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NASA/TM--2001-211301 Equations of Motion for the g-LIMIT Microgravity Vibration Isolation System Y.K. Kim and M.S. Whorton Marshall Space Flight Center, Marshall Space Flight Center, Alabama October 2001

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NASA/TM--2001-211301 Equations of Motion for the g-LIMIT Microgravity Vibration Isolation System Y.K. Kim and M.S. Whorton Marshall Space Flight Center, Marshall Space Flight Center, Alabama National Aeronautics and Space Administration Marshall Space Flight Center ° MSFC, Alabama 35812 October 2001

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TRADEMARKS Trade names and trademarks are used in this report for identification only. This usage does not constitute an official endorsement, either expressed or implied, by the National Aeronautics and Space Administration. Available NASA Center for AeroSpace Information 7121 Standard Drive Hanover, MD 21076 1320 (301) 621 0390 ii from: National Technical Information Service 5285 Port Royal Road Springfield, VA 22161 (703) 487_4650

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TABLE OF CONTENTS ......................................................................................................................... 1 1. INTRODUCTION EQUATIONS OF MOTION ............................. 2 2. FORMULATION OF SIX-DOF RIGID BODY 3. STATE-SPACE MODEL FORMULATION ................................................................................. 17 3.1 Acceleration Sensor Measurement Model ........................................................................... 17 3.2 Position Sensor Measurement Model ................................................................................... 19 .................................................................................................. 20 3.3 State and Output Equations DESIGN AND ANALYSIS .......................... 22 4. UNCERTAINTY MODELING FOR CONTROL 5. MATHEMATICAL MODEL VERIFICATION ............................................................................ 27 6. CONCLUSIONS ........................................................................................................................... 28 REFERENCES ................................................................................................................................... 29 iii

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LIST OF ACRONYMS cg center of gravity CM center of mass DOF degree of freedom g-LIMIT GLovebox Integrated Microgravity Isolation Technology IM isolator module MSG microgravity science glovebox PIP power and information processor STABLE supression of transient acceleration by levitation evaluation TM technical memorandum w.rot. with respect to V

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TECHNICAL MEMORANDUM EQUATIONS OF MOTION FOR THE g-LIMIT MICROGRAVITY VIBRATION ISOLATION SYSTEM 1. INTRODUCTION A desirable microgravity environment for experimental science payloads may require an active vibration isolation control system. A vibration isolation system named g-LIMIT (GLovebox Integrated Microgravity Isolation Technology) is being developed by NASA Marshall Space Flight Center to support microgravity science experiments using the microgravity science glovebox (MSG). 1 In order to provide a quiescent acceleration environment for an experiment, an active isolation system must sense and cancel the inertial accelerations applied to the experiment. With g-LIMIT, this is accomplished by six independent control actuation channels that provide six independent forces to a platform upon which the experiment resides, g-LIMIT is designed around three integrated isolator modules (IM's), each of which is comprised of a dual-axis actuator, two axes of acceleration sensing, two axes of position sensing, and control electronics. The base of the isolator is the power and information processor (PIP), which is attached to the MSG work volume floor. Flexible umbilicals transferring power and data are the only physical connection between the isolated payload mounting structure and the PIR In this technical memorandum (TM), the six-degree-of-freedom (DOF) lineaxized equations of motion for g-LIMIT are derived. Although the motivation for this model development is control design and analysis of g-LIMIT, the equations axe derived for a general configuration and may be used for other isolation systems as well. Since the translational motion of the isolation platform is constrained to 1 cm travel in any direction and hence the rotational motion is also small, small angle and small displacement assumptions axe used to derive linearized equations of motion. It was also assumed that the base has only translational motion that is transmitted to the platform.

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1. FORMULATION OF SIX-DOF RIGID BODY EQUATIONS OF MOTION In this section, lineaxized equations of motion for the six-DOF rigid body dynamic system of the platform axe derived using a Newtonian approach. The simplified configuration of the g-LIMIT system is shown in figure 1. From figure 1, the following position vectors in inertial coordinates system are defined: position vector from the origin of the inertial coordinates to the origin of the base coordinates, /_0 ; three initial position vectors from the origin of the base coordinates to three position sensors, /_pi (i = 1, 2, 3); two initial position vectors from the origin of the base coordinates to two umbilical attach points on the base,/_i (i = 1, 2); two initial position vectors from the umbilical attach points on the base to the umbilical attach points on the platform, Si (i = 1, 2); initial position vector from the origin of the base coordinates to the origin of the platform coordinates, /_b; and relative displacement vector, i of the platform at the origin of the platform coordinates, three components (x, y, z) of which axe translational degrees of freedom for the equations of motion of the platform. The following position vectors in a platform body coordinates system axe also defined: position vector from the origin of the platform coordinates to the center of mass (CM) of the platform, Yc; position vector from the origin of the platform coordinates to the extemal force's acting point on the platform, _'d; two position vectors from the origin of the platform coordinates to two umbilical attach points on the platform, _'_i (i = 1, 2); three position vectors from the origin of the platform coordinates to three position sensors, fPi (i = 1, 2, 3); three position vectors from the origin of the platform coordinates to three accelerometers, rai (i = 1, 2, 3) ; and three position vectors from the origin of the platform coordinates to three actuators, r¢i (i = 1, 2, 3). During the derivation of the equations of motion, vectors will be expressed by the product of a row matrix, whose elements axe its three components in chosen coordinates, and a column matrix, whose elements axe three orthogonal unit vectors of the coordinates. For example, /_0 and rai can be expressed as follows: ,00 =R 0 F r (1) and fai :rai AT, (i:1,2,3) , (2) where R o = [X o YO Zo ] is a row matrix of three components of/0 in the inertial coordinate system and F = [i ]/(] is a row matrix of three orthogonal unit vectors of the inertial coordinate system. rai = [xai Yai zai ] is a row matrix of three components of ?'ai in the platform body coordinate system and A = [t ]/] is a row matrix of three orthogonal unit vectors of the platform coordinate system. For the rotational motion of the platform, three rotational DOF (0 x , Oy, Oz) are chosen to represent three angles about x, y, z axis of the platform coordinates, respectively. With the rotational sequence

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Attach Point _ Umbilical No.1 on Platform ral rPl' rfl Attach Point Fixed in Base i_ IM No. 2 Umbilical No.2 Attach Point J3 on Platform _3 IM Ne. 3 Umbilical No. 2 Attach Point No.1 CM rc 7 Ro Fixed in Base K Figure 1. g-LIMIT coordinate frame and vector definitions. ofO x , Oy, 0 z, a transformation matrix C, that relates three orthogonal unit vectors of the inertial coordinates system to those of the platform coordinates A=FC and its transpose A T =cTF with c2.c3 -c2.s3 C sls2c3+s3cl -sls2s3+c3cl -cls2c3+s3sl cls2s3+c3sl system, is given by (3a) T (3b) s2 ] -slc2 / clc2J where cl=cosO x, sl=sinO x, c2=cosOy, s2=sinOy, c3=cosO z, and s3=sinO z.

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Assuming the rotational angles are small, the transformation matrix C may be simplified as C_ 0 z 1 - x (4) 1 -0 z O ] - Oy 0x Defining a rotational skew matrix as O= Oz 0 x (5) 0 -0 z o2y ] - Oy 0 x the rotational transformation matrix C and its transposed matrix can be rewritten C = I3x3 + O (6a) and 13x3 - (} , (6b) CT = where I3x3 is a 3 by 3 identity matrix. Then eq. (3) can also be rewritten A = F (I3x3 + 0) (7a) and AT = (I3x3 - 0) FT" (7b) A skew matrix representation of any row matrix is also defined similar to eq. (5). For example, the skew matrix of rai = [xai Yai zai ] is denoted as rai and defined by 0 -zai Yai (8) L-y i xai o In order to derive six-DOF equations of motion of the platform using a Newtonian approach, absolute translational and angular accelerations at the platform CM axe needed in the inertial coordinates

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system.Thepositionvectorfromtheorigin of inertialcoordinatessystemto theCM of theplatform, rcr_ is defined by (9) =R 0FT+RbFT+rFT+rcA T , where r = [x, y, z] is a row matrix whose three components are translational degrees of freedom for the equations of motion of the platform. The absolute linear velocity of the platform CM is given by differentiating cm w.r.t, time: crrl= [OFT +iV T +o AT xrc AT , (10) where o0=[0 x Oy 0 z ]is a row matrix whose three components axe angular velocities about x, y, and z axis of the platform coordinates. The absolute linear acceleration of the platform CM is given by differentiating hcm w.r.t, time: .,-,,,=Ro r:r + i; r:r +(bAr xr. A:r +mA r ×mA r xr. A:r (11) where 6) =[0 x Oy 0 z ] is a row matrix whose three components axe angular accelerations about x, y, and z axis of the platform coordinates. Equation (11) may be reduced to the following linearized equation by neglecting terms higher than first order under the assumption of small angles and displacements: rc•"_ = k 0 F T + F F T + (b rc F T (12) Therefore, the translational equation of motion for the platform becomes /_=M rc__ (13) =M R0 FT+M FFT +M (oTcF T , where M is mass of the platform and total acting force at the platform CM, P is defined by _'=FF T (14)

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with F = [FX Fy FZ ] whose three components axe acting force at the platform CM to the directions of X, Y, and Z axis of the inertial coordinate system. Defining a state X as a column matrix [x y z 0 x Oy 0 z ] T, the translational equation of motion of the platform may be rewritten as the following matrix form: +M[I3× 3 -rc]X. (15) F T = MI3×3f_O T The rotational equation of motion for the platform may be derived from H , (16) a) c = where the total acting moment vector at the platform CM is defined by a) c, = M c, F T with M C = [M x My M Z ] whose three elements are the components of the moment acting at the platform CM about X, Y, and Z axis of the inertial coordinates system. /q is the angular moment vector at the platform CM and is defined as o) Im T AT , (17) PI = where I m is the mass moment of inertia matrix about the platform CM and defined as I m = -Iy x Iyy ix x -Ixy -- IZX (18) ZZ & The time derivative of the angular moment vector at the platform CM, H, is given by fI = do IrJA T + oA T × oo IrJA T =((b Im T -oo ImT(o)(I3x3-O) F T -_ ((70Im T - COImT (o) F T (19) -_ (o Im T F T.

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Thus, by combining eqs. (16) and (19), the rotational equation of motion of the platform can be written M C F T = (o Im T F T . (20) Rewriting eq. (20) in matrix form using the state X, Mcf=[03x 3 Iv_]X • (21) Finally, combining the translational equation of motion (15) and rotational equation of motion (21) yields the following six-DOF rigid body equations of motion of the platform: ]I i3 31 0 +I 13 3 (22) Me,r [ 0>3 [ 03×3 The next step is to define the total acting force and moment at the platform CM. The total force acting at the platform CM, P is comprised of three actuator forces Pa_ (m = 1, 2, 3), two umbilical spring forces P_i (i = 1,2), two umbilical damping forces Pdui (i = 1,2), and a direct disturbing force Fd. As shown in figure 1, three actuators axe assumed to be located at the counterclockwise azimuths of 01,02 ,03 about the z axis from the positive x axis. Three row matrices of the unit vectors of each actuator coordinates are defined asA_ =[[r_ _ ]r_/r ] (m=1,2,3). The relationship between the unit vectors system of platform coordinates and the unit vectors system of three actuator coordinates is given by Av_ =ACv_ (m=1,2,3) , (23) where cos 0v_ C_= sin 0_ cos 0_ (24) 0 Transposing eq. (23) with eq. (3) yields Av_T = CrJ A T = C_zT C T F T 0 (m = 1, 2, 3) (25)

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Theforcethatis generatedby theruth actuator is defined by Fa,_z]Am T (m=1,2,3) , (26) /?a, =[Fa,x 0 where Fa_x and Fa,zz axe the two orthogonal x and z axis components of the ruth actuator force. These force components axe determined by the control system. Substitution of eqs. (6) and (25) into eq. (26) yields 0 (,3 3 (27) Rewriting eq. (27) in matrix form, equation of three actuator forces becomes (m = 1,2,3) . (28) [ii][F.z The spring force due to the umbilical may be determined as the product of the umbilical spring coefficient and the deformation vector of the umbilical. given by The deformation vector of the ith umbilical is = Rb FT + rF T + rui AT _ Rui F T _ SiF T = Rb FT + rFT + rui (I3x3 - O)FT - Ru iFT - Si FT T+FT+[0_0, 0]f_FT (i = 1, 2) (29) =(Rb+-R_i-Si)F

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Sincethefirst termof eq.(29)is zero,eq.(29)becomes j.--[ zlr_+[o_o_o_]_.__ Writing the above equation in the matrix form, d_t i lo=J = [I3x3 -r_i] X (i=1, 2) (30) (i = 1, 2) (31) Therefore, the spring force due to the ith umbilical, F_i r can be given by F_ti T=K_i d ti r =K i [I3x3 -r_i] X (i=1,2) , (32) and in the vector form x I9[r_i K_i (i= 1, 2) , (33) where K_i is a 3 by 3 stiffness coefficient matrix whose elements axe spring stiffness of the ith umbilical in the direction of the inertial coordinates. The damping force due to the umbilical may be determined by product of the umbilical damping coefficient and the time derivative of deformation vector of the umbilical. Neglecting higher order terms, the time derivative of deformation vector of the ith umbilical is given by

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Writing eq.(34)in thematrixform, {tiT = +i T 0y 10zl Therefore, the damping force due to the ith umbilical, F T di = C_i {tiT -,i]J (i=1,2) , (36) =C_i [I3x 3 and in the vector form TFT (i=1, 2) , (37) /d i = j_T ]I3x3_c l"i I "i (i=1, 2) (34) (35) F_d i T can be determined by where C_i is a 3 by 3 matrix whose elements axe damping coefficient of the ith umbilical in the directions of the inertial coordinates. A disturbance force Fd, assumed to be applied directly at the position fd of the platform, is defined as P_=f_ AT (I3x3- O)r T. (38) = f_ 10

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Rewritingeq.(38)in matrixform, Fd T = (I3x3 + O) fd T, (39) where fd is a row matrix whose three elements axe x, y, and z axis components of the disturbance force in the platform coordinates. Consequently, the total force acting on the CM of the platform, Frcan be determined by combining eqs. (28), (32), (36), and (39): 3 2 2 FT = Z Fa_T - Z F_iT - Z F_di r + Fd r. (40) m=l i=1 Define the actuator force input vector as and 03x2 03x2 gr 03×2 03×2 03×2 where 03× 2 is a 3 by 2 zero matrix. Then, i=1 (41) 03×2 (42) -- +0)[Cl c2 c3]ur If 2 2 - ZK_i [13x3 - i]x-2c i[,3x3 i=1 + (I3x3 + 0) fd T . i=1 (43) 11

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In orderto completethederivationof equationsof motionof theplatform,thetotalmoment actingatthe platformCM is determinednext.ThetotalmomentactingattheplatformCM consists of momentsdueto threeactuatorforces,Ma,_ (m = 1,2,3) ; moments due to two umbilical spring forces, M_i (i = 1,2); moments due to two umbilical damping to direct disturbing force, /l_td . forces, Md_i (i = 1,2); and the moment due The moment about the platform CM due to the ruth actuator force is given by ., =(% -)×L., = (% - rc)d x P_,., x P_,., (m = 1,2,3) , (44) -,-r_,.d whererfa,=[(xL,-xc) (yL,-yc) (zL,-zc) ] . Substitution of eqs. (7b) and (27) into eq. (44) yields [...z[;° ](,,x,)..F_ +[...z_[:° ]c..._-F_o -:,,, Written in matrix form, eq. (45) becomes (45) (m = 1, 2, 3) (46) where ( )~ is a skew matrix of the row matrix inside the parentheses. 12

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Themomentaboutthe platformCM dueto theith umbilical spring force can be determined by =(r_-r_)A r ×P_ - e_Ar × P_i where rF_i=[(x_i-xc) (y_i-yc) (z_i-zc) ] . Substituting eqs. (7b) and (33) into eq. (47) gives r f FT (i = 1, 2) (48) Writing eq. (48) in matrix form, (i=1,2) , (47) M ir = F_i K i [I3x3 -_] X (i = 1, 2). (49) The moment about the platform CM due to the ith umbilical damping force can be determined by _¢_di = rF_iAr ×/_di Substituting eqs. (7b) and (37) into eq. (50) gives -__, I)x31c l _i l (i = 1, 2) . (50) - r (51) rF_i (i = 1, 2) 13

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andin matrixform M_diT=F i C_i[I3x 3 -i]J (i=1,2) (52) The moment about the platform CM due to the direct disturbance force Fd can be determined by =(rd -rc )AT XFd AT X Fd , (53) - rFd whererFd=[(Xd-X c) (Yd-Yc) (Zd-Zc)]. Substituting eqs. (7b) and (38) into eq. (53) gives Md = red(I3×3- 0)rr x f_ (I3×3- 0)r r -f_ redr r +f_ (red0)~ r r + f_ 0 @drr ~ +0 ;Fd]F r (54) = fd[-;Fd + (rFd O) and in matrix form M_r =[_ +_ 0-(r_ 0)-]iS (55) Consequently, the total moment about the platform CM, McTcan be determined by combining eqs. (46), (49), (52), and (55): 14

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2 3 2 T -E MudiT + MdT m=l i=1 rrl_-I 2 I, x3 i=t 2 C_iII3x 3 i--1 +If:Fd + _FdO-(rfd Defining the following skew matrices, and eq. (56) can be rewritten as MC T = I( [_Fal C _) i=1 -fuil _ (56) (9)~lfd T (57a) (57b) ([_Fa3C3)IUT fa T _2_fFuiKuiII3x3 -_il X i=1 2 C_i i/3x 3 _T_i I , - E rFui i=1 + RFafar. (58) Finally, the equation of motion for six-DOF rigid body motion of the platform can be determined by substituting eqs. (43) and (58) into eq. (22): 15

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M I3x3 03x3 i=II_F. ' K.,[I3× 3 -., ] =-1 03×31k° Le_n + 1 (59) To express this equation of motion in concise form, the following definitions are introduced: MX= -M I3x3 03x3 cx=Z Fx =- 3×3] D°r + LR_d With these definitions, the equation of motion second order ordinary differential equation: MxX + CxX + 16 -M c (60a) ] Im (60b) (60c) fur (60d) of the platform can be written as the following KxX = Fx • (61)

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1. STATE-SPACE MODEL FORMULATION For many modem control design methods, the dynamics and measurements of the system to be controlled (the "plant," denoted by a subscripted "p") axe expressed in state-space form consisting of first order ordinary differential equations. A standard notation for the state space formulation of the plant dynamics and outputs is Jcp = Apxp + BlpW + B2pu yp = Cpxp + DlpW + D2pu , (62) where xp _ 9_YZisthe state vector, w _ 9_ Yzwis the disturbance input vector, u _ 9_mis the control force input vector, and yp _ 9_Pis the output vector. This section will develop the state-space formulation of the equations of motion in eq. (61) and the sensor measurements. With this application, the outputs consist of acceleration and position measurements at the sensor location. 3.1 Acceleration Sensor Measurement Model Each g-LIMIT isolator module has two sensors which measure acceleration at the location of the accelerometers in the x and z axis directions of the IM coordinates. The acceleration vector at the location of the accelerometer of the ruth IM, 6 m can be given by G_ = ko rr + frr + 6)Ar × %, Ar + b..Ar (m = 1, 2, 3) (63) : ko rr + fr r Combining eqs. (6) and (25) yields AT = CreAm T (64) and O)CmAm T (65) CT : (I3x3 + Substituting eqs. (64) and (65) into eq. (63), 17

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am = R0(I3x3 + 0)CreAm T + r(I3x3 +O)CmAm T + & ra_ CreAm T -_ R0(I3x 3 +O)CmAm T + [2 j) 5]CreAm T T (m: 1,2,3) (66) +[0 x Oy Oz]Fa_CmAm Writing eq. (66) in the matrix form using the state X, (67) Then, the acceleration output of two accelerometers of the ruth IM can be given by am x 0 [amz 1['o=01 am 0 T (re=k2,3) :[;o]o __ ['_ -,lx (68) Therefore, the total acceleration measurement vector, A = [alx alz a2x a2z a3. a3 ], can be determined by -; ; O1]c1T[I3x3 -ral] A T = -; ; O1]c2T[I3x3 -ra2] -,oo]o oc3_[,3x3-3] = TAx + A 1 R0 T . 18 0 0 C1T(I3x3-O) /o T -,oo]o ocJ(I3×3-0) -1°°1]o o cF(I×-0) (69)

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3.2 Position Sensor Measurement Model Each g-LIMIT IM has two position sensors which measure the relative displacement of the isolated platform with respect to the MSG-fixed base at the location of position sensor in the x and z axis directions of the IM coordinates. The relative position vector at the location of position sensor of the ruth IM, p is given by = RbF T + rF T + rp_ AT - Rp_ F T (m =1,2,3) = Rbrr + r r + rp,(I3×3- O)rr - Rp_rr +[xYzlr T +[OxOyOz]p_r T . (70) =(R b +rp-Rp_)F T Note that the first term of eq. (70) is zero. Substituting eq. (65) into eq. (70) and then obtaining first order terms yields 8_, =[xyzlF _ +[OxOyOz]p_r _ (m:1,2,3) x Oy Oz]Fp_Crr_A_f . (71) -[x y zlC_A_f +[0 Writing eq. (71) in matrix form using the state X, ]X (m = 1, 2, 3) . (72) (p_T = cmT[I3x 3 - Fp Then, the output of two position sensors of the ruth IM can be given by SP_,x 0 0 T 0 T [I3×3 rp_ . (73) 0 1 (m =1,2,3) 19

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Therefore,thetotal positionmeasurementvector,tiP = [tip1X 3Plz 3P2x 3P2z fiP3_ fiP3_]' can be determined by pT = o' ° X o o c3[I3X3-1 -T_X. (74) 3.3 State and Output Equations The state space equations may now be written. From eq. (61), the dynamics of the platform may be written as = Mx-1Fx - Mx-1Cx (75) - Mx-1Kx X . This second order differential equation can be written in state space form by defining the state, input, and output vectors as follows: • Statevector: xp=[ X X]_ • Disturbance input vector: w = [/_o fd ]T • Control force input vector: u = ff • Outputvector: yp=[_p 2O _T] T

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Theresultingstatespaceequationsaxe: kp=L_M_IK x M_Icx xp + [(,x+o)[q c c] -'" t O_x_'" [_ 06x3 06x3 n] JJw 06x3 ] 06x3 ]] + _[(,x+o)[q c c] .. The coefficient matrices can now be identified (76) by comparison of eq. (76) with eq. (62). 21

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1. UNCERTAINTY MODELING FOR CONTROL DESIGN AND ANALYSIS A key objective of control system design is robustness to variations between the actual system and the model on which control designs axe based. For microgravity vibration isolation systems, the primary uncertain parameters of interest axe the payload mass, umbilical stiffness, umbilical damping, and composite isolation system/payload center of gravity (cg). Although both mass and stiffness or mass and damping uncertainties are important for consideration, it is evident that the mass terms appear in the system A matrix (from which stability is determined) as products with the stiffness and damping. Hence, both uncertainties cannot be considered simultaneously Simultaneous mass and stiffness or mass and damping with standard linear robust control methods. uncertainty need not be considered, however, since mass uncertainty may be effectively accounted for in either stiffness and damping uncertainty or uncertainty in the product term itself. In the following section, the uncertain dynamics will be developed for parametric uncertainty in stiffness, damping, and cg location. For a treatment of uncertainty in the product term (system natural frequency and damping ratio), see references 2 and 3. Considering only one uncertain umbilical and treating the uncertainties as additive parametric uncertainty, the uncertain umbilical stiffness, damping, respectively, as and composite cg location may be defined, K M = (KM) 0 + z_r_7_ C_tI = (CM) 0 + AC_t rc = rco(1 + where the zero subscript indicates the nominal value. 5cg) , (77) The uncertain cg location implies uncertain moment arm for the application of umbilical and actuator forces as well: rFctl = r_t1 - rc = (r_l)0 and - 8cgrc0 (78) = (rva_) o - 8ogre.o (79) 22

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Theskewsymmetricmatricesbecome fFul = (fFul)0 - acgVcO and from eq. (57a), (80) (81) These uncertain terms are now substituted into the coefficient matrices of the state space equations of motion, eq. (76). From eq. (60a), -MI3x 3 -M_ c- -MI3x 3 MX= (82) 03x3 IM 03x3 IM with the inverse given by (ref. 5, p. 656): (rc0 +Scgrc0)IM 1- (83) MXI= -M-113x3 03x 3 Also from eq. (60c), for one umbilical IM 1 (84) KX = _(rFul)0 - (Kul)0(_cgrcO)((Kul)O+ AKu + zXKu) ] [13x3 - rul] 23

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andlikewisefromeq.(60b), (85) CX = ((rFul)0 - cgrc0)((Cul)0(Cul)0+ACu + ACu) ] [I3x3 - rul] • The first product term to be expressed is the product MTrlKx, given by (86) [ 03x3 IM1 ((fFul)O(_cgfcO)((Kul)o+AKu) [13x3 -rul], which after neglecting products of uncertainties becomes + [M-113x3 + rc0/Ml(rFul)0 ]AKu [13x3 Mx1Kx = (Mx1Kx)o L IMl(rFul)0 +[_c 0/M1 ((_Ful)0 1 fco)l(cg(gul)O[13x3 - rul] L -*d c0 J Similarly, the mass and damping product term is (87) Mx1Cx = (Mx1Cx)o + [M-113x3 + fc0IMl(rFul)011 [I3x3 [ I/(rrul)0 IAC rco)](_cg(CM)O[IBx3 - f_l] • (88) +[rcoId((rFM)OL_id_cO - Considering the uncertain component of the B iN matrix, 24

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c2 1 ] = d[(,)oC,(=)oC=()oC_] -I- -Rco)Cl__)cgiMl_co[C((RFa2)o-Rco)C21C2 C3 ] ((RFa3)o-Rco)C3]] . (89) The partitions of the Ap matrix in eq. (76) may now be evaluated by considering the nominal and uncertain components of the preceding uncertain product terms. The system Ap matrix may be written as the nominal portion plus the uncertain contributions, or + AA C + AAcg , (90) Ap = A O + AA K where the nominal portion, Ao, is the Ap matrix corresponding to zero uncertainty. By grouping the uncertain terms of the individual product terms, the uncertain components of Ap are [ 06x6 06x6] AAK = H -M-113x3 - TcOIMI(TFul)O ]AKu [I3x3 ] (TFul)O 1 06x6] LL -/M1 =/{ a-l [ 13x3 AKu[[/gx 3 -Tul ] 03x6] ,x31] [t-'v'x ;0[(r_l)0 = AAKL * AK u * AAKR , 06x6 (91) 06x6 =/ ,l['x Ac.[O_x_[,x o x31] Lt,-'v'x }o[(er_l)O = AAcL * AC_ * AAcR , (92) 25

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and rcOIM1((rF.1)0 - rco ) (cg(Kul)O[I3×3 -rul] a<.= - _i1_co _ 06x_6 _[cdl((.l)O - Tco)- _cgI3x3 [(Kul)0 [I3x3 =L - d co 06x3 = AAcgL *(_cgI3x3 * AAcgR • Finally, _2p =/[_o/d 03x3 (_cgl6x6 [ 06x6 ]] EL03x3 -IdRco = AB2p L * (_cgl6x6 * AB2p R • 06x6 _ [Tc0Id ((TFul)0- rc0)- (cg(Cul )O[I3×3 L - d co - Ful] (Cul)O[I3×3 - rul]] (93) ((ke_2)o - k,o)C2 ((Rva3)0 -/_co)C3 ]U T C2 C3 l (94) Note that uncertainties in the disturbance input axe not included herein as they axe treated directly in the weight selection for robust control design. A block diagram of the uncertain plant with these uncertainties is given in figure 2. w X l Figure 2. Uncertain plant block diagram. 26

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1. MATHEMATICAL MODEL VERIFICATION In the previous sections the mathematical model of the g-LIMIT dynamics and control system is derived to analyze the dynamics of g-LIMIT and to design control systems for g-LIMIT. This mathematical model was developed for an arbitrary configuration and mass properties, allowing easy adaptation to other isolation systems in addition to g-LIMIT. In order to verify this mathematical model, it was coded using MATLAB TM and simulated for various test cases using the configuration and mass properties of the suppression of transient acceleration by levitation evaluation (STABLE) vibration isolation system. 4 These simulation results were compared with those obtained from the STABLE TREETOPS model. 4 For these simulations, accelerometer bias and noises were not included. First, to check the validity of the six DOF equations of motion of the platform and mathematical models of position sensors and accelerometers, a direct disturbance force was given on the CM of the platform and then time-response simulation was performed without controllers on. Second, to check the validity of acceleration control logic and the interaction between the system dynamics and the acceleration controller, a sinusoidal base acceleration disturbance was given without any direct disturbance force and the time response simulation was performed with only acceleration controller on. Finally, to check the validity of position proportional-integral-derivative control logic and the interaction between the system dynamics, the acceleration controller and the position controller, an initial displacement was given to the platform without any other disturbance and then the time response simulation was performed with both acceleration and position controllers on. For all three test cases the output of six position sensors and six accelerometers obtained from the mathematical model derived herein and the STABLE TREETOPS model were matched. Therefore, rate under the assumption of small motions. this mathematical model is believed to be accu- 27

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1. CONCLUSIONS ThisTM documentsthemathematicalmodelingof theg-LIMIT systemthatwasdevelopedto providethedynamicequationsof motionin stateequationformfor controlsystemdesign.State-space equationsaxeprovidedfor accelerationandrelativepositionmeasurementsatboththeplatformCM andthesensorlocations.Disturbanceinputsconsistof baseaccelerationandadirectlyappliedforce. Thismathematicalmodelwill alsobeusedfor areferenceto verify a g-LIMIT TREETOPSmodelwhich will bedevelopedandusedasthetruthmodelto predicttheperformanceof theg-LIMIT systemwith the designedcontroller. Sincefinal configurationandmasspropertiesof the g-LIMIT systemaxenot yetdetermined,the equationsof motionwerederivedfor a generalconfigurationof a six-DOFrigid bodysystem.However, this mathematicalmodelwasverifiedagainsttheTREETOPSmodelfor theSTABLEconfigurationand canbeeasilymodifiedfor thefinal g-LIMIT configuration. 28

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REFERENCES 1. Whorton, M.S.: "Design Definition Document of Glovebox Integrated Microgravity Isolation Technology (g-LIMIT) Characterization Test," g-LIMIT-DOC-0001, September 30, 1999. 2. Balas, G.J.; and Young, RM.: "Control Design for Variations in Structural Natural Frequencies," Journal of Guidance, Control, and Dynamics, Vol. 18, No. 2, pp. 325-332, Maxch-April 1995. . Whorton, M.S.; Calise, A.J.; and Hsu, C.-C.: "A Study of Fixed-Order Mixed Norm Designs for a Benchmark Problem in Structural Control," Earthquake Engineering and Structural Dynamics, Vol. 27, pp. 1315-1330, 1998. 4. Nurre, G.S.; Kim, Y.K.; and Whorton, M.S.: "A TREETOPS Simulation of the STABLE Microgravity Vibration Isolation System," NASA TM--1999-209009, January 1999. 5. Kailath, T.: Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980. 29

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REPORT DOCUMENTATION PAGE FormApproved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operation and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and tothe Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503 1. AGENCY USE ONLY (Leave Blank) 2. REPORT DATE October 2001 4. TITLE AND SUBTITLE 3. REPORT TYPE AND DATES COVERED Technical Memorandum 5. FUNDING NUMBERS Equations of Motion for the g-LIMIT Microgravity Vibration Isolation System 6. AUTHORS Y.K. Kim and M.S. Whorton 7. PERFORMING ORGANIZATION NAMES(S) AND ADDRESS(ES) George C. Marshall Space Flight Center Marshall Space Flight Center, AL 35812 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeronautics and Space Administration Washington, DC 20546_0001 11. SUPPLEMENTARY NOTES 8. PERFORMING ORGANIZATION REPORT NUMBER M-1028 10. SPONSORING/MONITORING AGENCY REPORT NUMBER NASA/TM--2001-211301 Prepared by the Vehicle Control Systems Group of the Transportation Directorate. 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified-Unlimited Subject Category 18 Nonstandard Distribution 13. ABSTRACT (Maximum 200 words) 12b. DISTRIBUTION CODE A desirable microgravity environment for experimental science payloads may require an active vibration isolation control system. A vibration isolation system named g-LIMIT (GLovebox Integrated Microgravity Isolation Technology) is being developed by NASA Marshall Space Flight Center to support microgravity science experiments using the microgravity science glovebox. In this technical memorandum, the full six-degree-of-freedom nonlinear equations of motion for g-LIMIT are derived. Although the motivation for this model development is control design and analysis of g-LIMIT, the equations are derived for a general configuration and may be used for other isolation systems as well. 14. SUBJECT TERMS 15. NUMBER OF PAGES 36 microgravity, vibration isolation, dynamics, equations of motion 17, SECURITY CLASSIFICATION 18, SECURITY CLASSIFICATION OF REPORT OF THIS PAGE Unclassified Unclassified NSN 7540-01-280-5500 16. PRICE CODE 19, SECURITY CLASSIFICATION 20, LIMITATION OF ABSTRACT OF ABSTRACT Unclassified Unlimited Standard Form 298 (Rev. 2-89) Prescribed by ANSI Sld 239 18 298 102