AIM-90-3317-CB AN INNOVATIVE APPROACH TO THE MOMENTUM MANAGEMENT CONTROL FOR SPACE STATION FREEDOM Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 Jalal Mapar t G- u' man Space Station Engineering and Integration Contractor 1760 Business Drive Center Reston, Virginia 22090 Abstract In all the previous work, the space station Equations Of Motion (EOM) have been linearized about the Local-Vertical Local-horizontal (LVLH). It is assumed that the products of inertia remain small, i.e., principal axes nearly aligned with the LVLH, allowing the pitch axis to be decoupled while the roll/yaw axes remain coupled. These assumptions are used by Wie et al. [31 in the development of a continuous Momentum Management System (MMS) that uses the Linear Quadratic Regulator (LQR) technique for generating the feedback gains. The continuous MMS as proposed by Wie et aJ. [41 is shown in simple block diagram form in Figure 1. The weighting matrices associated with the LQR are obtained by trial and error. This process is often time consuming and does not always yield desirable closed-loop poles. More recently, Wie et nJ. I41 have presented the EOM for the case of large pitch TEA, which may be encountered during the assembly flights of the space station, but, in order to simplify the equations, they lhave neglected the cross products of inertia. Since the large TEASare chiefly due to large cross products of inertia, the applicability of the equations to early stages of the station is debatable. However, when applied to the assembly complete vehicle, the controller, which also includes disturbance rejection filters to minimize the steady state effects of the aerodynamic torques, is shown to stabilize the system by achieving the TEA. Sunkel and Shieh 151 have applied the regional pole placement and optimal control techniques to the linearized model of the space station and have solved directly for both the feedback gains and the weighting matrices. They use the matrix sign function algorithm for the solution of the Riccati equations and show drastic time savings associated with the manual assignment of the weighting matrices as used by Wie ef al. [3,41. Again, the momentum management algorithm is applied to the assembly complete vehicle and stability is obtained as before. A new approach to the Control Moment Gyro (CMG) momentum management and attitude control of the Space Station Freedom is presented. First, the nonlinear equations of motion are developed in terms of body attitude and attitude rate with respect to the Local Horizontal Local Vertical (LVLH); then, they are linearized about any arbitrary stable point via the use of perturbations techniques. It is shown that for some assembly flights, linearization of equations of motion about the LVLH may not be valid and that a better choice would be to to linearize about a Torque Equilibrium Attitude (TEA). Next, a three-axis-coupled control law is used and the controller gains are determined via a combination of the optimal control and regional pole placement techniques. Finally, It is shown that the proposed linearization process, together with the coupled control laws, can stabilize a previously uncontrollable space station assembly flight. Introducb'on The Space Station Freedom will require several space shuttle flights to complete. When activated after a few assembly flights, it will use Control Moment Gyros (CMGs) as the primary attitude control devices during the normal coasting flight operations. Since the CMGs are momentum exchange devices, external torques must be applied to prevent momentum saturation. Several momentum management methods, both discrete and continuous, have been developed [l-61. These methods are based on the use of the environmental torques to drive the vehicle to a Torque Equilibrium Attitude (TEA) thus minimizing the CMG momentum usage. Manager, flight mechanics branch, member AIAA. Copyright @ 1990 American Institute of Aeronautics and Astronautics, Inc. All rights reserved, 23 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 While the simplifying assumptions used in the previous papers [l-61 are valid for the assembly complete type space station configurations, they may not be applicable for the build-up stages. Holmes [71 has applied the momentum management algorithm developed by Wie et al. [3,4] to one of the station assembly flights and has shown that the large cross products of inertia produce a large TEA; thus, invalidating the assumptions used in the previous papers [1-61. This paper presents a new approach to modeling the rigid body dynamics by developing the EOM and linearizing them about any arbitrary point along the attitude trajectory. A suitable stable point is proposed for linearization of the EOM. It is shown that a three-axis-coupled control law, whose gains are determined by regional pole placement [5], can successfully stabilize an assembly flight which was not controllable previously. The proposed momentum management method is given in its generic form so that i t can be easily applied to the assembly complete type configurations as a special case. Finally, a comparison is made between the proposed method and the previous algorithms by presenting simulation results for a space station assembly vehicle. and the ( * ) represents the time derivative. The non-control torque, T,,, is the sum of the gravity gradient and aerodynamic torques which will be given later, and u is the control torque which is related to the CMG momentum, H , by The body rate can be expressed as where WB/L, is the body rate vector with respect to LVLH and w~ is the LVLH rate vector in the bodyaxes. For a pitch-yaw-roll (2-3-1) Euler rotation sequence, WB/L and q are given by [81 Mathematical Models In this section the rotational EOM are presented in matrix form and linearized about any arbitrary point. Next, a suitable stable point is proposed for linearization and the EOM are simplified. For simplicity, the station is assumed to be a single rigid body in a circular orbit. The nonlinear rotational EOM in terms of components along the fixed bodyaxes are given by the well-known Euler's equations. I c 3 + z I o = T,,+ u (1) where C=cos and S=sin; #/ 0, and ware the body attitude with respect to the LVLH and n is the orbital rate. Eqs. ( 3 a ) a n d (3b) can be used to express the total body rate as U=F&+G where F= 1 0 0 sv 0 C4Cw S# -S#Cy C# Here, the (7 represents the skew symmetric matrix and and 0 represents the roll, pitch, and yaw attitude angles with respect to LVLH. where ( wn / 4 , o, ) are the body-axis components of the angular rate with respect to the inertial frame; (I, I f, , I, 1 are the moments inertia, and (Ixy ,I,, I I y z ) are the negative products of inertia; hl The aerodynamic torque is modeled as bias plus cyclic terms [3,4,51 in the body axes as Note that the total body rate is now expressed in terms of F and G which are functions of the attitude and attitude rate with respect to LVLH and orbit rate. Eq. (4) can be differentiated to give the body angular acceleration as Cj = F& + F i where Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 0 + rn A ksin (k nt + tpk) (9) k= 1 +c W W 0 T,,,,=Bias where, usually, m=2. The cyclic components at once and twice the orbital rate are due to the diurnal bulge effect and the rotating solar panels, respectively. Substituting Eqs. (7)and (9) into Eq.. ( 6 ) and rewriting the EOM in a more compact functional form yields 1 0 &= f(@,i,u) Because the expansion of EOM will result in rather lengthy expressions, we will proceed with the matrix notation until the final result is obtained. Equation (10) represents the most general form of the nonlinear EOM in terms of the body attitude and rate with respect to LVLH. Linearization of the EOM is performed by applying small perturbations to the nominal solution of the Eq. (10). Let %(r) be such a nominal solution. Then, for small perturbations from a point on the nominal trajectory, the solution of Eq. (lo), to first order, is given by G=n Substituting Eqs. (4) and ( 5 ) into Eq. (1) and simplifying yields - & = (fF)-’(TnC- (F& + G ) f (Fcb + G ) ) - F - * ( F &+G) + ( f F ) - ’ u Eq. ( 6 ) represents the nonlinear rotational dynamics of a body with respect to the rotating LVLH coordinate system. By integrating Eq. (6), one can obtain both the body attitude and attitude rate with respect to LVLH directly. Note that in Eq. ( 6 ) no assumption on the cross products of inertia has been introduced. The gravity gradient torque, Tgg, is given by where 6 q t ) represents any small perturbation from the nominal trajectory. Applying Eqs. (11) to Eq. (IO), and noting that &D(t) would give rise to a perturbation in u , will yield the desired linearized EOM. u T,, = 3n2Rf R (7) where R is the unit vector from the center of earth to the center of mass of the vehicle. For a (2-3-1) rotation sequence, R can be expressed as where the subscript N means “at the nominal point.” Equation (12) can be written in first order form as -c#ce + s#se s~ 24 that minimizes the performance index in (21) is given by u = -Kx = R ' 1B TP X (21) 3 3 where K is the feedback gain and P, an nxn nonnegative definite symmetric matrix, is the solution of the Riccati equation * K,66+KAy6Hy+Kn 6 H y + K i 2 a z + 3 3 K~~ h2 +K& HY 3 &z +K& &r + K $ ~ Y + (19c) 1 T Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 P B R - B P - P A - A ~ -P Q = o (22) The regional pole placement used to generate the feedback gains in Eqs. (19)is adapted from the algorithm by Sunkel and Shieh [51. They solve for Q, R and K so that the closed loop system (A-BK)has eigenvalues on or within a specified region, as shown in Figure 2, without explicitly using the eigenvalues of the open-loop system. The process involves the use of matrix sign function techniques for the solution of the modified Riccati equation of (22). It is shown in [SI that the use of matrix sign function greatly reduces the computational time required for the solution of the Riccati equations. The design procedure is given in detail in [SI and will not be repeated here. The MMS presented in this paper is shown in simple block diagram form in Figure 3. The algorithm depends only on an estimate of the inertia properties of the vehicle. The estimated TEA angles are extracted from the transformation matrix from principal to body axes. The matrix is obtained by simply calculating the eigenvector matrix associated with the inertia matrix. Next, these angles are used to compute A and B . These matrices are then used to generate the gains for the control laws, as given in Eqs. (191, via the LQR and regional pole placement [SI. Note that, in contrast to the method shown in Figure 1, the above procedure does not require any trial and error iterations on the weighting matrices and can be automated easily. where the gain superscripts (1, 2, 3) refer to the roll, pitch, and yaw axes. ln order to avoid CMG momentum buildup, we have included the integral of the CMG momentum vector H I i.e., Note that, in order to minimize the steady state oscillations of roll, pitch, yaw attitude and CMG momentum, we have used the cyclic disturbance rejection filters as proposed by Wie et al. 131. The filter equations are given below for momentum rejection in roll and attitude rejection in pitch and yaw at frequencies n and 2n. Also, in the control laws we have chosen the attitude rate with respect to LVLH in all three axes. To compute the gains associated with the control laws, we use the LQR with the regional pole placement technique. Let the quadratic performance index for the system in (13)be Resulb In this section we present simulation results for a space station assembly flight and compare the proposed MMS with that of Reference [31. The vehicle under consideration, assembly flight #5 (MB-51, is shown in Figure 4. This vehicle was also used by Holmes [7]to demonstrate the limitations of the linearization techniques for vehicles with large cross products of inertia. Although the solar dynamic collectors will not be used on the station the vehicle was used as a test model to check the MMS algorithm. The vehicle properties together with a where the weighting matrices Q and R are nxn nonnegative and rnxm positive definite symmetric matrices, respectively. The feedback control law 25 representative aerodynamic torque are given in Table 1. For MB-5, the principal-to-body angles were computed to be Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 #, = -0.4' e, = -17.1' y-, = 2.4' For gain calculation, the parameter h (see Figure 2) was set at 0.375n. The gains for the three-axiscoupled controller are given in Table 2. In order to perform a consistent comparison with the method in [3], we applied the same pole placement method with the same parameters to the A and B matrices of [3]. However, since the control laws of 131 destabilized the vehicle in less than an orbit, we used the three-axis-coupled control laws presented in this paper. The resulting controller gains are also given in Table 2. The results, obtained from nonlinear simulations, are presented in Figures 4-9. For all the cases, the vehicle was aligned with the LVLH, i.e., zero attitude and attitude rates, and the initial filter states were set equal to zero. The roll, pitch, and yaw attitude histories, Figures 5a thru 7a, show that the proposed MMS converges to a TEA of approximately -330 in pitch and the CMG momentum, Figures 5b thru 7b, remains bounded after about 3 orbits. Note that, even though the aerodynamic torque has shifted the estimated pitch TEA ( principal-to-body pitch angle ) by about 14O, the controller has stabilized the system. The stability can be attributed to the fact that the linearized EOM about an estimated TEA provide a better representation of the total system dynamics. In contrast, the MMS based on [31 is not stable and diverges from the beginning. The instability is mainly due to the large pitch TEA (produced by large cross products of inertia and the aerodynamic torque) being well outside the linear-angle range. In fact, the instability suggests that the A and B matrices of [31, obtained from the EOM linearized about the LVLH, do not adequately describe the dynamics of the nonIinear system. As a result, omission of the products of inertia, and the use of small angle approximation, do not appear to be valid assumptions for MB-5 and other stages that follow the same trend on the inertia properties. The MMS algorithm presented in this paper has been developed such that the only inputs to it are the characteristics of the vehicle to be controlled, i.e, the inertia matrix, and the parameter h for the gain calculation. Since linearization about any arbitrary point is included in the EOM, the MMS algorithm can compute the A and B matrices and the controller gains automatically. The total process is computationally fast and can be automated so that the MMS can adapt to any configuration by simply acquiring the minimum amount of information about the system to be controlled. A new MMS based on linearizing the EOM about an estimated TEA has been presented. A set of threeaxis-coupled attitude control laws was also given. It was shown that, for space station vehicles that have strong inertia coupling, the use of small angle approximation and the omission of the moss products of inertia in the linearized EOM may not provide a good representation of the nonlinear model. Although a set of gains can be computed based on the linearized models of the space station, their use in nonlinear simulations has been shown to destabilize the vehicle. The proposed MMS is computationally fast and converges quickly. Since the algorithm depends only on the inertia properties and the desired region for the closed loop system poles, it can adapt to the system if the aforementioned inputs are provided. In fact, depending on the frequency of the gain calculation, the MMS can be used for control of the space station during the payload moving maneuvers. For such maneuvers, the inertias can change drastically. Instead of scheduling the controller gains at discrete intervals, the proposed MMS can be executed at the proper frequency so that the controller gains could be thought of as time varying adaptive gains. For specific changes in the inertias, a new estimated TEA and set of gains can be computed. It is anticipated that these time varying gains would stabilize the system because they will be based on the new linearized EOM presented in this paper. In this section the A and B matrices representing the linearized EOM about any arbitrary point are presented. Applying small perturbations to Eq. (1) will yield - Z 6 ; +( WZ - I @ ) 60 = 6Tnc+ 6~ (A1) Equations (4) and (5) are used to compute expressions for 6w and 6h. i W =aF [email protected], * aF aF 6F=-&+-c%b (A4) Note that the coefficients of [email protected] Sh, are evaluated at the desired reference trajectory. These coefficients form the components of the A and B matrices which are then complemented with CMG momentum, Eq. (141, and filter equations, Eqs. (1618), in order to compute the system matrices for gain computation. sG=-6Q,, aG s b = " 6 @ + - S iab (AS) Reference 6h =( 6F)6 + F6& + (66) + F6& + SC ( A 3 ) where [email protected] Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 [email protected] [email protected] [email protected] ai ai 1. Hattis, P. D., " Predictive Momentum of Guidance, Control, and Dynamics, Vol. 9, No. 4, JulyAug 1986, pp. 454461. It is easily shown that acab -- [email protected] a i The term 2. Bishop, L. R., et al., " Proposed CMG Momentum Management Scheme for Space Station, " Paper No. 87-2528, AIAA Guidance, Navigation, and Control Conference, August 1987. ST, is also provided below. N - GTnc= 3n2(RI - I R ) 6R 3. Wie, B., Byun, K. W., Warren, W., Geller, D., Long, D., and Sunkel, J., " A New Momentum Management Controller for the Space Station, " Journal of Guidance, Control, and Dynamics, Vol. 12, NO. 5, Sep-Oct 1989, pp. 714-722. where Equations (A2) a n d (A31 can be further simplified by noting that 6 and 6 vanish on thereference trajectory implying that F=O,G=O, 4. Warren, W., Wie, B., Geller, D., " PeriodicDisturbance Accommodating Control of the Space Station for Asymptotic Momentum Management, " Paper No. 89-3476, AIAA Guidance, Navigation, and Control Conference, August 1989. ad = O [email protected] and 5 . Sunkel, J. and Shieh, L., " An Optimal Momentum Management Controller for the Space Station, " Paper No. 89-3474, AIAA Guidance, Navigation, and Control Conference, August 1989. Equation (A101 is then solved to obtain the linearized EOM about any arbitrary point. 6. Woo, H. H., Morgan, H. D., and Falangas, E. T., "Momentum Management and Attitude Control Design for a Space Station, " Journal of Guidance, Control, and Dynamics, Vol. 11, No. 1, JanuaryFebruary 1988, pp. 19-25. Finally, Eqs. (A9) and (A10) are used in Eq. ( A l l ) to any On the [email protected] rates* - -z)@ - ( G I a# 7. ~ ~ E., l The Limitations ~ ~ of ~ the Linearization Techniques on Nonlinear Space Station Equations of Motion, " Grumman Space Station Engineering Integration Contractor, Report NO. PSH-341-RP89-OO2, August 1989. + (IF)-'6u 8. Junkins, J. L. ,Turner, J. D., Optimal Spacecraft Rotational Maneuvers, " Studies in Astronautics 3, Elsevier Scientific Publishing Company, New York, NY, 1985. '1 3n2(RI I' 26 , Table 1 Vehicle parameters Aerodynamic torque (ft-lbs) Inertia ( slugs-ft2 I, fy Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 I, 3.9676E7 2.5588E6 4.0544E7 I,, I,, I,, -1.5544E6 2.6162E5 -2.6641E5 Roll Pitch Yaw 0.2 + 0.1 sin(nf)+ 0.01 sin(2nt) 1.2 + 0.4 sin(nt) + 0.1 sinQnt) -0.02 + 0.02 sin(nt) + 0.05 sin(2nt) Table2 Controller gains for the propsed MMS and MMS of [31 Gain Gains for the promed MMS Roll Pitch Yaw Gains for the MMS of I31 Roll Pitch Yaw -4.2531E+3 -8.8315E+1 -5.9569E+3 -5.8531E+3 -9.4582E+2 -6.3198E+3 -6.0154E+6 2.1047E+5 -1.9527E+6 -6.3867E+6 8.8480E+l -2.5776E+6 -1.3474E-1 2.4400E-3 -4.8447E-2 -1.4449E-1 -2.7309E-3 -6.315OE-2 1.7773E-5 -1.7318E-6 -1.9954E-5 1.7274E-5 -4.7783E-6 -1.6951E-5 -5.1888E-8 3.6896E-9 3.5004E-8 -4.8944E-8 8.9199E-9 2.5976E-8 -4.2272E-5 3.8643E-7 -1.2175E-5 -5.2958E-5 -3.6524E-6 -1.9888E-5 1.8615E-9 9.3212E-10 8.4531E-9 2.4228E-9 -5.0495E-9 -9.3830E-9 -5.3360E-5 1.6374E-6 -4.4354E-5 -4.3641E-5 -4.6196E-6 -5.3872E-5 -3.4 605E+ 1 -2.6634E+2 -5.8056E+O 1.5639E+2 -2.2254E+2 2.4804E+2 -2.2558E+4 -1.7873E+5 -2.3200E+2 1.8017E+5 -1.7853E+5 1.4912E+5 -3.1247E-3 -5.6951E-2 2.4451E-3 -2.1484E-2 -5.8201E-2 1.5664E-2 -5.3356E-7 -1.1399E-5 5.1890E-7 -3.2993E-6 -1.1469E-5 2.9243E-6 1.0415E-5 -3.7762E-5 9.3871E-6 4.2359E-5 -2.3379E-5 2.9997E-5 1.9171E-3 -2.3821E-1 7.5065E-3 8.8315E-2 -2.2327E-1 3.0972E-2 2.5716E-5 -5.6787E-4 2.6766E-5 9.2073E-5 -5.1726E-4 2.1017E-5 -2.9007E-3 -2.6711E-1 -4.5491E-3 3.5036E-2 -2.6814E-1 2.8538E-2 7.2451E+3 -3.2708E+2 -1.2731E+3 7.2557E+3 -2.9491E+2 -8.6966E+2 4.1 226E+6 -3.4183E+5 -2.2137E+6 2.4071E+6 -5.3573E+5 -1.5240E+6 1.0292E-1 -8.9361E-3 -3.9890E-2 6.1373E-2 -1.3294E-2 -2.2266E-2 1.8243E-5 -2.1006E-8 6.1781E-6 2.2800E-5 -1.1865E-7 6.3630E-6 4.0537E-3 -1.7697E-4 9.8062E-4 2.6247E-3 -3.0929E-4 1.3267E-3 2.1789E+O -6.1987E-3 -2.5900E+0 4.1 198E+O -4.0102E-1 -2.1667E+O 6.6404E-3 -2.5999E-4 -5.2998E-3 5.3139E-3 - 1.4189E-3 -5.7521E-3 4.5404E-1 1.1749E-2 -4.3803E+O I .8650E+0 -1.0042E-1 -4.0506E+0 Compute theA and B matrice from the EOM linearized about the LVLH. 7 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 x = Ax :g -a !i rq i + Bu Find u to minimize I= TQx + uTR u) dr Controller gains - 7space Station Dynamics Solution: Fig. 1 Simple block diagram of the MMS. After Wie et al. [4]. P Im Re x openlooppole before design 0 Fig. 2 Pole assignment sector. 27 Closedl~poles after design Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 I A andB I 1 Define h for the Regional pole placement algorithm [51 Controllergains Fig. 3 Simple block diagram of the proposed MMS. Fig. 4 A representative assembly flight #5. lo 5 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317 -20 -I5 -25 c 1' ! - ............. L ..._........_.______........ ;............... 2 0 4oo00 _____________ I 4 6 T i (orb&) 8 I 10 2 0 Fig. 5a Roll attitude comparison. 4 6 Time (orbits) 8 10 Fig. 5b Roll momentum comparison. 20 15ooo 0 a 3 1 P g -20 loo00 i4 B -60 ii; -80 -100 -120 0 2 4 6 T i (orbits) 8 10 Fig. 6a Pitch attitude comparison. 0 2 4 6 Time (orbin) 8 2 0 4 6 T i w (orbits) 8 10 Fig. 6b Pitch momentum comparison. 10 0 Fig. 7a Yaw attitude comparison. 2 4 6 T i( d i t s ) 8 10 Fig. 7b Yaw momentum comparison. New method __________________ 28 Oldmethod

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