Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field


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1 Dynamics Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field Amit K. Sanyal, Jinglai Shen, N. Harris McClamroch 1 Department of Aerospace Engineering University of Michigan Ann Arbor, MI {asanyal, jinglais, Abstract The dynamics of a dumbbell shaped spacecraft a modeled as two identical mass particles connected by a linear elastic spring. The equations of motion of the spacecraft in a planar orbit in a central gravitational field a psented. The equations of motion characterize orbit, attitude, shape or elastic deformation deges of fedom the coupling between them. Relative equilibria, corsponding to circular planar orbits, a obtained from these equations of motion. Linear equations of motion that describe perturbations from these lative equilibria a psented. New dynamics control problems a introduced for these linear equations. Controllability sults a psented for various actuation assumptions, based on the linear equations. individual mass particle of the dumbbell spacecraft. The diffential gravity effects about the center of mass play a crucial role in the overall spacecraft dynamics. Although the dumbbell spacecraft model is exceedingly simple, it is effective in demonstrating complex dynamics control phenomena that can arise when a multibody spacecraft is in orbit about a massive central spherical body. It provides a framework for studying the orbital, attitude, shape deges of fedom, the coupling between them. The dynamics of such multibody systems psent significant analytical challenges. In this paper, we introduce new orbital, attitude, shape control problems that have not been pviously studied in the published literatu. We demonstrate that the spacecraft is completely controllable, based on the linear equations of motion, for several diffent actuation assumptions. 1 Introduction This paper models a dumbbellshaped spacecraft moving in a fixed orbital plane in a central gravitational field. The equations of motion describe the translational or orbit dynamics, the rotational dynamics, the shape or elastic deformation dynamics. The spacecraft model consists of two ideal mass particles of identical mass connected by a linear elastic spring. The spring is assumed to be massless, rigid in the transverse diction, flexible along its longitudinal axis. The dumbbell spacecraft can rotate translate in a plane. The shape of the spacecraft is given by the distance of each mass particle from its center of mass. We make the following assumptions throughout the paper: an inertial frame is chosen such that its origin is at the center of a large idealized spherical central body, e.g. the Earth. A gravitational force acts on each 1 This search has been supported in part by NSF under grants ECS ECS The dumbbell spacecraft can also be viewed as a model of a tethed spacecraft. The levant published literatu tats the dynamics control of tethed spacecraft. Deployment, station keeping, trieval of tethers have been studied in [1]. Attitude control issues for tethed spacecraft have been tated in []. Orbital dynamics control issues for tethed spacecraft have been tated in [3, 4]. The use of attitudeorbit coupling to alter spacecraft orbital motions by controlling the attitude was dealt with in [5, 6, 7]. The authors a not awa of any published papers that tat orbital, attitude shape dynamics issues in a unified way. This paper makes a contribution to this problem for the simplified dumbbell spacecraft. The psent paper can also be viewed as an extension of [8]. In that prior paper the coupling between translation, rotation, shape deges of fedom was studied, new control problems that exploit the coupling between them we suggested. However, [8] did not include a central body gravity field, so the sults in that paper a not applicable to the problems con
2 sided he. Finally, we contrast the sults in this paper with sults in our earlier papers that tat control problems involving translation, rotation, shape deges of fedom. Examples of our prior work in this diction include [9, 10, 11]. In those papers, gravitational effects we completely ignod, the coupling between translation, attitude shape was inhently nonlinear. In this paper, the coupling between translation or orbit, attitude shape deges of fedom is effectively linear, since it arises from the central body gravity assumption. The diffence between linear nonlinear coupling effects is very important in actually exploiting these mechanisms for control purposes. Equations of Motion Let r be the radial distance of the spacecraft center of mass from the origin of the inertial frame, φ be the angle between this radial vector the principal inertial axis, q be the distance of each mass particle from the spacecraft center of mass, θ be the attitude angle of the dumbbell spacecraft with spect to the radial diction. Let m be the mass of each mass particle in the dumbbell spacecraft. We approximate the gravitational potential energy up to second order in q r 1 as V g = m r q r 1 3cos θ. 1 The elastic potential energy is given by V s = kq l, whe l is half the distance between the two mass particles when the spring force is zero. Thus, the Lagrangian is given by L = m q +ṙ + q θ +q θ φ + q φ + r φ + q r r 1 3cos θ kq l. 3 Note that the Lagrangian is independent of φ, this corsponds to conservation of the total angular momentum L φ =mr + q φ + q θ, in the absence of external forces along the local horizontal. from the above Lagrangian, a: m r m r φ r + 3q r 1 4 3cos θ = f r, 4 mr φ + q φ + q θ +q q θ +q q φ +rṙ φ =fh, 5 mq θ + q 6m φ +q θ q +q q φ+ r 3 q cos θ sin θ = τ, 6 m q m q θ +4q θ φ +q φ q r 1 3 3cos θ +kq l =f s, 7 for the local radial, local horizontal, attitude, shape deges of fedom, spectively. He f r, f h, τ, f s a the generalized forces corsponding to the generalized coordinates r, φ, θ, q, spectively. 3 Dynamics Control of Orbit, Attitude Shape In this section, we study the dynamics control of orbit, attitude, shape deges of fedom of the dumbbell spacecraft. Relative equilibria a identified. Linear equations of motion for the orbit, attitude shape dynamics a obtained that describe the perturbations from a lative equilibria. Several controllability sults a psented based on the linear perturbation equations. 3.1 Relative Equilibria for the Orbit, Attitude Shape Dynamics We first identify natural lative equilibria that corspond to circular orbits for this dumbbell spacecraft. At a lative equilibrium: r =ṙ =0, φ =0, θ =0, q =0, fr = f h = τ = f s =0. We use the subscript e to denote quantities evaluated at a lative equilibrium. The local horizontal orbit equation of motion is trivially satisfied at a lative equilibrium. Equations 4, 6 7 give the following algebraic equations for lative equilibria: r e φ e = 6qe 3 3q e 4 1 3cos θ e, 8 cos θ e sin θ e =0, 9 q e φ e q e 3 1 3cos θ e = k m q e l. 10 We assume that q e 0. The EulerLagrange equations of motion, obtained From equations 8, 9, 10 we obtain the follow
3 ing two categories of lative equilibria: θ e = nπ, n Z, φ e = 3 + 3q e 5, 3q e 3 = k m q e l, 11 θ e =n + 1 π, n Z, φ e = 3 3q e 5, q e = l. 1 The class of lative equilibria given by equation 11 corsponds to the dumbbell spacecraft aligned with the local radial. The class of lative equilibria given by equation 1 corsponds to the dumbbell spacecraft aligned with the local horizontal. 3. Linearized Orbit, Attitude Shape Dynamics We now derive linear equations of motion for perturbations from a lative equilibrium. Let x =[δr δφ δθ δq] T denote the vector of configuration perturbations, let u = [f r f h τ f s ] denote the vector of inputs. We can expss the linear equations of motion about a lative equilibrium given by 11 with θ e =0as whe C 1 = M 1 ẍ + C 1 ẋ + K 1 x = Bu, 13 M 1 = m 0 0 m + qe mqe 0 0 mqe mqe 0 0 m 0 4mr eω 1 4mr eω 1 4mq eω 1 0 4mq eω 1 0 4mq eω 1 4mq eω 1 0 { m K 1 = ω 1 + } + 1q e mq e r 4 e 6mq e, 14, mq... e { }, k m ω1 + 3 He M 1 is symmetric positive definite, C 1 is skewsymmetric, K 1 is symmetric. The matrix B Now we provide the linear equations of motion for perturbations from a lative equilibrium given by 1 with θ e = π. We can expss the linearized equations of motion in the form whe M ẍ + C ẋ + K x = Bu, 17 C = K = M = m 0 0 m + qe mqe 0 0 mqe mqe 0 0 m 0 4mr eω 4mr eω 4mq eω 0 4mq eω 0 4mq eω 4mq eω 0 { m ω1 + 3 ω = 6q e r 5 e } 6mq e r 4 e mq e r 4 e 3q e 6mq e k He M is symmetric positive definite, C is skewsymmetric, K is symmetric. The matrix B Note that the linearized equations about both classes of lative equilibria form linear gyroscopic systems. The study of system theotic properties of such systems has been dealt with in prior literatu, a sample of which can be found in [1, 13, 14] the fences thein. We now study controllability under various actuation assumptions for these equations. 3.3 Controllability of the Linear Orbit, Attitude Shape Equations of Motion The linear equations of motion a expssed in a stard linear second order control vector form [13] Mẍ + Cẋ + Kx = Bu 1, 18, 19, 0 ω 1 = + 3q e whe x R n, M is a symmetric positive definite matrix, C is a skewsymmetric matrix flecting gyroscopic effects, K is a symmetric matrix. Necessary sufficient conditions for complete controllability for
4 such systems a developed in [14]. Equation 1 is completely controllable if only if the controllability rank condition rank[λ M + λc + K, B] =n holds for all λ that satisfies det[λ M + λc + K] =0. We fer to the matrix [λ M + λc + K, B] asthe controllability matrix. This controllability sult is now used to give the following sults. Proposition 1. If f h = 0, then the linear orbit, attitude shape equations of motion a not completely controllable. Proof. Suppose the is no orbital actuation along the local horizontal, but the other the deges of fedom a actuated. Since the second row of both K 1 K a zero, it follows that the controllability matrix has a row of zeros for the second row when λ = 0. The sult follows. The above sult is consistent with the fact that the angular momentum of the spacecraft in orbit about the central body is conserved if the is no orbital actuation in the horizontal diction. Proposition. The linear orbit, attitude shape equations of motion a completely controllable if the orbit deges of fedom a actuated. Proof. In this case the controllability matrix has rank 4 for all values of λ. Proposition 3. The linear orbit, attitude shape equations of motion a completely controllable if the horizontal orbit dege of fedom the attitude dege of fedom a actuated. Proof. In this case, the controllability matrix has rank 4 for all values of λ. Proposition 4. The linear orbit, attitude shape equations of motion a completely controllable if the horizontal orbit dege of fedom the shape dege of fedom a actuated. fedom is actuated. This is possible due to the linear coupling between the orbit, attitude shape deges of fedom. Although these coupling effects may be weak, they may still be used to control the spacecraft using established linear control methods. 4 Dynamics Control of Orbit Attitude In this section, we study the dynamics control of the orbit attitude deges of fedom of the dumbbell spacecraft. The shape of the dumbbell spacecraft is assumed fixed. Relative equilibria for the orbit attitude dynamics a identified. Linear equations of motion a obtained that describe the perturbations from a lative equilibria. Several controllability sults a psented based on the linear perturbation equations. 4.1 Relative Equilibria for the Orbit Attitude Dynamics When the spacecraft has a fixed shape q e 0, the nonlinear equations of motion 47 become: m r m r φ r + 3q e r 4 1 3cos θ = f r, mr φ + q e φ + q e θ +rṙ φ =fh, 3 mqe θ 6m + φ+ r 3 q e cos θ sin θ = τ, 4 mq e θ + φ q e r 3 1 3cos θ +kq e l =f s. 5 Equation 5 defines the generalized shape force f s quid to maintain a constant shape. In the mainder of this section, equation 5 is assumed to be satisfied, it is not consided further. In this case, the lative equilibria satisfy r =ṙ =0, φ =0, θ =0, fr = f h = τ =0. The lative equilibria a thus characterised by equations 8 9. As in the general case consided in section 3, the a two categories of lative equilibria: Proof. In this case, the controllability matrix has rank 4 for all values of λ. θ e = nπ, n Z, φ e = + 3q e 5, 6 Propositions, 3 4 show that the dumbbell spacecraft is completely controllable for perturbations about its natural lative equilibria under various cases of underactuation, provided the horizontal orbit dege of θ e =n + 1 π, n Z, φ e = 3q e 5, 7 They corspond to the dumbbell spacecraft aligned with the local radial the local horizontal, spec
5 tively. 4. Linearized Orbit Attitude Dynamics We first consider a lative equilibrium with θ e =0, which belongs to the first category given by equation 6. Let x =[δr δφ δθ] T denote the vector of configuration perturbations, let u =[f r f h τ] denote the vector of control inputs. We can expss the linearized equations of motion as M 1 ẍ + C 1 ẋ + K 1 x = Bu, 8 whe M 1 = C 1 = K 1 = ω 1 = m 0 m + qe mqe 0 mqe mqe 0 4mr eω 1 0 4mr eω 1 0 m ω q e, 9, q e 5 0 6mq e,31 He M 1 is symmetric positive definite, C 1 is skewsymmetric, K 1 is symmetric. The matrix B Now we consider a lative equilibrium given by 7 with θ e = π, which belongs to the second category of lative equilibria. We can expss the linearized equations of motion in the form M ẍ + C ẋ + K x = Bu, 3 whe M = C = K = m 0 m + qe mqe 0 mqe mqe 0 4mr eω 0 4mr eω 0 { m ω 1 +, 33, 34 } 6q e 5 0 6mq e,35 He M is symmetric positive definite, C is skewsymmetric, K is symmetric. The matrix B The linear perturbation equations about both classes of lative equilibria have the form of linear gyroscopic systems, as in the case when the shape is not fixed. We study controllability under various actuation assumptions for these equations. 4.3 Controllability of the Linear Orbit Attitude Equations of Motion We now study controllability of the linear orbit attitude equations of motion given by Corollary 1. If f h = 0, then the linear orbit attitude equations of motion a not completely controllable. This is a corollary of Proposition 1. As in the case when the shape is not fixed, it is necessary to actuate the horizontal orbital dege of fedom. In the following we look at two cases with such actuation. Corollary. The linear orbit attitude equations of motion a completely controllable if the orbit deges of fedom a actuated. This is a corollary of Proposition. Corollary 3. The linear orbit attitude equations of motion a completely controllable if the horizontal orbit dege of fedom the attitude dege of fedom a actuated. This is a corollary of Proposition 3. Corollaries 3 show that the dumbbell spacecraft with fixed shape is completely controllable for perturbations about its lative equilibria under some cases of underactuation, provided the horizontal orbit dege of fedom is actuated. This is possible due to the linear coupling between the orbital attitude deges of fedom, which may be used to control the spacecraft, using established linear control methods. ω = 3q e 5 Conclusion Equations of motion of an elastic dumbbell spacecraft in a planar orbit about a massive spherical central body
6 have been developed. Although the dumbbell spacecraft model, consisting of an elastic interconnection of two mass particles, is extmely simple, the model does illustrate the coupling between orbit deges of fedom, attitude deges of fedom, shape deges of fedom. Relative equilibria, corsponding to circular orbits, a determined, linear equations of motion that describe the perturbations from the lative equilibria a obtained. Our development tats the general case of orbit, attitude shape coupling. For completeness, we also tat the special case that the shape is constrained to be fixed. This leads to equations of motion that describe the orbit attitude dynamics of a dumbbell spacecraft of fixed shape. The control inputs consist, in the most general case, of horizontal vertical forces on the dumbbell spacecraft, a moment about an axis normal to the orbital plane, a force between the two mass particles of the dumbbell spacecraft. The full actutation assumption is that horizontal vertical control forces a used to control the orbit, a control moment is used to control the attitude, a shape control force is used to control the shape. However, we identify several meaningful control problems corsponding to underactuation assumptions. Our controllability analysis suggests that in many cases effective control can be achieved using coupling between the orbit, attitude shape deges of fedom. Many of the key coupling terms, such as the gravity gradient moment, a small their control influence only acts over long time periods. Thus, the practical utility of exploiting these coupling mechanisms, at least in some cases, mains to be demonstrated. Dynamics control problems have been suggested based on the models for a dumbbell spacecraft that have been developed. The linear controllability sults imply that orbital actuation is necessary to achieve complete controllability. If the is no orbital actuation, the models studied in this paper cannot be completely controllable since angular momentum is necessarily conserved. However, it is important to mention that, if the is no orbital actuation, a simplified model, ferd to as the Routh duced model, can be developed that incorporates the conservation law. This duced model also leads to new dynamics control problems that differ from the problems consided in this paper. Control problems for Routh duced models a of significant practical importance, they a tated in our forthcoming publications. Refences [1] A. K. Misra, V. J. Modi, A Survey on the Dynamics Control of Tethed Satellite Systems, Advances in Astronautical Science, vol. 6, pp , [] S. B. Cho, N. H. McClamroch, Attitude Control of a Tethed Spacecraft, Proceedings of the American Control Confence, pp , Denver, CO, 003. [3] V. J. Modi, A. K. Misra, Orbital Perturbations of Tethed Satellite Systems, Journal of Astronautical Sciences, pp , [4] S. B. Cho, N. H. McClamroch, Optimal Orbit Transfer of a Spacecraft with Fixed Tether Length, submitted for publication. [5] J.P.Moran, EffectsofPlaneLibrationsontheOrbital Motion of a Dumbbell Satellite, ARS Journal, vol. 31, no. 8, pp , [6] K.Y. Lian, L.S. Wang, L.C. Fu, Controllability of Spacecraft Systems in a Central Gravitational Field, IEEE Transactions on Automatic Control, vol. 39, no. 1, pp , [7] D. J. Schees, Stability in the Full TwoBody Problem, Celestial Mechanics Dynamical Astronomy, vol. 83, pp , 00. [8] S. Cho, N. H. McClamroch M. Reyhanoglu, Dynamics of multibody vehicle their formulation as nonlinear control systems, Proceedings of the American Control Confence, pp , Chicago, IL, 000. [9] M. Reyhanoglu N. H. McClamroch, Reorientation Maneuvers of Planar Multibody Systems in Space using Internal Controls, AIAA Journal of Guidance, Control Dynamics, vol. 15, no. 6, pp , 199. [10] C. Rui, I. Kolmanovsky, N. H. McClamroch, Nonlinear Attitude Shape Control of Spacecraft with Articulated Appendages Reaction Wheels, IEEE Transactions on Automatic Control, vol. 45, no. 8, pp , 000. [11] J. Shen N. H. McClamroch, Translational Rotational Maneuvers of an Underactuated Space Robot using Prismatic Actuators, International Journal of Robotics Research, vol. 1, no. 56, pp , 00. [1] J. E. Marsden, T. S. Ratiu, Introduction to Mechanics Symmetry, nd. ed., SpringerVerlag Inc., New York, [13] R. E. Skelton, Dynamic Systems Control: Linear Systems Analysis Synthesis, John Wiley & Sons Inc., New York, [14] A.J. Laub W.F. Arnold, Controllability Observability Criteria for Multivariable Linear Second Order Models, IEEE Transactions on Automatic Control, Vol. 9, No., pp , Acknowledgement The authors thank Dr. Sangbum Cho for help in formulating the concepts tated in this paper. Our discussions with him we instrumental in the development of the psent paper.
Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field
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