Stability and Control of Radial Deployment of Electric Solar Wind Sail

Nonlinear Dyn (2021) 103:481–501

https://doi.org/10.1007/s11071-020-06067-7 (0123456789().,-volV)( 0123456789().,-volV)

ORIGINAL PAPER

Stability and control of radial deployment of electric solar wind sail

Gangqiang Li . Zheng H. Zhu . Chonggang Du

Received: 27 June 2020 / Accepted: 30 October 2020 / Published online: 5 January 2021 Ó Springer Nature B.V. 2021

Abstract The paper studies the stability and control deployment of tethers by using optimal control. of radial deployment of an electric solar wind sail with Finally, the analysis results show that radial deploy- the consideration of high-order modes of elastic ment is advantageous due to the isolated deployment tethers. The electric solar wind sail is modeled by mechanism, and a jammed tether can be isolated from combining the flexible tether dynamics, the rigid-body affecting the deployment of rest tethers. dynamics of central spacecraft, and the flexible-rigid kinematic coupling. The tether deployment process is Keywords Space tether Á Electric solar wind sail Á modeled by the nodal position finite element method Multibody dynamics Á Rigid-flexible coupling Á in the arbitrary Lagrangian–Eulerian framework. A Flexible structural stability Á Nodal position finite symplectic-type implicit Runge–Kutta integration is element method Á Arbitrary Lagrangian–Eulerian proposed to solve the resulting differential–algebraic equation. A proportional–derivative control strategy is applied to stabilize the central spacecraft’s attitudes to ensure tethers’ stable deployment with a constant 1 Introduction spinning rate. The results show the electric solar wind sail requires thrust at remote units in the tangential Electric solar wind sail (E-sail) is an innovative direction to counterbalance the Coriolis forces acting propellantless propulsion technology for interplane- on the tethers and remote units to deploy tethers tary exploration [1–4]. Typically, an E-sail consists of radially successfully. The parametric analysis shows a central spacecraft connected with multiple long and the tether deployment speed and the thrust magnitude thin conductive tethers in a hub-spoke like configura- significantly impacts deployment stability and tether tion [5]. The geometrical configuration is maintained libration, which opens the possibility of successful and stabilized by pulling tethers radially with the centrifugal forces resulting from spinning tethers around the central spacecraft. These spinning tethers & G. Li Á Z. H. Zhu ( ) are positively charged by the central spacecraft to form Department of Mechanical Engineering, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada an electrostatic field over a large circular area—the e-mail: [email protected] spin plane, which deflects the trajectory of incident protons in the solar wind to generate thrust [5–7]. C. Du Although the E-sail has been studied extensively Department of Earth and Space Science and Engineering, York University, 4700 Keele Street, Toronto, [2, 4, 8–10], less attention has been paid to the ON M3J 1P3, Canada 123 482 G. Li et al. dynamic process of tether deployment from the central freedom (DOF) rigid body, while the tethers are spacecraft. There are two deployment schemes pro- modeled with 2-noded bar elements with variable posed in Ref. [11] to deploy the tethers either lengths and zero compressive stiffness. Recently, three tangentially or radially. Each deployment different beam models of the tether for the E-sail have scheme has its pros and cons. For the tangential been tested based on the Abaqus software, and its deployment of the flexible tethers, the tethers are pre- impact on the transient response of the tether is wound up on the exterior of the central spacecraft and investigated [19]. In the current paper, all tethers are then deployed by spinning the central spacecraft in a treated as flexible elastic tensile members in the predefined spin trajectory. The deployment dynamics deployment process. Two approaches deal with the has been well studied in Refs. [12, 13]. It is noted that variation of tether length in discretized tether models the jamming of any one of the tethers will cause the [20, 21]. In the first approach, the total numbers of tether tangling leading to the failure of the deploy- elements in the model are kept constant to ease ment. In case of the radial deployment, each tether is programming and implementation. The tether deploy- stored in an individual spool. The tethers could be ment is represented by increasing the lengths of either either pulled out by the thruster at the remote units at all elements at the same rate [22] or selected elements at the end of tethers with an initial push by the spring different rates [23]. In the second approach, the total forces [14] or be pushed out by individually controlled numbers of elements in the model vary as the tethers are active spool and then pulled the tethers straight by deployed. As the tether length increases in the deploy- spinning the central spacecraft [11]. Substantial engi- ment process, the element connected to the central neering analysis has recently shown that the radial spacecraft increases. Once the length of this variable- deployment scheme has the advantage of low failure length element is longer than a preset element length, risk of tether tangling over the tangential deployment this element is divided into two elements: a variable- scheme. However, the Coriolis force, induced by length element and a new constant-length element either the orbital motion of E-sail or the spin of E-sail [24–26]. This process is achieved in the Arbitrary in the latter case of radial deployment, may cause the Lagrangian–Eulerian (ALE) framework [24, 25, 27]. tether librate about its equilibrium positions and The second approach is superior to the first approach for eventually tangle each other. Many control laws for the current study due to the easy implementation of the space tether deployment have been proposed to kinematic constraints of the connection relationship suppress the libration motion caused by the Coriolis between the central spacecraft and multiple tethers. force that is induced by the deployment velocity Recently, the authors proposed a nodal position finite [15–17]. Furthermore, the thruster in the remote unit at element method in the ALE description (NPFE- the tip of tether is proposed to deploy the tethers to M_ALE) to study the variable-length tether problem their equilibrium positions faster [18]. It is noted that in the space elevator with moving climbers [21, 28, 29]. the central spacecraft is assumed as a lumped mass Furthermore, the finite element model of variable- without the consideration of attitude dynamics. How- length tether embedded with the second approach is ever, for an E-sail system, studies suggest that the also used for the space tether systems [24, 27]. attitude motion of central spacecraft should be regu- However, it is found that the previous models cannot lated to make it rotating around its principal axis in a be directly applied in the current study due to the predefined trajectory [6, 11]. In the current work, we following limitations: the neglection of orbital motion assume the tethers are deployed by reeling them out of and gravity gradient along the tether [24], the neglec- the central spacecraft. The deployed tethers are kept tion of the transverse flexural motion of tether [27], and straight in the radial direction by spinning the central the use of an inappropriate time integration spacecraft. The reel and spin rates controlled individ- scheme leading to the violation of constraint conditions ually to ensure the tethers are deployed in the radial at the interface of central spacecraft and tether [25, 27]. direction safely. For the E-sail, the orbital motion of the central The novelty of current work is the dynamic model- spacecraft around the Sun is at the order of ing and characterization of the radial tether deployment 10À7 rad=s when the E-sail is at 1 Astronomical Unit process considering high-order modes of elastic tethers. (AU) from the Sun, which is at the same numerical The central spacecraft is modeled as a six-degree-of- 123 Stability and control of radial deployment of electric solar wind sail 483 order of the convergence tolerance of implicit solvers axis at a constant spinning rate. The central spacecraft employed in the program, such as the first-order is modeled as a 6-DOF rigid body with attitude Backward Euler and the second-order generalized dynamics, while the remote units located at the tip of alpha methods. This leads to the violation of constraint each tether are assumed as lumped masses without equations in the differential equations and generates attitude dynamics. The motion of the E-sail is unreliable results [27]. To address these mentioned described in two coordinate systems, the heliocen- limitations, the NPFEM_ALE model of the space tric–ecliptic inertial frame OgXgYgZg and the body- elevator [21, 25, 28, 29] is extended and applied to the fixed frame ObXbYbZb. The definition of the helio- tether deployment problem of E-sail. Furthermore, a centric–ecliptic inertial frame can be found in our framework of an implicit Runge–Kutta implicit time previous works [6, 30]. The origin of the body-fixed integration with s stages and a symplectic property for frame (ObXbYbZb) is located at the center of mass of solving the differential–algebraic equations is the central spacecraft with the ObZb axis pointing to developed. the normal direction of spinning plane of the E-sail, the ObYb axis pointing to the direction of the cross product of ObZb and OgZg, and the ObXb axis 2 Mathematic formulation of electric solar wind completing a right-handed frame. Also, two additional sail frames are introduced. One is to describe the libration motion of flexible tethers with the origin at each 2.1 Finite element discretization of flexible tethers tether’s anchor points on the central spacecraft. Another is used to apply the thrust conveniently with Consider the central spacecraft shown in Fig. 1 that the origin at the remote unit. Their definition will be deploy flexible tethers by rotating about its principal given in the following sections.

Fig. 1 Schematic diagram of radial deployment of E-sail tethers a coordinate deﬁnition, b radial deployment mechanism of an E-sail

123 484 G. Li et al.

The equations of motion (EOM) of the tethers are length of the kth element measured from the beginning derived from the generalized D’Alembert principle of each element, I is a diagonal identity matrix with [20, 29]. From the virtual work principle, the sum of 3 Â 3 dimension, Le is the stretched length, q and A are virtual work of all the applied and inertial forces on an the material density and cross-section area of element, arbitrary virtual displacement of the tethers should be r and e are the vectors of stress and strain, respec- zero at an arbitrary moment. The virtual work of the tively, f g;k is the vector of gravitational force per unit kth element is written as, : :: length. The overhead dots ðÞ and ðÞ denote the first- dWk ¼ dWe;k þ dWg;k þ dWi;k ¼ 0 ð1Þ and second-order time derivatives. Substituting Eqs. (2)–(7) into Eq. (1) yields, where d is the variational operator, and dWe;k, dWg;k, € and dWi;k denote the virtual work done by the elastic, Me;kXe;k ¼ Fe;k þ Fg;k À Fp;k ð8Þ gravitational, and inertial forces, respectively. Z Z with pkþ1 8 Z T T 1 dWe;k ¼À de rAkdp ¼À Akde rdp > L > p;k T pk > Me;k ¼ q AkN Ne;kdn > 2 k e;k dXT F 2 > Z À1 ¼À e;k e;k ð Þ > 1 o T > Lp;k e Z Z < Fe;k ¼ EkAke dn pkþ1 2 oXe;k T T ZÀ1 9 dW dX f A dp dX f A dp > 1 ð Þ g;k ¼ e;k g;k k ¼ e;k g;k k > Lp;k p > T k > Fg;k ¼ Ne;kf g;kdn ¼ dXT F > 2 ZÀ1 e;k g;k > 1 :> Lp;k T ð3Þ Fp;k ¼ qkAkN ap;kdn 2 e;k Z À1 ÀÁ T € where Me is the mass matrix of element, Fe and Fg are dWi;k ¼ dXe;k ÀqkAkX dp ¼ Z the force vectors of the elastic and gravitational forces, pkþ1 respectively, F is the force vector caused by the À dXT q A X€ dp ð4Þ p;t e;k k k mass transportation across the element boundaries due pk ÀÁ T to tether deployment, which vanishes when the ¼ÀdX Me;kX€e;k þ Fp;k e;k material coordinate of the tether is fixed [21, 28, 29, 31]. Lp;k ¼ pkþ1 À pk is the element length X€ ¼ Ne;kX€e;k þ ap;k ð5Þ due to variation of material coordinate. In the current 1 À nðÞt oNa;k 1 þ nðÞt oNa;k paper, the tether damping is not considered due to the Ne;k ¼ I; Xa;k; I; Xa;k lack of data in space [25]. 2 opk 2 opkþ1 For an E-sail containing m 2 tethers as shown in 1 À nðÞt 1 þ nðÞt and N ¼ I; I Fig. 1, the tether is divided into n elements with n ? 1 a;k 2 2 nodes. Thus, there are a total of m Â n tether elements. ð6Þ By assembling the EOM of all elements with the standard procedure in the finite element method oNa;k oNa;k ap;k ¼ 2 p_ þ p_ X_ a;k [32, 33], the EOMs of all tethers become, op k op kþ1 k kþ1 2 2 2 o Na k o Na k o Na k M X€ ¼ F þ F À F ð10Þ þ ; p_2 þ 2 ; p_ p_ þ ; p_2 X t t e g p o 2 k o o k kþ1 o 2 kþ1 a;k pk pk pkþ1 pkþ1 where Mt is the mass matrix of tethers with detailed ð7Þ form in Ref. [25], Xt is the position vector of tethers, T F F F where Xa;k ¼ ðÞXk; Yk; Zk; Xkþ1; Ykþ1; Zkþ1 and e, g, and p are the force vectors acting on tethers T [21, 28]. The lumped masses of the remote units are Xe;k ¼ ðÞXk; Yk; Zk; pk; Xkþ1; Ykþ1; Zkþ1; pkþ1 are the position vectors of element nodes and material point, added into the mass matrix Mt of the tethers in the corresponding places for the connecting nodes [25]. respectively, pjðÞj ¼ k; k þ 1 is the material coordinate with the subscript indicating the connecting node, nðÞ¼t s Le;kðÞÀ1 nðÞt 1 with s being the arc- 123 Stability and control of radial deployment of electric solar wind sail 485

2.2 Translation and attitude motions of the central b C1 ¼ Xs þ Tb2gðÞh Xanc;j À Xt;j ¼ 0ðj ¼ 1; ...; mÞ spacecraft ð14Þ 2 3 The central spacecraft is assumed as a 6 DOF rigid cwch cwshs/ À c/sw c/cw þ s/sw body. First, the EOM of translation dynamics of the 4 5 Tb2gðÞ¼h chsw c/cw þ shs/sw Àcws/ þ c/shsw central spacecraft is derived in the global inertial Àsh chs/ chc/ coordinate system by Newton’s second law directly, ð15Þ

MsX€s ¼ Fs;g þ Fs;p ð11Þ b where Xanc;j is the position vector of anchor points with the superscript b representing the body-fixed where Ms ¼ diagðÞms; ms; ms is the mass matrix of the frame and the subscript j ¼ ðÞ1 m representing the central spacecraft with ms being the mass of the central number index of the tether. They will be given in spacecraft, X ¼ ðÞX ; Y ; Z T is the position vector of s s s s numerical simulation section. T ðÞh is the transfor- the center of mass of the central spacecraft with b2g mation matrix of the central spacecraft from the body- overhead dot representing time derivative, while F s;g fixed frame O X Y Z to the global inertial frame and F are the external force vectors due to the b b b b s;p O X Y Z , X is the position vector of anchor point gravitational and other perturbative forces, respectively. g g g g t;j for the jth tether in the global inertial frame. The equation of attitude motion of the central spacecraft is described in the body-fixed coordinate 2.4 Constraint equations for tethers system ObXbYbZb via the Euler’s equations of motion [27], There are two types of tether nodes used in the h_ ¼ JðÞh x proposed model: the moving node and the normal ð12Þ finite element node. The moving node represents the Hsx_ þ x Â ðÞ¼Hsx Ms;g þ Ms;p þ Ms;c 0 1 tether deployment process [21, 29]. As shown in ch s/sh c/sh Fig. 1, a drum-type deployment mechanism is 1 JðÞ¼h @ 0c/ch Às/ch A ð13Þ assumed here, where two drums are used to control a ch 0s/ c/ tether’s deploying speed. It is assumed (1) there is no

T slippage between the drum and the tether, (2) the where h ¼ ðÞ/; h; w is the Euler angle vector with /, momentum of the rotating drum does not affect the h, and w denoting the roll, pitch, and yaw angle, attitudes of the central spacecraft, and (3) all tethers respectively. It is noted that the singularity condition, ÀÁ are deployed at the same speed. The tether deployment T h ¼ 90 , will never happens. x ¼ xx; xy; xz is the process is modeled as the following: (1) the nodes of angular velocity vector, Hs ¼ diagðÞIX; IY ; IZ is the tethers connecting to the central spacecraft are defined angular inertial matrixÀÁ of the central spacecraft with as the moving nodes, (2) the material coordinate p of 1 2 2 1 2 the moving node follows a prescribed trajectory or IXb ¼ IYb ¼ 12 ms 3rs þ h and IZb ¼ 2 msrs , rs and h are the radius and height of the cylindrical central deploying speed. The constraint equations for the spacecraft, respectively. Ms;g, Ms;p and Ms;t are the moving nodes are defined as, torque vectors due to the gravitational, perturbative, C2 ¼ pj À pj;pre ¼ 0 ð16Þ and control input forces, respectively. The symbols c and s in Eq. (13) represent the cosine and sine where pj;pre ¼ pj;ini þ p_j;preDt is the material coordi- functions. nate of a predefined trajectory with the subscript j representing the jth node. Here, for simplicity, the

2.3 Coupling constraint equations tether is deployed at a constant speed p_j;pre, where between the tether and central spacecraft p_j;pre\0 represents the tether deployment, and p_j;pre [ 0 represents the tether retrieval. The constraint equations due to the kinematic coupling For the self-content and remodeling purpose for the between the tethers and the central spacecraft are interested readers, the necessary process of dividing an written as, element is given here. The interested reader can ﬁnd

123 486 G. Li et al. detailed information from our previous works of the new variable-length element is L1 À Ls, and the [21, 25, 29]. As shown in Fig. 2a, where the E-sail is second new element is a constant-length element with connected with four tethers, the nodes of anchor points a standard length Ls. of the tethers are defined as the moving nodes, and the Moreover, the mass conservation at the central elements connected to the moving nodes at one end are spacecraft should satisfy the following equation to defined as the variable-length elements. The lengths of account for the loss or gain of mass by the deployment variable-length elements increase at the spacecraft of tether, such that, side as the tethers are deployed out from the central Xm spacecraft. Once the variable-length element’s length ms ¼ ms;init þ p_j;preAjqjDt ð18Þ exceeds a preset value, the element will be divided into j¼1 a variable-length element and a constant-length ele- where m is the mass of the central spacecraft before ment by adding one node. s;init the deployment, m represents the number of tethers In the deployment process, only the dividing of while A and q are the cross-section area and material elements happens. Two parameters, the standard L s density of tethers. and upper bound (L ) lengths are defined for this max The process of dividing of element happens simul- process. The variable-length element will be divided taneously for each tether when the deployed speed of into two elements if its length exceeds the upper bound each tether is the same. So the standard lengths of each L . For example, as shown in Fig. 2b, the first node max tether L ðÞj ¼ 1; ...; m are set slightly different L ¼ is a moving node representing the tether deployed out sj sj ½1 þ 0:01ðÞj À 1 L ðÞj ¼ 2; ...; m to avoid the from the central spacecraft in the arrow direction. If s1 phenomenon. the condition Except for the moving nodes, all other nodes of L1 ¼ p2 À p1 Lmax ð17Þ tethers are the normal nodes in the finite element method with constant material coordinates [24, 25]. is satisfied, then the variable-length element is divided The constraint equations of these normal nodes are, into two new elements by adding a new node between the first and second nodes. The position, velocity, and C3 ¼ pj À pj;ini ¼ 0 ð19Þ acceleration of the newly added node are obtained by where p is the material coordinate of jth normal linear interpolation. After that, the nodes and elements j;ini node with the subscript ‘‘ini’’ indicating the constant. after the second node must be renumbered. The length

Fig. 2 Illustration of dividing of tether elements 123 Stability and control of radial deployment of electric solar wind sail 487 8 2.5 Equation of motion of the E-sail > _ > Xs ¼ Vs > _ T > MsVs þ C k1 ¼ Fs;g þ Fs;p > 1;Xs The EOM of the E-sail is obtained by combining > _ > h ¼ JðÞh x Eqs. (8)–(19) by the Lagrangian multiplier method, > T <> Hsx_ þ x Â ðÞþHsx C1;hk1 ¼ Ms;g þ Ms;p þ Ms;c that is, _ 8 Xt ¼ Vt € T > P3 > MsXs þ C1;X k1 ¼ Fs;g þ Fs;p > _ T > s > MtVt þ Cl;X kl ¼ Fe þ Fg À Fp > _ > t > h ¼ JðÞh x > l¼1 > T > b > H x_ þ x Â HIðÞþx C k ¼ M þ M þ M > C1 ¼ Xs þ Tb2gðÞh Xanc;j À Xt;j ¼ 0 <> s s 1;h 1 s;g s;p s;c > P3 :> C2 ¼ pj À pj;pre ¼ 0 € T > MtXt þ Ci;X kk ¼ Fe þ Fg À Fp C3 ¼ pj À pj;ini ¼ 0 > t > k¼1 > C ¼ X þ T ðÞh Xb À X ¼ 0 ð21Þ > 1 s b2g anc;j t;j > C ¼ p À p ¼ 0 : 2 j j;pre where Vs and Vt represent the velocity vectors of the C3 ¼ pj À pj;ini ¼ 0 central spacecraft and tether, respectively. ð20Þ Second, define a new vector Z ¼ T T T where C ¼ ðÞoC1=oXs is the Jacobian matrix of ðÞXs; Vs; h; x; Xt; Vt to simplify Eq. (21) as, 1;Xs constraint equation with respect to the position vector AZðÞZ ¼ fZðÞ ð22Þ of the central spacecraft, the superscript T represents CZðÞ¼0 T the transpose of a matrix, C1;h is the Jacobian matrix of the constraint equation with respect to the attitude of where 0 1 the central spacecraft, CT k;Xs ¼ I00000 B C o o T B 0M 0000C ðÞCk= Xt ðÞk ¼ 1; 2; 3 is the Jacobian matrix of B s C B C constraint equations with respect to the position B 00I000C A ¼ B C and vectors of the tether, and k ðÞk ¼ 1; 2; 3 are the B 000H00C k B s C Lagrangian multiplier vectors. @ 0000I0A 8 00000Mt > > f 1ðÞ¼Z Vs 3 Time integration method > T > f ðÞ¼Z Fs;g þ Fs;p À C k1 > 2 1;Xs > <> f ðÞ¼Z JðÞh x The differential–algebraic equations in Eq. (20) are 3 f ðÞ¼Z M þ M þ M À x Â ðÞÀH x CT k solved numerically by an implicit s-stage Runge– > 4 s;g s;c s;p s 1;h 1 > > f 5ðÞ¼Z Vt Kutta integration scheme with symplectic property, > > P3 which ensures the numerical model’s stability or :> f Z F F F CT k 6ðÞ¼ e þ g À p À l;Xt l energy conservation for the long-term integration l¼1 process [20]. However, Ref. [20] applied the implicit ð23Þ s-stage Runge–Kutta integration scheme to solve the It should be pointed out that the matrix A in Eq. (23) differential equations of tethers without constraint is a singular matrix due to the rank-deficient submatrix equations and requires the inverse of the tether’s mass M [28]. Thus, the matrix A is not invertible in a matrix. With the introduction of material coordinate p, t traditional way [20], and the implicit s-stage Runge– the mass matrix of tethers is rank-deficient and not Kutta integrator is used, that is, invertible. Thus, a Newton–Raphson iteration algorithm is used to solve these algebraic equations [32]. The scheme is explained as follows. First, the second-order differential equations in Eq. (20) is reduced to the first-order differential equations, that is,

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8 ! ! > Ps Ps > n n n n >g1 ¼A t þc1Dt;Z þDt a1;jKj K1 Àf t þcsDt;Z þDt a1;jKj ¼0 where biðÞi ¼ 1; ÁÁÁ; s are the weight coefﬁcients, and > > j j > they are taken from Eq. (25)[20]. >. >. ! ! > Ps Ps < n n n n gs ¼A t þcsDt;Z þDt as;jKj Ks Àf t þcsDt;Z þDt as;jKj ¼0 > j j > o > C 4 Attitude controller for the central spacecraft >gsþ1 ¼ K1 ¼0 > oZ >. >. >. :> oC In this section, a proportional–derivative (PD) control gs s ¼ Ks ¼0 þ oZ strategy is developed to ensure the central spacecraft’s ð24Þ spinning rate remains stable in the tether deployment process. Rewrite the ﬁrst equation in Eq. (12) as, where Dt is the time step size, ai;jðÞi;j¼1;...;s is the integration matrix, and ciðÞi¼1;...;s is the integration _ x ¼ J1ðÞh h ð29Þ nodes [20]. Their values are taken from the Butcher À1 tableau form, i.e. where J1ðÞ¼h J ðÞh . Substituting Eq. (29) into the second equation of

c1 Eq. (12) gives, a11 a12 ÁÁÁ a1s c 1 a21 a22 ÁÁÁ a2s ~ € ~ _ MðÞh h þ NðÞh h ¼ u ð30Þ ...... with as1 as2 ÁÁÁ ass cs 8 ð25Þ < M~ ðÞ¼h JTðÞh H J ðÞh b b ÁÁÁ b 1 s 1 ÀÁ 1 2 s ~ T _ T _ : NðÞ¼h J1ÀÁðÞh HsJ1ðÞþh J1 ðÞh SJ1ðÞh h HsJ1ðÞh T The algebraic equation (24) with unknown B ¼ u ¼ J1 ðÞh Ms;g þ Ms;p þ Ms;c ðÞK1; ÁÁÁ; Ks is solved by the Newton–Raphson iter- ð31Þ ative algorithm. Denote Bq as an approximate solution after the qth iteration, the true solution can be written where SðÞ is used to represent the skew-symmetric as, matrix [34]. q q Deﬁne the attitude errors of the central spacecraft Btrue ¼ B þ DB ð26Þ as, 8 where the DBq is the correction to the approximate < eh ¼ h À hd solution. _ _ e_h ¼ h À hd ð32Þ Substituting Eq. (26) into (24) and expanding it into : € € e€h ¼ h À hd a Taylor series by ignoring higher-order terms yield, where the subscript d denotes the desired value. 0 gB gBq DBq gBq JqDBq 27 ¼ ðÞ¼ðÞþ ðÞþ ð Þ Substituting Eq. (32) into Eq. (30) yields, q where J ¼ og=oBj q is the Jacobian matrix of these B Me~ ðÞe€þ Ne~ðÞe_ ¼ u0 ð33Þ algebraic equations with respect to the vector Bq. The q 0 ~€ ~_ detailed expressions of the Jacobian matrix J are where u ¼ u À Mhd À Nhd. given in ‘‘Appendix A’’. Solve for the correction DBq Deﬁne the PD controller for Eq. (33) is as for Ref. iteratively until the residual converges e ¼ [34], that is . 0 qþ1 u ¼ÀKPe À KDe_ ð34Þ gB24ðÞqnðÞþþ1 24 ðÞ24ðÞqnðÞþþ 1 24 emax or the iteration number reaches the maximum allowed where KP and KD are gain matrices of the proportional and derivative terms, respectively. The detailed iteration qmax. expression will be given in the numerical simulation Once the solution B ¼ ðÞK1; ...; Ks is obtained, the state in the next time step Znþ1 becomes, part. Substituting Eq. (34) into Eq. (31) yields the control Xs nþ1 n torque Ms;c of the central spacecraft, Z ¼ Z þ Dt bjKj ð28Þ j

123 Stability and control of radial deployment of electric solar wind sail 489

Table 2 Position vectors Name Values of anchor points in the body-ﬁxed frame b Xanc;1 (m) ðÞ2; 0; 0 b Xanc;2 (m) ðÞ0; 2; 0 b Xanc;3 (m) ðÞÀ2; 0; 0 b Xanc;4 (m) ðÞ0; À2; 0

ÀÁ ÀÁ n _ _ n hd;j ¼ hd t ÀÁþ cjDt ; hd;j ¼ hd t þ cjDt ; and € € n hd;j ¼ hd t þ cjDt ð36Þ

where cjðÞj ¼ 1; ...; s represent the integration nodes of the s-stage Runge–Kutta method [20].

Fig. 3 Libration motion of flexible tethers of the E-sail and their simplified expressions 5 Libration motion of flexible tethers hiÀÁ ~ ~ M J K h_ h_ K h h Mh€ Nh_ The libration motion of flexible tethers of an E-sail is s;c ¼ À v À d À pðÞþÀ d d þ d 0 described in the coordinate system O Xb0 Yb0 Zb0 as À Ms;g À Ms;p shown in Fig. 3. Different from the rigid tether or ð35Þ dumbbell model of tether, there will be n sets of libration (in-plane and out-of-plane) angles for each Here, the desired attitude trajectory of the central tether if it is divided into n elements [33, 35]. This _ € spacecraft (hd, hd, hd) is calculated for the given makes libration control difficult for the tether. To desired angular velocity xd at the internal stages of the address this challenge, virtual in-plane and out-of- s-stage Runge–Kutta integration method, plane angles are introduced here by a virtual straight tether AB (red dot line) in Fig. 3, which connects the first and last nodes of each flexible tether. Thus, the libration motion of the tether is approximated by the

Table 1 Physical Parameters Values parameters of the E-sail system Mass of central spacecraft (kg) 400 Shape of central spacecraft Cylinder Height of central spacecraft (m) 2.0 Radius of central spacecraft (m) 2.0 Initial orbit of E-sail Circle

Position of central spacecraft Ob (AU) 1.0 Number of tethers of E-sail 4 Density of tether material (kg/m3) 1440.0 Diameter of tether (m) 7.38 9 10-5 Elastic module of tether (GPa) 70.0 Initial location in solar system (AU) (1,0,0) Initial length of each tether (m) 10.0 Initial angular velocity of central spacecraft (deg/s) xXb 1:141 Â 10À5 Initial angular velocity of central spacecraft (deg/s) xZb 0.48

123 490 G. Li et al.

Table 3 Parameters of numerical simulation cases Case Deployment velocity Thrust at remote unit High-order ﬂexural modes of tether included Tether 1 jammed (after name (m/s) (N) (yes/no) 100 s*)

Case A 0.00 0.000 No (one element per tether) No Case B 0.20 0.000 No (one element per tether) No Case C 0.20 0.000 Yes (multiple elements per tether) No Case D 0.20 0.005 No (one element per tether) No Case E 0.20 0.005 Yes (multiple elements per tether) No Case F 0.10 0.005 Yes (multiple elements per tether) No Case G 0.40 0.005 Yes (multiple elements per tether) No Case H 0.20 0.010 Yes (multiple elements per tether) No Case I 0.20 0.002 Yes (multiple elements per tether) No Case J 0.20 0.005 Yes (multiple elements per tether) Yes *100 s is chosen arbitrayly

( libration motion of the virtual tether AB in terms of in- R 0 T 0 h T h R AB b ;j¼ÀÁb2b ðÞ1 j g2bðÞ AB g;j plane (in-plane angle, rotating around the Z0 axis) and T b R ¼ X À X ;Y À Y ;Z À Z 00 AB g;j B g;j A g;j B g;j A g;j B g;j A g;j out-of-plane angle (rotating around the Y b that is the 0 ðÞj ¼ 1;2;...;m Y b axis after the ﬁrst rotation), that is, ÀÁ ð38Þ À1 y z aj ¼ tan ÂÃRAB b0 RAB b0 À1 x z y where m represents the total number of tether of an E- bj ¼ tan RAB b0 RAB b0 cos aj þ RAB b0 sin aj T ðÞj ¼ 1; 2; ...; m x y z sail, and RAB b0;j ¼ RAB b0;j;RAB b0;j;RAB b0;j is the ð37Þ vector of line AB expressed in the coordinate system 0 T O Xb0 Yb0 Zb0 , and Tg2bðÞ¼h Tb2gðÞh . In addition,

Tb2b0 ðÞh1 j is the transformation matrix from the coordinate system at CM ObXbYbZb to the coordinate system with the origin of the coordinate system 0 ðÞO Xb0 Yb0 Zb0 jðÞj ¼ 1;2;...;m at the anchor point of each tether as shown in Fig. 3, 2 3 cb0 casb0 sa0sb0 4 0 0 0 0 0 5 Tb2b0 ðÞ¼h1 sb ca cb sa cb ð39Þ 0 Àsa0 cb0 T b b b b where Xanc;j ¼ Xanc;j; Yanc;j; Zanc;j is the position vector of the. anchor point of the jth tether, a0 ¼ cosÀ1 Xb Yb , and anc;j,anc;j ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 0 À1 b b b b ¼ sin Zanc;j Xanc;j þ Yanc;j .

6 Simulation results and discussion

Fig. 4 Relative error and iteration numbers a e, b iteration The system parameters of the E-sail in the simulation number in Case A are given in Table 1 [6, 30]. The E-sail is initially 123 Stability and control of radial deployment of electric solar wind sail 491

Fig. 5 Angular velocity of central spacecraft a xx, b xy, c xz in Case A

assumed in a circular orbit with orbital radius listed in Table 2. Other perturbative forces except for

R ¼ 1 AU. Its initial position is at (Xg, Yg, Zg)=(1 the gravity of the Sun are not considered in the study. AU, 0, 0) in the global frame. It orbits in the positive Besides, the two stages Gauss–Legendre Runge–Kutta direction of OgZg axis in the global framepffiffiffiffiffiffiffiffiffiffiffiffi with an with fourth-order accuracy is applied, and the Butcher Xb 3 table is listed in Eq. (40). orbital angular velocity, xorb ¼ x ¼ lS=R ¼ 20 3 ffiffiffi ffiffiffi 1:141 Â 10À5 =s. Here, l = 1.3271244 9 10 m / p p S 1=2 À 3 6 1=414 À 3 6 s2 is the gravitational constant of the Sun. At the same pffiffiffi pffiffiffi 1=2 þ 3 6 1 4 þ 3 61=4 time, the E-sail also spins about the ObZb axis in the pffiffiffi pffiffiffi ð40Þ positive direction at 0.48 deg/s. Thus, the initial 1=2 þ 3 21=2 À 3 2 angular velocity vector of the central spacecraft is ÀÁ T T Based on the trial and error, the iteration control x ¼ ðÞxXb ; xYb ; xZb ¼ 1:141 Â 10À5; 0; 0:48 . The -10 parameters, emax and qmax, are defined as 10 and position vectors of the anchor points of tethers are 100, respectively. The iteration will stop if either the 123 492 G. Li et al.

direction. The detailed process can be found in the authors’ work in [6].

6.1 Validation of proposed method

The proposed method is first validated by simulating an E-sail with constant-length tethers (deployed speed is zero), which is the Case A in Table 3. The PD controller is applied to stabilize the central spacecraft’s attitude to avoid the loss of attitude stability of the central spacecraft caused by the oscillation of the tether [34]. The detailed implementation of the PD controller for the central spacecraft’s attitude in the implicit Runge–Kutta integrator can be found in Section IV. The diagonal terms of the gain matrices 6 KP and KD are chosen as 10 . The desired angular velocity of the central spacecraft is x ¼ ÀÁ d T 1:141 Â 10À5; 0; 0:48 deg/s. For simplicity, each tether is modeled by one element only in the validation case, and the time step size is 0.0005 s. The total simulation period is 750 s. The results of residual error e and the iteration number m over the time are shown in Fig. 4, where the maximum iteration number never exceeds 18, much less than the maximum allowed iteration number mmax ¼ 100, while the residual error is controlled below 10-10 successfully. This indicates the proposed implicit s-stage Runge–Kutta time integrator works well. Figure 5 shows that the PD controller works successfully to control the central spacecraft spinning along the ObZb-axis with a constant rate of 0:48 deg/s. It also shows the large gains in the PD controller are Fig. 6 Case A a tether length, b tether tension, c calculated necessary to decrease the response time of the E-sail, angular velocity of remote unit (tether 1), d in-plane libration see in Fig. 5a and c. As expected, the angular velocity angle a1 of tether 1, and e out-of-plane libration angle b1 of of remote units is in phase with that of the central tether 1 spacecraft if no tether is deployed, see Fig. 6. For example, Fig. 6c shows the calculated angular veloc- conditions e e or q q are satisfied. If both max max ity of the remote unit in the O Z axis is equal to the conditions are not satisfied, the simulation is deemed b b spinning rate of the central spacecraft in the O Z axis. failed and stops. The unit of the central spacecraft’s b b The tether experiences a low tension, which suggests a angular velocity is in rad/s in the simulation and is heavy remote unit may be needed to increase the presented as deg/s in the results for reading tether’s tension resulting from the centrifugal effect convenience. for stability if the tether is getting slack [30]. Also, as The initial conditions for the tethers and remote shown in Fig. 6d and e, the tether’s libration angles are units are calculated by a special process by assuming small in out-of-plane without tether deployment, the same angular velocities of the remote units and the which is expected. In conclusion, the proposed high- central spacecraft initially about the O Z axis b b fidelity model of E-sail with the PD controller is validated for the central spacecraft’s attitude control.

123 Stability and control of radial deployment of electric solar wind sail 493

Fig. 7 Snap shots of geometrical conﬁguration of E-sail in Cases B and C

The kinematic constraints for the coupling of the central spacecraft leading to the failure of tether flexible tether and the rigid body of central spacecraft deployment. work very well. The simulation results are shown in Figs. 7 and 8. Figure 7 shows the snapshots of the tether configura- 6.2 Tether deployment without thrust at remote tions in the deployment process. The tethers start to units bend immediately after deployment due to the Coriolis forces acting on the deployed tethers and remote units. Two numerical simulation cases (B and C listed in It is because the tether does not have the bending Table 3) are conducted to examine the effect of the rigidity to resist the bending moment caused by the tether’s transverse flexural motion in the deployment Coriolis forces. However, the Coriolis forces are process. The tethers are modeled with one element counter affected by the centrifugal forces on the same only in Case B and multiple elements in Case C. In bodies to form a dynamic balance, which prevents the Case C, the process of the dividing of an element is tether from immediately wrapping around the central activated when the tether deployment causes the spacecraft. When the length of the deployed tether is length of the variable-length element longer than the short, say less than or equal to the upper bound of the upper bound of the elemental length (Lmax). Based on elemental length Lmax, the results of these two cases the trial and error, the standard element length and are the same. After the length of deployed tether is upper bound Lmax are set as Ls = 20 m and greater than the Lmax, a new element is generated and Lmax ¼ 1:65Ls. added to the tether discretization. The model in case C Each tether’s deployment speed is assumed the shows the flexural mode of the tether, which cannot be same as 0.2 m/s for all tethers in both cases. The time captured in case B with a single element as expected. It step size and total simulation time are 0.0005 s and shows the multiple elements are needed to capture the 750 s, respectively. The simulation terminates when flexural modes in the deployment dynamics of tether. the libration angles (in-plane and out-of-plane angles) The simulation stops at 231 s when the tether wraps exceed 90°, which implies the tether wraps around the around the central spacecraft. The corresponding 123 494 G. Li et al.

force slows down the spin velocity of the remote units in the tether deployment process. The moment of Coriolis force increases as the tether length increases, which leads to tethers wrapping around the central spacecraft eventually. The results indicate that the external force, such as thrust at the remote unit, is needed to cancel the Coriolis effect to ensure successful tether deployment [36].

6.3 Tether deployment with thrust at remote unit

A tangential thrust is applied at the remote unit to avoid the tether wrapping of the central spacecraft. The direction of the thrust is along the opposite direction of the Coriolis force in the spin plane. A 000 00 00 body-fixed frame OBXb Y bZ b is introduced at the location of the remote units to implement the application of thrust. As shown in Fig. 3, the body-fixed 000 00 00 frame OBXb Y bZ b is parallel with the body-fixed 000 00 00 frame OAXb Y bZ b and is attached at point A. The tangential thrust (0.005 N) is applied in the positive 00 direction of the OBY b axis at point B. To examine the effect of the transverse flexural motion of tether in the deployment dynamics, two cases (D and E in Table 3) Fig. 8 Results of cases B and C (a) in-plane libration angle a1, are considered. Here, the standard-length Ls of case E b out-of-plane libration angle b1, c spin velocity of remote unit (tether 1) is set as 50 m to reduce the computational loads. The standard element length and upper bound Lmax are set deployed tether length is 46.2 m, which shows that the as Ls = 20 m and Lmax ¼ 1:65Ls, which is the same as tether deployment fails. that in Case C. The results are shown in Figs. 9 and 10 Next, Fig. 8 shows the in-plane and out-of-plane and Table 4. libration angles and the calculated spin velocity of As shown in Fig. 9, the E-sail can deploy more remote units relative to the central spacecraft. It is tether with wrapping the central spacecraft when a noted that the results of one and multiple elements constant tangential thrust is applied at the remote models of the tether are remarkably close. Figure 8a, b units. Refs. [5, 11] show that a typical E-sail with shows that the in-plane libration is the dominated tether length is ranging from 10 to 25 km. Based on libration mode. Two reasons can be attributed: (1) the the results of the deployed 160 m tether, it can be Coriolis and centrifugal forces are in the plane of estimated that the tether will eventually wrap (in-plane rotation, and (2) the perturbative forces that generate libration angle reaches 90°) the central spacecraft in the out-of-plane component force are not considered cases of D and E, see in Table 4. It is because the in the current study. As a result, the tethers’ out-of- resultant Coriolis force from the deployed tether and plane libration angles will not be plotted in the remote unit increases continuously as the tether length following analysis because it is negligible when the increases. The moment of the Coriolis force will other perturbative forces are not considered. Figure 8a eventually exceed the counterbalance moment by the and c show the in-plane angle of the first tether, and the constant tangential thrust. For instance, the in-plane spin velocity of the tip node of tether gradually angle reaches 90° when the tether is 7515 m long in increases and decreases, respectively, as the tether is the case D (single element per tether) and 9104 m long deployed. It illustrates the remote units cannot keep in in the case E (multiple elements per tether). This is phase with the central spacecraft because the Coriolis because the effective spin radius of the remote unit is

123 Stability and control of radial deployment of electric solar wind sail 495

Fig. 9 Snap shots of geometric conﬁguration of E-sail in cases D and E

shorter in the case E due to the tether’s flexural bending than that in case D (one element per tether). It indicates the flexural mode of tether should be considered for precision analysis and control. Also, a radial thrust is needed to attenuate the wave oscillation along the tether [10]. The results of the in-plane libration angle, the spin velocity of the remote unit, and tether tension are shown in Fig. 10. Figure 10a shows the in-plane angles in cases D and E gradually increase and reach to 1.7° and 1.4° in the given period, respectively. The results of these two cases are close initially but eventually deviate due to the tether’s flexural motion in case E becoming significant. Figure 10b and c show the high-order modes in the case E due to the multiple elements per tether used in the analysis.

6.4 Parametric study of the E-sail deployment

To further investigate the effects of tether deployment speed and thrust magnitude on the E-sail’s deployment Fig. 10 a Results of cases D and E (tether 1) in-plane libration dynamics, ﬁve numerical simulation cases (E–I) are angle a1, b spin velocity of remote unit, and c tether tension at the remote unit conducted, see Table 3. In all cases, the tethers are discretized into multiple elements per tether. The

standard-length Ls, upper bound Lmax, and time step

123 496 G. Li et al.

Table 4 Deployed length versus libration angle (average value of four tethers) Deployed length (m) 110 130 160 1600 (estimation) 7515 (estimation) Case D libration (°) - 1.14 - 1.38 - 1.74 - 19.00 - 90.00 Deployed length (m) 110 130 160 1600 (estimation) 9104 (estimation) Case E libration (°) - 0.93 - 1.16 - 1.39 - 15.79 - 90.00

Fig. 11 Results of parametric analysis a in-plane libration angle a1, b spin velocity of remote unit (tether 1) size are set the same as those in Case D. In all cases, the magnitudes of the thrust are kept constant, such that 0.005 N in cases (E, F, G), 0.010 N in case H, and 0.002 N in case I. Figure 11 shows the variation of the deployed tether length versus the in-plane libration angle and the Fig. 12 Tether tension of the tether 1 at the central spacecraft spin velocity of the remote unit. The results reveal that a proper choice of the thrust and deployment speed is 6.5 Tether deployment with tether jammed critical in suppressing the libration motion of flexible tethers to achieve the E-sail’s successful deployment. As listed in Table 3, a numerical simulation case J is Figure 12 shows the tension of the variable-length conducted. It is assumed that the deploying mecha- element of the first tether. It illustrates that the tether is nism for tether one is jammed at 100 s after the central in the low-tension region, and the phenomenon of spacecraft starts to deploy tethers. The thrust on tether tether slack occurs. For example, Fig. 12a shows the one is stopped accordingly. The other parameters in variable-length element’s tension varies between 0 case E are assumed the same as those in case E. The and 0.2 N in the tether deployment process. It infers results are shown in Figs. 13 and 14. that the central spacecraft’s spin rate controller should Figure 13 shows the changes caused by the jammed be applied to avoid the tether slack. tether in the libration angle. The spin velocity of the remote unit associated with the jammed and unjammed tethers varies significantly from the case

123 Stability and control of radial deployment of electric solar wind sail 497

Fig. 13 Results of cases E and J (a) in-plane libration angle a1, b spin velocity of remote unit (tether 1), c spin velocity of remote unit (tether 2), d tension at remote unit (tether 1), e tension at remote unit (tether 2)

123 498 G. Li et al.

Fig. 14 Geometric conﬁguration of E-sail in cases E and J

E, where the tether is not jammed. For example, as coupling between the flexible tethers and the rigid- shown in Fig. 13b and d, the spin velocity of the body central spacecraft are included. Our results show remote unit and tension associated tether one drops that the proposed implicit Runge–Kutta integration sharply when the tether is jammed. It is because the scheme with a symplectic property can ensure the additional external forces, such as the Coriolis and kinematic constraint conditions is precisely satisfied. thrust at the remote unit, disappears when the tether is The proportional–derivative controller is applied to jammed. As a result, the first tether’s response in case J stabilize the attitudes of the central spacecraft by is quite different from the case E, where the first tether rotating at a constant spin rate. The results show the one is not jammed. For the second tether, Fig. 13c and E-sail cannot deploy long tethers without thrust at the e shows the second tether’s responses for the cases E remote units in the radial deployment mode. Tangen- and J are remarkably similar as expected because the tial thrust is required at the remote units to counter- second tether is not jammed. Finally, Fig. 14 shows balance the Coriolis forces acting on the tethers and the E-sail successfully deploys the targeted tether remote units to deploy tethers in the radial deployment length for the rest three tethers. It shows the failure of a mode successfully. Moreover, the parametric analysis single tether could be isolated from the rest tethers shows the tether deployment speed and the thrust when each tether has an isolated deployment mech- magnitude significantly affect deployment stability, anism, and the failure of tether deployment is isolated which opens the possibility of successfully deploying from other tethers. Therefore, the radial deployment tethers by advanced control strategies such as optimal method’s deployment success rate is high because control. Finally, the analysis shows the radial deploy- each tether has an isolated deployment mechanism. ment is advantageous in isolating the failure of a single tether jammed in the deployment process from the successful deployment of the rest tethers. 7 Conclusions Acknowledgements This work is supported by the Discovery In this study, the dynamics and control of radial tether Grant (RGPIN-2018-05991) and Discovery Accelerate Supplement Grant (RGPAS-2018-522709) of Natural Sciences deployment of a spinning E-sail are studied by a high- and Engineering Research Council of Canada. fidelity three-dimensional model. In the model, the effects of flexible elastic tether dynamics, rigid-body dynamics of the central spacecraft, and the kinematic

123 Stability and control of radial deployment of electric solar wind sail 499 hi hi Compliance with ethical standards o o J s h s x o o J s h s x g6 g6 Conﬂict of interest The authors declare no conﬂict of interest ¼À ; ¼À ; oK ;x oK ;x oK ;x oK ;x in this article. 1 1 ÀÁ2 2 s n and x ¼ x þ Dta21K1;x þ a22K2;x ðA:8Þ Appendix: Expression of Jacobian matrix o o g7 ½x Â ðÞHsx ¼ Hs þ ; and The Jacobian matrix J of Eq. (27) are as, oK1;x oK1;x o o ðA:9Þ og og g7 ½x Â ðÞHsx 1 I ; 1 Dta I ; and ¼ ¼ 3Â3 ¼À 11 3Â3 oK2;x oK2;x oK1;X oK1;V os s ðA:1Þ g1 ¼ÀDta12I3 3 og og oK Â 7 Dta CT and 7 Dta CT 2;Vs o ¼ 11 1;h o ¼ 12 1;h K1;k1 K2;k1 og og ðA:10Þ 2 ¼ I ; 1 ¼ÀDta I ; and oK 3Â3 oK 21 3Â3 hi 2;Xs 2;Vs A:2 o ð Þ s s g2 o x H x ¼ÀDta I og Â s oK 22 3Â3 8 ¼ ; and 2;Vs oK oK 1;x 1hi;x ðA:11Þ o o o s x H s x g3 g3 T og Â s ¼ Ms; ¼ Dta C ; and 8 11 1;Xs ¼ Hs þ oK1;V oK1;k o o s 1 : K2;x K2;x og ðA 3Þ 3 Dta CT ¼ 12 1;Xs oK2;k og og 1 8 ¼ Dta CT and 8 ¼ Dta CT oK 21 1;h oK 22 1;h o o 1;k1 2;k1 g4 g4 T ¼ Ms; ¼ Dta21C ; and ðA:12Þ oK oK 1;Xs 2;Vs 1;k1 : og ðA 4Þ 4 T og ¼ Dta22C1;X 9 o s ¼ I4mnðÞÂþ1 4mnðÞþ1 ; K2;k1 o K1;Xt o og o½JðÞh x og o½JðÞh x g9 5 ¼ I À ; 5 ¼À ; ¼ÀDta11I4mnðÞÂþ1 4mnðÞþ1 ; ðA:13Þ 3Â3 oK1;V oK1;h oK1;h oK2;h oK2;h t ÀÁ o n g9 and h ¼ h þ Dta11K1;h þ a12K2;h and ¼ÀDta12I4mnðÞÂþ1 4mnðÞþ1 oK2;V ðA:5Þ t og o o o o 10 g5 ½JðÞh x g5 ½JðÞh x ¼ I4mnðÞÂþ1 4mnðÞþ1 ; ¼À ; ¼À ; oK2;X oK oK oK oK t 1;x 1;x ÀÁ2;x 2;x og n 10 ¼ÀDta I ; : and x ¼ x þ Dta11K1;x þ a12K2;x o 21 4mnðÞÂþ1 4mnðÞþ1 ðA 14Þ K1;Vt ðA:6Þ o g10 hi hi and o ¼ÀDta22I4mnðÞÂþ1 4mnðÞþ1 K2;Vt o o J s h s x o o J s h s x g6 g6 ¼À ; ¼ I3 3 À ; o o oK oK oK Â oK g11 g11 T 1;h 1;h 2;h 2;h ¼ Mt; ¼ Dta11C ; and ÀÁ o o 1;Xt s n K1;Vt K1;k1 and h ¼ h þ Dta21K1;h þ a22K2;h o ðA:15Þ g11 T ¼ Dta12C o 1;Xt ðA:7Þ K2;k1

123 500 G. Li et al.

o o o o g11 T g11 T g17 g17 ¼ Dta11C ; ¼ Dta12C ; ¼ Dta11C3;X ; ¼ Dta12C3;X ; o 2;Xt o 2;Xt o s o s K1;k2 K2;k2 K1;Xt K2;Xt o o g11 T g18 ¼ Dta11C ; ; ðA:16Þ ¼ Dta21C3;X ; ; ðA:22Þ o 3;Xt o s K1;k3 K1;Xt o o g11 T g18 and ¼ Dta12C and ¼ Dta22C3;X o 3;Xt o s K2;k3 K2;Xt

og og 12 M ; 12 Dta CT ; and ¼ t ¼ 21 1;Xt oK2;V oK1;k o t 1 ðA:17Þ g12 T ¼ Dta22C o 1;Xt References K2;k1 o o 1. Janhunen, P.: Electric sail for spacecraft propulsion. g12 T g12 T ¼ Dta21C ; ¼ Dta22C ; J. Propuls. Power 20(4), 763–764 (2004). https://doi.org/10. o 2;Xt o 2;Xt K1;k2 K2;k2 2514/1.8580 o 2. Quarta, A.A., Mengali, G.: Electric sail mission analysis for g12 T ¼ Dta21C ; ; ðA:18Þ outer solar system exploration. J. Guid. Control Dyn. 33(3), oK 3;Xt 1;k3 740–755 (2010). https://doi.org/10.2514/1.47006 og 3. Bassetto, M., Quarta, A.A., Mengali, G.: Locally-optimal and 12 ¼ Dta CT 22 3;Xt electric sail transfer. Proc. Inst. Mech. Eng. Part G J. oK2;k 3 Aerosp. Eng. 233(1), 166–179 (2019). https://doi.org/10. og og 1177/0954410017728975 13 ¼ Dta C ; 13 ¼ Dta C ; 4. Bassetto, M., Mengali, G., Quarta, A.A.: Stability and o 11 1;Xs o 12 1;Xs K1;Xs K2;Xs control of spinning electric solar wind sail in heliostationary o orbit. J. Guid. Control Dyn. 42(2), 425–431 (2019). https:// g13 ¼ Dta11C1;h; doi.org/10.2514/1.g003788 oK1;h ðA:19Þ 5. Janhunen, P., Quarta, A., Mengali, G.: Electric solar wind og og sail mass budget model. Geosci. Instrum. Methods Data 13 ¼ Dta C ; 13 ¼ Dta C ; oK 12 1;h oK 11 1;Xs Syst. 2(1), 85–95 (2013). https://doi.org/10.5194/gi-2-85- 2;h 1;Xs 2013 og and 13 ¼ Dta C 6. Li, G., Zhu, Z.H., Du, C., Meguid, S.A.: Characteristics of o 12 1;Xs coupled orbital-attitude dynamics of flexible electric solar K2;Xs wind sail. Acta Astronaut. 159(June), 593–608 (2019). og og https://doi.org/10.1016/j.actaastro.2019.02.009 14 ¼ Dta C ; 14 ¼ Dta C ; 7. Mengali, G., Quarta, A.A., Janhunen, P.: Electric sail per- o 21 1;Xs o 21 1;h K1;Xs K1;h formance analysis. J. Spacecr. Rockets 45(1), 122–129 o (2008). https://doi.org/10.2514/1.31769 g14 ¼ Dta21C1;Xs ; 8. Simmons, M., Montalvo, C.: Vibration modes of an electric oK1;X s ðA:20Þ sail. J. Guid. Control Dyn. 43(7), 1393–1398 (2020). https:// og og doi.org/10.2514/1.g004924 14 Dta C ; 14 Dta C ; ¼ 22 1;Xs ¼ 22 1;h 9. Liu, F., Hu, Q., Liu, Y.: Attitude dynamics of electric sail oK2;X oK2;h s from multibody perspective. J. Guid. Control Dyn. 41(12), og and 14 ¼ Dta C 2633–2646 (2018). https://doi.org/10.2514/1.g003625 o 22 1;Xs K2;Xs 10. Montalvo, C., Wiegmann, B.: Electric sail space flight dynamics and controls. Acta Astronaut. 148(July), 268–275 o o (2018). https://doi.org/10.1016/j.actaastro.2018.05.009 g15 g15 ¼ Dta C ; ¼ Dta C ; 11. Fulton, J., Schaub, H.: Fixed-axis electric sail deployment o 11 2;Xs o 12 2;Xs K1;Xt K2;Xt dynamics analysis using hub-mounted momentum control. og Acta Astronaut. 144(March), 160–170 (2018). https://doi. 16 ¼ Dta C ; ; ðA:21Þ oK 21 2;Xs org/10.1016/j.actaastro.2017.11.048 1;Xt 12. Wang, R., Wei, C., Wu, Y., Zhao, Y., Cui, H.: Dynamic og and 16 ¼ Dta C analysis of the spinning deployment for flexible tether o 22 2;Xs electric sail spacecraft. J. Harbin Eng. Univ. 40(4), 724–729 K2;Xt (2019). https://doi.org/10.11990/jheu.201709101 13. Wang, R., Wei, C., Wu, Y., Zhao, Y.: The study of spin control of flexible electric sail using the absolute nodal coordinate formulation. In: 2017 IEEE International Con- ference on Cybernetics and Intelligent Systems and IEEE 123 Stability and control of radial deployment of electric solar wind sail 501

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