Space Debris Removal using a Tether: A Model

Texte intégral

(1).

(2)

(3)

(4)

(5)

(6)

(7)

(8) . an author's

(9)

(10)

(11) https://oatao.univ-toulouse.fr/19592

(12)

(13) . http://dx.doi.org/10.1016/j.ifacol.2017.08.1373.

(14) . Velez, Pedro and Atie, Tala and Alazard, Daniel and Cumer, Christelle Space Debris Removal using a Tether: A Model.. (2017) In: IFAC World Congress 2017, 9 July 2017 - 14 July 2017 (Toulouse, France). .

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23) .

(24) Proceedings of the 20th World Congress The International Federation ofCongress Automatic Control Proceedings of 20th Proceedings of the the 20th World World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Congress Toulouse, France, July 2017 The International Federation of Automatic The International Federation of Automatic Control Control Toulouse, France,Federation July 9-14, 2017 The International of Automatic Control Toulouse, France, Toulouse, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017. Space Debris Removal using Tether: Space Debris Debris Removal Removal using using aa a Tether: Tether: Space A Model Space Debris Removal A Model Modelusing a Tether: A A Model ∗ ∗∗ ∗∗∗. G Gcc G G Gccttc G Pctt G ct G PP P cc t P t PP R tc tGc R P c R RtGG Gcc R G c R t RG t R RGG Gtt R G t R RO R ROO O R InO I0IInnnn×m n×m n 000Isn×m n×m n×m s0ssdX dX sdX dX dt R dX dt R dt dt R R dt [.] R R [.] R [.] [.]R [.]R R a A ,, V a VA A aaAA,, V V AA A a , A ττAB V A AB ττAB τAB AB. (∗AB) (∗AB) (∗AB) (∗AB) (∗AB). Pedro Velez Tala Atie ∗∗ , Daniel Alazard ∗∗∗ , Pedro Velez Velez∗∗∗,,, Tala Tala Atie Atie∗∗ Daniel Alazard∗∗∗ ∗∗∗∗ Alazard ∗∗,, Daniel ∗∗∗ , Pedro Cumer Pedro Velez Christelle , Tala Atie ∗∗∗∗ Alazard ∗∗∗,, ∗ ∗∗ , Daniel Christelle Cumer ∗∗∗∗ Pedro VelezChristelle , Tala Atie , Daniel Alazard , ∗∗∗∗ Christelle Cumer Cumer Christelle Cumer ∗∗∗∗ ∗ ∗ MSc. Student, ISAE-SUPAERO, 10 Avenue Edouard Belin, 31055 ∗∗ MSc. Student, ISAE-SUPAERO, 10 Avenue Edouard Belin, 31055 MSc. Student, ISAE-SUPAERO, 10pedro.garcia@isae.fr) Avenue Toulouse, France (e-mail: ISAE-SUPAERO, Avenue Edouard Edouard Belin, Belin, 31055 31055 ∗ MSc. Student, Toulouse, France (e-mail: (e-mail:10 pedro.garcia@isae.fr) MSc. ISAE-SUPAERO, 10 Avenue Edouard Belin, 31055 ∗∗ Student, Toulouse, France pedro.garcia@isae.fr) Student, ISAE-SUPAERO (e-mail: tala.atie@isae.fr) Toulouse, France (e-mail: pedro.garcia@isae.fr) ∗∗ MSc. MSc.Toulouse, Student, ISAE-SUPAERO ISAE-SUPAERO (e-mail: tala.atie@isae.fr) tala.atie@isae.fr) ∗∗ France (e-mail: pedro.garcia@isae.fr) ∗∗∗ ∗∗ MSc. Student, (e-mail: Professor, (e-mail: MSc. Student,ISAE-SUPAERO ISAE-SUPAERO (e-mail:alazard@isae.fr) tala.atie@isae.fr) ∗∗ ∗∗∗ Professor, ISAE-SUPAERO (e-mail: alazard@isae.fr) MSc. Student, ISAE-SUPAERO (e-mail: tala.atie@isae.fr) ∗∗∗∗ ∗∗∗ ∗∗∗ Professor, ISAE-SUPAERO (e-mail: alazard@isae.fr) Research Scientist, ONERA, 2 Avenue Edouard Belin, 31055 Professor, ISAE-SUPAERO (e-mail: alazard@isae.fr) ∗∗∗∗ ∗∗∗ Research Scientist, ONERA, 2 Avenue Edouard Belin, 31055 31055 ∗∗∗∗ Professor, ISAE-SUPAERO (e-mail: alazard@isae.fr) ∗∗∗∗ Research Scientist, ONERA, 2 Avenue Edouard Belin, Toulouse, France (e-mail: christelle.cumer@onera.fr) Research Scientist, ONERA, 2 Avenue Edouard Belin, ∗∗∗∗ Toulouse, France (e-mail: christelle.cumer@onera.fr) 31055 Research Scientist, ONERA, 2 Avenue Edouard Belin, 31055 Toulouse, France Toulouse, France (e-mail: (e-mail: christelle.cumer@onera.fr) christelle.cumer@onera.fr) Toulouse, France (e-mail: christelle.cumer@onera.fr) Abstract: Focused on the removal of space debris, this paper studies the modeling of target Abstract: Focused Focused on on the the removal removal of of space space debris, debris, this this paper paper studies studies the the modeling modeling of of aaa target target Abstract: satellite connected to a chaser satellite by a tether. All dynamic couplings between the flexible Abstract: Focused on the removal of space debris, this paper studies the modeling of a target satellite connected to a chaser satellite by a tether. All dynamic couplings between the flexible Abstract: Focused on the removal of space debris, this paper studies the modeling of a target satellite connected tothe aa chaser satellite by AllThe dynamic couplings between the and rigid modes of satellites are accounted for. tether attached to both satellites is satellite chaser satellite by aa tether. tether. dynamic between the flexible flexible and rigid rigidconnected modes of ofto the satellites are accounted accounted for.All The tether couplings attached to to both both satellites satellites is satellite connected to aspring chaser satellite by a tether. All dynamic couplings between the flexible and modes the satellites are for. The tether attached is modeled as a massless when stretched and non-existent when compressed. The objective and rigid modes of the satellites are accounted for. The tether attached to both satellites is modeled as a massless spring when stretched and non-existent when compressed. The objective and rigid modes of the satellites are accounted for. The tether attached to both satellites is modeled as massless spring when stretched and non-existent when The objective of this work to model the behaviour of the three bodies: chaser, tether, and target in the modeled as aa is massless spring stretched and non-existent when compressed. compressed. objective of this this work work is to model model the when behaviour of the the three bodies: chaser, chaser, tether, and and The target in the the modeled as ais massless spring when stretched and non-existent whensatellite. compressed. The objective of to the behaviour of three bodies: tether, target in orbital reference frame for arbitrary initial conditions of the target Simulations of of this work is to model the behaviour of the three bodies: chaser, tether, and target in the orbital reference frame for arbitrary initial conditions of the target satellite. Simulations of of this work is to model the behaviour of the three bodies: chaser, tether, and target in the orbital reference frame arbitrary initial ofControl the satellite. Simulations of whole system including the chaser Attitude and Orbit System (AOCS) are performed orbital reference frame for for initial conditions conditions the target target satellite. Simulations of the the whole system system including thearbitrary chaser Attitude Attitude and Orbit Orbitof Control System (AOCS) are performed performed orbital reference frametofor arbitrary initialtumbling conditions ofControl theand target satellite. Simulations of the whole including the chaser and System (AOCS) are to evaluate its ability damp the debris motion to tow the debris. whole system including the chaser Attitude and Orbit Control System (AOCS) are performed to evaluate its ability ability to the damp the debris tumbling motion and System to tow the the debris. whole system chaser Attitude and Orbit Control (AOCS) to its the tumbling motion and debris. to evaluate evaluate itsincluding ability to to damp damp the debris debris tumbling motion and to to tow tow the debris.are performed to evaluate its ability to damp the debris tumbling motion and to tow the debris. Keywords: Keywords: Satellite, Satellite, Space, Space, Debris, Debris, Tether, Tether, Modeling, Modeling, Attitude, Attitude, Control. Control. Keywords: Keywords: Satellite, Satellite, Space, Space, Debris, Debris, Tether, Tether, Modeling, Modeling, Attitude, Attitude, Control. Control. Keywords: Satellite, Space, Debris, Tether, Modeling, Attitude, Control. 1. INTRODUCTION NOMENCLATURE 1. INTRODUCTION INTRODUCTION NOMENCLATURE NOMENCLATURE 1. NOMENCLATURE 1. INTRODUCTION NOMENCLATURE 1. INTRODUCTION In order to maintain future for satellite activity, it :: Center of mass of the chaser satellite. In order order to to maintain maintain aaa future future for for satellite satellite activity, activity, it it Center of mass of the chaser satellite. In is imperative to preserve space environment of Earth. In order to maintain a future for satellite activity, it :: Center of mass of the chaser satellite. target Center of mass of the chaser satellite. is imperative to preserve space environment of Earth. :: Center Center of of mass mass of of the the target target satellite. In order to maintain a future for satellite activity, it chaser is imperative to preserve space environment of As more debris been accumulating in space from is imperative to have preserve space environment of Earth. Earth. satellite. Tether connection point with the chaser. : Center of mass of the target satellite. As more debris have been accumulating in space from Tether connection connection point with the chaser. is imperative to preserve space environment of Earth. ::: Tether Center of mass of the target satellite. As more have accumulating in from dysfunctional satellites and pieces of destroyed satellites, As more debris debris have been been accumulating in space space from target. Tether connection point point with with the the chaser. chaser. dysfunctional satellites and pieces pieces of destroyed destroyed satellites, target. As more debris have have been accumulating in space from Tether connection point with the chaser. dysfunctional satellites and of satellites, functioning satellites become endangered by the dysfunctional satellites and pieces of destroyed satellites, ::: Tether connection point with the target. Body reference frame attached to the Tether connection point with the target. functioning satellites satellites have becomeof endangered endangered by the the :: Body Body reference frame attached to the dysfunctional satellites and pieces destroyed satellites, Tether connection point with the target. functioning have become by threat of collision imposed by the debris clouds (Johnson, functioning satellites have become endangered by the reference frame attached to the satellite, with origin at point G . : chaser Body reference frame attached to the c threat of collision imposed by the debris clouds (Johnson, chaser satellite, with origin at point G . functioning satellites have become endangered by the c :: chaser Body reference frame attached to the threat of collision imposed by the debris clouds (Johnson, 2010). This was the case in 2009 when an active American threat of collision imposed by the debris clouds (Johnson, satellite, with origin at point G . Body reference attached to the chaser satellite, frame with origin at point Gcctar. 2010). This was the case in 2009 when an active American :: Body Body reference reference frame attached to the tarthreat of collision imposed by the debris clouds (Johnson, chaser satellite, with origin at point G . 2010). This was case when an ctarsatellite (Iridium) with a dysfunctional Russian 2010). This was the thecollided case in in 2009 2009 an active active American American frame attached toGthe satellite, with origin at point : get Body reference frame attached t .. tarsatellite (Iridium) collided with when dysfunctional Russian get satellite, with origin at point pointto Gthe 2010). This was the case in 2010). 2009 when an active American t :: get Body reference frame attached toorigin the tarsatellite (Iridium) collided with aaaThus dysfunctional Russian military satellite (Ansdell, has risen the need satellite (Iridium) collided with dysfunctional Russian satellite, with origin at G Orbital reference frame, with at get satellite, with origin at point G tt.. military satellite (Ansdell, 2010). Thus has risen the need :: Orbital Orbital reference frame, with origin at satellite (Iridium) collided with a dysfunctional Russian satellite, with origin atwith pointorigin Gt . at military satellite (Ansdell, 2010). Thus has risen the need to remove debris from the vicinity of Earth. Among many military satellite (Ansdell, 2010). Thus has risen the need reference frame, point O. : get Orbital reference frame, with origin at to remove debris from the vicinity of Earth. Among many point O. military satellite (Ansdell, 2010). Thus has risen the need :: point Orbital reference frame, with origin at to remove debris from the vicinity of Earth. Among many studies, methods and developments in that field, one way to remove debris from the vicinity of Earth. Among many O. Identity of dimension n × n. point O. matrix studies, methods and developments in that field, one way :: Identity Identity matrix of dimension n × n. to remove debris from the vicinity of Earth. Among many point O. studies, methods and developments in that field, one way to deal with this problem is through a chaser towing the studies, methods and developments in that field, one way matrix of dimension nnm. × n. Null matrix of dimension n × :: Identity matrix of dimension × n. to deal with this problem is through a chaser towing the Null matrix of dimension n × m. studies, methods and developments in(Fig. field, onefocus way Identity matrix of dimension n × n. to deal with this problem is through aathat chaser towing the debris a tether into re-entry 1). The to deal with this problem is through chaser towing the :: Null matrix of dimension nn × m. Laplace variable. Null matrix of dimension × m. debris with a tether into re-entry (Fig. 1). The focus :: Laplace Laplace variable. to deal with this problem is through a chaser towing the matrix of dimension n × m. debris with aa is tether into re-entry (Fig. 1). The focus of this paper to model the system of two satellites debris with tether into re-entry (Fig. 1). The focus variable. : Null Laplace variable. of this paper is to model the system of two satellites debris with a tether into re-entry (Fig. 1). The focus :: Laplace variable. Time-derivation of vector X in frame R. of this istether to the of satellites connected by as accurately as possible. work of this paper paper to model model the system system of two twoThis satellites Time-derivation of of vector X X in frame frame R. connected by aaa is tether as accurately accurately as possible. possible. This work of this paper is to the model the system of two satellites :: Time-derivation Time-derivation of vector vector X in in frame R. R. connected by tether as as This work can be compared to past work of (Jasper et al., 2012), connected by a tether as accurately as possible. This work : Time-derivation of vector X in frame R. can be compared to the past work of (Jasper et al., 2012), connected by a tether as accurately as possible. This work can be compared to the past work of (Jasper et al., 2012), who considered chaser and target as point masses, and of can be compared to the past work of (Jasper et al., 2012), :: Matrix or vector projected in frame R. who considered chaser and target as point masses, and of can be compared to the past work of (Jasper et al., 2012), Matrix or or vector projected projected in frame frame R. who considered chaser and target as point masses, and of (Aslanov and Yudintsev, 2015), who modeled the chaser who considered chaser and target as point masses, and of :: Matrix Matrix or vector vector projected in in frame R. R. (Aslanov and Yudintsev, 2015), who modeled the chaser who considered chaser and target as point masses, and of (Aslanov and Yudintsev, 2015), who modeled the chaser :: Matrix or vector projected in frame R. as a rigid point mass. Their models are a special case (Aslanov and Yudintsev, 2015), who modeled the chaser inertial acceleration and velocity of point as a rigid rigidand point mass. Their Their models are aa special special case (Aslanov Yudintsev, 2015), who modeled the chaser :: inertial inertial acceleration acceleration and and velocity velocity of of point point as a point mass. models are case of the model of this paper which takes into any a rigid point mass. Their models a account special case : A. inertial acceleration and velocity of point as of the the model of this this paper which takesare into account any as a of rigid point mass. Their models are a account specialpoint case A. of model of paper which takes into any :: A. inertial acceleration and velocity of point kind flexible modes. Furthermore, the connection of the model of this paper which takes into account any Kinematic model between the point A and A. kind of flexible modes. Furthermore, the connection point of the model of this paper which takes into account any Kinematic model model between between the the point point A A and and kind of modes. Furthermore, the point A. between the chaser (respectively target) and the tether is kind of flexible flexible modes. Furthermore, the connection connection point ::: Kinematic the point B. Kinematic model between the point A and between the chaser (respectively target) and the tether is kind of flexible modes. Furthermore, the connection point the point B. between the chaser (respectively target) and the tether is : the Kinematic model between the point A and not located at the center of mass of the satellites, and between the chaser (respectively target) and the tether is B. the point point B. not located at the center of mass of the satellites, and between the chaser (respectively target) and the tether is not located at the center of mass of the satellites, and the point B. will thus create couplings between the translational not located at the center of mass of the satellites, and will thus thus create create couplings between the translational not located at the center of mass of the satellites, and II33 (∗AB) will couplings between the translational and (∗AB) rotational motions. This requires modeling the tether’s ττAB = will thus create couplings between modeling the translational and rotational motions. This requires requires the tether’s tether’s = 0I3×3 II3 I33 (∗AB) (∗AB) AB = will thusatcreate couplings betweenmodeling the translational and rotational motions. This the ττAB 03×3 motion the tether’s tips. rotational motions. This requires modeling the tether’s 3 I (∗AB) AB = 0 3 I motion at the tether’s tips. I33 rotational motions. This requires modeling the tether’s τAB = 03×3 3×3 motion motion at at the the tether’s tether’s tips. tips. 03×3 I3 A toolbox to model linear behavior of the couplings :: Antisymmetric matrix associated to AB. motion at the tether’sthe tips. A toolbox to model the linear behavior behavior of of the the couplings couplings Antisymmetric matrix associated to AB. T A toolbox to model the linear :: Antisymmetric matrix associated to AB. between the rigid and flexible modes of a satellite was If AB = [x, y, z] , then A toolbox to model the linear behavior of the couplings Antisymmetric matrix associated to AB. T R , then between the rigid and flexible modes of a satellite was If AB = [x, y, z] TT A toolbox to model the linear behavior of the couplings : If Antisymmetric matrix associated to AB. R between the rigid and flexible modes of a satellite was y, z] , then established in (Alazard et al., 2008). It takes as input between the rigid and flexible modes of a satellite wasa If AB AB = = [x, [x, y, z] , then R R T established in (Alazard et al., 2008). It takes as input between thein rigid andand flexible modes oftakes a as satellite was If AB = [x, y, z]−z R , then established (Alazard et al., 2008). It as input aa vector of applied forces torques and gives output the 0 y established in (Alazard et al., 2008). It takes as input a 0 −z y vector of applied forces and torques and gives as output the established in (Alazard et al., 2008). It takes as input a vector of applied forces and torques and gives as output the 0z0 −z yy translational and rotational accelerations of the satellite. 0 −x vector of applied forces and torques and gives as output the (∗AB) = −z translational and rotational accelerations of the satellite. z 0 −x (∗AB) = vector of applied forces and torques and gives as output the 0 −z y translational and accelerations ofsatellites. the zz 0x (∗AB) This paper adopts that model for the The 0 R translational and rotational rotational accelerations the satellite. satellite. 0 −x −x (∗AB) = = −y This paper adopts adopts that model model for the the of satellites. The −y translational and rotational accelerations of the satellite. z xx 0 −x (∗AB) = −y R This paper that for satellites. The 00 This paper adopts that model for the satellites. The −y x 0 R R This paper adopts that model for the satellites. The −y x 0 R. Copyright © 2017 IFAC Copyright © 2017 IFAC Copyright Copyright © © 2017 2017 IFAC IFAC Copyright © 2017 IFAC 10.1016/j.ifacol.2017.08.1373. 7518 7518 7518 7518 7518.

(25) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. . aG ω˙. . RG. −1 = Dsat (s) G. . F ext T ext,G. . (1) RG. Although very practical and easy to use, one must remember that the SDT only provides a linear model. Frame Transformation: The target body frame (RGt ) and the chaser body frame (RGc ) are required since the SDT uses inputs and outputs in these frames, while the orbital frame (RO ) will be used as a global reference in order to compare the positions and obtain the dynamics of the tether. The frame transformations are the usual Euler angles transformations 3-2-1, using the attitude angles φ, θ and ψ of each satellite. TRO →RG denotes the frame transformation from the orbital frame to the body frame. Fig. 1. Chaser and target satellites connected by a tether tether is modeled as an elastic, when stretched. It then combines the dynamics of the three bodies to determine the behavior of the entire system in response to an applied force or torque at the chaser. The paper models the linear behaviour of each flexible satellite since the non-linear terms are negligible for small satellite angular velocity. However, it accounts the non-linear coriolis and centrifugal terms in the double integration of the acceleration at the connection points of the tether and satellites. This gives the true attitude of the satellites and the exact motion of points Pc and Pt at the tips of the tether. Three frames of reference are needed to correctly model this system: • two body reference frames: RGc and RGt , the first one attached to the chaser, the other one to the target. Their origins are the centers of mass of the hubs (Gc and Gt ) of the corresponding satellites. • one orbital reference frame: RO , in relation to which all other body frames, positions and velocities will be defined. This reference frame is assumed to be inertial in this study. In the sequel, subscripts c and t will be omitted for general formulae, which are valid for both satellites. 2. MODELING 2.1 Satellite Dynamics Toolbox (SDT). Translation of the Model: In (Alazard et al., 2008) it was demonstrated that the linear dynamics model of the whole system can be modeled by the feedbacks of the direct dynamic models of each appendage on the inverse dynamic model of the hub. The whole model can then be built using substructuring techniques and is appropriated for parametric sensitivity analysis (Guy et al., 2014). A multi-body modeling approach was chosen and, therefore, in order to add the dynamics of several parts together, they all need to be modeled point. Thus the at the same I3 (∗P G) kinematic model τP G = allows the dynamic 0 I3 model to be translated from point G (or any other point) to P in the same reference frame. It neglects all non-linear terms. Since the tether applies a force on each satellite at its connection point (P ) and yet each satellite is modeled at the center of mass of the hub (G), τP G is used to transport the wrench applied by the tether at point P to point G : F ext F ext = τPT G (2) T ext,G T ext,P In the SDT, τP G is also used to transport the acceleration twist from G to P : aP a = τP G G (3) ω˙ ω˙ but non-linear terms in the computation of accelerations are neglected. 2.2 Connection of Satellites. In order to model the connected system the satellite models are obtained by the Satellite Dynamics Toolbox. This toolbox is explained in (Alazard et al., 2008) and can be found in (Alazard and Cumer, 2014). The SDT assumes the satellite to be formed of 2 kinds of components: a rigid body (the hub), and flexible (or rigid) appendages (could be 1 or more or even none). The satellite dynamics results in the couplings of flexible modes coming from the appendage(s) and rigid modes coming from the hub. From geometrical and mechanical data of the satellite, the toolbox gives a minimal state-space representation of the overall dynamic model at the center of mass G and projected in the body frame. The 6x1 input vector of this model is composed by external forces and torques T (wrench vector) F Text T Text,G R and the 6x1 output G vector is composed by the linear and angular accelerations T (derivative of the twist vector) aTG ω˙ T R : G. Tether Model: As the model is about studying the interaction between the two satellites, it is crucial to conveniently model the tether connecting them and its dynamics. The tether is cylindrical and treated as a massless spring when it is extended and as non-existent otherwise. It applies a symmetrical force on each extremity, being each one connected to one of the satellites. The orientation of the two forces is along Pc Pt and these forces have opposite senses. The force delivered by the tether on the target is:  Pc Pt F tether = −k(|Pc Pt | − l0 ) |Pc Pt | , if |Pc Pt | > l0  F tether = 0, if |Pc Pt | ≤ l0 (4). where l0 is the natural length of the tether. The parameters influencing k will be the natural length (l0 ), the area. 7519.

(26) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. (A)/diameter (d) of the cross section and the Young Modulus (E) of the material of the tether. The choice of the material and geometry of the tether is crucial because: • There is a maximum force that the tether could take without breaking or losing its elasticity. • As the control inputs of the two connected satellites are only located on the chaser, the tether must be conceived in order to transmit as quickly as possible the desired outputs on the target satellite. • The dynamics of the tether creates vibrations on the system. The interaction of the stiffness with the flexible modes of the satellites and its effects must be assessed. Inertial position and velocity of point Pc (resp. Pt ): As mentionned in the introduction, the tether model requires the exact motions of points Pc and Pt . Equation (3) is no more valid. From the inertial acceleration twist at point Gc (resp. Gt ) projected in frame RGc (resp. RGt ), provided by the SDT, the objective is to develop a general procedure (a generic ”block”) to obtain : • [OPc ]RO and [OPt ]RO , required by the tether model, • [OGc ]RO and [OGt ]RO , required to monitor the satellites motion, • [V Pc ]RO , [V Gc ]RO , [V Pt ]RO , [V Gt ]RO , • p, q and r, the three components of the angular velocity vector of each satellite (expressed in each satellite frame) and the attitude angles φ, θ and ψ of each satellite, • and TRO →RG .. This block is depicted in Fig. 2 and is based on the following kinematic equations: dV G dV G aG = = +ω×VG (5) dt RO dt RG [V G ]RG =. . dV G dt dt RG R. VP. dOP dOP = = + ω × OP dt R0 dt RG [OP ]RG =. . dOP dt dt RG R.   φ˙ p 1 sin φ tan θ cos φ tan θ  θ˙  = 0 q cos φ − sin φ 0 sin φ/ cos θ cos φ/ cos θ r ψ˙ f (φ,θ,ψ). G. O. 2. Translating the system from point P (connection with tether) to point G (center of mass of the platform) in using (2). 3. For the chaser, adding the external force Fcth and torque Tcrw,Gc , due to the attitude and orbit control system : thruster (th) and reaction wheels (rw). 4. Computing with the Satellite Dynamics Toolbox the acceleration vector due to the forces applied to each satellite (in the body frame) (see (1)). 5. Integrating the accelerations to obtain the position and attitude of each satellite (as explained in the previous section). 6. Using the position of points P of the two satellites, compute the difference between the position of the two connection points in order to compute the distance between them. [Pc Pt ]RO = [OPt ]RO − [OPc ]RO 7. Computing the force the tether applies (if stretched) see (4). 8. Using the force computed (see (4)) as input for the target and its symmetrical as input for the chaser and looping to step 1. 3. CLOSED-LOOP SIMULATION 3.1 Chaser Attitude Controller. (7). (8). (9). G. [OP ]RO = TTRO →RG [OP ]RG. 1. Transforming the tether force from the orbital frame (RO ) to the satellite body frame (RGc or RGt ). F tether F tether = TRO →RG 03×1 R 03×1 R. (6). G. V P = V G + ω × GP. Overview of the Connected Model: A complete scheme of the model used to represent the two satellites connected by a tether is presented in Fig. 3. In a nutshell, the flow of the model consists of (see corresponding numbers in Fig. 3):. (10). (11). Controllers will now be added to control the behavior of the system. In a first phase, an attitude controller for the chaser satellite will be designed to control the behavior of the overall system. The target satellite is assumed to be passive and non cooperative, while the chaser is fitted with an active attitude and orbit control system (reaction wheels, thruster). For this attitude control, a decentralized 3-axes PD controller was chosen, where Kp will be the 3 component vector (Kpx , Kpy , Kpz ) containing the proportional gains and Kv the 3 component vector (Kvx , Kvy , Kvz ) containing the derivative gains. The tuning of these parameters will be done assuming a rigid body dynamics and a dynamic decoupling between axes. Later, in the event of a flexible chaser satellite, a finer tuning can be done resorting to simulations. For this section only, we will also assume the linearization: ˙ θ, ˙ ψ). ˙ Under these assumptions, the following (p, q, r) = (φ, decoupled transfer functions can be deduced:. 7520.

(27) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. [aG ]RG. . + −. dV G dt |RG. . . I.C.. RG. 1 s. [V G ]RG. [V P ]RG. + +. + −. dOP dt |RG. . I.C.. RG. 1 s. [OP ]RG. +. TTRO →RG. −. I.C.. ˙ RG [ω]. [ω]RG =. 1 s. . p q r. . [GP ]RG f (φ, θ, ψ). . TTRO →RG. ˙ φ θ˙ ˙ ψ. . . I.C.. 1 s. φ θ ψ. . [OP ]RO [OG]RO. TRO →RG. Fig. 2. Acceleration integration block. (1). TRO →RGt. (4). (2). [τ TPt Gt ]RGt [F tether ]RO. (8). [DtGt ]−1 (s). (2). TRO →RGc . F cth. T crw,Gc. . [τ TPc Gc ]RGc. (3). − +. aG t ω˙ t. . (5) RGt. (7). (4). [DcGc ]−1 (s). . Target acceleration integration. [OP t ]RO. [P c P t ]RO. F tether (k, l0 ) see: Eq. (4). (1). . aG c ω˙ c. . (5) RGc. Chaser acceleration integration. [OGt ]RO. + −. [OP c ]RO. (6). [OGc ]RO φ θ ψ. . RGc. p q r. . Fig. 3. Model of target and chaser satellites connected by a tether.  Kpx     φ Ixx   =   Kv Kpx φ x  ref  s2 + s+   Ixx Ixx         Kpy    θ Iyy = Kv Kpy  θref y   s2 + s+   Iyy Iyy        Kpz      ψ Izz   =   ψref Kv Kpz z   s2 + s+ Izz Izz. (12).  Kpx = ω02 .Ixx      Kpy = ω02 .Iyy    Kpz = ω02 .Izz  Kvx = 2.ω0 .ξ.Ixx     Kvy = 2.ω0 .ξ.Iyy    Kvz = 2.ω0 .ξ.Izz. (14). To obtain a proper response, it was used, as a first reference, ξ = 0.7 and ω0 = 1rad/s. Such a tuning is very stiff for a satellite but realistic in an active debris removal scenario. Indeed, an objective of this study is to check if the tumbling motion of the target can be damped by the transmission from the target to the chaser via the tether and by the attitude control system of the chaser (except, of course, for the rotation around the tether axis).. where Ixx , Iyy and Izz are the diagonal inertia terms of the chaser at point Gc , in frame RGc .. In the next section such a controller will be applied to the whole model.. It is desired, for all rotations, a second order response characterized by the generic parameters ω0 and ξ:. 3.2 Simulation Scenario. φ φref. =. θ θref. =. ψ ω02 = 2 ψref s + 2ξω0 s + ω02. Thus, one can deduce the gains must be:. (13). In the next section the simulation results will be presented. The chaser satellite has the attitude controller described previously. The references for this controller were kept at zero during the whole simulation, i.e., the chaser was servolooped on its orbit. Also, the thrusters of the chaser were. 7521.

(28) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. Parameter l0 k mc Ixx Iyy Izz Gc Pc. . Gt Pt. RGc. RGt OGc (t = 0) RO OGt (t = 0) RO ωt. RO. (t = 0). value (SI) 10 7.85 107 100 10 10 20 [1, 0, 0]T [−5, 0, 0]T [0, 0, 0]T [10, 0, 0]T [0, 10π/180, 0]T. Table 1. Main model data and non-null initial conditions.. Fig. 4. Position of the two satellites in RO. turned on, as to create a negative force along the x-axis (in the chaser body frame), in order to tow the target satellite. The target satellite was set to have a tumbling motion, rotating along its y-axis. The chaser and the target were aligned along the x-axis in RO , with the target positioned in front of the chaser. The length of the cable was set to be initially unstretched. The tether will be tensioned due to the tumbling of the target and the backwards (along the xaxis in RO ) motion of the chaser. However the connecting points between the tether and the satellites were set at: [Gc Pc ]RG = [1, 0, 0]T and [Gt Pt ]RG = [−5, 0, 0]T . c. t. The chaser used here was a purely rigid body of mass mc , therefore having no flexible effects. It was also set to be an inertially symmetrical body, and thus to have null cross products of inertia. The target was set to be a spacecraft made of a rigid hub of mass 100kg with two flexible symmetric appendages, one of mass 50kg and the other 20kg. The appendages are placed geometrically symmetric with respect to the x-z plane in RGt . The center of mass of the whole target shifts from that of the hub, and nondiagonal inertia moments are expected for this satellite. The data relative to the target can be found in (Alazard and Cumer, 2014) in the tutorial entitled ” Example 1: Spacecraft1.m”. Other data and initial conditions are summarized in Table 1.. Fig. 5. Attitude of the chaser satellite in RO. 3.3 Simulation Figures 4, 5, 6, 8 show values in the orbital frame. Hereafter, all references to axes will be in the orbital frame (RO ). All units are according to the SI. The initial conditions place the two attachment points at a distance of 4m while the unstretched tether length is 10m. This explains why in Fig. 4 there is no motion of the debris in the x-axis for the first 12 seconds (detumbling starts at around 6 seconds). The distance between the attachment points of the satellites (Pt and Pc ) can be seen in Fig.7 It can be seen, in Fig. 8, the action of the tether was mostly done by ”impulses”, trying to tow and realign the target satellite when it goes off the desired path along the x-axis. It can also be seen, in Fig. 4 that there was a negative acceleration of the system on the x-axis as expected, meaning the towing was successful. However, it can also be seen some residual non-null position values on the yaxis and z-axis. This is due to the fact that the thrusters. Fig. 6. Attitude of the target satellite in RO. Fig. 7. Distance between the attachment points. 7522.

(29) Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017. 4. CONCLUSIONS AND PERSPECTIVES The objective of this study was to develop a modeling tool to study and design a tethered space tug during preliminary design. It models the non-linear behavior of (any) two satellites with possible flexible appendages, connected by a tether. Moreover the system is roughly controlled resorting to forces and torques applied on one of the satellites, enough to tow the other passive satellite. An important aspect of this simulator is that almost all the non-linear terms were accounted, while no simplifications were done when connecting the system. The only nonlinear terms missing are due to each of the satellites models, obtained by the Satellite Dynamics Toolbox (SDT). For future perspectives, further controllers can be designed in order to, for example, maintain a constant position or a constant movement in only one axis or even to try to obtain a constant distance between the two satellites. Since here the satellites are already assumed to be exactly on the same orbit, the Clohessy-Wiltshire equations can be added to improve the model, as they were developed to govern rendez-vous maneuvers (Gladun, 2005). Sensitivity analysis could be assessed to evaluate the interaction of the tether’s stiffness with the flexible modes of the satellite and to suggest recommendations on the tether’s stiffness.. Fig. 8. Force applied by the tether in RO. REFERENCES. Fig. 9. Vibrations seen on the target acceleration along y-axis in RGt , for E=100 MPa of the chaser are aligned with its own x-axis and not the orbital frame x-axis. Therefore, during the transient response of the AOCS to counteract the disturbances due to the tether, the thrusters will create an acceleration on the orbital y-axis and z-axis. In Fig. 5 it can be seen that the AOCS stabilizes the attitude of the chaser. In Fig. 6 it can be seen that the initial tumbling around the y-axis (θ increasing linearly) is counteracted on the first stretch of the tether. It can also be observed that the attitude of the target satellite is not entirely controlled. Around the y-axis and z-axis the attitude motion is bounded, since the tether will allow the target to deviate until it is tensioned. Around the x-axis (φ), however, the control has no action because the tether only applies tension forces and is not modeled to apply torsion torques. Furthermore, it can be stated that the stiffness of the tether is determinant on the amount of vibrations existent on the system. This result was previously showed in (Aslanov and Yudintsev, 2015). Tested for a modulus of Young of 100kPA, 100MPa and 100GPa, the results show much greater vibrations for the 100MPa scenario (Fig. 9).. Alazard, D. and Cumer, C. (2014). Satellite dynamics toolbox: Principles, user guide and tutorials. URL http://personnel.isae.fr/daniel-alazard/matlab -packages/satellite-dynamics-toolbox.html?lang =fr. Alazard, D., Cumer, C., and Tantawi, K. (2008). Linear dynamic modeling of spacecraft with various various flexible appendages and on-board angular momentums. In 7th International ESA Conference on Guidance, Navigation and Control Systems. ESA, Tralee, County Kerry, Ireland. Ansdell, M. (2010). Active space debris removal: Needs, implications, and recommendations for today’s geopolitical environment. Journal of Public and International Affairs. Aslanov, V. and Yudintsev, V. (2015). Dynamics, analytical solutions and choice of parameters for towed space debris with flexible appendages. Advances in Space Research, 55, 660–667. Das, R., Sen, S., and Dasgupta, S. (2007). Robust and fault tolerant controller for attitude control of a satellite launch vehicle. IET Control Theory Apply, 1(1). Gladun, S. (2005). Investigation of Close Proximity Operations of an Autonomous Robotic On-Orbit Servicer Using Linearized Orbital Mechanics. Master’s thesis, University of Florida. Guy, N., Alazard, D., Cumer, C., and Charbonnel, C. (2014). Dynamic modeling and analysis of spacecraft with variable tilt of flexible appendages. Journal of Dynamic Systems, Measurement, and Control, 136(2). Jasper, L.E.Z., Seubert, C.R., Schaub, H., Valery, T., and Yutkin, E. (2012). Tethered tug for large low earth orbit debris removal. AAS/AIAA Astrodynamics Specialists Conference. Johnson, N. (2010). Orbital Debris: The Growing Threat To Space Operations.. 7523.

(30)