Citation
Reverse Kinematic Analysis and Uncertainty Analysis of the Space Shuttle Aft Propulsion System (APS) Pod Lifting Fixture

Material Information

Title:
Reverse Kinematic Analysis and Uncertainty Analysis of the Space Shuttle Aft Propulsion System (APS) Pod Lifting Fixture
Creator:
BRINK, JEFFREY S. ( Author, Primary )
Copyright Date:
2008

Subjects

Subjects / Keywords:
Coordinate systems ( jstor )
End effectors ( jstor )
Geometric angles ( jstor )
Kinematics ( jstor )
Mechanical measurement ( jstor )
Mechanical systems ( jstor )
Orbital mechanics ( jstor )
Orbiters ( jstor )
Space mechanics ( jstor )
Tolerance for uncertainty ( jstor )

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Jeffrey S. Brink. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
5/1/2005
Resource Identifier:
436098675 ( OCLC )

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Full Text












REVERSE KINEMATIC ANALYSIS AND UNCERTAINTY ANALYSIS
OF THE SPACE SHUTTLE AFT PROPULSION SYSTEM (APS) POD LIFTING
FIXTURE















By

JEFFREY S. BRINK


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Jeffrey S. Brink

































This thesis is dedicated to the Kennedy Space Center workers who have done their best to
help this effort succeed, in hopes of making operations better and safer.















ACKNOWLEDGMENTS

I would like to thank NASA Orbiter Handling's Rob Summers and United Space

Alliance (USA) Orbiter Handling's Glenn Roberts for their invaluable help. As the

experts on this hardware and installation process, they responded promptly and patiently

to my constant barrage of questions. Their willingness to do anything they can to make

operations safer is commendable. They shoot a pretty good game of pool, too.

Boeing Orbiter Handling's Will Judd was also very helpful. He answered several

technical questions and verified the accuracy of some information I had found on my

own.

NASA Payload Mechanical Engineering's Doug Lenhardt and NASA Mechanical

Design Engineer Paul Schwindt were instrumental in the Pro/E portion of this study.

Doug and Paul answered questions that helped my knowledge of Pro/E grow from

beginner level to intermediate.

After I experienced quite a bit of difficulty getting the integrated C program to run,

NASA Senior Software Engineer Dan Nieten was kind enough to teach me several very

helpful debugging techniques. These techniques enabled me to figure out what was

wrong, and how to make it work.

NASA Thermal Protection Systems' Lisa Huddleston provided guidance on

performing my literature review and gave tips on how to use numerical methods to

reduce error in the integrated program. I also frequently consulted Lisa about details of









this study and (even though robotics is not her field of expertise) she never seemed to run

out of good questions.

I would also like to thank my supervisory committee for their contribution. Dr.

Carl Crane III (my supervisory chair) was particularly helpful. The concepts taught in his

textbook formed the backbone to this solution method. Dr. John Scheuller and Dr. Ashok

Kumar provided good insight as this thesis reached its conclusion.

NASA Launch Accessories' Kristina Morace and NASA Orbiter Handling's Ryan

Holmes checked the technical content of this thesis and helped make it more clear and

understandable.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ......... .. .. ....................................... .... .... .............. viii

LIST OF FIGURES ............................... ... ...... ... ................. .x

LIST OF ABBREVIATIONS ............................................................. .............. xi

A B S T R A C T .............................................. ..........................................x iii

CHAPTER

1 B A C K G R O U N D ................................................................ ....... ................ .

In tro du ctio n ............................................ ...................... ... .................... .
H ardw are F am iliarization ................................................................... ......... ...........2
Installation Procedure and M ethodology ................................................. ............... 4

2 PROPOSED ALIGNMENT METHOD ...............................................................10

O v erv iew ............... ........ ..... ... ........... ... ....................................... 10
Determination of Desired Joint Angles ................................................. ................. 10
Alignment of APS Pod Attach Points with Orbiter Attach Points....................11
Adjustment Mechanism Reverse Kinematic Analysis .....................................17
Adjustm ent mechanism parameters .................................. ............... 17
Close-the-loop variable calculations ........... ... ............... ... ............ 18
Reverse kinematic analysis of a PPPS mechanism .....................................19
Rotator B ar Length Calculation...................................... ......................... 26
N om in al S solution .......... ............................................................................. . ....... 2 7

3 UNCERTAINTY ANALYSIS ........................................................ ............. 31

Off-Nominal Conditions within Tolerance.............................................31
Adjustment Mechanism Spherical Joint Socket Locations .............. ...............31
Orbiter Attach Point Locations..... ................................31
The APS Pod Fitting Locations.......................... .......... ..................... 32
Lifting Fixture Attach Point 3 Location....... ..........................................32
Lifting Fixture Adjustment at Attach Point 3........................... .............32









U uncertainty of Input V alues............................................... ............................. 33
C alculation-R elated U ncertainties.................................... ......................... .. ......... 33
H ardw are Positioning U uncertainty ........................................ ........................ 34
Uphill/Downhill Joint Offset Measurement ................ ............... ........34
Off-the-Deck/On-the-Deck Joint Offset Measurement.................................. 35
Forward/Aft Joint Offset M easurem ent............................................................35
Rotator Bar Joint Offset Measurement.....................................................35
C om pliance in Joints .................................................... ........ ....... ............36
Total Uncertainty Calculation.......................................................... 36
One Hundred Percent Covariance Method........................................................36
R oot Sum Squared M ethod ........................................ ........................... 37

4 EVALUATION OF PROPOSED ALIGNMENT METHOD...............................52

D iscu ssion of R results ........ .. ...... .. ............ ....................... ....................52
Recommendations........ ........ ....... ... .................. ........ 53
Summary of Recommendations................. .......... .......................... 56
C o n c lu sio n s........................................................................................................... 5 6

APPENDIX

A SHUTTLE COORDINATE SYSTEMS................................. ...................... 58

B REVERSE KINEMATIC ANALYSIS NOTATION......................................63

L IST O F R E F E R E N C E S ........................................................................ .....................65

B IO G R A PH IC A L SK E TCH ..................................................................... ..................66
















LIST OF TABLES


Table page

2-1 Adjustment mechanism parameters. ............................................. ............... 30

2-2 Comparison of joint offsets calculated by the program to measured using the
C A D m odel. ..................................................................30

3-1 Forward spherical joint socket tolerances. ............................... ..... ...........38

3-2 Aft spherical joint socket tolerances. .................................................................... 38

3-3 Orbiter Attach Point 1 tolerances. ........................................ ......................... 39

3-4 Orbiter Attach Point 2 tolerances. ........................................ ......................... 39

3-5 Orbiter Attach Point 3 tolerances. ........................................ ......................... 40

3-6 APS pod Attach Point 1 tolerances. .............................................. ............... 40

3-7 APS pod Attach Point 2 tolerances. .............................................. ............... 41

3-8 APS pod Attach Point 3 tolerances. .............................................. ............... 41

3-9 Lifting fixture Attach Point 3 tolerances ....................................... ............... 42

3-10 Lifting fixture Attach Point 3 adjustment. .................................... .................42

3-11 M isalignm ent of the rotator bar base .................................................. .. .. ............ 43

3-12 Com puter program uncertainty. ........................................ ......................... 43

3-13 Forward adjustment mechanism uphill measurement uncertainty.........................44

3-14 Aft adjustment mechanism uphill measurement uncertainty ................................45

3-15 Forward adjustment mechanism off-the-deck measurement uncertainty ...............45

3-16 Aft adjustment mechanism off-the-deck measurement uncertainty.......................46

3-17 Fwd and aft adjustment mechanism measurement uncertainty in the forward
direction ................................................... .................. ..............46









3-18 Rotator bar length measurement uncertainty. .................................. .................47

3-19 Joint com pliance uncertainty ................................ ...................... ............... 47

3-20 Summary of Attach Point 1 uncertainty sources............................................... 48

3-21 Summary of Attach Point 2 uncertainty sources ................................................49

3-22 Summary of Attach Point 3 uncertainty sources............................................... 50

3-23 Total uncertainty using the 100% covariance method. .........................................50

3-24 Total uncertainty using the root sum squared method. .........................................51
















LIST OF FIGURES


Figure pge

1-1 A left APS pod is being removed from the space shuttle orbiter Atlantis ...............7

1-2 The twelve APS pod attach point locations, left pod shown
(right pod m irror) ................................................ .................. 7

1-3 G SE used to install an APS pod. ........................................ ........................... 8

1-4 Forward and aft adjustment mechanisms allow motion in forward/aft,
uphill/downhill, and off-the-deck/on-the-deck directions ........................................ 8

1-5 R otator bar joint axes. ............................................................. ................ .9

1-6 A left APS pod is transported by crane to Atlantis for installation .......................9

2-1 Joint offset calculation procedure....................................... .......................... 28

2-2 Three points are needed to determine each adjustment mechanism's position and
orientation (aft adjustment mechanism shown, typical of all adjustment
m mechanism s)......... ... .................................................. ............... ... 29

2-3 Adjustment mechanism joint axis vectors and link vectors ................................29

3-1 Adjustment mechanism joint axis vectors and link vectors.............. .......... 44

A -i O rbiter coordinate system ............................................. .............................. 61

A-2 Right APS pod coordinate system .......... ................. ....... ................ ..... .......... 62
















LIST OF ABBREVIATIONS


Term or acronym

Adjustment
mechanisms
APS

APS pod

Cylinder joint

FEA


GSE


Definition

Two PPPS manipulators used for OMS pod alignment.

Aft Propulsion System

Orbiter component that houses the Orbital Maneuvering
System and the aft Reaction Control System.
A joint that allows rotational and translational motion about the
same axis.
Finite Element Analysis


Ground Support Equipment


Hydraulic jack


Hypergol


KSC

Lifting fixture

Move director

NASA

Nominal solution


OMS


OPF


A force-output device used to overcome the binding condition
experienced by the aft adjustment mechanism and cause
forward/aft motion along a prismatic joint axis.
A rocket fuel that spontaneously ignites when mixed with an
oxidizer. Monomethyl hydrazine is the hypergol used by the
OMS.
Kennedy Space Center

A large structure that attaches to the OMS pod and is
manipulated by the adjustment mechanisms and rotator bar.
The USA technician that uses information from technicians and
engineers to determine the next manipulation.
National Aeronautics and Space Administration

The joint offsets required to align the lifting fixture.

Orbiter Maneuvering System, propulsion system that provides
thrust for orbital insertion, orbit circularization, orbit transfer,
rendezvous, and deorbit. [1]
Orbiter Processing Facility










Term or acronym


Orbiter





Orbiter deck


Plucker coordinates


Prismatic joint

Pro/E

RCS

Revolute joint

Rotator bar


Space shuttle


Spherical joint

USA


The Orbiter is a double-delta winged reentry vehicle capable of
carrying both passengers and cargo to low-earth orbit and back
to a controlled gliding landing. [1] The Orbiter is the only
Space Shuttle element to reach orbit. NASA's Space Shuttle
fleet consists of three orbiters: Atlantis, Endeavour, and
Discovery.
The orbiter surface that mates to the OMS pod.

Homogeneous coordinates used to describe points, lines, or
planes. Plticker coordinates can be used to simplify equations
so that calculations can be performed more efficiently.
A joint that allows translational motion only (also known as a
slider joint).
Pro/ENGINEER, a CAD software package used for modeling
and finite element analysis.
Reaction Control System, propulsion system used as the
primary flight control at altitudes greater than 70,000 feet. [1]
A joint that allows rotational motion only (also known as a
hinge joint).
A RRPRRR manipulator used for OMS pod alignment.

NASA's only manned spaceflight vehicle. The Space Shuttle is
comprised of an orbiter, external tank, and two solid rocket
boosters.
A joint that allows rotational motion in x, y, and z directions
(also known as a ball-and-socket joint).
United Space Alliance, a Boeing and Lockheed Martin joint
venture. USA is the prime contractor for space shuttle
operations.


Definition















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering

REVERSE KINEMATIC ANALYSIS AND UNCERTAINTY ANALYSIS
OF THE SPACE SHUTTLE AFT PROPULSION SYSTEM (APS) POD LIFTING
FIXTURE

By

Jeffrey S. Brink

May 2005

Chair: Carl D. Crane III
Major Department: Mechanical and Aerospace Engineering

The space shuttle Aft Propulsion System (APS) pod requires precision alignment to

be installed onto the orbiter deck. The Ground Support Equipment (GSE) used to

perform this task cannot be manipulated along a single Cartesian axis without causing

motion along the other Cartesian axes. As a result, manipulations required to achieve a

desired motion are not intuitive. My study calculated the joint angles required to align

the APS pod, using reverse kinematic analysis techniques. Knowledge of these joint

angles will allow the ground support team to align the APS pod more safely and

efficiently. An uncertainty analysis was also performed to estimate the accuracy

associated with this approach and to determine whether any inexpensive modifications

can be made to further improve accuracy.














CHAPTER 1
BACKGROUND

Introduction

The Kennedy Space Center (KSC) is NASA's operations center for the space

shuttle. This role includes mission configuration and deconfiguration, vehicle

modifications, repair, and routine maintenance (essentially everything that happens to a

space shuttle orbiter, from landing through launch). The installation of an Aft Propulsion

System (APS) pod onto an orbiter is one example of a KSC operation.

My study aimed to improve the APS pod installation operation by calculating the

joint offsets required to align the APS pod with the orbiter deck. These calculations can

be performed before the operation begins, which will reduce the operational time spent

performing the alignment. APS pod installation is classified as a hazardous operation,

which makes reducing the operational time particularly desirable. Throughout the

operation, the potential exists for a hypergol leak to develop, which could be lethal to

nearby personnel.

The Ground Support Equipment (GSE) used to perform an APS pod installation

consists of a lifting fixture, two adjustment mechanisms, and rotator bar. There are a

total of 13 joints that change position during the operation. My analysis showed that they

can be considered as three separate manipulators connected by a large structure. After

the desired position of the large structure has been ascertained, the end effector position

and orientation of each manipulator is found. A reverse kinematic analysis is then

performed for each manipulator, to determine joint angles.









An uncertainty analysis was then conducted. Both 100% covariance and root sum

squared methods were used to quantify alignment accuracy. This analysis was used to

determine whether the solution method described above is accurate enough to aid APS

pod-installation operations; and to determine which of the error sources can be eliminated

for significant improvements in accuracy.

Hardware Familiarization

Each space shuttle orbiter has two APS pods, one on each side of the vertical tail.

Each APS pod houses the Orbital Maneuvering System (OMS) and the aft Reaction

Control System (RCS). OMS is the propulsion system that provides thrust for orbital

insertion, orbit circularization, orbit transfer, rendezvous, and deorbit [1]. The RCS

propulsion system is used as the primary flight control at altitudes greater than 70,000

feet [1]. An APS pod is shown in Figure 1-1.

The APS pods were not designed to be maintenance while installed on an orbiter.

As a result, APS pods must be removed for inspections and repairs, generally after every

three or four flights. On completion of these tasks, the APS pod is installed onto the

orbiter.

The APS pod is attached to the orbiter deck at 12 locations (called attach points).

During the installation operation, all efforts are focused on aligning attach points 1, 2,

and 3. If those three attach points are aligned, then attach bolts can be installed at all

twelve attach-point locations. Attach Point 1 is the forward inboard attach point, Attach

Point 2 is the forward outboard attach point, and Attach Point 3 is the aft outboard attach

point (Figure 1-2).

The APS pod Attach Point 3 fitting must be aligned within 0.0033" (+/- 0.0018"

depending on tolerances) of the orbiter bushing or else the attach bolt cannot be installed.









The APS pod Attach Point 2 bushing resides in a slotted hole, which prevents pod-to-pod

variation and other factors from making an APS pod "not fit" onto an orbiter. The APS

pod Attach Point 2 fitting must be aligned within 0.0063" +/- 0.0048" in the non slotted

direction. Since attach points 2 and 3 are 12' 3.3" apart, this accuracy requirement is

equivalent to positioning Attach Point 3 within 0.0033" on the orbiter deck and then

orienting the APS pod to within 0.00250 of the desired position. The orbiter Attach Point

1 bushing can accommodate misalignments of up to 0.2231" along the orbiter deck plane.

The alignment process is made significantly more challenging by the presence of a

bulb seal on the bottom surface of the APS pod. The bulb seal is basically a hollow

flexible tube that forms an environmental seal when compressed against the orbiter deck.

When the bulb seal is compressed, the APS pod cannot be moved along the orbiter deck

plane without risking damage to the bulb seal (the bulb seal is more likely to tear than

slide along the deck). As a result, if the APS pod is lowered onto the orbiter deck and

discovered to be misaligned, then it cannot simply be adjusted until it is aligned. Instead,

it must be raised off the deck (until the bulb seal is no longer compressed) before it can

be adjusted and then lowered onto the orbiter deck. The ground support equipment used

to install APS pods is a lifting fixture, forward adjustment mechanism, aft adjustment

mechanism, and rotator bar (Figure 1-3).

The lifting fixture is a large structure that connects to the APS pod at APS pod

Attach Point 1, 2, 3, and 4 fittings. The lifting fixture can be adjusted at attach points 1,

2, 3 and 4 to accommodate pod-to-pod dimensional variation. As a result of this

adjustment capability, not every APS pod will have the exact same position and

orientation relative to the lifting fixture.









The forward and aft adjustment mechanisms structurally support the lifting fixture

and allow its position to be adjusted. Forward and aft adjustment mechanisms are nearly

identical. Temporary supports are mounted to the orbiter and the adjustment mechanisms

each form a spherical joint with a support. Adjustment mechanisms have three

orthogonal prismatic joints that allow motion in the forward/aft, uphill/downhill, and off-

the-deck/on-the-deck directions. The lifting fixture is connected to the adjustment

mechanisms by the forward/aft prismatic joint. Figure 1-4 shows adjustment mechanism

prismatic joint axis directions.

The center of gravity of the lifting fixture and APS pod is not directly above the

spherical joints for the duration of APS pod installation. As a result, the rotator bar is

used to control the rotation of the lifting fixture about the two adjustment mechanism

spherical joints (the two spherical joints essentially form a hinge). One end of the rotator

bar is rigidly attached to the lifting fixture and the other end is mounted to a large

stationary beam. Revolute joints are used to ensure the rotator bar does not restrict the

lifting fixture motion. Figure 1-5 shows the rotator bar's revolute joint axes. It should be

noted that none of the revolute and spherical joints can be actuated. Their role in the

alignment process is purely passive.

Installation Procedure and Methodology

The APS pod installation operation begins with the APS pod in the Orbiter

Processing Facility (OPF) transfer aisle. The lifting fixture, forward adjustment

mechanism, and aft adjustment mechanism assembly have already been attached to the

APS pod. A bridge crane lifts the APS pod and GSE to a location near the orbiter (Figure

1-6). Adjustment mechanism prismatic joints have been adjusted to provide maximum

clearance to the orbiter as the spherical joints are connected. The crane transfers the









weight of the APS pod, lifting fixture, and adjustment mechanisms to the spherical joints.

The rotator bar is then extended until it can be connected to the lifting fixture and the

alignment portion of this operation begins.

According to the APS pod operations and maintenance manual, the rotator bar is

extended until the APS pod mating surface is approximately 2.5" above the orbiter deck

and the surfaces are parallel. The lifting fixture is moved forward or aft to the "install

position" (a marking on the lifting fixture structure). Additional adjustments are made to

align attach points 2 and 3. The APS pod is then lowered to 1/8" above the orbiter deck

with mating surfaces parallel. The bulb seal is now contacting the orbiter deck but it is

not compressed. Adjustments are made again to align attach points 2 and 3. Once

aligned, the APS pod is lowered onto the orbiter deck and observers check to make sure

the inboard side of the APS pod is also seated. If the APS pod is misaligned, it must be

raised off the deck 1/8" or more, adjusted as required, and lowered back onto the deck.

Attach point bolts are installed upon successful completion of the APS pod alignment

procedure [2].

APS pod alignment is not accomplished as easily as the operations and

maintenance manual describes. It is not possible to adjust the position of one attach point

without affecting the position of the other attach points. This is particularly evident when

either Attach Point 2 or 3 has been aligned-adjustments intended to align one attach

point usually misalign the other. The person who determines the next manipulation and

commands that it be executed is known as the move director. Different move directors

have differing philosophies about whether Attach Point 2 or Attach Point 3 should be

aligned first.









Move directors also use different techniques to align an APS pod. One technique is

to align the APS pod 1/8" above the orbiter deck, then simultaneously retract the rotator

bar and actuate both adjustment mechanisms in the on-the-deck direction. Another

technique is to slightly misalign the APS pod, then allow the rotator bar to correct the

misalignment as both adjustment mechanisms are simultaneously adjusted in on-the-deck

direction. It should be noted that simultaneous motion is accomplished by manual

start/stop and velocity control.

Additionally, not all equipment operates as designed. Field experience has shown

that the forward/aft install position does not necessarily align the APS pod. The aft

adjustment mechanism's forward/aft prismatic joint was not designed with an actuator -

it was designed to "follow" the motion of the forward adjustment mechanism. However,

it binds rather than follows so a hydraulic jack is used to force it to follow.


































Figure 1-1. A left APS pod is being removed from the space shuttle orbiter Atlantis [3].


VHAPTO-O


Figure 1-2. The twelve APS pod attach point locations, left pod shown (right pod mirror)
[2].




























Mechanism


Forward Adluslment Mecnanism


Rotator Ba.


Figure 1-3. GSE used to install an APS pod.


Forward Uphill

Off-the-deck -On-the-deck


downhill A

Figure 1-4. Forward and aft adjustment mechanisms allow motion in forward/aft,
uphill/downhill, and off-the-deck/on-the-deck directions [3].


~~i~ii~t'F~a~uP





























Revolute Joint Axis


S Revolute Joint Axis


Figure 1-5. Rotator bar joint axes.


Figure 1-6. A left APS pod is transported by crane to Atlantis for installation [3].














CHAPTER 2
PROPOSED ALIGNMENT METHOD

Overview

The geometry of all relevant hardware and the desired position and orientation of

the APS pod is known. As a result, a reverse kinematic analysis was conducted to

determine sets of joint angles that would align the APS pod with the orbiter.

Due to the large number of joints associated with this hardware, the rotator bar and

adjustment mechanisms were treated as three independent robots connected by a large

rigid structure. Since the large structure (lifting fixture) has adjustment capability to

attach to the APS pod, it is assumed that these adjustments will be measured before the

operation begins and is incorporated into the analysis.

The lifting fixture CAD model (Figure 1-3) shows it in the position required to

align the APS pod. As discussed previously, the displayed alignment position does not

necessarily align the APS pod with the orbiter. The coordinates given in the CAD model

were used as a starting point in this analysis. They were used to determine the position

and orientation of the three robot end effectors. End effector position and orientation,

partnered with known geometry, was used to perform reverse kinematic analyses. The

output of these analyses was the joint angles required to align the APS pod. Figure 2-1

shows the process used to determine the manipulator joint offsets.

Determination of Desired Joint Angles

An existing Pro/E model of the lifting fixture, rotator bar, and adjustment

mechanisms was refined for this analysis. The location of each end effector and APS pod









attach point, in orbiter coordinates, was ascertained from the model and input into a C

program. The C program translated and rotated the input points until the APS pod attach

points were aligned with orbiter attach points. The resulting end effector points were

used in reverse kinematic analyses to calculate joint angles. These joint angles can be

described as the joint angles required to align the APS pod with the orbiter attach points.

The orbiter coordinate system and APS pod coordinate systems are discussed in detail in

Appendix A.

Alignment of APS Pod Attach Points with Orbiter Attach Points

The initial position and orientation of the lifting fixture in the CAD model is

arbitrary. The lifting fixture's adjustments at attach points 1, 2, 3 and 4 have been

physically measured and incorporated into the CAD model. The CAD model also

contains seven points on the lifting fixture needed to describe end effector positions and

orientations. Attach points 1, 2, and 3 are located in the CAD model of the left APS pod

and their desired locations on the orbiter are also included in the model. Adjustment

mechanism points can be seen in Figure 2-2.

Attach Point 3 has the most stringent accuracy requirements so it is aligned first.

The alignment algorithm begins by calculating the misalignment of APS pod Attach

Point 3, PPod, relative to orbiter Attach Point 3, pOrbte
p pOrbte pPod (2.1)
Translate AttachPt3 AttachPt3

The APS pod and lifting fixture can be translated such that APS pod Attach Point 3

is aligned by adding PTram,,,, to all points located on either object








12


pPod pPod + p
AttachPtI A=ttachPtl Translate
pPod = pPod P
AttachPt2 AttachPt2 Translate
pPod = PPod + p
AttachPt3 AttachPt3 Translate

pLftrngFzxture pLfhngFzxture + P
RotatorBar RotatorBar Translate

pLfrngFzxture _= pLfhngFxture + P
FwdAdjMechOng FwdAdjMechOng Translate

pLftrngFxture pLfhngFixture + (2.2)
Fwd4djMechX Fwd4djMechK Translate

pLftIngFixture p PLftIngFixture +,
FwdAdjMechZ FwdAdjMechZ Translate
pLfthngFzxture p_ PLngFzxture +P
AftAdjMechOng AftAdMechOng Translate
plLftrngFzxture PLfhngFzxture + PT
AfAdjMechX AfAdMechX Translate

pLzftrngFxture pLfhngFixture +
AftAdjMechZ AftAdjMechZ Translate


Attach Point 2 has the next most stringent accuracy requirements and therefore is


aligned second. The unit vectors from pod Attach Point 3 to pod Attach Point 2 and pod


Attach Point 3 to orbiter Attach Point 2 are described as


pPod _PPod
V AttachPt2 AttachPt3
32 Pod
pPod _PPod
AttachPt2 AttachPt3
(2.3)
pOrbiter PPod
V AttachPt2 AttachPt3
32 Orb
POrbiter PPod
AttachPt2 AttachPt3



The unit vector mi perpendicular to both V32 pod and V2 Orb can be calculated as



3m V32 Pod XV32 Orb (2.4)
min = -- (2.4)
32 Pod 32Orb


and the angle 02 between V2 pod and V32 Orb is the unique solution to Equations 2.5 and


2.6.


2 =cos 1 32 Orb (2.5)
2 \~32 Pod 32 Orb) 25










0 =sin 1- 2 Pod 32 Orb (2.6)


Pod Attach Point 2 will become aligned and pod Attach Point 3 will remain

aligned if lifting fixture and APS pod points are rotated by angle 02 about an axis

containing vector mi and passing through pod Attach Point 3. This transformation

matrix T, is then calculated. [4]


mIxmlxn1 + cos 02 minmxl in, sin n2 minxmliz + il sin 02

T m imnIl + mlZ sin 02 mnmilyv + cos 02 mymny, mI sin 02
mmz1u, n ly sin 02 inmin, + mix sin 02 minz i + cos 02
O 0 0
pPod
AttachPt3,X
pPod
AttachPt3,Y (2.7)
pPod
AttachPt3,Z


01 cos02

This rotation is then performed on all APS pod and lifting fixture points.









pPod PPod pPod

AttachPtL 2 AttachPt2 AttachPt3

1 1
pPod 7 Pod pPod
SAttachPt3 = T AttachPt3 AttachPt3

pLfngFxture pLifingFxture pPod
1 1
pLfhngFxture pLfhngFxture pPod
FwdAdjMechOng T FwdAdjMechOng AttachPt3
1 1 1
pLifhngFixture pLfhngFixture pPod
FwdAdjMechX FwdAdjMechX AttachPt3
1 L 1
pLifngmFxture -pLfnngFtxture pPod
FwdAdMech T FwdAdjMechZ AttachPt3
1 1 1
pLfngFixture pLfngFixture pPod
AAdMechng T AfAdMechng AttachPt3
1 1
pLifngFixture pLifngFixture pPod
LAftAdjMechX I T AftA djMechX AttachPt3
1 1 1
pLfhngFixture pLfhngFixture PPod
SAfAdjMechZ T AfAdjMechZ AttachPt3
1 1 1


(2.8)


Another rotation must be performed to align pod Attach Point 1. The axis of

rotation must pass through pod Attach Point 2 and pod Attach Point 3 so that these points

do not become misaligned. The Plicker coordinates of this line are

{S2; SOL2} {32 Orb, AttachPt3 32 Orb} (2.9)

and the parallel line passing through pod Attach Point 1 is given as

{S; SOL } = 2 OrP X 2 O} (2.10)
{S1;SOL1} { Orb' AttachPtl 32 Orb (2.10)









Vectors p, and p, are perpendicular to each line and originate at the orbiter coordinate

system origin.

P = S SOL1 (2.11)
P2 = S2 x SOL2

When aligned, pod Attach Point 1 will reside on the ypod=100 plane. The ypod coordinate

of pod Attach Point 1 can be calculated using part of the LefPdT transformation matrix:

-0.0526981625
-0.7183402679 pPod
y .4AttachPt1 (212
pod 0.6936931332 1(
-194.96530381

The required angle of rotation 01 can now be calculated.


= sin pod -100 (2.13)
2P2 p


I _)- 2 od100)2
0, = cos-1 2 A- (Ypod l)- (2.14)
P2 P


The transformation matrix equation can be used again to calculate transformation

matrix T,. [4]

V32 Pod,x 32 Pod,xU2 + COS 1 V32 Pod,x 32 Pod,y 2 32 Pod,zin
T 3 P 32 Pod,x32 Pod,y 2 32 Pod,z sin 1 32 Pod,y~3,2 Pody 2+cos 1
32 Pod,x 32 Podz odsin 2 od, 32 od, 2 + V32 odxsin
0 0
SVPodx 2Pod,z2 2T Pod, S Vin 2 P ht3x(2.15)
(2.15)

V32 Pod,r 32 Pod 32 Pod,y sin 01 PAhPt3x
1Pod
V32-Pod, y 32-Podzt)2 32_ Pod,x sin 1 01 |
V32 Pod,zv32 Pod,.U2 + COS 1 PAttachPt3,y
0 1






16


where v2 1 cos 01

Transformation matrix T2 is then used to perform the rotation of all points about

the line passing through pod Attach Point 2 and pod Attach Point 3.

pPod pPod pPod
AttachPt1 T AttachPt1 AttachPt3 I
1 1 1
pPod P Pod pPod
AttachPt2 T AttachPt2 AttachPt3
1 1
P -Pod pPod pPod

1 1


pLifngFixture pLifngFixture p PPod
RotatorBar T RotatorBar AttachPt3
1 ] 1
PliFFngFMxture pLifngFixture p Pod
LFwdAdMechOng FwdAdjMechOng AttachPt3
1 1
pfng Fxture 7 pLIfngFxture -pPod
LFwdAdMechX -_T 2 FwdAdjMechX AttachPt3
1 1
PLftlngFmxture 7 pLffingFixture -PPod
LFwdAdjMechZ T 2 FwdAdjMechZ AttachPt3
1 1

PLftngFmxture PLftngFxture -PPod
1 1

AftAdjMechX T AftAdjMechX AttachPt3
1 1
pLftngFixture 7 pLftngFxture PPod
AftAdMechZ AdMechZ A AttachPt3
1 1


(2.16)


APS pod attach points 1, 2, and 3 have been aligned. A reverse kinematic analysis

will now be conducted to determine the adjustment mechanism joint angles with the APS

pod aligned.









Adjustment Mechanism Reverse Kinematic Analysis

Adjustment mechanism parameters

Joint axis vectors S, and link vectors aY must be chosen for the adjustment

mechanisms. These selections are shown in Figure 2-3. It should be noted that the

spherical joints are treated as "three noncoplanar cointersecting revolute joints". [4]

Joint angle Oj is defined as the angle from a to ajk about the vector S Similarly,


twist angle aij is defined as the angle from S to S about the vector a .


S, sin O = a x ak
(2.17)
a sin a =S x S


Joint offset SJ is the distance from aY to ajk along S Link length ay is the distance


from S to S, along a The first joint angle, (i, describes the angle from fixed


coordinate system X- axis and vector a2 .

S, sin 1 = XFxed xa12 (2.18)

Adjustment mechanism joint angles, twist angles, joint offsets, and link lengths are

defined in Table 2-1. The constant parameters are exactly the same for both forward and

aft adjustment mechanisms. Parameters marked as 'variable' are not necessarily the

same for both adjustment mechanisms because adjustment mechanism end effector

positions are not identical. Close-the-loop variables are created by the hypothetical

closure link. [4] a67 and a67 are user-specified values (rather than a function of

manipulator geometry) since the seventh joint is hypothetical.










Close-the-loop variable calculations

Close-the-loop variables can be calculated using the constant mechanism

parameters listed in Table 2-1. Flxed is defined as [0 0 1] since the vector S is


exactly aligned with the fixed coordinate system Z- axis. Fixed S is given by the

expression

Fixed 7Fixed I- Fxd g (2.19)


Unit vectors Fixed a and Fixed are the X- axis and Z- axis of the adjustment mechanism

6th coordinate system, respectively. They can be calculated as

Fixed pLifhngFixture PLifhngFixture
a67 FwdAdjMechX AdjMechOng
Fixeds = pLifhngFixture -pLifhngFixture
6 FwdAdjMechZ AdjMechOng

These definitions allow the unit vector Fixeda71 to be determined from

Fixed So Fixed o
a7e 1 Fixed S Fixed


The close-the-loop variables can now be calculated. A unique value for the twist

angle a71 between vectors S7 and S, is given by


cos(a71) =Fixed S7 *Fxed S
si* n(71) (Fixed s Fixed Fixed (2.22)
sin(al) = S x S, 7 al

Similarly, the joint angle 07 can be found

cos(07) Fixed a7 *Fxed a1
sin(,) =Fixed -> Fixed ) F7xed s (2.23)
sin(07) ( r a67 x^ a71^ 7









The angle yi is defined as the angle between a7 and the X- axis of the fixed coordinate

system.

cos(,)=Fixed a7 .[1 0 0]O

sin(y) = (Fx ed 0 )Fxed (2.24)


The joint offset S7 along the S, vector can be calculated by

(Fixed S X j pLfhngFixture Fixed
Z1 AdjMechOng 71
S7 = echo (2.25)
sin(al)

The link length a71 along the a71 vector is given by

(pLLfhingFixture Fixed S *Fixed S
'AdjMechOng X7
a71 -- (2.26)
sin(a71)

Joint offset S1 along the S, vector is

(pLifhngFixture Fixed S *Fixed --
S AdjMechOng a7 71
S,= =.(2.27)
sin(a'7)

The close-the-loop variables a71, 07, yi, S7, a71, and Si will be used in the reverse

kinematic analysis of this PPPS mechanism.

Reverse kinematic analysis of a PPPS mechanism

As previously stated, the adjustment mechanisms are RPPPS spatial mechanisms

but will be analyzed as an equivalent RPPPRRR mechanism. They are categorized as

group 1 mechanisms since the spatial mechanisms and equivalent spherical mechanisms

have a single degree-of-freedom. Spherical equations generally contain a high number of

terms. However, these terms exist in patterns that allow a shorthand notation to describe

the equation more concisely. Notation variables are defined in Appendix B. 01 is the









first unknown joint angle that will be calculated. The fundamental spherical heptagon

equation

Z45671 = 23 (2.28)

can be expanded to

S12 (X4567S1 + Y4567C )+ 12Z4567 = 23 (2.29)

X4567 and Y4567 are defined by notation variables X456 and Y456

X4567 X456C7 -456S7
Y4567 =71 (X456S7 + Y456c7 S71Z456 (2.30)
Z4567 = S71 (X456S7 + g456C7 )+ C71456

07 and a71 are close-the-loop variables that have already been calculated, so X456, Y456,

and Z456 are the only unknown terms that must be calculated. They can be defined as

X456 X45C6 -45S6
Y456 C67 (X45S6 -45c6)- S67Z45 (2.31)
Z456 S= 67 (X45S6 + 45C6) + C67Z45

06 is a constant mechanism parameter and a67 is a user-specified value, so both are known

quantities. X45, Y45, and Z45 must now be defined.

X45 X4c5 Y45
Y45 C56 (X4S5 + Y )- S56Z4 (2.32)
Z45 S56 (X4S5 Y4C5 +56Z4

05 and a56 are constant mechanism parameters. X4, Y4, and Z4 are defined as

X4 S S34S4
4 -(S45C34 + C45S34C4) (2.33)
Z4 C45C34-S45S34C4









04, a34 and a45 are constant mechanism parameters so X4, Y4, and Z4 can be calculated.

Using substitution, X4567 and Y4567 can now be calculated. This results in an equation of

the form Ac, + Bs, + D = 0. Using the trigonometric solution method [4], 01 can be

calculated


S= cos-1 C23 124567 + (2.34)
1 1 (s12Y4567)2 +(s12X4567)2

where y is the unique solution of


-sin 1 S12X4567
s124567)2 +(s12X4567)2 (2.35)
(2.35)
-1 S12Y4567
7/ = Cco s __ _
J(s12Y4567) +(s12X4567)2

It should be noted that 01 has two solutions, designated as 01A and 01B. Other joint

angles are a function of 01, so it is necessary to solve each joint angle using 01A and 01B.

There are two sets of joint angles, solution set A and solution set B, that satisfy the input

end effector position and orientation for the specified manipulator geometry.

Unknown joint angle 62 can now be calculated using O1A and O1B. The fundamental

spherical heptagon equations

X45671 = 23S236)
Y45671 = S23C2

will be used. Since X4567 Y4567, and Z4567 were determined in the 01 derivation, X45671

and Y4671 can be calculated immediately using O1A and O1B.









X45671 X4567C1 -4567S1 (
Y45671 C12 (X4567S1 + Y45671 )-S12Z4567

02 is the unique solution to equation 2.38 for the solution set containing OlA and the set

containing O1B.


02 = sin-'1 X45671
S s23
(2.38)

02 = cos-1 45671


The reverse kinematic analysis continues with the calculation of unknown joint

angle 63. The fundamental spherical heptagon equations

X56712 34(2.39)
Y56712 = S34C3

can be used because O1A, O1B, 02A and 62B have been calculated. X56712 and Y56712 are

defined as

X56712 X5671C2 -5671S2 .
Y56712 C23 (X5671S2 Y5671c2)-S23Z5671

Similar to the solution for 01, the solution for 63 proceeds by solving for the

notation variables that comprise X56712 and Y6712-.









X5671 X567 1 567S1
S671 = c12 (X5671 + Y567C1)- 12Z567
Z5671 12 (X567s1 + Y567C) + C12Z567


X567 X56 6 -56S6
-567 71 (X56S7 + 56C7 ) 71Z56
Z567 71 (X56S7 + 56C7 ) + ) 71Z56


X56 X56 C 5S6
56 = 67 (X5s6 + Y6)- S67Z5
Z56 S67 (X5 + Y56)+ 67Z5


X, 5 45S5
(56C 5 C56S45C5)
Z5 c56c45 -s56S45C5 (2.41)

A unique solution for joint angle 63 can now be calculated for solution set A and solution

set B using their associated joint angles.


03= sin 1 X56712
(2.42)

03= cos-1 Y5712
SS34

Joint offset S4 will be calculated using the vector loop equation. The vector loop

equation is given by

S S + 2 a 2 + S2S2 + a23a23 +S3S3 +a34 a34 + S4S4 a45 a45
(2.43)
+S5 S + a56 a, + S6 + a67 a + S, S7 + 6 a71a = 0

Since several of these offsets are zero, the vector loop equation can be reduced to

S SI + a34 34 +S4 S4 + 67 67 +S7S7 + a71a71 = 0 (2.44)









Using spatial heptagon direction cosines set 5, the vector loop equation [4]

becomes

SX76 + a34W45 + S4X5 + a676 + S7X6 aW76 = 0 (2.45)

Several notation terms in Equation 2.45 must be calculated

X76 X7C6 Y7S6
W45 5 4 S5S4c45
W76 = 6C7 S6S7C67
(2.46)
X6 6766
X7 71S 7
Y7 (S67C71 + C67S7IC7)

Unknown joint offset S4 can now be calculated by solving Equation 2.45 for S4.

S4 -S1X76 34W45 a676 a7176 (2.47)
X5

Joint offset S6 is calculated by substituting spatial heptagon direction cosines set 4

into the vector loop equation. Equation 2.44 then reduces to

SX765 +344 + S6 X5 + a67W65 + 7X65 a71W765 = 0 (2.48)

Equation 2.48 can be further reduced by noting that c4=0.

S,X765 + S6 X, + a67W65 + SX65 + a7,W765 = 0 (2.49)

The unknown notation variables can be calculated from









X765 = X76 5 76S5

Y76 56 (X7s6 +7C6) S56 Z7
Z7 C67c71 S71c7
X5 = s56S5

X65 X6C5 65 (2.50)
Y6 (S56C67 + C56S67C6)
W65 C5C6 S5S656
W765 S5 (U76S56 + 76 56)+ 5W76
U76 S S767
V76 (S67 + C6SC67)

Solving Equation 2.49 for S6 yields

S -SX765 67W65 65 -71765 (2.51)
S6 = (2.51)
X5

The only joint offset that has not yet been calculated is S5. Spatial heptagon

direction cosines set 3 are substituted into the vector loop equation given in Equation

2.44.

SX23 34 + S5 X4 67654 + SX654 +7123 = 0 (2.52)

Once again, notation variables must be calculated before the unknown joint offset can be

determined.









X23 Xc, Ys
X2 s 12s2
Y2 (12C12 + C12S12C2)
X4 S 45S4
X54 Xc4 YS4
X654 X65C4 65S4
(2.53)
65 C45 X6S5 +Y C5 S 45Z6(2.53)
z6 56C67 s- 56S67C6
W123 3 (U12z23 + V23)+) +W2


V12 =-( C1 + CS1C12
W, 2 C2cI ss 2,S

Finally, joint offset S5 can be calculated

S-SX23 a34 S6X54 S7X654 71 23 .54)
S5 = (2.54)
X4

Both solution sets have now been calculated. In the case of the adjustment

mechanism, all three joint offsets are the same in both solution sets. These joint offsets

will be used to orient the lifting fixture so that a finite element analysis can be performed.

Rotator Bar Length Calculation

The rotator bar can be described as a RRRCRR mechanism. A reverse kinematic

analysis could be conducted to ascertain joint angles and joint offsets that result in the

APS pod being aligned. To conduct this analysis, the position and orientation of the end

effector relative to the base must be input. The position of the rotator bar 6th coordinate

system origin has been previously defined as P,'TauB The orientation of the end

effector can be determined by also inputting points on the a67 and S6 axes and


subtracting PLngFd e from each to determine the unit vectors a7 and Sg










Before conducting a reverse kinematic analysis of the rotator bar, it was noted that

unique geometry makes it possible to calculate the joint offset of the cylinder joint. With

the exception of the cylinder joint offset, all rotator bar link lengths and joint offsets are

zero. As a result, the cylinder joint offset can be calculated using the distance equation

LifhngFxture LifngFiF~xre LfngFxture LfngFxture 2
S\ RotatorBarX RotatorBarBaseX I PRotatorBarY RotatorBarBaseY (2.55)
S, (2.55)
+p PLfngFmxture PLft,,ngFxture
SRotatorBarZ RotatorBarBaseZ

Although unknown joint angles could be calculated using S3, they are not needed for this

analysis.

Nominal Solution

A program was written to perform pod alignment and subsequent joint offset

calculations as described in this chapter. The mechanism parameters specified in


Table 2-1 and the input coordinates of points pPod PP o PPod pRfngFxture


pLzh ngFzxture pLzh ngFzxture pLzhngFzxture pLiftngFixture pLingFxture pLand pLngFxture w
FwdA4djMechOng FwdA4djMechX > FwdAdjMechZ AftAdjMechOng > AftAdjMechX ,> AftAdjMechZ were

used.

Table 2-2 compares the joint offsets calculated by the program to the joint offsets

measured using a perfectly aligned APS pod CAD model. The similar results show that

the program is able to accurately calculate joint offsets for a known lifting fixture

location. The two solutions, solution A and solution B, have nearly identical joint offsets

but differing joint angles (not shown).

The program is also able to perform a reverse kinematic analysis for right APS

pods. This is accomplished by inverting the sign of the input Yorbiter coordinates then

proceeding with the rest of the solution method. Right APS pods, lifting fixtures and









orbiter attach points are mirror images of left APS pods, lifting fixtures, and orbiter deck

attach points.

The CAD model of the
lifting fixture is positioned
and oriented arbitrarily.




The APS pod attach point
and end effector positions
are input into an
alignment algorithm.


Figure 2-1. Joint offset calculation procedure.


The algorithm aligns APS pod
attach points with orbiter attach
points and calculates the
resulting end effector position
and orientation for each
manipulator.


The rotator bar length is
calculated and a reverse
kinematic analysis is performed
on both adjustment
mechanisms.




























Figure 2-2. Three points are needed to determine each adjustment mechanism's position
and orientation (aft adjustment mechanism shown, typical of all adjustment
mechanisms).


- -


i i i i i 23
Figure 2-3. Adjustment mechanism joint axis vectors and link vectors.









Table 2-1. Adjustment mechanism parameters.
Link length, inches Twist angle, degrees Joint offset, inches Joint angle,
degrees
a12 = 0.000 a12 = 270.0 S1 = Close-the-loop (p = variable
variable
a23 = 0.000 a23 = 270.0 S2 = 0.000 02 = variable
a34 = 4.147 a34 = 270.0 S3 = 0.000 03 = variable
a45 = 0.000 a45 = 270.0 S4 = variable 04 = 270.0
a56 = 0.000 a56 = 90.0 S5 = variable 05 = 270.0
a67 = 0.000 a67 = 90.0 S6 = variable 06 = 180.0
a71 = Close-the-loop a7l = Close-the-loop S7 = Close-the-loop 07 = Close-the-
variable variable variable loop variable

Table 2-2. Comparison of joint offsets calculated by the program to measured using the
CAD model.
Forward adjustment Aft adjustment mechanism Rotator
,echs .Aft adjustment mechanism ,
mechanism bar
Measurement 4 S
method S4 s5 S6 S4 s5 S6 S3
CAD model 13.6655" 14.5243" 8.7968" 12.3174" 14.8512" 8.8823" 94.5707"
Programodel 13.6655" 14.5244" 8.7968" 12.3174" 14.8512" 8.8823" 94.5707"
so tonram 13.6655" 14.5244" 8.7968" 12.3174" 14.8512" 8.8824"
solution A 94.5706"
94.5706"
Program 13.6655" 14.5243" 8.7968" 12.3174" 14.8512" 8.8824"
solution B














CHAPTER 3
UNCERTAINTY ANALYSIS

Due to the stringent accuracy requirements associated with the APS pod installation

operation, an uncertainty analysis was conducted to assess the validity of the proposed

solution method. Two different uncertainty calculation methods were used to create an

upper and lower bound for total uncertainty. The 100% covariance method produces

very conservative results since it assumes all errors are at their maximum value. The root

sum squared method provides more optimistic results. The 100% covariance and root

sum squared uncertainty calculations provide an upper and lower bound respectively for

the total error that can be reasonably expected.

Off-Nominal Conditions within Tolerance

Manufacturing tolerances can have a significant effect on the overall accuracy of a

manipulator. As a result, precision manipulators are often manufactured with very tight

tolerances.

Adjustment Mechanism Spherical Joint Socket Locations

There are manufacturing tolerances associated with the spherical joint sockets and

the position of their mounting holes on the orbiter. These tolerances yield an uncertainty

of +0.0349" in the position of each spherical joint (both Xorbiter and Zorbiter directions).

The resulting misalignment of the APS pod can be seen in Tables 3-1 and 3-2.

Orbiter Attach Point Locations

There is also uncertainty associated with the position of the orbiter attach points on

the orbiter. According to drawing tolerances, the orbiter attach points are located within









0.010" of their intended position in the Xpod, Ypod, and Zpod directions. The effect of a

0.0104" attach point misalignment in all directions on the alignment of an APS pod can

be seen for each attach point individually in Tables 3-3, 3-4, and 3-5.

The APS Pod Fitting Locations

There is also uncertainty about the locations of the fittings on the APS pod. As

with the orbiter attach point locations, the tolerance associated with the APS pod fitting

locations is 0.010". Therefore, an uncertainty of 0.0104" in all directions is associated

with each pod fitting location. The APS pod misalignment caused by this uncertainty is

given in Tables 3-6, 3-7, and 3-8.

Lifting Fixture Attach Point 3 Location

The position of the APS pod relative to the lifting fixture is largely determined by

the location of the lifting fixture's attachment to the APS pod Attach Point 3 fitting.

Drawing tolerances affect the position of the lifting fixture's attachment location relative

to the adjustment mechanism end effectors. According to drawing tolerances, the lifting

fixture Attach Point 3 location can be up to (0.0349", 0.0698", 0.0349") from the

intended position in (Xpod, Ypod, Zpod). It may be surprising to note that this tolerance

single-handedly prevents the Attach Point 3 accuracy requirement from being met.

However, it is probable that engineers did not expect to apply robot kinematics to the

lifting fixture when it was designed in 1977. The pod misalignment resulting from this

uncertainty can be simply calculated. Results are presented in Table 3-9.

Lifting Fixture Adjustment at Attach Point 3

There also exists the capability to adjust the position of the APS pod at the Attach

Point 3 location. The magnitude of this adjustment capability is 0.418" in the +Xpod









directions. Since this adjustment potentially shifts the entire APS pod, all three attach

points are affected by this source of uncertainty as shown in Table 3-10.

Uncertainty of Input Values

Location of rotator bar base. At first glance, it might appear that the base of the

rotator bar can be considered "ground" at a known position relative to the spherical joint

sockets and orbiter attach points. Unfortunately, this is not the case. One factor is the

dimensional variance between each of the three OPFs. The rotator bar base is mounted to

a beam in each OPF and the location of that beam might not be identical in each OPF. A

much more significant factor is that the position of the orbiter relative to the OPF is not

always the same. After each mission, the orbiter is towed into the OPF and jacked off the

floor. Per specification, the orbiter must be towed to +1" forward/aft, 1.5"

port/starboard, and +0.25" up/down of a specified nominal position. Therefore, the

position of the rotator bar base can be significantly different from nominal.

As a result, the position of the rotator bar base relative to the spherical joint sockets

must be measured before each APS pod installation. Preliminary indications are that this

position can be measured to a total accuracy of 0.010" or better. The uncertainty can be

determined by positioning the rotator bar base 0.0104" from the nominal position and

using nominal joint offsets. The effect of this measurement error is greatest if it occurs

along the prismatic joint axis. Table 3-11 shows the effect of this error on attach point

alignment.

Calculation-Related Uncertainties

Computer program uncertainty. Ideally, the computer program calculates the

exact joint offsets required to position the lifting fixture as desired. However, roundoff

errors can propagate and potentially become a significant error source.









To determine error associated with the program, the lifting fixture CAD model was

positioned in a known location. Using Pro/E's mechanism application, connections

between components were defined as joints rather than rigid connections. This

automatically positioned the adjustment mechanisms and rotator bar joints as needed to

properly connect to the lifting fixture and mechanism base. The position of specific

points in the model was then input into the program. The program calculated the joint

offsets required to align the lifting fixture. These offsets were compared to the joint

offsets measured in the CAD model.

Inaccuracies were initially experienced due to errors in the CAD model.

Additionally, using only three decimal places for input values caused significant errors.

After these problems had been remedied, numerical methods were not required to further

refine the calculations. Results can be viewed in Table 3-12.

Hardware Positioning Uncertainty

Although joint offsets can be accurately calculated to several decimal places, the

ability of the pod installation team to adjust the rotator bar and adjustment mechanisms is

limited. These limitations are largely due to measurement device uncertainty, mounting

inaccuracies, and the tolerances of the components being measured. Adjustment

mechanism measurement directions are shown in Figure 3-1.

Uphill/Downhill Joint Offset Measurement

The joint offset S4 has been previously defined as the distance along the S4 vector

between the a34 and a45 vectors. The pod installation team can adjust this joint offset to

match the value calculated by the alignment program.









The desired joint offset reading on the measurement device can be achieved.

However, the accuracy of this measurement is affected by factors such as measurement

device uncertainty, mounting inaccuracies, and hardware tolerances. A preliminary

assessment of these factors suggests that an accuracy of +0.010" can be achieved. This

uncertainty analysis will determine the alignment error resulting from a misalignment of

0.0104" uphill for the forward and aft adjustment mechanism. The results of this analysis

are stated below in Tables 3-13 and 3-14.

Off-the-Deck/On-the-Deck Joint Offset Measurement

The accuracy of the S5 joint offset measurement is also affected by measurement

device uncertainty, mounting inaccuracies, and hardware tolerances. The total

measurement uncertainty is approximated as 0.010". An analysis has been performed

to ascertain attach point misalignment due to a 0.0104" misalignment in the off-the-deck

direction for each adjustment mechanism. The results of this analysis can be found in

Tables 3-15 and 3-16.

Forward/Aft Joint Offset Measurement

As with the measurement uncertainties for joint offsets S4 and S5, it is assumed that

measurements of joint offset S6 are accurate to within 0.010". The effect of a 0.0104"

misalignment in the forward direction was studied. In order for this misalignment to

occur, the entire lifting fixture must be 0.0104" forward which means both the forward

and aft S6 are misaligned by the same value. The resulting misalignment of attach points

is listed in Table 3-17.

Rotator Bar Joint Offset Measurement

The rotator bar S3 prismatic joint offset is affected by the same uncertainty sources

as the adjustment mechanism joint offsets. Measurement uncertainty is also









approximately 0.010". The misalignment of a rotator bar that is extended 0.0104"

more than the measurement indicates is shown in Table 3-18.

Compliance in Joints

It is stated that "80% of the flexibility of industrial robots comes from the joint."

[5] Quantifying this compliance is a difficult task. It will be estimated by considering

each joint on an individual basis. Joints exist where two components are attached. The

dimensional difference between those two components was first identified. Loading was

then considered while determining the relative position of the components under the

assumption they are in contact. One component is translated and, in some cases also

rotated, to the determined position. The resulting APS pod misalignment is calculated

from these translations and rotations. The total uncertainty related to compliance is

documented in Table 3-19.

Total Uncertainty Calculation

There are different methods for calculating total system uncertainty for a specified

set of individual uncertainties. Individual uncertainties at each attach point are

summarized in Tables 3-20, 3-21, and 3-22. This information will be used to calculate

total uncertainty using the 100% covariance method and the least squared method. The

total uncertainty will then be compared to the accuracy requirements for APS pod

alignment.

One Hundred Percent Covariance Method

The 100% covariance method assumes that each individual uncertainty is at a

maximum at the same time. Assuming that all significant sources of uncertainty have

been found and reasonably approximated, the 100% covariance method provides the

"worst case scenario".









The total uncertainty is calculated by summing all individual uncertainties. The

Xpod and Zpod accuracy requirements at attach points 1 and 3 are not specified

individually. Since it is desired to compare total uncertainty to accuracy requirements,

the Xpod and Zpod total uncertainty has been combined for attach points 1 and 3. Table 3-

23 allows the uncertainty calculated using the 100% covariance method to be compared

to the accuracy requirement.

Root Sum Squared Method

The root sum squared method provides the lower bound for total uncertainty

projections. The root sum squared uncertainty is calculated by


U ot = = ,2 (3.1)

where UTotal is the total uncertainty and the individual uncertainties are given by Ui

through Un.

The Xpod and Zpod uncertainties at attach points 1 and 3 have been combined for

comparison to accuracy requirements. This comparison is presented in Table 3-24.









Table 3-1. Forward spherical joint socket tolerances.
dine N in Fwd socket misaligned Resulting
Coordinate Nominal .
Attach point solution by 0.0349" in the +Xorbiter attach point
axis solution
and +Zorbiter directions misalignment
Xpod 106.5000" 106.5335" -0.0335"
Attach Point 1 Ypod 100.0000" 100.0353" -0.0353"
Zpod 152.3460" 152.3745" -0.0285"
Xpod 106.5000" 106.5181" -0.0181"
Attach Point 2 Ypod 100.0000" 100.0224" -0.0224"
Zpod 49.1790" 49.2075" -0.0285"
Xpod 253.5000" 253.5167" -0.0167"
Attach Point 3 Ypod 100.0000" 100.0078" -0.0078"
Zpod 39.7680" 39.7747" -0.0067"

Table 3-2. Aft spherical joint socket tolerances.
Aft socket misaligned by Resulting attach
Coordinate Nominal .
Attach point ai option 0.0349" in the +Xorbiter point
axis solution
and +Zorbiter directions misalignment
Xpod 106.5000" 106.5016" -0.0016"
Attach Point 1 Ypod 100.0000" 100.0062" -0.0062"
Zpod 152.3460" 152.3440" 0.0020"
Xpod 106.5000" 106.5141" -0.0141"
Attach Point 2 Ypod 100.0000" 99.9962" 0.0038"
Zpod 49.1790" 49.1770" 0.0020"
Xpod 253.5000" 253.5153" -0.0153"
Attach Point 3 Ypod 100.0000" 100.0155" -0.0155"
Zpod 39.7680" 39.7839" -0.0159"









Table 3-3. Orbiter Attach Point 1 tolerances.
Attach Point 1
Nominal misalignment of Resulting attach
.Nominal misalignment of
Attach point Coordinate axis 1 point
A h p t solution 0.0104" in all point
misalignment
three directions mis
Xpod 106.5000" 106.5104" -0.0104"
Attach Point 1 Ypod 100.0000" 100.0104" -0.0104"
Zpod 152.3460" 152.3564" -0.0104"
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 2 Ypod 100.0000" 100.0000" 0.0000"
Zpod 49.1790" 49.1790" 0.0000"
Xpod 253.5000" 253.5000" 0.0000"
Attach Point 3 Ypod 100.0000" 100.0000" 0.0000"
Zpod 39.7680" 39.7680" 0.0000"

Table 3-4. Orbiter Attach Point 2 tolerances.
Attach Point 2
*Resulting attach
Nominal misalignment of
Attach point Coordinate axis nominal mialinnt of point
solution 0.0104" in all poi
three directions misalignment
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 1 Ypod 100.0000" 100.0000" 0.0000"
Zpod 152.3460" 152.3460" 0.0000"
Xpod 106.5000" 106.5104" -0.0104"
Attach Point 2 Ypod 100.0000" 100.0104" -0.0104"
Zpod 49.1790" 49.1894" -0.0104"
Xpod 253.5000" 253.5000" 0.0000"
Attach Point 3 Ypod 100.0000" 100.0000" 0.0000"
Zpod 39.7680" 39.7680" 0.0000"









Table 3-5. Orbiter Attach Point 3 tolerances.
Attach Point 3
Nominal misalignment of Resulting attach
.Nominal misalignment of
Attach point Coordinate axis 1 point
A h p t solution 0.0104" in all point
misalignment
three directions mis
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 1 Ypod 100.0000" 100.0000" 0.0000"
Zpod 152.3460" 152.3460" 0.0000"
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 2 Ypod 100.0000" 100.0000" 0.0000"
Zpod 49.1790" 49.1790" 0.0000"
Xpod 253.5000" 253.5104" -0.0104"
Attach Point 3 Ypod 100.0000" 100.0104" -0.0104"
_Zpod 39.7680" 39.7784" -0.0104"

Table 3-6. APS pod Attach Point 1 tolerances.
Attach Point 1 Fitting Resulting
iResulting
Nominal misalignment of
Attach point Coordinate axis Nominal misalignment of attach point
solution 0.0104" in all three n
directions misalignment
directions
Xpod 106.5000" 106.5104" -0.0104"
Attach Point 1 Ypod 100.0000" 100.0104" -0.0104"
Zpod 152.3460" 152.3564" -0.0104"
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 2 Ypod 100.0000" 100.0000" 0.0000"
Zpod 49.1790" 49.1790" 0.0000"
Xpod 253.5000" 253.5000" 0.0000"
Attach Point 3 Ypod 100.0000" 100.0000" 0.0000"
Zpod 39.7680" 39.7680" 0.0000"









Table 3-7. APS pod Attach Point 2 tolerances.
Attach Point 2 Fitting Resulting
N.T i Resulting
Nominal misalignment of
Attach point Coordinate axis Nominal misalignment of attach point
Attah p t solution 0.0104" in all three atah pint
directions misalignment
directions
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 1 Ypod 100.0000" 100.0000" 0.0000"
Zpod 152.3460" 152.3460" 0.0000"
Xpod 106.5000" 106.5104" -0.0104"
Attach Point 2 Ypod 100.0000" 100.0104" -0.0104"
Zpod 49.1790" 49.1894" -0.0104"
Xpod 253.5000" 253.5000" 0.0000"
Attach Point 3 Ypod 100.0000" 100.0000" 0.0000"
Zpod 39.7680" 39.7680" 0.0000"

Table 3-8. APS pod Attach Point 3 tolerances.
Attach Point 3 Fitting Re
N.T i Resulting
Attach point Coordinate axis Nominal misalignment of attach point
solution 0.0104" in all three n
directions misalignment
directions
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 1 Ypod 100.0000" 100.0000" 0.0000"
Zpod 152.3460" 152.3460" 0.0000"
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 2 Ypod 100.0000" 100.0000" 0.0000"
Zpod 49.1790" 49.1790" 0.0000"
Xpod 253.5000" 253.5104" -0.0104"
Attach Point 3 Ypod 100.0000" 100.0104" -0.0104"
Zpod 39.7680" 39.7784" -0.0104"









Table 3-9. Lifting fixture Attach Point 3 tolerances.
Coordinate Nominal Lifting fixture Attach Resulting
Attach point axis solution Point 3 misalignment due attach point
axis solution .
to tolerances misalignment
Xpod 106.5000" 106.5349" -0.0349"
Attach Point 1 Ypod 100.0000" 100.0698" -0.0698"
Zpod 152.3460" 152.3809" -0.0349"
Xpod 106.5000" 106.5349" -0.0349"
Attach Point 2 Ypod 100.0000" 100.0698" -0.0698"
Zpod 49.1790" 49.2139" -0.0349"
Xpod 253.5000" 253.5349" -0.0349"
Attach Point 3 Ypod 100.0000" 100.0698" -0.0698"
Zpod 39.7680" 39.8029" -0.0349"

Table 3-10. Lifting fixture Attach Point 3 adjustment.
Maximum Adjustment of Resulting
h p Coordinate Nominal 0.4184" in the +Xpod resulting
Attach point attach point
axis solution direction at Attach Point aah int
3 misalignment

Xpod 106.5000" 106.9184" -0.4184"
Attach Point 1 Ypod 100.0000" 100.0000" 0.0000"
Zpod 152.3460" 152.3460" 0.0000"
Xpod 106.5000" 106.9184" -0.4184"
Attach Point 2 Ypod 100.0000" 100.0000" 0.0000"
Zpod 49.1790" 49.1790" 0.0000"
Xpod 253.5000" 253.9184" -0.4184"
Attach Point 3 Ypod 100.0000" 100.0000" 0.0000"
Zpod 39.7680" 39.7680" 0.0000"










Table 3-11. Misalignment of the rotator bar base.
0.0104" rotator Resulting attach
Nominal
Attach point Coordinate axis olion base point
solution
misalignment misalignment
Xpod 106.5000" 106.4989" 0.0011"
Attach Point 1 Ypod 100.0000" 99.9836" 0.0164"
Zpod 152.3460" 152.3563" -0.0103"
Xpod 106.5000" 106.5001" -0.0001"
Attach Point 2 Ypod 100.0000" 99.9974" 0.0026"
Zpod 49.1790" 49.1893" -0.0103"
Xpod 253.5000" 253.5002" -0.0002"
Attach Point 3 Ypod 100.0000" 99.9983" 0.0017"
_Zpod 39.7680" 39.7800" -0.0120"

Table 3-12. Computer program uncertainty.
Joint offsets
Joint offs Joint offsets from Resulting
Coordinate from the CAD
Attach point iae om e the program are attach point
axis model are
S d input into the model misalignment
used
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 1 Ypod 100.0000" 100.0000" 0.0000"
Zpod 152.3460" 152.3460" 0.0000"
Xpod 106.5000" 106.5000" 0.0000"
Attach Point 2 Ypod 100.0000" 100.0000" 0.0000"
Zpod 49.1790" 49.1790" 0.0000"
Xpod 253.5000" 253.5000" 0.0000"
Attach Point 3 Ypod 100.0000" 100.0000" 0.0000"
Zpod 39.7680" 39.7680" 0.0000"









- -


Figure 3-1.


Adjustment mechanism joint axis vectors and link vectors.


Table 3-13. Forward adjustment mechanism uphill measurement uncertainty.
Nominal Misalignment of the Resulting
Attach point Coordinate axis olion Fwd S4 0.0104" attach point
solution
Uphill misalignment
Xpod 106.5000" 106.5071" -0.0071"
Attach Point 1 Ypod 100.0000" 100.0080" -0.0080"
Zpod 152.3460" 152.3686" -0.0226"
Xpod 106.5000" 106.5011" -0.0011"
Attach Point 2 Ypod 100.0000" 100.0005" -0.0005"
Zpod 49.1790" 49.2015" -0.0225"
Xpod 253.5000" 253.5005" -0.0005"
Attach Point 3 Ypod 100.0000" 99.9996" 0.0004"
Zpod 39.7680" 39.7819" -0.0139"












Table 3-14. Aft adjustment mechanism uphill measurement uncertainty.
Nominal Misalignment of the Resulting
Attach point Coordinate axis olion Aft S4 0.0104" attach point
solution
Uphill misalignment
Xpod 106.5000" 106.4929" 0.0071"
Attach Point 1 Ypod 100.0000" 100.0058" -0.0058"
Zpod 152.3460" 152.3550" -0.0090"
Xpod 106.5000" 106.4992" 0.0008"
Attach Point 2 Ypod 100.0000" 100.0003" -0.0003"
Zpod 49.1790" 49.1880" -0.0090"
Xpod 253.5000" 253.4999" 0.0001"
Attach Point 3 Ypod 100.0000" 99.9994" 0.0006"
Zpod 39.7680" 39.7860" -0.0180"

Table 3-15. Forward adjustment mechanism off-the-deck measurement uncertainty.
Nominal Misalignment of the Resulting
Attach point Coordinate axis olion Fwd S5 0.0104" attach point
solution
Off-the-Deck misalignment
Xpod 106.5000" 106.5002" -0.0002"
Attach Point 1 Ypod 100.0000" 100.0103" -0.0103"
Zpod 152.3460" 152.3566" -0.0106"
Xpod 106.5000" 106.5006" -0.0006"
Attach Point 2 Ypod 100.0000" 100.0113" -0.0113"
Zpod 49.1790" 49.1895" -0.0105"
Xpod 253.5000" 253.5007" -0.0007"
Attach Point 3 Ypod 100.0000" 100.0023" -0.0023"
_Zpod 39.7680" 39.7792" -0.0112"









Table 3-16. Aft adjustment mechanism off-the-deck measurement uncertainty.
Nomil Misalignment of the Resulting
Attach point Coordinate axis solution Aft S5 0.0104" Off- attach point
solution
the-Deck misalignment
Xpod 106.5000" 106.4993" 0.0007"
Attach Point 1 Ypod 100.0000" 99.9961" 0.0039"
Zpod 152.3460" 152.3570" -0.0110"
Xpod 106.5000" 106.4996" 0.0004"
Attach Point 2 Ypod 100.0000" 99.9978" 0.0022"
Zpod 49.1790" 49.1900" -0.0110"
Xpod 253.5000" 253.4996" 0.0004"
Attach Point 3 Ypod 100.0000" 100.0063" -0.0063"
_Zpod 39.7680" 39.7795" -0.0115"

Table 3-17. Fwd and aft adjustment mechanism measurement uncertainty in the forward
direction.
Misalignment of the Resulting
T AResulting
Nominal Fwd S6 and Aft S6
Attach point Coordinate axis nominal Fd 6 ad At S6 attach point
solution by 0.0104" in the misalignment
Forward Direction mis
Xpod 106.5000" 106.4997" 0.0003"
Attach Point 1 Ypod 100.0000" 99.9992" 0.0008"
Zpod 152.3460" 152.3565" -0.0105"
Xpod 106.5000" 106.5002" -0.0002"
Attach Point 2 Ypod 100.0000" 99.9995" 0.0005"
Zpod 49.1790" 49.1895" -0.0105"
Xpod 253.5000" 253.5002" -0.0002"
Attach Point 3 Ypod 100.0000" 99.9992" 0.0008"
Zpod 39.7680" 39.7792" -0.0112"









Table 3-18. Rotator bar length measurement uncertainty.
Nomil Misalignment of the Resulting
Attach point Coordinate axis olion Rotator bar S3 attach point
solution
0.0104" Extend misalignment
Xpod 106.5000" 106.4989" 0.0011"
Attach Point 1 Ypod 100.0000" 99.9836" 0.0164"
Zpod 152.3460" 152.3563" -0.0103"
Xpod 106.5000" 106.5001" -0.0001"
Attach Point 2 Ypod 100.0000" 99.9974" 0.0026"
Zpod 49.1790" 49.1893" -0.0103"
Xpod 253.5000" 253.5002" -0.0002"
Attach Point 3 Ypod 100.0000" 99.9983" 0.0017"
_Zpod 39.7680" 39.7800" -0.0120"


Table 3-19. Joint compliance uncertainty.
Nominal Attach point
Attach point Coordinate axis nominal ta ont
solution misalignment
Xpod 106.5000" 0.0022"
Attach Point 1 Ypod 100.0000" 0.3914"
Zpod 152.3460" 0.1172"
Xpod 106.5000" 0.0002"
Attach Point 2 Ypod 100.0000" 0.0886"
Zpod 49.1790" 0.0628"
Xpod 253.5000" 0.0004"
Attach Point 3 Ypod 100.0000" 0.0955"
Zpod 39.7680" 0.0620"









Table 3-20. Summary of Attach Point 1 uncertainty sources.
Attach Point 1
Uncertainty Description ich. P
(inches)
Xpod gpod Zpod
Fwd spherical joint socket location 0.0335 0.0353 0.0285
Aft spherical joint socket location 0.0016 0.0062 0.0020
Orbiter Attach Point 1 location 0.0104 0.0104 0.0104
Orbiter Attach Point 2 location 0.0000 0.0000 0.0000
Orbiter Attach Point 3 location 0.0000 0.0000 0.0000
APS pod Attach Point 1 location 0.0104 0.0104 0.0104
APS pod Attach Point 2 location 0.0000 0.0000 0.0000
APS pod Attach Point 3 location 0.0000 0.0000 0.0000
Lifting fixture nominal Attach Point 3 location 0.0349 0.0698 0.0349
Adjustment capability of Attach Point 3 0.4184 0.0000 0.0000
Measurement of rotator bar base location 0.0011 0.0164 0.0103
Calculations by computer program 0.0000 0.0000 0.0000
Iterative process acceptance criteria 0.0005 0.0005 0.0004
Measurement of fwd adjustment mechanism uphill/downhill 0.0071 0.0080 0.0226
0.0071 0.0080 0.0226
position
Measurement of aft adjustment mechanism uphill/downhill 0.0071 0.0058 0.0090
position
Measurement of fwd adjustment mechanism off-the-deck/on- 0.0002 0.0103 0.0106
0.0002 0.0103 0.0106
the-deck position
Measurement of aft adjustment mechanism off-the-deck/on-
0.0007 0.0039 0.0110
the-deck position
Measurement of adjustment mechanism forward/aft position 0.0003 0.0008 0.0105
Measurement of rotator bar joint offset 0.0011 0.0164 0.0103
Compliance in joints 0.0022 0.3914 0.1172









Table 3-21. Summary of Attach Point 2 uncertainty sources.
Attach Point 2
Uncertainty Description ich. P
(inches)
Xpod gpod Zpod
Fwd spherical joint socket location 0.0181 0.0224 0.0285
Aft spherical joint socket location 0.0141 0.0038 0.0020
Orbiter Attach Point 1 location 0.0000 0.0000 0.0000
Orbiter Attach Point 2 location 0.0104 0.0104 0.0104
Orbiter Attach Point 3 location 0.0000 0.0000 0.0000
APS pod Attach Point 1 location 0.0000 0.0000 0.0000
APS pod Attach Point 2 location 0.0104 0.0104 0.0104
APS pod Attach Point 3 location 0.0000 0.0000 0.0000
Lifting fixture nominal Attach Point 3 location 0.0349 0.0698 0.0349
Adjustment capability of Attach Point 3 0.4184 0.0000 0.0000
Measurement of rotator bar base location 0.0001 0.0026 0.0103
Computer program roundoff error 0.0000 0.0000 0.0000
Iterative process acceptance criteria 0.0001 0.0005 0.0004
Measurement of fwd adjustment mechanism uphill/downhill 0.0011 0.0005 0.0225
position
Measurement of aft adjustment mechanism uphill/downhill 0.0008 0.0003 0.0090
position
Measurement of fwd adjustment mechanism off-the-deck/on- 0.0006 0.0113 0.0105
0.0006 0.0113 0.0105
the-deck position
Measurement of aft adjustment mechanism off-the-deck/on-
0.0004 0.0022 0.0110
the-deck position
Measurement of adjustment mechanism forward/aft position 0.0002 0.0005 0.0105
Measurement of rotator bar joint offset 0.0001 0.0026 0.0103
Compliance in joints 0.0002 0.0886 0.0628









Table 3-22. Summary of Attach Point 3 uncertainty sources.
Attach Point 3
Uncertainty Description ich. P
(inches)
Xpod gpod Zpod
Fwd spherical joint socket location 0.0167 0.0078 0.0067
Aft spherical joint socket location 0.0153 0.0155 0.0159
Orbiter Attach Point 1 location 0.0000 0.0000 0.0000
Orbiter Attach Point 2 location 0.0000 0.0000 0.0000
Orbiter Attach Point 3 location 0.0104 0.0104 0.0104
APS pod Attach Point 1 location 0.0000 0.0000 0.0000
APS pod Attach Point 2 location 0.0000 0.0000 0.0000
APS pod Attach Point 3 location 0.0104 0.0104 0.0104
Lifting fixture nominal Attach Point 3 location 0.0349 0.0698 0.0349
Adjustment capability of Attach Point 3 0.4184 0.0000 0.0000
Measurement of rotator bar base location 0.0002 0.0017 0.0120
Computer program roundoff error 0.0000 0.0000 0.0000
Iterative process acceptance criteria 0.0001 0.0005 0.0001
Measurement of fwd adjustment mechanism uphill/downhill 0.0005 0.0004 0.0139
position
Measurement of aft adjustment mechanism uphill/downhill 0.0001 0.0006 0.0180
position
Measurement of fwd adjustment mechanism off-the-deck/on- 0.0007 0.0023 0.0112
the-deck position
Measurement of aft adjustment mechanism off-the-deck/on-
0.0004 0.0063 0.0115
the-deck position
Measurement of adjustment mechanism forward/aft position 0.0002 0.0008 0.0112
Measurement of rotator bar joint offset 0.0002 0.0017 0.0120
Compliance in joints 0.0004 0.0955 0.0620

Table 3-23. Total uncertainty using the 100% covariance method.
Attach Point 1 Attach Point 2 Attach Point 3
(inches) (inches) (inches)
Description Xpod/Zpod Ypod Xpod Ypod Zpod Xpod/Zpod Ypod
Total
otal 0.6028 0.5856 0.5099 0.2259 0.2335 0.5585 0.2237
Uncertainty
Accuracy
Accuracy 0.2231 0.0010 2.0000 0.0010 0.0063 0.0033 0.0010
Requirement
Remaining -0.3797 -0.5846 1.4901 -0.2249 -0.2272 -0.5552 -0.2227
Margin






51


Table 3-24. Total uncertainty using the root sum squared method.
Attach Point 1 Attach Point 2 Attach Point 3
(inches) (inches) (inches)
Description Xpod/Zpod Ypod Xpod Ypod Zpod Xpod/Zpod Ypod
Total
otal 0.4414 0.4004 0.4207 0.1166 0.0856 0.4287 0.1207
Uncertainty
Accuracy 0.2231 0.0010 2.0000 0.0010 0.0063 0.0033 0.0010
Requirement
Remaining -0.2183 -0.3994 1.5793 -0.1156 -0.0763 -0.4254 -0.1197
Margin














CHAPTER 4
EVALUATION OF PROPOSED ALIGNMENT METHOD

Both uncertainty calculation methods used in Chapter 3 indicate that the proposed

solution method is not as accurate as desired. If used, it would not be able to align the

APS pod with the orbiter deck accurately enough to eliminate the need for additional

manipulations. The additional manipulations required for alignment can not be

calculated before the operation because the direction and magnitude of the misalignment

can not be predicted. Additional manipulations are highly undesirable because the

motion resulting from mechanism adjustments is not intuitive. As a result, the

uncertainty analysis will be further examined and recommendations will be made to

reduce total uncertainty.

Discussion of Results

Uncertainty due to tolerances can be significantly reduced by measuring "as-built"

dimensions. The uncertainty associated with those measurements is significantly less

than the uncertainty due to tolerances, in some cases by more than one order of

magnitude. Measurements of specific points on the lifting fixture, APS pod, and orbiter

can be taken before APS pod installation.

Uncertainty can be further reduced by more accurately measuring rotator bar joint

offset S5 and adjustment mechanism joint offsets S4, S5, and S6. Improvements to reduce

uncertainty might require significant modifications to rotator bar and adjustment

mechanism hardware. However, these modifications are highly desirable because the









uncertainty associated with each joint offset measurement exceeds the total allowable

uncertainty at Attach Point 3 by one order of magnitude.

The estimated uncertainty due to compliance in rotator bar and adjustment

mechanism joints is one of the most significant error sources studied. Although joint

compliance can not be reduced without significant modifications to hardware, its effect

can be minimized. There are only two lifting fixtures, one for right APS pods and one for

left APS pods, and those lifting fixtures are in approximately the same position and

orientation during each APS pod installation. The geometry of the lifting fixture

configuration indicates that each joint is under load during this operation and those loads

determine the direction affected by compliance. As a result, joint compliance has only a

minimal effect on precision. To obtain accurate results, a correction factor can be used to

compensate for joint compliance error.

A correction factor might also be used to compensate for uncertainties related to

lifting fixture, APS pod, and orbiter geometry. For a given lifting fixture, APS pod, and

orbiter it might be discovered that the calculated solution results in a misalignment that is

consistent in magnitude and direction. This precise solution could be made more

accurate by implementing a correction factor. Since there are only two lifting fixtures,

three orbiters, and a small number of APS pods, a database could be created relatively

quickly to facilitate the calculation of a correction factor for each scenario.

Recommendations

The lifting fixture, adjustment mechanisms, and rotator bar are not currently

outfitted with a means of measuring joint offsets. Therefore, it is suggested that

measurement devices for each joint offset be installed. Laser rangefinders should be

considered because they provide high accuracy and can be installed without modifying









load-bearing components. The proposed solution method will allow the ground support

team to align the APS pod by manipulating joints to precalculated positions. Without

measurement devices, the ground support team will not know when the precalculated

positions have been reached. Based on an uncertainty analysis, an accuracy of +0.010"

is insufficient for measurement devices. Overall system accuracy would be greatly

improved if measurement device accuracy approached 0.001". Crude measurement

techniques such a rulers or tape measures would provide highly inaccurate results.

The lifting fixture design includes adjustment capability at all attach points to

accommodate APS pod dimensional variation. This adjustment capability is large

enough that it must be accounted for when considering the location of the APS pod

relative to reference points on the lifting fixture. The simplest and most accurate method

of determining the position of the APS pod relative to the lifting fixture is to make high-

accuracy measurements before each APS pod is installed. APS pod attach points 1, 2,

and 3 must be measured and the rotator bar and adjustment mechanism end effector

locations. Since the orientation of the adjustment mechanism end effectors must be

known, a second point along each adjustment mechanism S6 and a6 vectors must also

be measured.

Total uncertainty can be further reduced by making additional high-accuracy

measurements of the rotator bar base and spherical joint socket locations. Orbiter attach

point positions can be similarly determined. These measurements yield only marginal

accuracy improvements but are desirable nonetheless. In general, high-accuracy

measurements can be used to compensate for insufficiently loose tolerances.









As previously mentioned, a hydraulic jack is used to ensure the aft adjustment

mechanism "follows" the forward adjustment mechanism during forward/aft motion of

the lifting fixture. The hydraulic jack is not a precise method for positioning the aft

adjustment mechanism. The adjustment mechanisms could be more accurately

positioned in the forward/aft direction before they are installed in the sockets. Since the

adjustment mechanisms will not be supporting the weight of the lifting fixture or APS

pod at that point, they can be adjusted to the precalculated solution position without

experiencing binding. If the S6 joint offset calculations are correct, the adjustment

mechanisms will not need to be adjusted in the forward/aft direction during operations.

One advantage of the proposed solution method is that joint offset adjustments do

not need to be made in any particular sequence. Therefore, it is not imperative that

motions be simultaneous with one exception. As the APS pod is lowered to the orbiter

deck, the bottom of the APS pod should be approximately parallel to the orbiter deck

surface. If one bulb seal compresses before the others make contact, then the resulting

frictional force might cause a slight misalignment. To minimize this effect, a reverse

kinematic analysis should also be performed to position the APS pod at a waypoint 1/8"

above the orbiter deck and aligned in Xpod and Zpod. The APS pod can be slowly lowered

onto the deck from that waypoint by simultaneously adjusting the rotator bar and

adjustment mechanisms.

It would also be beneficial to calculate the joint angle adjustments needed to move

the APS pod in small increments along each axis. If the lifting fixture is manipulated to

the precalculated joint angles and found to be misaligned, the ground support team will









know how to manipulate the lifting fixture to align the APS pod rather than rely on

intuition.

A Finite Element Analysis (FEA) should be performed on the lifting fixture with

the APS pod in the installed position. This will allow the rigid body assumption to be

validated. If the FEA shows that lifting fixture deflection is significant, then the

deflected lifting fixture shape will be used throughout reverse kinematic analysis

calculations.

Summary of Recommendations

The following recommendations should be implemented to ensure success of the

proposed alignment method:

* Install measurement devices to measure all variable joint offsets.

* Make high-accuracy measurements (+0.001") of all critical relative positions.

* Position the adjustment mechanisms in the forward/aft direction per S6 joint offset
solutions before they begin supporting lifting fixture and APS pod weight.

* Position the APS pod at a waypoint 1/8" above the orbiter deck and aligned along
the plane of the orbiter deck. The joint offsets needed to position the pod at the
waypoint are calculated by reverse kinematic analysis.

* Use joint angle adjustments for small misalignments along each pod coordinate
axis.

* Perform FEA to validate the rigid body assumption.

Conclusions

It should be evident at this point that the design of the lifting fixture, adjustment

mechanisms, and rotator bar is not adequate for the precision positioning task it is

required to perform. Given the shuttle program's limited resources, it is highly unlikely

that this GSE will be redesigned to improve operations in lieu of upgrades to flight

hardware. The reverse kinematic analysis presented in my study does not add the desired






57


level of accuracy to the APS pod alignment operation. However, the level of accuracy it

does provide is a substantial improvement to the current process. Implementation of this

solution method adds a relatively small amount of manpower and cost to operations

compared to the projected benefit. This solution method is a viable aid to APS pod

alignment during installation onto the orbiter.














APPENDIX A
SHUTTLE COORDINATE SYSTEMS

Several different coordinate systems are commonly used to describe locations on

the space shuttle. Two of these coordinate systems are the orbiter coordinate system and

the APS pod coordinate system.

The orbiter coordinate system is essentially the orbiter's version of a body-fixed

coordinate system. The origin is located forward of the orbiter's nose. The positive X-

direction is aft, positive Y- direction is outboard through the starboard wing, and positive

Z- direction is out through the vertical tail. The location and orientation of the orbiter

coordinate system can be seen in Figure A-1. Orbiter coordinates are used throughout

this analysis except when the location of the orbiter deck is specifically needed.

The APS pod coordinate systems are local coordinate systems used to describe

orbiter locations relative to the orbiter deck planes. In both right APS pod coordinates

and left APS pod coordinates, the plane of the appropriate orbiter deck can be described

by the equation Ypod = 100. The origin is located forward of the APS pod and below the

orbiter deck. The positive X- direction is aft and slightly outboard, positive Y- direction

is perpendicular to the orbiter deck and includes an outboard component, and positive Z-

direction is angled inboard along the plane of the orbiter deck. The right APS pod

coordinate system can be seen in Figure A-2.

The transformation matrices between orbiter coordinates and right APS pod

coordinates are documented as









0.9986104865 -0.0526981625 0 1212.4087676
Orbiter 0.0379078857 0.7183402679 0.6946583705 59.30094352
RightPod T 0.0366072197 0.6936931332 0.7193398003 311.74989638

0 0 0 1
0.9986104865 0.0379078857 0.0366072197 -1224.3843795 (A
RightPod -0.0526981625 0.7183402679 0.6936931332 -194.96530381
Orbiter 0 -0.6946583705 0.7193398003 -183.06021143

0 0 0 1

Drawings containing the APS pod and its ground support equipment show

hardware associated with the left APS pod. The CAD model is based on these drawings

and therefore also shows left APS pod hardware. Unfortunately, transformation matrices

for the left APS pod coordinate system are not documented.

The location and orientation of the APS pod coordinate systems are symmetric

about the orbiter coordinate XZ plane. Using this fact, transformation matrices for the

left APS pod coordinate system can be calculated.

It should be noted that the left APS pod coordinate system does not obey the right

hand rule. In the interest of avoiding calculation errors due to the use of a left hand

coordinate system, the derived left APS pod coordinate system will be made to obey the

right hand rule by inverting its Z- axis.

First, the rotation angle 0 about unknown unit vector m can be calculated from


S= cos ll r22 -3 = 44.0984373537 (A.2)


where rj, r22, and r33 are the first three diagonal terms in oRit d T. The unit vector m


can be determined from











mn =+ -cos = +0.997532008229
-1 cos

my = 2m1 +r2 = +0.026302324052
2y m (1-cosO)
r, +r,
m = 13 = 3 +0.065100540004
2m (- cos0)

The sign of mx and subsequent signs of my and mz are determined from


m = -0.997532008229
2sin
m = -0.026302324052


(A.3)


(A.4)


m = -0.065100540004


To calculate the left APS pod coordinate system, a rotation of -0 about the mirror of

vector m will be conducted. The resulting transformation matrix is


mnxnu + cos 0

Orbiter -my + m sin 0
LeftPod T
LeflPod mxu + m sin

0


-mnmug-m sin8
mymyU + cos 0
-mymu + m sin
0


mm u -m sin
-mym U mx sin 0
n mmu + cos
0


-194.965
183.060 (A.5)
1


v cos = 0.2818547227

After altering the matrix to invert the Z- axis, the result is


0.9986104865
-0.0526981625
0
0


-0.0379078857
-0.7183402679
-0.6946583705
0


0.0366072197
0.6936931333
-0.7193398003
0


1224.380
-194.965
(A.6)
183.060
1


The inverse of this matrix is


0.9986104864
-0.0379078857
0.0366072197
0


-0.0526981625
-0.7183402680
0.6936931333
0


0
-0.6946583705
-0.7193398003
0


1212.40876751
-59.3009435119
(A.7)
311.749896399
1


Orbiter T =
LeftPod


Orbiter T =
LeftPod T








In retrospect, the simplest solution would have been to have the program invert all

negative orbiter Y- coordinates input into the program. This would have eliminated the

need for the left APS pod coordinate derivation because the right APS pod coordinate

system could be used instead.


--Y.


Figure A-1. Orbiter coordinate system.




































Figure A-2. Right APS pod coordinate system. [2]














APPENDIX B
REVERSE KINEMATIC ANALYSIS NOTATION

The reverse kinematic analysis notation used in my study is defined as follows.

Subscripts h, i, j, k, and 1 are used where f=i-3, g=i-2, h=i-1, j=i+l, k=i+2, and l=i+3.


cos O
sin 8
SCOS a'
- sin a
- S1 jS
S( SjkCj + Cjk SCj )
- CjkCj SjkS1jCj
S--k S jk

- -SCjk + CjSjkCj )
- CjCjk -SjSjkCj
SXc YS

-^-
-Ck j + c)- kZ
i Sjk (X sj + cj + CkZ,
XkC kS
c- (XkSJ +YkC) S Zk

Ss(xk, X YkC )+ CZk
- Xck Sk
SCkI ( Xsk + -Ck) SklZU

=Sk (Xsk +Ck k)+ CklZ
- XkC -Y kS
Ch, (Xch S + S kC,) -Sh Zk

S=, (XkS, + Y, C)+ ChZ


(B.1)




(B.2)




(B.3)



(B.4)




(B.5)




(B.6)




(B.7)









Xhjk Xh jk h Sk
Yhu(k Ckl (Xhjsk +YhCk)- SklZhj (B.8)
ZhUk Skl ( XhUSk + YiCk )+ CklZhy
Xk7zh Xk C h Aki Sh
Yk1,h Cgh (X +Ysh + i h) SghZk (B.9)
ZIgth Sgh (Xk Sh + YCh) + CghZk7,
XhjUki XhujkCI hjkSl
Yhuki Cim (XhukSl + YhUkCI Sl)lZhuk (B. 10)
ZhUikl sIm- (XhjkS, + YhzijkcI ) + clmZhUk
Xllekh ~ Xllki Ch Iji, Sh
Ylk'h Cgh (xlksh +Y ch)- SghZlkJ, (B.11)
ZlIkzh Sgh (XlkgrSh + I,Ch ) + CghZilkl
u, s s
V =-(sc +csc^) (B.12)
Wl Cc S C'O
U CC S Si

Vi s (c +cs c J) (B.13)
W cci -S, s C
U Uk U c -k jSk = U,,
Vik Ck (UU+Sk cJ )- Sk = Vkz (B.14)
WKk Sk (Usjk + Jcjk)+ Ckj Wi,,
UhUk Uhc jk VhoSjk = Uk7,h
Vhk hCk (UhSVk + VhJk SkWh= Vkjh (B.15)

WhiSkSk (UhijSk + VhUJsCk U CkWhj W~,kh
Ughjk Ughi jk VghjSjk = Ulkhg
Vghk Ck (UghzSJk + VghUJk)- SkWghz =Vjlhg (B.16)
Wghik Sk (UghUSJyk + VghUCJk) k+ CkWghj kWihg
Ufghjk UfghU jk fghuSjk = Ukjghgf
VfgJhk Ck (UfghUSYk +Vfgh k- SkWfghj VA,,hgf (B.17)
Wfghjk Sk (UfghiSJk +Vfghj Ck )CkWfghj kjWhgf















LIST OF REFERENCES


[1] D. R. Jenkins, The History of the National Space Transportation System: The
First 100 Missions, 3rd ed. Stillwater, MN: Voyageur Press, 2001.

[2] Rockwell International, "Lifting Fixture Set-OMS/RCS Pod," National
Aeronautics and Space Administration, Houston, TX H70-0679, 1978.

[3] D. Armstrong, "John F. Kennedy Space Center Multimedia Gallery," n.d.;
http://mediaarchive.ksc.nasa.gov/index.cfm, 23 Oct. 2004.

[4] C. D. Crane III and J. Duffy, Kinematic Analysis of Robot Manipulators. New
York: Cambridge University Press, 1998.

[5] W. H. ElMaraghy, H. A. ElMaraghy, A. Zaki, and A. Massoud, "Design and
Control of Robots with Flexibilities," Annals of the CIRP, vol. 43, pp. 359-362,
1994.















BIOGRAPHICAL SKETCH

Jeff Brink graduated from the University of Michigan in December 2000 with

Bachelor of Science degrees in aerospace engineering and mechanical engineering. Jeff

started work at NASA in February 2001 as a systems engineer in shuttle processing at the

Kennedy Space Center in Florida. In September 2001, he enrolled part-time in graduate

school at the University of Central Florida. One year later, Jeff transferred to the

University of Florida to study robotics. With the help of the University of Florida's

distance learning program, FEEDS, Jeff will graduate in May 2005 with a Master of

Engineering degree.




Full Text

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REVERSE KINEMATIC ANALYSIS AND UNCERTAINTY ANALYSIS OF THE SPACE SHUTTLE AFT PROPULSI ON SYSTEM (APS) POD LIFTING FIXTURE By JEFFREY S. BRINK A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Jeffrey S. Brink

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This thesis is dedicated to the Kennedy Space Center workers who have done their best to help this effort succeed, in hopes of making operations better and safer.

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ACKNOWLEDGMENTS I would like to thank NASA Orbiter Handlings Rob Summers and United Space Alliance (USA) Orbiter Handlings Glenn Roberts for their invaluable help. As the experts on this hardware and installation process, they responded promptly and patiently to my constant barrage of questions. Their willingness to do anything they can to make operations safer is commendable. They shoot a pretty good game of pool, too. Boeing Orbiter Handlings Will Judd was also very helpful. He answered several technical questions and verified the accuracy of some information I had found on my own. NASA Payload Mechanical Engineerings Doug Lenhardt and NASA Mechanical Design Engineer Paul Schwindt were instrumental in the Pro/E portion of this study. Doug and Paul answered questions that helped my knowledge of Pro/E grow from beginner level to intermediate. After I experienced quite a bit of difficulty getting the integrated C program to run, NASA Senior Software Engineer Dan Nieten was kind enough to teach me several very helpful debugging techniques. These techniques enabled me to figure out what was wrong, and how to make it work. NASA Thermal Protection Systems Lisa Huddleston provided guidance on performing my literature review and gave tips on how to use numerical methods to reduce error in the integrated program. I also frequently consulted Lisa about details of iv

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this study and (even though robotics is not her field of expertise) she never seemed to run out of good questions. I would also like to thank my supervisory committee for their contribution. Dr. Carl Crane III (my supervisory chair) was particularly helpful. The concepts taught in his textbook formed the backbone to this solution method. Dr. John Scheuller and Dr. Ashok Kumar provided good insight as this thesis reached its conclusion. NASA Launch Accessories Kristina Morace and NASA Orbiter Handlings Ryan Holmes checked the technical content of this thesis and helped make it more clear and understandable. v

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES.............................................................................................................x LIST OF ABBREVIATIONS............................................................................................xi ABSTRACT.....................................................................................................................xiii CHAPTER 1 BACKGROUND..........................................................................................................1 Introduction...................................................................................................................1 Hardware Familiarization.............................................................................................2 Installation Procedure and Methodology......................................................................4 2 PROPOSED ALIGNMENT METHOD.....................................................................10 Overview.....................................................................................................................10 Determination of Desired Joint Angles......................................................................10 Alignment of APS Pod Attach Points with Orbiter Attach Points......................11 Adjustment Mechanism Reverse Kinematic Analysis........................................17 Adjustment mechanism parameters.............................................................17 Close-the-loop variable calculations............................................................18 Reverse kinematic analysis of a PPPS mechanism......................................19 Rotator Bar Length Calculation...........................................................................26 Nominal Solution........................................................................................................27 3 UNCERTAINTY ANALYSIS...................................................................................31 Off-Nominal Conditions within Tolerance.................................................................31 Adjustment Mechanism Spherical Joint Socket Locations.................................31 Orbiter Attach Point Locations............................................................................31 The APS Pod Fitting Locations...........................................................................32 Lifting Fixture Attach Point 3 Location..............................................................32 Lifting Fixture Adjustment at Attach Point 3......................................................32 vi

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Uncertainty of Input Values........................................................................................33 Calculation-Related Uncertainties..............................................................................33 Hardware Positioning Uncertainty.............................................................................34 Uphill/Downhill Joint Offset Measurement........................................................34 Off-the-Deck/On-the-Deck Joint Offset Measurement.......................................35 Forward/Aft Joint Offset Measurement...............................................................35 Rotator Bar Joint Offset Measurement................................................................35 Compliance in Joints...........................................................................................36 Total Uncertainty Calculation.....................................................................................36 One Hundred Percent Covariance Method..........................................................36 Root Sum Squared Method.................................................................................37 4 EVALUATION OF PROPOSED ALIGNMENT METHOD....................................52 Discussion of Results..................................................................................................52 Recommendations.......................................................................................................53 Summary of Recommendations..................................................................................56 Conclusions.................................................................................................................56 APPENDIX A SHUTTLE COORDINATE SYSTEMS.....................................................................58 B REVERSE KINEMATIC ANALYSIS NOTATION.................................................63 LIST OF REFERENCES...................................................................................................65 BIOGRAPHICAL SKETCH.............................................................................................66 vii

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LIST OF TABLES Table page 2-1 Adjustment mechanism parameters.........................................................................30 2-2 Comparison of joint offsets calculated by the program to measured using the CAD model..............................................................................................................30 3-1 Forward spherical joint socket tolerances................................................................38 3-2 Aft spherical joint socket tolerances........................................................................38 3-3 Orbiter Attach Point 1 tolerances.............................................................................39 3-4 Orbiter Attach Point 2 tolerances.............................................................................39 3-5 Orbiter Attach Point 3 tolerances.............................................................................40 3-6 APS pod Attach Point 1 tolerances..........................................................................40 3-7 APS pod Attach Point 2 tolerances..........................................................................41 3-8 APS pod Attach Point 3 tolerances..........................................................................41 3-9 Lifting fixture Attach Point 3 tolerances..................................................................42 3-10 Lifting fixture Attach Point 3 adjustment................................................................42 3-11 Misalignment of the rotator bar base........................................................................43 3-12 Computer program uncertainty................................................................................43 3-13 Forward adjustment mechanism uphill measurement uncertainty...........................44 3-14 Aft adjustment mechanism uphill measurement uncertainty...................................45 3-15 Forward adjustment mechanism off-the-deck measurement uncertainty.................45 3-16 Aft adjustment mechanism off-the-deck measurement uncertainty.........................46 3-17 Fwd and aft adjustment mechanism measurement uncertainty in the forward direction....................................................................................................................46 viii

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3-18 Rotator bar length measurement uncertainty...........................................................47 3-19 Joint compliance uncertainty....................................................................................47 3-20 Summary of Attach Point 1 uncertainty sources......................................................48 3-21 Summary of Attach Point 2 uncertainty sources......................................................49 3-22 Summary of Attach Point 3 uncertainty sources......................................................50 3-23 Total uncertainty using the 100% covariance method.............................................50 3-24 Total uncertainty using the root sum squared method.............................................51 ix

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LIST OF FIGURES Figure page 1-1 A left APS pod is being removed from the space shuttle orbiter Atlantis .................7 1-2 The twelve APS pod attach point locations, left pod shown (right pod mirror).................................................................................................... ...7 1-3 GSE used to install an APS pod.................................................................................8 1-4 Forward and aft adjustment mechanisms allow motion in forward/aft, uphill/downhill, and off-the-deck/on-the-deck directions..........................................8 1-5 Rotator bar joint axes.................................................................................................9 1-6 A left APS pod is transported by crane to Atlantis for installation ...........................9 2-1 Joint offset calculation procedure.............................................................................28 2-2 Three points are needed to determine each adjustment mechanism's position and orientation (aft adjustment mechanism shown, typical of all adjustment mechanisms).............................................................................................................29 2-3 Adjustment mechanism joint axis vectors and link vectors.....................................29 3-1 Adjustment mechanism joint axis vectors and link vectors.....................................44 A-1 Orbiter coordinate system........................................................................................61 A-2 Right APS pod coordinate system............................................................................62 x

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LIST OF ABBREVIATIONS Term or acronym Definition Adjustment mechanisms Two PPPS manipulators used for OMS pod alignment. APS Aft Propulsion System APS pod Orbiter component that houses the Orbital Maneuvering System and the aft Reaction Control System. Cylinder joint A joint that allows rotational and translational motion about the same axis. FEA Finite Element Analysis GSE Ground Support Equipment Hydraulic jack A force-output device used to overcome the binding condition experienced by the aft adjustment mechanism and cause forward/aft motion along a prismatic joint axis. Hypergol A rocket fuel that spontaneously ignites when mixed with an oxidizer. Monomethyl hydrazine is the hypergol used by the OMS. KSC Kennedy Space Center Lifting fixture A large structure that attaches to the OMS pod and is manipulated by the adjustment mechanisms and rotator bar. Move director The USA technician that uses information from technicians and engineers to determine the next manipulation. NASA National Aeronautics and Space Administration Nominal solution The joint offsets required to align the lifting fixture. OMS Orbiter Maneuvering System, propulsion system that provides thrust for orbital insertion, orbit circularization, orbit transfer, rendezvous, and deorbit. [1] OPF Orbiter Processing Facility xi

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Term or acronym Definition Orbiter The Orbiter is a double-delta winged reentry vehicle capable of carrying both passengers and cargo to low-earth orbit and back to a controlled gliding landing. [1] The Orbiter is the only Space Shuttle element to reach orbit. NASAs Space Shuttle fleet consists of three orbiters: Atlantis, Endeavour, and Discovery. Orbiter deck The orbiter surface that mates to the OMS pod. Plcker coordinates Homogeneous coordinates used to describe points, lines, or planes. Plcker coordinates can be used to simplify equations so that calculations can be performed more efficiently. Prismatic joint A joint that allows translational motion only (also known as a slider joint). Pro/E Pro/ENGINEER, a CAD software package used for modeling and finite element analysis. RCS Reaction Control System, propulsion system used as the primary flight control at altitudes greater than 70,000 feet. [1] Revolute joint A joint that allows rotational motion only (also known as a hinge joint). Rotator bar A RRPRRR manipulator used for OMS pod alignment. Space shuttle NASAs only manned spaceflight vehicle. The Space Shuttle is comprised of an orbiter, external tank, and two solid rocket boosters. Spherical joint A joint that allows rotational motion in x, y, and z directions (also known as a ball-and-socket joint). USA United Space Alliance, a Boeing and Lockheed Martin joint venture. USA is the prime contractor for space shuttle operations. xii

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering REVERSE KINEMATIC ANALYSIS AND UNCERTAINTY ANALYSIS OF THE SPACE SHUTTLE AFT PROPULSION SYSTEM (APS) POD LIFTING FIXTURE By Jeffrey S. Brink May 2005 Chair: Carl D. Crane III Major Department: Mechanical and Aerospace Engineering The space shuttle Aft Propulsion System (APS) pod requires precision alignment to be installed onto the orbiter deck. The Ground Support Equipment (GSE) used to perform this task cannot be manipulated along a single Cartesian axis without causing motion along the other Cartesian axes. As a result, manipulations required to achieve a desired motion are not intuitive. My study calculated the joint angles required to align the APS pod, using reverse kinematic analysis techniques. Knowledge of these joint angles will allow the ground support team to align the APS pod more safely and efficiently. An uncertainty analysis was also performed to estimate the accuracy associated with this approach and to determine whether any inexpensive modifications can be made to further improve accuracy. xiii

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CHAPTER 1 BACKGROUND Introduction The Kennedy Space Center (KSC) is NASAs operations center for the space shuttle. This role includes mission configuration and deconfiguration, vehicle modifications, repair, and routine maintenance (essentially everything that happens to a space shuttle orbiter, from landing through launch). The installation of an Aft Propulsion System (APS) pod onto an orbiter is one example of a KSC operation. My study aimed to improve the APS pod installation operation by calculating the joint offsets required to align the APS pod with the orbiter deck. These calculations can be performed before the operation begins, which will reduce the operational time spent performing the alignment. APS pod installation is classified as a hazardous operation, which makes reducing the operational time particularly desirable. Throughout the operation, the potential exists for a hypergol leak to develop, which could be lethal to nearby personnel. The Ground Support Equipment (GSE) used to perform an APS pod installation consists of a lifting fixture, two adjustment mechanisms, and rotator bar. There are a total of 13 joints that change position during the operation. My analysis showed that they can be considered as three separate manipulators connected by a large structure. After the desired position of the large structure has been ascertained, the end effector position and orientation of each manipulator is found. A reverse kinematic analysis is then performed for each manipulator, to determine joint angles. 1

PAGE 15

2 An uncertainty analysis was then conducted. Both 100% covariance and root sum squared methods were used to quantify alignment accuracy. This analysis was used to determine whether the solution method described above is accurate enough to aid APS pod-installation operations; and to determine which of the error sources can be eliminated for significant improvements in accuracy. Hardware Familiarization Each space shuttle orbiter has two APS pods, one on each side of the vertical tail. Each APS pod houses the Orbital Maneuvering System (OMS) and the aft Reaction Control System (RCS). OMS is the propulsion system that provides thrust for orbital insertion, orbit circularization, orbit transfer, rendezvous, and deorbit [1]. The RCS propulsion system is used as the primary flight control at altitudes greater than 70,000 feet [1]. An APS pod is shown in Figure 1-1 The APS pods were not designed to be maintenanced while installed on an orbiter. As a result, APS pods must be removed for inspections and repairs, generally after every three or four flights. On completion of these tasks, the APS pod is installed onto the orbiter. The APS pod is attached to the orbiter deck at 12 locations (called attach points). During the installation operation, all efforts are focused on aligning attach points 1, 2, and 3. If those three attach points are aligned, then attach bolts can be installed at all twelve attach-point locations. Attach Point 1 is the forward inboard attach point, Attach Point 2 is the forward outboard attach point, and Attach Point 3 is the aft outboard attach point ( Figure 1-2 ). The APS pod Attach Point 3 fitting must be aligned within 0.0033 (+/0.0018 depending on tolerances) of the orbiter bushing or else the attach bolt cannot be installed.

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3 The APS pod Attach Point 2 bushing resides in a slotted hole, which prevents pod-to-pod variation and other factors from making an APS pod not fit onto an orbiter. The APS pod Attach Point 2 fitting must be aligned within 0.0063 +/0.0048 in the non slotted direction. Since attach points 2 and 3 are 12 3.3 apart, this accuracy requirement is equivalent to positioning Attach Point 3 within 0.0033 on the orbiter deck and then orienting the APS pod to within 0.0025 of the desired position. The orbiter Attach Point 1 bushing can accommodate misalignments of up to 0.2231 along the orbiter deck plane. The alignment process is made significantly more challenging by the presence of a bulb seal on the bottom surface of the APS pod. The bulb seal is basically a hollow flexible tube that forms an environmental seal when compressed against the orbiter deck. When the bulb seal is compressed, the APS pod cannot be moved along the orbiter deck plane without risking damage to the bulb seal (the bulb seal is more likely to tear than slide along the deck). As a result, if the APS pod is lowered onto the orbiter deck and discovered to be misaligned, then it cannot simply be adjusted until it is aligned. Instead, it must be raised off the deck (until the bulb seal is no longer compressed) before it can be adjusted and then lowered onto the orbiter deck. The ground support equipment used to install APS pods is a lifting fixture, forward adjustment mechanism, aft adjustment mechanism, and rotator bar ( Figure 1-3 ). The lifting fixture is a large structure that connects to the APS pod at APS pod Attach Point 1, 2, 3, and 4 fittings. The lifting fixture can be adjusted at attach points 1, 2, 3 and 4 to accommodate pod-to-pod dimensional variation. As a result of this adjustment capability, not every APS pod will have the exact same position and orientation relative to the lifting fixture.

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4 The forward and aft adjustment mechanisms structurally support the lifting fixture and allow its position to be adjusted. Forward and aft adjustment mechanisms are nearly identical. Temporary supports are mounted to the orbiter and the adjustment mechanisms each form a spherical joint with a support. Adjustment mechanisms have three orthogonal prismatic joints that allow motion in the forward/aft, uphill/downhill, and off-the-deck/on-the-deck directions. The lifting fixture is connected to the adjustment mechanisms by the forward/aft prismatic joint. Figure 1-4 shows adjustment mechanism prismatic joint axis directions. The center of gravity of the lifting fixture and APS pod is not directly above the spherical joints for the duration of APS pod installation. As a result, the rotator bar is used to control the rotation of the lifting fixture about the two adjustment mechanism spherical joints (the two spherical joints essentially form a hinge). One end of the rotator bar is rigidly attached to the lifting fixture and the other end is mounted to a large stationary beam. Revolute joints are used to ensure the rotator bar does not restrict the lifting fixture motion. Figure 1-5 shows the rotator bars revolute joint axes. It should be noted that none of the revolute and spherical joints can be actuated. Their role in the alignment process is purely passive. Installation Procedure and Methodology The APS pod installation operation begins with the APS pod in the Orbiter Processing Facility (OPF) transfer aisle. The lifting fixture, forward adjustment mechanism, and aft adjustment mechanism assembly have already been attached to the APS pod. A bridge crane lifts the APS pod and GSE to a location near the orbiter ( Figure 1-6 ). Adjustment mechanism prismatic joints have been adjusted to provide maximum clearance to the orbiter as the spherical joints are connected. The crane transfers the

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5 weight of the APS pod, lifting fixture, and adjustment mechanisms to the spherical joints. The rotator bar is then extended until it can be connected to the lifting fixture and the alignment portion of this operation begins. According to the APS pod operations and maintenance manual, the rotator bar is extended until the APS pod mating surface is approximately 2.5 above the orbiter deck and the surfaces are parallel. The lifting fixture is moved forward or aft to the install position (a marking on the lifting fixture structure). Additional adjustments are made to align attach points 2 and 3. The APS pod is then lowered to 1/8 above the orbiter deck with mating surfaces parallel. The bulb seal is now contacting the orbiter deck but it is not compressed. Adjustments are made again to align attach points 2 and 3. Once aligned, the APS pod is lowered onto the orbiter deck and observers check to make sure the inboard side of the APS pod is also seated. If the APS pod is misaligned, it must be raised off the deck 1/8 or more, adjusted as required, and lowered back onto the deck. Attach point bolts are installed upon successful completion of the APS pod alignment procedure [2]. APS pod alignment is not accomplished as easily as the operations and maintenance manual describes. It is not possible to adjust the position of one attach point without affecting the position of the other attach points. This is particularly evident when either Attach Point 2 or 3 has been alignedadjustments intended to align one attach point usually misalign the other. The person who determines the next manipulation and commands that it be executed is known as the move director. Different move directors have differing philosophies about whether Attach Point 2 or Attach Point 3 should be aligned first.

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6 Move directors also use different techniques to align an APS pod. One technique is to align the APS pod 1/8 above the orbiter deck, then simultaneously retract the rotator bar and actuate both adjustment mechanisms in the on-the-deck direction. Another technique is to slightly misalign the APS pod, then allow the rotator bar to correct the misalignment as both adjustment mechanisms are simultaneously adjusted in on-the-deck direction. It should be noted that simultaneous motion is accomplished by manual start/stop and velocity control. Additionally, not all equipment operates as designed. Field experience has shown that the forward/aft install position does not necessarily align the APS pod. The aft adjustment mechanisms forward/aft prismatic joint was not designed with an actuator it was designed to follow the motion of the forward adjustment mechanism. However, it binds rather than follows so a hydraulic jack is used to force it to follow.

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7 Figure 1-1. A left APS pod is being removed from the space shuttle orbiter Atlantis [3]. Figure 1-2. The twelve APS pod attach point locations, left pod shown (right pod mirror) [2].

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8 Figure 1-3. GSE used to install an APS pod. Figure 1-4. Forward and aft adjustment mechanisms allow motion in forward/aft, uphill/downhill, and off-the-deck/on-the-deck directions [3].

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9 Figure 1-5. Rotator bar joint axes. Figure 1-6. A left APS pod is transported by crane to Atlantis for installation [3].

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CHAPTER 2 PROPOSED ALIGNMENT METHOD Overview The geometry of all relevant hardware and the desired position and orientation of the APS pod is known. As a result, a reverse kinematic analysis was conducted to determine sets of joint angles that would align the APS pod with the orbiter. Due to the large number of joints associated with this hardware, the rotator bar and adjustment mechanisms were treated as three independent robots connected by a large rigid structure. Since the large structure (lifting fixture) has adjustment capability to attach to the APS pod, it is assumed that these adjustments will be measured before the operation begins and is incorporated into the analysis. The lifting fixture CAD model ( Figure 1-3 ) shows it in the position required to align the APS pod. As discussed previously, the displayed alignment position does not necessarily align the APS pod with the orbiter. The coordinates given in the CAD model were used as a starting point in this analysis. They were used to determine the position and orientation of the three robot end effectors. End effector position and orientation, partnered with known geometry, was used to perform reverse kinematic analyses. The output of these analyses was the joint angles required to align the APS pod. Figure 2-1 shows the process used to determine the manipulator joint offsets. Determination of Desired Joint Angles An existing Pro/E model of the lifting fixture, rotator bar, and adjustment mechanisms was refined for this analysis. The location of each end effector and APS pod 10

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11 attach point, in orbiter coordinates, was ascertained from the model and input into a C program. The C program translated and rotated the input points until the APS pod attach points were aligned with orbiter attach points. The resulting end effector points were used in reverse kinematic analyses to calculate joint angles. These joint angles can be described as the joint angles required to align the APS pod with the orbiter attach points. The orbiter coordinate system and APS pod coordinate systems are discussed in detail in Appendix A. Alignment of APS Pod Attach Points with Orbiter Attach Points The initial position and orientation of the lifting fixture in the CAD model is arbitrary. The lifting fixtures adjustments at attach points 1, 2, 3 and 4 have been physically measured and incorporated into the CAD model. The CAD model also contains seven points on the lifting fixture needed to describe end effector positions and orientations. Attach points 1, 2, and 3 are located in the CAD model of the left APS pod and their desired locations on the orbiter are also included in the model. Adjustment mechanism points can be seen in Figure 2-2 Attach Point 3 has the most stringent accuracy requirements so it is aligned first. The alignment algorithm begins by calculating the misalignment of APS pod Attach Point 3, 3Pod A ttachPtP relative to orbiter Attach Point 3, 3Orbiter A ttachPtP 3OrbiterPodTranslateAttachPtAttachPtPPP 3 (2.1) The APS pod and lifting fixture can be translated such that APS pod Attach Point 3 is aligned by adding to all points located on either object TranslateP

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12 112233PodPodAttachPtAttachPtTranslatePodPodAttachPtAttachPtTranslatePodPodAttachPtAttachPtTranslateRotatorPPPPPPPPPP LiftingFixtureLiftingFixtureBarRotatorBarTranslateLiftingFixtureLiftingFixtureFwdAdjMechOrigFwdAdjMechOrigTranslateFwdAdPPPPPP LiftingFixtureLiftingFixturejMechXFwdAdjMechXTranslateLiftingFixtureLiftingFixtureFwdAdjMechZFwdAdjMechZTranslateAftAdjMechOPPPPPP LiftingFixtureLiftingFixturerigAftAdjMechOrigTranslateLiftingFixtureLiftingFixtureAftAdjMechXAftAdjMechXTranslateAftAdjMecPPPPPP LiftingFixtureLiftingFixturehZAftAdjMechZTranslatePP (2.2) Attach Point 2 has the next most stringent accuracy requirements and therefore is aligned second. The unit vectors from pod Attach Point 3 to pod Attach Point 2 and pod Attach Point 3 to orbiter Attach Point 2 are described as 2332_232332_2PodPodAttachPtAttachPtPodPodPodAttachPtAttachPtOrbiterPodAttachPtAttachPtOrbOrbiterAttachPtAPPVPPPPVPP 3PodttachPt (2.3) The unit vector perpendicular to both 1m 32_PodV and 32_OrbV can be calculated as 32_32_132_32_ P odOrb P odOrbVVmVV (2.4) and the angle 2 between and 32_PodV 32_OrbV is the unique solution to Equations 2.5 and 2.6. 1232_32_cos P odOrbVV (2.5)

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13 1232_32sin P odOrbVV (2.6) Pod Attach Point 2 will become aligned and pod Attach Point 3 will remain aligned if lifting fixture and APS pod points are rotated by angle 2 about an axis containing vector and passing through pod Attach Point 3. This transformation matrix is then calculated. [4] 1m 1T 11121111211112111121112111121111121111211123,3,3,cossinsinsincossinsinsincos000xxxyzxzyxyzyyyzxxzyyzxzzPodAttachPtXPodAttachPtYPoAttachPtZmmmmmmmmmmmmmmmmTmmmmmmmmPPP 1211cosd (2.7) This rotation is then performed on all APS pod and lifting fixture points.

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14 11312231311111PodPodPodAttachPtAttachPtAttachPtPodPodPodAttachPtAttachPtAttachPtPodAttachPtPPPTPPPTP33131111PodPodAttachPtAttachPtLiftingFixtureLiftingFixturePodRotatorBarRotatorBarAttachPtLifFwdAdjMechOrigPPTPPPTP311111tingFixtureLiftingFixturePodFwdAdjMechOrigAttachPtLiftingFixtureLiftingFixtureFwdAdjMechXFwdAdjMechXPPTPPT 3311111PodAttachPtLiftingFixtureLiftingFixturePodFwdAdjMechZFwdAdjMechZAttachPtLiftingFixtureAftAdjMechOrigPPPPTP3131111LiftingFixturePodAftAdjMechOrigAttachPtLiftingFixtureLiftingFixturePodAftAdjMechXAftAdjMechXAttachPtPPTPPPT3111LiftingFixtureLiftingFixturePodAftAdjMechZAftAdjMechZAttachPtPPPT (2.8) Another rotation must be performed to align pod Attach Point 1. The axis of rotation must pass through pod Attach Point 2 and pod Attach Point 3 so that these points do not become misaligned. The Plcker coordinates of this line are 2232_332_{;}{;}PodOLOrbAttachPtOrbSSVPV (2.9) and the parallel line passing through pod Attach Point 1 is given as 1132_132{;}{;}PodOLOrbAttachPtOrbSSVPV (2.10)

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15 Vectors 1 p and 2 p are perpendicular to each line and originate at the orbiter coordinate system origin. 11122OLOLpSS 2 p SS (2.11) When aligned, pod Attach Point 1 will reside on the ypod=100 plane. The ypod coordinate of pod Attach Point 1 can be calculated using part of the transformation matrix: LeftPodOrbiterT 1pod-0.0526981625-0.7183402679y=0.69369313321-194.96530381TPodAttachPtP (2.12) The required angle of rotation 1 can now be calculated. pod1121y100sinpp (2.13) 22211121100cospodppypp (2.14) The transformation matrix equation can be used again to calculate transformation matrix [4] 2T 32_,32_,2132_,32_,232_,132_,32_,232_,132_,32_,21232_,32_,232_,132_,32_,232_,1cossinsincossinsin00PodxPodxPodxPodyPodzPodxPodyPodzPodyPodyPodxPodzPodyPodyPodzPodxVVVVVVVVVVTVVVVVV 32_,32_,232_,13,32_,32_,232_,13,32_,32_,213,sinsincos01Pod P odxPodzPodyAttachPtxPod P odyPodzPodxAttachPtyPod P odzPodzAttachPtzVVVPVVVPVVP (2.15)

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16 where 211cos Transformation matrix is then used to perform the rotation of all points about the line passing through pod Attach Point 2 and pod Attach Point 3. 2T 11322232311111PodPodPodAttachPtAttachPtAttachPtPodPodPodAttachPtAttachPtAttachPtPodAttachPtPPPTPPPTP33232111PodPodAttachPtAttachPtLiftingFixtureLiftingFixturePodRotatorBarRotatorBarAttachPtLifFwdAdjMechOrigPPTPPPTP322111tingFixtureLiftingFixturePodFwdAdjMechOrigAttachPtLiftingFixtureLiftingFixtureFwdAdjMechXFwdAdjMechXPPTPPT 3321111PodAttachPtLiftingFixtureLiftingFixturePodFwdAdjMechZFwdAdjMechZAttachPtLiftingFixtureAftAdjMechOrigPPPPTP3232111LiftingFixturePodAftAdjMechOrigAttachPtLiftingFixtureLiftingFixturePodAftAdjMechXAftAdjMechXAttachPtPPTPPPT3211LiftingFixtureLiftingFixturePodAftAdjMechZAftAdjMechZAttachPtPPPT (2.16) APS pod attach points 1, 2, and 3 have been aligned. A reverse kinematic analysis will now be conducted to determine the adjustment mechanism joint angles with the APS pod aligned.

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17 Adjustment Mechanism Reverse Kinematic Analysis Adjustment mechanism parameters Joint axis vectors and link vectors jS ija must be chosen for the adjustment mechanisms. These selections are shown in Figure 2-3 It should be noted that the spherical joints are treated as three noncoplanar cointersecting revolute joints. [4] Joint angle j is defined as the angle from ija to jka about the vector Similarly, twist angle ij is defined as the angle from jS iS to jS about the vector ija sinsinjjijjkijijijSaaS aS (2.17) Joint offset is the distance from jS ija to jka along jS Link length is the distance from to along The first joint angle, 1, describes the angle from fixed coordinate system Xaxis and vector ija iS jS ija 12a 11sinFixedSX 12a (2.18) Adjustment mechanism joint angles, twist angles, joint offsets, and link lengths are defined in Table 2-1 The constant parameters are exactly the same for both forward and aft adjustment mechanisms. Parameters marked as variable are not necessarily the same for both adjustment mechanisms because adjustment mechanism end effector positions are not identical. Close-the-loop variables are created by the hypothetical closure link. [4] a67 and 67 are user-specified values (rather than a function of manipulator geometry) since the seventh joint is hypothetical.

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18 Close-the-loop variable calculations Close-the-loop variables can be calculated using the constant mechanism parameters listed in Table 2-1 1FixedS is defined as 001T since the vector 1S is exactly aligned with the fixed coordinate system Zaxis. 7FixedS is given by the expression 767FixedFixedFixedSa 6S (2.19) Unit vectors and 67Fixeda 6FixedS are the Xaxis and Zaxis of the adjustment mechanism 6th coordinate system, respectively. They can be calculated as 676FixedLiftingFixtureLiftingFixtureFwdAdjMechXAdjMechOrigFixedLiftingFixtureLiftingFixtureFwdAdjMechZAdjMechOrigaPPSPP (2.20) These definitions allow the unit vector a71Fixed to be determined from 77171aFixedFixedFixedFixedFixedSSS 1S (2.21) The close-the-loop variables can now be calculated. A unique value for the twist angle 71 between vectors and 7S 1S is given by 7171717171cos()sin()FixedFixedFixedFixedFixedSSSS a 7S (2.22) Similarly, the joint angle 7 can be found 7677176771cos()sin()FixedFixedFixedFixedFixedaaaa (2.23)

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19 The angle 1 is defined as the angle between 71a and the Xaxis of the fixed coordinate system. 171171cos()100sin()100TFixedTFixedFixedaaS 1 (2.24) The joint offset S7 along the 7S vector can be calculated by 1771sin()FixedLiftingFixtureFixedAdjMechOrigSPaS 71 (2.25) The link length along the 71a 71a vector is given by 17171sin()LiftingFixtureFixedFixedAdjMechOrigPSa 7S (2.26) Joint offset S1 along the vector is 1S 7171sin()LiftingFixtureFixedFixedAdjMechOrigPSS 71a (2.27) The close-the-loop variables 71, 7, 1, S7, a71, and S1 will be used in the reverse kinematic analysis of this PPPS mechanism. Reverse kinematic analysis of a PPPS mechanism As previously stated, the adjustment mechanisms are RPPPS spatial mechanisms but will be analyzed as an equivalent RPPPRRR mechanism. They are categorized as group 1 mechanisms since the spatial mechanisms and equivalent spherical mechanisms have a single degree-of-freedom. Spherical equations generally contain a high number of terms. However, these terms exist in patterns that allow a shorthand notation to describe the equation more concisely. Notation variables are defined in Appendix B. 1 is the

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20 first unknown joint angle that will be calculated. The fundamental spherical heptagon equation 4567123 Z c (2.28) can be expanded to 12456714567112456723sXsYccZc (2.29) 4567X and are defined by notation variables and 4567Y 456X 456Y 45674567456745677145674567714564567714567456771456XXXYZXYZcYsYcscsZsscc (2.30) 7 and 71 are close-the-loop variables that have already been calculated, so , and 456X 456Y 456 Z are the only unknown terms that must be calculated. They can be defined as 456456456456674564566745456674564566745XXXYZZcYsYcscsZsXsYcc (2.31) 6 is a constant mechanism parameter and 67 is a user-specified value, so both are known quantities. , and 45X 45Y 45 Z must now be defined. 4545454556454556445564545564XXXYZZcYsYcscsZsXsYcc (2.32) 5 and 56 are constant mechanism parameters. , and 4X 4Y 4 Z are defined as 43444453445344453445344Xsc-sssYcsZccsc 4c (2.33)

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21 4, 34 and 45 are constant mechanism parameters so , and 4X 4Y 4 Z can be calculated. Using substitution, and can now be calculated. This results in an equation of the form Using the trigonometric solution method [4], 1 can be calculated 4567X 4567Y 110AcBsD 123124567122124567124567cossYsccZX (2.34) where is the unique solution of 112456722124567124567112456722124567124567sXsinsYsXsYcossYsX (2.35) It should be noted that 1 has two solutions, designated as 1A and 1B. Other joint angles are a function of 1, so it is necessary to solve each joint angle using 1A and 1B. There are two sets of joint angles, solution set A and solution set B, that satisfy the input end effector position and orientation for the specified manipulator geometry. Unknown joint angle 2 can now be calculated using 1A and 1B. The fundamental spherical heptagon equations 4567123245671232 X ssYsc (2.36) will be used. Since , and 4567X 4567Y 4567 Z were determined in the 1 derivation, and can be calculated immediately using 1A and 1B. 45671X 45671Y

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22 45671456714567145671124567145671124567XY-sZXXcYsYcsc (2.37) 2 is the unique solution to equation 2.38 for the solution set containing 1A and the set containing 1B. 145671223145671223sincosXsYs (2.38) The reverse kinematic analysis continues with the calculation of unknown joint angle 3. The fundamental spherical heptagon equations 5671234356712343 X ssYs c (2.39) can be used because 1A, 1B, 2A and 2B have been calculated. and are defined as 56712X 56712Y 56712567125671256712235671256712235671XY-sZXXcYsYcsc (2.40) Similar to the solution for 1, the solution for 3 proceeds by solving for the notation variables that comprise and 56712X 56712Y

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23 567156715671567112567156711256756711256715671125675675665665677156756771565677156756771565656565667565667556675656XXXYZXYZXXXYZZXXXYZcYsYcscsZsscccYsYcscsZsXsYcccYsYcscsZsXsYcc675545555645564555564556455ZXsc-sssYcscZccsc (2.41) A unique solution for joint angle 3 can now be calculated for solution set A and solution set B using their associated joint angles. 156712334156712334=sinsY=cossX (2.42) Joint offset S4 will be calculated using the vector loop equation. The vector loop equation is given by 1112122223233334344445455556566667677771710SSaaSSaaSSaaSSaaSSaaSSaaSSaa (2.43) Since several of these offsets are zero, the vector loop equation can be reduced to 1134344467677771710SSaaSSaaSSaa (2.44)

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24 Using spatial heptagon direction cosines set 5, the vector loop equation [4] becomes 1763445456767671760SXaWSXacSXaW (2.45) Several notation terms in Equation 2.45 must be calculated 7676764554544576676767667677177677167717XXcYsWccsscWccsscXssXssYsccsc (2.46) Unknown joint offset S4 can now be calculated by solving Equation 2.45 for S4. 176344567676717645SXaWacSXaWSX (2.47) Joint offset S6 is calculated by substituting spatial heptagon direction cosines set 4 into the vector loop equation. Equation 2.44 then reduces to 1765344656765765717650SXacSXaWSXaW (2.48) Equation 2.48 can be further reduced by noting that c4=0. 1765656765765717650SXSXaWSXaW (2.49) The unknown notation variables can be calculated from

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25 765765765765676765677677171755656565656566756676655656567655765676565767676776676767XZXXWXcYsYcXsYcsZccscssXcYsYsccscWccsscsUsVccWUssVsccsc (2.50) Solving Equation 2.49 for S6 yields 176567657657176565SXaWSXaWSX (2.51) The only joint offset that has not yet been calculated is S5. Spatial heptagon direction cosines set 3 are substituted into the vector loop equation given in Equation 2.44. 1233454676547654711230SXaSXaWSXaW (2.52) Once again, notation variables must be calculated before the unknown joint offset can be determined.

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26 232323212221212121224454545454654654654654565654566566756676123312231223312121121221211212212112XXYXYZWUXcYssssccscssXXcYsXXcYscXsYcsZccsscsUsVccWssVsccscWccssc (2.53) Finally, joint offset S5 can be calculated 1233465476547112354SXaSXSXaWSX (2.54) Both solution sets have now been calculated. In the case of the adjustment mechanism, all three joint offsets are the same in both solution sets. These joint offsets will be used to orient the lifting fixture so that a finite element analysis can be performed. Rotator Bar Length Calculation The rotator bar can be described as a RRRCRR mechanism. A reverse kinematic analysis could be conducted to ascertain joint angles and joint offsets that result in the APS pod being aligned. To conduct this analysis, the position and orientation of the end effector relative to the base must be input. The position of the rotator bar 6th coordinate system origin has been previously defined as LiftingFixtureRotatorBarP The orientation of the end effector can be determined by also inputting points on the 67a and axes and subtracting from each to determine the unit vectors 6S LiftingFixtureRotatorBarP 67a and 6S

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27 Before conducting a reverse kinematic analysis of the rotator bar, it was noted that unique geometry makes it possible to calculate the joint offset of the cylinder joint. With the exception of the cylinder joint offset, all rotator bar link lengths and joint offsets are zero. As a result, the cylinder joint offset can be calculated using the distance equation 2232LiftingFixtureLiftingFixtureLiftingFixtureLiftingFixtureRotatorBarXRotatorBarBaseXRotatorBarYRotatorBarBaseYLiftingFixtureLiftingFixtureRotatorBarZRotatorBarBaseZPPPPSPP (2.55) Although unknown joint angles could be calculated using S3, they are not needed for this analysis. Nominal Solution A program was written to perform pod alignment and subsequent joint offset calculations as described in this chapter. The mechanism parameters specified in Table 2-1 and the input coordinates of points 1Pod A ttachPtP 2Pod A ttachPtP 3Pod A ttachPtP , , LiftingFixtureRotatorBarP LiftingFixtureFwdAdjMechOrigP LiftingFixtureFwdAdjMechXP LiftingFixtureFwdAdjMechZP LiftingFixture A ftAdjMechOrigP LiftingFixture A ftAdjMechXP and LiftingFixture A ftAdjMechZP were used. Table 2-2 compares the joint offsets calculated by the program to the joint offsets measured using a perfectly aligned APS pod CAD model. The similar results show that the program is able to accurately calculate joint offsets for a known lifting fixture location. The two solutions, solution A and solution B, have nearly identical joint offsets but differing joint angles (not shown). The program is also able to perform a reverse kinematic analysis for right APS pods. This is accomplished by inverting the sign of the input Yorbiter coordinates then proceeding with the rest of the solution method. Right APS pods, lifting fixtures and

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28 orbiter attach points are mirror images of left APS pods, lifting fixtures, and orbiter deck attach points. The CAD model of the lifting fixture is positioned and oriented arbitrarily. The APS pod attach point and end effector positions are input into an ali g nment al g orithm. The algorithm aligns APS pod attach points with orbiter attach points and calculates the resulting end effector position and orientation for each mani p ulator. The rotator bar length is calculated and a reverse kinematic analysis is performed on both adjustment mechanisms. Figure 2-1. Joint offset calculation procedure.

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29 Figure 2-2. Three points are needed to determine each adjustment mechanism's position and orientation (aft adjustment mechanism shown, typical of all adjustment mechanisms). a45 S3,a12 S4 S5, S7 S2, a 34 a56 a67 S6 Figure 2-3. Adjustment mechanism joint axis vectors and link vectors. S1,a23

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30 Table 2-1. Adjustment mechanism parameters. Link length, inches Twist angle, degrees Joint offset, inches Joint angle, degrees a12 = 0.000 12 = 270.0 S1 = Close-the-loop variable 1 = variable a23 = 0.000 23 = 270.0 S2 = 0.000 2 = variable a34 = 4.147 34 = 270.0 S3 = 0.000 3 = variable a45 = 0.000 45 = 270.0 S4 = variable 4 = 270.0 a56 = 0.000 56 = 90.0 S5 = variable 5 = 270.0 a67 = 0.000 67 = 90.0 S6 = variable 6 = 180.0 a71 = Close-the-loop variable 71 = Close-the-loop variable S7 = Close-the-loop variable 7 = Close-the-loop variable Table 2-2. Comparison of joint offsets calculated by the program to measured using the CAD model. Forward adjustment mechanism Aft adjustment mechanism Rotator bar Measurement method S4 S5 S6 S4 S5 S6 S3 CAD model 13.6655 14.5243 8.7968 12.3174 14.8512 8.8823 94.5707 Program solution A 13.6655 14.5244 8.7968 12.3174 14.8512 8.8824 Program solution B 13.6655 14.5243 8.7968 12.3174 14.8512 8.8824 94.5706

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CHAPTER 3 UNCERTAINTY ANALYSIS Due to the stringent accuracy requirements associated with the APS pod installation operation, an uncertainty analysis was conducted to assess the validity of the proposed solution method. Two different uncertainty calculation methods were used to create an upper and lower bound for total uncertainty. The 100% covariance method produces very conservative results since it assumes all errors are at their maximum value. The root sum squared method provides more optimistic results. The 100% covariance and root sum squared uncertainty calculations provide an upper and lower bound respectively for the total error that can be reasonably expected. Off-Nominal Conditions within Tolerance Manufacturing tolerances can have a significant effect on the overall accuracy of a manipulator. As a result, precision manipulators are often manufactured with very tight tolerances. Adjustment Mechanism Spherical Joint Socket Locations There are manufacturing tolerances associated with the spherical joint sockets and the position of their mounting holes on the orbiter. These tolerances yield an uncertainty of in the position of each spherical joint (both Xorbiter and Zorbiter directions). The resulting misalignment of the APS pod can be seen in 0.0349" Tables 3-1 and 3-2 Orbiter Attach Point Locations There is also uncertainty associated with the position of the orbiter attach points on the orbiter. According to drawing tolerances, the orbiter attach points are located within 31

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32 0.010 of their intended position in the Xpod, Ypod, and Zpod directions. The effect of a 0.0104 attach point misalignment in all directions on the alignment of an APS pod can be seen for each attach point individually in Tables 3-3 3-4 and 3-5 The APS Pod Fitting Locations There is also uncertainty about the locations of the fittings on the APS pod. As with the orbiter attach point locations, the tolerance associated with the APS pod fitting locations is 0.010. Therefore, an uncertainty of 0.0104 in all directions is associated with each pod fitting location. The APS pod misalignment caused by this uncertainty is given in Tables 3-6 3-7 and 3-8 Lifting Fixture Attach Point 3 Location The position of the APS pod relative to the lifting fixture is largely determined by the location of the lifting fixtures attachment to the APS pod Attach Point 3 fitting. Drawing tolerances affect the position of the lifting fixtures attachment location relative to the adjustment mechanism end effectors. According to drawing tolerances, the lifting fixture Attach Point 3 location can be up to (0.0349, 0.0698, 0.0349) from the intended position in (Xpod, Ypod, Zpod). It may be surprising to note that this tolerance single-handedly prevents the Attach Point 3 accuracy requirement from being met. However, it is probable that engineers did not expect to apply robot kinematics to the lifting fixture when it was designed in 1977. The pod misalignment resulting from this uncertainty can be simply calculated. Results are presented in Table 3-9 Lifting Fixture Adjustment at Attach Point 3 There also exists the capability to adjust the position of the APS pod at the Attach Point 3 location. The magnitude of this adjustment capability is 0.418 in the podX

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33 directions. Since this adjustment potentially shifts the entire APS pod, all three attach points are affected by this source of uncertainty as shown in Table 3-10 Uncertainty of Input Values Location of rotator bar base. At first glance, it might appear that the base of the rotator bar can be considered ground at a known position relative to the spherical joint sockets and orbiter attach points. Unfortunately, this is not the case. One factor is the dimensional variance between each of the three OPFs. The rotator bar base is mounted to a beam in each OPF and the location of that beam might not be identical in each OPF. A much more significant factor is that the position of the orbiter relative to the OPF is not always the same. After each mission, the orbiter is towed into the OPF and jacked off the floor. Per specification, the orbiter must be towed to 1" forward/aft, port/starboard, and up/down of a specified nominal position. Therefore, the position of the rotator bar base can be significantly different from nominal. 1.5" 0.25" As a result, the position of the rotator bar base relative to the spherical joint sockets must be measured before each APS pod installation. Preliminary indications are that this position can be measured to a total accuracy of 0.010 or better. The uncertainty can be determined by positioning the rotator bar base 0.0104 from the nominal position and using nominal joint offsets. The effect of this measurement error is greatest if it occurs along the prismatic joint axis. Table 3-11 shows the effect of this error on attach point alignment. Calculation-Related Uncertainties Computer program uncertainty. Ideally, the computer program calculates the exact joint offsets required to position the lifting fixture as desired. However, roundoff errors can propagate and potentially become a significant error source.

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34 To determine error associated with the program, the lifting fixture CAD model was positioned in a known location. Using Pro/Es mechanism application, connections between components were defined as joints rather than rigid connections. This automatically positioned the adjustment mechanisms and rotator bar joints as needed to properly connect to the lifting fixture and mechanism base. The position of specific points in the model was then input into the program. The program calculated the joint offsets required to align the lifting fixture. These offsets were compared to the joint offsets measured in the CAD model. Inaccuracies were initially experienced due to errors in the CAD model. Additionally, using only three decimal places for input values caused significant errors. After these problems had been remedied, numerical methods were not required to further refine the calculations. Results can be viewed in Table 3-12 Hardware Positioning Uncertainty Although joint offsets can be accurately calculated to several decimal places, the ability of the pod installation team to adjust the rotator bar and adjustment mechanisms is limited. These limitations are largely due to measurement device uncertainty, mounting inaccuracies, and the tolerances of the components being measured. Adjustment mechanism measurement directions are shown in Figure 3-1 Uphill/Downhill Joint Offset Measurement The joint offset S4 has been previously defined as the distance along the vector between the and vectors. The pod installation team can adjust this joint offset to match the value calculated by the alignment program. 4S 34a 45a

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35 The desired joint offset reading on the measurement device can be achieved. However, the accuracy of this measurement is affected by factors such as measurement device uncertainty, mounting inaccuracies, and hardware tolerances. A preliminary assessment of these factors suggests that an accuracy of 0.010" can be achieved. This uncertainty analysis will determine the alignment error resulting from a misalignment of 0.0104 uphill for the forward and aft adjustment mechanism. The results of this analysis are stated below in Tables 3-13 and 3-14 Off-the-Deck/On-the-Deck Joint Offset Measurement The accuracy of the S5 joint offset measurement is also affected by measurement device uncertainty, mounting inaccuracies, and hardware tolerances. The total measurement uncertainty is approximated as 0.010" An analysis has been performed to ascertain attach point misalignment due to a 0.0104 misalignment in the off-the-deck direction for each adjustment mechanism. The results of this analysis can be found in Tables 3-15 and 3-16 Forward/Aft Joint Offset Measurement As with the measurement uncertainties for joint offsets S4 and S5, it is assumed that measurements of joint offset S6 are accurate to within 0.010. The effect of a 0.0104 misalignment in the forward direction was studied. In order for this misalignment to occur, the entire lifting fixture must be 0.0104 forward which means both the forward and aft S6 are misaligned by the same value. The resulting misalignment of attach points is listed in Table 3-17 Rotator Bar Joint Offset Measurement The rotator bar S3 prismatic joint offset is affected by the same uncertainty sources as the adjustment mechanism joint offsets. Measurement uncertainty is also

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36 approximately The misalignment of a rotator bar that is extended 0.0104 more than the measurement indicates is shown in 0.010" Table 3-18 Compliance in Joints It is stated that % of the flexibility of industrial robots comes from the joint. [5] Quantifying this compliance is a difficult task. It will be estimated by considering each joint on an individual basis. Joints exist where two components are attached. The dimensional difference between those two components was first identified. Loading was then considered while determining the relative position of the components under the assumption they are in contact. One component is translated and, in some cases also rotated, to the determined position. The resulting APS pod misalignment is calculated from these translations and rotations. The total uncertainty related to compliance is documented in Table 3-19 Total Uncertainty Calculation There are different methods for calculating total system uncertainty for a specified set of individual uncertainties. Individual uncertainties at each attach point are summarized in Tables 3-20 3-21 and 3-22 This information will be used to calculate total uncertainty using the 100% covariance method and the least squared method. The total uncertainty will then be compared to the accuracy requirements for APS pod alignment. One Hundred Percent Covariance Method The 100% covariance method assumes that each individual uncertainty is at a maximum at the same time. Assuming that all significant sources of uncertainty have been found and reasonably approximated, the 100% covariance method provides the worst case scenario.

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37 The total uncertainty is calculated by summing all individual uncertainties. The Xpod and Zpod accuracy requirements at attach points 1 and 3 are not specified individually. Since it is desired to compare total uncertainty to accuracy requirements, the Xpod and Zpod total uncertainty has been combined for attach points 1 and 3. Table 3-23 allows the uncertainty calculated using the 100% covariance method to be compared to the accuracy requirement. Root Sum Squared Method The root sum squared method provides the lower bound for total uncertainty projections. The root sum squared uncertainty is calculated by 21nTotaliiU U (3.1) where UTotal is the total uncertainty and the individual uncertainties are given by Ui through Un. The Xpod and Zpod uncertainties at attach points 1 and 3 have been combined for comparison to accuracy requirements. This comparison is presented in Table 3-24

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38 Table 3-1. Forward spherical joint socket tolerances. Attach point Coordinate axis Nominal solution Fwd socket misaligned by 0.0349 in the +Xorbiter and +Zorbiter directions Resulting attach point misalignment Xpod 106.5000 106.5335 -0.0335 Ypod 100.0000 100.0353 -0.0353 Attach Point 1 Zpod 152.3460 152.3745 -0.0285 Xpod 106.5000 106.5181 -0.0181 Ypod 100.0000 100.0224 -0.0224 Attach Point 2 Zpod 49.1790 49.2075 -0.0285 Xpod 253.5000 253.5167 -0.0167 Ypod 100.0000 100.0078 -0.0078 Attach Point 3 Zpod 39.7680 39.7747 -0.0067 Table 3-2. Aft spherical joint socket tolerances. Attach point Coordinate axis Nominal solution Aft socket misaligned by 0.0349 in the +Xorbiter and +Zorbiter directions Resulting attach point misalignment Xpod 106.5000 106.5016 -0.0016 Ypod 100.0000 100.0062 -0.0062 Attach Point 1 Zpod 152.3460 152.3440 0.0020 Xpod 106.5000 106.5141 -0.0141 Ypod 100.0000 99.9962 0.0038 Attach Point 2 Zpod 49.1790 49.1770 0.0020 Xpod 253.5000 253.5153 -0.0153 Ypod 100.0000 100.0155 -0.0155 Attach Point 3 Zpod 39.7680 39.7839 -0.0159

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39 Table 3-3. Orbiter Attach Point 1 tolerances. Attach point Coordinate axis Nominal solution Attach Point 1 misalignment of 0.0104 in all three directions Resulting attach point misalignment Xpod 106.5000 106.5104 -0.0104 Ypod 100.0000 100.0104 -0.0104 Attach Point 1 Zpod 152.3460 152.3564 -0.0104 Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 2 Zpod 49.1790 49.1790 0.0000 Xpod 253.5000 253.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 3 Zpod 39.7680 39.7680 0.0000 Table 3-4. Orbiter Attach Point 2 tolerances. Attach point Coordinate axis Nominal solution Attach Point 2 misalignment of 0.0104 in all three directions Resulting attach point misalignment Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 1 Zpod 152.3460 152.3460 0.0000 Xpod 106.5000 106.5104 -0.0104 Ypod 100.0000 100.0104 -0.0104 Attach Point 2 Zpod 49.1790 49.1894 -0.0104 Xpod 253.5000 253.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 3 Zpod 39.7680 39.7680 0.0000

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40 Table 3-5. Orbiter Attach Point 3 tolerances. Attach point Coordinate axis Nominal solution Attach Point 3 misalignment of 0.0104 in all three directions Resulting attach point misalignment Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 1 Zpod 152.3460 152.3460 0.0000 Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 2 Zpod 49.1790 49.1790 0.0000 Xpod 253.5000 253.5104 -0.0104 Ypod 100.0000 100.0104 -0.0104 Attach Point 3 Zpod 39.7680 39.7784 -0.0104 Table 3-6. APS pod Attach Point 1 tolerances. Attach point Coordinate axis Nominal solution Attach Point 1 Fitting misalignment of 0.0104 in all three directions Resulting attach point misalignment Xpod 106.5000 106.5104 -0.0104 Ypod 100.0000 100.0104 -0.0104 Attach Point 1 Zpod 152.3460 152.3564 -0.0104 Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 2 Zpod 49.1790 49.1790 0.0000 Xpod 253.5000 253.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 3 Zpod 39.7680 39.7680 0.0000

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41 Table 3-7. APS pod Attach Point 2 tolerances. Attach point Coordinate axis Nominal solution Attach Point 2 Fitting misalignment of 0.0104 in all three directions Resulting attach point misalignment Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 1 Zpod 152.3460 152.3460 0.0000 Xpod 106.5000 106.5104 -0.0104 Ypod 100.0000 100.0104 -0.0104 Attach Point 2 Zpod 49.1790 49.1894 -0.0104 Xpod 253.5000 253.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 3 Zpod 39.7680 39.7680 0.0000 Table 3-8. APS pod Attach Point 3 tolerances. Attach point Coordinate axis Nominal solution Attach Point 3 Fitting misalignment of 0.0104 in all three directions Resulting attach point misalignment Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 1 Zpod 152.3460 152.3460 0.0000 Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 2 Zpod 49.1790 49.1790 0.0000 Xpod 253.5000 253.5104 -0.0104 Ypod 100.0000 100.0104 -0.0104 Attach Point 3 Zpod 39.7680 39.7784 -0.0104

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42 Table 3-9. Lifting fixture Attach Point 3 tolerances. Attach point Coordinate axis Nominal solution Lifting fixture Attach Point 3 misalignment due to tolerances Resulting attach point misalignment Xpod 106.5000 106.5349 -0.0349 Ypod 100.0000 100.0698 -0.0698 Attach Point 1 Zpod 152.3460 152.3809 -0.0349 Xpod 106.5000 106.5349 -0.0349 Ypod 100.0000 100.0698 -0.0698 Attach Point 2 Zpod 49.1790 49.2139 -0.0349 Xpod 253.5000 253.5349 -0.0349 Ypod 100.0000 100.0698 -0.0698 Attach Point 3 Zpod 39.7680 39.8029 -0.0349 Table 3-10. Lifting fixture Attach Point 3 adjustment. Attach point Coordinate axis Nominal solution Maximum Adjustment of 0.4184 in the +Xpod direction at Attach Point 3 Resulting attach point misalignment Xpod 106.5000 106.9184 -0.4184 Ypod 100.0000 100.0000 0.0000 Attach Point 1 Zpod 152.3460 152.3460 0.0000 Xpod 106.5000 106.9184 -0.4184 Ypod 100.0000 100.0000 0.0000 Attach Point 2 Zpod 49.1790 49.1790 0.0000 Xpod 253.5000 253.9184 -0.4184 Ypod 100.0000 100.0000 0.0000 Attach Point 3 Zpod 39.7680 39.7680 0.0000

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43 Table 3-11. Misalignment of the rotator bar base. Attach point Coordinate axis Nominal solution 0.0104 rotator base misalignment Resulting attach point misalignment Xpod 106.5000 106.4989 0.0011 Ypod 100.0000 99.9836 0.0164 Attach Point 1 Zpod 152.3460 152.3563 -0.0103 Xpod 106.5000 106.5001 -0.0001 Ypod 100.0000 99.9974 0.0026 Attach Point 2 Zpod 49.1790 49.1893 -0.0103 Xpod 253.5000 253.5002 -0.0002 Ypod 100.0000 99.9983 0.0017 Attach Point 3 Zpod 39.7680 39.7800 -0.0120 Table 3-12. Computer program uncertainty. Attach point Coordinate axis Joint offsets from the CAD model are used Joint offsets from the program are input into the model Resulting attach point misalignment Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 1 Zpod 152.3460 152.3460 0.0000 Xpod 106.5000 106.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 2 Zpod 49.1790 49.1790 0.0000 Xpod 253.5000 253.5000 0.0000 Ypod 100.0000 100.0000 0.0000 Attach Point 3 Zpod 39.7680 39.7680 0.0000

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44 a45 S3,a12 S4 S5, S7 S2, a 34 a56 a67 S6 S1,a23 Figure 3-1. Adjustment mechanism joint axis vectors and link vectors. Table 3-13. Forward adjustment mechanism uphill measurement uncertainty. Attach point Coordinate axis Nominal solution Misalignment of the Fwd S4 0.0104 Uphill Resulting attach point misalignment Xpod 106.5000 106.5071 -0.0071 Ypod 100.0000 100.0080 -0.0080 Attach Point 1 Zpod 152.3460 152.3686 -0.0226 Xpod 106.5000 106.5011 -0.0011 Ypod 100.0000 100.0005 -0.0005 Attach Point 2 Zpod 49.1790 49.2015 -0.0225 Xpod 253.5000 253.5005 -0.0005 Ypod 100.0000 99.9996 0.0004 Attach Point 3 Zpod 39.7680 39.7819 -0.0139

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45 Table 3-14. Aft adjustment mechanism uphill measurement uncertainty. Attach point Coordinate axis Nominal solution Misalignment of the Aft S4 0.0104 Uphill Resulting attach point misalignment Xpod 106.5000 106.4929 0.0071 Ypod 100.0000 100.0058 -0.0058 Attach Point 1 Zpod 152.3460 152.3550 -0.0090 Xpod 106.5000 106.4992 0.0008 Ypod 100.0000 100.0003 -0.0003 Attach Point 2 Zpod 49.1790 49.1880 -0.0090 Xpod 253.5000 253.4999 0.0001 Ypod 100.0000 99.9994 0.0006 Attach Point 3 Zpod 39.7680 39.7860 -0.0180 Table 3-15. Forward adjustment mechanism off-the-deck measurement uncertainty. Attach point Coordinate axis Nominal solution Misalignment of the Fwd S5 0.0104 Off-the-Deck Resulting attach point misalignment Xpod 106.5000 106.5002 -0.0002 Ypod 100.0000 100.0103 -0.0103 Attach Point 1 Zpod 152.3460 152.3566 -0.0106 Xpod 106.5000 106.5006 -0.0006 Ypod 100.0000 100.0113 -0.0113 Attach Point 2 Zpod 49.1790 49.1895 -0.0105 Xpod 253.5000 253.5007 -0.0007 Ypod 100.0000 100.0023 -0.0023 Attach Point 3 Zpod 39.7680 39.7792 -0.0112

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46 Table 3-16. Aft adjustment mechanism off-the-deck measurement uncertainty. Attach point Coordinate axis Nominal solution Misalignment of the Aft S5 0.0104 Off-the-Deck Resulting attach point misalignment Xpod 106.5000 106.4993 0.0007 Ypod 100.0000 99.9961 0.0039 Attach Point 1 Zpod 152.3460 152.3570 -0.0110 Xpod 106.5000 106.4996 0.0004 Ypod 100.0000 99.9978 0.0022 Attach Point 2 Zpod 49.1790 49.1900 -0.0110 Xpod 253.5000 253.4996 0.0004 Ypod 100.0000 100.0063 -0.0063 Attach Point 3 Zpod 39.7680 39.7795 -0.0115 Table 3-17. Fwd and aft adjustment mechanism measurement uncertainty in the forward direction. Attach point Coordinate axis Nominal solution Misalignment of the Fwd S6 and Aft S6 by 0.0104 in the Forward Direction Resulting attach point misalignment Xpod 106.5000 106.4997 0.0003 Ypod 100.0000 99.9992 0.0008 Attach Point 1 Zpod 152.3460 152.3565 -0.0105 Xpod 106.5000 106.5002 -0.0002 Ypod 100.0000 99.9995 0.0005 Attach Point 2 Zpod 49.1790 49.1895 -0.0105 Xpod 253.5000 253.5002 -0.0002 Ypod 100.0000 99.9992 0.0008 Attach Point 3 Zpod 39.7680 39.7792 -0.0112

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47 Table 3-18. Rotator bar length measurement uncertainty. Attach point Coordinate axis Nominal solution Misalignment of the Rotator bar S3 0.0104 Extend Resulting attach point misalignment Xpod 106.5000 106.4989 0.0011 Ypod 100.0000 99.9836 0.0164 Attach Point 1 Zpod 152.3460 152.3563 -0.0103 Xpod 106.5000 106.5001 -0.0001 Ypod 100.0000 99.9974 0.0026 Attach Point 2 Zpod 49.1790 49.1893 -0.0103 Xpod 253.5000 253.5002 -0.0002 Ypod 100.0000 99.9983 0.0017 Attach Point 3 Zpod 39.7680 39.7800 -0.0120 Table 3-19. Joint compliance uncertainty. Attach point Coordinate axis Nominal solution Attach point misalignment Xpod 106.5000 0.0022 Ypod 100.0000 0.3914 Attach Point 1 Zpod 152.3460 0.1172 Xpod 106.5000 0.0002 Ypod 100.0000 0.0886 Attach Point 2 Zpod 49.1790 0.0628 Xpod 253.5000 0.0004 Ypod 100.0000 0.0955 Attach Point 3 Zpod 39.7680 0.0620

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48 Table 3-20. Summary of Attach Point 1 uncertainty sources. Attach Point 1 (inches) Uncertainty Description Xpod Ypod Zpod Fwd spherical joint socket location 0.0335 0.0353 0.0285 Aft spherical joint socket location 0.0016 0.0062 0.0020 Orbiter Attach Point 1 location 0.0104 0.0104 0.0104 Orbiter Attach Point 2 location 0.0000 0.0000 0.0000 Orbiter Attach Point 3 location 0.0000 0.0000 0.0000 APS pod Attach Point 1 location 0.0104 0.0104 0.0104 APS pod Attach Point 2 location 0.0000 0.0000 0.0000 APS pod Attach Point 3 location 0.0000 0.0000 0.0000 Lifting fixture nominal Attach Point 3 location 0.0349 0.0698 0.0349 Adjustment capability of Attach Point 3 0.4184 0.0000 0.0000 Measurement of rotator bar base location 0.0011 0.0164 0.0103 Calculations by computer program 0.0000 0.0000 0.0000 Iterative process acceptance criteria 0.0005 0.0005 0.0004 Measurement of fwd adjustment mechanism uphill/downhill position 0.0071 0.0080 0.0226 Measurement of aft adjustment mechanism uphill/downhill position 0.0071 0.0058 0.0090 Measurement of fwd adjustment mechanism off-the-deck/on-the-deck position 0.0002 0.0103 0.0106 Measurement of aft adjustment mechanism off-the-deck/on-the-deck position 0.0007 0.0039 0.0110 Measurement of adjustment mechanism forward/aft position 0.0003 0.0008 0.0105 Measurement of rotator bar joint offset 0.0011 0.0164 0.0103 Compliance in joints 0.0022 0.3914 0.1172

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49 Table 3-21. Summary of Attach Point 2 uncertainty sources. Attach Point 2 (inches) Uncertainty Description Xpod Ypod Zpod Fwd spherical joint socket location 0.0181 0.0224 0.0285 Aft spherical joint socket location 0.0141 0.0038 0.0020 Orbiter Attach Point 1 location 0.0000 0.0000 0.0000 Orbiter Attach Point 2 location 0.0104 0.0104 0.0104 Orbiter Attach Point 3 location 0.0000 0.0000 0.0000 APS pod Attach Point 1 location 0.0000 0.0000 0.0000 APS pod Attach Point 2 location 0.0104 0.0104 0.0104 APS pod Attach Point 3 location 0.0000 0.0000 0.0000 Lifting fixture nominal Attach Point 3 location 0.0349 0.0698 0.0349 Adjustment capability of Attach Point 3 0.4184 0.0000 0.0000 Measurement of rotator bar base location 0.0001 0.0026 0.0103 Computer program roundoff error 0.0000 0.0000 0.0000 Iterative process acceptance criteria 0.0001 0.0005 0.0004 Measurement of fwd adjustment mechanism uphill/downhill position 0.0011 0.0005 0.0225 Measurement of aft adjustment mechanism uphill/downhill position 0.0008 0.0003 0.0090 Measurement of fwd adjustment mechanism off-the-deck/on-the-deck position 0.0006 0.0113 0.0105 Measurement of aft adjustment mechanism off-the-deck/on-the-deck position 0.0004 0.0022 0.0110 Measurement of adjustment mechanism forward/aft position 0.0002 0.0005 0.0105 Measurement of rotator bar joint offset 0.0001 0.0026 0.0103 Compliance in joints 0.0002 0.0886 0.0628

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50 Table 3-22. Summary of Attach Point 3 uncertainty sources. Attach Point 3 (inches) Uncertainty Description Xpod Ypod Zpod Fwd spherical joint socket location 0.0167 0.0078 0.0067 Aft spherical joint socket location 0.0153 0.0155 0.0159 Orbiter Attach Point 1 location 0.0000 0.0000 0.0000 Orbiter Attach Point 2 location 0.0000 0.0000 0.0000 Orbiter Attach Point 3 location 0.0104 0.0104 0.0104 APS pod Attach Point 1 location 0.0000 0.0000 0.0000 APS pod Attach Point 2 location 0.0000 0.0000 0.0000 APS pod Attach Point 3 location 0.0104 0.0104 0.0104 Lifting fixture nominal Attach Point 3 location 0.0349 0.0698 0.0349 Adjustment capability of Attach Point 3 0.4184 0.0000 0.0000 Measurement of rotator bar base location 0.0002 0.0017 0.0120 Computer program roundoff error 0.0000 0.0000 0.0000 Iterative process acceptance criteria 0.0001 0.0005 0.0001 Measurement of fwd adjustment mechanism uphill/downhill position 0.0005 0.0004 0.0139 Measurement of aft adjustment mechanism uphill/downhill position 0.0001 0.0006 0.0180 Measurement of fwd adjustment mechanism off-the-deck/on-the-deck position 0.0007 0.0023 0.0112 Measurement of aft adjustment mechanism off-the-deck/on-the-deck position 0.0004 0.0063 0.0115 Measurement of adjustment mechanism forward/aft position 0.0002 0.0008 0.0112 Measurement of rotator bar joint offset 0.0002 0.0017 0.0120 Compliance in joints 0.0004 0.0955 0.0620 Table 3-23. Total uncertainty using the 100% covariance method. Attach Point 1 (inches) Attach Point 2 (inches) Attach Point 3 (inches) Description Xpod/Zpod Ypod Xpod Ypod Zpod Xpod/Zpod Ypod Total Uncertainty 0.6028 0.5856 0.5099 0.2259 0.2335 0.5585 0.2237 Accuracy Requirement 0.2231 0.0010 2.0000 0.0010 0.0063 0.0033 0.0010 Remaining Margin -0.3797 -0.5846 1.4901 -0.2249 -0.2272 -0.5552 -0.2227

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51 Table 3-24. Total uncertainty using the root sum squared method. Attach Point 1 (inches) Attach Point 2 (inches) Attach Point 3 (inches) Description Xpod/Zpod Ypod Xpod Ypod Zpod Xpod/Zpod Ypod Total Uncertainty 0.4414 0.4004 0.4207 0.1166 0.0856 0.4287 0.1207 Accuracy Requirement 0.2231 0.0010 2.0000 0.0010 0.0063 0.0033 0.0010 Remaining Margin -0.2183 -0.3994 1.5793 -0.1156 -0.0763 -0.4254 -0.1197

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CHAPTER 4 EVALUATION OF PROPOSED ALIGNMENT METHOD Both uncertainty calculation methods used in Chapter 3 indicate that the proposed solution method is not as accurate as desired. If used, it would not be able to align the APS pod with the orbiter deck accurately enough to eliminate the need for additional manipulations. The additional manipulations required for alignment can not be calculated before the operation because the direction and magnitude of the misalignment can not be predicted. Additional manipulations are highly undesirable because the motion resulting from mechanism adjustments is not intuitive. As a result, the uncertainty analysis will be further examined and recommendations will be made to reduce total uncertainty. Discussion of Results Uncertainty due to tolerances can be significantly reduced by measuring as-built dimensions. The uncertainty associated with those measurements is significantly less than the uncertainty due to tolerances, in some cases by more than one order of magnitude. Measurements of specific points on the lifting fixture, APS pod, and orbiter can be taken before APS pod installation. Uncertainty can be further reduced by more accurately measuring rotator bar joint offset S5 and adjustment mechanism joint offsets S4, S5, and S6. Improvements to reduce uncertainty might require significant modifications to rotator bar and adjustment mechanism hardware. However, these modifications are highly desirable because the 52

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53 uncertainty associated with each joint offset measurement exceeds the total allowable uncertainty at Attach Point 3 by one order of magnitude. The estimated uncertainty due to compliance in rotator bar and adjustment mechanism joints is one of the most significant error sources studied. Although joint compliance can not be reduced without significant modifications to hardware, its effect can be minimized. There are only two lifting fixtures, one for right APS pods and one for left APS pods, and those lifting fixtures are in approximately the same position and orientation during each APS pod installation. The geometry of the lifting fixture configuration indicates that each joint is under load during this operation and those loads determine the direction affected by compliance. As a result, joint compliance has only a minimal effect on precision. To obtain accurate results, a correction factor can be used to compensate for joint compliance error. A correction factor might also be used to compensate for uncertainties related to lifting fixture, APS pod, and orbiter geometry. For a given lifting fixture, APS pod, and orbiter it might be discovered that the calculated solution results in a misalignment that is consistent in magnitude and direction. This precise solution could be made more accurate by implementing a correction factor. Since there are only two lifting fixtures, three orbiters, and a small number of APS pods, a database could be created relatively quickly to facilitate the calculation of a correction factor for each scenario. Recommendations The lifting fixture, adjustment mechanisms, and rotator bar are not currently outfitted with a means of measuring joint offsets. Therefore, it is suggested that measurement devices for each joint offset be installed. Laser rangefinders should be considered because they provide high accuracy and can be installed without modifying

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54 load-bearing components. The proposed solution method will allow the ground support team to align the APS pod by manipulating joints to precalculated positions. Without measurement devices, the ground support team will not know when the precalculated positions have been reached. Based on an uncertainty analysis, an accuracy of is insufficient for measurement devices. Overall system accuracy would be greatly improved if measurement device accuracy approached 0.010" 0.001" Crude measurement techniques such a rulers or tape measures would provide highly inaccurate results. The lifting fixture design includes adjustment capability at all attach points to accommodate APS pod dimensional variation. This adjustment capability is large enough that it must be accounted for when considering the location of the APS pod relative to reference points on the lifting fixture. The simplest and most accurate method of determining the position of the APS pod relative to the lifting fixture is to make high-accuracy measurements before each APS pod is installed. APS pod attach points 1, 2, and 3 must be measured and the rotator bar and adjustment mechanism end effector locations. Since the orientation of the adjustment mechanism end effectors must be known, a second point along each adjustment mechanism 6S and 67a vectors must also be measured. Total uncertainty can be further reduced by making additional high-accuracy measurements of the rotator bar base and spherical joint socket locations. Orbiter attach point positions can be similarly determined. These measurements yield only marginal accuracy improvements but are desirable nonetheless. In general, high-accuracy measurements can be used to compensate for insufficiently loose tolerances.

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55 As previously mentioned, a hydraulic jack is used to ensure the aft adjustment mechanism follows the forward adjustment mechanism during forward/aft motion of the lifting fixture. The hydraulic jack is not a precise method for positioning the aft adjustment mechanism. The adjustment mechanisms could be more accurately positioned in the forward/aft direction before they are installed in the sockets. Since the adjustment mechanisms will not be supporting the weight of the lifting fixture or APS pod at that point, they can be adjusted to the precalculated solution position without experiencing binding. If the S6 joint offset calculations are correct, the adjustment mechanisms will not need to be adjusted in the forward/aft direction during operations. One advantage of the proposed solution method is that joint offset adjustments do not need to be made in any particular sequence. Therefore, it is not imperative that motions be simultaneous with one exception. As the APS pod is lowered to the orbiter deck, the bottom of the APS pod should be approximately parallel to the orbiter deck surface. If one bulb seal compresses before the others make contact, then the resulting frictional force might cause a slight misalignment. To minimize this effect, a reverse kinematic analysis should also be performed to position the APS pod at a waypoint 1/8 above the orbiter deck and aligned in Xpod and Zpod. The APS pod can be slowly lowered onto the deck from that waypoint by simultaneously adjusting the rotator bar and adjustment mechanisms. It would also be beneficial to calculate the joint angle adjustments needed to move the APS pod in small increments along each axis. If the lifting fixture is manipulated to the precalculated joint angles and found to be misaligned, the ground support team will

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56 know how to manipulate the lifting fixture to align the APS pod rather than rely on intuition. A Finite Element Analysis (FEA) should be performed on the lifting fixture with the APS pod in the installed position. This will allow the rigid body assumption to be validated. If the FEA shows that lifting fixture deflection is significant, then the deflected lifting fixture shape will be used throughout reverse kinematic analysis calculations. Summary of Recommendations The following recommendations should be implemented to ensure success of the proposed alignment method: Install measurement devices to measure all variable joint offsets. Make high-accuracy measurements ( 0.001" ) of all critical relative positions. Position the adjustment mechanisms in the forward/aft direction per S6 joint offset solutions before they begin supporting lifting fixture and APS pod weight. Position the APS pod at a waypoint 1/8 above the orbiter deck and aligned along the plane of the orbiter deck. The joint offsets needed to position the pod at the waypoint are calculated by reverse kinematic analysis. Use joint angle adjustments for small misalignments along each pod coordinate axis. Perform FEA to validate the rigid body assumption. Conclusions It should be evident at this point that the design of the lifting fixture, adjustment mechanisms, and rotator bar is not adequate for the precision positioning task it is required to perform. Given the shuttle programs limited resources, it is highly unlikely that this GSE will be redesigned to improve operations in lieu of upgrades to flight hardware. The reverse kinematic analysis presented in my study does not add the desired

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57 level of accuracy to the APS pod alignment operation. However, the level of accuracy it does provide is a substantial improvement to the current process. Implementation of this solution method adds a relatively small amount of manpower and cost to operations compared to the projected benefit. This solution method is a viable aid to APS pod alignment during installation onto the orbiter.

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APPENDIX A SHUTTLE COORDINATE SYSTEMS Several different coordinate systems are commonly used to describe locations on the space shuttle. Two of these coordinate systems are the orbiter coordinate system and the APS pod coordinate system. The orbiter coordinate system is essentially the orbiters version of a body-fixed coordinate system. The origin is located forward of the orbiters nose. The positive Xdirection is aft, positive Ydirection is outboard through the starboard wing, and positive Zdirection is out through the vertical tail. The location and orientation of the orbiter coordinate system can be seen in Figure A-1 Orbiter coordinates are used throughout this analysis except when the location of the orbiter deck is specifically needed. The APS pod coordinate systems are local coordinate systems used to describe orbiter locations relative to the orbiter deck planes. In both right APS pod coordinates and left APS pod coordinates, the plane of the appropriate orbiter deck can be described by the equation Ypod = 100. The origin is located forward of the APS pod and below the orbiter deck. The positive Xdirection is aft and slightly outboard, positive Ydirection is perpendicular to the orbiter deck and includes an outboard component, and positive Zdirection is angled inboard along the plane of the orbiter deck. The right APS pod coordinate system can be seen in Figure A-2 The transformation matrices between orbiter coordinates and right APS pod coordinates are documented as 58

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59 0.9986104865-0.052698162501212.40876760.03790788570.71834026790.694658370559.300943520.03660721970.69369313320.7193398003311.7498963800010.99861048650.037OrbiterRightPodRightPodOrbiterTT90788570.0366072197-1224.3843795-0.05269816250.71834026790.6936931332-194.965303810-0.69465837050.7193398003-183.060211430001 (A.1) Drawings containing the APS pod and its ground support equipment show hardware associated with the left APS pod. The CAD model is based on these drawings and therefore also shows left APS pod hardware. Unfortunately, transformation matrices for the left APS pod coordinate system are not documented. The location and orientation of the APS pod coordinate systems are symmetric about the orbiter coordinate XZ plane. Using this fact, transformation matrices for the left APS pod coordinate system can be calculated. It should be noted that the left APS pod coordinate system does not obey the right hand rule. In the interest of avoiding calculation errors due to the use of a left hand coordinate system, the derived left APS pod coordinate system will be made to obey the right hand rule by inverting its Zaxis. First, the rotation angle about unknown unit vector m can be calculated from 11122331cos44.09843735372rrr (A.2) where , and are the first three diagonal terms in The unit vector 11r 22r 33r OrbiterRightPodT m can be determined from

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60 1112211331cos0.9975320082291cos0.02630232405221cos0.06510054000421cosxyxzxrmrrmmrrmm (A.3) The sign of mx and subsequent signs of my and mz are determined from 32230.9975320082292sin0.0263023240520.065100540004xyzrrmmm (A.4) To calculate the left APS pod coordinate system, a rotation of about the mirror of vector m will be conducted. The resulting transformation matrix is cossinsin1224.380sincossin194.965sinsincos183.0600001cos0.2818547227xxxyzxzyxyzyyyzxOrbiterLeftPodxzyyzxzzmmmmmmmmmmmmmmmmTmmmmmmmm 1 (A.5) After altering the matrix to invert the Zaxis, the result is 0.99861048650.03790788570.03660721971224.3800.05269816250.71834026790.6936931333194.96500.69465837050.7193398003183.060000OrbiterLeftPodT 1 (A.6) The inverse of this matrix is 0.99861048640.052698162501212.408767510.03790788570.71834026800.694658370559.30094351190.03660721970.69369313330.7193398003311.7498963990001OrbiterLeftPodT (A.7)

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61 In retrospect, the simplest solution would have been to have the program invert all negative orbiter Ycoordinates input into the program. This would have eliminated the need for the left APS pod coordinate derivation because the right APS pod coordinate system could be used instead. Figure A-1. Orbiter coordinate system.

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62 Figure A-2. Right APS pod coordinate system. [2]

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APPENDIX B REVERSE KINEMATIC ANALYSIS NOTATION The reverse kinematic analysis notation used in my study is defined as follows. Subscripts h, i, j, k, and l are used where f=i-3, g=i-2, h=i-1, j=i+1, k=i+2, and l=i+3. cossincossiniiiijijijijcscs i (B.1) jijjjjkijjkijjjkijjkijjXssYsccscZccssc j (B.2) jjkjjijjkijjkjijjkijjkjXssYsccscZccssc j (B.3) ijijijijjkijijjkiijjkijijjkiXXcYsYcXsYcsZ Z sXsYccZ (B.4) kjkjkjkjijkjkjijkkjijkjkjijkXXcYsYcXsYcsZ Z sXsYccZ (B.5) ijkijkijkijkklijkijkklijijkklijkijkklijXXcYsYcXsYcsZ Z sXsYccZ (B.6) kjikjikjikjihikjikjihikjkjihikjikjihikjXXcYsYcXsYcsZ Z sXsYccZ (B.7) 63

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64 hijkhijkhijkhijkklhijkhijkklhijhijkklhijkhijkklhijXXcYsYcXsYcsZ Z sXsYccZ (B.8) kjihkjihkjihkjihghkjihkjihghkjikjihghkjihkjihghkjiXXcYsYcXsYcsZ Z sXsYccZ (B.9) hijklhijklhijklhijkllmhijklhijkllmhijkhijkllmhijklhijkllmhijkXXcYsYcXsYcsZ Z sXsYccZ (B.10) lkjihlkjihlkjihlkjihghlkjihlkjihghlkjilkjihghlkjihlkjihghlkjiXXcYsYcXsYcsZ Z sXsYccZ (B.11) ijiijijjijiijijjijiijUssVsccscWccssc (B.12) jijijjiijijijjiijijijUssVsccscWccssc (B.13) ijkijjkijjkkjiijkkijjkijjkkijkjiijkkijjkijjkkijkjiUUcVsUVcUsVcsWVWsUsVccWW (B.14) hijkhijjkhijjkkjihhijkkhijjkhijjkkhijkjihhijkkhijjkhijjkkhijkjihUUcVsUVcUsVcsWVWsUsVccWW (B.15) ghijkghijjkghijjkkjihg g hijkkghijjkghijjkkghijkjihg g hijkkghijjkghijjkkghijkjihgUUcVsUVcUsVcsWVWsUsVccWW (B.16) fghijkfghijjkfghijjkkjihgf f ghijkkfghijjkfghijjkkfghijkjihgf f ghijkkfghijjkfghijjkkfghijkjihgfUUcVsUVcUsVcsWVWsUsVccWW (B.17)

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LIST OF REFERENCES [1] D. R. Jenkins, The History of the National Space Transportation System: The First 100 Missions, 3rd ed. Stillwater, MN: Voyageur Press, 2001. [2] Rockwell International, Lifting Fixture Set-OMS/RCS Pod, National Aeronautics and Space Administration, Houston, TX H70-0679, 1978. [3] D. Armstrong, John F. Kennedy Space Center Multimedia Gallery, n.d.; http://mediaarchive.ksc.nasa.gov/index.cfm, 23 Oct. 2004. [4] C. D. Crane III and J. Duffy, Kinematic Analysis of Robot Manipulators. New York: Cambridge University Press, 1998. [5] W. H. ElMaraghy, H. A. ElMaraghy, A. Zaki, and A. Massoud, Design and Control of Robots with Flexibilities, Annals of the CIRP, vol. 43, pp. 359-362, 1994. 65

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BIOGRAPHICAL SKETCH Jeff Brink graduated from the University of Michigan in December 2000 with Bachelor of Science degrees in aerospace engineering and mechanical engineering. Jeff started work at NASA in February 2001 as a systems engineer in shuttle processing at the Kennedy Space Center in Florida. In September 2001, he enrolled part-time in graduate school at the University of Central Florida. One year later, Jeff transferred to the University of Florida to study robotics. With the help of the University of Floridas distance learning program, FEEDS, Jeff will graduate in May 2005 with a Master of Engineering degree. 66