DIFFERENTIAL DRAG SPACECRAFT RENDEZVOUS USING AN ADAPTIVE LYAPUNOV CONTROL STRATEGY

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1 (Preprint) IAA-AAS-DyCoSS1-XX-XX DIFFERENIAL DRAG SPACECRAF RENDEZVOUS USING AN ADAPIVE LYAPUNOV CONROL SRAEGY Daid Pérez * and Riccardo Beilacqua I. INRODUCION his paper introduces a noel Lyapuno-based adaptie control strategy for spacecraft maneuers using atmospheric differential drag. he control forces required for rendezous maneuers at low Earth orbits can be generated by arying the aerodynamic drag affecting each spacecraft. his can be accomplished, for example, by rotating dedicated sets of drag panels. hus, the relatie spacecraft motion can be controlled without using any propellant since the motion of the panels can be powered by solar energy. A noel Adaptie Lyapuno Controller is designed, and a critical alue for the relatie drag acceleration that ensures Lyapuno stability is found. he critical alue is used to adapt the Lyapuno controller, enhancing its performances. he method is alidated using simulations in the Analytical Graphics Incorporated's Satellite ool Kit software. he results show that the Adaptie Lyapuno technique outperforms preious control strategies for differential drag based spacecraft maneuering. his work presents a noel adaptie Lyapuno control strategy to perform spacecraft rendezous maneuers at Low Earth Orbits (LEO), exploiting atmospheric differential drag forces. C.L.Leonard (see reference 1) introduced an alternatie method for generating the control forces required by the rendezous maneuers at LEO. his method consists of arying the aerodynamic drag experienced by different spacecraft, thus generating differential accelerations between them. he interest towards this methodology comes from the decisie role that efficient and autonomous spacecraft rendezous maneuering will hae in future space missions. In order to increase the efficiency and economic iability of such maneuers, propellant consumption must be optimized. Employing the differential drag based methodology allows for irtually propellant-free control of the relatie orbits, since maneuerable dedicated drag surfaces can be powered by solar energy. he ORBCOMM constellation formation keeping (see reference 2) is the first application of these ideas, while, the JC2Sat project deeloped by the Canadian and Japanese Space Agencies (see references 3 and 4) is an enisioned application of these ideas. he ariation in the drag can be induced, for example, by closing or opening flat panels attached to the spacecraft, hence effectiely modifying its ballistic coefficient. he reference frame commonly employed for spacecraft relatie motion representation is the Local Vertical Local * Ph.D. Candidate, Mechanical, Aerospace and Nuclear Engineering Department, Rensselaer Polytechnic Institute, JEC th Street roy, NY Assistant Professor, Mechanical, Aerospace and Nuclear Engineering Department, Rensselaer Polytechnic Institute, JEC th Street roy, NY

2 Horizontal (LVLH) reference frame, where x points from Earth to the reference satellite (irtual or real), y points along the track (direction of motion), and z completes the right-handed frame (see Figure 1). In such a frame we enision a target and a chaser spacecraft, and the following three cases for the configurations of the panels can be considered (see Figure 1): 1. All the panels of the chaser are deployed generating the maximum possible drag while those of the target are not deployed to achiee the opposite, hence generating a negatie acceleration of the chaser relatie to the target. 2. All the panels of the target are deployed generating the maximum possible drag while those of the chaser are not deployed to achiee the opposite, thus generating a positie acceleration of the chaser relatie to the target. 3. Both chaser and target hae a couple of panels deployed, which means that there is no relatie acceleration between the spacecraft. his can be also achieed by haing both spacecraft closing all of their panels. Figure 1. Drag panels concept to generate differential drag (obtained from 5). It is worth to underline that atmospheric differential drag is expected to proide an effectie control only in the orbital plane (x and y); for this reason we will limit our discussions to the inplane motion, assuming that no out-of-plane (z) motion is present, or that it is controlled with different means. he magnitude of the differential drag acceleration fluctuates during the maneuer as the spacecraft encounters regions of the thermosphere with different atmospheric densities. In the thermosphere, atmospheric density can change significantly due to solar and geomagnetic actiities. hese ariations are difficult to model and measure accurately on board; hence, robust control strategies must be designed to increase the reliability of spacecraft maneuering using differential drag. 2

3 he problem of designing a control system for the rendezous maneuer using differential drag becomes the problem of designing a real-time logic to command the closed and open positions for the flat panels, with the intent of forcing the satellites to follow the desired rendezous trajectory, or simply regulate to a final desired state (rendezous). It is worth underlying that the sought-for logic needs to be based on the assumption that the control is either positie maximum, negatie maximum, or zero (Figure 1) neglecting the time required by the panel to rotate. his logic is here designed using a Lyapuno approach. In essence, a Lyapuno function of the tracking error is selected, and the control signal is chosen so that the tracking error conerges to zero (i.e. the first order time deriatie of the Lyapuno function is negatie), thus, the nonlinear dynamics of the system are forced to follow a desired trajectory. his significantly simplifies the control problem, since the desired trajectory can be designed using controlled linear dynamics approximating the reality of spacecraft relatie motion. he method proposed here builds upon and improes preious work presented and tested in references 6 and 7. In particular, a stable linear reference model is introduced; this model tracks the desired rendezous trajectory. he Lyapuno controller can then be used to either directly track the desired trajectory or track the dynamics of the linear reference model. In this work the Lyapuno controller is directly tracking the final desired position, that is the zero relatie position and elocity between chaser and target spacecraft, which is the rendezous condition. he stable linear reference model is still needed for the regulation case, een if the linear reference is not tracked. In fact, the linear stable model enables the definition of the correct Lyapuno function that in turns dries the nonlinear dynamics to behae in a desired fashion in terms of time response. In order to enhance the performance and robustness of the Lyapuno controller, a way of adapting the Lyapuno function, in terms of the drag acceleration critical alue necessary for stability, is deeloped. he definition of appropriate Lyapuno functions is a challenge that aries from problem to problem, and a widely studied theory exists (see references 8, 9, and 10). In this work, we define a quadratic Lyapuno function of the tracking error, and we change its positie definite matrix in an adaptie fashion, effectiely changing the Lyapuno function in real time to achiee the best performances during the rendezous maneuers. he adaptation is achieed through analytical expressions giing the dependence of the differential drag acceleration critical alue from the chosen stable linear model. By means of these relationships, the differential drag critical alue is maintained minimal during the maneuer, achieing the best possible control authority margin in real time. All the deriaties necessary to obtain the analytical results are computed and presented in the paper. he foremost contributions of this work are: 1) An analytical expression for the differential drag acceleration critical alue that ensures stability in the sense of Lyapuno for the system. 2) Analytical expressions for the partial deriaties of the critical alue of the differential drag acceleration in terms of Q (Lyapuno equation matrix), and A d (reference linear dynamics matrix). 3) A control strategy, based on the Lyapuno approach, for two spacecraft rendezous using differential drag, which uses adaptation to choose in real time an appropriate positie definite matrix P in a quadratic Lyapuno function. 4) Demonstration of feasibility of the approach ia Satellite ool Kit (SK) numerical simulations. 5) Assessment of the performances of the designed adaptie Lyapuno control strategy in terms of the duration of the rendezous maneuer and the number of switches in the dif- 3

4 ferential drag (control effort), in comparison with the non adaptie Lyapuno control strategy preiously presented by the authors in reference 7. 6) Oerall, this paper proides a aluable strategy that can be implemented onboard real spacecraft, een small spacecraft with limited computing capabilities. In fact, the adaptie Lyapuno-based control methodology does not require numerical iterations and can run in real time, requiring onboard measurements that would be aailable during flight. he paper is organized as follows. Section II introduces the concept of atmospheric differential drag and its mathematical expression. Section III illustrates the spacecraft relatie motion linear and nonlinear dynamics employed in the following deelopments. Section IV is dedicated to the Lyapuno function definition, the panels actiation strategy, as well as the analytical deriaties of the differential drag critical alue with respect to the independent ariable matrices. In section V we use such deriaties to create the noel adaptie Lyapuno controller. Section VI alidates the approach ia Satellite ool Kit numerical simulations, and Section VII draws the conclusions. II. DIFFERENIAL DRAG he drag acceleration experienced by a spacecraft at LEO is a function of the atmospheric density, atmospheric winds, elocity of the spacecraft relatie to the medium, and the geometry, attitude, drag coefficient and mass of the spacecraft. he interdependence of these parameters (e.g. the drag coefficient is affected by the temperature of the medium which also determines the density of the medium) and the lack of knowledge in some of their dynamics make the modeling of the drag force a challenging and still largely unsoled problem. his results in large uncertainties regarding the control forces aailable for maneuers using drag forces. Consequently the control systems used for drag maneuers must be able to cope with these uncertainties. he aerodynamic acceleration experienced by a spacecraft is typically decomposed into the lift (lift forces are negligible at LEO) and drag forces, the latter usually expressed as: 1 2 ad BCs (1) 2 Where ρ is the atmospheric density, s is the elocity of the spacecraft relatie to the atmospheric particles. he ballistic coefficient BC is gien by : C A BC m Where C D is the drag coefficient of the spacecraft, A is the cross-wind surface area of the spacecraft and m is the mass of the spacecraft. D (2) Using equation (1), magnitude of the relatie acceleration caused by the differential aerodynamic drag for the spacecraft system (target and chaser) is gien as: 1 2 ad rel BCs (3) 2 Where ΔBC is the difference in ballistic coefficients between the target and chaser. he acceleration expressed aboe is described earlier in Figure 1. In the thermosphere, the solar actiity creates large ariations of temperature, which drie ariations of the atmospheric density. hese ariations produce significant changes in the aailable magnitude of drag acceleration for a gien ballistic coefficient. 4

5 III. LINEAR REFERENCE AND NONLINEAR MODELS Hill s groundbreaking work on lunar motion (see reference 11), which described the motion of the Moon relatie to the Earth, was the first study on the relatie motion of bodies in space. Afterwards, inspired by Hill s work, Clohessy and Wiltshire (see reference 12) deeloped a linear model, which describes the motion of a chaser spacecraft relatie to a target spacecraft. his model has been widely used in applications inoling low thrust proximity maneuers. Unfortunately, this model does not account for the differential effects on the spacecraft motion due to nonlinearities such as the J2 perturbation, caused by the earth s flattening. he effect of the J2 perturbation and other nonlinearities is more significant in maneuers with longer times of execution such as those performed using differential drag. For this reason the use of a linear model that partially accounts for aeraged effects of these nonlinearities is desired, as the one described in the following section. he model described in the following section is used for the deriation of the Lyapuno controller. Linear Reference Model A linearized model which represents the relatie motion of spacecraft under the influence of the J2 was deeloped by Schweighart and Sedwick (see reference 13). Adding the control acceleration ector (u) to the Schweighart and Sedwick equations, the following system of linear differential equations in the LVLH frame is obtained. 0 d 02 x2 I2 x2 0 y d b 0 0 a B d 0 x d x A Bu, A,, x, d x 0 0 a 0 1 yd 3JR a 2 nc, b 5c 2 n, c 1 [1 3 cos(2 i )], d a b t 8r Where n is the mean motion of the target, J 2 is the second zonal harmonic, R is the Earth mean radius, r t is the target s orbit radius and i t is the target s inclination. Noteworthy, the control action is only along the y direction, as depicted in Figure 1. Since the dynamics of the Schweighart and Sedwick model are unstable, a Linear Quadratic Regulator (LQR) feedback controller is used to stabilize them and obtain the necessary linear model for Lyapuno deelopments. he resulting reference model is described by: xd Ad x d, Ad A - BK, (5) Where K is a constant matrix found by soling the LQR problem for the Schweighart and Sedwick model, thus ensuring A d to be Hurwitz. he Q LQR matrix and R LQR alue used to sole the LQR problem are: 19 Q LQR I 4x4, RLQR 1.5*10 (6) It is worth mentioning that the state ector x d is the desired reference trajectory, and control action is along the y direction only (since the drag force acts always opposite to the direction of motion). his stable linear reference system can be regulated or forced to track a desired guidance trajectory. (4) 5

6 Nonlinear Model he dynamics of spacecraft relatie motion are nonlinear due to effects such as the J 2 perturbation, and the nonlinear ariations on the atmospheric density at LEO. he adaptie Lyapunobased approach suggested here intends to cope with these unmodeled effects, by minimizing the differential drag critical alue at all times. In this section the model that will be used for the nonlinear dynamics is presented. he general expression for the real world nonlinear dynamics, including nonlinearities such as the J 2 perturbation, is defined as: adrel x f x Bu, x x y x y, u 0 (7) adrel where a Drel acts along the y direction only, as explained earlier. All the nonlinearities are accounted for in the nonlinear function f(x). For the deelopment of the Lyapuno controller f(x) is assumed to be a function of the spacecraft position only, and is assumed (for a single spacecraft in the inertial frame) to be: r f r, r [ x ] 3 in yin zin (8) r Where r is the position ector of the spacecraft in the inertial frame, r is its magnitude and μ is the graitational parameter. IV. LYAPUNOV APPROACH In this section the nonlinear adaptie control law based on the Lyapuno approach is described. he approach is inspired by preious work from one of the authors (see reference 6) and was further deeloped by the authors in reference 7. A Lyapuno function is defined as: V e Pe, e = x - xd, e = x - xd, P 0 (9) Where P is a positie definite matrix, e is the tracking error ector, x and x d are defined as the actual spacecraft relatie state ector and the reference state ector respectiely. he first order time deriatie of the Lyapuno function can be manipulated, using some algebra, to obtain (see also reference 6): V e (A P + PA )e 2 e P( f x - A x + Ba uˆ - Bu ) (10) d d d Drel d If the matrix A d is Hurwitz, a symmetric positie definite matrix Q is chosen such that the Lyapuno equation (11) is satisfied. For this reason the reference model must be stable, which is the case for the stabilized Schweighart and Sedwick model shown in equation (5). For a gien set of matrices Q, and A d the following Lyapuno equation must be soled to find P -Q A P + PA (11) d d Choosing a positie definite Q matrix, results in the following expression for the time deriatie of the Lyapuno function: where is gien by: V e Qe 2 (12) û (13) 6

7 and β, δ and û (the command sent to the panel actuators) are gien by the following expressions: 1 e PB a, ˆ Drel u 0 (14) 1 = e P A x f x Bu (15) he three states for û represent the commands: all panels open on chaser and all closed on target (-1), no differential between spacecraft, i.e. same panels deployed or no panels deployed on both spacecraft (0), and the opposite of the first configuration, i.e. all panels open on target and all closed on chaser (1). Equations (10), (12) and (15) represent the case in which the nonlinear system is tracking the reference model (state ector x tracks state ector x d ); howeer, if the nonlinear system, directly tracks a desired guidance, equations (10), (12) and (15) simplify to: d 2 V e P( f x x Ba u ˆ) 2, = e P x f x (16) t Drel t Furthermore, if the desired guidance is assumed to be a constant zero state ector, (the controller acts as a regulator), which is the desired final state for a rendezous maneuer (zero relatie positions and elocity between chaser and target), equations (10), (12) and (15) are reduced to: d 2 V e P( f x Ba u ˆ) 2, = -e P f x (17) It must be mentioned that een though the matrices A d and Q are not present in equations (16) and (17), they still affect the behaior of V since they are used to obtain P. his causes the matrix P to still contain information on the linearized dynamics of the system initially contained in matrix A d. he P matrix enforces certain relationships among the system s states, and also weights their contribution within the Lyapuno function, in such a way to obtain a desired controlled relatie motion between the two spacecraft. he characteristics of this desired motion (e.g. time response) are chosen by selecting the linear dynamics. Drag panels actiation strategy Guaranteeing 0would imply that the tracking error (e) conerges to zero, since the term inoling Q is already negatie. In other words, as long as 0, the system dynamics of the spacecraft will conerge to the desired state. Howeer, since the control ariable û does not affect δ, the system cannot be guaranteed to be Lyapuno stable for the chosen Lyapuno function if δ is positie, and has a higher magnitude than β. he magnitude of β is linearly dependent on the atmospheric density ρ, which indicates that if ρ is too small the system is unstable since β will not hae a magnitude large enough to oercome a positie alue of δ of greater magnitude. In others words, the motion of the spacecraft cannot be controlled if ρ is not large enough, that is there is not enough control authority (not enough drag force for controlling the spacecraft motion). he actiation strategy for the control (as proposed in reference 7) is designed such that the chosen alue of û forces the product βû to be negatie, thus û can be expressed as: Drel uˆ sign( ) sign( e PB ) (18) Due to the low magnitude of the relatie accelerations that are attainable at LEO, this actiation strategy is applied eery 10 minutes (see the simulations section). his allows for lower frequencies of actuation, and for the drag forces to hae enough time to change the orbits of the 7

8 spacecraft. It is worth emphasizing that all the components in the aboe actiation strategy would be aailable in real time on board the spacecraft. Critical alue for the magnitude of differential drag acceleration As it can be seen in equations (12) and (13), the control signal û is only present in one of the three elements constituting V. Consequently, the product βû is the only one that can be used to influence the behaior of V which must be always negatie to insure that the system is stable in the sense of Lyapuno. his suggests that there must be a minimum alue for a Drel that allows for V to be negatie for gien alues of β and δ. his alue is found analytically by soling the following inequality: e PB a uˆ (19) 0 Drel which results from the inequality 0. Soling this expression for a Drel yields: a Drel (20) e PB he absolute alue comes from replacing the û for sign(e P B). his inequality indicates that if the differential drag acceleration between the spacecraft is larger than the right hand side of the inequality, the deriatie of the Lyapuno function will be negatie, and consequently the tracking error will go to zero. he right hand side of the inequality then constitutes an analytical critical alue for the magnitude of the differential drag, which can be calculated in real time during the maneuer, and it proides a proxy alue for the necessary atmospheric density to ensure stability for the controller. his lower bound or critical alue is defined as a Dcrit e (21) PB For the simplest case in which the controller acts as a regulator (see equation (17)) equation (21) becomes: adcrit e P f x e PB As it can be obsered from equation (22), the critical a Dcrit depends on the tracking error ector e and the matrices P and B for the cases in which the nonlinear system is directly tracking the desired guidance or being regulated. B, and e cannot be changed by design; howeer, the matrix P, (determined in equation (11)) depends only on the matrices Q, and A d, which are chosen proided that they satisfy some requisites, namely, Q being positie definite and A d being Hurwitz and being an approximation of the systems dynamics. Matrix deriaties Choosing appropriate alues for the entries of Q and A d will unequiocally modify the behaior of a Dcrit. o gain a better understanding of the relationship between the entries of matrices Q and A d and a Dcrit, the partial deriaties of a Dcrit in terms of the matrices Q, and A d are calculated. (22) 8

9 he deelopment of these partial deriaties requires the definition the following four matrix deriatie representations defined in reference 14 (equations (23), (26), (27) and(28)): Y, X Y, X 11 1n Y YX, X Y, X 1 Y, X n nn Equation (23) shows the general representation for the partial deriatie of an n-by-n matrix Y in terms of the n-by-n matrix X. A ery useful operator when dealing with matrix deriaties is the ec operator which for an n-by-n Z matrix is defined in reference 14 as: ec( Z) Z Z Z Z Z 11 n1 1n nn (24) Its inerse operator is defined as: Z Z 11 1n unec( Z ) Z (25) Z Z n1 nn he following different representations of the general matrix deriatie can be used for finding complicated deriaties (matrix deriatie chain rule, matrix deriatie product rule, etc.): Y X Y X Y X ( ec( Y)), X ( ec( Y)), X ec( Y), ec( X) ( ec( Y)), X ( ec( Y)), X ec( Y), X ec( Y), X 11 n1 1n nn ec( Y), X ec( Y), X 11 1n ec( Y), X 1 ec( Y), X n nn Y Y Y n1 Y ec( ), X ec( ), X 11 1n ec( ), X ec( ), X Where the ectors Y and X are the ectorized (ec) ersions of the matrices Y and X. By obsering the representations in equations (26), (27) and (28), it is readily concluded that these three matrix deriaties, containing ectorized forms of the matrices X and Y, contain the same entries as the general matrix deriatie (equation (23)), but organized in different structures. By rearranging the entries in these three matrix deriaties, it is possible to obtain the original matrix deriatie. nn (23) (26) (27) (28) 9

10 hree reersible transformations from the representations in equations (26), (27) and (28) to the general matrix deriatie representation in equation (23) are defined as follows: ransformation 1: Y X 1 Y X o Input: n 2 -by-n 2 matrix Y,X o Output: n 2 -by-n 2 matrix Y,X o ranspose each of the rows of the input matrix Y,X to obtain n ectors o Unectorize each one of these n ectors to obtain n n-by-n sub-matrices o hese sub-matrices are the blocks that form the output matrix Y,X Inerse ransformation 1: Y 1 Y 1 X X o Input: n 2 -by-n 2 matrix Y,X o Output: n 2 -by-n 2 matrix Y,X o Diide the input matrix Y,X into n blocks, each composed of one n-by-n submatrix o Vectorize all of these sub-matrices to form n ectors o hese ectors are rows of the n 2 -by-n 2 matrix Y,X (29) (30) ransformation 2: Y Y 2 X X o Input: n 3 -by-n matrix Y,X o Output: n 2 -by-n 2 matrix Y,X o Diide the input matrix Y,X into n blocks each composed of an n 2 -by-1 ectors o Unectorize each one of these n ectors to obtain n n-by-n sub-matrices o hese sub-matrices are the blocks that form the output matrix Y,X Inerse ransformation 2: Y 1 Y 2 X X o Input: n 2 -by-n 2 matrix Y,X o Output: n 3 -by-n matrix Y,X o Diide the input matrix Y,X into n blocks, each composed of an n-by-n submatrix o Vectorize all of these sub-matrices to form n ectors n 2 -by-1 o hese n 2 -by-1 ectors are the blocks that form the n 3 -by-n output matrix Y,X (31) (32) ransformation 3: Y X 3 Y X (33) 10

11 o Input: n 3 -by-n matrix [Y ],X o Output: n 2 -by-n 2 matrix Y,X o Diide the input matrix [Y ],X into n blocks each composed of a 1-by-n 2 ectors o ranspose these ectors to obtain n 2 -by-1 ectors o Unectorize each one of these n ectors to obtain n n-by-n sub-matrices o hese sub-matrices are the blocks that form the output matrix Y,X Inerse ransformation 3: Y 1 Y 3 X X o Input: n 2 -by-n 2 matrix Y,X o Output: n-by-n 3 matrix [Y ],X o Diide the input matrix Y,X into n blocks, each composed of a n-by-n submatrices o Vectorize all of these sub-matrices to form n ectors n 2 -by-1 o ranspose these ectors to obtain 1-by-n 2 ectors o hese 1-by-n 2 ectors are the blocks that form the n-by-n 3 output matrix [Y ],X (34) he first step in the deelopment of the desired partial deriaties (a Dcrit, Q and a Dcrit, A d ) is to find the partial deriatie of a Dcrit in terms of the matrix P. his is accomplished by using equation (22) in which a Dcrit is explicitly expressed in terms of P. Equation (22) can be rewritten as: a Dcrit e P f x e PB num den he partial deriaties of the numerator and the denominator in terms of the matrix P are found to be: num den e PB e f x, eb P P e PB After using the deriatie quotient rule and some algebra the resulting expression is found: a Dcrit x e f e PB e P f x eb 3 P e PB e PB he second step is to obtain the partial deriaties of the matrix P in terms of the matrices Q, and A d. o obtain these deriaties equation (11) is rewritten as: A P Q, A I A A I, P ec( P), Q ec( Q) 4 x4 d d 4 x4 (35) (36) (37) (38) 11

12 Where represent the Kronecker product defined in reference 14 for the same n-by-n Y and X matrices as: XY X Y X Y 11 1n X Y X Y n1 nn (39) he transformation in equation (38) (found in reference 14) allows for the formulation of an explicit equation of the elements of the matrix P in terms of the elements of the matrices Q, and A d, which is found to be: P A Q (40) 1 Using equation (40) it is possible to find the partial deriaties of the ector P in terms of the ector Q : P 1 A Q Subsequently, transformation 1 is used to obtain the partial deriatie of the matrix P in terms of the matrix Q, which yields: P 1 A Q 1 Again, using equation (40) it is possible to find the partial deriaties of the ector P in terms of the matrix A : P A Where U nxn is an n-by-n permutation matrix defined as: 1 I x A U x I16 x16 A I16 x16 Q n n nxn rs rs r s (41) (42) (43) U E (44) E Where the matrix E rs is an elementary matrix with a one at position r,s and zeros elsewhere. Afterwards, the deriatie of A in terms of A d is found to be: A I U U I I U U I Ad Where U 1 is a permutation matrix defined as: 4 x4 1 4 x4 4 x4 4 x4 1 4x4 4 x r s rs rs (45) U E (46) E he matrix chain rule, as defined by reference 14, is used to obtain the partial deriatie of P in terms of the matrix A d : P ec( A ) P I16 x16 I4 x4 Ad Ad ec( A ) (47) 12

13 Using inerse transformation 1, inerse transformation 3, and equations (45) and (43) the results of equation (47) become: P 1 A 1 P 3 I16 x16 I4 x4 1 Ad Ad A ransformation 2 is used to obtain the partial deriatie of the matrix P in terms of the matrix A d, which yields: P P 2 Ad Ad Next, the matrix chain rule is used again to obtain the partial deriatie of the scalar a Dcrit in terms of the matrix Q: a ec( P) Dcrit adcrit I1 x1 I4 x4 Q ec( ) Q P Using inerse transformation 1, inerse transformation 3 and equations (42) and (37) the results of equation (50) become: a Dcrit 1 P 1a Dcrit 3 I4 4 x 1 Q Q P Finally, the matrix chain rule is used again to obtain the partial deriatie of the scalar a Dcrit in (48) (49) (50) (51) terms of the matrix A d a ec( P) Dcrit adcrit I1 x1 I4 x4 A ec( ) d Ad P Using inerse transformation 1, inerse transformation 3 and equations (49) and (37) the results of equation (52) become: adcrit 1 P 1a Dcrit 3 I4 x 4 1 Ad Ad P Equations (51) and (53) proide analytical expressions that can be calculated in real time, describing the behaior of a Dcrit in terms of each and eery one of the entries of the matrices Q and A d, which can be changed proided that they satisfy their restrictions. (52) (53) V. ADAPIVE LYAPUNOV CONROL SRAEGY By calculating the partial deriaties defined in equations (51) and (53), the entries of the matrices Q and A d, to which a Dcrit is the most sensitie are identified (those entries which hae the largest partial deriatie). Once these entries are identified, the one with the highest partial deriatie is selected, and slightly modified by a small alue (δ A =10-6 for A d, and δ Q =10-6 for Q). he sign of this modification is chosen such that it reduces the deriatie of a Dcrit, thus inducing a downward trend in the behaior of the critical alue for the magnitude of the differential acceleration. By reducing this critical alue the oerall robustness of the controller is improed. he adaptie ariations in the Q and A d are expressed as: 13

14 da ij a dq Dcrit ij a Dcrit A sign( ) A, Q sign( ) Q dt Aij dt Qij Where κ A and κ Q are defined as: adcrit adcrit adcrit adcrit 1 if for i, j k, l 1 if for i, j k, l A A ij A kl, Q Q ij Qkl 0 else 0 else hese were designed such that the modified Q and A d matrices still satisfy their requirements of positie definiteness for Q and for A d being Hurwitz. he adaptations of the Q and A d matrices also affect the panels actiation strategy since they cause ariations in P which is used for the panel actiation strategy (see equation (18)). In other words, the adaptations of the matrices Q and A d result in an adaptation of the quadratic Lyapuno function shown in equation (9). he adaptations are applied at the same time that the drag panels actiation strategy is applied, that is eery 10 minutes. he control strategy is summarized in Figure 2. (54) (55) Obtain state ector from SK Propagate in SK for 10 minutes Calculate matrix deriaties Run panel actiation strategy Adapt matrices Q, and A d VI. NUMERICAL SIMULAIONS Figure 2. Control Strategy diagram. he proposed technique was alidated using computer numerical simulations. he initial orbital elements of the target and other parameters for the numerical simulations are shown in able 1. he target and chaser spacecraft are assumed to be identical, therefore drag coefficient and frontal areas for all panel configurations are the same. he initial relatie position and elocity of the chaser in the LVLH frame are shown in able 2 (the same initial state was used in preious work in references 7 and 15). 14

15 able 1. Spacecraft Parameters Parameter Value Second zonal harmonic J E-03(from reference 16) Radius of the Earth R (km) Graitational parameter μ (km 3 /sec 2 ) arget s inclination (deg) 98 arget s semi-major axis (km) 6778 arget s right ascension of the ascending node (deg) 262 arget s argument of perigee (deg) 30 arget s true anomaly (deg) 25 arget s eccentricity 0 s (km/sec) 7.68 m(kg) 10 S min (m 2 ) 0.5 S 0 (m 2 ) 1.3 S max (m 2 ) 2.5 C Dmin 1.5 C D0 2 C Dmax 2.5 able 2. Initial conditions in the LVLH frame Parameter Value x (km) -1 y (km) 2 x (km/sec) -8.40E-06 y (km/sec) -1.70E-04 An SK scenario with full graitational field model, ariable atmospheric density (using NRLMSISE-00 aailable in SK) and solar pressure radiation effects is used. SK s High- Precision Orbit Propagator (HPOP) is used for simulating the maneuers. he control strategy is implemented in MALAB, which interacts with the SK scenario using SK Connect commands. he nonlinear dynamics of the system in the inertial frame are propagated in SK. At each time step, SK sends the state ariables to MALAB where they are transformed into the LVLH frame. he partial deriaties of a Dcrit, in terms of the matrices Q, and A d are computed (see equations (51) and (53)), and the adaptation of these matrices is performed (see equation (54) ). his allows for the recalculation of the matrix P and the calculation of the control signal. his signal is fed into SK in the form of panel configurations for the spacecraft. o reduce the frequency of actuation and allow the drag forces enough time to change the orbits, the adaptie Lyapuno controller was actiated eery 10 minutes. Moreoer, the rendezous maneuer was assumed to be finalized when the inter spacecraft distance was below 10m. Noteworthy, both the adaptie Lyapuno approach here deeloped, as well as the non adaptie Lyapuno approach suggested earlier by the authors (7), are able to drie the spacecraft to relatie distances in the order of a few meters, with no propellant required, outperforming by orders of magnitude the results presented in reference

16 Simulations of the adaptie Lyapuno controller are compared with simulations of the non adaptie Lyapuno controller presented by the authors in reference 7. he results of the simulations are presented in Figure 3, Figure 4, Figure 5 and Figure 7, comparing the maneuer trajectories, control sequences, critical alues for the magnitude of the differential acceleration, and errors between the non-adaptie Lyapuno controller and the adaptie Lyapuno controller. Figure 3. Numerical simulation result maneuer trajectory in the x-y plane: (Left) Complete maneuer, (Right top, and bottom) Final Stages of the maneuer. Figure 4. Numerical simulation result control signals: (op) non adaptie Lyapuno controller, (Bottom) Adaptie Lyapuno controller. 16

17 Figure 5. Numerical simulation result: Critical alue for the magnitude of the differential acceleration (a Dcrit ) oer the entire maneuer. Figure 6. Numerical simulation result: Comparison between the critical alue for the magnitude of the differential acceleration (a Dcrit ) and the actual alue of the differential acceleration for the adaptie controller. Figure 7. Numerical simulation result error oer the entire maneuer: (Left top) x error, (Left bottom) y error, (Right) Normalized error. 17

18 he adaptie Lyapuno controller was able to reach the rendezous state after 29hr, which represent a reduction on the maneuer duration of 24% oer the non adaptie Lyapuno controller (which took 38hr). Furthermore, the adaptie Lyapuno controller required 56 changes in the panels configurations, which signifies a reduction on the control effort of 50% oer the non adaptie Lyapuno controller (which needed 113). he aerage alue for the critical alue for the magnitude of the differential acceleration (shown in Figure 5) for the adaptie Lyapuno controller was 4.43*10-5 m/sec 2, which means a reduction on this alue of 39% oer the non adaptie Lyapuno controller (which had 7.36*10-5 m/sec 2 ). It is worth underlying that a reduction in the critical alue implies increased margin on control authority during the maneuer. he non adaptie Lyapuno controller needs more time and a higher control effort since it approaches the rendezous state performing more persistent and higher oscillations (these oscillation can be seen in Figure 7 after six hours hae elapsed, and Figure 3 Right). he reduction on the maneuer time and the control effort is caused by the adaptation of the matrix P which allows the adaptie Lyapuno control to tune itself as the error eoles; this results in lower oscillations on the errors behaior (see Figure 7) and in the trajectory itself (see Figure 3 Right) in the later stages of the maneuer, which is when smoother/finer action is required. Figure 6 Right also shows an increase of the error during the first portion of the manueer, which also means an increase of the Lyapuno function, before the system can finally drie itself towards the zero error desired state. his is due to two main factors. he first reason is that the control is allowed to change state only eery 10 minutes, ignoring required changes in the panels configurations within that time frame. his was imposed to enable realistic simulations, and remoe the possibility of chattering. In addition, despite the aerage reduction of the differential drag critical alue obtained with the adaptie Lyapuno controller, there are still interals where the actual differential drag acceleration is lower than the critical acceleration (see Figure 6). Oerall, the proposed approach still enables rendezous maneuers that are realistic from the actuation point of iew, in terms of duration, and it holds a potential for straightforward implementation on real spacecraft. VII. CONCLUSIONS In this work a noel adaptie Lyapuno controller for spacecraft autonomous rendezous maneuers using atmospheric differential drag is presented. An analytical expression for the critical alue for the magnitude of the differential acceleration that ensures stability in the sense of Lyapuno for the system is found. Based on this, analytical expressions for the partial deriaties of the critical alue with respect to the independent ariable matrices required by the Lyapuno controller are deried. hese partial deriaties are used for the deelopment of the adaptation strategy for the Lyapuno function. he quadratic Lyapuno function is modified in real time, during flight, minimizing the alue of the differential drag critical acceleration, thus maximizing the control authority margin. he adaptie control method is alidated using numerical simulations in Satellite ool Kit. he performance of the adaptie control method is assessed in terms of the number of switches in the differential acceleration (control effort), and maneuer duration in comparison with a non adaptie Lyapuno controller. he resulting behaior of the adaptie Lyapuno controller is an improement oer the non adaptie Lyapuno controller since it presents a significantly lower control effort (50% less actuation) and it takes less time to reach the desired rendezous state (24% less time). he implementation of the reference model on the adaptie Lyapuno controller (in this work the system tracked a constant final state) is expected to allow the method to track a desired path and 18

19 not only a constant state, opening to the possibility of potentially performing any type of spacecraft relatie maneuer using atmospheric differential drag. ACKNOWLEDGMENS he authors wish to acknowledge the United States Air Force Office of Scientific Research (AFOSR) for sponsoring this inestigation under the Young Inestigator Program. he authors would also like to thank Professor Fengyan Li from Rensselaer Polytechnic Institute for her aluable suggestions on the mathematical deelopments presented in this paper. 19

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