Full report

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NASA/TM—2005–213848 A “Kane’s Dynamics” Model for the Active Rack Isolation System Part Three: Addition of Umbilicals to the Nonlinear Model J.K. Rupert Dynetics, Inc., Huntsville, Alabama R.D. Hampton United States Military Academy, West Point, New York G.S. Beech Marshall Space Flight Center, Marshall Space Flight Center, Alabama February 2005

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NASA/TM—2005–213848 A “Kane’s Dynamics” Model for the Active Rack Isolation System Part Three: Addition of Umbilicals to the Nonlinear Model J.K. Rupert Dynetics, Inc., Huntsville, Alabama R.D. Hampton United States Military Academy, West Point, New York G.S. Beech Marshall Space Flight Center, Marshall Space Flight Center, Alabama Natonal Aeronautcs and Space Admnstraton Marshall Space Flght Center • MSFC, Alabama 35812 February 2005 

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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. Avalable from: NASA Center for AeroSpace Informaton 7121 Standard Drve Hanover, MD 21076–1320 301–621–0390  Natonal Techncal Informaton Servce 5285 Port Royal Road Springfield, VA 22161 703–487–4650

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TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................................................ 1 2. COORDINATE SYSTEMS .......................................................................................................... 3 3. GENERALIZED COORDINATES AND GENERALIZED SPEEDS ........................................ 5 4. ANGULAR VELOCITIES OF REFERENCE FRAMES AND RIGID BODIES ....................... 6 5. GENERALIZED ACTIVE FORCE CONTRIBUTIONS DUE TO THE UMBILICALS .......... 8 5.1 Generalized Active Force Equations ..................................................................................... 8 S1 F U j 5.2 Partial Velocities v ........................................................................................................ 9 r S1 F1 5.3 Partial Angular Velocities ω ......................................................................................... 11 r 5.4 General Form for Umbilical Forces and Moments ................................................................ 11 5.5 Umbilical Elongations and Elongation Rates ........................................................................ 12 5.6 Umbilical Angles of Twist and Twist Rates ........................................................................... 14 6. IMPLEMENTATION IN AUTOLEV .......................................................................................... 17 7. MODEL VERIFICATION ............................................................................................................ 18 7.1 Onboard Impulsive-Disturbance Force, No Damping ........................................................... 19 7.2 Onboard Sinusoidal-Disturbance Force, With Damping ....................................................... 19 7.3 Onboard Impulsive-Disturbance Moment, No Damping ...................................................... 21 7.4 Onboard Disturbance Moment, With Damping ..................................................................... 22 7.5 Off-Board Translational Disturbance .................................................................................... 24 7.6 Off-Board Rotational Disturbance ......................................................................................... 27 7.7 Comparisons Using Nondiagonal Stiffness and Damping Matrices ..................................... 29 8. CONCLUSION ............................................................................................................................ 30 REFERENCES .................................................................................................................................. 31 

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LIST OF FIGURES 1. Detailed diagram of the umbilical assembly ...................................................................... 1 2. Kinematic diagram, including the ith actuator assembly and the umblcal ...................... 3 3. CAD-based technique for determining umbilical attachment locations ............................ 17 4. Translational single-degree-of-freedom truth model ......................................................... 18 5. Translational displacement due to onboard impulsive-disturbance force .......................... 19 6. Translational displacement due to onboard sinusoidal-disturbance force ω with = 1 and ζ = 1 ........................................................................................................ 20 ωn 7. Translational displacement due to onboard sinusoidal-disturbance force ω with = 5 and ζ = 1 ........................................................................................................ 20 ωn 8. Translational displacement due to onboard sinusoidal-disturbance force ω with = 1 and ζ = 0 25. ................................................................................................... 21 ωn 9. Rotational displacement due to onboard impulsive-disturbance moment ......................... 22 10. Rotational displacement due to onboard sinusoidal-disturbance moment ω with = 1 and ζ = 1 ........................................................................................................ 23 ωn 11. Rotational displacement due to onboard sinusoidal-disturbance moment ω with = 5 and ζ = 1 ........................................................................................................ 23 ωn 12. Rotational displacement due to onboard sinusoidal-disturbance moment ω with = 1 and ζ = 0 25. ................................................................................................... 24 ωn 13. Translational single-degree-of-freedom truth model, with base motion ............................ 25 v

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LIST OF FIGURES (Continued) 14. Translational displacement due to off-board sinusoidal position disturbance ω with = 0 1. and ζ = 0 707. .............................................................................................. 25 ωn 15. Translational displacement due to off-board sinusoidal position disturbance ω with = 1 and ζ = 0 25. .................................................................................................... 26 ωn 16. Translational displacement due to off-board sinusoidal position disturbance ω with = 10 and ζ = 0 707. ............................................................................................... 26 ωn 17. Rotational displacement due to off-board sinusoidal rotation disturbance ω with = 10 and ζ = 0 707. ............................................................................................... 27 ωn 18. Rotational displacement due to off-board sinusoidal rotation disturbance ω with = 0 1. and ζ = 0 707. .............................................................................................. 28 ωn 19. Rotational displacement due to off-board sinusoidal rotation disturbance ω with = 1 and ζ = 0 25. .................................................................................................... 28 ωn v

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LIST OF ACRONYMS ARIS actve rack solaton system CAD computer-aided design ISPR nternatonal standard payload rack ISS International Space Station TM Techncal Memorandum v

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NOMENCLATURE x vector of arbitrary quantity x x first time derivative of arbitrary quantity x xˆ unit length of arbitrary quantity x  x arbtrary reference frame x x arbtrary dynamcal system x lx linearized arbitrary quantity x * x arbitrary rigid-body x center of mass T x transpose of matrx x d d t ordinary (or total) derivative with respect to time t ∂ ∂u partial derivative with respect to scalar u tr A trace of arbtrary matrx A Scalars Uppercase i A1 ntersecton pont on ith actuator arm (fig. 2) i A2 pont locatng ith upper stinger (fig. 2) i A3 pont locatng ith Lorentz coil (fig. 2) C system dampng matrx: lbf⋅s/ft E system dsturbance nput matrx v

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NOMENCLATURE (Continued) Ci Ci F magntude of ith Lorentz coil force vector F : lbf Fi pont locatng ith cross-flexure (fig. 2) Fr system holonomc generalzed actve force, for the rth generalzed speed: ft⋅lbf * Fr system holonomc generalzed nerta force, for the rth generalzed speed: ft⋅lbf Fr system nonholonomc generalzed actve force, for the rth generalzed speed: ft⋅lbf  * Fr system nonholonomc generalzed nerta force, for the rth generalzed speed: ft⋅lbf Fu umbilical attachment point at the flotor end Fuh stator-fixed reference position of Fu I dentty matrx * A A/ d body A, for body-fixed coordinate directions aˆ I jk central nerta scalar of rg and aˆ : ft⋅lbf⋅s2 k K system stffness matrx: lbf/ft M system mass matrx: slug, lbf⋅s2/ft O zero matrx  1 j  1 Q rotaton matrx from the Si coordnate system to the Fi coordnate system Qij ijth element of rotaton matrx Q B Qr contrbuton to the system holonomc generalzed actve force for the rth general- zed speed, due to rgd body B: ft⋅lbf B * (Qr) contrbuton to the system holonomc generalzed nerta force for the rth general- zed speed, due to rgd body B: ft⋅lbf Si pont locatng ith lower stinger (fig. 2) Su umblcal attachment pont at the stator end v

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NOMENCLATURE (Continued) S total dynamical system (stator, flotor, actuators, umbilical) XFu, YFu, ZFu geometrc lengths: ft Lowercase i a j geometrc length for ith actuator assembly: ft ξˆ ci umblcal dampng n the drecton: lbf⋅s/ft i i c j cosne of angle qj for ith actuator assembly i f j geometrc length for ith actuator assembly: ft i f jk rotaton matrx element  g drecton cosnes for nˆ n the F coordnate system i φ ξˆ 1 ki umblcal stffness n the drecton: lbf/ft i i l j geometrc length for ith actuator assembly: ft : slug, lbf⋅s2/ft mA mass of arbtrary rgd body A i p2 geometrc length for ith actuator assembly: ft i q j jth generalzed coordnate for ith actuator: rad rjk rotaton matrx element i s j sne of angle qj for ith actuator assembly i u j jth generalzed speed for ith actuator: rad/s i v j geometrc length for ith actuator assembly: ft x0, y0, z0 geometrc lengths: ft x

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NOMENCLATURE (Continued) ξˆ xi umblcal elongaton n the drecton: ft i xFu, yFu, zFu geometrc lengths: ft xSu, ySu, zSu geometrc lengths: ft ξˆ κi torsonal umblcal stffness about the axs: ft⋅lbf/rad i φ flotor angle of twist relative to stator: rad ξˆ φi angle-of-twist component in the drecton: rad i ξˆ γi torsonal umblcal dampng about the axs: ft⋅lbf⋅s/rad Vectors Uppercase i Fb umblcal bas force n the reference poston: lbf Ci F force exerted on the ith Lorentz coil by the flotor: lbf FD unknown disturbance force acting directly on the flotor: lbf Fi F force exerted on the flotor by the ith actuator arm: lbf FU force exerted on the flotor by the umbilical: lbf HA/A* angular momentum of arbtrary rgd body A with respect to its mass center A*: ft⋅lbf⋅s Ai M moment exerted on the ith actuator arm through the upper stnger, due to the ith pushrod: ft⋅lbf Mb umblcal bas moment n the reference poston: ft⋅lbf Ci Ci th Lorentz coil by the flotor, with F assumed to act M moment exerted on the i at the ith cross flexure: ft⋅lbf x

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NOMENCLATURE (Continued) MD unknown disturbance moment acting directly on the flotor, with F assumed to act at the flotor mass center: ft⋅lbf Fi D Fi M moment exerted on the flotor by the ith actuator, with F assumed to act at the ith cross-flexure: ft⋅lbf Lowercase AaB acceleraton of arbtrary pont B, with arbitrary reference frame A assumed fixed: ft/s2 aI translational acceleration of stator (due to g-jitter): ft/s 2 d system dsturbance vector: lbf and ft⋅lbf elements i system control current vector: amp nˆ unit vector in direction of rotation axis for stator-to-flotor rotation φ n φ ˆ φ q Vector of generalized coordinates: rad rAB poston vector from arbtrary pont A to arbtrary pont B: ft u vector of generalzed speeds: rad/s uI vector of ndependent generalzed speeds: rad/s AvB velocty of arbtrary pont B, with arbitrary reference frame A assumed fixed: ft/s AαB angular acceleraton of arbtrary reference frame B with respect to arbitrary reference frame A: rad/s2 Aω B angular velocty of arbtrary reference frame B with respect to arbitrary reference frame A: rad/s x

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x

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TECHNICAL MEMORANDUM A “KANE’S DYNAMICS” MODEL FOR THE ACTIVE RACK ISOLATION SYSTEM PART THREE: ADDITION OF UMBILICALS TO THE NONLINEAR MODEL 1. INTRODUCTION As the only active rack-level isolation system for the International Space Station (ISS), the actve rack isolation system (ARIS) is the central component of the stationwide strategy to meet ISS solation requirements.1 It serves to attenuate the varous dsturbances that unavodably accompany manned space flight. Umbilicals, as shown in figure 1, are to be used in support of many experiments planned for space, providing services such as cooling, power, vacuum, and data transmission. ARIS is nominally equipped with thirteen umbilicals.1 Vacuum Resource Safing Power Main Power Video High Rate Date Fire Detection/Maintenance 1553 Bus A Vacuum Exhaust Moderate Temperature Cooling Supply Moderate Temperature Cooling Return Gaseous Nitrogen 1553 Bus B Figure 1. Detailed diagram of the umbilical assembly. Unfortunately, the ARIS umbilicals are not only the primary transmitters of indirect; i.e., offboard, dsturbances from ISS to the nternatonal standard payload rack (ISPR), they are also typcally nonlinear, hysteretic, and poorly characterized.2 Snce the respectve umblcals are attached at dfferent locatons to ISS at the bottom plate and to the base of the ISPR at the z-panel, it is desirable to include this attachment-point information as part of the ARIS model to improve model fidelity and reduce the amount of uncertainty for which the controller must compensate. 1

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In part one of this series, NASA/TM—2001–211063, a high-fidelity, linearized, analytical model of ARIS was derived using traditional, hand-calculation methods.3 The model was developed directly, using Thomas Kane’s method, without intermediate development of the full nonlinear model.4 Part one briefly outlined an approach for incorporating an ARIS umbilical into the model, using diagonalized stiffness and damping matrices. Part two, NASA/TM—2004–213552, presented four computer-based, numerical models of ARIS, one of which was purely kinematical, and the remaining three, dynamical.5 These numercal models were used collectively to verify and simplify the linearized analytical model developed in part one. This Technical Memorandum (TM) completes the nonlinear ARIS model of part two, by adding multiple Hookean umbilicals with full stiffness and damping matrices. Comparisons against simple single-degree-of-freedom truth models indicate that the completed nonlinear model has input responses that follow those expected given the laws of physics. This TM (1) briefly summarizes the existing Kane’s model, (2) describes the addition of an arbitrary number of massless umbilicals with specified arbitrary attachment points and arbitrary (parallel) stiffness and dampings, to that model, (3) provides a brief description of Autolev™ in which the underlying untethered model was developed, and (4) presents the process by which verification of the enhanced model was accomplished along with some verification results. To develop the enhanced model, it was assumed that the umbilicals were to be included individually in the model, rather than as a combined, or effective, umbilical. For the sake of simplicity and since the range of moton s small, each umblcal s assumed to be Hookean and massless snce an algebrac state-space model, one that is useful for controller design using state-space-based methods, is desired. Also, each is modeled as a three-dimensional (six-by-six, constant) stiffness matrix in parallel with a three-dimensional (six-by-six, constant) damping matrix. In the verification stage, the model of the tethered ARIS system is compared with two baseline, or truth, models. One truth model consists of a translational, one-dimensional, second-order springmass-damper system, and the other consists of a corresponding rotational system.6 The verification checks indicate that the proposed Kane’s model, when properly calibrated against experimental data, will provide a high-fidelity representation of ARIS for simulation and controller-design purposes. 2

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1. COORDINATE SYSTEMS The coordnate systems, generalzed coordnates, generalzed speeds, and angular veloctes used for the Kane’s model were presented previously in part one.3,7,8 They are reviewed in this section and in sections 3 and 4 for the reader’s convenience. With the ISPR (flotor) in the reference position; i.e., centered in its ISS-fixed rattle space, fix eight right-handed, orthogonal coordinate systems in the ISPR, one at each of the cross-flexure centers (fig. 2). Let the ith coordnate system have orgn Fi, located at the center of the ith cross flexure, i fˆ (i=1,…,8), with axis directions determined by an orthonormal set of unit vectors, j (j=1,2,3). The overhat ndcates unt length, the ndex, i, corresponds to the ith actuator assembly, and the ndex, j, i fˆ distinguishes the three vectors. Orient the unit vectors such that s along the ith arm, toward the ith i fˆ 2 voice coil; s drected parallel to the other segment of the ith arm and toward the upper stinger, which 1 i i i i fˆ fˆ × fˆ s located at A2 ; and s n the drecton along the intersection of the two crosspieces of the 3 1 2 ith cross flexure. Coil i i A l 3 4 i Arm l 2 i i l A 1 1 F i Flotor Cross flexure i (angle q ) FUj 1 Rattlespace Umbilical SUj Stator Upper Stinger i i (angles q , q ) 2 3 i A 2 Pushrod i P* l i 3 S i Lower Stinger i i i (angles q , q , q ) 4 5 6 Figure 2. Kinematic diagram, including the ith actuator assembly and the umbilical. 3

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i Fix a similar right-handed coordinate system, αˆ (j=1,2,3), in the arm of each actuator. Locate i j i fˆ each system, αˆ , such that it is coincident with the corresponding flotor-fixed coordinate system, , j when the flotor is in the reference position. j i Locate eight more arm-fixed coordinate systems, described by aˆ , at the respectve upper stngi j ers, ponts A2 . At the respective lower stingers, points Si, place eight pushrod-fixed coordinate systems i i pˆ , and eight stator-fixed (ISS-fixed) coordinate systems, s . Orient these 24 coordinate systems such j ˆ j that when the stingers are relaxed; i.e., with the flotor in the reference position, the coordinate directions, i i i i i i aˆ , pˆ , and sˆ , are coalgned for the ith actuator, with pˆ (along with a and s , n the reference j j j ˆ ˆ 2 2 2 i i i poston) drected from Si towards A2 . Directions pˆ and pˆ (n the reference poston) are determned 1 3 i i by fixed rotation matrices relating the aˆ to the αˆ coordinate systems. j j Finally, define a primary, central, flotor-fixed, reference coordinate system with coordinate direcfˆ tons, j . All other flotor-fixed coordinate systems, by known direction cosine angles, to this system. j i fˆ j , are assumed capable of being referenced; e.g., Define a stator-fixed coordinate system, ξˆ , that is associated with the jth umbilical. Define the i 1 j rotation matrix from the stator-fixed coordinate system, sˆ , to the new coordinate system, ξˆ , by  ˆ j  ξ  1   j  i i  ˆ1  s 1    1  ξˆ = ℜ s ,    2   j ξˆ   3  where  j r11  j ℜ  = r  j  21  j r  31   ˆ  (1) j  22  1  sˆ   3  j j  r12 r13  j j r r  . (2) 22 23  j j r r  32 33 Note that ℜ  is a constant matrix, and it could differ for different umbilical circumstances.  j  4

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1. GENERALIZED COORDINATES AND GENERALIZED SPEEDS i i fˆ Let the αˆ coordnate system rotate, relatve to the coordnate system, through the postve j i i fˆ j i angle, q1, about the 3 axis against the cross-flexure stiffness, k1. Similarly, let the orientation of the i i aˆ coordnate system, relatve to the pˆ coordnate system, be descrbed by consecutve postve rotaj j i i i i tons— q about the pˆ axs and q about the moved 3 axis. Also, let the orientation of the p coord- 2 1 3 i ˆ j i nate system, relatve to the sˆ coordnate system, be descrbed by consecutve postve rotatons— q j i i i 4 about the sˆ axs, q about the moved 2 axs, and q about the moved 1 axis. The six generalized 3 5 6 speeds are defined as the time rates of change of the respective generalized coordinates: i i ( , ;8 j = 1,,6) . (3) u j = q j i = 1, i i i Fnally, let c j and s j represent the cosnes and snes of the respectve angles, q j . 5

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1. ANGULAR VELOCITIES OF REFERENCE FRAMES AND RIGID BODIES Desgnate the reference frames correspondng to the stator, the ith pushrod, the ith arm, and the       flotor, as the symbols S P,i, Ai, and F , respectively. Let Si and Fi represent, respectvely, the coordnate   i i  ˆi ˆi ˆi   ˆi ˆ systems n S and F, respectively defined by s s T T sˆ  and f f f . Two intermediate  1 2 3  1 2 3 reference frames were introduced previously to permit describing the angular velocity of each pushrod relative to the stator. Designate those intermediate frames corresponding to the ith actuator assembly as     Ri and Qi. Another intermediate reference frame was previously introduced between frames Pi and Ai;  desgnate that frame as Ti . Finally, designate the stator-fixed reference frame corresponding to the jth  umblcal as Ξ j . Let each intermediate reference frame have a frame-fixed, dextral set of unit vectors. Indicate the unit vectors for each of these frame-fixed coordinate systems by using the corresponding lowercase  i letter; i.e., rˆ correspondng to Ri, etc. The following gives the expressions for the angular velocities j of the varous reference frames and rgd bodes of S : Fi Ai = u f (4) ω Pi Ti = u pˆ , (5) ω Ti Ai = u tˆ , (6) ω Si Ri = u sˆ , (7) ω Ri Qi = u rˆ , (8) ω and Qi Pi = u qˆ . (9) ω i ˆi 1 3 , i i 2 1 i i 3 3 i i 4 3 i i 5 2 i i 6 1 Using the addition theorem for angular velocities, the angular velocities of the rigid bodies of S are S A i i i i i i i i i i i ω i = u pˆ + u tˆ + u s + u r + u q , 2 1 3 3 6 ˆ ˆ ˆ (10) 4 3 5 2 6 11

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Si Pi i i i i i i ω = u sˆ + u r + u q , 4 3 and Si Fi i i ˆ ˆ (11) 5 2 6 1 i i i i ω = u sˆ + u r + u q . 4 3 ˆ ˆ (12) 5 2 6 1 7

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1. GENERALIZED ACTIVE FORCE CONTRIBUTIONS DUE TO THE UMBILICALS 5.1 Generalized Active Force Equations Equations of motion developed using Kane’s method consist of contributions called generalized actve forces (GAFs), related to system forces and moments, and contrbutons called generalzed nerta forces (GIFs), related to the time derivatives of linear and angular momenta.4 For ARIS, the development of the GAF and GIF contributions due to the actuator arms and pushrods, the flotor, and a single umbilical with diagonal stiffness and damping matrices is detailed in part one.3 Because the umbilicals are assumed to be massless, they make no contrbutons to the GIFs, and thus, they contrbute only to the GAFs. In prevous treatments (part one of ths seres and a paper by Rupert and Hampton), the umblcal stiffness and damping matrices were assumed to be diagonal with no coupling between translation and rotation, or between axes.3,8 However, coupling will always exist between translation and rotation and typically between axes as well.2,9 In other words, for an arbitrary set of orthonormal axes, one expects that the six-by-six umbilical stiffness and damping matrices will be full. Consequently, full matrices are assumed in the following treatment. Desgnate the rth partal velocty of the jth umbilical’s flotor-attachment point (with reference  S FUj assumed to be fixed, for purposes of differentiation) as 1 v and the rth partal angular frame S S F r velocity of the flotor relative to the stator as ω . Designate the force and moment that the jth umbilir cal exerts on the flotor as FUj and MUj, respectively, where the force is assumed to act on the flotor at the jth umblcal attachment pont FUj. Let all direct disturbance forces and moments that are exerted on the flotor—such as those due to air currents, to direct contact by astronauts, or to moving parts on flotor-mounted experiments—be designated collectively by the respective symbols FD and MD. Finally, let –FCi and –MCi represent, respectvely, the force and moment exerted by the ith Lorentz coil, located i at A3, on the flotor, where Ci F aˆ , (13) F = assumed to act at pont Fi and i C F A Ci M i = r i 3 × F Ci i 1 Ci i i i = −F (l + l ) ˆa . (14) 1 4 3 Then, assuming a nominal complement of 13 umbilicals, the GAF contribution due to the flotor, for the rth generalized speed, can be written as 8

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13 F  S FU j U S F  8  U  S F D i i ˆi Q := 1 v ⋅ F j + ω ⋅ M ω ⋅  MM + k q f  r ∑ r r  j=1 8 * S1 F D S1 Fii j  + r ∑ 1   1 3  i=1  Ci S1 F Ci (15) + v ⋅ F −∑( v ⋅ F + ω ⋅ M ) . r r i=1 r The jth umbilical affects only the first two terms, with the contribution j F S1 FUj F + ω ⋅ M . (16) Qr := v ⋅ r These two terms will be treated in sections 5.2 and 5.3. Uj S F Uj r S1 F U j 5.2 Partial Velocities v r Represent by rAB the poston vector from arbtrary pont A to arbtrary pont B, and define the following position vectors using the indicated scalars: i F Ai2 i i i l aˆ + l aˆ , (17) r = 1 i S Ai2 i 2 2 1 i ˆ i (18) r = l p3 , * S Pii r = 2 i * 2 i i p pˆ , (19) 2 A A2i i i i a aˆ + a aˆ , (20) r = 1 and * i F Fi i ˆ i 1 2 2 i i i i f fˆ + f fˆ . r = f1 f + 2 3 (21) 1 2 3 Represent the position vector from the flotor center of mass to the jth umbilical’s flotor-attachment pont as * j 1 F FU j ˆ + Y fˆ + Z fˆ (22) r = X f FU 1 j 1 j 1 FU 2 FU 3 for appropriately defined measure numbers. Then, referring to figure 2, the position vector from the first-lower-stinger attachment point to the flotor center of mass is seen to be * 1 1 1 1 1 1 1 S F1 1 1 1 1 1 r = l pˆ − l aˆ − l aˆ + f fˆ + f fˆ + f fˆ . (23) 3 2 1 2 2 1 1 22 3 1 2 3 9

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Snce * * S F1U j S F1 F FU j r = r the rth partial velocity of the flotor-attachment point U FU j *  ∂ S1 S1 F v = v +  r r  ∂ur  ∗ ∗ S F1 F FU j + r , (24) F s6 j *  S1 F1 F FU j ( ω × r ) . (25) S1 F1 The poston vectors r and r , and the angular velocty vector ω , can be expressed 1 n terms of the coordnates, q j (j=1,…,6), and the associated generalized speeds of actuator number 1. Ths means that the velocty of each attachment pont, FU , can be expressed n terms of these same j coordinates and generalized speeds so that the partial velocities defined in equation (25) are nonzero for only those six generalized speeds; i.e., for r=1,…,6. th stator-fixed coordinate system, ξˆ , (associated with Expanding equation (25) in terms of the j j i the jth umblcal) and collectng terms, the rth partal velocty for FU can be expressed as 3 S1 F U j v = ∑ ri r i=1 U j j U j j ˆ (26) v ξ , i where the scalars, vri , are simply the measure numbers in analytical (algebraic) form. In matrix form, the sx nonzero partal veloctes for the jth umbilical’s flotor-attachment point are  3  U j  j ˆ  ∑v1i ξ i   i=1   3  U ˆ j  F j  S U j   v ξ  1 v ∑ 2ii  1  i  i=1   FU j    S1 3  v  2  U ˆ j  j  F  ∑v3i ξ S U j  i   1 v  3  i=1   U j U j U j  v11 v12 v113    U j U j U j  j v v v  ˆ   21 22 23  ξ  1   U j U j U j  v v v  j   31 32 33  ˆ (27)   =   = ξ  . FU j 3  S1   U j  v j ˆ 4 v ξ   ∑ 4i i  FU j  S1   i=1  v  5    3  F   U ˆ j  S U j j 1 v v ξ  6  ∑ 5ii i   i=1   3  U j  j ˆ  ∑v6i ξ  i   i=1  10  U j UU j U j  2 v v v    41 42 43  j ξˆ   U j U j U j  v v v  3   51 52 53   U U U  j j j v61 v62 v63 

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S1 F1 5.3 Partial Angular Velocities ω r From equation (12), the angular velocity of the flotor relative to the stator can be expressed as S F 1 1 1 1 1 u sˆ + u rˆ + u qˆ − u fˆ . (28) 1ω 1 = u pˆ + u tˆ + 1 1 1 1 1 1 1 2 1 3 3 4 3 2 11 3 5 6 1 Following the procedure and using a notation analogous to that of the previous subsection, the nonzero partal angular veloctes can be expressed n the matrx form,  3  U j  j ˆ  ∑ω1i ξ i   i=1   3  U ˆ j  j ∑ω2i ξ    S F i  1ω    U j U j U j ω11 ω12 ω13 1 i=1         S F 3  1ω  j U j U j U j j ω ω ω  ˆ  2 ˆ  21 22 233  ξ  U jj    ∑ω3i ξ  1  S F i  U j U j U j     1ω  ω ω ω  j  3  i=1      =   = 31 32 33 ˆ (29) ξ  . S1 F 3  j j j   ω   U j  U U U 2 ω ω ω   4 j ˆ     ∑ω4i ξ  41 42 43 j ξˆ  S1 F i  U U U   ω    5 i=1  51 52 53  j jj j 3 ω ω ω    S1 F   3     ω  j 6  U j ˆ  U j U j U j ω ξ ω61 ω62 ω63  ∑ 5i i   i=1   3  U j  jj ˆ  ∑ω6i ξ  i   i=1  5.4 General Form for Umbilical Forces and Moments Uj U j With full stiffness and damping matrices, the jth-umbilical force, F , and moment, M , will each be expressed in terms of umbilical-elongation-, elongation-rate-, angle-of-twist-, and twist-rate- FUh j FU j component measure numbers. In vector form, the umbilical elongation is r , where FUh desigj nates the stator-fixed location of the umbilical attachment point, F , in the reference position. Let φnˆ Uj φ represent, in vector form, the rotation of the flotor, relative to the stator, from the reference position. The term nˆ s n the postve drecton of the rotaton axs, and φ is the positive angle of twist about φ that axis. Then, for the jth umbilical, the elongations; i.e., the elongation measure numbers, are F j j FUhj U j xi = r j the elongaton rates are x , the angles of twist are i ⋅ξˆ , (30) i 11

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j φ = φnˆ i φ i  j and the twist rates are φi . j ⋅ξˆ , (31) U j U j Let the bias force and moment exerted on the flotor by the jth umblcal be F and M , respectively. Next, for i=1,2,3, define U j b b U j j ˆ (32) F = F ⋅ξ , bi U j b i U j j ˆ (33) M = M ⋅ξ , bi U j F j ⋅ξˆ , (34) Fi = and U j M j ⋅ξˆ . (35) Mi = b i U j i U j i Then, in matrix form, the combined force and moment equations are as follows:  U j   U j  F1 x1      U j   U j  F2 x2    U U    j j  U  k11  k16  U  j   j  F3  x3   U j   U jj  x F  1   b1   U j   U j  x F  2   b2   U j U j  c11  c16  U   U     j j x3   F  b3     +        +   , (36) =      U j U j M    φ  1 U j U j 1   k  k    U 6166 U j j M  φ  2 K 2     U j U j M  φ   3   3  U jj U j   φ  M  U j U j 1 b1 c  c      61666 U U   j   j  φ2 M C b2     U j U j φ  M   3   b3  where the stiffness and damping elements, obtained by measurements or other means, and the stiffness and dampng matrces are ndcated by the desgnatons k, c, K, and C, respectively. The stiffness and damping element superscripts and subscripts have their obvious meanings. To use the left-hand elements of equation (36) in equation (16), these elongations and angles of twist, along with their rates, must now be reexpressed in terms of the system coordinates and generalized speeds. 5.5 Umbilical Elongations and Elongation Rates Recall equation (30), for i=1,2,3; but FUhj FU j S F11 F F1U j 1 U j U j Uhj r = r + r 12 S S S F − r − r , (37)

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where the right-hand-side terms can be expressed by S F11 1 1 1 1 1 1 r = l pˆ − l a − l a , 3 2 F F j 1 ˆ ˆ (38) 2 1 1 2 j 1 j 1 r 1 U j = x fˆ + y f + z f , FU 1 S S j 1 ˆ ˆ (39) FU 2 FU 3 j 1 j 1 r 1 U j = x sˆ + y s + z s , S 1 U and ˆ ˆ (40) S 2 S 3 U U SU j FUhj j 1 1 1 j j r = x sˆ + y s + z s 0 1 2 3 for appropriately defined coefficients. ˆ ˆ (41) 0 0 j Using equation (1), equation (37) can now be written in terms of the ξˆ coordnate system n lnearzed form F F j j i j j j j r Uhj U j = x ξˆ + x ξ + x ξ , 1 1 where  j j 1 1 ˆ ˆ (42) 2 2 3 3 1 j 1 1 1 1 1 1   j  C + y (q − qq − q ) + z q − l q + l (q + q ) x  1 F 1 3 1 U    j   j j 1 1 4 F 5 3 4 1 3 4 U  j 1 1 1 1 1 1  x  = ℜ  ⋅ C − z (q + q ) − ( ) − ( )) 2  j  2 FF 2 6 U  j   x   j 1 j x q − q − q l q + q  (43) F 1 3 4 2 3 4 U  1 j 1 1 1 1 1   3  C +(l − x )q +(y − l )(q + q ) + l qq 3 2 F  U for j j x − l − x , (44) C1 = x − FU j j j C2 = y − y FU SU and j j − z − z . (46) C3 = z FU Differentiating equation (43),6 5 F 1 2 6 3 6 U  j 1 j S 2 0 U 1 1 j + l − l − y , (45) 3 1 0 j j S 0 U 13

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 j   j 1 1 1 1 1 1 1 1 1  j x y (u1 − u33 − u4) + 5 3 4 1( 3 4) 1  FU    j   j 1 1 z u − l u + l u + u FU  j 1 1 1 1 1 1  x  = ℜ  ⋅ −z (u + u ) − x (u − u − u ) − l (u + u ) . 2  j  F 22 6 U    j 1 j 1 (47) F 1 3 4 2 3 4 U  j 1 1 1 1 1  x   l −− x u + y − l u + u l u  3  ( 2 F ) 5  U ( F 1)( 2 6) + 3 6 U  Equations (43) and (47) give the umbilical elongations and elongation rates in terms of the six independent generalized speeds. It is now required to do the same for the umbilical angles and angles of twist. 5.6 Umbilical Angles of Twist and Twist Rates Express nˆ as φ 1 g sˆ + g sˆ . (48) nˆ = g sˆ + φ 1 1 Define rotation matrix Q by  ˆ1  f  1   1  1 1 2 2 3 3  ˆ1  s 1    1  fˆ = [Q] s ,   2   1  fˆ   3   ˆ  (49) 2  1  sˆ   3  and then the linearized three-by-three rotation matrix, lQ, has elements, lQij, defined as  ˆ1  f  1 1 + q3 + q4 −q5 s  1  1 −q    1 1 1 1 fˆ = q − q − q   1 3 4 2    1 1 1 1 11   ˆ1  1    1 1  1  1 q + q  sˆ  . (50) 2 6 2    1 1 1  ˆ   5 q2 − q6 1  sˆ  q − f   3    3  Let the postsuperscript T ndcate matrx transposton and let tr lQ represent the trace of lQ. For small angles it can be shown that10  0 g3 −g2   φ −g3 0 g1   g2 −g1 0  14 T  lQ − lQ = . (51)  1 2/ (1 + tr lQ)) 

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Substtuton for lQ, as defined in equation (50), into equation (51) and simplification yield 1 1 1 g1 = ⋅(q2 + q6) , (52) φ 1 1 g2 = φ and 1 1 1 1 ⋅ q , (53) 5 − q − q ) . (54) g3 = − ⋅(q1 3 4 φ 1 Substituting from equations (52)–(54) into equation (48) and transforming into the sˆ coordnate sysi tems by use of lQ yield the following linearized expression for the flotor rotation: 1 1 1 1 1 11 1 1 1 φ nˆ =(q + q )sˆ + q s −(q − q − q )s . ˆ ˆ (55) φ 2 6 1 5 2 3 1 3 4 Use of equations (31) and (48) leads at last to the linearized forms for angular position and rotation rate:  j   φ1     j   φ  = ℜ   2  j   j   1 1  q2 + q6  1  q  (56) 5 11 1 1  φ3  −(q1 − q3 − q4)    and   j   φ1      j   φ  = ℜ   2  j   j    1 1  u2 + u6  1  u  . (57) 5 1 1 1  φ −− u − u − u  3   ( 1 3 4) The jth umbilical portion of the GAF contribution due to the flotor for the rth generalzed speed can now be expressed in terms of the coordinates and independent generalized speeds: 15

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FU j j F S1 U j S F T  6  U j ∑v  r1   r=1   6  U  j ∑vr2   U  j   F1 r=11      U  6 j  U  F2 j   ∑vr3   U j   r=1   F3  U j (58) Qr = v ⋅ F + ω ⋅ M =     , r r where  U j   U j  F1 x1      U j   U j  F2 x2    U U    j j  U  k11  k16  U  j   j  F3  x3  6 U j  U  M  j 1 ∑ωr1     U j r=1  M  2  6    U j  U j  M  ∑ω  3   r2  r=1   6  U j  ω  ∑∑ r3   r=1   U j   U j  x F  1   b1   U j   U j  x F  2   b2   U j U j  c11  c16  U   U     j j x3   F  b3     +        +   , (59) =      U j U j M    φ  1 U j U j 1   k  k    U  61 66  U j j M  φ  2 2     U j U j M  φ   3   3  U j U j   φ  M  U j U j 1 b1 c  c       61 66  U U   j   j  φ2 MM b2     U j U j φ  M   3   b3  and the elongations, elongation rates, angles of twist, and twist rates are as defined in equations (43), (47), (56), and (57). The model is now ready for calibration and computer implementation. 16

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1. IMPLEMENTATION IN AUTOLEV A nonlinear, rigid-body model for the untethered ARIS; i.e., without umbilicals, with full actuator dynamics (masses and inertias), was previously developed using OnLine Dynamics’ Autolev, a DOS-based interpreter specifically designed to solve dynamics problems using Kane’s method.11 The umbilical contributions developed above (eqs. (58) and (59)) were incorporated into that model by appropriately adding those contributions to the untethered-model GAFs. In Autolev, this required entering the locations of the umbilical attachment points, the orientations of the stator-fixed coordinate systems associated with the umbilicals, and the umbilical forces and moments determined above. The geometric data required to include the umbilicals were determined from CAD models, such as in figure 3. Since umbilical stiffness and damping values were not available, representative estimates were entered and varied as needed for the process of model verification (described in section 7). Figure 3. CAD-based technique for determining umbilical attachment locations. After adding necessary terms to the Autolev model of ARIS, the code was compiled in the Autolev workspace. Autolev then auto-generated a C-program file containing the equations of motion and a numerical integration routine for their solution. Results of executing the C-program were plotted usng Mcrosoft Excel. 17

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1. MODEL VERIFICATION Numerous simulations were run to verify the model for special cases. The basic procedure was to reduce the actuator stffness to very small levels and apply selected test forces (moments) through (about) the flotor center of mass. Corresponding simulations were compared between the full nonlinear model and appropriate (translational or torsional) single-degree-of-freedom truth models. Representative comparison checks are described in sections 7.1–7.8. For each of the following comparisons, a single umbilical is assumed to be attached between the stator and the flotor center of mass. Umbilical biases are assumed to be zero. The umbilical coordinate system, ξˆ , is assumed to be aligned with the flotor-fixed coordinate system, , when the flotor is in the i fˆ i reference position. Stiffness and damping terms that couple translation and rotation are all assumed to be zero. For forces (moments) applied through (about) the flotor mass center and for small-enough actuator stiffness, the ARIS system dynamics should approximate those of a simple second-order spring-massdamper system (fig. 4). k Translational Stiffness c Translational (Viscous) Damping c m Mass F Applied Force k x Translational Displacement kx m x m F F • cx Figure 4. Translational single-degree-of-freedom truth model. The flotor mass is assumed to be 55.6 slugs (1,790 lbm, 811.94 kg), and the central principal moments of inertia, 166.8 slug-ft2 (5,371 lbm-ft2, 226.34 kg-m ). The umbilical stiffness matrix is 2 assumed to be diagonal with each diagonal element set at 30 (lbf/ft for translational elements and lbf-ft/rad for rotational elements). The damping matrix is also assumed diagonal with all diagonal terms numerically equal but varied as needed to control the damping ratios, depending on the particular simulation. 18

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7.1 Onboard Impulsive-Disturbance Force, No Damping For the first test of the ARIS model, an impulsive disturbance force was applied directly to the flotor mass center along the ξˆ direction, with zero damping. The disturbance was also applied 2 to a simple spring-mass truth model with the same mass and stiffness. The impulse was approximated by a rapdly decayng exponental functon: −d t ˆ (60) F = Ae ξ , 2 where d=100 s–1 and A=100 lbf (444.82 N). The two systems had identical sinusoidal responses (fig. 5) at the expected natural frequency of 0.7346 rad/s (0.1169 Hz). 0.03 ARIS and Truth Model Results 0.02 0.01 0 Position (ft) – 0.01 – 0.02 – 0.03 0 5 10 15 20 25 30 Time (s) Figure 5. Translational displacement due to onboard impulsive-disturbance force. 7.2 Onboard Sinusoidal-Disturbance Force, With Damping In a second test, a sinusoidal disturbance force with an amplitude of 3 lbf (13.34 N) and a frewas applied to the flotor mass center, again in the ξˆ direction, using various frequency quency of ω 2 ratos, ω/ωn, and dampng factors, ζ. As before, the disturbance was also applied to a corresponding truth model. The two system responses were identical (figs. 6–8). 19

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0.06 ARIS and Truth Model Results 0.04 0.02 0 Displacement (ft) – 0.02 – 0.04 – 0.06 0 5 10 15 20 25 Time (s) Figure 6. Translational displacement due to onboard sinusoidal-disturbance force ω with = 1 and ζ = 1. ωn 0.012 0.01 ARIS and Truth Model Results 0.008 0.006 0.004 Displacement (ft) 0.002 0 – 0.002 – 0.004 0 5 10 15 20 Time (s) Figure 7. Translational displacement due to onboard sinusoidal-disturbance force ω with = 5 and ζ = 1. ωn 20

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0.2 0.15 ARIS and Truth Model Results 0.1 0.05 0 Displacement (ft) – 0.05 – 0.1 – 0.15 – 0.2 0 5 10 15 20 25 Time (s) Figure 8. Translational displacement due to onboard sinusoidal-disturbance force ω with = 1 and ζ = 0 25.. ωn 7.3 Onboard Impulsive-Disturbance Moment, No Damping To check the ARIS model rotational response, an impulsive disturbance moment was applied ξˆ directly to the flotor at its mass center about the direction with zero damping. The disturbance was 2 also applied to a simple rotational spring-mass truth model, with the same inertia and rotational stiffness. Again, the impulse was approximated by a rapidly decaying exponential function: −dt ˆ M = Ae ξ , (61) 2 where d=100 s–1 and A=100 lbf-ft/s (135.58 N-m/s). The two systems had identical sinusoidal responses (fig. 9) at the expected natural frequency of 0.4241 rad/s (0.0675 Hz). 21

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0.02 ARIS and Truth Model Results 0.015 0.01 0.005 0 Angle (rad) – 0.005 – 0.01 – 0.015 – 0.02 0 20 40 60 80 100 120 Time (s) Figure 9. Rotational displacement due to onboard impulsive-disturbance moment. 7.4 Onboard Disturbance Moment, With Damping To test the damped rotational response, a sinusoidal disturbance moment with amplitude ξˆ 0.09 lbf-ft/s (0.122 N-m/s) and frequency, ω, was applied at the flotor mass center, about the direc- 2 tion, using various frequency ratios, ω/ωn, and dampng factors, ζ. The disturbance was also applied to a corresponding truth model. Again, the two system responses were identical (figs. 10–12). 22

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0.002 ARIS and Truth Model Results 0.0015 0.001 0.0005 0 Angle (rad) – 0.0005 – 0.001 – 0.0015 – 0.002 0 5 10 15 20 25 30 35 40 Time (s) Figure 10. Rotational displacement due to onboard sinusoidal-disturbance moment ω with = 1 and ζ = 1. ωn 0.0004 0.0003 ARIS and Truth Model Results 0.0002 0.0001 Angle (rad) 0 – 0.0001 – 0.0002 0 5 10 15 20 25 30 Time (s) Figure 11. Rotational displacement due to onboard sinusoidal-disturbance moment ω with = 5 and ζ = 1. ωn 23

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0.008 ARIS and Truth Model Results 0.006 0.004 0.002 0 Angle (rad) – 0.002 – 0.004 – 0.006 – 0.008 0 10 20 30 40 50 Time (s) Figure 12. Rotational displacement due to onboard sinusoidal-disturbance moment ω with = 1 and ζ = 0 25.. ωn 7.5 Off-Board Translational Disturbance In normal operation, the flotor will also be subject to off-board disturbances, which are transmitted via the ARIS umbilicals and actuators. With negligible actuator stiffness, the system response should match that of a corresponding truth model (fig. 13). A translational sinusoidal disturbance of amplitude 0.25 in (0.635 cm) was applied to the stator (base) in the ξˆ direction with the ARIS umbilical oriented 2 so that its stator attachment point was collinear with the flotor mass center, along the ξˆ direction. (The 2 flotor attachment point was assumed to be at the mass center, as before.) Various frequency ratios, ω/ωn, and dampng factors, ζ, were used with the flotor-to-stator relative displacements compared to those of the truth model. The responses of the two systems were identical (figs. 14–16). 24

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M k 2 C Base y k 2 x Figure 13. Translational single-degree-of-freedom truth model, with base motion. 0.00025 0.0002 0.00015 0.0001 Angle (rad) 0.00005 0 – 0.00005 0 5 10 ARIS and Truth Model Results 15 20 25 30 Time (s) Figure 14. Translational displacement due to off-board sinusoidal position disturbance ω with = 0 1. and ζ = 0 707.. ωn 25

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0.05 ARIS and Truth Model Results 0.04 0.03 0.02 0.01 0 – 0.01 Displacement (ft) – 0.02 – 0.03 – 0.04 – 0.05 0 5 10 15 20 25 30 Time (s) Figure 15. Translational displacement due to off-board sinusoidal position disturbance ω with = 1 and ζ = 0 25.. ωn 0.12 0.1 0.08 0.06 ARIS and Truth Model Results 0.04 Displacement (ft)0.02 0 – 0.02 – 0.04 0 2 4 6 8 10 12 14 Time (s) Figure 16. Translational displacement due to off-board sinusoidal position disturbance ω with = 10 and ζ = 0 707.. ωn 26

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7.6 Off-Board Rotational Disturbance A rotational sinusoidal disturbance, of amplitude 0.001 rad (0.0573º) was applied to the stator (base) about the ξˆ direction with the ARIS umbilical attached and oriented as in the previous section. 2 Various frequency ratios, ω/ωn, and dampng factors, ζ, were used with the flotor-to-stator relative rotations compared to those of the truth model. The responses of the two systems were identical (figs. 17–19). 0.006 0.005 0.004 ARIS and Truth Model Results 0.003 0.002 Angle (rad) 0.001 0 – 0.001 – 0.002 0 5 10 15 20 25 Time (s) Figure 17. Rotational displacement due to off-board sinusoidal rotation disturbance ω with = 10 and ζ = 0 707.. ωn 27

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0.000015 0.00001 0.000005 ARIS and Truth Model Results 0 Angle (rad) – 0.000005 – 0.00001 – 0.000015 0 10 20 30 40 50 60 70 80 90 100 Time (s) Figure 18. Rotational displacement due to off-board sinusoidal rotation disturbance ω with = 0 1. and ζ = 0 707.. ωn 0.0025 ARIS and Truth Model Results 0.002 0.0015 0.001 0.0005 0 Angle (rad) – 0.0005 – 0.001 – 0.0015 – 0.002 – 0.0025 0 10 20 30 40 50 60 Time (s) Figure 19. Rotational displacement due to off-board sinusoidal rotation disturbance ω with = 1 and ζ = 0 25.. ωn 28

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7.7 Comparisons Using Nondiagonal Stiffness and Damping Matrices Suppose now that the stiffness and damping matrices, [K] and [C], respectvely, used for the ˆ'  '  above simulations are changed by selecting a new coordinate system, ξ , rotated through ℜ from the prevous coordnate system, ξˆ : i  ˆ'   ˆ  ξ ξ  1   1       ˆ'  '  ˆ  i    1  s 1    '    1 (62) ξ  = ℜ  ξ  = ℜ ℜ s  . 2   2      ˆ'   ˆˆ  ξ ξ  3   3       2  1  s   3  Then the upper left and lower right block diagonal three-by-three submatrices, [K11] and [K22], of the (ntally dagonal) stffness matrx, [K], become, respectvely, T  '   '      K11 = ℜ ' K11 ℜ (63)       and T  '   '      K22 = ℜ ' K22 ℜ . (64)       Smlar relatonshps hold for the dampng matrx, [C], and its corresponding submatrices. If the same disturbance force or moment is now applied as in any of the preceding simulations, then the resulting system motion should remain unmodified, which was shown to be the case.12 Ths ndcates that the ARIS model correctly handles cross-coupling among axes; i.e., full stiffness and damping submatrices, [K11] , [K22], [C11] , and [C22]. 29

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1. CONCLUSION This TM has shown how multiple ARIS umbilicals, modeled as massless three-dimensional Hookean springs in parallel with three-dimensional viscous dampers, are to be included in the nonlinear Kane’s model of ARIS. The umbilical force, moment, partial-velocity, and partial-angular-velocity terms were determined for the flotor contributions to the GAFs in Kane’s dynamical equations. Next, these modifications were incorporated into the ARIS Autolev simulation code. Simplifying assumptions were made to permit the comparison of the Autolev model’s input responses to those of simple second-order truth models for selected test inputs. Numerous simulations, involving onboard and off-board force and moment disturbances, showed that the ARIS-model responses match those of the truth models. This indicates that the proposed model, when properly calibrated against experimental data, will provide a high-fidelity representation of ARIS for simulation and controller-design purposes. 30

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REFERENCES 1. Bushnell, G.S.; Anderson, T.M.; Becraft, M.D.; and Jacot, A.D.: “Active Rack Isolation System Development for the International Space Station,” AIAA 1997–1203, April 1997. 2. Bushnell, G.S.; and Becraft, M.D.: “Microgravity Performance Flight Characteristics of an International Space Station Actve Rack Isolaton Prototype System,” IEEE–0–7803–5276–9, September 1999. 3. Hampton, R.D.; and Beech, G.S.: “A ‘Kane’s Dynamics’ Model for the Active Rack Isolation System,” NASA/TM—2001–2110063, 36 pp., Marshall Space Flight Center, AL, June 2001. 4. Kane, T.R.; and Levinson, D.A.: Dynamics: Theory and Applications, 379 pp., McGraw-Hill, Inc., New York, 1985. 5. Beech, G.S.; Hampton, R.D.; and Rupert, J.K.: “A ‘Kane’s Dynamics’ Model for the Active Rack Isolation System: Part Two: Nonlinear Model Development Verification and Simplification,” NASA/TM—2004–213552, 32 pp., Marshall Space Flight Center, AL, November 2004. 7. Beech, G.S.: A High Fidelity Model for the Active Rack Isolation System, Master’s Thesis, 69 pp., The University of Alabama in Huntsville, Huntsville, AL, December 2000. 6. Thomson, W.T.: Mechanical Vibrations, 2nd Ed., Prentice-Hall, Inc., New York, 1953. 8. Rupert, J.K.; and Hampton, R.D.: “An Improved Umbilical Model for The Active Rack Isolation System,” AIAA 2000–0573, January 2000. 9. Johnson, T.L.; and Tolson, H.R.: “Development of a Simulation Capability for the Space Station Actve Rack Isolaton System,” Specification NASA/CR—1998–206942, Langley Research Center, March 1998. 10. Greenwood, D.T.: Principles of Dynamics, 2nd Ed., Prentice-Hall, Inc., New Jersey, 1988. Online: Theory and Implementation with Autolev™, 11. Kane, T.R.; and Levinson, D.A.: Dynamics 408 pp., Sunnyvale, CA, 1996. 12. Rupert, J.K.: An Umbilical Model for the Active Rack Isolation System, Masters Thesis, The University of Alabama in Huntsville, 2001. 31

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Form Approved REPORT DOCUMENTATION PAGE 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 to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503 1. AGENCY USE ONLY (Leave Blank) 2. REPORT DATE February 2005 4. TITLE AND SUBTITLE 3. REPORT TYPE AND DATES COVERED Techncal Memorandum 5. FUNDING NUMBERS A “Kane’s Dynamics” Model for the Active Rack Isolation System Part Three: Addition of Umbilicals to the Nonlinear Model 6. AUTHORS J.K. Rupert,* R.D. Hampton,** and G.S. Beech 7. PERFORMING ORGANIZATION NAME(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) Natonal Aeronautcs and Space Admnstraton Washington, DC 20546–0001 11. SUPPLEMENTARY NOTES 8. PERFORMING ORGANIZATION REPORT NUMBER M–1138 10. SPONSORING/MONITORING AGENCY REPO NUMBER NASA/TM—2005–213848 Prepared by the Engineering Systems Department, Engineering Directorate *Dynetics, Inc., **United States Military Academy 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified-Unlimited Subject Category 88 Avalablty: NASA CASI 301–621–0390 13. ABSTRACT (Maximum 200 words) 12b. DISTRIBUTION CODE In the late 1980s, microgravity researchers began to voice their concern that umbilical-transmitted energy could significantly degrade the acceleration environment of microgravity space science experiments onboard manned spacecraft. Since umbilicals are necessary for many experiments, control designers began to seek ways to compensate for these “indirect” disturbances. Hampton, et al., used the Kane’s method to develop a model of the active rack isolation system (ARIS) that includes (1) actuator control forces, (2) direct disturbance forces, and (3) indirect, actuator-transmitted disturbances. Their model does not, however, include the indirect, umbilical-transmitted disturbances. Since the umbilical stiffnesses are not negligible, these indirect disturbances must be included in the model. Until the umbilicals have been appropriately included, the model will be incomplete. This Technical Memorandum presents a nonlinear model of ARIS with umbilicals included. Model verification was achieved by utilizing two commercial-off-the-shelf software tools. Various forces and moments were applied to the model to yield simulated responses of the system. Plots of the simulation results show how various critical points on an ARIS-outfitted international standard payload rack behave under the application of direct disturbances, indirect disturbances, and control forces. Simulations also show system response to a variety of initial conditions. 14. SUBJECT TERMS ARIS, umblcals, dynamcs, control, math model 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION OF REPORT OF THIS PAGE Unclassified Unclassified NSN 7540-01-280-5500 32 15. NUMBER OF PAGES 44 16. PRICE CODE 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF ABSTRACT Unclassified Unlimited Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. 239-18 298-102

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National Aeronautics and Space Administration IS04 George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama 35812