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Dynamics and control of multibody tethered systems
PII:
Acta Astronautica Vol. 42, No. 9, pp. 503±517, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/98 $19.00 + 0.00 S0094-5765(98)00041-1
DYNAMICS AND CONTROL OF MULTIBODY TETHERED SYSTEMS S. KALANTZIS, V. J. MODI{, S. PRADHAN Department of Mechanical Engineering, University of British Columbia, Vancouver BC, V6T 1Z4 Canada
and A. K. MISRA Department of Mechanical Engineering, McGill University, Montreal, Quebec, H3A 2K6 Canada (Received 29 September 1997) AbstractÐThe equations of motion for a multibody tethered satellite system in a three dimensional Keplerian orbit are derived. The model considers a multi-satellite system connected in series by ¯exible tethers. Both tethers and subsatellites are free to undergo three dimensional attitude motion, together with longitudinal and transverse vibration for the tether. The elastic deformations of the tethers are discretized using the assumed-mode method. In addition, the tether attachment points to the subsatellites are kept arbitrary and time varying, with deployment and retrieval degrees of freedom. The governing equations of motion are derived using an Order (N) Lagrangian formulation. Next, two independent controllers, i.e. an attitude and vibration controller, are designed to regulate the rigid and ¯exible motion present in the system, excited from various maneuvres performed during the course of a mission. The former controller utilizes the thrusters and momentum-wheels located on the rigid satellites with a control algorithm based on the feedback linearization technique. On the other hand, the latter is designed using the robust linear quadratic Gaussian-loop transfer recovery method actuating the variable tether attachment point, or oset position. Both controllers are successful in suppressing unwanted disturbances in the system in a acceptable amount of time. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
system must be controlled. This, of course, would be essential in the case of a satellite system intended for scienti®c experiments such as the micro-gravity facilities aboard the proposed International Space Station. Vibration of the ¯exible members will have to be checked if they aect the integrity of the onboard instrumentation. Thruster as well as momentum-wheel approaches [2] have been considered to regulate the rigid-body motion of the end-satellite, as well as the swinging motion of the tether. The procedure is particularly attractive due to its eectiveness over a wide range of tether lengths as well as ease of implementation. Other methods include tension and length rate control which regulate the tether's tension and nominal unstretched length, respectively [3,4]. It is usually implemented at the deployment spool of the tether. More recently, an oset strategy involving time dependent motion of the tether attachment point to the platform has been proposed [5]. It overcomes the problem of plume impingement created by the thruster control and the ineectiveness of tension control at shorter lengths. However, the eectiveness of such a con-
With the ever increasing demands by today's consumers to eciently exploit the resources available in space, a growing number of researchers and engineers are looking to tethered satellite systems to expand the current design capabilities of satellites. With that said, investigators have attempted to gain insight into the complex dynamics and control issues of tethered systems using a variety of models. Some of the results from these eorts have been reviewed by Misra and Modi [1] from which the major conclusions of these ®ndings can be summarized as follows: stationkeeping is only marginally stable; deployment can become unstable if a critical speed is exceeded; retrieval is always unstable. Furthermore, tether vibrations can also be excited to unacceptable levels during these maneuvres. In view of these conclusions, it is clear that an appropriate control strategy is needed to regulate the dynamics of the system, i.e. attitude and tether vibrations. First, the attitude motion of the entire {Author to whom correspondence should be addressed. 503
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troller can become limited with an exceedingly long tether due to a practical limit on the permissible oset motion. In response to these two control issues, a hybrid thruster/oset scheme has been proposed to combine the best features of the two methods [6,7]. In addition to attitude control, these schemes can be used to attenuate the ¯exible response of the tether. Tension and length rate control [8] as well as thruster based algorithms [9,10] have been proposed to this end. Modi et al. [11] have successfully demonstrated the eectiveness of an oset strategy to damp undesired vibrations. Passive energy dissipative devices, e.g. viscous dampers, are also another viable solution to the problem. The development of various control laws to implement the above mentioned strategies has also received much attention. An eigenstructure assignment in conjunction with an oset controller for vibration attenuation and momentum wheels for platform liberation control has been developed [12], in addition to the controller design from a graph theoretic approach [11]. Also, non-linear feedback methods, such as the feedback linearization technique (FLT), that are more suitable for controlling highly nonlinear, non-autonomous, coupled systems have also been considered [2,7]. It is important to point out that several linear controllers, including the classic state feed-back linear quadratic regulator (LQR), have received considerable attention [6,13]. Moreover, robust methods such as the linear quadratic Gaussin/loop transfer recovery (LQG/LTR) method have also been developed and implemented on tethered systems [2]. A more complete review of these algorithms as well as others applied to tethered systems has been presented by Pradhan [2]. The present analysis extended the current understanding of tethered system control to the case of N-body chain link systems capable of threedimensional motion [14], as shown in Fig. 1. The mathematical model is ®rst derived resulting in the nonlinear, coupled equations of motion with tether ¯exibility included. Next, the attitude controller is developed using the feedback linearization technique where thrusters and momentum-wheels are used to regulate the attitude motion of the individual links and hence of the entire system. In addition, a robust vibration controller is designed and implemented using the variable tether attachment point, or oset motion. The eectiveness of both the controllers is studied, through numerical simulation, at ®rst individually, and then in combination with each other.
Fig. 1. A schematic diagram of the space platform based N-body tethered satellite system.
nate system Fi is assigned to each link i. This frame is capable of three-dimensional rotation with respect to the inertial frame F0, located at the centre of Earth, and is composed of three axes: xi, along the length of the link; yi, perpendicular to the link; and zi, normal to the orbital plane. The rotation is described by three Euler angles: ai, bi and gi representing the inplane, pitch; out-of-plane, roll; and spin or yaw motion, respectively, by the
2. DYNAMICS FORMULATION
From Fig. 2, the analysis of the multibody system begins with the motion description of each link (tether or satellite) separately. A body ®xed coordi-
Fig. 2. Vector components of the ith and (i ÿ 1)th chain links.
Multibody tethered systems
transformation matrix 2
3
Cbi Cai 6 7 Cbi Sai 6 7 6 7 ÿSbi 6 7 6 7 6 7 6 ÿCgi Sai Sgi Sbi Sai 7 6 7 Ti 6 Cgi Cai Sgi Sbi Sai 7 6 7 Sgi Cbi 6 7 6 7 6 7 6 Sg Sai Cg Sb Cai 7 i i 6 i 7 4 ÿSg Cai Cg Sb Sai 5 i i i Cgi Cbi
1
iÿ1 ÿ X
Tkÿ1 dk Tk KAk d k Tk {^ lk Tiÿ1 di 8
k1
where ri is the rigid component of the position of dmi from Fi and ffi (ri) is the ¯exible deformation vector at ri. Tether ¯exibility, in this analysis, is discretized using the assumed-mode method. The mode shapes considered in the formulation correspond to the shape functions for a string [15], which are, for the jth longitudinal mode, 2jÿ1 xi ; 3 F xj i xi ; li li where li is the ith tether length. In the case of inplane and out-of-plane transverse defections, the admissible functions are given as p jpxi F yj i xi ; li F zji xi ; li 2 sin ; 4 li where Z2 is added as a normalizing factor. Recasting the mode shape functions into a matrix form and assigning a corresponding generalized modal coordinate di, the ¯exible position vector f fi can be de®ned as 5
where Fi contains the mode shape functions given by eqns (3) and (4). Thus inserting eqn (5) into eqn (2) and de®ning gi=ri+Fidi Rdmi Di Ti gi :
duces an additional variable di which represents the position vector of the frame Fi from the previous frame Fi ÿ 1. For the case of where link i represents a tether link and link i ÿ 1 a rigid satellite, di={dxi, dyi, dzi}T is the oset of the tether attachment point to the rigid satellite which is used to control the tether vibration. De®ning KAx ÿ 1=Fi ÿ 1(li ÿ 1+dxi) and applying eqn (7) to recursively de®ne Di ÿ 1, then it can be shown that Di
which relates the body-®xed frame Fi to the frame F0. Here, Cx and Sx are abbreviations for cos(x) and sin(x), respectively. Thus, from Fig. 2, the inertial position of the elemental mass of the ith link can now be given as h i 2 Rdmi Di Ti ri f i f ri :
f i f ri F i xi ; li dd i t
505
6
thus completely de®ning the position vector of dmi with respect to the set of generalized coordinates given by q = {qT1 . . . qTN}T where 0 1 di B Zi C B qi @ C 9 di A li and Zi={ai, bi, gi}T. Note that by properly manipulating eqn (6) and eqns (7) and (8) then a factorized expression of the equations of motion of the entire system can be derived resulting in a Order (N) formulation of the governing equations of motion [14]. This has tremendous computational advantages when the equations are numerically solved for a large number of links (N large). Using eqns (6) and (8) the kinetic, potential and strain energies of the entire N-body system are given by N 1X T Ke 10 R_ R_ dmi dmi ; 2 i1 mi dmi Vg ÿ m
N X mi i1
"
Di
N mX 1 2 i1 D3i
3 gTi gi dmi 2DTi Ti gi dmi ÿ 2 DTi Ti Di # gi gTi dmi Ti Di ; 11
N 1X E i Ai Ve 2 i1
li 0
e2i dxi ;
12
Di in eqn (6), represents the inertial position of Fi from F0 and it serves to completely describe the orbital motion of the ith link with respect to Earth. However, from Fig. 2, it is clear that Di Diÿ1 Tiÿ1 liÿ1 {^ di ÿ 7 Tiÿ1 F iÿ1 liÿ1 dxi d iÿ1 ;
respectively. Here m = 3.986 105 km3/s2 is Earth's gravitational coecient, Ei and Ai are the Young's modulus and cross-sectional area of the ith link, respectively and ei is the ith link strain function given by " # @ui 1 @vi 2 @wi 2 ; 13 ei @xi 2 @xi @xi
which is de®ned with respect to the orbital position of the previous link i ÿ 1. In addition, eqn (7) intro-
where ui, vi and wi are the ¯exible deformations along the xi, yi and zi directions, respectively.
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The eect of structural damping is incorporated by introducing Rayleigh's dissipation function [16]. Application of the Lagrange procedure results in the nonlinear, non-autonomous and coupled equations of motion. As can be expected, the complete set of equations is indeed lengthy and hence is not presented here for brevity. Essentially, they have the form ÿ ÿ ÿ _ t Q q; q; _ t M q; t q F q; q; 14 where M(q, t) is the nonlinear symmetric mass _ t) is the forcing vector function and matrix; F(q, q, _ t) is the vector of non-conservative generalQ(q, q, ized external forces, including control inputs, acting on the system. 2.1. Speci®cation of the oset position The derivation of the equations of motion using the O (N) formulation requires that the oset coordinate di be treated as an generalized coordinate. However, this oset must later be constrained or be controlled to follow a prescribed motion dictated by either the designer or the controller. This results in an additional term that is added to the generalized acceleration vector giving
respectively [16]. Fem is the mth external force acting on the system at a position Rm from the inertial frame and nu is the total number of actuators (forces). 2.2.1. Generalized thruster forces. One set of thrusters is placed on all subsatellites and on the end-body, i.e. all rigid bodies, except the ®rst (platform), as shown in Fig. 3. They are capable of ®ring in all three directions. From the schematic, the inertial position of the thruster, located at the centre of mass of body 3, is given as ÿ R3 d1 T1 d2 T2 l2 {^ KA2 d 2 T3 rcm3 d1 df 1 T3 rcm3 :
17
De®ning
ÿ df j Tj dj1 Tj1 lj1 {^ KAj1 d j1
18
then the thruster position of body 5 is given as R5 d1 df 1 df 3 T5 rcm5 :
19
In general, the inertial position of the thruster on the ith body can now be given as Ri d1
iÿ2 X
df 2kÿ1 Ti rcmi :
20
k1
q S f; Mjdc h iÿ1 ÿ Mÿ1 f Mÿ1 P c P cT Mÿ1 P c dc P cT Mÿ1 f
15
which is the new constrained vector equation of motion with the oset speci®ed by dc [17]. Note that pre-multiplying Mÿ1 by PcT is the equivalent of extracting the rows of Mÿ1 corresponding to the d equations whereas post-multiplying by Pc is equivalent to extracting the columns of the matrix corresponding to the d equations.
We next de®ne the column matrix " # h iT @ i Ri Qt j @qj " # @ @ @ @ Ri Ri Ri Ri @dj @Z Zj @Z Zj @lj
21
where i = 3, 5, . . . , N and j = 1, 2, . . . , i. The vector derivative of Ri with respect to the scaler
2.2. Generalized control forces The treatment of non-conservative external forces acting on the system is considered next. In reality, their are numerous environmental forces present on the system, e.g. atmospheric drag, solar radiation pressure as well as inertial dampers. However, in this model, only the eect of the active control thrusters and momentum-wheels is considered. These forces will be used to control the attitude motion of the system. When deriving the equation of motion of a system using the Lagrange equations, the contribution of each force must be properly distributed among all the generalized coordinates. This is done using the relation Qk
nu X m1
Fem
@Rm @qk
16
where Qk and qk are the kth element of Q and q,
Fig. 3. Inertial position of subsatellite thruster forces.
Multibody tethered systems
components qk is stored in the kth column of [Qitj]T. Thus, for the case of 3-bodies, eqn (16) for thrusters becomes 2 3 3 Qt 1 6 6 3 7 7 22 Q 6 Qt2 7T3 Tt2 4 5 Q3t3 where Tti={0, Ttai , Ttbi }T is the thrust vector for the ith link (tether i ÿ 1). Here Ttai and Ttbi are the thrust in the in-plane yi and out-of-plane, zi directions, respectively. For the case of 5-bodies, 2 3 3 Qt1 Q5t1 6 6 Q3 Q5 7 7 6 t2 t2 7 6 7 6 7 T3 Tt2 : 23 Q 6 Q3t3 Q5t3 7 6 7 T5 Tt4 6 7 5 6 0 Qt4 7 4 5 0 Q5t5 The result for seven or more bodies follows the pattern established by eqn (23). 2.2.2. Generalized momentum-wheel torques. When deriving the generalized moments arising from the momentum-wheels on each satellite, including the platform, it is easier to view the torques as coupled forces. From Fig. 4, we see that the generalized force arising from these coupled forces, for the ith link, are Qk
3 X
Fei
i1
ÿ
3 X i1
@ fRi Ei g @qk
Fei
@ f R i ÿ Ei g @qk
24
507
where |^ acting at E1 Exi {^;
Fe1 Fai Fe2 ÿ Fbi Fe3 Fgi
k^ acting at E2 Exi {^;
k^ acting at E3 Eyi |^:
Expanding eqn (24), it becomes clear that Qk 2
3 X i1
Fei
@ f Ei g @qk
25
which is independent of Ri and therefore quite simpler. Thus de®ning the moments Mmai=2FaiExi, Mmbi=2FbiExi and Mmgi=2FgiEyi and, transforming the coordinates to the body ®xed frame using the rotation matrix, the ith link generalized force due to the momentum-wheels on link i, is @ @ Ti {^ Mmai ÿ Ti k^ Ti {^ Mmbi Qi Ti |^ @qi @qi @ Ti k^ Ti |^ Mmgi QM i M mi @qi where Mmi={Mmai, Mmbi, Mmgi}T and @ @ Ti |^ Ti {^ ÿ Ti k^ Ti {^ QM i @qi @qi @ Ti |^ : Ti k^ @qi
26
27
When combining the thruster forces and momentum-wheel torques together, a compact expression for the generalized force vector, for 5 bodies, can be expressed as 2 M 3 0 Q51 T5 0 Q1 Q31 T3 6 7 Q32 T3 0 Q52 T5 0 7 6 0 6 7 Q 6 Q53 T5 0 7 Q33 T3 QM 3 6 0 7 6 7 0 0 Q54 T5 0 5 4 0 0 0 0 Q55 T5 QM 5 0 1 Mm1 B B Tt2 C C B C C Qu u: M 28 B m 3 B C B C @ Tt4 A Mm5 The pattern of eqn (28) is retained for the case of N links.
3. ATTITUDE CONTROL
Fig. 4. Coupled force representation of rigid body momentum-wheels.
The design of an attitude controller, with objective to regulate the librational motion of the system, is now undertaken. As discussed earlier, several methods are available to accomplish the controller's objective, including length rate and tension control; oset, or tether attachment position, control; and
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thruster and momentum-wheel based controllers. All of these methods have their relative advantages and drawbacks, however, the thruster/momentumwheel approach is chosen for its good overall performance over a wide range of tether lengths. This control method utilizes the on-board thrusters located on each rigid satellite, excluding the ®rst one (platform), to regulate the pitch and roll motion of the previous tether, i.e. the thrusters located on satellite 2 (link 3) regulate the attitude motion of tether 1 (link 2). On the other hand, the rigid body motion of each rigid link, including the platform, is controlled using a set of three momentum wheels placed mutually perpendicular to each other. The choice of control algorithms, from which the controller is designed, is based on several important factors. Firstly, the controller scheme has to perform well for many dierent time-varying con®gurations, as is the case during tether deployment or retrieval. In addition, it would have to be computationally ecient in order to be implemented in realtime. Finally, simplicity of implementation is also desired. Several control algorithms exist in practice. The implementation of a ®nite number of linear time invariant (LTI) controllers, scheduled discretely over a range of system con®gurations (gain scheduling [2]), is one possible solution. An alternative approach involves the use of adaptive control techniques in conjunction with linear time varying (LTV) systems theory, from which the design of the controller would include the time-varying tether length case. One such technique is on-line parametric identi®cation [18] where the system's ®xed parameters are estimated in real-time during the course of the simulation. However, the controller that is eventually chosen is based on the nonlinear feedback linearization technique (FLT). This controller is well suited for highly time-varying systems whose dynamics can be accurately modelled, as is the case for space systems in general. The FLT method is based on transforming the nonlinear, time-varying governing equations into a LTI system using a nonlinear time-varying feedback [2,19]. Once the system is transformed, the resulting LTI system can be controlled with any of the numerous linear control algorithms available in the literature. In this case, a simple PD controller is adopted and shown to have good performance. The choice of a FLT control scheme satis®es one of the criteria of selection mentioned earlier, namely being valid over a wide range of tether lengths. The computational eciency of the method is addressed next. In order to implement this controller in realtime, the computation of the system's inverse dynamics must be executed quickly, hence a simpler model, that performs well, is in order hence, a model based on the rigid nonlinear equations of motion is chosen.
3.1. Controller design using feedback linearization technique The control model used is based on the rigid equations of motion given by Mr qr f r Qu u
29
where the left hand side is the equations of motion of the rigid model. The right hand side is the generalized external forcing term due to the thrusters and momentum-wheels and is given by eqn (28). Letting, f~r f r ÿ Qu u and substituting in eqn (15) gives ~ Mr j dc : qr S f;
30
31
Expanding eqn (31), it can be shown that ÿ qr S f r ; Mr j dc ÿ S Qu ; Mr j 0 u Fr ÿ Qr u
32
where S(Qu, Mrvv0) is the column-vector matrix given by ÿ h ÿ S Qu ; j 0 S Qu :; 1; Mr j 0 ; ÿ S Qu :; 2; Mr j 0 ; . . .
ÿ i . . . ; S Qu :; nu; Mr j 0
33
and Qu(:, i) is the ith column of Qu. Extracting only the controlled equations from eqn (32), i.e. the attitude equations, we get qrc Frc ÿ Qrc u vrc
34
where vrc is the new control input required to regulate the decoupled linear system. At this point, a simple PD controller can be applied, i.e., vrc qrcd K q_ rcd ÿ q_ rc Kp qrcd ÿ qrc : 35 Eqn (35) is known as the secondary controller whereas eqn (32) is the primary controller. Kp and Kv are the proportional and derivative gain matrices and qÈrcd, qÇ rcd and qrcd are the desired acceleration, velocity and position vectors for the attitude angles of each controlled body, respectively. Solving for u from eqn (34) leads to u Qÿ1 rc Frc ÿ vrc
36
and the new controlled equations of motion are given by eqns (15) and (28). 3.2. Simulation results The FLT controller is implemented on the 3body STSS tethered system de®ned by the following set of parameters:
Multibody tethered systems
2
1091430 ÿ8135 6 . I1 4 ÿ8135 8646050 328108 27116
3 328108 7 27116 5 8286760
kg m2 platform inertia; 3 200 0 0 6 7 . I2 4 0 400 0 5 0 0 400 2
kg m2 endÿsatellite inertia; . m1=90,000 kg (mass of the space station platform); . m2=500 kg (mass of the end-satellite); . EtAt=61,645 N (tether elastic stiness); . rt=4.9 kg/km (tether density); . Zd=0.5% (tether structural damping coecient); . rcm3={1,0,0}T m (satellite 2 tether attachment point). The response variables are de®ned as follows: . a1, b1 satellite 1 (platform) pitch and roll angle; . a2, b2 satellite 2 (end-satellite) pitch and roll angle; . at, bt tether pitch and roll angle; . lt tether length; . d2 = {dx, dy, dz}T tether attachment position relative to satellite 1; . d2={dx, dy, dz}T tether ¯exible modal coordinates in the longitudinal x, in-plane transverse y, and out-of-plane transverse z directions, respectively. All the attitude angles, ai and bi , are measured with respect to the local vertical, local horizontal (LVLH) frame. The system is taken to be in a nominal circular orbit, 289 km, in altitude, with an orbital period of 90.3 min. The choice of proportional and derivative matrix gains is based on a desired settling time of tsj=0.5 orbits and a damping factor of zj=0.7 for both the pitch and roll actuators in each body. Given zj and tsj , it can be shown [20] that q ÿ1 ln 0:05 1 ÿ z2j onj 37 zj tsj and hence kpj o2nj ; kvj 2z j onj
38
where kpj and kvj are the diagonal elements of the matrices Kp and Kv de®ned by eqn (35), respectively. Note that due to the fact that each body-®xed frame Fi is referred directly to the inertial frame F0, the nominal pitch equilibrium angle is zero degrees only when measured from the LVLH frame (aiÿy1).
509
However, it is y1 (true anomaly) when referred to F0, thus the desired position vector qrcd is set to be equal to y1(t), i.e. the time-varying true anomaly, for the pitch motion; and zero for the roll and yaw motion, such that 00 1 1 y1 t B B@ 0 A C C B C 0 B 1 C B C B C .. C 2 <3N1 : 39 qrcd B . B C 1 C B0 B y1 t C B C @@ 0 A A 0 N The desired velocity and acceleration are given as 00 1 1 y_ 1 t B C C BB C B@ 0 A C B C 0 B 1 C B C B C .. _q qrcd B C 2 <3N1 ; . B C 1 C B0 B y_ 1 t C B C C C BB @ A 0 @ A 0 N 00 1 1 y1 t B C C BB C B@ 0 A C B C 0 B 1 C B C B C .. qrcd B 40 C 2 <3N1 ; . B C 1 C B0 B C B y1 t C C C BB A @ 0 @ A 0 N respectively. When the system is in a nominal circular orbit, qrcd=0. Figure 5 presents the controlled response of the STSS system with oset d2={1, 0, 1}T m during the retrieval maneuvre from a tether length of 20 km to 200 m in 3.5 orbits. With the FLT controller, based on the non-linear rigid model, in place, both the pitch and roll angles of the rigid bodies and of the tether are now attenuated in less than 1 orbit. The control requirement for the pitch motion of the platform and the tether are maximum during the constant velocity phase of retrieval when the Coriolis force acting on the system is maximum. However, for the roll case, where there is no Coriolis force present, the maximum control requirement occurs during the initial phases of retrieval. Moreover, there is clear evidence of ¯exible deformations along the transverse directions of the tether, again due to the same Coriolis force. Although, these vibrations stabilize during the terminal phase of retrieval, they remain present for the duration of the mission.
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Fig. 5. Retrieval dynamics of the three body STSS, using the non-linear, rigid FLT controller; with oset along the local vertical and orbit normal: (a) attitude and vibration response; (b) control actuator time histories. 4. VIBRATION CONTROL
The vibration control strategy chosen in this analysis is based on oset control, i.e. control of the tether's attachment point to the platform (satellite 1). All three degrees of freedom of the oset position are used to control both the longitudinal and the in-plane and out-of-plane transverse modes of vibration. In practice, the oset controller can represent the motion of a dedicated manipulator or robotic arm attached to the space station or space shuttle from which the tether is deployed. Tether elastic deformations arising from deployment or retrieval of the tether can be controlled
during the maneuvre or, once the maneuver is completed and the tether length remains constant (stationkeeping). The latter method is adopted here. Vibration control during the stationkeeping phase is generally sucient since this phase is where most of the mission objectives are carried out. 4.1. System linearization and state-space realization The design of the oset controller begins with the linearization of the equations of motion of the system about their equilibrium position. However, this can present a problem when considering the current form of the equations of motion namely, the pitch
Multibody tethered systems
angle, ai. As mentioned in the previous section, the pitch angle is not referred to the LVLH frame thus its equilibrium is not a constant, but is equal to y1(t). Two methods can be used to resolve this problem. The ®rst method corresponds to using the nonstationary equations of motion in their present form and deriving a controller based on the linear time varying (LTV) system. The second method is based on designing a linear time invariant (LTI) controller based on a new set of reduced governing equations representing the motion of the three body tethered system. The latter method is chosen here. Furthermore, the LTI controller design is done completely o-line and thus is computationally more ecient. The reduced model is derived using the Lagrangian approach with the following generalized coordinates, qred a1 ; b1 ; a2 ; b2 ; dx ; dy ; dz ; a3 ; b3
41
where ai and bi are the rigid pitch and roll angles of link i, relative to the LVLH frame. The symbols dx, dy and dz are the ¯exible modal coordinates for the longitudinal, in-plane and out-of-plane transverse tether ¯exible deformations respectively, de®ned in this case, for the ®rst mode of vibration only (note, the sux ``2'' of the modal coordinates d2 is dropped for conciseness). The non-linear, nonautonomous, coupled equations of motion of the tethered system can now be given as Mred qred f r 0
511
constant. Inverting Ms and multiplying through we get _ r ÿ Mÿ1 qr ÿ Mÿ1 s Cs q s Ks qr ÿ1 _ ÿ Mÿ1 s Cd d22 ÿ Ms Kd d2 ÿ1 ÿ Mÿ1 s Md d22 ÿ Ms f s :
De®ning ud=dÈ2 and v = {qTr, dTr}T then eqn (44) can be rewritten as " # ÿ1 ÿMÿ1 s Cs ÿMs Cd v v_ 0 0 " # ÿ1 ÿMÿ1 s Ks ÿMs Kd v 0 0 " # ! ÿMÿ1 ÿMÿ1 s Md s fs ud ; Id 0 v MCv_ MKv MIud Fs : Finally, de®ning x01 x x02
Ms qred Cs q_ red Ks qred Md d2 Cd d_ 2 Kd d2 f s 0 43 where Ms, Cs, Ks, Md, Cd, Kd and fs in eqn (43) are
!
0
v_ v
45
! 46
then the LTI equations of motion can be recast into the state-space form x_ 0 Ax0 Bud Fd where
MC MK 2 <2424 ; A 0 Id12
47
42
where Mred and fred are the reduced mass matrix and forcing term of the systems, and are functions of qred and qÇ red in addition to the time varying oset position, d2, and its time derivatives d_ 2 and dÈ2. An additional consequence of referring the pitch motion to a local frame is that the new reduced equations are now independent of y1, and under the further assumption of the system negotiating a circular orbit, y_ 1 remains constant. The non-linear reduced equations can now be linearized about their equilibrium point. For all the generalized coordinates, the equilibrium position is zero with the exception of dx which has a non-zero equilibrium, dx0 and d2, which is linearized about d20. Finally, linearizing eqn (42) and recasting into matrix form gives
44
B and
MI 0
Fd
Fs 0
48
2 <243 ;
49
2 <241 :
50
Let x0 xeq x;
51
where x is the perturbation vector from the constant equilibrium state vector xeq. Substituting eqn (51) into eqn (47), we get the linear, perturbation state vector equation of motion modelling the reduced tethered system as ÿ 52 x_ 0 x_ Ax Bud Fd Axeq : It can be shown that for ideal control [21], Fd+Axeq=0 in eqn (52) thus leaving the familiar state-space equation x_ Ax Bud :
53
The selection of the output vector completes the state space realization of the system. The output
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vector consists of the longitudinal deformation from the equilibrium position, dx0, of the tether at xt=lt; the slope of the tether due to the transverse deformation at xt=0; and the oset position d2, from its initial position d20. Thus, the output vector y is given by 0
1 dp x ÿ dx0 B C F u lt dx ÿ dx0 B 2p dy C 0 B B C C F v 0dy B C B plt C 0 B B C C F 0d z 2 p w B B C yB B dz C C C d ÿ d x x 0 B C B lt C @ B A C dy ÿ dy0 B dx ÿ dx0 C @ dy ÿ dy A dz ÿ dz 0
1
0
54
the longitudinal and transverse modes of vibration which also allows the decoupling of the ¯exible subsystem into a set of longitudinal (dx and dx) and transverse (dy , dz , dy and dz) subsystems. The appended oset system, d, must also be included since it acts as the control actuator. It is important to note however that the inplane and out-ofplane transverse modes can not be decoupled because their oscillation frequencies are of the same order. The two ¯exible subsystems are summarized below. 4.2.1. Longitudinal subsystem. xu Au xu Bu udu ; yu Cu xu
0
dz ÿ dz0
where F'v(0) = @/@xi[Fv(xi)]xi=0 and F'w(0) = @/ @xi[Fw(xi)]xi=0. 4.2. Linear quadratic Gaussian-loop transfer recovery Now that the linear state space model is de®ned by eqns (53) and (54), the design of the control algorithm can commence. The control algorithm chosen is the linear quadratic Gaussian (LQG) estimator based optimal controller [21,22]. The LQG controller is a widely used optimal controller whose theory has been well developed by many authors in the last 25 years. It consists of the design of a Kalman±Bucy Filter (KBF) which provides an estimate of the states xÃ, and a linear quadratic regulator (LQR), which is separately designed assuming all the states of x are known. Both the LQR and KBF designs independently have good robustness properties, i.e. retain good performance when disturbed, due to model uncertainty. However the combined LQR and KBF designs, i.e. the LQG design, has poor stability margins in the presence of model uncertainties. This limitation has led to the development of an LQG design procedure that improves the performance of the compensator, by recovering the full state feedback robustness properties at the plant input or output. This procedure is known as the linear quadratic Gaussian-loop transfer recover (LQG/LTR) control algorithm. A detailed development of its theory is available in the literature [22] and hence not repeated here. As mentioned earlier, the main objective of this oset controller is to regulate the tether vibration described by the d equations. However, from the response of the uncontrolled system [14] there is a large dierence between the magnitude of the librational and vibrational frequencies. This separation of frequencies allows for the separate design of the vibration controller and that of the attitude controller. Thus, only the ¯exible subsystem, composed of the d and d equations, is required in the oset controller design. Similarly, there is also a wide separation of oscillation frequencies between
55
where xu={d_ x , d_ x2 , dx, dx2}T, udu=dÈx2 and, from eqn (54), Cu
0 0 0 0 F u lt 0 0 0 0 1 0 0
1 0 : 0 1
56
Au and Bu are the rows and columns of A and B, respectively and correspond to the components of xu. The state LQR weighting matrix Qu is chosen as 2
1 6 0 6 Qu 4 0 0
0 1 0 0
3 0 0 0 0 7 7 1 0 5 0 10
57
and the input weighting Ru=Id2. The state noise covariance matrix is given as 2
1 6 0 Xu 6 40 0
0 1 0 0
0 0 4 0
3 0 07 7 05 1
58
while the measurement noise covariance matrix is
1 0 : Yu 0 15
59
Given the above mentioned matrices, the design of the LQG/LTR compensator is computed in MATLAB returning the state space representation of the LQG/LTR dynamic compensator ^_ u Aku x^ u ÿ Bku yu ; x udu Cku x^ u
60
where xÃu is the state estimate vector of xu. Loop transfer recovery is performed at the system output, i.e. the return ratio at the output approaches that of the KBF loop given by Cu(sIdÿAu)Kfu [22]. This is done by choosing a suciently large scaler value, r, such that the singular values of the return ratio
Multibody tethered systems
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Fig. 6. Singular value plot for the LQG and LQG/LTR compensator with respect to target return ratio: (a) longitudinal design; (b) transverse design.
approach those of the target design. For the longitudinal controller design, r= 5 105. Figure 6(a) shows a plot of the singular values of the recovered compensator design and the non-recovered, LQG, compensator design with respect to the target design. Unfortunately, perfect recovery is not possible, especially at higher frequencies, because the system is non-minimal, i.e. it has transmission zeros with positive real parts [23]. 4.2.2. Transverse subsystem. xv Av xv Bv udv ; yv Cv xv
61
where xv={d_ y , d_ z , d_ y2 , d_ z2 , dy , dz , dy2 , dz2 ,}T, È È udv={dy2, dz2}T and
2
0 60 Cv 6 6 40 2
0
6 60 6 6 6 6 60 6 6 40 0
0 0 0 F 0v 0 0 0 0 0 0 0 0 0 0 0 0
0 p 2p 0 0 0 lt 0 0 0
0
0 0 0 0 0 0
0 0
0 0 F w 0 0 0
3 0 0 0 07 7 7 1 05 0 1 3
0 0 07 7 7 p 7 7 2p 0 07 7: lt 7 7 0 1 05 0 0 1
62
Av and Bv are the rows and columns of the A and B corresponding to the components of xv.
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The state LQR weighting matrix Qv is given as 2
10 0 0 6 0 10 0 6 6 0 0 104 6 6 0 0 0 Qv 6 6 0 0 0 6 6 0 0 0 6 4 0 0 0 0 0 0
0 0 0 104 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
3 0 07 7 07 7 07 7 07 7 07 7 05 1
63
and the input weighting Rv=Id4. The state noise covariance matrix is given as
2
1 60 6 60 6 60 Xv 6 60 6 60 6 40 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 9 104 0 0
3 0 0 7 7 0 7 7 0 7 7 64 0 7 7 0 7 7 0 5 4 9 10
while the measurement noise covariance matrix is 2 3 1 0 0 0 60 1 0 0 7 7: 65 Yv 6 4 0 0 5 104 0 5 4 0 0 0 5 10
Fig. 7. Stationkeeping dynamics of the three body STSS, using the non-linear, rigid FLT attitude controller and LQG/LTR offset vibration: (a) attitude and libration controller response.
Multibody tethered systems
515
Fig. 7. Stationkeeping dynamics of the three body STSS, using the non-linear, rigid FLT attitude controller and LQG/LTR oset vibration: (b) vabration and oset response.
Similar to the design approach of the longitudinal controller, the transverse oset controller can be designed with r = 50. The result is the transverse compensator system given by ^_ v Akv x^ v ÿ Bkv yv ; x udv Ckv x^ v
66
where xÃv is the state estimate of xv. The singular value plot of the target transfer function with respect to the ones achieved by the LQG and LQG/
LTR are presented in Fig. 6(b). As can be seen, the recovery is not as good as that for the longitudinal case, again due to the non-minimal system. Moreover, the low value of r indicates that only a little recovery in the transverse subsystem is possible, suggesting that the robustness of the LQG design is almost the maximum that can be achieved, under these conditions. De®ning xf={xTu, xTv}T, xÃf={xÃTu, xÃTv}T and ud={udu, uTdv}T, the longitudinal compensator can be combined to give
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" ^_ f x " ud
yf
A ku 0
0 A kv
#
"
Bku x^ f ÿ 0
# 0 y Ak x^ f ÿBk yf ; Bkv f
# 0 C ku x^ f Ck x^ f ; 0 C kv ! " # Cu 0 yu xf Cf xf : 0 Cv yv
67
De®ning a permutation matrix Pf such that xf=Pfx where x is the state vector of the full non-linear system de®ned by eqn (15), then the compensator and full nonlinear system equations can be combined as ÿ q S f; M j dc ; ^_ f Ak x^ f ÿ Bk Cf Pf x; x ud Ck x^ f :
(FLT) with the thrusters and momentum-wheels located on the rigid satellites. The controller is successful in regulating the pitch and roll motion of both the tether and end-satellites in a reasonable amount of time. Furthermore, a vibration controller making use of the variable tether attachment point or oset position is successful in suppressing any unwanted ¯exible deformations arising from various mission maneuvers. In this case, the robust linear quadratic Gaussian-loop transfer recovery method is employed to design the controller for the stationkeeping phase.
REFERENCES
68
4.3. Simulation results The response for the stationkeeping STSS with a tether length of 20 km and initial oset position of 1 m in each direction is presented in Fig. 7 where both the FLT attitude control and the LQG/LTR oset controller are activated. As expected, the oset controller is successful in quickly damping out the vibration in both the longitudinal and transverse direction. However, it is clear that the presence of oset control requires a relatively large control moment to regulate the attitude of the platform. This is due to the additional torque created by the larger moment arm, over 5 m and 2 m in the in-plane and out-of-plane direction respectively, introduced by the oset control. In addition, the control moment for the platform is modulated by the tether's transverse vibration through oset coupling. However, this does not signi®cantly aect the motion of the tether whose thruster force remains relatively unchanged from the uncontrolled vibration case.
5. CONCLUDING REMARKS
The equations of motion of an N-body tethered system undergoing three-dimensional orbital and attitude motion are derived using a computationally ecient Order (N) Lagrangian formulation. The tether, which is also free to vibrate in the longitudinal as well as inplane and out-of-plane transverse directions, is described using the assumed-mode methods with an arbitrary number of modes. The rigid satellites are also capable of rigid-body rotation about their equilibrium. The governing equations are then integrated numerically in the presence of an attitude and ¯exible vibration controller. The attitude controller is designed using the feedback linearization technique
1. Misra A. K. and Modi V. J., A survey on the dynamics and control of tethered satellite systems, NASA/AIAA/PSN International Conference on Tethers, Arlington, VA, U.S.A., 1986, Paper No. AAS-86-246, In: Advances in the astronautical sciences, Bainum PM et al. editors, American Astronautical Society, 62, pp. 667±719. 2. Pradham, S., Planar Dynamics and Control of Tethered Satellite Systems. Ph.D. Thesis, Department of Mechanical Engineering, The University of British Columbia, 1994. 3. Rupp, C. C., A tether tension control law for tethered sub-satellite deployed along local vertical, NASA TM X-64963, 1975. 4. Misra, A. K. and Modi, V. J., Acta Astronautica, 1992, 26(2), 77±84. 5. Pradhan, S., Modi, V. J. and Misra, A. K., in AIAA/ AAS Astrodynamics Specialist Conference, Paper No. 96-3570. San Diego, CA, U.S.A., 1996, pp. 1±11. 6. Modi, V. J., Lakshmanan, P. K. and Misra, A. K., Acta Astronautica, 1990, 21(5), 283±94. 7. Pradhan, S., Modi, V. J. and Misra, A. K., International Journal of Control, 1996, 64(2), 175±93. 8. Misra, A. K. and Modi, V. J., Journal of Guidance, Control, and Dynamics, 1982, 5(3), 278±85. 9. Xu, D. M., Misra, A. K. and Modi, V. J., in Proceedings of the NASA/JPL Workshop on applications of Distributed Systems Theory to the Control of Large Space Structures. Pasadena, CA, 1982, pp. 317± 27. 10. Xu, D. M., Misra, A. K. and Modi, V. J., Journal of Guidance, Control, and Dynamics, 1986, 9(6), 663±72. 11. Modi, V. J., Pradham, S. and Misra, A. K., Acta Astronautica, 1995, 35(6), 373±84. 12. Pradham, S., Modi, V. J., Bhat, M. S. and Misra, A. K., Journal of Guidance, Control, and Dynamics, 1994, 17(5), 983±9. 13. Bainum, P. M. and Kumar, V. K., Acta Astronautica, 1980, 7(12), 1333±48. 14. Kalantzis, S., Modi, V. J., Pradham, S. and Misra, A. K., in AIAA/AAS Astrodynamics Specialist Conference, Paper no. 96-3571. San Diego, CA, U.S.A., 1996, pp. 12±26. 15. Xu, D. M., The Dynamics and Control of the Shuttle Supported Tethered Subsatellite System. Ph.D. Thesis, Department of Mechanical Engineering, McGill University, 1984. 16. Greenwood, D. T., Principles of dynamics, Englewood Clis: Prentice Hall, 1965. 17. Kalantzis, S., Dynamics and Control of Multibody Tethered Systems Using an Order-N Formulation. M.A.Sc. Thesis, Department of Mechanical
18. 19. 20. 21. 22. 23.
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Mred, fred, qred
mass matrix, force vector and generalized coordinate vector for the reduced model, respectively total number of links actuator coupling matrix or longitudinal LQR state weighting matrix inertial position of the ith link mass element dmi generalized acceleration vector of coupled system, q, for the full non-linear, ¯exible system ith link rotation matrix control thrust in the pitch and roll direction for the ith link, respectively position vector to the frame Fi from the frame Fi ÿ 1 desired oset acceleration vector (ith link oset position) in®nitesimal mass element of the ith link Cartesian components of di along the local vertical, local horizontal and orbit normal directions, respectively FÿQ rigid and ¯exible position vectors of dmi, ri+Fidi length of the ith link mass of the ith link {qT1, . . . , qTN}T rigid position of dmi in the frame Fi position of centre of mass of the ith link relative to the frame Fi desired settling time of the jth attitude actuator actuator force vector for entire system ¯exible deformation of the ith link along the xi, yi and zi directions, respectively control input for ¯exible subsystem Cartesian components of ri actual and estimated state of ¯exible subsystem, respectively output vector of ¯exible subsystem.
Engineering, The University of British Columbia, 1996. Tsakalis, K. S. and Ioannou, P. A., Linear time-varying systems: control and adaptation, Englewood Clis, NJ: Prentice Hall, 1993. Su, R., Systems and Control Letter, 1982, 2(1), 48±52. Kuo, B. C., Automatic control systems, Englewood Clis, NJ: Prentice Hall, 1987. Athans, M., IEEE Transactions on Automatic Control, 1971, AC-16(6), 529±52. Maciejowski, J. M., Multivariable feedback design, Wokingham: Addison-Wesley, 1989. Stein, G. and Athans, M., IEEE Transactions on Automatic Control, 1987, AC-32(2), 105±114.
6. NOMENCLATURE Xu, Yu Fi(xi, li) ai, bi, gi di(t) zj Zi y1 Af, Bf, Cf, Df Di Di KAi M(q,t) Mmai, Mmbi, Mmgi M r , fr
longitudinal state and measurement noise covariance matrices, respectively matrix containing mode shape functions of the ith ¯exible link pitch, roll and yaw angles of the ith link time-varying modal coordinate for the ith ¯exible link damping factor of the jth attitude actuator set of attitude angles (Zi={ai, bi, gi}T) true anomaly state-space representation of ¯exible subsystem inertial position vector of frame Ff magnitude of Di Fi(li+dxi+ 1) system's coupled mass matrix control moments in the pitch, roll and yaw directions, respectively for the ith link rigid mass matrix and force vector, respectively
N Qu Rdmi S(f,M/dc) Ti Ttai, Ttbi di dc dmi dxi, dyi, dzi f gi li mi q ri rcmi tsj u ui, vi, wi ud xi, yi, zi xf, xÃf yf
Acta Astronautica Vol. 42, No. 9, pp. 503±517, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/98 $19.00 + 0.00 S0094-5765(98)00041-1
DYNAMICS AND CONTROL OF MULTIBODY TETHERED SYSTEMS S. KALANTZIS, V. J. MODI{, S. PRADHAN Department of Mechanical Engineering, University of British Columbia, Vancouver BC, V6T 1Z4 Canada
and A. K. MISRA Department of Mechanical Engineering, McGill University, Montreal, Quebec, H3A 2K6 Canada (Received 29 September 1997) AbstractÐThe equations of motion for a multibody tethered satellite system in a three dimensional Keplerian orbit are derived. The model considers a multi-satellite system connected in series by ¯exible tethers. Both tethers and subsatellites are free to undergo three dimensional attitude motion, together with longitudinal and transverse vibration for the tether. The elastic deformations of the tethers are discretized using the assumed-mode method. In addition, the tether attachment points to the subsatellites are kept arbitrary and time varying, with deployment and retrieval degrees of freedom. The governing equations of motion are derived using an Order (N) Lagrangian formulation. Next, two independent controllers, i.e. an attitude and vibration controller, are designed to regulate the rigid and ¯exible motion present in the system, excited from various maneuvres performed during the course of a mission. The former controller utilizes the thrusters and momentum-wheels located on the rigid satellites with a control algorithm based on the feedback linearization technique. On the other hand, the latter is designed using the robust linear quadratic Gaussian-loop transfer recovery method actuating the variable tether attachment point, or oset position. Both controllers are successful in suppressing unwanted disturbances in the system in a acceptable amount of time. # 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
system must be controlled. This, of course, would be essential in the case of a satellite system intended for scienti®c experiments such as the micro-gravity facilities aboard the proposed International Space Station. Vibration of the ¯exible members will have to be checked if they aect the integrity of the onboard instrumentation. Thruster as well as momentum-wheel approaches [2] have been considered to regulate the rigid-body motion of the end-satellite, as well as the swinging motion of the tether. The procedure is particularly attractive due to its eectiveness over a wide range of tether lengths as well as ease of implementation. Other methods include tension and length rate control which regulate the tether's tension and nominal unstretched length, respectively [3,4]. It is usually implemented at the deployment spool of the tether. More recently, an oset strategy involving time dependent motion of the tether attachment point to the platform has been proposed [5]. It overcomes the problem of plume impingement created by the thruster control and the ineectiveness of tension control at shorter lengths. However, the eectiveness of such a con-
With the ever increasing demands by today's consumers to eciently exploit the resources available in space, a growing number of researchers and engineers are looking to tethered satellite systems to expand the current design capabilities of satellites. With that said, investigators have attempted to gain insight into the complex dynamics and control issues of tethered systems using a variety of models. Some of the results from these eorts have been reviewed by Misra and Modi [1] from which the major conclusions of these ®ndings can be summarized as follows: stationkeeping is only marginally stable; deployment can become unstable if a critical speed is exceeded; retrieval is always unstable. Furthermore, tether vibrations can also be excited to unacceptable levels during these maneuvres. In view of these conclusions, it is clear that an appropriate control strategy is needed to regulate the dynamics of the system, i.e. attitude and tether vibrations. First, the attitude motion of the entire {Author to whom correspondence should be addressed. 503
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troller can become limited with an exceedingly long tether due to a practical limit on the permissible oset motion. In response to these two control issues, a hybrid thruster/oset scheme has been proposed to combine the best features of the two methods [6,7]. In addition to attitude control, these schemes can be used to attenuate the ¯exible response of the tether. Tension and length rate control [8] as well as thruster based algorithms [9,10] have been proposed to this end. Modi et al. [11] have successfully demonstrated the eectiveness of an oset strategy to damp undesired vibrations. Passive energy dissipative devices, e.g. viscous dampers, are also another viable solution to the problem. The development of various control laws to implement the above mentioned strategies has also received much attention. An eigenstructure assignment in conjunction with an oset controller for vibration attenuation and momentum wheels for platform liberation control has been developed [12], in addition to the controller design from a graph theoretic approach [11]. Also, non-linear feedback methods, such as the feedback linearization technique (FLT), that are more suitable for controlling highly nonlinear, non-autonomous, coupled systems have also been considered [2,7]. It is important to point out that several linear controllers, including the classic state feed-back linear quadratic regulator (LQR), have received considerable attention [6,13]. Moreover, robust methods such as the linear quadratic Gaussin/loop transfer recovery (LQG/LTR) method have also been developed and implemented on tethered systems [2]. A more complete review of these algorithms as well as others applied to tethered systems has been presented by Pradhan [2]. The present analysis extended the current understanding of tethered system control to the case of N-body chain link systems capable of threedimensional motion [14], as shown in Fig. 1. The mathematical model is ®rst derived resulting in the nonlinear, coupled equations of motion with tether ¯exibility included. Next, the attitude controller is developed using the feedback linearization technique where thrusters and momentum-wheels are used to regulate the attitude motion of the individual links and hence of the entire system. In addition, a robust vibration controller is designed and implemented using the variable tether attachment point, or oset motion. The eectiveness of both the controllers is studied, through numerical simulation, at ®rst individually, and then in combination with each other.
Fig. 1. A schematic diagram of the space platform based N-body tethered satellite system.
nate system Fi is assigned to each link i. This frame is capable of three-dimensional rotation with respect to the inertial frame F0, located at the centre of Earth, and is composed of three axes: xi, along the length of the link; yi, perpendicular to the link; and zi, normal to the orbital plane. The rotation is described by three Euler angles: ai, bi and gi representing the inplane, pitch; out-of-plane, roll; and spin or yaw motion, respectively, by the
2. DYNAMICS FORMULATION
From Fig. 2, the analysis of the multibody system begins with the motion description of each link (tether or satellite) separately. A body ®xed coordi-
Fig. 2. Vector components of the ith and (i ÿ 1)th chain links.
Multibody tethered systems
transformation matrix 2
3
Cbi Cai 6 7 Cbi Sai 6 7 6 7 ÿSbi 6 7 6 7 6 7 6 ÿCgi Sai Sgi Sbi Sai 7 6 7 Ti 6 Cgi Cai Sgi Sbi Sai 7 6 7 Sgi Cbi 6 7 6 7 6 7 6 Sg Sai Cg Sb Cai 7 i i 6 i 7 4 ÿSg Cai Cg Sb Sai 5 i i i Cgi Cbi
1
iÿ1 ÿ X
Tkÿ1 dk Tk KAk d k Tk {^ lk Tiÿ1 di 8
k1
where ri is the rigid component of the position of dmi from Fi and ffi (ri) is the ¯exible deformation vector at ri. Tether ¯exibility, in this analysis, is discretized using the assumed-mode method. The mode shapes considered in the formulation correspond to the shape functions for a string [15], which are, for the jth longitudinal mode, 2jÿ1 xi ; 3 F xj i xi ; li li where li is the ith tether length. In the case of inplane and out-of-plane transverse defections, the admissible functions are given as p jpxi F yj i xi ; li F zji xi ; li 2 sin ; 4 li where Z2 is added as a normalizing factor. Recasting the mode shape functions into a matrix form and assigning a corresponding generalized modal coordinate di, the ¯exible position vector f fi can be de®ned as 5
where Fi contains the mode shape functions given by eqns (3) and (4). Thus inserting eqn (5) into eqn (2) and de®ning gi=ri+Fidi Rdmi Di Ti gi :
duces an additional variable di which represents the position vector of the frame Fi from the previous frame Fi ÿ 1. For the case of where link i represents a tether link and link i ÿ 1 a rigid satellite, di={dxi, dyi, dzi}T is the oset of the tether attachment point to the rigid satellite which is used to control the tether vibration. De®ning KAx ÿ 1=Fi ÿ 1(li ÿ 1+dxi) and applying eqn (7) to recursively de®ne Di ÿ 1, then it can be shown that Di
which relates the body-®xed frame Fi to the frame F0. Here, Cx and Sx are abbreviations for cos(x) and sin(x), respectively. Thus, from Fig. 2, the inertial position of the elemental mass of the ith link can now be given as h i 2 Rdmi Di Ti ri f i f ri :
f i f ri F i xi ; li dd i t
505
6
thus completely de®ning the position vector of dmi with respect to the set of generalized coordinates given by q = {qT1 . . . qTN}T where 0 1 di B Zi C B qi @ C 9 di A li and Zi={ai, bi, gi}T. Note that by properly manipulating eqn (6) and eqns (7) and (8) then a factorized expression of the equations of motion of the entire system can be derived resulting in a Order (N) formulation of the governing equations of motion [14]. This has tremendous computational advantages when the equations are numerically solved for a large number of links (N large). Using eqns (6) and (8) the kinetic, potential and strain energies of the entire N-body system are given by N 1X T Ke 10 R_ R_ dmi dmi ; 2 i1 mi dmi Vg ÿ m
N X mi i1
"
Di
N mX 1 2 i1 D3i
3 gTi gi dmi 2DTi Ti gi dmi ÿ 2 DTi Ti Di # gi gTi dmi Ti Di ; 11
N 1X E i Ai Ve 2 i1
li 0
e2i dxi ;
12
Di in eqn (6), represents the inertial position of Fi from F0 and it serves to completely describe the orbital motion of the ith link with respect to Earth. However, from Fig. 2, it is clear that Di Diÿ1 Tiÿ1 liÿ1 {^ di ÿ 7 Tiÿ1 F iÿ1 liÿ1 dxi d iÿ1 ;
respectively. Here m = 3.986 105 km3/s2 is Earth's gravitational coecient, Ei and Ai are the Young's modulus and cross-sectional area of the ith link, respectively and ei is the ith link strain function given by " # @ui 1 @vi 2 @wi 2 ; 13 ei @xi 2 @xi @xi
which is de®ned with respect to the orbital position of the previous link i ÿ 1. In addition, eqn (7) intro-
where ui, vi and wi are the ¯exible deformations along the xi, yi and zi directions, respectively.
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The eect of structural damping is incorporated by introducing Rayleigh's dissipation function [16]. Application of the Lagrange procedure results in the nonlinear, non-autonomous and coupled equations of motion. As can be expected, the complete set of equations is indeed lengthy and hence is not presented here for brevity. Essentially, they have the form ÿ ÿ ÿ _ t Q q; q; _ t M q; t q F q; q; 14 where M(q, t) is the nonlinear symmetric mass _ t) is the forcing vector function and matrix; F(q, q, _ t) is the vector of non-conservative generalQ(q, q, ized external forces, including control inputs, acting on the system. 2.1. Speci®cation of the oset position The derivation of the equations of motion using the O (N) formulation requires that the oset coordinate di be treated as an generalized coordinate. However, this oset must later be constrained or be controlled to follow a prescribed motion dictated by either the designer or the controller. This results in an additional term that is added to the generalized acceleration vector giving
respectively [16]. Fem is the mth external force acting on the system at a position Rm from the inertial frame and nu is the total number of actuators (forces). 2.2.1. Generalized thruster forces. One set of thrusters is placed on all subsatellites and on the end-body, i.e. all rigid bodies, except the ®rst (platform), as shown in Fig. 3. They are capable of ®ring in all three directions. From the schematic, the inertial position of the thruster, located at the centre of mass of body 3, is given as ÿ R3 d1 T1 d2 T2 l2 {^ KA2 d 2 T3 rcm3 d1 df 1 T3 rcm3 :
17
De®ning
ÿ df j Tj dj1 Tj1 lj1 {^ KAj1 d j1
18
then the thruster position of body 5 is given as R5 d1 df 1 df 3 T5 rcm5 :
19
In general, the inertial position of the thruster on the ith body can now be given as Ri d1
iÿ2 X
df 2kÿ1 Ti rcmi :
20
k1
q S f; Mjdc h iÿ1 ÿ Mÿ1 f Mÿ1 P c P cT Mÿ1 P c dc P cT Mÿ1 f
15
which is the new constrained vector equation of motion with the oset speci®ed by dc [17]. Note that pre-multiplying Mÿ1 by PcT is the equivalent of extracting the rows of Mÿ1 corresponding to the d equations whereas post-multiplying by Pc is equivalent to extracting the columns of the matrix corresponding to the d equations.
We next de®ne the column matrix " # h iT @ i Ri Qt j @qj " # @ @ @ @ Ri Ri Ri Ri @dj @Z Zj @Z Zj @lj
21
where i = 3, 5, . . . , N and j = 1, 2, . . . , i. The vector derivative of Ri with respect to the scaler
2.2. Generalized control forces The treatment of non-conservative external forces acting on the system is considered next. In reality, their are numerous environmental forces present on the system, e.g. atmospheric drag, solar radiation pressure as well as inertial dampers. However, in this model, only the eect of the active control thrusters and momentum-wheels is considered. These forces will be used to control the attitude motion of the system. When deriving the equation of motion of a system using the Lagrange equations, the contribution of each force must be properly distributed among all the generalized coordinates. This is done using the relation Qk
nu X m1
Fem
@Rm @qk
16
where Qk and qk are the kth element of Q and q,
Fig. 3. Inertial position of subsatellite thruster forces.
Multibody tethered systems
components qk is stored in the kth column of [Qitj]T. Thus, for the case of 3-bodies, eqn (16) for thrusters becomes 2 3 3 Qt 1 6 6 3 7 7 22 Q 6 Qt2 7T3 Tt2 4 5 Q3t3 where Tti={0, Ttai , Ttbi }T is the thrust vector for the ith link (tether i ÿ 1). Here Ttai and Ttbi are the thrust in the in-plane yi and out-of-plane, zi directions, respectively. For the case of 5-bodies, 2 3 3 Qt1 Q5t1 6 6 Q3 Q5 7 7 6 t2 t2 7 6 7 6 7 T3 Tt2 : 23 Q 6 Q3t3 Q5t3 7 6 7 T5 Tt4 6 7 5 6 0 Qt4 7 4 5 0 Q5t5 The result for seven or more bodies follows the pattern established by eqn (23). 2.2.2. Generalized momentum-wheel torques. When deriving the generalized moments arising from the momentum-wheels on each satellite, including the platform, it is easier to view the torques as coupled forces. From Fig. 4, we see that the generalized force arising from these coupled forces, for the ith link, are Qk
3 X
Fei
i1
ÿ
3 X i1
@ fRi Ei g @qk
Fei
@ f R i ÿ Ei g @qk
24
507
where |^ acting at E1 Exi {^;
Fe1 Fai Fe2 ÿ Fbi Fe3 Fgi
k^ acting at E2 Exi {^;
k^ acting at E3 Eyi |^:
Expanding eqn (24), it becomes clear that Qk 2
3 X i1
Fei
@ f Ei g @qk
25
which is independent of Ri and therefore quite simpler. Thus de®ning the moments Mmai=2FaiExi, Mmbi=2FbiExi and Mmgi=2FgiEyi and, transforming the coordinates to the body ®xed frame using the rotation matrix, the ith link generalized force due to the momentum-wheels on link i, is @ @ Ti {^ Mmai ÿ Ti k^ Ti {^ Mmbi Qi Ti |^ @qi @qi @ Ti k^ Ti |^ Mmgi QM i M mi @qi where Mmi={Mmai, Mmbi, Mmgi}T and @ @ Ti |^ Ti {^ ÿ Ti k^ Ti {^ QM i @qi @qi @ Ti |^ : Ti k^ @qi
26
27
When combining the thruster forces and momentum-wheel torques together, a compact expression for the generalized force vector, for 5 bodies, can be expressed as 2 M 3 0 Q51 T5 0 Q1 Q31 T3 6 7 Q32 T3 0 Q52 T5 0 7 6 0 6 7 Q 6 Q53 T5 0 7 Q33 T3 QM 3 6 0 7 6 7 0 0 Q54 T5 0 5 4 0 0 0 0 Q55 T5 QM 5 0 1 Mm1 B B Tt2 C C B C C Qu u: M 28 B m 3 B C B C @ Tt4 A Mm5 The pattern of eqn (28) is retained for the case of N links.
3. ATTITUDE CONTROL
Fig. 4. Coupled force representation of rigid body momentum-wheels.
The design of an attitude controller, with objective to regulate the librational motion of the system, is now undertaken. As discussed earlier, several methods are available to accomplish the controller's objective, including length rate and tension control; oset, or tether attachment position, control; and
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thruster and momentum-wheel based controllers. All of these methods have their relative advantages and drawbacks, however, the thruster/momentumwheel approach is chosen for its good overall performance over a wide range of tether lengths. This control method utilizes the on-board thrusters located on each rigid satellite, excluding the ®rst one (platform), to regulate the pitch and roll motion of the previous tether, i.e. the thrusters located on satellite 2 (link 3) regulate the attitude motion of tether 1 (link 2). On the other hand, the rigid body motion of each rigid link, including the platform, is controlled using a set of three momentum wheels placed mutually perpendicular to each other. The choice of control algorithms, from which the controller is designed, is based on several important factors. Firstly, the controller scheme has to perform well for many dierent time-varying con®gurations, as is the case during tether deployment or retrieval. In addition, it would have to be computationally ecient in order to be implemented in realtime. Finally, simplicity of implementation is also desired. Several control algorithms exist in practice. The implementation of a ®nite number of linear time invariant (LTI) controllers, scheduled discretely over a range of system con®gurations (gain scheduling [2]), is one possible solution. An alternative approach involves the use of adaptive control techniques in conjunction with linear time varying (LTV) systems theory, from which the design of the controller would include the time-varying tether length case. One such technique is on-line parametric identi®cation [18] where the system's ®xed parameters are estimated in real-time during the course of the simulation. However, the controller that is eventually chosen is based on the nonlinear feedback linearization technique (FLT). This controller is well suited for highly time-varying systems whose dynamics can be accurately modelled, as is the case for space systems in general. The FLT method is based on transforming the nonlinear, time-varying governing equations into a LTI system using a nonlinear time-varying feedback [2,19]. Once the system is transformed, the resulting LTI system can be controlled with any of the numerous linear control algorithms available in the literature. In this case, a simple PD controller is adopted and shown to have good performance. The choice of a FLT control scheme satis®es one of the criteria of selection mentioned earlier, namely being valid over a wide range of tether lengths. The computational eciency of the method is addressed next. In order to implement this controller in realtime, the computation of the system's inverse dynamics must be executed quickly, hence a simpler model, that performs well, is in order hence, a model based on the rigid nonlinear equations of motion is chosen.
3.1. Controller design using feedback linearization technique The control model used is based on the rigid equations of motion given by Mr qr f r Qu u
29
where the left hand side is the equations of motion of the rigid model. The right hand side is the generalized external forcing term due to the thrusters and momentum-wheels and is given by eqn (28). Letting, f~r f r ÿ Qu u and substituting in eqn (15) gives ~ Mr j dc : qr S f;
30
31
Expanding eqn (31), it can be shown that ÿ qr S f r ; Mr j dc ÿ S Qu ; Mr j 0 u Fr ÿ Qr u
32
where S(Qu, Mrvv0) is the column-vector matrix given by ÿ h ÿ S Qu ; j 0 S Qu :; 1; Mr j 0 ; ÿ S Qu :; 2; Mr j 0 ; . . .
ÿ i . . . ; S Qu :; nu; Mr j 0
33
and Qu(:, i) is the ith column of Qu. Extracting only the controlled equations from eqn (32), i.e. the attitude equations, we get qrc Frc ÿ Qrc u vrc
34
where vrc is the new control input required to regulate the decoupled linear system. At this point, a simple PD controller can be applied, i.e., vrc qrcd K q_ rcd ÿ q_ rc Kp qrcd ÿ qrc : 35 Eqn (35) is known as the secondary controller whereas eqn (32) is the primary controller. Kp and Kv are the proportional and derivative gain matrices and qÈrcd, qÇ rcd and qrcd are the desired acceleration, velocity and position vectors for the attitude angles of each controlled body, respectively. Solving for u from eqn (34) leads to u Qÿ1 rc Frc ÿ vrc
36
and the new controlled equations of motion are given by eqns (15) and (28). 3.2. Simulation results The FLT controller is implemented on the 3body STSS tethered system de®ned by the following set of parameters:
Multibody tethered systems
2
1091430 ÿ8135 6 . I1 4 ÿ8135 8646050 328108 27116
3 328108 7 27116 5 8286760
kg m2 platform inertia; 3 200 0 0 6 7 . I2 4 0 400 0 5 0 0 400 2
kg m2 endÿsatellite inertia; . m1=90,000 kg (mass of the space station platform); . m2=500 kg (mass of the end-satellite); . EtAt=61,645 N (tether elastic stiness); . rt=4.9 kg/km (tether density); . Zd=0.5% (tether structural damping coecient); . rcm3={1,0,0}T m (satellite 2 tether attachment point). The response variables are de®ned as follows: . a1, b1 satellite 1 (platform) pitch and roll angle; . a2, b2 satellite 2 (end-satellite) pitch and roll angle; . at, bt tether pitch and roll angle; . lt tether length; . d2 = {dx, dy, dz}T tether attachment position relative to satellite 1; . d2={dx, dy, dz}T tether ¯exible modal coordinates in the longitudinal x, in-plane transverse y, and out-of-plane transverse z directions, respectively. All the attitude angles, ai and bi , are measured with respect to the local vertical, local horizontal (LVLH) frame. The system is taken to be in a nominal circular orbit, 289 km, in altitude, with an orbital period of 90.3 min. The choice of proportional and derivative matrix gains is based on a desired settling time of tsj=0.5 orbits and a damping factor of zj=0.7 for both the pitch and roll actuators in each body. Given zj and tsj , it can be shown [20] that q ÿ1 ln 0:05 1 ÿ z2j onj 37 zj tsj and hence kpj o2nj ; kvj 2z j onj
38
where kpj and kvj are the diagonal elements of the matrices Kp and Kv de®ned by eqn (35), respectively. Note that due to the fact that each body-®xed frame Fi is referred directly to the inertial frame F0, the nominal pitch equilibrium angle is zero degrees only when measured from the LVLH frame (aiÿy1).
509
However, it is y1 (true anomaly) when referred to F0, thus the desired position vector qrcd is set to be equal to y1(t), i.e. the time-varying true anomaly, for the pitch motion; and zero for the roll and yaw motion, such that 00 1 1 y1 t B B@ 0 A C C B C 0 B 1 C B C B C .. C 2 <3N1 : 39 qrcd B . B C 1 C B0 B y1 t C B C @@ 0 A A 0 N The desired velocity and acceleration are given as 00 1 1 y_ 1 t B C C BB C B@ 0 A C B C 0 B 1 C B C B C .. _q qrcd B C 2 <3N1 ; . B C 1 C B0 B y_ 1 t C B C C C BB @ A 0 @ A 0 N 00 1 1 y1 t B C C BB C B@ 0 A C B C 0 B 1 C B C B C .. qrcd B 40 C 2 <3N1 ; . B C 1 C B0 B C B y1 t C C C BB A @ 0 @ A 0 N respectively. When the system is in a nominal circular orbit, qrcd=0. Figure 5 presents the controlled response of the STSS system with oset d2={1, 0, 1}T m during the retrieval maneuvre from a tether length of 20 km to 200 m in 3.5 orbits. With the FLT controller, based on the non-linear rigid model, in place, both the pitch and roll angles of the rigid bodies and of the tether are now attenuated in less than 1 orbit. The control requirement for the pitch motion of the platform and the tether are maximum during the constant velocity phase of retrieval when the Coriolis force acting on the system is maximum. However, for the roll case, where there is no Coriolis force present, the maximum control requirement occurs during the initial phases of retrieval. Moreover, there is clear evidence of ¯exible deformations along the transverse directions of the tether, again due to the same Coriolis force. Although, these vibrations stabilize during the terminal phase of retrieval, they remain present for the duration of the mission.
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Fig. 5. Retrieval dynamics of the three body STSS, using the non-linear, rigid FLT controller; with oset along the local vertical and orbit normal: (a) attitude and vibration response; (b) control actuator time histories. 4. VIBRATION CONTROL
The vibration control strategy chosen in this analysis is based on oset control, i.e. control of the tether's attachment point to the platform (satellite 1). All three degrees of freedom of the oset position are used to control both the longitudinal and the in-plane and out-of-plane transverse modes of vibration. In practice, the oset controller can represent the motion of a dedicated manipulator or robotic arm attached to the space station or space shuttle from which the tether is deployed. Tether elastic deformations arising from deployment or retrieval of the tether can be controlled
during the maneuvre or, once the maneuver is completed and the tether length remains constant (stationkeeping). The latter method is adopted here. Vibration control during the stationkeeping phase is generally sucient since this phase is where most of the mission objectives are carried out. 4.1. System linearization and state-space realization The design of the oset controller begins with the linearization of the equations of motion of the system about their equilibrium position. However, this can present a problem when considering the current form of the equations of motion namely, the pitch
Multibody tethered systems
angle, ai. As mentioned in the previous section, the pitch angle is not referred to the LVLH frame thus its equilibrium is not a constant, but is equal to y1(t). Two methods can be used to resolve this problem. The ®rst method corresponds to using the nonstationary equations of motion in their present form and deriving a controller based on the linear time varying (LTV) system. The second method is based on designing a linear time invariant (LTI) controller based on a new set of reduced governing equations representing the motion of the three body tethered system. The latter method is chosen here. Furthermore, the LTI controller design is done completely o-line and thus is computationally more ecient. The reduced model is derived using the Lagrangian approach with the following generalized coordinates, qred a1 ; b1 ; a2 ; b2 ; dx ; dy ; dz ; a3 ; b3
41
where ai and bi are the rigid pitch and roll angles of link i, relative to the LVLH frame. The symbols dx, dy and dz are the ¯exible modal coordinates for the longitudinal, in-plane and out-of-plane transverse tether ¯exible deformations respectively, de®ned in this case, for the ®rst mode of vibration only (note, the sux ``2'' of the modal coordinates d2 is dropped for conciseness). The non-linear, nonautonomous, coupled equations of motion of the tethered system can now be given as Mred qred f r 0
511
constant. Inverting Ms and multiplying through we get _ r ÿ Mÿ1 qr ÿ Mÿ1 s Cs q s Ks qr ÿ1 _ ÿ Mÿ1 s Cd d22 ÿ Ms Kd d2 ÿ1 ÿ Mÿ1 s Md d22 ÿ Ms f s :
De®ning ud=dÈ2 and v = {qTr, dTr}T then eqn (44) can be rewritten as " # ÿ1 ÿMÿ1 s Cs ÿMs Cd v v_ 0 0 " # ÿ1 ÿMÿ1 s Ks ÿMs Kd v 0 0 " # ! ÿMÿ1 ÿMÿ1 s Md s fs ud ; Id 0 v MCv_ MKv MIud Fs : Finally, de®ning x01 x x02
Ms qred Cs q_ red Ks qred Md d2 Cd d_ 2 Kd d2 f s 0 43 where Ms, Cs, Ks, Md, Cd, Kd and fs in eqn (43) are
!
0
v_ v
45
! 46
then the LTI equations of motion can be recast into the state-space form x_ 0 Ax0 Bud Fd where
MC MK 2 <2424 ; A 0 Id12
47
42
where Mred and fred are the reduced mass matrix and forcing term of the systems, and are functions of qred and qÇ red in addition to the time varying oset position, d2, and its time derivatives d_ 2 and dÈ2. An additional consequence of referring the pitch motion to a local frame is that the new reduced equations are now independent of y1, and under the further assumption of the system negotiating a circular orbit, y_ 1 remains constant. The non-linear reduced equations can now be linearized about their equilibrium point. For all the generalized coordinates, the equilibrium position is zero with the exception of dx which has a non-zero equilibrium, dx0 and d2, which is linearized about d20. Finally, linearizing eqn (42) and recasting into matrix form gives
44
B and
MI 0
Fd
Fs 0
48
2 <243 ;
49
2 <241 :
50
Let x0 xeq x;
51
where x is the perturbation vector from the constant equilibrium state vector xeq. Substituting eqn (51) into eqn (47), we get the linear, perturbation state vector equation of motion modelling the reduced tethered system as ÿ 52 x_ 0 x_ Ax Bud Fd Axeq : It can be shown that for ideal control [21], Fd+Axeq=0 in eqn (52) thus leaving the familiar state-space equation x_ Ax Bud :
53
The selection of the output vector completes the state space realization of the system. The output
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vector consists of the longitudinal deformation from the equilibrium position, dx0, of the tether at xt=lt; the slope of the tether due to the transverse deformation at xt=0; and the oset position d2, from its initial position d20. Thus, the output vector y is given by 0
1 dp x ÿ dx0 B C F u lt dx ÿ dx0 B 2p dy C 0 B B C C F v 0dy B C B plt C 0 B B C C F 0d z 2 p w B B C yB B dz C C C d ÿ d x x 0 B C B lt C @ B A C dy ÿ dy0 B dx ÿ dx0 C @ dy ÿ dy A dz ÿ dz 0
1
0
54
the longitudinal and transverse modes of vibration which also allows the decoupling of the ¯exible subsystem into a set of longitudinal (dx and dx) and transverse (dy , dz , dy and dz) subsystems. The appended oset system, d, must also be included since it acts as the control actuator. It is important to note however that the inplane and out-ofplane transverse modes can not be decoupled because their oscillation frequencies are of the same order. The two ¯exible subsystems are summarized below. 4.2.1. Longitudinal subsystem. xu Au xu Bu udu ; yu Cu xu
0
dz ÿ dz0
where F'v(0) = @/@xi[Fv(xi)]xi=0 and F'w(0) = @/ @xi[Fw(xi)]xi=0. 4.2. Linear quadratic Gaussian-loop transfer recovery Now that the linear state space model is de®ned by eqns (53) and (54), the design of the control algorithm can commence. The control algorithm chosen is the linear quadratic Gaussian (LQG) estimator based optimal controller [21,22]. The LQG controller is a widely used optimal controller whose theory has been well developed by many authors in the last 25 years. It consists of the design of a Kalman±Bucy Filter (KBF) which provides an estimate of the states xÃ, and a linear quadratic regulator (LQR), which is separately designed assuming all the states of x are known. Both the LQR and KBF designs independently have good robustness properties, i.e. retain good performance when disturbed, due to model uncertainty. However the combined LQR and KBF designs, i.e. the LQG design, has poor stability margins in the presence of model uncertainties. This limitation has led to the development of an LQG design procedure that improves the performance of the compensator, by recovering the full state feedback robustness properties at the plant input or output. This procedure is known as the linear quadratic Gaussian-loop transfer recover (LQG/LTR) control algorithm. A detailed development of its theory is available in the literature [22] and hence not repeated here. As mentioned earlier, the main objective of this oset controller is to regulate the tether vibration described by the d equations. However, from the response of the uncontrolled system [14] there is a large dierence between the magnitude of the librational and vibrational frequencies. This separation of frequencies allows for the separate design of the vibration controller and that of the attitude controller. Thus, only the ¯exible subsystem, composed of the d and d equations, is required in the oset controller design. Similarly, there is also a wide separation of oscillation frequencies between
55
where xu={d_ x , d_ x2 , dx, dx2}T, udu=dÈx2 and, from eqn (54), Cu
0 0 0 0 F u lt 0 0 0 0 1 0 0
1 0 : 0 1
56
Au and Bu are the rows and columns of A and B, respectively and correspond to the components of xu. The state LQR weighting matrix Qu is chosen as 2
1 6 0 6 Qu 4 0 0
0 1 0 0
3 0 0 0 0 7 7 1 0 5 0 10
57
and the input weighting Ru=Id2. The state noise covariance matrix is given as 2
1 6 0 Xu 6 40 0
0 1 0 0
0 0 4 0
3 0 07 7 05 1
58
while the measurement noise covariance matrix is
1 0 : Yu 0 15
59
Given the above mentioned matrices, the design of the LQG/LTR compensator is computed in MATLAB returning the state space representation of the LQG/LTR dynamic compensator ^_ u Aku x^ u ÿ Bku yu ; x udu Cku x^ u
60
where xÃu is the state estimate vector of xu. Loop transfer recovery is performed at the system output, i.e. the return ratio at the output approaches that of the KBF loop given by Cu(sIdÿAu)Kfu [22]. This is done by choosing a suciently large scaler value, r, such that the singular values of the return ratio
Multibody tethered systems
513
Fig. 6. Singular value plot for the LQG and LQG/LTR compensator with respect to target return ratio: (a) longitudinal design; (b) transverse design.
approach those of the target design. For the longitudinal controller design, r= 5 105. Figure 6(a) shows a plot of the singular values of the recovered compensator design and the non-recovered, LQG, compensator design with respect to the target design. Unfortunately, perfect recovery is not possible, especially at higher frequencies, because the system is non-minimal, i.e. it has transmission zeros with positive real parts [23]. 4.2.2. Transverse subsystem. xv Av xv Bv udv ; yv Cv xv
61
where xv={d_ y , d_ z , d_ y2 , d_ z2 , dy , dz , dy2 , dz2 ,}T, È È udv={dy2, dz2}T and
2
0 60 Cv 6 6 40 2
0
6 60 6 6 6 6 60 6 6 40 0
0 0 0 F 0v 0 0 0 0 0 0 0 0 0 0 0 0
0 p 2p 0 0 0 lt 0 0 0
0
0 0 0 0 0 0
0 0
0 0 F w 0 0 0
3 0 0 0 07 7 7 1 05 0 1 3
0 0 07 7 7 p 7 7 2p 0 07 7: lt 7 7 0 1 05 0 0 1
62
Av and Bv are the rows and columns of the A and B corresponding to the components of xv.
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The state LQR weighting matrix Qv is given as 2
10 0 0 6 0 10 0 6 6 0 0 104 6 6 0 0 0 Qv 6 6 0 0 0 6 6 0 0 0 6 4 0 0 0 0 0 0
0 0 0 104 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
3 0 07 7 07 7 07 7 07 7 07 7 05 1
63
and the input weighting Rv=Id4. The state noise covariance matrix is given as
2
1 60 6 60 6 60 Xv 6 60 6 60 6 40 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 9 104 0 0
3 0 0 7 7 0 7 7 0 7 7 64 0 7 7 0 7 7 0 5 4 9 10
while the measurement noise covariance matrix is 2 3 1 0 0 0 60 1 0 0 7 7: 65 Yv 6 4 0 0 5 104 0 5 4 0 0 0 5 10
Fig. 7. Stationkeeping dynamics of the three body STSS, using the non-linear, rigid FLT attitude controller and LQG/LTR offset vibration: (a) attitude and libration controller response.
Multibody tethered systems
515
Fig. 7. Stationkeeping dynamics of the three body STSS, using the non-linear, rigid FLT attitude controller and LQG/LTR oset vibration: (b) vabration and oset response.
Similar to the design approach of the longitudinal controller, the transverse oset controller can be designed with r = 50. The result is the transverse compensator system given by ^_ v Akv x^ v ÿ Bkv yv ; x udv Ckv x^ v
66
where xÃv is the state estimate of xv. The singular value plot of the target transfer function with respect to the ones achieved by the LQG and LQG/
LTR are presented in Fig. 6(b). As can be seen, the recovery is not as good as that for the longitudinal case, again due to the non-minimal system. Moreover, the low value of r indicates that only a little recovery in the transverse subsystem is possible, suggesting that the robustness of the LQG design is almost the maximum that can be achieved, under these conditions. De®ning xf={xTu, xTv}T, xÃf={xÃTu, xÃTv}T and ud={udu, uTdv}T, the longitudinal compensator can be combined to give
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" ^_ f x " ud
yf
A ku 0
0 A kv
#
"
Bku x^ f ÿ 0
# 0 y Ak x^ f ÿBk yf ; Bkv f
# 0 C ku x^ f Ck x^ f ; 0 C kv ! " # Cu 0 yu xf Cf xf : 0 Cv yv
67
De®ning a permutation matrix Pf such that xf=Pfx where x is the state vector of the full non-linear system de®ned by eqn (15), then the compensator and full nonlinear system equations can be combined as ÿ q S f; M j dc ; ^_ f Ak x^ f ÿ Bk Cf Pf x; x ud Ck x^ f :
(FLT) with the thrusters and momentum-wheels located on the rigid satellites. The controller is successful in regulating the pitch and roll motion of both the tether and end-satellites in a reasonable amount of time. Furthermore, a vibration controller making use of the variable tether attachment point or oset position is successful in suppressing any unwanted ¯exible deformations arising from various mission maneuvers. In this case, the robust linear quadratic Gaussian-loop transfer recovery method is employed to design the controller for the stationkeeping phase.
REFERENCES
68
4.3. Simulation results The response for the stationkeeping STSS with a tether length of 20 km and initial oset position of 1 m in each direction is presented in Fig. 7 where both the FLT attitude control and the LQG/LTR oset controller are activated. As expected, the oset controller is successful in quickly damping out the vibration in both the longitudinal and transverse direction. However, it is clear that the presence of oset control requires a relatively large control moment to regulate the attitude of the platform. This is due to the additional torque created by the larger moment arm, over 5 m and 2 m in the in-plane and out-of-plane direction respectively, introduced by the oset control. In addition, the control moment for the platform is modulated by the tether's transverse vibration through oset coupling. However, this does not signi®cantly aect the motion of the tether whose thruster force remains relatively unchanged from the uncontrolled vibration case.
5. CONCLUDING REMARKS
The equations of motion of an N-body tethered system undergoing three-dimensional orbital and attitude motion are derived using a computationally ecient Order (N) Lagrangian formulation. The tether, which is also free to vibrate in the longitudinal as well as inplane and out-of-plane transverse directions, is described using the assumed-mode methods with an arbitrary number of modes. The rigid satellites are also capable of rigid-body rotation about their equilibrium. The governing equations are then integrated numerically in the presence of an attitude and ¯exible vibration controller. The attitude controller is designed using the feedback linearization technique
1. Misra A. K. and Modi V. J., A survey on the dynamics and control of tethered satellite systems, NASA/AIAA/PSN International Conference on Tethers, Arlington, VA, U.S.A., 1986, Paper No. AAS-86-246, In: Advances in the astronautical sciences, Bainum PM et al. editors, American Astronautical Society, 62, pp. 667±719. 2. Pradham, S., Planar Dynamics and Control of Tethered Satellite Systems. Ph.D. Thesis, Department of Mechanical Engineering, The University of British Columbia, 1994. 3. Rupp, C. C., A tether tension control law for tethered sub-satellite deployed along local vertical, NASA TM X-64963, 1975. 4. Misra, A. K. and Modi, V. J., Acta Astronautica, 1992, 26(2), 77±84. 5. Pradhan, S., Modi, V. J. and Misra, A. K., in AIAA/ AAS Astrodynamics Specialist Conference, Paper No. 96-3570. San Diego, CA, U.S.A., 1996, pp. 1±11. 6. Modi, V. J., Lakshmanan, P. K. and Misra, A. K., Acta Astronautica, 1990, 21(5), 283±94. 7. Pradhan, S., Modi, V. J. and Misra, A. K., International Journal of Control, 1996, 64(2), 175±93. 8. Misra, A. K. and Modi, V. J., Journal of Guidance, Control, and Dynamics, 1982, 5(3), 278±85. 9. Xu, D. M., Misra, A. K. and Modi, V. J., in Proceedings of the NASA/JPL Workshop on applications of Distributed Systems Theory to the Control of Large Space Structures. Pasadena, CA, 1982, pp. 317± 27. 10. Xu, D. M., Misra, A. K. and Modi, V. J., Journal of Guidance, Control, and Dynamics, 1986, 9(6), 663±72. 11. Modi, V. J., Pradham, S. and Misra, A. K., Acta Astronautica, 1995, 35(6), 373±84. 12. Pradham, S., Modi, V. J., Bhat, M. S. and Misra, A. K., Journal of Guidance, Control, and Dynamics, 1994, 17(5), 983±9. 13. Bainum, P. M. and Kumar, V. K., Acta Astronautica, 1980, 7(12), 1333±48. 14. Kalantzis, S., Modi, V. J., Pradham, S. and Misra, A. K., in AIAA/AAS Astrodynamics Specialist Conference, Paper no. 96-3571. San Diego, CA, U.S.A., 1996, pp. 12±26. 15. Xu, D. M., The Dynamics and Control of the Shuttle Supported Tethered Subsatellite System. Ph.D. Thesis, Department of Mechanical Engineering, McGill University, 1984. 16. Greenwood, D. T., Principles of dynamics, Englewood Clis: Prentice Hall, 1965. 17. Kalantzis, S., Dynamics and Control of Multibody Tethered Systems Using an Order-N Formulation. M.A.Sc. Thesis, Department of Mechanical
18. 19. 20. 21. 22. 23.
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517
Mred, fred, qred
mass matrix, force vector and generalized coordinate vector for the reduced model, respectively total number of links actuator coupling matrix or longitudinal LQR state weighting matrix inertial position of the ith link mass element dmi generalized acceleration vector of coupled system, q, for the full non-linear, ¯exible system ith link rotation matrix control thrust in the pitch and roll direction for the ith link, respectively position vector to the frame Fi from the frame Fi ÿ 1 desired oset acceleration vector (ith link oset position) in®nitesimal mass element of the ith link Cartesian components of di along the local vertical, local horizontal and orbit normal directions, respectively FÿQ rigid and ¯exible position vectors of dmi, ri+Fidi length of the ith link mass of the ith link {qT1, . . . , qTN}T rigid position of dmi in the frame Fi position of centre of mass of the ith link relative to the frame Fi desired settling time of the jth attitude actuator actuator force vector for entire system ¯exible deformation of the ith link along the xi, yi and zi directions, respectively control input for ¯exible subsystem Cartesian components of ri actual and estimated state of ¯exible subsystem, respectively output vector of ¯exible subsystem.
Engineering, The University of British Columbia, 1996. Tsakalis, K. S. and Ioannou, P. A., Linear time-varying systems: control and adaptation, Englewood Clis, NJ: Prentice Hall, 1993. Su, R., Systems and Control Letter, 1982, 2(1), 48±52. Kuo, B. C., Automatic control systems, Englewood Clis, NJ: Prentice Hall, 1987. Athans, M., IEEE Transactions on Automatic Control, 1971, AC-16(6), 529±52. Maciejowski, J. M., Multivariable feedback design, Wokingham: Addison-Wesley, 1989. Stein, G. and Athans, M., IEEE Transactions on Automatic Control, 1987, AC-32(2), 105±114.
6. NOMENCLATURE Xu, Yu Fi(xi, li) ai, bi, gi di(t) zj Zi y1 Af, Bf, Cf, Df Di Di KAi M(q,t) Mmai, Mmbi, Mmgi M r , fr
longitudinal state and measurement noise covariance matrices, respectively matrix containing mode shape functions of the ith ¯exible link pitch, roll and yaw angles of the ith link time-varying modal coordinate for the ith ¯exible link damping factor of the jth attitude actuator set of attitude angles (Zi={ai, bi, gi}T) true anomaly state-space representation of ¯exible subsystem inertial position vector of frame Ff magnitude of Di Fi(li+dxi+ 1) system's coupled mass matrix control moments in the pitch, roll and yaw directions, respectively for the ith link rigid mass matrix and force vector, respectively
N Qu Rdmi S(f,M/dc) Ti Ttai, Ttbi di dc dmi dxi, dyi, dzi f gi li mi q ri rcmi tsj u ui, vi, wi ud xi, yi, zi xf, xÃf yf