Advances in Spacecraft Technologies Part 13 ppt

Advances in Spacecraft Technologies 470 Sating, R.R., "Development of an analog MIMO Quantitative Feedback Theory (QFT) CAD Package," Air Force Institute of Technology, Wright Patterson AFB, OH, AFIT/GE/ENG/92J-04, Ms Thesis, 1992. Shaked, U., Horowitz, I. and Golde, S., "Synthesis of multivariable, basically non-interacting systems with significant plant uncertainty.," Automatica, 12(1): pp. 61-71, 1976 Sidi M., “Design of Robust Control Systems: From classical to modern practical approaches”. Krieger Publishing, 2002. Skogestad, S. and Havre, K., "The use of RGA and condition number as robustness measures," Computers & Chemical Engineering, 20: pp. S1005-S1010, 1996. Skogestad, S. and Morari, M., "Implications of large RGA elements on control performance," Industrial & Engineering Chemistry Research, 26(11): pp. 2323-2330, 1987a. 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Walke, J.G., Horowitz, I.M. and Houpis, C.H., "Quantitative synthesis of highly uncertain, multiple input-output, flight control system for the forward swept wing X-29 aircraft," IEEE 1984 National Aerospace and Electronics Conference. NAECON 1984 (IEEE Cat. No. 84CH2029-7), 21-25 May 1984, Dayton, OH, USA, 1984. Wall, J.E., Doyle, J.C. and Harvey, C.A., "Tradeoffs in the design of multivariable feedback systems," 18th Allerton Conference, 1980. Witcher, M.F. and McAvoy, T.J., "Interacting control systems: steady state and dynamic measurement of interaction," ISA Transactions, 16(3): pp. 35-41, 1977. Wolovich, W.A., “Linear multivariable systems”, vol. 11. New York [etc.]: Springer-Verlag, 1974. Yu, C C. and Fan, M.K.H., "Decentralized integral controllability and D-stability," Chemical Engineering Science , 45(11): pp. 3299-3309, 1990. Yaniv, O., "MIMO QFT using non-diagonal controllers," International Journal of Control, 61(1): pp. 245-253, 1995. Yaniv O., “Quantitative Feedback Design of Linear and Non-linear Control Systems”. Kluver Academic Publishing , 1999. Yaniv, O. and Horowitz, I.M., "A quantitative design method for MIMO linear feedback systems having uncertain plants," International Journal of Control, 43(2): pp. 401-421, 1986. 21 Fuzzy Attitude Control of Flexible Multi-Body Spacecraft Siliang Yang and Jianli Qin Beijing University of Aeronautics and Astronautics, China 1. Introduction In order to complete the flexible multi-body spacecraft attitude control, this chapter will research on the dynamics and attitude control problems of flexible multi-body spacecraft which will be used in the future space missions. Through investigating plentiful literatures, it is known that some important progress has been obtained in the research of flexible multi-body spacecraft dynamic modeling and fuzzy attitude control technologies. In the aspect of dynamic modeling, most models were founded according to spacecrafts with some special structures. In order to satisfy the requirement of modern project design and optimization, acquire higher efficiency and lower cost, researching on the dynamic modeling problem of flexible multi-body spacecraft with general structures and founding universal and programmable dynamic models are needed. In the aspect of attitude control system design, the issues encountered in flexible spacecraft have increased the difficulties in attitude control system design, including the high stability and accuracy requirements of orientation, attitude control and vibration suppression, high robustness against the different kinds of uncertain disturbances. At the present time, classical control theory and modern control theory are often used in flexible multi-body spacecraft attitude control. These two methods have one common characteristic which is basing on mathematics models, including control object model and external disturbance model. It is usually considered that the models are already known or could be obtained by identification. But those two methods which are based on accurate math models both have unavoidable defects for large flexible multi-body spacecraft. Until this time, the most advanced and effective control system is the human itself. Therefore, researching on the control theory of human being and simulating the control process is an important domain of intelligent control. If we consider the brain and the nerve center system as a black box, we only investigate the relationship between the inputs and the outputs and the behavior represented from this process, that is called fuzzy control. The fuzzy control doesn’t depend on the accurate math models of the original system. It controls the complicated, nonlinear, uncertainty original system through the qualitative cognition of the system dynamic characteristics, intuitional consequence, online determination or changing the control strategies. This control method could more easily be realized and ensured its real time characteristic. It is especially becoming to the control problem of math models unknown, complicated, uncertainty nonlinear system. Accordingly, large flexible multi-body spacecraft attitude control using fuzzy control theory is a problem which is worth researching. Advances in Spacecraft Technologies 472 2. Attitude dynamic modeling of flexible multi-body spacecraft Mathematics model is the basement of most control system design. Dynamic modelling is to describe the real system in physics world using models in mathematics world. Mathematics model provide the mapping from input to response, the coincidence extent between the response and the real object being controlled represent the quality of the model. Mathematics model world is totally different from physics system world, so a real physics object being controlled can not be constructed exactly by mathematics models. Therefore, engineers intend to establish a model which can reflect dynamic characteristics of spacecraft system, as well as the controller design based on the model can be applied into the real system. In this section, the attitude dynamic equations of flexible multi-body spacecraft with topological tree configuration have been derived based on the Lagrange equations in terms of quasi-coordinates. The dynamic equations are universal and programmable due to the information of system configuration being introduced into the modelling process. 2.1 Description of system configuration 2.1.1 Coordinate system definition The movement of spacecraft is always described in a reference coordinate, several coordinate systems used in the attitude dynamic modeling process are as follows: 1. Inertial coordinate system () ii ii oxyz i f This inertial frame is defined as its origin at the mass center of the earth, the third axis i z perpendicular to the Earth’s equatorial plane pointing to the arctic, axis i x and i y lying in the Earth’s equatorial plane, axis i x pointing to the direction of the vernal equinox, axis i y forms this coordinate system as a right-handed one. 2. Orbit coordinate system () oooo oxyz o f Its origin at the mass center of the spacecraft, axis o z pointing to the geocenter, axis o x lying in the orbit plane perpendicular to axis o z , pointing along the direction of spacecraft velocity, axis o y forms this coordinate system as a right-handed one. 3. Central body coordinate system () bb bb oxyz b f Its origin at the mass center of the spacecraft which has not been deformed, axis b x pointing along the direction of spacecraft velocity, axis b z pointing to the geocenter，axis b y forms this coordinate system as a right-handed one. 4. Floating coordinate system () ai ai ai ai oxyz ai f , 2, 3, , = in Floating coordinate system is the body frame of the flexible body i, its origin usually at the mass center of the flexible body i which has not been deformed. 5. Gemel coordinate system () ck ck ck ck oxyz ck f Gemel coordinate system is the body frame of the gemel k between the flexible bodies, its origin usually at the connection point between the gemel k and its inboard connected flexible body. 2.1.2 Description of spacecraft system Considering a flexible multi-body spacecraft with topological tree configuration which contains a central body and 1n − flexible appendages, there are several accesses, objects are connected by gemels, ignoring collision and friction at gemels, access l of the system can be shown in figure 1: Fuzzy Attitude Control of Flexible Multi-Body Spacecraft 473 i z i y i x i o o x o y o z o o o R  b R  j R  i R  oi R  p r  aj o ai o i p  io u  i u  if u  i B j B () j Li = i b  i h  ci o cj o i a  i ρ  i s  j b  j h  j p  j a  j s  () L j h  () L j b  () L j p  () L j B 1 B ()aL j o () L j a  () L j ρ  () L j s  b ρ  ()cL j o 1 o R  ob R  b a  b o j ρ  Fig. 1. Access l of flexible multi-body spacecraft system In this figure, ii ii oxyz is the inertial frame, oo oo oxyzis the orbit frame. 1 B is the central body, origin of bb bb oxyz is at the mass center of the spacecraft which has not been deformed, as well as axises are parallel to the principle axises of inertia. Radius vecter b a  and b ρ  respectively are rigid and flexible displacement of mass center b o , ob R  is the radius vector of mass center b o in orbit coordinate system, ob b b Ra ρ = +    . Gemel frames are founded at each gemel k h , ck o is the origin of gemel coordinate system of k h , its radius vector in flexible body () L k B is k s  , kkk sa ρ = +    , k a  and k ρ  respectively are rigid and flexible displacement of k h in flexible body () L k B . ak o is the origin of floating coordinate system of object k B , its radius vector in gemel coordinate system of k h is k h  , kk k hbp = +    , k b  and k p  respectively are rigid and flexible displacement of origin ak o in gemel coordinate system of k h , 2, ,kn=  . If the central body of the spacecraft system is rigid, the origin of orbit coordinate system is coinsident with the origin of central body coordinate system, according to the analysis result of mass matrix (Lu, 1996), choosing frames like this can eliminate the coupling term of rigid body translation and rotation., as well as 0 ob b b Ra ρ = ==    . 2.2 Description of flexible multi-body system using graph theory Graph theory (Wittenburg & Roberson, 1977) is a useful tool to describe topological configuration, here several relative concept were given before we using it to do more research. Oriented graph description: multi-body system can be described using gemel and its adjacent objects, if we express the objects in system using the vertex, express gemels using arc, then topological configuration of multi-body can be expressed as a oriented graph ,DVA= . There is a bijection between the collection of vertex V and the collection of objects and also a bijection between the collection of arc A and the collection of gemels. A Advances in Spacecraft Technologies 474 description of multi-body system with topological tree configuration using oriented graph is shown in figure 2. i H represent Gemels, and i B represent objects. 1 B 6 B 4 B 3 B 2 B 5 B 7 B 2 H 5 H 7 H 6 H 4 H 3 H 1 H 0 B Fig. 2. Description of multi-body system using oriented graph Regular labelling: Regular labelling approach is specified as follows: 1. The adjacency object of the root object 0 B is defined as 1 B , relative gemel defined as 1 H ; 2. Each object has the same serial number with its inboard connected gemel; 3. Each object has a bigger serial number than its inboard connected object; 4. Each gemel has a deviated direction from the root object 0 B . Multi-body system shown in figure 2 is numbered in accordance with rules Inboard connected object array: according to the regular labelling approach, label the N objects of spacecraft system. Define a N order one dimension integer array () Li , 1, ,iN=  , i is the subscript of object i B , () Li is the subscript of the inboard connected object of i B . System topological configuration can be described by array () Li which is called inboard connected object array of system. A graph can be conveniently expressed by matrix, its advantage is that structural features and character can be studied using of kinds of operation in matrix algebra. Access matrix: Supposed that ,DVA= is an oriented graph, { } 12 ,,, n Vuu u=  , name matrix () ij Tt= is the access matrix of the ortiented graph D , if: when there is a connectivity between and otherwise 1, 0, ji ij uu t = ⎧ ⎨ ⎩ (1) Access matrix of the system with topological tree configuration shown in figure 2 can be written as: 1000000 1100000 1110000 1101 000 10001 00 1000110 1000001 T = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (2) Fuzzy Attitude Control of Flexible Multi-Body Spacecraft 475 2.3 Recursion relationship of adjacency bodies kinematics Considering the kinematics relationship between flexible body i and flexible body j , ()jLi= . According to figure 2, we know that : ijii RRsh = ++     (3) where ρ =+   iii sa , = +    ii i hbp. The time derivative of equation (3) in inertial coordinate system is obtained as follows : i ijajiiciii DR RR s hp Dt ωρω = =+×++×+           , 2, 3, ,iN =  (4) where () iD D t expresses the time derivative of vector “ i “in inertial coordinate system, the“ i “and“ ii “ above vectors respectively express the 1-order and 2-order time derivative in their own body coordinate systems. aj ω  is the angular velocity vector of the floating coordinate system of flexible body j B , ci ω  is the angular velocity vector of the gemel coordinate system of gemel i h . Suppose that the deformation of flexible bodies is always in the range of elastic deformation, translation and rotation modal matrix of gemel i h respectively are ci φ and ci ψ , corresponding modal coordinate is ci q , translation modal matrix of flexible body i B is ai φ , cooresponding modal coordinate is i q , we have: icici q ρ =   φ , iaii pq=   φ , ci aj ci ci q ωω =+    ψ (5) Write equation 4 in form of matrix for convenience like: TT TT T×× =+−+     iR as a ch cp ωρ ω ijjiajjiiiciii R- (6) where T [] xyz iii=  i is the unit base vector of i f ; [] T j xj yj zj aaa=   a , 2, 3, ,jN =  , is the unit base vector of aj f ; [] T ixiyizi ccc=   c , is the unit base vector of ci f , aj ω , i s and i ρ respectively are component column arrays of corresponding vector in aj f , ci ω , i h and i p respectively are component column arrays of corresponding vector in ci f . Vector equation (5) can be written in form of matrix as follows: T i i ci ci ρ =  cq φ , T iiaii p =  aq φ , () T ci j aj ajci ci ci ω =+  aAq ωψ (7) From Eq. (6) and Eq. (7), we obtain TTT T T T () i j j i i i ciaj aj i ci ci i i ci ci i ai i R ×× × =− + + − +     iR as chA c q ch q a q ωφ ψ φ (8) Advances in Spacecraft Technologies 476 where ci q is the component column array of corresponding vector in ci f ; i q is the component column array of corresponding vector in ai f ; ciaj A is the coordinate conversion matrix from aj f to ci f , besides T ciaj ajci =AA. For central body 1 B , we have boobobb RRR Ra ρ = +=++      (9) where o ω  is the angular velocity of orbit coordinate system; bbb q ρ =     φ , b φ is the translation modal matrix of the central body, b q is the corresponding modal coordinate. Upon that matrix form of absolute velocity vector of the spacecraft can be written as TT T TT T booobobbooobobb R ×× =− + =− +     iR iR b q iv iR b q ω φωφ (10) where T [] o oxoyoz iii=  i is the unit base vector of orbit coordinate system o f ; T [] xyz bbb=  b is the unit base vector of cantral body coordinate system b f . From Eq.(8) and Eq. (10), we get TT T T T T T T TT T T T () () i o o ob o b b k k k ckb b k ck ck k k ck ck k ak k j i i i ciaj aj i ci ci i i ci ci i ai i iD R ××× × ×× × ∈ =− + − + + − + +− + + − + ⎡⎤ ⎣⎦ ∑     iv iR b q bs chA c q ch q a q as chA c q ch q a q ωφ ωφ ψ φ ωφ ψ φ (11) where k is the serial number of the outboard connected object of the central body 1 B in the access from object 1 B to i B ; D is the serial number collection of the objects in the access from object 1 B to i B except for 1 B and its outboard connected object; ( ) jLi= ; b ω is the component column array of angular velocity of the central body 1 B in b f . If we define that TT 1 = ba, 1cib cia =AA, 1ba = ω ω , then the Eq.(11) can be written in more concise form like: TT T TT T T T () iooobobb j i i i ciaj aj i ci ci i i ci ci i ai i iE R × ×× × ∈ =− + +− + + − + ⎡ ⎤ ⎣ ⎦ ∑     iv iR b q as chA c q ch q a q ωφ ωφ ψ φ (12) where is the serial number collection of objects in the access from object 1 B to i B except for the central body 1 B . The projection of the vector i R   in the central body coordinate system can be written as: () () ib bio bo ob o bb baj i bci i ciaj aj bci ci ci bci i ci ci bai ai i iE R × ×× × ∈ =− + +− + + − + ⎡ ⎤ ⎣ ⎦ ∑    Av A R q As AhA A q Ah q A q ωφ ωφ ψ φ (13) where bi A is the coordinate conversion matrix from i f to b f , besides, T bi ib =AA; bo A is the coordinate conversion matrix from o f to b f , besides, T bo ob =AA; baj A is the coordinate conversion matrix from aj f to b f , besides, T baj ajb = A A ; bci A is the coordinate conversion Fuzzy Attitude Control of Flexible Multi-Body Spacecraft 477 matrix from ci f to b f , besides, T bci cib =AA; bai A is the coordinate conversion matrix from ai f to b f , besides, T bai aib =AA. We make several definitions as follows: 1 2 2 [( ) ] [( ) ] bi o bo ob o b b baj i bci i ciaj aj bci ci ci bci i ci ci bai ai i i n baj i bci i ciaj aj bci ci ci bci i ci ci bai ai i i n × ×× × = ×× × = −+ =− + + − + =− + + − +     E=Av AR q EAsAhA AqAhqAq EAsAhA AqAhqAq ωφ ωφ ψ φ ωφ ψ φ (14) T 12 [] n = EEE E (15) Upon that the origin velocity of arbitrary flexible body coordinate system can be expressed using access matrix as follows: () ()=  ib i RTE (16) where () i T represents the i row of access matrix T . 2.4 Dynamics modeling based on quasi-Lagrange equations 2.4.1 Quasi-Lagrange equations Quasi-Lagrange equations is a kind of improvement of classical Lagrange equations, on one hand they have the advantage of normalized derivation, on the other hand they can reserve the presentation form of dynamics equations of rigid body. Therefore they are applicable for researching the dynamics problem of large spacecraft. Using quasi-Lagrange equations system dynamics can be expressed as follows: () bbt bb bb br bbb T aib i aib b aib ai air ai i ai vb bbb vi iii dL L dt dL L L dt dL L L dt dL dΦ dL dt d d dL dΦ dL dt d d × ×× ××× ∂∂ += ∂∂ ∂∂∂ ++ = ∂∂∂ ∂∂ ∂ ++ += ∂∂ ∂ ∂ +−= ∂ ∂ +−= ∂ ⎧⎛ ⎞ ⎜⎟ ⎪ ⎝⎠ ⎪ ⎪ ⎛⎞ ⎪ ⎜⎟ ⎝⎠ ⎪ ⎪ ⎛⎞ ⎪ ⎨ ⎜⎟ ⎝⎠ ⎪ ⎪ ⎛⎞ ⎪ ⎜⎟ ⎪ ⎝⎠ ⎪ ⎛⎞ ⎪ ⎜⎟ ⎩⎝ ⎠ ω   ω Q vv v ω Q ω v ω Av Aω A ω Q ω v Q qqq Q qqq ⎪ (17) where L is the Lagrange function of system, LTU = − , T is the kinetic energy of system, is the potential energy of system, is the dissipated energy of system; b v , b ω respectively are Advances in Spacecraft Technologies 478 the spacecraft central body coordinate system velocity and angular velocity coordinates in bb bb oxyz relative to the inertial coordinate system; ai v is the velocity coordinate in bb bb oxyz of the floating coordinate system of flexible body i relative to the inertial coordinate system; ai ω is the angular velocity coordinate in bb bb oxyz of the floating coordinate system of flexible body i relative to the inertial coordinate system; b q is the modal coordinate of the central body 1 B ; i q is the modal coordinate of the flexible body i B ; bt Q is the generalized force corresponding to the translation of spacecraft central body; br Q is the generalized moment corresponding to the rotation of spacecraft central body; air Q is the generalized moment corresponding to the rotation of flexible body i B ; vb Q is the generalized force corresponding to the modal coordinate b q ; vi Q is the generalized force corresponding to the modal coordinate i q ; aib A is the conversion matrix from bb bb oxyz to ai ai ai ai oxyz ; for an arbitrary 31 × column array [] 123 T x xx=x , × x represents the skew symmetric matrix as follows: 32 31 21 0 0 0 × − =− − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ x x x x xx x (18) 2.4.2 Lagrange function Using Lagrange equations to found the system dynamics model, firstly, we should calculate the kinetic energy and potential energy of each body in the system, then add them together in order to get the total kinetic and potential energy of the system, finally obtain the Lagrange function. 2.4.2.1 Kinetic energy of system The kinetic energy of flexible body i expressed by generalized velocity is as follows: 1 2 T iiii T = M υ υ (19) The generalized velocity i υ in the above formula is defined as: () T TT i i ai i =   Rq υω (20) where i  R is the floating coordinate system origin velocity of the flexible body i ; ai ω is the floating coordinate system origin angular velocity of the flexible body i ; i q is the deformation modal coordinate of the flexible body i . i M is the mass matrix of the flexible body i , which is defined as: ii i RR R Rf iii iR f ii i Rf f ff ω ωωωω ω = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ mmm Mmmm mm m (21) [...]... discretizing constrained nonlinear programming problem and search for the numerical solution by employing SNOPT as a solver (Gil et al., 2002) 1.5 Optimal trajectory planning Point-to-point optimal trajectory planning is usually classified into two main categories: minimum travelling time and actuation effort Optimization tasks are generally aimed to minimize the performance index that is defined in the... cos θ3 (m4 θ˙3 sin θ3 L2 + L2 m4 θ˙2 sin θ2 L3 3 m3 θ˙3 sin θ3 r2 + L2 m3 θ˙2 sin θ2 r3 ) 3 (m3 (2r3 θ˙3 cos θ1 cos θ3 (r3 θ˙1 cos θ2 sin θ1 + r3 θ˙3 cos θ1 sin θ3 2r3 θ˙3 cos θ3 sin θ1 ( L2 θ˙2 sin θ1 sin θ2 + r3 θ˙3 sin θ1 sin θ3 L2 θ˙1 cos θ1 cos θ2 − r3 θ˙1 cos θ1 cos θ2 )))/2 (m4 (2L3 θ˙3 cos θ1 cos θ3 ( L2 θ˙1 cos θ2 sin θ1 L2 θ˙2 cos θ1 sin θ2 + L3 θ˙1 cos θ2 sin θ1 L3 θ˙3 cos θ1 sin θ3 ) + 2L3... (2) ∑ Dk,j j =0 where Dk,j represents the differentiation matrix x (ζ j ) (23) 6 502 Advances in Spacecraft Technologies Advances in Spacecraft Technologies Since the Chebyshev polynomials are defined over the interval [-1,1], it is required to perform a linear transformation for the expression in physical time domain t t = [(t f − t0 )ζ + t f + t0 ]/2 (24) The derivatives with respect to physical time... path planning for the space manipular was illustrated by using virtual robotic test-bed (VRT) (Yoshida et al., 2008) The development of Real-time adaptive motion planning (RAMP) also provides optimal trajectory planning for robotic arm control in different dynamic environments and for various scenarios (Vannoy & Xiao, 2008) 2 498 Advances in Spacecraft Technologies Advances in Spacecraft Technologies. .. = Pob − Pi Fig 1 Coordinate system of the 3D robotic manipulator (8) 4 500 Advances in Spacecraft Technologies Advances in Spacecraft Technologies Pi+1,ob = Pob − Pi+1 (9) The projection from the center of mass of the obstacle normal to the link is given by li = Pi,ob · Pi,i+1 / Pi,i+1 (10) The distance between the center of mass of the obstacle and the link can then be determined by di = Pi,ob − li... 2 (m3 r3 θ˙1 sin(2θ2 ))/2 + (m3 r3 θ˙3 sin(2θ3))/2 + + 2 2 L2 L3 m4 θ˙1 sin(2θ2 ) + L2 m3 r3 θ˙1 sin(2θ2 ) L2 L3 m4 θ˙2 θ˙3 sin(θ2 − θ3 ) + L2 m3 r3 θ˙2 θ˙3 sin(θ2 − θ3 ) (5) Applications of Optimal Trajectory Planning Applications of Optimal Trajectory Planning and Invariant Manifold Based Control for Robotic Systems in Space and Invariant Manifold Based Control for Robotic Systems in Space η3 = +... ignore the high order nonlinear coupling item caused by them Through all above simplification, we obtain the finally spacecraft dynamics equations with uncertain moment of inertial as follows: ( I + ΔI ) ω + ω [( I + ΔI ) ω + Cη ] + Cη = u + w × η + Dη + Kη + C ω = 0 T (63) where I is the moment of inertial matrix of spacecraft; ΔI is the uncertain increment of moment of inertial caused by rotation... free trajectory planning for two-link robotic arm manipulators in the presence of morphing mobile obstacles by minimizing the actuation efforts and performing payload capture in two-dimensional formulation (Williams et al., 2009) It has been further extended to the optimal control system, which continuously evaluates the change of the position, velocity and shape of tumbling objects in the three-dimensional... order modal coordinate 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 0 1000 2000 3000 time（ t/s ） 4000 5000 6000 (b) Simulation results of the modal coordinates Fig 8 The control effect of the fixed universe controller when the moment of inertia of the spacecraft increases 20 percent 496 Advances in Spacecraft Technologies 4 Conclusion In this chapter, we research on the flexible multi-body spacecraft attitude... been done using different techniques and strategies Xiong Luo deployed evolutionary programming algorithms to search for the optimal solution (Luo et al., 2004) Minimum travelling time path planning was also developed by using polytope method with penalty function (Cao et al., 1997) Other work involved collision avoidance and minimum-energy path planning (MEPP) or minimum-fuel path planning (MFPP) was . uncertainty nonlinear system. Accordingly, large flexible multi-body spacecraft attitude control using fuzzy control theory is a problem which is worth researching. Advances in Spacecraft Technologies. moment of inertial matrix of spacecraft; I Δ is the uncertain increment of moment of inertial caused by rotation of the solar panel; C is the coupling coefficient Advances in Spacecraft Technologies. coordinate system velocity and angular velocity coordinates in bb bb oxyz relative to the inertial coordinate system; ai v is the velocity coordinate in bb bb oxyz of the floating coordinate