Detumbling and Nutation Canceling Maneuvers with Complete Analytic Reduction for Axially Symmetric Spacecraft

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2010 Detumbling and nutation canceling maneuvers with complete analytic reduction for axially symmetric spacecraft

Romano, Marcello

Acta Astronautica, Vol. 6, 2010, pp. 989-998 http://hdl.handle.net/10945/46462 ARTICLE IN PRESS

Acta Astronautica 66 (2010) 989–998

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Acta Astronautica

journal homepage: www.elsevier.com/locate/actaastro

Detumbling and nutation canceling maneuvers with complete analytic reduction for axially symmetric spacecraft

Marcello Romano Ã

Mechanical and Astronautical Engineering Department, and Space Systems Academic Group, Naval Postgraduate School, 700 Dyer Road, Monterey, CA, USA article info abstract

Article history: A new method is introduced to control and analyze the rotational motion of an axially Received 9 April 2009 symmetric rigid-body spacecraft. In particular, this motion is seen as the combination of Accepted 18 September 2009 the rotation of a virtual sphere with respect to the inertial frame, and the rotation of the Available online 7 November 2009 body, about its symmetry axis, with respect to this sphere. Two new exact solutions are Keywords: introduced for the motion of axially symmetric rigid bodies subjected to a constant Rigid-body dynamics and kinematics external torque in the following cases: (1) torque parallel to the angular momentum and Artiﬁcial satellites (2) torque parallel to the vectorial component of the angular momentum on the plane Rotation perpendicular to the symmetry axis. By building upon these results, two rotational Integrable cases of motion maneuvers are proposed for axially symmetric spacecraft: a detumbling maneuver and a nutation canceling maneuver. The two maneuvers are the minimum time maneuvers for spherically constrained maximum torque. These maneuvers are simple and elegant, as they reduce the control of the three degrees-of-freedom nonlinear rotational motion to a single degree-of-freedom linear problem. Furthermore, the complete (both for the dynamics and for the kinematics) and exact analytic solutions are found for the two maneuvers. An extended survey is reported in the introduction of the paper of the few cases where the rotation of a rigid body is fully reduced to an exact analytic solution in closed form. Published by Elsevier Ltd.

1. Introduction as an effect of either external disturbance or reorientation control torque. This maneuver cancels the transversal compo- Spacecraft detumbling maneuvers and spacecraft nutation nents of the angular velocity, which causes the associated canceling maneuvers have critical importance and are often characteristic coning motion, and brings the spacecraft back to performed in astrodynamic applications. The detumbling its nominal pure spinning condition about the symmetry axis. maneuver is typically performed by most spacecraft after their The results presented in this paper apply to axially separation from the rocket launcher, once the orbital ﬂight symmetric spacecraft, which constitute a large sub-class of state has been reached. This maneuver cancels any residual all spacecraft. In particular, virtually all of the spinning angular velocity and prepare the spacecraft for its nominal stabilized spacecraft are axially symmetric. rotational motion condition. On the other hand, the nutation While a vast literature exists regarding spacecraft canceling maneuver is typically needed for spinning stabilized rotational maneuvers, (see, for instance, [1,2]) no exact spacecraft whenever a transversal (with respect to the analytic reduction is generally found for those maneuvers. spinning axis) component of the angular velocity builds up The analysis of spacecraft rotational maneuvers is a practical application of the theory of the rotational motion of a rigid body. Ã Tel.: þ1831656 2885; fax: þ1831656 2238. The problem of the rotational motion of a rigid body can E-mail address: [email protected] be divided into two parts. The dynamic problem aims to

0094-5765/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.actaastro.2009.09.015 ARTICLE IN PRESS

990 M. Romano / Acta Astronautica 66 (2010) 989–998 obtain the angular velocity of the body with respect to an 8. Rigid body with axially symmetric ellipsoid of inertia inertial reference by starting from the knowledge of the subjected to a torque constant in magnitude, perpendi- initial angular velocity and the history of the applied cular to the symmetry axis and rotating at a specific torque. On the other hand, the kinematic problem focuses on constant rate about the symmetry axis [18]. determining the current orientation of the body from the knowledge of the initial orientation and the history of the angular velocity. No other exact analytic solutions exist besides the one The exact analytic solution for the rotational motion of a listed above, to the best knowledge of the author. The rigid body exists only for a small number of special cases, limitation in the number of these solutions constitutes a which are surveyed here below. critical knowledge gap in the field of rigid body mechanics, When no external torque acts on the rigid body as previously noticed by Ivanova [14]. (Euler–Poinsot case) the rotational motion of a generic In order to find the solutions to the problems listed as triaxial body has an exact analytic solution for both the items 2–4 above, Ivanova [14,15] utilize an original view of dynamic and kinematic problems [3–8]. the turn-tensor (rotation matrix) as a function of right and When only the torque due to gravity acts on the rigid left angular velocities vector, introduced by Ivanova [19],and body, exact solutions exist for both the dynamic and introduce a theorem which is analogous to the reduction kinematic problems only in the following two special cases: principle previously established by Hestenes [20]. In order to find the solution for the problem listed as 1. Lagrange–Poisson heavy-top case: Rigid body under item five above, Romano [17] exploits the parametrization gravity force with two equal principal moments of of the rotation by three complex numbers in order to inertia at the fixed point and the center of mass along reduce the kinematic differential equation to an equation of the third axis of inertia [6,9–12]. Riccati which is then solved through appropriate choices of 2. Kovalevskaya heavy-top case: Rigid body under gravity substitutions, thereby yielding a completely reduced force with two equal principal moments of inertia at the analytic solution in terms of confluent hypergeometric fixed point, value of the third moment of inertia equal to functions. The parametrization of the rotation used by [17] half of the value of the moment inertia about the other was introduced by [21], and is very similar to one two axes, and the center of mass in the plane of equal previously introduced by Darboux [22] (see also [23]). moments of inertia [6,13]. Before Romano [17] and Ivanova [24] derives a partially reduced solution for the same problem while studying the motion of a ball on a rough plate. This partial solution When the rigid body is subjected to an external torque includes not reduced integrals spanning over the time which is dependent at most on time (self-excited body), the duration of the maneuver. complete (i.e. for both the kinematics and the dynamics) In order to find the solutions for the problem listed as exact analytic solution exists, in a form totally reduced to item six o eight above, Romano [18] exploits the Hestenes’ elementary functions, only in the following cases: reduction principle and the solution of Romano [17].An incomplete solution for the problem in item six was 1. Single degree-of-freedom rotation of a generic rigid previously sketched by Lurie [23]. In particular, Lurie [23] body with the applied torque along one of the three uses the Cayley–Klein’s parameters in order to express the principal axes of inertia and initial angular velocity dynamics and kinematics equations as a Darboux differ- along the same axis. ential equation problem, then transforms the Darboux 2. Rigid body with axially symmetric ellipsoid of inertia problem into a Weber differential equation problem, which subjected to a torque proportional to the current angular is nevertheless left unsolved. momentum vector [14]. Finally, a partially reduced analytic solution exists 3. Rigid body with axially symmetric ellipsoid of inertia (limited for the dynamic problem) for the case of an axially subjected to the viscous friction modeled as a torque symmetric body subjected to a constant torque [25].In equal to the opposite of the angular velocity vector left particular, Tsiotras and Longuski [25], building upon the multiplied by a matrix of constant coefficients [14,15]. development of Bodewadt¨ [26], use a complex form 4. Rigid body with axially symmetric ellipsoid of inertia subje- expression of the Euler’s dynamic equations and give a cted to the superposition of a viscous friction torque dire- solution for the angular velocity involving a not reduced cted as the angular momentum vector and a second torque, Fresnel integral which has to be evaluated over the which is either constant and inertially fixed or constant and duration of the maneuver. fixed with the axis of symmetry of the body [16]. Many researchers have proposed approximate solutions 5. Rigid body with spherical ellipsoid of inertia subjected for the motion of a rigid body. For instance, as regards the to a constant torque fixed with the body and arbitrary kinematic problem, Iserles and Nørsett [27] study the initial angular velocity [17]. solution in terms of series expansion for the more general 6. Rigid body with axially symmetric ellipsoid of inertia subje- problem of solving linear differential equation in Lie cted to a constant torque parallel to the symmetry axis [18]. Groups, building upon the work of Magnus [28]. Celledoni 7. Rigid body with axially symmetric ellipsoid of inertia and Saefstroem [29] propose ad hoc numerical integration subjected to a torque constant in magnitude, perpendicular algorithms. Finally, Livneh and Wie [30] introduce approx- to the symmetry axis and fixed with the body, and initial imate results for the motion of a triaxial rigid body angular velocity perpendicular to the symmetry axis [18]. subjected to a constant torque. 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This paper introduces for the first time, to the best where 2 3 knowledge of the author, the following original contributions: 0 rq 6 7 OðBo Þ¼4 r 0 p 5: ð3Þ 1. A version which is simplified, without loss of generality, BN qp 0 with respect to the one proposed in [17] is introduced for the exact analytic solution of the motion of a rigid In general, Eqs. (1) and (2) do not have an exact analytic body with spherical ellipsoid of inertia and subjected to solution. When the matrix OðtÞ commutes with its time a constant torque. integral, Eq. (2) has the solution [32] Z 2. A new exact analytic solution is introduced for the t B motion of a rigid body with axially symmetric ellipsoid RNBðtÞ¼RNBð0Þexp Oð oBNðxÞÞ dx ; ð4Þ of inertia subjected to a torque constant in magnitude 0 and parallel to the angular momentum vector. where expðÞ indicates the matrix exponential. 3. A new exact analytic solutions is introduced for the motion of a rigid body with axially symmetric ellipsoid 3. Exact analytic solutions for the motion of a rigid body of inertia subjected to a torque constant in magnitude with spherical ellipsoid of inertia and parallel to the vectorial component of the angular momentum vector on the plane perpendicular to the Let us assume, in this section, that the rigid body has a axis of the body. spherical ellipsoid of inertia, with 4. A detumbling maneuver with complete analytic reduc- I ¼ I ¼ I ¼ I: ð5Þ tion is introduced for axially symmetric spacecraft. 1 2 3 5. A nutation canceling maneuver with complete analytic This section summarizes the results introduced by Romano reduction is introduced for axially symmetric spacecraft. [17] for the motion of a rigid body with spherical ellipsoid of inertia and subjected to a constant torque. In particular, the mathematical developments here presented are simplified The results presented in this paper have both a with respect to Romano [17], without loosing generality, by theoretical importance for their basic mechanics implica- choosing a particular body-fixed coordinate system such that tions and a practical importance for the astrodynamic one component of the initial angular velocity is zeroed. applications related to spacecraft rotational maneuvering. The developments reported in this section will be used The paper is organized as follows: Section 2 introduces in the following sections of this paper in order to analyze the dynamic and kinematic equations for the rotational the motion of an axially symmetric rigid body. motion of a generic rigid body, Section 3 introduces the For a rigid body having a spherical ellipsoid of inertia, simplified solution for the motion of a spherically sym- and subjected to a constant body-fixed torque, the kine- metric body, Section 4 introduces the equation of motion matic differential equations in terms of the stereographic for an axially symmetric rigid body, Section 5 presents the complex rotation variables w [21,33] are [17] new exact analytic solutions, Section 6 introduces the k _ 1 2 1 detumbling and nutation canceling maneuvers. Finally, wk ¼ 2 p0wk iðr0 þ UtÞwk þ 2p0; k ¼ 1; 2; 3: ð6Þ Section 7 concludes the paper. where i is the imaginary unit and, without loosing generality (see Corollary 1 here below), we assume the 2. Rotation of a triaxial rigid body initial angular velocity and the acting torque (normalized by the value of the moment of inertia) to have components

For a generic rigid body the Euler’s dynamic equations of fp0; 0; q0g and f0; 0; Ug, respectively, in the chosen body- rotation are [31] ﬁxed coordinate system B.

I1p_ ¼ðI2 I3Þqr þ m1; Theorem 1. Given p0, r0, and U real numbers, the general _ I2q ¼ðI3 I1Þrp þ m2; solution for each one of Eq. (6), governing the rotational I3r_ ¼ðI1 I2Þpq þ m3; ð1Þ kinematics of a rigid body with spherical ellipsoid of inertia, initial angular velocity components pð0Þ¼p , qð0Þ¼0, and where fIi : i ¼ 1; 2; 3g are the principal moments of inertia, 0 fp; q; rg are the components of the absolute angular velocity rð0Þ¼r0 along the three body fixed axes s1, s2, and s3 (constituting the coordinate system B), and subjected to a along the principal coordinate system B, and fmi : i ¼ 1; 2; 3g are the components of the resultant external torque. constant torque U, normalized by the value of the moment of inertia and directed along the axis , is the following: The rotation matrix RNB 2 SOð3Þ from the principal body s3 pffiffiffiffi fixed coordinate system B to the inertial coordinate system ð1 þ iÞ U N obeys the following kinematic differential equation [6] wðt; cÞ¼ ½6z þ Gðz; cÞ; ð7Þ 3p0 _ B RNB ¼ RNBOð oBNÞ; ð2Þ with 3 n 5 n 3 1 n 3 2 F ; ; z2 ðn 1Þz2 þ 6 F 1 ; ; z2 cnz 3 F ; ; z2 1 1 2 2 1 1 2 2 1 1 2 2 Gðz; cÞ :¼ ; ð8Þ n 1 1 n 3 F ; ; z2 c þ F ; ; z2 z 1 1 2 2 1 1 2 2 ARTICLE IN PRESS

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where 1F1 denotes the confluent hypergeometric function [34], body-fixed coordinate system F by the unit vector c 2 C is the constant of integration and 8 9 <> u1 => 2 F ð1 þ iÞðr þ UtÞ ip u^ ¼ u2 ð14Þ z :¼ p0ffiffiffiffi ; n :¼1 0 : ð9Þ :> ;> 2 U 4U u3

F and that the initial angular velocity o0 is expressed in F by o0. Corollary 1. The solution in terms of the rotation matrix, Finally, let us assume that the initial orientation of F with which corresponds to the solution given by Theorem 1 in terms respect to the inertial coordinate system N is given by RFNð0Þ. of stereographic rotation variables, is Then, by exploiting the property of successive rotations, it results RNBðtÞ¼½rkjðtÞ; k; j ¼ 1; 2; 3; ð10Þ R ðtÞ¼R R ðtÞR R ð0Þ; ð15Þ with FN FB BN BF FN where R is a time-independent rotation matrix which iðw w Þ w þ w FB ¼ k k ¼ k k rk1 2 ; rk2 2 ; describes the orientation of the generic body-ﬁxed coordinate 1 þjwkj 1 þjwkj system F with respect to the particular body-ﬁxed coordinate 1 jw j2 ¼ k ¼ ð Þ system B, in which the torque coincides with the third axis and rk3 2 ; k 1; 2; 3; 11 1 þjwkj the initial angular velocity has zero component along the where wk ¼ wðt; ckÞ, being wðt; ckÞ given by Eq. (7) with the second axis. following values for the initial conditions (obtained by In particular, the matrix RFB is considering the rotation matrix RNB to be equal to the identity F F F matrix at the initial time t ¼ 0, without loss of generality): RFB ¼½ s1; s2; s3; ð16Þ

8 ! 9 > 2 > > 1 n 3 ir0 2 > <> 1F1 ; ; ½6r0 þ 3p0r0 þ 6iUþ2g => ð1 þ iÞ 2 2 2U c ¼ pffiffiffiffi ! ! 1 > 2 2 > 6 U > n 3 ir n 1 ir > :>2 F 1 ; ; 0 nr þ F ; ; 0 ðp þ 2r Þ;> 1 1 2 2 2U 0 1 1 2 2 2U 0 0 8 ! 9 > 2 > > 1 n 3 ir0 2 > <> 1F1 ; ; ½6ir0 3p0r0 6U2g => ð1 iÞ 2 2 2U c ¼ pffiffiffiffi ! ! 2 > 2 2 > 6 U > n 3 ir n 1 ir > :>2 F 1 ; ; 0 nr þ F ; ; 0 ðip þ 2r Þ;> 1 1 2 2 2U 0 1 1 2 2 2U 0 0

! 8 9 fF ¼ g > 2 > where si : i 1; 2; 3 are the column matrices of compo- > 1 n 3 ir0 2 > <> g 31F1 ; ; ðr0 þ iUÞ => nents in the coordinate system F of the unit vectors fsi : i ¼ ð1 þ iÞ 2 2 2U c ¼ pffiffiffiffi ! !; 1; 2; 3g identifying the axes of the coordinate system B. These 3 > 2 2 > 6r0 U > n 3 ir n 1 ir > :> F 1 ; ; 0 n þ F ; ; 0 ;> column matrices are given by 1 1 2 2 2U 1 1 2 2 2U F F ð12Þ s3 ¼ u^ ; where n is defined as in Eq. (9), and F ^ F F ð uÞ o0 ! s2 ¼ ; 2 F ^ F 3 n 5 ir jð uÞ o0j g :¼ F ; ; 0 ðn 1Þr2: ð13Þ 1 1 2 2 2U 0 F F F s1 ¼ð s2 Þ s3; ð17Þ where ðÞ indicates the matricial form of the vector In the case of having a general body-fixed coordinate product (skew symmetric matrix having the same system which is not coincident with a given inertial B coordinate system N at the initial time, and whit respect structure as the matrix Oð oBNÞ in Eq. (3)). Consequently, to which the constant torque is not directed along the third it yields axis and the initial angular velocity has non-zero compo- 8 9 8 9 > > > p > nents along the three axes, the results of Theorem 1 and < 0 = < 0 = B F B F Corollary 1 are still applicable, with the modification u ¼ RBF u ¼ 0 ; o0 ¼ RBF o0 ¼ 0 : ð18Þ :> ;> :> ;> reported in the following corollary. U r0

Corollary 2. Let us consider that the direction of the Finally, RBNðtÞ is obtained by transposing the resulting normalized external torque u ¼ m=I is identiﬁed in a generic matrix of Eq. (10) of Corollary 1, and by using Eq. (18). ARTICLE IN PRESS

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4. Rotation of an axially symmetric rigid body Furthermore, by inserting the ﬁrst equality of Eq. (23) into the Euler’s equation, it gives Let us assume, for the rest of this paper, that the rigid h_ ¼ Io_ ¼ m: ð24Þ body has a revolution ellipsoid of inertia, with h Eqs. (22) and (24) are the mathematical expression of a I1 ¼ I2 ¼ I I3: ð19Þ the Reduction Theorem [20] which can be stated as follows: The absolute angular momentum of the body ðhÞ can the evolution in time of the absolute angular momentum of therefore be expressed as an axially symmetric body subjected to an external torque m is equal to the evolution in time of the angular h ¼ Io þ I o ; ð20Þ ? 3 J momentum of a ‘‘virtual’’ homogeneous spherical body where o? is the orthogonal vectorial component of the (here doubled as Virtual Sphere) with the same value of the angular velocity o on the plane normal to the body axis e transversal inertia of the axially symmetric body and and oJ ¼ðo eÞe is the orthogonal vectorial component of subjected to the same external torque. the angular velocity along the body axis. In order to analyze in more details the motion of the Alternatively, by following the development of Hestenes axially symmetric body, we consider the following three [20], the absolute angular momentum of the body can be Cartesian coordinate systems: also expressed by 1. A principal coordinate system B attached to the axially h ¼ Io þðI3 IÞðo eÞe: ð21Þ symmetric body and centered at its center of mass. Conversely, the angular velocity can be seen, at any 2. A coordinate system S attached to the Virtual Sphere and instant of time, as the sum of two vectorial components, centered at its center of mass. whose one is parallel to the angular momentum vector ðohÞ 3. An inertially ﬁxed coordinate system N. and the other one is parallel to the symmetry axis ðoeÞ, i.e. o ¼ o þ o : ð22Þ h e Assumptions 1. Without loosing generality we assume

Notably, the vectorial components oh and oe are not that the coordinate system B has its third axis parallel to orthogonal. Therefore, the magnitudes of oh and oe are not the axis of inertial symmetry of the body ðeÞ, and that the given by the scalar product of the vector o with the unit coordinate systems B and S have superimposed axes at the a vectors h and e. In other words, it yields oe oJ. See also initial time ðt ¼ 0Þ, i.e. that RBSð0Þ is an identity matrix. Fig. 1. Because of Eqs. (22) and (24) and Assumptions 1, we can In particular, from Eqs. (21) and (22), by taking into see oh as the absolute angular velocity of the coordinate account that o e ¼ðh eÞ=I3, it yields system S with respect to N, and oe ¼ Aðoh eÞ as the h ðI I3Þ h angular rate characterizing the relative spinning of the oh ¼ ; oe ¼ e e ¼ Aðoh eÞe; ð23Þ I I3 I coordinate system B with respect to the coordinate system S about their common third axis. with A is a constant deﬁned as A ¼ðI I3Þ=I3. This is an interesting result: the vectorial component of the Consequently, we can rewrite Eq. (22) in the following angular velocity along e is function of the vectorial angular matrix form velocity along h. Therefore, if is known, the full angular B S B oh oBNðtÞ¼RBSðtÞ oSNðtÞþ oBSðtÞ; ð25Þ velocity of the axially symmetric rigid body is also known. B where oBNðtÞ is the column vector of components of the angular velocity of B with respect to N resolved in B, and

R ðtÞBhðtÞ So ðtÞ¼R ðtÞBo ¼ SB ; Bo ðtÞ¼Bo : ð26Þ SN SB h I BS e h e The kinematic solution associated to Eq. (25) is R ðtÞ¼R ðtÞR ðtÞ; ð27Þ BN BS SN h || = as it can be immediately demonstrated. h I Furthermore, Eq. (24) can be rewritten in matrix form as 8 9 <> p_ => 1 1 S _ ¼ _ ¼ S ¼ B ð Þ oSN > q > m RSB m; 28 = : ; I I e A ( h ⋅ e e ) r_ ⊥ with the initial condition 8 9 > > 1 1 > p0 > B < = I I RSBð0Þ h q 3 So ð0Þ¼ 0 ¼ 0 ; ð29Þ SN I > I > :> 3r ;> I 0

Fig. 1. Angular momentum, angular velocity and axis of symmetry for a where ðp; q; rÞ are the components of o along B and ðp; q; rÞ generic axially symmetric body, represented by its inertia ellipsoid. are the components of oh along S. ARTICLE IN PRESS

994 M. Romano / Acta Astronautica 66 (2010) 989–998

In particular, for all of the cases when the external acting 5.1. Case of constant torque along the angular momentum torque depends at most on time (self-excited rigid body), vector from Eqs. (28) and (23) it results Z The result introduced by the following theorem regards I 1 t o ðtÞ¼ArðtÞ¼A 3 r þ m ðxÞ dx ; ð30Þ the case of applied constant torque along the initial angular e I 0 I e 0 momentum vector of the axially symmetric body, and where me ¼ m e. therefore also along the angular momentum vector at Finally, the evolution in time of RBS, describing the any time during the motion, since the direction of the elementary rotation of coordinate system B with respect to angular momentum vector cannot change in this case. S about the axis e, is given by (see also Eqs. (2) and (4)) Furthermore, the direction of the angular momentum Z t vector is also the direction of the angular velocity of the S Virtual Sphere, because of Eq. (23). The development in this RBSðtÞ¼RBSð0Þexp Oð oSBÞðxÞdx 2 0 3 section will be exploited later in this paper for the analysis cosðf ðtÞÞ sinðf ðtÞÞ 0 of the detumbling maneuver of an axially symmetric 6 7 ¼ 4 sinðf ðtÞÞ cosðf ðtÞÞ 0 5; ð31Þ spacecraft. 001 Theorem 2. Given p0, q0, r0, and U real numbers, the solution with of the dynamic problem of determining, at any time t, the Z t absolute angular velocity of a rigid body having two equal f ðtÞ¼ oeðxÞ dx ð32Þ principal moments of inertia about the principal body axes s1 0 and s2, initial angular velocity components pð0Þ¼p0, and qð0Þ¼q0, and rð0Þ¼r0 along the three principal axes, and 8 9 subjected to an external torque of constant magnitude ðmÞ and > 0 > < = always parallel to the angular momentum vector of the body So ¼ Bo ¼ 0 : ð33Þ SB SB :> ;> (and, therefore, to the angular velocity of the Virtual Sphere), is oeðtÞ the following: 8 9 In conclusion, Eqs. (25) and (27) determine the <> p0cosðf ðtÞÞ þ q0sinðf ðtÞÞ => dynamics and the kinematics, respectively, of the motion B Ut o ðtÞ¼ p0sinðf ðtÞÞ þ q0cosðf ðtÞÞ 1 þ ; BN > > S of an axially symmetric rigid body (with moments of : ; j oSNð0Þj r0 inertia I and I3). In particular, the overall motion of the axially symmetric body (i.e. the motion of B w.r.t. N)is ð35Þ decomposed into the combination of the absolute rotation of an homogeneous Virtual Sphere with principal moment where U ¼ m=I, of inertia I (i.e. the motion of S w.r.t. N), and the relative "# I I Ut2 spinning of the axially symmetric rigid body with respect to ð Þ¼ 3 þ ð Þ f t r0 t S 36 the Virtual Sphere about the inertial symmetry axis e (i.e. I 2j oSNð0Þj the motion of B w.r.t. S). The critical advantage of the proposed approach is that and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the two composing motions can be solved independently in I 2 several significant cases, by using Eqs. (25)–(33), and jSo ð0Þj ¼ p2 þ q2 þ 3 r : ð37Þ SN 0 0 I 0 determining the remaining unknown quantities. These unknown quantities are, namely, the angular velocity Furthermore, the solution of the correspondent kinematic components pðtÞ; qðtÞ, obtained by integrating Eq. (28), problem of determining, at any time t, the orientation of the and the rotation matrix RNSðtÞ, obtained by integrating the body with respect to the inertial frame, i.e. of the coordinate following kinematic differential equation system B with respect to N, is given by _ S RNS ¼ RNSOð oSNÞ: ð34Þ RBNðtÞ¼RBSðtÞRSNðtÞð38Þ

where RBSðtÞ is given by Eq. (31) with f ðtÞ as in Eq. (36), and Z 5. New exact analytic solutions for the motion of an t S axially symmetric rigid body RNSðtÞ¼RNSð0Þexp Oð oSNðxÞÞ dx : ð39Þ 0 The exact analytic solution for the complete dynamic and kinematic problems of an axially symmetric rigid body Proof. This theorem directly follows from the analytical were introduced by Romano [18] for several signiﬁcant developments of Section 4. In particular the angular cases, by exploiting the Reduction Theorem approach. velocity of the Virtual Sphere is given by 8 9 Those cases are listed in the introduction of this paper. 8 9 > p0 > The following two sections introduce two new results <> pðtÞ => <> => q Ut with respect to [18]. This results will be exploited later in So ðtÞ¼ qðtÞ ¼ 0 1 þ ; ð40Þ SN > > > > jS ð Þj this paper for the analysis of the detumbling and nutation : ; > I3 > oSN 0 rðtÞ : r0 ; canceling maneuvers of an axially symmetric spacecraft. I ARTICLE IN PRESS

M. Romano / Acta Astronautica 66 (2010) 989–998 995 as obtained by integrating Eq. (28), with initial conditions subjected to an external torque of constant magnitude ðmÞ and given by Eq. (29) and always parallel to the transversal vectorial component of the angular momentum vector of the axially symmetric body (and, 1 Bm So ð0Þ R Bm ¼ ¼ U SN ; ð41Þ therefore, to the transversal component of the angular velocity I SB I jSo ð0Þj SN of the Virtual Sphere), is the following: which corresponds to the theorem’s hypothesis of having a Bo ðtÞ constant external torque m parallel to the initial angular BN8 8 92 3 9 > <> p0cosðf ðtÞÞ þ q0sinðf ðtÞÞ => > velocity of the sphere. <> 6 Ut 7 => p0sinðf ðtÞÞ þ q0cosðf ðtÞÞ 41 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 Moreover, Eq. (35) is found through the use of Eq. (25), ¼ :> ;> 2 2 ; > p0 þ q0 > with So ðtÞ given by Eq. (40), and Bo given by Eq. (33) :> ;> SN BS r with 0 ð46Þ I I3 Ut oe ¼ ArðtÞ¼ r0 1 þ : ð42Þ I jS ð0Þj where U ¼ m=I, and oSN I I3 Finally, the rotation matrix RNSðtÞ (given by Eq. (39)) is f ðtÞ¼ r t: ð47Þ I 0 obtained by applying the method of Eq. (4), which can be used here as the commutation condition holds, with the Furthermore, the solution of the correspondent kinematic angular velocity components given by Eq. (40). problem of determining, at any time t, the orientation of the

A perhaps simpler alternative to obtain RNSðtÞ is to use the body with respect to the inertial frame, i.e. of the coordinate Euler’s Theorem of rotation (see, for instance, [31]) system B with respect to N, is given by T R ðtÞ¼R ðtÞR ðtÞ; ð48Þ RSNðtÞ¼fcosðfðtÞÞI þ½1 cosðfðtÞÞaa BN BS SN sinðfðtÞÞOðaÞgRSNð0Þ; ð43Þ where RBSðtÞ is given by Eq. (31) with f ðtÞ as in Eq. (47), and where OðÞ is the skew-symmetric matrix function appear- RSNðtÞ is obtained from Eq. (15) of the Corollary 2 of Theorem ing also in Eq. (3), I indicates the three by three identity 1, by substituting the subindex F by S and the time- matrix, a is the axis of rotation independent matrix RFB by the following matrix, computed through Eq. (16), So ð0Þ a ¼ SN ð44Þ 2 3 jSo ð0Þj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip0 SN 6 0 7 6 2 þ 2 2 þ 2 7 6 p0 q0 p0 q0 7 and, finally, the Euler’s angle is 6 7 R ¼ 6 p0 q0 7: ð49Þ U FB 6 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 S 2 6 2 2 2 2 7 fðtÞ¼joSNð0Þjt þ t : ð45Þ p þ q p þ q 2 4 0 0 0 0 5 10 0 Interestingly, RBNðtÞ (correctly found through Eq. (38)) B cannot be directly obtained by using Eq. (4), with oBNðtÞ Proof. This theorem follows directly from the analytical from Eq. (35), as the commutation condition is not verified developments of Section 4. In particular the angular in this case. & velocity of the Virtual Sphere is given by 2 3 8 () 9 8 9 > > 5.2. Case of constant torque along the transversal component > p0 6 Ut 7 > > pðtÞ > > 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 > of the angular momentum vector < = < 1 þ = S q0 2 2 o ðtÞ¼ qðtÞ ¼ p0 þ q0 ; ð50Þ SN :> ;> > > rðtÞ > > The result introduced by the following theorem regards :> I3 ;> r0 the case of applied constant torque along the transversal I vectorial component of the angular momentum vector of as obtained by integrating Eq. (28), with initial conditions the axially symmetric body, i.e. the component on the plane given by Eq. (29) and perpendicular to the symmetry axis e. The torque direction 2 3 8 () 9 is also equal to the direction of the transversal component > > <> p0 6 U 7 => S 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 of the angular velocity of the Virtual Sphere, because of 1 B m RSB m ¼ ¼ q0 2 2 ; ð51Þ Eq. (29), and also to the direction of the transversal I I > p0 þ q0 > :> ;> component of the angular velocity of the body, because of 0 Eq. (20). The development in this section will be exploited later in this paper for the analysis of the nutation canceling which corresponds to the theorem’s hypothesis of having a maneuver of an axially symmetric spacecraft. constant external torque m parallel to the transversal vectorial component of the angular velocity of the sphere. Theorem 3. Given p0, q0, r0, and U real numbers, the solution Moreover, Eq. (46) is found through the use of Eq. (25), of the dynamic problem of determining, at any time t, the S B with oSNðtÞ given by Eq. (50), and oBS given by Eq. (33) absolute angular velocity of a rigid body having two equal with principal moments of inertia about the principal body axes s1 I I and s2, initial angular velocity components pð0Þ¼p0, o ¼ ArðtÞ¼ 3 r : ð52Þ e I 0 qð0Þ¼q0, and rð0Þ¼r0 along the three principal axes, and ARTICLE IN PRESS

996 M. Romano / Acta Astronautica 66 (2010) 989–998

Notably, in this case, neither RBN nor RSN can be found h through Eq. (4), as the commutation condition is not e satisﬁed in these cases. &

6. Controlling the rotation of an axially symmetric h = = spacecraft by the virtual sphere approach SN h I

This section introduces two new maneuvers to control the rotation of an axially symmetric spacecraft, namely a detumbling maneuver and a nutation canceling maneuver. Building upon the results presented in Section 5, it is shown here that the two proposed maneuvers have a complete exact analytic reduction, for the determination of both the 1 1 absolute angular velocity and the rotation matrix giving the I I attitude of the spacecraft with respect to the inertial frame. 3 For both maneuvers we assume that the initial absolute B attitude and angular velocity (i.e. RBNð0Þ and oBNð0Þ) are known. Furthermore, without loss of generality, we con- m sider the Assumptions 1 to be valid. Therefore, RBNð0Þ¼ RSNð0Þ and the initial angular velocity of the Virtual Sphere Fig. 2. Conceptual sketch of the proposed detumbling maneuver for an S ð oSNð0ÞÞ is given by Eq. (29). axially symmetric spacecraft, represented by its inertia ellipsoid. The gray In particular, for the detumbling maneuver, a torque is area represents the Virtual Sphere. The applied torque m has direction applied parallel to the angular momentum vector; on the opposite to the angular momentum vector. The torque is stationary w.r.t. the inertial frame and to the Virtual Sphere, while it is moving w.r.t. the other hand, for the nutation canceling maneuver, a torque is spacecraft body. applied parallel to the transversal vectorial component of the angular momentum of the spacecraft. In practical astronautical applications, for both maneu- metric spacecraft ver cases, the torque can be generated by on-board B S actuators (e.g. thrusters). mðtÞ¼RBSðtÞ m; ð54Þ

where RBSðtÞ is given by Eq. (31) with f ðtÞ as in Eq. (36) and S 6.1. Detumbling maneuver m is given by Eq. (53). By considering Eq. (35) with U ¼ m=I, the maneuver The proposed detumbling maneuver consists in apply- duration Dt, i.e. the time of application of the constant ing to the axially symmetric spacecraft a torque m which is torque in order to zeroing the angular velocity, yields constant in magnitude and kept ﬁxed with the Virtual jSo ð0ÞjI Dt ¼ SN ; ð55Þ Sphere, in the direction opposite to the angular momentum, m and therefore also to the initial angular velocity of the Virtual Sphere (see Fig. 2). which is a positive constant, since m is negative. The applied torque is therefore expressed in the Notably, the proposed detumbling maneuver is also the coordinate system S as optimal detumbling maneuver of minimum time if the maximum available torque is used (in case of spherically So ð0Þ constrained maximum torque). Indeed, in this case, the S ¼ SN ð Þ m m S ; 53 j oSNð0Þj problem is equivalent to the linear one-dimensional problem of bringing the Virtual Sphere to a rest condition where m is a negative constant, equal in magnitude to the starting from an arbitrary initial angular rate, by applying value of the acting torque. the maximum available torque parallel and opposite to the The torque is applied for a time duration such that the angular velocity vector. angular velocity of the Virtual Sphere is zeroed. By zeroing the angular velocity of the Virtual Sphere also the angular velocity of the axially symmetric spacecraft is zeroed, as 6.2. Nutation canceling maneuver it follows from the Reduction Theorem (see Eqs. (22) and (23)). The proposed nutation canceling maneuver consists in The proposed detumbling maneuver constitutes a applying to the axially symmetric spacecraft a torque m particular case of the motion analyzed in Theorem 2. which is constant in magnitude, ﬁxed with the Virtual Therefore the exact analytic solution introduced by Sphere, and having direction opposite to the vectorial Eqs. (35) and (38) (for the dynamics and the kinematics component of the angular momentum on the plane normal respectively) hold true. In particular, the proposed detum- to the body symmetry axis e, and, therefore, also opposite bling maneuver is conducted by applying the following to the transversal vectorial component of the angular external torque, e.g. with thrusters, to the axially sym- velocity of the Virtual Sphere (see Fig. 3). ARTICLE IN PRESS

M. Romano / Acta Astronautica 66 (2010) 989–998 997

In particular, the proposed nutation canceling maneuver h is conducted by applying the following external torque, e.g. e with thrusters, to the axially symmetric spacecraft

B S mðtÞ¼RBSðtÞ m; ð58Þ

where RBSðtÞ is given by Eq. (31) with f ðtÞ as in Eq. (47), and Sm is given by Eq. (56). By considering Eq. (46) with U ¼ m=I, the maneuver duration Dt, i.e. the time of application of the constant h⊥ torque in order to zeroing the transversal angular velocity, yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 p0 þ q0I Dt ¼ ; ð59Þ ⊥ 1 m I which is a positive constant, since m is negative. Notably, the proposed nutation canceling maneuver is m also the optimal nutation canceling maneuver of minimum 1 time if the maximum available torque is used (in case of

I3 spherically constrained maximum torque). Indeed, in this case, the problem is equivalent to the linear one-dimen-

Fig. 3. Conceptual sketch of the proposed nutation canceling maneuver sional problem of bringing to zero the transversal compo- for an axially symmetric spacecraft, represented by its inertia ellipsoid. nent of the angular velocity of the Virtual Sphere starting The gray area in the figure represents the Virtual Sphere. The applied from its arbitrary initial value, by applying the maximum torque m has direction opposite to the vectorial component of the available torque parallel and opposite to the angular angular momentum on the plane normal to the axis e. The torque is velocity vector component. stationary with respect to the Virtual Sphere, while it is moving with respect to both the inertial frame and the spacecraft body. 7. Conclusions The applied torque is therefore expressed in the coordinate system S as For the first time, to the knowledge of the author, the 2 3 8 () 9 exact analytic solutions, for both the kinematic and the > > > p0 6 1 7 > dynamic problems, have been introduced in this paper for < 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 = S m ¼ m q0 2 2 ; ð56Þ the rotational motion of a rigid body having revolution > p0 þ q0 > :> ;> ellipsoid of inertia, in the following two cases 0 where m is a negative constant. 1. External torque constant in magnitude and parallel to The torque is applied for a time duration such that the the angular momentum vector. transversal component of the angular velocity of the Virtual 2. External torque constant in magnitude and parallel to Sphere is zeroed. the vectorial component of the angular momentum The proposed nutation canceling maneuver constitutes vector on the plane perpendicular to the axis of the a particular case of the motion analyzed in Theorem 3. body. Therefore the exact analytic solution introduced by Eqs. (46) and (48) (for the dynamics and kinematics The proposed analytic solutions are valid for any length respectively) hold true. of time and rotation amplitude. The kinematic solutions are The conceptual rationale for this maneuver is the presented in terms of the rotation matrix. following: by zeroing the vectorial component of the angular In particular, the results of this paper are obtained by velocity of the Virtual Sphere perpendicular to the symmetry building upon the recently found exact analytic solution for axis of the spacecraft (i.e. the axis z of both the coordinate the motion of a rigid body with a spherical ellipsoid of inertia, systems S and B), also the transversal component of the and by decomposing the motion of an axially symmetric rigid angular momentum of the axially symmetric spacecraft is body into the combination of the motion of a Virtual Sphere zeroed, as it follows from the Reduction Theorem (Eq. (22)). with respect to the inertial frame and that of the axially Therefore, the nutation angle between the spacecraft symmetric body with respect to this Virtual Sphere. symmetry axis and the angular momentum vector is Based on these analytical developments, two new space- canceled. Consequently, the precessing motion of the spacecraft control maneuvers are proposed which have a craft symmetry axis about the angular momentum vector complete analytic reduction: namely, a detumbling maneu- stops and the spacecraft is left in a pure spinning motion ver and a nutation canceling maneuver. In particular, for the about its axis of symmetry, with angular velocity detumbling maneuver the applied torque is kept parallel to 8 9 the initial angular velocity of the virtual sphere; on the other <> 0 => hand, for the nutation cancelation maneuver, the applied Bo ðtÞ¼ 0 : ð57Þ BN :> ;> torque is kept parallel to the transversal vectorial component r0 of the angular velocity of the spacecraft. In practice, the ARTICLE IN PRESS

998 M. Romano / Acta Astronautica 66 (2010) 989–998 requested torque can be generated by on-board actuators, as, [15] E.A. Ivanova, A new approach to solution of some problems of the for instance, body-fixed thrusters commanded through a rigid body dynamics, Zeitschrift fuer Angewandte Mathematik und Mechanik 81 (9) (2001) 613–622. modulation and a mapping scheme in order to produce an [16] E.A. Ivanova, Rotation of rigid body under the action of constant approximately linear duty cycle (see, for instance, [2]). motor moment and friction moment, in: Proceedings of the The analytical results presented in the paper have been XXX Summer School ‘‘Advanced Problems in Mechanics’’, 2003, pp. 292–296. verified by numerical experiments with sample numerical [17] M. 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