Linear Dynamic Modeling of Spacecraft With Various Flexible Appendages

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1 roceedings of the 7th World Congress he International ederation of utomatic Control Linear Dynamic Modeling of Spacecraft With Various lexible ppendages Khalid H.M. antawi Daniel lazard Christelle Cumer erospace masters student, ISE, 0, v. Edouard Belin, oulouse RNCE. ( Supervising rofessor at Université de oulouse-ise, 0, v. Edouard Belin, oulouse RNCE. ( Supervising scientist at ONER, 2, v. Edouard Belin, oulouse RNCE. ( bstract: We present here a method and some tools developed to build linear models of multi-body systems for space applications (typically satellites). he multi-body system is composed of a main body (hub) fitted with rigid and flexible appendages (solar panels, antennas, propellant tanks,...etc). Each appendage can be connected to the hub by a cantilever joint or a pivot joint. More generally, our method can be applied to any open mechanical chain. In our approach, the rigid six degrees of freedom (d.o.f) (three translational and three rotational) are treated all together. hat is very convenient to build linear models of complex multi-body systems. hen, the dynamics model used to design OCS, i.e. the model between forces and torques (applied on the hub) and angular and linear position and velocity of the hub, can be derived very easily. his model can be interpreted using block diagram representation. Keywords: Modeling, Dynamics, lexible modes, Effective mass. INRODUCION Spacecraft are very complex mechanical multi-body systems including flexible and/or rotating appendages. he design of the OCS requires a linear model taking into account all the rigid and flexible couplings between the hub (where the OCS acts) and the various appendages. Note that the linear assumption is quite realistic for such systems since perturbations and so motions are very small (except for very derous observation satellites). his linear assumption is furthermore valid in the field of future missions for deep space exploration involving formation flying of several spacecraft. or this kind of formation flying mission, it is more and more accepted that the 3 rotation d.o.f. and the 3 translation d.o.f. must be treated all together (aulocher et al. 2005). herefore, a 6 d.o.f. model including couplings between rotations and translations must be developed. Lots of multibody software are available to build such kind of models but they address the nonlinear behavior and they are too much loud to be handled at the early prototyping phase. So a tool is required to develop quickly the dynamic model and to prototype the OCS or to analyze and to optimize the main dynamic parameters of the mechanical structure or OCS and finally to assess the global performance of the system. Here we propose some tools developed with Matlab/SI- MULINK to built efficiently the linear dynamic model of any open mechanical chain. More precisely, the linear multi-body model considered here is depicted on igure. his model, called inverse dynamic model, gives the relationship between the inputs, which are composed of: the six ernal forces and torques,o applied on the hub (base) by the ttitude and Orbit Control System(OCS) at a reference point O, the n drive torques C m (i) applied at the pivot joint i (i,, n, n is the number of pivot joints) between an appendage and the hub, and the output, which is composed of: the six linear and angular accelerations of the hub at point O (resp. a O and ), the angular acceleration θ(i) of the pivot point i (for i,, n). ig.. eneral inverse dynamic model his paper introduces gradually each complexity of the modeling problem. he first section concerns the simplest case of two interconnected rigid bodies. Let us recall that the approach assumes that the hub (or central body or base) is rigid. In the second section flexible appendages are taken into account using effective mass representation /08/$ IC / KR

2 7th IC World Congress (IC'08) he way how a motorized pivot joint between an appendage and the hub can be taken into account is described in section 3. model validation is proposed in section 4.. INERCONNECED RIID BODIES MODEL Let us consider a spacecraft composed of a rigid main body or hub (called here after the base B) with its center of mass at point, and a rigid appendage cantilevered to the base B at point (see igure 4). Let us denote R (, x, y, z) the reference frame rigidly attached to the hub at and R (, x, y, z) the same frame translated to point. In the sequel the dynamic model of the appendage will be obviously given in the frame R. ig. 2. simple spacecraft model, two rigid bodies connected at point. Dynamic model of the base B at point Let us consider the base B alone (without appendage), according to the Newton s and Euler s equations, the dynamic model of the base B at its center of mass reads as follows: D B where a mi3 0 a 0 J B () m is the mass of the body B, I n is the n n identity matrix, J B is the inertia matrix (in kg.m2 ) at point of the body B in the frame R, is the absolute angular velocity vector of the body B (i.e. the angular velocity of the frame R w.r.t the inertial frame R i in (rad/s)). and d R d Ri, since has the same coordinates in R and R i. In (), the three translational accelerations and the three angular accelerations are considered together. Note that for the rotation dynamics, the relation, J B is a linear approximation, the actual non-linear dynamic equation reads:, J B J B where is the cross product. he nonlinear term ( J B ) on the right hand side of the equation above can be neglected if angular velocity is small enough (linear assumption)..2 ransport of the dynamic model of B from point to point Let us recall that the relation between the velocities at points and is : V V #» V ( ) (2) where ( ) is the antisymmetric matrix associated with the vector #.» hat is, if x, y, z R c is the coordinate vector of #» projected in any frame R c then ( ) reads: 0 z y 0 z y ( ) z 0 x, ( ) z 0 x. y x 0 R c y x 0 R c Note that equation (2) allows a vector product to be transformed into a matrix-vector product and can be projected in any frame. hen, the six d.o.f. kinematic vectors ν and ν of the body B respectively at points and are given by : V I3 ( V ) 0 I 3 (3) }{{}}{{}}{{} ν τ ν τ is called the (6 6) kinematic model between the points and. Now, let us consider the inertial accelerations at points and : a d V Ri and It is well-known that : a a #» or a rigid body, (d #» ) R 0 a d V Ri ((d #» ) ) R #» and, as explained before, all nonlinear terms can be neglected. he acceleration at point is then deduced from the acceleration at point by the linear relation : a a ( ) (4) rom equation (4) one can derive the following kinematic relationship : a a I3 ( ) a τ 0 I 3. (5) o obtain the relationship between the 6 d.o.f ernal force vectors at point and at point, it is interesting to express the ernal force power computed along a virtual velocity field : V V, Combining (3) and (6), one can easily obtain : I τ 3 0, ( ) I 3 (6) (7) 49

3 7th IC World Congress (IC'08) rom (5) and (7), the direct dynamic model D B base B at point becomes:, of the mi3 0 a a τ τ 0 J B D B (8) hus the transport of the direct dynamic model of a body B from a point to a point reads: mi 3 m( ) D B τ Dτ B. m( ) J B m( ) 2.3 Connection with a rigid appendage If we consider now that a rigid appendage is cantilevered to the base B at point, the reaction force B/ and torque B/, at point between the base and the appendage must be taken into account in the dynamic model of the base. hus equation (8) becomes: a B/, B/, D B (9). (0) he appendage is characterized by its own dynamic model D at point. If we assume that the only force and torque applied on the appendage are the reaction force and torque with the base B, then one can write: a B/ D () B/, Substituting () in (0) we get the equation of motion of the whole system at point : a (D D B ), a (D τ DB τ ) (2) It could be more interesting to express the whole dynamic model at the center of mass of the base B, since the ernal forces and torques will correspond to the OCS (reaction wheel and thrust) which are mounted on the base. hen, one can directly write: a (τ D τ D) B, a (D D) B (3) If # C» is the vector between and the center of mass C of the appendage in the frame R, we can also write: mi3 D τ 0 C τ C 0 JC his equation introduces the dynamic model of the appendage at the point : (D ). We can also compute the inverse dynamic model (which will be used for designing the OCS): a (D B D ) It can be shown that: (D B D ) D B (D Satellite ) (4) ) (I 6 τ D τ D B (5) Equation (4) can be expressed with the block diagram presented in igure 3 which highlights that the dynamic model of appendage acts as a feedback between acceleration and forces at point on the base B. Such a block diagram representation will be very useful to introduce uncertainties in the various geometric or dynamic parameters. B/ B/, τ - D B - D τ a a ig. 3. Block Diagram of the inverse Dynamic Model.4 Rigid Connection With Rotation: In the general case, a rotation matrix R 3 3 between the frame R (, x, y, z ) (in which the dynamic model D will be described) and the frame R (, x, y, z) (parallel to R at point ) must be taken into account. hat is illustrated in igure 4 in the special case where the appendage is rotated with an angle θ around z axis, that is: cos(θ) sin(θ) 0 R 3 3 sin(θ) cos(θ) 0. (6) 0 0 o write the equation of motion of the whole system at point (equation (4)), that is to compute D, we have to take into account the rotation on the dynamic model D before to transport this dynamic model from to : R3 3 0 D τ 0 R 3 3 }{{} R 6 6 D R3 3 0 s 0 R 3 3 s τ (7) 50

4 7th IC World Congress (IC'08) with : M (s) D L L L K L D 6 k sik I k K (si k D) 0k 6. L k 6 ig. 4. Rigid connection between the hub and a rotated solar array. 2. CONNECION O LEXIBLE ENDE lexibility of an appendage will be represented by the effective mass approach (Imbert and Mamode 977). his representation is very useful when one want to study dynamic couplings between the flexible modes of the appendage and the rigid modes of the whole system without analysis of internal deformations (or loads) of the appendage. he so called " Cantilever Hybrid Model" (see Cumer and Chrétien 200) will be used: at point, the static dynamic model of the appendage (equation ()) is now removed by the following differential equations: a B/ D L η (8) B/, a η diag(2ξ i w i ) η diag(wi 2 )η L where L l l 2 l k. (9) l i ( 6), w i, ξ i are the modal contribution 2 at point, the frequency, and the damping ratio of the flexible mode i respectively, for i,..., k (k is the number of flexible modes taken into account). η is the vector of flexible modal coordinates. he direct dynamic model of the appendage can also be described by the state-space representation: η 0k k I k η 0k 6 a η η K k k D k k L k 6 B/ L K L D η a (D B/, η L L ) (20) where D diag(2ξ i w i ) and K diag(wi 2). his state-space representation allows the direct transfer matrix M (s) between force and acceleration of the appendage at point (also called dynamic mass matrix) to be computed: B/ B/, a M (s) (2) In the case where flexible mode damping ratios are neglected (D 0), this transfer matrix can be re-arranged in the following way: k M (s) D 0 D wi 2 i s 2 wi 2 where: i D 0 is the residual mass matrix rigidly cantilevered to the base B at point and is given by: D 0 D L L D k i li l i, D i l i l i is rank- effective-mass matrix of the ith mode, D is the static gain (DC gain) of M (s). o build the dynamic model of the whole system (rigid base flexible appendage), we only have to remove D by M (s) (defined by state space representation (20)) in equations (2) to (5) and (7) and in the block diagram depicted in igure IVO JOIN BEWEEN BSE ND ENDE In the case where the base B and the appendage are linked by a pivot joint (around the z axis 3 ), the reaction torque about the z axis is null. hen (2) projected in the frame (, x, y, z ) becomes: B/x B/y B/z B/,x B/,y 0 a x a y a M z (s) x y z θ (22) where θ is the relative angular acceleration, along the pivot z -axis, of the appendage w.r.t the base B. If the pivot joint is motorized with a motor applying a torque C m around z axis (i.e. a torque applied by the base B on the appendage ), the dynamic model of the appendage at point becomes: 3 It is assumed here that the pivot joint is along z axis in the frame R attached to the appendage. It is always possible to meet this assumption using a rotation matrix R 3 3 (see section.4). 2 if modal contribution matrix is given at point C (the appendage center of mass) and denoted L C, then one can write: L L C τ C. 5

5 7th IC World Congress (IC'08) B/x B/y B/z B/,x B/,y C m a x a y a M z (s) x y z θ (23) herefore, a new input C m and a new output θ are introduced to the whole inverse dynamic model which is depicted in igure 5 and replaces the model described by equation (4). B/ B/, C m M (s) (:5,:) τ M (s) (6,6) - - B - D M (s) (6,:5) eye(5) 0 τ z - θ a ig. 5. schematic illustration of the inputs and outputs when pivot joints are added. rom (23), the equation for the last row reads 4 : a x a y C m M (s)(6, : 5) a z M (s)(6, 6) ( z θ) (24) x y thus the pivot angular acceleration θ is equal to: θ M a x a y C m M (s) (6, : 5) a z z. (s) (6, 6) x y (25) he inverse dynamic model D Satellite can be described by the functional diagram depicted in igure ENERLISION ND VLIDION he purpose of this section is to validate the inverse dynamic model (D Satellite see igure 5) tacking into account some pivot joints between the base and some appendages by comparison with the direct dynamic model assuming that all appendages are cantilevered on the base D,cantilever Satellite. If we consider the direct model of the rigid base D B and n flexible appendages i (defined by dynamic mass matrix M i i (s), i,, n) cantilevered to the base at point i through a rotation matrix R i6 6, then using previous 4 Matlab syntax is used to defined by (s)(i : j, k : l) the subsystem between outputs i to j and inputs k to l in the system (s). ig. 6. Block Diagram of the Inverse Dynamic Model with pivot joint results of sections and 2, one can compute the 6 6 direct dynamic model of the whole system: n D,cantilever Satellite D B τ ir i6 6 M i i (s)r i6 6 τ i. i If we assume now that these n appendages are not cantilevered but are connected with pivot joints, then the result of section 3 allows to build the whole inverse dynamic model D Satellite (see igure 5). his model can be detailed in the following way: a n, (26) θ n 2n 6 22n n C mn a is the transfer function between and. a 2 is the transfer function between and C mn. 2 is the transfer function between θ n and. 22 is the transfer function between θ n and C mn. rom model D Satellite, one can lock pivot joints by nulling the pivot acceleration: θ n 0 n, (27) and then emulate cantilevered joints: indeed from equation (26) and (27) and eliminating C m, one can derive: a (28) hen one can verify that the direct dynamic model D,cantilever Satellite is recovered: D,cantilever Satellite n 22 n n 2n 6. 52

6 7th IC World Congress (IC'08) n other way to lock pivot joints is to feedback the pivot positions to pivot torques through a very significant stiffness according to igure 7. hen the new inverse dynamics model exhibits a high frequency flexible modes which can be reduced to provide the inverse cantilevered dynamic model. ig. 7. ivot joints are locked using a feedback though an infinite stiffness k. R.C. Dorf and R. Bishop. Modern Control Systems. rentice Hall, 200. S. aulocher, Ch. ittet, and J.-. Chrétien. Six-DO formation flying modeling and control with an application to space interferometry. In 6th International ES Conference on uidance, Navigation and Control Systems, Loutraki, reece, 7-20 October ES. J.. Imbert and. Mamode. he effective mass concept in base motion dynamics and application to solar array dynamics. pages , Munich, ermany, 977. Nastran User s Conference. J.-. Magni. Extension of the linear fractional representation toolbox. In IEEE, editor, roceedings of the IEEE International Coference on CCSD, aipei, 2-4 September IEEE. K. Ogata. Modern Control Engineering. rentice Hall, D.. Wells. heory and roblems of Lagrangian Dynamics. Wiley, CONCLUSION ND ERSECIVES In this paper, the linear dynamic model of a spacecraft composed of a rigid base and various flexible and rigid appendages connected to the base by a cantilever joint or a pivot joint has been developed. he build of this model lies on basic operations: transportation of a dynamic model to one point to an other, connection of 2 dynamic models, use of effective masses to handle dynamic mass matrix for a flexible appendage, subdivision of the dynamic mass matrix to take into account a pivot joint. ll these operations can be simply represented by a block diagram and can be performed recursively to model any kind of open mechanical chain. he reader will find in /demos/sd a Matlab package called Spacecraft Dynamics oolbox to develop such models and an illustrative example. he short-term perspectives of this work are the following: to take into account an on-board kinetic momentums, to take into account a metrological model between the accelerations at the reference point and what is measured by the sensors (linear and angular accelerometers), to interface our toolbox with the Linear ractional Representation (LR) toolbox to handle uncertain dynamic parameters in the modeling process (Magni 2004). REERENCES D. lazard and J.. Chrétien. Commande ctive Des Structures lexibles. Lecture notes, SUERO, S.C. Chapra and R. Canale. Numerical Methods or Engineers. Mcrow Hill, C. Cumer and J.. Chrétien. Minimal lft form of a spacecraft built up from two bodies. In I uidance, Navigation, and Control Conference. ONER/DCSD, I,