian one, it is transformed following the approach of schauner and Hempel (965) to yield a closed form relative motion. his SM given in Yamanaka and Ankersen (22) is an improvement made to Carter (998). Here the independent variable is the true anomaly. he time-explicit SM is easier to adopt in the control system. he method of Gim and Alfriend (23) has solved the problem of SM, having matched the relative motion about an elliptical orbit with J 2 effect. hough the time-explicit expression is analytical and accurate, it is laborious for on-board implementation. he model includes short and long periodic effects and is in terms of non-singular elements. It is important to mention that there is no need to include the long-period effects under J 2 perturbation. In Hamel and de Lafontaine (27), this is explored, and an alternative SM was presented, being valid for all cases when the reference orbit is not circular. It is based on mean orbit element differences between 2 satellites. When the eccentricity is small, then a lesser complex version is given in Alfriend and Yan (25). he SM formulated in Yan et al. (24) is based on the unit-sphere method (Vadali 22). he approximate SM in suda (2) is in Cartesian frame and involves a series for on-board implementation. Here the approach is geometric, that is, it applies a geometric transformation between the relative states of the deputy with respect to the chief and the orbital frame of the chief satellite. his was adopted to describe the relative motion in Balaji and atnall (23) which is similar to the unit-sphere approach (Vadali 22) and appeared at the same time. he relative motion described in Vadali (22) as well as in Balaji and atnall (23) includes the J 2 effect, is applicable for eccentric orbits, and is valid for larger relative distances, using Keplerian elements. he present geometric approach is in Cartesian frame. he relative SM obtained here is approximate and explores optimized computation. Navigation plan suggested here is suited for on-board real-time usage of SM in leader/follower coordination (Alfriend et al. 2). he SM is applicable for both osculating and mean elements of the 2 satellites. It also provides a scope of including secular effects due to higher-order perturbation. Finally, numerical experiments bring out the accuracy of state updated by the SM in terms of varying eccentricity and inclination. REFERENCE ORBI AND SAE RANSIION MARIX he orbital motion of the deputy and chief satellite with J 2 effect is discussed next. he absolute SM of the 2 satellites is then derived. his is required in each of the satellites in controlling their reference orbits. he equation of motion of an individual satellite is (Roy 982): = U where R denotes the second derivative with respect to time of R = (X,Y,Z). he disturbing potential is: U(R,ψ,Z) = ( /R) + F with F = - ( /R)(R e /R) 2 J 2 (/2){ 3 sin 2 δ -}, where: μ is the Earth s gravitational parameter, equal to 398,6.64 km 3 /s 2 ; J 2 =.826; δ is the instantaneous declination; R e is the Earth s radius, which is 6, km; R is the magnitude of the satellite position vector, R. he position vector of the deputy and the chief satellites is denoted by R d and R c, respectively, in the Earth Centred Inertial Frame (ECIF). he potential is axi-symmetric about the Z-axis and is independent of the azimuth angle ψ. F has 2 types of expressions (Roy 982): () (2) (3)

3 33 Ramachandran MP F = F or F 2 (4) where: F contains only the secular component while F 2 includes the short-period perturbation. F = ( /R) (R e /R) 2 (J 2 /2) (-(3/2)sin 2 (i)) (5.) F 2 = ( /R) (R e /R) 2 ( J 2 /2) [(-(3/2)sin 2 (i))+ (5.2) +((3/2) sin 2 (i) cos(2(f+ω)) ] where f is the true anomaly. he Lagrange s planetary equation of motion (Roy 982) is: da/dt = (2/na) / de/dt = ((-e 2 )/(na 2 e)) / - ((-e 2 ) /2 /(n a 2 e)) ( / ) di/dt = (cos(i)/(na 2 (-e 2 ) /2 sin (i))) ( / ) (/(na 2 (-e 2 ) /2 )sin (i))) / (6) d /dt = (/(na 2 (-e 2 ) /2 sin (i))) / dω/dt = - (cos(i) /(na 2 (-e 2 ) /2 sin (i)) / + + (-e 2 ) /2 /( na 2 e) / dm/dt = n ((-e 2 ) /2 /( na 2 e)) / - (2/na) / where: n 2 a 3 = (7) where: a, e, and i are, respectively, the semi-major axis, eccentricity, and inclination; ω, Ω, and M are, respectively, argument of perigee, longitude of the ascending node and mean anomaly. here are 2 kinds of reference orbits for on-board implementation on both satellites. It is either (a) osculating and obtained by substituting F 2 in Eq. 6 or (b) the mean motion obtained by substituting F in Eq. 6. he latter is the more familiar first-order secular perturbation with a, e, and i remaining constant, as it is at the start epoch. When mapping the orbit elements to the mean ones, only 3 orbit elements are found to exhibit secular growth due to the J 2 effect and are ω = ω o + (3/2) J 2 (R e /p) 2 ň {2 (5/2)sin 2 (i)} t = (3/2) J 2 (R e /p) 2 ň cos(i) t (8) ň = n [ + (3/2) J 2 (R e /p) 2 [-(3/2)sin 2 (i))(-e 2 ) /2 ] M= M + ňt where p = a( e 2 ); ň is the perturbed mean motion. his has been used in the unit-sphere approach (Vadali 22). It is more meaningful to use reference orbit with periodic terms for close formations to avoid any collision. On the other hand, having the reference to be a mean motion fuel is saved by having impulse maneuver used once or twice appropriately in the orbit besides maximizing data collection duration (Gill et al. 27). Let X j (t) = (X j,y j,z j,x j,y j,z ) be in orbital state in the ECIF. j At the epoch (t ), the state is X j (t ), where j = denotes the state of the chief and j = 2 that of the follower. he absolute SM formulation follows the method of Markley (986). Equation 9 describes the potential and is in Cartesian frame when using periodic and secular terms: U = ( /R j ) ( /R j ) (R e /R j ) 2 J 2 (/2)((3Z j 2 /R j 2 ) -) 2 2 where: R j = (X j + Y j + Z 2 j ) /2. On the other hand, the potential for secular effects alone is: U = ( /R j ) ( /R j )(R e /R j ) 2 J 2 (/2) {(3/2) sin 2 (i) -} (9) () he acceleration derived from the potential for the case (a) with secular and short periodic motion is presented as in Chairadia et al. (22): a x = -( X j /R j 3 ) [ + (3/2) (R e /R j ) 2 J 2 (-(5Z j 2 /R j 2 )] a a y = (Y (Y/X j /X)a j )a x a Z z = -( Z j /R 3 j ) [ + (3/2) (R/R j ) 2 J 2 (3-(5Z 2 j /R 2 j )]. () On the other hand, for (b), when only secular effects are considered, the accelerations can be seen in Ramachandran (25) as: a x = (αx j /R j 5 ) X j /R j 3 a y = (α Y ( j /R /R 5 j ) Y Y j /R /R j 3 a z = ( (αz Z j /R /R 5 j ) Z Z j /R /R j 3 (2) where the constant α = μr 2 e J 2 (3/2) ( (3/2) sin 2 (i)). It can be noted that the inclination i is invariant in the motion described in Eq. 8. hese accelerations are then used to derive the partial derivative and obtain the absolute SM matching the 2 types of reference motions: (a) or (b). Details are given in Markley (986), as well as in Chairadia et al. (22). RELAIVE SAE RANSIION MARIX Formulation Next, it is described the relative motion of 2 satellites by the geometric principle. he motion of the deputy satellite is

4 Relative State ransition Matrix Using Geometric Approach for Onboard Implementation 33 in the chief satellite s local-vertical-local-horizontal coordinate frame (LVLH). he relative position vector r, with respect to the chief, has 3 components: x, y, and z, as illustrated in Fig.. Here, (x) is the component in the chief s radial direction, (z) is in the direction of the chief instantaneous angular momentum, and (y) is along the direction that completes the triad (Alfriend et al. 2). he relative distance is: Figure. Relative motion of deputy satellite in the frame of the chief. r = C oi [R d R c ] R d R c / R c C oi = (H c x R c )/( H c x R c ) r H c /( H c ) y (3.) (3.2) he transformation matrix C oi (3 3) relates the LVLH frame of the satellite to the ECIF. he position and velocity vectors are, respectively, R c = (X j,y j,z j ) and V c = (X j,y j,z ), where j = j denotes the chief satellite while R d and V d for j = 2 are used to denote the deputy satellite. he magnitudes are R c and R d for the chief and deputy satellites, respectively. he instantaneous angular momentum vector of the chief is H c = (R c V c ), where (x) is the usual vector cross-product. In Eq. (3.2), the second row is the normalized cross-product of the first and third rows. aking the derivative of Eq. 3., the relative velocity is: R c ECIF v = C oi [R d R c ] + C oi [R d R c ] Combining to get x rel = [P ] S x = [P ] S x rel = [ r,v] r,v r = [ x,y,z] and v = [x,y,z ] ] and v = [x,y,z ] -C oi C oi [ P ] = -C oi -C oi C oi C oi z x (4) (5) where S = [R c R c R d R d ]. Here [P] is a (6 2) matrix. In case the mean elements of the state Š are used, then another transformation needs to be carried out as in Gim and Alfriend (23), in addition to that of using Cartesian transformation. S = D Š (6) he transformation matrix D is reversible and has been sufficiently discussed by Gim and Alfriend (23). he following discussion uses osculating element. he formulation is analogous while using mean elements after applying the transformation as in Eq. 6. he relative SM Φ r (t,t ) follows: x rel (t) = Φ r (t,t ) x rel (t ) with Φ r (t,t ) = [ P ] [ Q ] [ ] (7) (8) he matrix P is known in Eq. 5. he second term on the right-hand side of Eq. 8 is the matrix Q and it uses the absolute SM derived earlier for both satellites. If the input is the reference state S(t ), then the update is obtained by using S(t) = [ Q ] S(t ) (9.) Using Φ a to individually (t,t ) denote the SM of the chief and deputy S(t) satellites, = [ Q ] S(t we have ) Φ a chief (t,t ) (6x6) (6x6) [ Q ] = (9.2) (9. (6x6) Φ a deputy (t,t )(6x6) he matrices are the absolute SM obtained either (a) using periodic accelerations or (b) admitting only secular accelerations as mentioned in Eqs. and 2, respectively. In case the relative SM for mean elements is needed, then the transformation for the osculating to mean, as in Eq. 8, is required. In order to derive the elements of C oi in P, a general relation is used = - (2) Substituting the 3 rows of C oi in place of (u/ u ) in the above equation, it is obtained

5 332 Ramachandran MP R c / R c - R c (R c.r c )/( R c 3 ) C oi = [(H c x R c ) / (H c x R c ) ] [(H c xr c )((H c xr c ).(H c xr c ) )/( H c xr c 3 ) (2) H c / H c - H c (H c.h c )/( H c 3 ) Again the second row is computed as a cross product of the first and third rows instead of the algebraic expression. he matrix [] in Eq. 8 is the inverse of matrix [P], and its new form is -C oi = (/2) -N -C oi [α] = C oi (3x3) [β] = C oi (3x3) C oi -C oi [γ] = Φ [ ] chief a (t,t (t,t ) (6x6) (6x6) [ he ] = relative deputy SM (t,t )(6x6) Φ r (t,t ) in Eq. 8 can be rewritten as (22) where: N = C oi C oi C oi. he relative SM Φ r (t,t ) in Eq. 9 is approximate due to truncation in Q (Markley 986). Computational Reduction he computation reduction aspect is discussed next. he absolute SM are anyway needed on-board for individual orbit control. he additions for relative control are the matrices [P] and [] and subsequently the matrix multiplication. It is now examined how this computation of the relative SM in Eq. 8 can be decomposed for computation efficiency. It may be noted that there are 4 matrices. Φ r (t,t ) = -α α γ -α -β -α β α λ N -α -α -α (23) (24) he matrix multiplication can be further decomposed as: [ f ] α [-f] Φ r (t,t ) = N α (25) [-g ] α [g] α where: [f] = [ α ] [γ] and [g] = [β α] [λ]. he repetition of matrices and (3 3) matrices involved in matrix multiplication helps to optimise the computation. he following navigation architecture further eases the computation. he chief or target reference orbit is available in the deputy or follower satellite due to inter-satellite communication (Gill et al. 27). he thruster on the deputy then controls the orbit to maintain the formation in a leader/follower coordination (Alfriend et al. 2). he chief reference orbit is undisturbed during this operation. Using the states at t and t, the elements of the matrix in Eq. 25 can be computed, ahead of that time. his management ensures that the elements of the first 3 rows of the matrix can be obtained beforehand. he orbit of the deputy or main satellite and the matrix [λ] alone varies due to orbit control thrusters and needs to be updated. his computational reduction in real time is the advantage of the present approach. On the other hand, in the case of unit-sphere approach (Vadali 22), the transformation matrix in Eq. 5 is constructed using both chief and deputy states. Hence, all the elements of the relative SM need to be computed in real time and are task-intensive. he SM in Gim and Alfriend (23) is more difficult for implementation. While using mean motion, the secular accelerations given in Eq. are used to construct mean relative SM. his, however, calls for conversion of the measurement transformed to first-order mean elements using Eq. 6 and that further to Cartesian frame. he orbit model in Eq. 8 can be enhanced to accommodate secular effects of other perturbation for near circular orbits. owards this, the orbit model suggested in Ramachandran (25) can be adopted. It means that the motion

6 Relative State ransition Matrix Using Geometric Approach for Onboard Implementation 333 is more realistic to account for secular variation which is along the track between the satellites in formation, and the SM is useful to match the realistic motion of the satellite. NUMERICAL ANALYSIS he experiment carried out here intended to estimate the accuracy of the SM given in Eq. 7. he mean motion is selected as input for each equation. From Eq. 8, the initial states at times t and t are achieved, and the matrix [Q] is obtained. Using the known relative position at time t as input and the approximate SM in Eq. 7, the update of the relative state at time t 2 is obtained. his appears on the left-hand side of Eq. 7. he time step selected is s. Subsequently, it is obtained the difference between this output x rel (t 2 ) and the actual relative state at that instant computed using Eqs. 3 and 4. he error in position and velocity in the LVLH frame is related to the reference orbit. his error is gauged against variations in inclination and eccentricity. It is known that the J 2 effect brings an out-of-plane (z) effect which is absent in CW equations. In the first experiment, the orbital eccentricity is.5, and the semi-major axis a is 7, km. he deputy satellite is at a distance of 2 km along the track. he plot in Fig. 2 shows the error while varying inclination from up to 9 degrees when retaining the eccentricity as.5. he numerical accuracy of the methods from Gim and Alfriend (23) and Yan et al. (24) shall be no different as they are mathematically similar and differ in formulation. Also, the comparison of propagation error will be limited by the frame of reference. A case of higher eccentricity when e =. is carried out. he error in the position and velocity was observed to be greater than that shown in Fig. 2, though the variation with inclination is similar. he errors in position and velocity are.68 m and.4 m/s, respectively. his increase in the error is as expected for higher values of e and due to the approximation made in Q of Eq It is seen that x and x (see Eqs. 3 and 4) have more perceivable error than (y, y, z, z ). he position error is in the top plot while the velocity error is that at the bottom. he (y, z) values agree to 4 decimal places while (y, z ) ones agree beyond 5 decimals. his implies that the update error along these axes is negligible. o understand the effect of eccentricity, the experiment is repeated by changing the eccentricity to.5. he units now are in cm and cm/s yet again the error is large in (x, x ). he error in the other axes agrees in 2 decimal places in position and 3 in velocity. When the distance of the satellites is km, then the error gets proportionally scaled to that in Fig. 3. his is due to the approximation in the absolute SM. he next test is carried out by varying the eccentricity from.6 to.6. he inclination is fixed and equal to 2 degrees. In Fig. 4, the error along (x, x ) is greater again and increases with eccentricity. It can be seen in Markley (986) that the error in absolute SM depends upon the position in the non-circular orbit. In this example, the argument of perigee is retained while the eccentricity varies. As a summary, it is clear that the error, using Eqs. 3 and 4, is the least along the track and across the track (y, z) these are very important in the control of the satellite. he radial error along (x, x ) was found more agreeable for smaller eccentricity. Position error [m] Velocity error [m/s] Position and velocity error, e = Figure 2. Error variation with respect to inclination when e =.5.! -.4 Figure 2. Error variation with respect to inclination when e =.5.! 3 6 9! Position error [cm] Velocity error [cm/s] Position and velocity error, e= Figure 3. Error variation with respect to inclination when e =.5.

7 334 Ramachandran MP Position error [m] Velocity error [m/s] Variation in position error with eccentricity, inclination = 2 o Variation in velocity error Eccentricity Figure 4. Error variation with respect to eccentricity when inclination is 2 degrees. CONCLUSION Based on the geometric approach, a relative SM was obtained, and it includes the oblate Earth effect. It is simpler for on-board implementation as it is decomposable in terms of the orbital states of the 2 satellites in formation. he navigation plan, as suggested in this paper, reduces the real-time computation when applying the proposed relative SM for orbit control purposes. A numerical experiment was carried out to measure the accuracy of the proposed method for all inclinations. It shows that the approach is more accurate when the distance between the satellites is below 2 km and when the eccentricity is.5. For larger baselines and eccentricity, the accuracy of the relative SM is seen limited due to the first-order approximation in the absolute SM. REFERENCES Alfriend K, Vadali SR, Gurfil P, How JP, Breger LS (2) Spacecraft formation flying. Oxford: Butterworth-Heinemann. Alfriend K, Yan H (25) Evaluation and comparison of relative motion theories. J Guid Contr Dynam 28(2): doi:.254/.669 Balaji SK, atnall A (23) System design issues of formationflying spacecraft. Proceedings of IEEE Aerospace Conference; Big Sky, USA. Carter E (998) State transition matrices for terminal rendezvous studies: brief survey and new example. J Guid Contr Dynam 2(): doi:.254/2.42 Chairadia APM, Kuga HK, Prado AFBA (22) Comparison between two methods to calculate the transition matrix of orbital motion. Mathematical of Problems in Engineering 22(22):Article ID doi:.55/22/ De Brujin F, Gill E, How J (2) Comparative analysis of Cartesian and curvilinear Clohessy-Wiltshire equations. Journal of Aerospace Engineering, Sciences and Applications 3(2):-5. Gill E, Montenbruck O, D Amico S (27) Autonomous formation flying for PRISMA Mission. J Spacecraft Rockets 44(3): doi:.254/.235 Gim DW, Alfriend K (23) State transition matrix of relative motion for perturbed noncircular reference orbit. J Guid Contr Dynam 26(6): doi:.254/ Hamel J, de Lafontaine J (27) Linearised dynamics of formation flying spacecraft on a J2 perturbed elliprical orbit. J Guid Contr Dynam 3(6): doi:.254/ Markley FL (986) Approximate Cartesian state transition matrix. J Astronaut Sci 34(2):6-69. Melton RG (2) ime-explicit representation of relative motion between elliptical orbits. J Guid Contr Dynam 23(4):64-6. doi:.254/2.465 Ramachandran MP (25) Approximate state transition matrix and secular orbit model. International Journal of Aerospace Engineering 25(25):Article ID doi:.55/25/ Roy AE (982) Orbital motion. Bristol: Adam Hilger Ltd. schauner J, Hempel P (965) Rendezvous with a targets in an elliptical orbit.astronautica Acta (2):4-9. suda Y (2) State transition matrix approximation with geometry preservation for general perturbed orbits. Acta Astronautica 68(7-8):5-6. doi:.6/j. actaastro Vadali SR (22) An analytical solution for relative motion of satellites. Proceedings of the 5th dynamics and control of systems and structures in space conference; Cranfield, UK. Yamanaka K, Ankersen F (22) New state transition matrix for relative motion on an arbitrary elliptical orbit. J Guid Contr Dynam 25():6-66. doi:.254/ Yan H, Sengupta P, Vadali SR, Alfriend K (24) Development of a state transition matrix for relative motion using the Unit-Sphere Approach. Proceedings of the AAS/AIAA Space Flight Mechanics Conference; Maui, USA.