Introduction Measurements of the acceleration environment on the U.S. Space Shuttle have demonstrated that the on-orbit environment will exceed the requirements for micro-gravity experiments [1]. To meet the required level of microgravity isolation many space-science experiments will likely require some attenuation of the nominal ISS acceleration environment. The expected acceleration levels over a mid-range of frequencies above 0.01 Hz and below 10 Hz. are particularly high occasionally reaching milli-g levels [1]. Three orders-of-magnitude attenuation of the induced accelerations on the experiment platform, with frequency roll-off of 20 db/decade over a range flom 0.01 Hz. to 10 Hz, has been established as a design requirement for a vibration isolation system [2], To meet this relatively stringent requirement it has been established that active vibration control is necessary [1]. The Glovebox integrated Microgravity Isolation Technology (g-LIMIT) is designed to isolate experiments from the medium flequency (>0.01 Hz) vibrations on the ISS, while passing the quasi-static (<0.01 Hz) accelerations to the experiment [21. The acceleration-attenuation capability of g-LIMIT is limited primarily by two factors: (1) the character of the umbilical required between the g-LIMIT base (stator) and the g-LIMIT experiment platform (flotor), and (2) the allowed stator-to-flotor rattlespace. A primary goal in g-LIMIT design was to isolate at the individual experiment, rather than entire rack level; ideally g-LIMIT isolates only the sensitive elements of an experiment. This typically results in a stator-to-flotor umbilical that can be greatly reduced in size and in the services it must provide. In the current design, g-LIMIT employs three umbilicals to provide experiments with power, and with data-acquisition and control services [3]. In orderto designcontrollersfor g-LIMIT it wasnecessaryto developanappropriate dynamicmodelof the system. The designmethodsemployedin the presentpaperrequirea linearizedsystemmodelin state-spaceform. A six-degree-of-fieedom(6-DOF) statemodel, augmentedwith absoluteaccelerationstates,wasdevelopedin aform appropriateforanoptimal controldesignfor g-LIMIT [I]. A setof representativeparametersusedin thestatespacemodel forcontrollerdesignisprovidedinthefollowing section. g-LIMIT State Space Model The linearized state-space equations of motion for g-LIMIT were used to develop lincar optimal controller designs [1]. To construct the state space model a set of representative flotor and umbilical parameters, shown in Table 1, were used in the controller design study. There are three umbilicals included in this model of g-LIMIT. The translational and rotational stiffness matrices for each umbilical were assumed to be diagonal along an umbilical-fixed set of coordinate directions. These diagonal stiffness values are included in Table 2. Similarity transformations of these diagonal matrices were performed assuming a coordinate transformation flom each local umbilical-fixed reference fi'ame to the stator-fixed frame. First, a coordinate rotation about the stator-fixed +Z axis of 120 deg and 240 deg was performed to align umbilcal #2 and #3, in their respective home locations. Then, for each umbilical, a 20 deg rotation about each coordinate axis was used to represent an arbitrary misalignment of the diagonal-stiffness directions to the stator-fixed directions.. The translational and rotational damping matrices were assumed to be proportional to the stiffness matrices with a damping ratio of 3% used for all of the vibrational modes. The resulting umbilical stiffness and damping matricies are given in reference [1]. They are not included in this paper but can easily be computed via coordinate transformations of the diagonal stiffness terms given in Table 2. All stiffness and damping translation/rotational cross-terms,i.e. K,,., K,.,, Q, andC,,, were considered to be zero. In addition to the parameters listed in Table #1 the actuator cun'ents were set to initial bias values. These bias currents were required to produce a bias force and moment to move the flotor from its assumed relaxed position to the home location. The flotor relaxed-position was assumed to be 2 mm fiom the home- position and misaligned by approximately 2 deg. about each stator-fixed coordinate axis. This resulted in the following set of bias current values; I_=-0.264A, It¢ =-0.159 A, and lt_= 0.123 A. Table1- g-LIMIT Parameters Parameter Symbol Value FlotorMass III 15.12 kg FlotorMomentsofInertia 0.50 kg m- 0.62 kg me 0.18 kg me ]__ FlotorProductsofInertia le-4 kg m 2 .W -le-4 kg m- IA. -8e-4 kg m2 Umbilical Locations (F) [0.0 -0.12 -0.032] m (3 Umbilicals) [0.1 0.06 -0.032] m [-0.1 0.06 -0.032] m Actuator Cun'ent Vectors (S) {S} ^ [0.0 0.0 1.0] L, (6 Actuator Coils) [-1.0 0.0 0.0] [o.o o.o 1.o1 [0.5 0.866 0.01 [0.0 0.0 1.0] [0.5 -0.866 o.0] Actuator Magnet B-Field Vectors _t'! ^ [0.0 1.0 0.0] (F) [o.o 1.0 O.Ol (3 Actuator Magnets) [0.866 -0.5 0.0] [0.866 -0.5 0.0] [-o.866-0.5 O.Ol [-0.866-0.5 0.01 Actuator Constant (_ Bi) 1.0 N/Amp Table 2 - Diagonal Stiffness Parameters Translational Rotational [N/ml [N-m/rad] Umbilical X-axis 25.0 3.0 Umbilical Y-axis 25.0 3.0 I Umbilical Z-axis ] 50.0 3.0 H2 Control Design An optimal controller design using a frequency weighted linear quadratic regulator (LQR) along with a full order Kalman filter was chosen as a candidate design methodology. This facilitates the design of robust controllers for the case of multi-input multi-output (MIMO) multi- degree-of-freedom (MDOF) systems. Before proceeding to the controller design for g-LIMIT a brief summary of recent research into the implementation of the H2 methodology to a microgravity isolation problem for a single-degree-of-freedom (SDOF) system will be presented. SDOF Case Study using Frequency Weighted H2 A SDOF case study has demonstrated the utility of the frequency weighted LQR approach applied to the microgravity vibration isolation problem [3]. The design of this class of linear optimal controllers requires a suitable choice of the fiequency weighting design filters[3]. The inherent kinematic coupling of the state variables complicates the choice of appropriate weighting functions. Indeed, certain combinations of state frequency weighting can lead to conflicting requirements for the controller optimization. This may result in poorly conditioned regulator and/or estimator Ricatti equations [3]. Recent results of a SDOF controller design study has developed a method that provides guidance in selecting state weighting filters [3]. In this research the frequency weighting filters have been related to the weighting _ of the pseudo- sensitivity function S and the weighting Vr of the pseudo-complementary-sensitivity function T. This technique leads to an intuitive weighting filter selection process for loop-shaping. This intuition arises from the fact that the performance index (for cheap control) can be expressed in terms of the pseudo-sensitivity and pseudo-complementary-sensitivity functions for a system having, as output, the flotor acceleration, and input, the stator acceleration (indirect disturbance). Thuschoosingappropriateweightingstrategiesfor S and T leads the designer to consider corresponding weighting-filter choices yielding a rational approach to filter selection. Equations (1) through (3), obtained fiom reference [3], show the relationship of S and T to the quadratic performance index J and the relationship of the state weights W_ (weighting on a;,), WB (weighting on x_,), and Wc (weighting on ._i) to S and T. (Note: In this section the vector notation has been omitted since this development pertains to the SDOF case) Assuming that J does not contain control weighting (i.e. cheap control) one has the following equation for the integrand of the quadratic performance index Ij [3] .... T" g,g.T]ai,,(s) Eq (1) lj:ai,,(.; ) Is* V5, V_.. s -1-- * Thus 1/, and hence J, is determined by the sum of a weighting _, on the relative acceleration (i.e. stator relative to flotor) and a weighting Vr on the absolute acceleration of the flotor. It was also shown in reference [3] that the pseudo-sensitivity function may be expressed in terms of the state weightings as: I Vs.=[(_-/* (_) ( s_-i (_B ) ] 5, Eq. (2) • LtS-) and the pseudo complementary-sensitivity function may be expressed in terms of the state weightings as: = Eq.(3) Thus, the above equations provide a basis for the choice of the state weighting filters (Wa, W_,andW c ) when considering the requirement trade-off between minimizing relative and absolute acceleration. The next section will discuss this trade-off in acceleration attenuation requirementsasit appliesto microgravityisolationcontrollerdesign.This will leada rational approachforselectedtheweightingfilters. Micro,wal'i O, Vibr_,lion Isolaliotz Design Criteria The microgravity vibration isolation controller design problem is summarized by (1) consideration of the rattlespace requirements (i.e. bumping of flotor against stator) at low frequencies (<0.01 Hz) and, (2) attenuation of the absolute acceleration of the flotor at mid-range frequencies (Dom 0.01 Hz up to 10.0 Hz), and (3) a "turning oft"' of the control effort at some high frequency say around 20-30 Hz. The rattlespace requirement conesponds to a tracking of the flotor's motion relative to the stator for low frequencies. Since the stator motion will be significant at low frequencies (i.e. ISS motion is very large at orbital frequency), the relative acceleration between the stator and flotor should have a unit closed-loop transmissibility to indirect disturbances over the low frequency range to avoid flotor to stator contact. To meet the science requirements the absolute flotor acceleration transmissibility should be attenuated by three orders-of-magnitude with frequency roll-off of 20 db/decade over a range from 0.01 Hz. to 10 Hz. Above this frequency the controller should "turn-off' and the closed-loop transmissibility should rejoin the open-loop transmissibility. This will avoid excitation of any high frequency vibrational modes. The above design criteria should be meet while limiting the actuator control effort to less than 40 amps/mirco-g over the entire frequency range [3]. Using the above design criteria along with Equations (1) through (3) a rational approach to selecting the state weighting filters can be developed: [3]. A summary of this criteria from reference [3] follows: State Weighting Design Criteria (1) Use the relative-position and relative-velocity state weighting to shape the low fiequency closed-loop acceleration transmissibility to control relative acceleration. (2) Use the acceleration state weighting to shape the closed-loop acceleration transmissibili_y to attenuate mid-range frequencies. (3) All state weighting filters should be chosen to cause adequate roll-off of S and T at high frequencies, thus forcing the control to "turn-off". The state frequency weighting LQR approach described above was applied to the SDOF system in a case study to determine the performance of the H2 methodology [3]. Four different scenarios involving selection of the state weighting filters, consistent with the approach described herein, were investigated in the study. Numerous observations of the effects of state wieghting, measurement noise, and process noise parameters on the fiequency shaping of the acceleration transmissibility as they relate to the regulator and observer designs are described in this reference. Explainations are provided that relate the observations to the effect on S and T. These case studies provides good examples of the logical application of the methodology and demonstrate good performance in meeting the the design. Turn nov,, to the application of this state weighting filter selection process to the 6DOF vibration control for g-LIMIT using the H2 methodology. ,_-LIMIT 6DOF Case Study A rational approach to state frequency weighting filter selection for the H2 control design method, summarized in the previous section, has been developed, justified, and demonstrated for the SDOF system in the design case studies [3]. This method will be applied to the vibration isolationof6DOFsystemfor g-LIMIT in anattempttoascertainthefeasibilityof extendingthis techniqueto MDOF controllerdesign.Accelerationresponsesto rotationalandtranslational, direct and indirect disturbances will be investigated. Closed-loop system acceleration transmissibility will be compared to the open-loop responses to demonstrate fulfillment of the design criteria for the nominal plant characteristics. The system robustness to modeling errors will be analyzed by investigating the effects of changes in the umbilical stiffness on closed-loop performance. Measurentent Selection Feed back of the absolute acceleration of the flotor will be used for the 6DOF controller designs. As stated in reference [3], any controller which uses only acceleration feedback to attenuate indirect disturbances will cause attenuation of direct disturbances. This is a result of the increase effective mass fiom an acceleration-only feedback controller. Thus, the choice of acceleration-only feedback allows the designer to focus on attenuation of indirect disturbances while attenuation of direct disturbances will be realized as a consequence. This increase in effective mass has an additional advantage of improving the stability robustness since the damping ratio increases with mass. Addition of a separate low frequency (<0.01 Hz) relative- position controller can be used to meet the rattle-space requirement. The controller design presented herein will focus on attenuation of acceleration transmitance and will not include a separate controller to meet the rattle-space objective. Weightin,;, Filter Selection The rational for selecting the state weighting filters for the H2 optimization is to select the desired loop shaping of the pseudo-sensitivity function S and the pseudo-complementary- sensitivity function T. This desired shaping is then related to the state weighting filter selection and to the acceleration control objectives as specified by design criteria #1, #2, and #3 in the above section. To meet design criteria #1 the specification of relative-position and relative- velocity state weighting is selected to provide a unit acceleration transmissibility at low fiequencies (<0.01 Hz.). This corresponds to relatively high weighting on S at low frequencies which diminishes considerably at mid-range and high frequencies. To meet design criteria #2 the specification of acceleration state weighting is selected to provide the required attenuation of the acceleration transmissibility at mid-range frequencies (between 0.01 Hz. and 10 Hz.). This conesponds to low weighting on T at low frequencies which increases over the mid-range and diminishes considerably over high frequencies. Design criteria #3 is accomplished by selecting all state weightings to attenuate S and T over the high frequency range. With these objectives in mind a good choice of state weightings would result in a low-pass shaping of S with corner fi'equency about 0.01 Hz., and a band-pass frequency shaping of S with pass band over the mid- range frequencies. Alternatively, an integrating-type frequency shaping of S could be used in lieu of the low-pass shaping since the desired corner frequency is very low. The advantage would be that weighting of S over quasi-static frequencies would be increased substantially. This would aide the controller in meeting criteria #1 while additionally creating fiequency separation of the inherently conflicting criteria #1 and #2. In the SDOF case studies [3] the above strategy for selecting the state weighting was used in controller design scenarios #3 and #4. Relatively high weighting on T was used in these cases in which case it was found that the regulator dominated the closed-loop response (i.e. the