n mcnamara - espace.library.uq.edu.au
TRANSCRIPT
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Abstract
This report investigates and compares steering laws for a control moment gyroscope (CMG)
attitude control system for a spacesuit. If there is an inadequate level of control in a CMG
system then the angular momentum vectors of individual CMGs may become aligned. This
state, referred to as a βsingularity stateβ, prevents the system from being able to produce
torque in one or more directions. The steering law problem has been defined and the
momentum envelope is presented to visualize system singularities. A literature review has
been conducted to find applicable steering laws that may be used to control a spacesuit sized
system and avoid singularities in the momentum envelope. Six potential steering laws are
identified as being potentially applicable to a spacesuit sized system; the Moore-Penrose
pseudoinverse (MPI), offline planning, preferred gimbal angle, linearly constrained, null
motion, and the singularity-robust inverse (SRI).
A conceptual pyramid configuration system of 4 CMGs has been constructed to analyse the
selected steering laws through a qualitative and quantitative comparison. The qualitative
assessment investigates several criteria; the singularity handling method, induced torque
error, system efficiency and full utilization of hardware, convergence to singularity
arrangements, and the potential for generalization to other CMG arrangements. The results
of the qualitative comparison conclude that the SRI control law is the most viable for use with
a spacesuit. A computational model was used to compare the behaviour of the conceptual
system using the MPI and the SRI control laws. The simulation results support the qualitative
analysis. The SRI control law successfully navigates around singularities by inducing a small,
proportional torqueing error. It is concluded that a CMG control system can potentially be
used with a spacesuit and that the SRI control law is the most viable method to avoid
singularities during operation.
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Contents Abstract ..................................................................................................................................................... i
List of Tables and Figures ......................................................................................................................... v
1. Introduction .................................................................................................................................... 1
1.1. Motivation and Problem Definition ........................................................................................ 3
1.2. Astronaut Musculoskeletal Injury Rates ................................................................................. 4
1.3. Project Aims ............................................................................................................................ 5
2. Scope ............................................................................................................................................... 6
3. Literature Review ............................................................................................................................ 7
3.1. Overview of Microgravity Attitude Control and Dynamics ..................................................... 7
3.2. Propellant Based Systems ....................................................................................................... 7
3.3. Gyroscopic Attitude Control Systems ..................................................................................... 8
3.3.1. Torque Reaction Wheels ..................................................................................................... 9
3.3.2. Control Moment Gyroscopes .............................................................................................. 9
3.3.3. Vehicle Characteristics and Gyroscopic Control System Efficiency ................................... 11
3.4. CMG System Arrangements and Singularities ...................................................................... 12
3.4.1. Singularity Avoidance and Null Motion ............................................................................. 15
4. Project Methodology and Process ................................................................................................ 16
5. System Model ................................................................................................................................ 17
5.1. Dynamics and Equations of Motion ...................................................................................... 17
5.2. Pseudoinverse and Minimum Norm Solution ....................................................................... 22
5.3. Momentum Envelope and CMG Gain ................................................................................... 24
5.4. Elliptical and Parabolic Singularities ...................................................................................... 25
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6. Steering Laws ................................................................................................................................. 29
6.1. No Singularity Avoidance ....................................................................................................... 31
6.1.1. Moore-Penrose Pseudoinverse ......................................................................................... 31
6.2. Singularity-Avoidance ............................................................................................................ 32
6.2.1. Offline Planning ................................................................................................................. 32
6.2.2. Preferred Gimbal Angle/Gimbal Reorientation ................................................................. 33
6.2.3. Linearly Constrained .......................................................................................................... 34
6.3. Singularity-Robust ................................................................................................................. 36
6.3.1. Gradient/Null Motion Weighted Matrix ............................................................................ 36
6.3.2. Singularity-Robust Inverse ................................................................................................. 37
6.4. Qualitative Summary ............................................................................................................. 39
7. Simulation and Results .................................................................................................................. 40
8. Discussion ...................................................................................................................................... 43
9. Conclusion ..................................................................................................................................... 45
10. References ................................................................................................................................. 46
11. Appendices ................................................................................................................................ 49
11.1. Appendix A β CMG Torque Derivation from First Principles ............................................. 49
11.2. Appendix B β Simulation Data ........................................................................................... 51
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List of Tables and Figures
Figure 1 - Astronaut affixed to Canadarm2 on the ISS ............................................................................ 2
Figure 2 - Astronaut with MMU attached ............................................................................................... 4
Figure 3 - Attitude errors, propellant-only attitude control system (left), N number of CMG controllers
added to system (right) (Source: Carpenter, et al., 2013) ...................................................................... 8
Figure 4 - Reaction wheel assembly momentum vectors ..................................................................... 10
Figure 5 - Single-gimbal CMG momentum vectors ............................................................................... 10
Figure 6 - RWA and CMG comparison of momentum potential envelopes (Source: Hamilton, et al.,
2015) ..................................................................................................................................................... 11
Figure 7 - CMG pyramid configuration .................................................................................................. 12
Figure 8 - CMG roof configuration ........................................................................................................ 13
Figure 9 - Control moment gyroscope diagram (Source: Yoon, 2004) .................................................. 17
Figure 10 - Pyramid CMG system diagram ............................................................................................ 18
Figure 11 - Momentum envelope of conceptual system (Source: Yoon, 2004) ................................... 25
Figure 12 - Hyperbolic singularity momentum envelope surface (Source: Yoon, 2004) ...................... 26
Figure 13 - Hyperbolic singularity CMG arrangement .......................................................................... 26
Figure 14 β Parabolic singularity momentum envelope surface (Source: Yoon, 2004) ........................ 27
Figure 15 - Parabolic singularity CMG arrangement ............................................................................. 27
Figure 16 - Momentum envelope of pyramid configuration CMG system (Source: Yoon, 2004) ........ 28
Figure 17 β Constrained steering law applied to system of 3 CMGs (Source: Jones, et al., 2012) ....... 35
Figure 18 - Effects of constrained steering law on momentum envelope (Source: Kurokawa, 1997) . 35
Figure 19 - Comparison of MPI and SRI Solutions ................................................................................. 42
Figure 20 - SRI Solution Induced Torque Error from Input Command .................................................. 42
Table 1 - Scope of the project ................................................................................................................. 6
Table 2 - Qualitative comparison summary .......................................................................................... 39
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1. Introduction
Control moment gyroscopes (CMGs) are momentum-exchange actuators which are used to
control the attitude of spacecraftβs and other terrestrial vehicles. Gyroscopic control systems
are especially useful for space environments due to the independence of the system from
fuel-based energy sources and propellants. Using electrics motors and actuators a gyroscopic
system can be used indefinitely provided that adequate electrical power be provided, the
most prevalent source is solar energy.
Arrays of CMGs can be used in tandem to control a system in multiple planes of motion.
Different configurations of these arrays present various benefits and restrictions depending
on the applicable scenario. There are various methods of controlling these CMGs. The control
system must account for both the degree of actuation possible for a single CMG, and the
relative moment of inertia between CMGs in an array. If there is an inadequate level of control
in a CMG system then the angular momentum vectors of individual CMGs may become
aligned. This state, referred to as a βsingularity stateβ, prevents the system from being able to
produce torque in one or more directions. An analog can be made between the singularity
state of a CMG system and that of a mechanical system, for example a delta-type parallel
manipulator (Bedrossian, Paradiso, & Bergmann, 1990).
Mechanical Manipulator
CMG System
Position π = π(π) Momentum π = π(π½) Velocity οΏ½ΜοΏ½ = π½(π)οΏ½ΜοΏ½ Torque π = π½(π½)οΏ½ΜοΏ½ Acceleration οΏ½ΜοΏ½ = π½(π)οΏ½ΜοΏ½ + π½(π)οΏ½ΜοΏ½ Rotatum (
π
ππ‘ Torque) οΏ½ΜοΏ½ = π½(π½)οΏ½ΜοΏ½ + π½(π½)οΏ½ΜοΏ½
System singular when no motion possible in a certain direction.
System singular when no torque possible in a certain direction.
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Modern day astronaut suits employ propellant based systems for principle motion and
attitude control during EVA missions. With the world moving rapidly towards spaceflight and
operations further away from Earth, spacesuits need to be lighter, more mobile, and safer
than what is currently available. The design and research into spacesuit technology has be
largely stagnant since the shuttle program. Current extravehicular activity (EVA) missions are
only carried out near a vehicle in microgravity and thus astronauts primarily rely on operator
controlled robot arms connected to space vehicles (Figure 1), and manual movement about a
vehicle by grabbing onto external handles to conduct operations in outer space.
When astronauts are required to venture further from the parent space vehicle, external
booster packs are affixed to an EVA suit. Propellant is used for principle motion and for
attitude control of the astronaut. The use of propellant for attitude control reduces the
available mission time of the suit for a given mass, and increases the mass of the system
unnecessarily. A new form of attitude stabilization is required in such a scenario.
Figure 1 - Astronaut affixed to Canadarm2 on the ISS
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1.1. Motivation and Problem Definition
The development of the spacesuit has been predominantly stagnant since the inception of the
Apollo program in 1967 (Harland, 2010). Spacesuits are categorized into operational areas and
provide different functionality depending on the scenario. The Extravehicular Mobility Unit
(EMU) was developed as an independent anthropomorphic spacesuit for deployment in outer-
and low-atmospheric environments (Thomas & McMann, 2006). The EMU system provides
environmental protection, mobility, life support, and communications to operators, an EMU
is essentially the smallest human space vehicle ever created (West & Witt, 2010). During
extravehicular activities astronauts use the EMU in conjunction with an array of separate,
modular units which provide different modes of position and attitude control in zero gravity.
The main unit used is the Manned Manoeuvring Unit (MMU), shown in Figure 2. The MMU
attaches to the back of the EMU and is worn like a backpack. The unit consists of a small one-
man nitrogen-propellant based propulsion system that the user can use for attitude control
and translational motion (Cheng, 2010). Counterintuitively, during a spacewalk an astronaut
does very little with their legs. The principle mode of traversing the exterior of a space vehicle
is by way of manually clambering along the hull using hand holds and bracing. In the vacuum
of space the pressurized, rubber composite spacesuit gloves expand and harden making
simple tasks difficult to perform. Current design constraints and a lack of alternative spacesuit
designs force astronauts to perform EVA operations in far more difficult circumstances than
immediately apparent (Danaher, Tanaka, & Hargens, 2005). When fine motor control is
required to complete tasks during an operation an astronaut must constantly control their
attitude and this inevitably leads to errors when dealing with precision tooling or scientific
sample collection.
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Figure 2 - Astronaut with MMU attached
1.2. Astronaut Musculoskeletal Injury Rates
One of the most prominent fields of study relating to human spaceflight is the potential risk
of musculoskeletal injury and minor trauma, more specifically, in the hands and forearms, of
astronauts (Opperman, et al., 2010). Jennings and Bagianβs (1996) study investigating
terrestrial-based musculoskeletal injuries of astronauts in the period of 1987 β 1995 found
astronauts sustained numerous minor injuries because of EVA-specific operations and
training. A total of 28 orthopedic surgical procedures were conducted in this period due to
sustained fractures and serious ligament, cartilage, and soft tissue injuries (Jennings & Bagian,
1996). Furthermore, post-flight medical debrief statistics between the period of 1996 β 2006
of all American and Russian spaceflight operations found a total of 219 in-flight
musculoskeletal injuries (Scheuring, et al., 2009). The study found that hand and forearm
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injuries dominate throughout the space program accounting for 47% of spacewalk related
injuries in the period of 2002 β 2004. It was also shown that EVA workers have a 19.6% chance
of incurring an injury during operations. The most common injuries include fingernail
delamination, minor muscle tears, tendon shortening, and minute fractures (Opperman,
Waldie, Natapoff, Newman, & Jones, 2010).
Current injury rates of astronauts are directly correlated with EVA operational practices. The
manual traversing and stabilization of astronauts about the parent space vehicle results in
numerous medical issues and hinders further operations (Danaher, Tanaka, & Hargens, 2005).
A new form of stabilization which will allow astronauts to work unhindered in zero-gravity
vacuum environments while maintaining a given attitude is required.
1.3. Project Aims
Before a new method of attitude control can be implemented into operation on astronaut
EVA suits, the behavioural characteristics of various passive and reactionary systems that
could potentially fill the role must be understand. Current literature on the topic of gyroscopic
attitude control falls outside the size-envelope of a one-man, spacesuit-sized system. As will
discussed below in Section 3, micro- and larger scale satellites use gyroscopes for attitude
control. These systems, however, are not applicable to a manned unit. More specifically the
various control laws used to control the system of CMGs on these vehicles must be
investigated for viability for use with a spacesuit. The focus of this thesis is to contribute to
the advancement of knowledge in the field of astronautics and system dynamics engineering.
The aim of the project is the qualitative and quantitative comparison of control moment
gyroscope system control laws for use with a spacesuit sized vehicle.
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To evaluate the performance and applicability of gyroscopes for the stated purpose the
project will investigate several topics to develop a final proposed system. The investigated
topics include;
- Zero gravity astronautics and dynamics of EVA suit sized systems,
- Propellant based attitude control packages,
- The functionality and characteristics of control moment gyroscopes, and
- The momentum envelope capabilities of CMG configurations.
2. Scope
Table 1 details aspects of the project which are inside and outside of the scope for this thesis.
Table 1 - Scope of the project
In Scope;
- Investigation of current technology and components.
- Computational models of the system using software packages such as Matlab, Simulink,
Solidworks, Ansys, and Python.
- Quantitative and qualitative comparison of CMG system control laws as found in literature.
- Theoretical and computer modelling of the proposed system.
Out of Scope;
- Bespoke software package coding or extensive computational work that requires a power
unit greater than that of a consumer grade PC.
- Computational fluid dynamics on rotor effects, these are assumed to be negligible given that
the rotors are enclosed in a small housing in zero gravity.
- Experimentation on designed system/s.
- Prototyping of designed components.
- Development of new steering law theory.
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3. Literature Review
3.1. Overview of Microgravity Attitude Control and Dynamics
Modern satellites and space vehicles use a range of methods for orientation and momentum-
management. There are three industry accepted technologies which allow this; high-torque
Reaction Wheel Assemblies (RWA), Control Moment Gyroscopes (CMG), and propellant based
systems (Leve, Hamilton, & Peck, 2015). The first two technologies are forms of energy and
momentum management within a system, the latter is a form of reactionary-control and
results in a non-zero net force on the system. Gyroscopic systems are continuous and present
a much finer degree of control as compared to the discrete propellant-based systems.
Although propellant based-systems can be used for small attitude adjustments the discrete
nature of the impulses from the jets result in a constant oscillation about the desired attitude
vector. When compared to gyroscopic systems there is a constant attitude error when using
propellant systems, as illustrated in Figure 3. The system of equations used to describe each
type of control is therefore different (Tewari, 2011). Unlike atmospheric flight vehicles,
attitude and orbital dynamics of a space vehicle are uncoupled allowing for independent
design and analysis (Leve, Hamilton, & Peck, 2015). This report deals exclusively with the
design and investigation into attitude control for a spacesuit. Translation and attitude-
translation combination manoeuvres fall outside the scope.
3.2. Propellant Based Systems
Propellant based systems have been used since the inception of the space program. Consisting
of a number of gimbaled or fixed nozzles, propellant is ejected from the parent body in order
to change or control the attitude of the system. This form of control is referred to as a reaction-
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control system whereby the system must react with another body or its surrounds to create a
change in the momentum of the spacecraft. Carpenter, et al. (2013) modelled an experimental
MMU with additional attitude control specific jet units. The designed system was then
analyzed with and without the addition of a set of π control moment gyroscopes. The results
show that the propellant-control only system developed large perturbations and uncertainties
when stabilizing the system to a set attitude. The propellant jets result in faster slew rates
however the total manoeuvre time without CMGs is extended due to the convergence rate
(Carpenter, et al., 2013).
Figure 3 - Attitude errors, propellant-only attitude control system (left), N number of CMG controllers added to system (right) (Source: Carpenter, et al., 2013)
3.3. Gyroscopic Attitude Control Systems
Reactionary systems require the use of finite propellant and therefore present large
inefficiencies for missions over ever-increasing duration and distance from Earth. Managing a
spacecraftβs momentum with rotors allows for the system to be controlled for an infinite
period given the required energy to maintain sufficient velocity of the rotors. There are two
different ways of managing the momentum of a system using gyroscopes, torque reaction
wheel assemblies (RWAs) and control moment gyroscopes (CMGs). The first uses a change in
flywheel velocity to enact a desired torque on the system, whereas the latter uses a changes
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the spin-moment vector of a constant speed fly-wheel to achieve the same. Both technologies
are discussed further in the next sections. A combination of the two technologies can also be
used, a variable speed control moment gyroscope (VSCMG). VSCMG systems are mentioned
in this report however they fall outside the scope of the project.
3.3.1. Torque Reaction Wheels
An RWA consists of a brushless motor attached to a high-inertia flywheel. The vector direction
of the flywheel is fixed about the spacecraft. The magnitude of the momentum vector varies
as the flywheel is sped up or slowed down. The mechanical shaft power is equal to the torque
times the speed of the rotor: π· = π β π (Leve, et al., 2015). Therefore, the resultant torque
applied to a spacecraft (ππΊπͺ) because of input power π· is thus; π =π·
π= βππΊπͺ. This type of
control can be viewed as a form of βmomentum exchangeβ whereby the momentum of the
flywheel is either increased or decreased by using electrical motors. The momentum energy
is transferred to electrical energy and stored, or vice versa. The spacecraft then experiences
an equal and opposite torque to that imparted on/by the flywheel. Although ideally this type
of system would be entirely reversible inefficiencies are introduced when converting work to
energy, and energy to work. Large energy storage capacity and electrical motors are also
required in addition to the flywheel.
3.3.2. Control Moment Gyroscopes
A CMG utilizes a secondary axis, or tertiary axis in some cases, to change the direction of the
momentum vector. In most applications, a CMG consists of a relatively constant speed
flywheel, however, variable-speed systems have become more prevalent (Leve, et al., 2015).
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The spacecraft and the CMG system exchange momentum to control and/or maintain the
attitude of the spacecraft. In contrast to an RWA system, the momentum of the entire system
is always maintained. Small amounts of energy are used to drive the gimbal motors however
the momentum of the flywheel and spacecraft as a whole is constant. The rotor with angular
momentum vector π, with magnitude π±πππ precesses dependent on applied torque π, such
that π = π Γ π = βππΊπͺ (Leve, et al., 2015).
Figure 4 - Reaction wheel assembly momentum vectors
Figure 5 - Single-gimbal CMG momentum vectors
π
π
π
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3.3.3. Vehicle Characteristics and Gyroscopic Control System Efficiency
As stated in Section 3.3.1 the torque potential of an RWA is limited by the mechanical power
of the motor. CMGs do not face the same restrictions. Hamilton, et al. (2015) presents the
differing capability envelopes of RWA, single-gimbal CMG, and double-gimbal CMG designs
(Figure 6). Similarly, Votel and Sinclair (2013) conducted evaluations of commercially available
gyroscopic control systems for small scale satellites. The authors present the results of their
analysis comparing the power consumption contours for each system type dependent on the
slew angle potential/requirements of the spacecraft. They concluded that for any satellite
exceeding a mass of 30kg should strongly consider a CMG system, and above 100kg there is
very little choice but to use a CMG (Votel & Sinclair, 2013).
Figure 6 - RWA and CMG comparison of momentum potential envelopes (Source: Hamilton, et al.,
2015)
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3.4. CMG System Arrangements and Singularities
There are infinite number of CMG configurations possible for a given system, these can be
categorized based on the net angular momentum of the system in the nominal position (Leve,
Hamilton, & Peck, 2015). More modern CMG configurations include the pyramid (Figure 7)
and scissor (or roof) (Figure 8) configurations. The configuration of the CMGs changes the
momentum envelope potential of the system, that is, the maximum potential torque that can
be applied in a certain direction. The details of the momentum envelope for a given system
are investigated mathematically below in Section 5.3. Although these configurations are more
generally used with single-gimbal CMGs, the limiting saturation alignment profiles are also
used in dual-gimbal CMG systems (Leve, Hamilton, & Peck, 2015).
Figure 7 - CMG pyramid configuration
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Figure 8 - CMG roof configuration
A singularity refers to the instance where two-or-more CMGs in a system have aligned or
exactly opposite momentum vectors. For systems with a cluster of π CMGs, when πΎ CMGs
are in a singular state then the system is only capable of producing torque in (π β πΎ)
dimensions. That is to say; for a common system of 4 CMGs, if 2 CMGs are in a singular state
then the system is only capable of producing a torque in 2 dimensions (McMahon & Schaub,
2009).
If the system is not controlled adequately then singularities within a CMG system can lead to
limited operational capabilities, total inoperability, and ultimately, system failure. Consider
the case where the required spacecraft attitude change results in an ever-increasing storage
of momentum in the CMG system. As momentum is exchanged between the spacecraft and
the gyroscopic control system the CMGs approach the maximum angular momentum state,
the point at which no further angular momentum can be stored in the system. Figure 6
illustrates this theoretical circumstance. Starting in a net-zero state where all CMGs have equal
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and opposite momentum vectors, and progressing to a saturation state where all momentum
vectors are aligned.
The system must then be desaturated to allow the CMG control system to dissipate angular
momentum. A torque must be applied to the spacecraft which the CMG system can react
against to realign the gyroscopes into the nominal position. This can be achieved by firing
propellant thrusters, deploying a magnetic torque rod which will react against the Earthβs
magnetic field, or any other means which produces an outside torque on the system (Russel,
Spencer, & Metrocavage, 2008).
The saturation limit is dictated by component and configuration characteristics and exceeding
this limit can cause catastrophic failure (Kim, 2011). The International Space Station (ISS) uses
a system of four dual-gimbal CMGs with a life expectancy of 10 years and an output torque of
258 Nm each (Gurrisi, et al., 2010). Two of the gyroscopes, designated CMG1 and 3, failed
prematurely. Gurrisi, et al. (2010) conducted a failure analysis and determined that CMG1 had
a βhardβ, catastrophic failure due to excessive lateral forces on the gimbal bearing, CMG3 had
a βsoftβ failure due to fretting on a bearing sleeve that resulted in excessive perturbation and
error. The initial failure began because of a large angular-momentum transfer prior to a
Shuttle docking procedure. Although manufacturing and assembly defects were the cause of
the failure the excessive strain on the supporting structure due to the saturated-nominal
position aggravated the defects (Gurrisi, et al., 2010).
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3.4.1. Singularity Avoidance and Null Motion
Avoiding singularities has been at the forefront of astronautic research in the recent past as
satellites become ever-smaller and the need for a control system utilizing three or less CMGs
becomes more apparent. Several authors have investigated steering laws for constant-speed
CMG clusters and other authors have investigated the potential of variable-speed CMG
(VSCMG) clusters. The avoidance of singularity states is achieved by means of either efficiency
path-planning, or null motion. Path-planning is widely used on satellites when attitude control
is pre-determined before-the-fact. This method of singularity avoidance is not applicable to a
spacesuit given the user operated nature of the conceptual system. Null motion refers to the
controlled momentum cancelling of two or more CMGs against each other.
Imagine that a user controlled manoeuvre results in two CMGs approaching a singularity state.
As is the case with a constant speed, single-gimbal CMG system, there is no way to increase
or decrease the momentum of any singular gyroscope. Only the gimbal angles can be
controlled by actuating the gimbal motors. The only way to avoid a singularity is to react the
individual gyroscopes against each other in a manner which results in a net-zero torque on the
system. In plain terms, two or more gimbals are actuated slightly to βbleedβ some momentum
energy away from the system, the result of which is the avoidance of a singularity while the
overall system can achieve the desired attitude change. The term null motion refers to the
comparative motion between the individual CMGs in a manner that results in a net-zero, null,
torque on the system.
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4. Project Methodology and Process
As outlined in Section 1.3, the aim of the project is the qualitative and quantitative comparison
of CMG system control laws for use with a spacesuit sized vehicle. To achieve this the project
has been broken into two separate stages, an initial qualitative comparison through literature,
and a quantitative comparison using a Python model of a conceptualized system. The
following steps outline the methodology used to achieve the desired project outcomes and to
meet the stated aim;
1. Defined the generalized CMG system from first principles.
2. Designed a conceptual system that will be used to compare steering laws.
3. Defined the singularity problem and thus the steering law problem, mathematically.
4. Found steering laws in the literature that address the steering law problem and that
are applicable to the conceptual system.
5. Investigated the chosen steering laws individually for qualitative characteristics and
compared.
6. Using the defined conceptual system, designed a Python model that computes the
system response for different control laws.
7. Implemented the different control laws being tested into the model.
8. Tested the modelled system response for a user-defined attitude manoeuvre and
compared the results.
9. Made an overall conclusion on the most appropriate control law for a spacesuit CMG
system and made comments regarding potential changes that would be required for
implementation.
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5. System Model
5.1. Dynamics and Equations of Motion
Consider a system with a single CMG device as depicted in Figure 9. By torqueing the gimbal,
the flywheel orientation is changed, thereby re-directing the rotorβs angular momentum. The
derivation of Equation 1 from first principles is shown in Appendix A. The resultant torqueing
on the overall system is equal and opposite to that imparted by the internally mounted CMG;
ππ = ππ Γ πΎοΏ½ΜοΏ½ = βπππΆ
where;
ππ = Torqueing vector of CMG π
ππππ = Spacesuit torqueing vector
πππ= Flywheel angular momentum vector
πΎοΏ½ΜΜοΏ½ = CMG gimbal angle actuation rate
Figure 9 - Control moment gyroscope diagram (Source: Yoon, 2004)
(1)
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Jones, Zeledon & Peck (2012) investigated potential configurations for single-gimbal CMG
systems. The authors concluded that for a system with constant speed-flywheels, π CMGs is
the most efficient system size for (π β 1)-dimensional attitude control. Therefore, a
conceptual system consisting of 4 CMGs has been used for this report. Equation 1 can be
adapted for a system of π = 4 CMGs;
πTOT = β(ππ Γ πΎοΏ½ΜοΏ½)
π=4
π=1
Applying the condition that all CMGs be identical in moment of inertia and angular velocity
the system model allows for simplification in design and modelling. Consider the conceptual
system depicted in Figure 10. The system has been designed using a typical pyramid
configuration as discussed in Section 3.4.
Figure 10 - Pyramid CMG system diagram
(2)
π1
π2
π3
π4 πΈ1
πΈ2
πΈ3
πΈ4
π
π½
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The system model must account for several angular momentum components related to;
β’ Point mass of the spacesuit,
β’ Moment of inertia of the spacesuit,
β’ Angular velocity of the spacesuit,
β’ Point masses of the CMGs,
β’ Flywheel angular velocity, and
β’ Gimbal angular velocity.
The rotational equation of motion of a rigid spacecraft equipped with a momentum-exchange
device can be described as;
οΏ½ΜοΏ½ + π Γ π― = π»EXT
where π― = (π»π₯, π»π¦ , π»π§) is the angular momentum vector of the spacesuit relative to the
origin, π = (π1, π2, π3) is the spacesuit angular velocity vector, and π»EXT is the external
torque vector applied to the spacesuit system (Wie, 2001). The cross product π Γ π― in matrix
notation takes the form;
πΓπ― = [
0 βππ§ ππ¦
ππ§ 0 βππ₯
βππ¦ ππ₯ 0] [
π»π₯
π»π¦
π»π§
]
Assuming that the system is rigid and that there are negligible losses in energy between the
CMGs and the parent spacesuit, the equation for angular momentum by Gui, Vukovich & Xu
(2016) can be used. The authors state that the angular momentum of the system about centre
of mass, M, for a system with π CMGs can be described as the sum of the spacecraft angular
moment and that of the individual CMGs;
π― = π±π + π΄ππΌπποΏ½ΜοΏ½ + π΄π πΌπ€π π
(3)
(4)
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Where πΈ = (πΎ1, β¦ , πΎπ)π β βπ and π = (Ξ©1, β¦ , Ξ©π)π β βπ represent the gimbal angles and
flywheel speed vectors of each CMG in the individual gimbal reference frames, respectively.
The matrix π½ is the Jacobian inertia matrix for the whole spacesuit system defined as;
π± = πΌπ΅ + π΄π πΌππ π΄π π + π΄π‘πΌππ‘π΄π‘
π + π΄ππΌπππ΄ππ
Where πΌπ΅ is the inertia matrix of the spacesuit system and the point masses of each CMG. The
matrices πΌπβ and πΌπ€β are diagonal with elements of inertia for the gimbal plus flywheel
structure and flywheel-only of each CMG. Therefore, πΌπβ = πΌπ€β + πΌπβ where πΌπ€β =
diag[πΌπ€β1, β¦ , πΌπ€βπ] and πΌπβ = diag[πΌπβ1, β¦ , πΌπβπ] for β= οΏ½ΜοΏ½, οΏ½ΜοΏ½ or οΏ½ΜοΏ½.
Matrices π¨β β β3Γπ represent the directional unit vectors in the roll (π₯), pitch (π¦) and yaw (π§)
directions relative to the body-frame (as defined in Figure 10). Therefore, π¨π = [οΏ½ΜοΏ½1, β¦ , οΏ½ΜοΏ½π],
π¨π = [οΏ½ΜοΏ½1, β¦ , οΏ½ΜοΏ½π], and π¨π = [οΏ½ΜοΏ½1, β¦ , οΏ½ΜοΏ½π], such that;
π¨β = [π¨π
π¨π
π¨π
] = [
("Roll Axis Component")
("Pith Axis Component")
("Yaw Axis Component")]
Because the matrices π΄β components are dependent on the individual gimbal angles, πΎπ, ie
π΄π = π΄π(πΎ) and π΄π = π΄π (πΎ), by definition π΄β = π΄β(πΎ) and therefore π± = π±(πΎ). Note that the
inertial matrix πΌπ΅ is constant, ie πΌπ΅ β πΌπ΅(πΎ).
Equation 4 can be further simplified by assuming the angular momentum contribution from
the gimbal structure is negligible compared to that of the flywheel and the overall system such
that π΄β = π΄. Due to the comparative nature of the final system model this assumption does
not adversely affect the project results. Therefore, eliminating the angular momentum
contribution due to gimballing, (π΄ππΌπποΏ½ΜοΏ½), Equation 4 becomes;
π― = π±π + β ππ
π=4
π
= π±π + π1 + π2 + π3 + π4 = π±π + π
(5)
(6)
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Carpenter et al. (2013) outlines the time derivative of Equation 6 with the assumption that the
point mass change of the CMGs relative to the overall system is negligible. Although this
assumption may be appropriate for a larger scale spacecraft system, such as the International
Space Station (ISS), this assumption does not hold true for the conceptual system. With
implementation on a spacesuit sized system the point masses of the CMGs have an
attributable effect on the angular momentum of the system. The methodology of the authors
has been used to find the time derivative of the angular momentum function with
modification to account for the CMG point mass angular velocity.
Introducing π΅ and π denoting the body and inertial origin frames of reference respectively
and taking the time derivative;
π― = π±Γππ΅/π + β ππ
π=4
π
οΏ½ΜοΏ½ = π±οΏ½ΜοΏ½π΅/π + β ππΜ
π=4
π
where ππ΅/π denotes the system angular velocity vector, π, in the body frame relative to the
inertial origin frame. The angular momentum of the CMGs for the conceptual system is;
π = β ππ(πΎπ)
π=4
π=1
= [βcos π½ sin πΎπ
cos πΎ1
sin π½ sin πΎ1
] β1 + [
β cos πΎ2
β cos π½ sin πΎ2
sin π½ sin πΎ2
] β2 + [cos π½ sin πΎ3
β cos πΎ3
sin π½ sin πΎ3
] β3 + [
cos πΎ4
cos π½ sin πΎ4
sin π½ sin πΎ4
] β4
Noting that the CMG flywheels have equal angular velocity, β = β1 = β2 = β3 = β4;
π = β ([βcos π½ sin πΎπ
cos πΎ1
sin π½ sin πΎ1
] + [
β cos πΎ2
β cos π½ sin πΎ2
sin π½ sin πΎ2
] + [cos π½ sin πΎ3
β cos πΎ3
sin π½ sin πΎ3
] + [
cos πΎ4
cos π½ sin πΎ4
sin π½ sin πΎ4
])
(7)
(8)
(9)
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= β ([
βcos π½ sin πΎπ β cos πΎ2 + cos π½ sin πΎ3 + cos πΎ4
cos πΎ1 β cos π½ sin πΎ2 β cos πΎ3 + cos π½ sin πΎ4
sin π½ sin πΎ1 + sin π½ sin πΎ2 + sin π½ sin πΎ3 + sin π½ sin πΎ4
])
Combining Equations 3, 6 and 8 the equation of motion of the system is shown to be;
(π±οΏ½ΜοΏ½ + οΏ½ΜοΏ½) + π Γ(π±π + π) = π»EXT
Equation 11 is then represented as two simultaneous equations relating to the system and the
CMG control torque separately. Introducing the control torque vector term, π;
π±οΏ½ΜοΏ½ + π Γ π±π = π + π»EXT
οΏ½ΜοΏ½ + π Γ π = βπ
For a desired attitude manoeuvre the required control torque, π, can be found. Rearranging
Equation 13 further, the non-linear equation describing the control system dynamics, referred
to as the torque equation, is;
οΏ½ΜοΏ½ = βπ β π Γ π
5.2. Pseudoinverse and Minimum Norm Solution
The most rudimentary steering law is that of the pseudoinverse solution. By inverting the
torque equation (Equation 14) the required gimbal angle rates for a desired system angular
momentum state can be calculated. The time derivative of the CMG angular momentum
vector, π, is found by taking the derivative of Equation 9 such that;
οΏ½ΜοΏ½ = β ππΜ (πΎπ)
π=4
π=1
= π¨οΏ½ΜοΏ½
where πΈ = (πΎ1, πΎ2, πΎ3, πΎ4) is the gimbal angle vector, and π΄ is shown to be;
(10)
(11)
(12)
(13)
(14)
(15)
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π΄ = [βcos π½ cos πΎπ
βsin πΎ1
sin π½ cos πΎ1
sin πΎ2
β cos π½ cos πΎ2
sin π½ cos πΎ2
cos π½ cos πΎ3
sin πΎ3
sin π½ cos πΎ3
β sin πΎ4
cos π½ cos πΎ4
sin π½ cos πΎ4
]
Such that;
οΏ½ΜοΏ½ = [βcos π½ cos πΎπ
βsin πΎ1
sin π½ cos πΎ1
sin πΎ2
β cos π½ cos πΎ2
sin π½ cos πΎ2
cos π½ cos πΎ3
sin πΎ3
sin π½ cos πΎ3
β sin πΎ4
cos π½ cos πΎ4
sin π½ cos πΎ4
] [
πΎ1Μ
πΎ2Μ
πΎ3Μ
πΎ4Μ
]
Therefore, adding Equations 14 and 15 and letting οΏ½ΜοΏ½ β‘ π, the pseudoinverse solution is shown
to be;
π¨οΏ½ΜοΏ½ = π β‘ βπ β π Γ π
β΄ οΏ½ΜοΏ½ = π¨π(π¨π¨π)β1π
Solving Equation 19 produces the minimum 2-norm vector of gimbal rates, οΏ½ΜοΏ½, that will result
in the desired attitude change. This solution drives the system towards a singularity state
which will cause the system to reach a singularity state. This can be viewed in terms of energy
to better understand why the system converges to a singular state.
The most energy efficient way to achieve an attitude change is to use angular momentum
contribution from all available sources. If the individual CMGs attribute equal angular
momentum changes to the overall system, the CMGs inherently approach a singular state
where β1 = Β±β2. As explained by Paradiso, 1991, βthe minimum norm property of the
pseudoinverse encourages the formation of singular statesβ. The pseudoinverse is only a
particular solution to the torque equation, other steering laws either use different approaches
to solve the equation or use the pseudoinverse solution and apply some form of modifying
scalar.
(16)
(17)
(18)
(19)
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5.3. Momentum Envelope and CMG Gain
The momentum envelope of a given system represents the potential torque output of the
system in all planes of operation. More broadly it is used as a representation of the potential
total angular momentum for a given system along any given axis (Paradiso, 1991). It is a 3D
surface that represents the saturation limits of a system and any other internal singularity
states that arise. Previous findings from the literature detail the effects that different
configurations of CMG systems on the momentum envelope. Referring to Figure 10, the angle
of the pyramid sides, π½, affects the overall shape of the outer surface of the systemβs
momentum envelope. For the conceptual pyramid system, setting π½ = 54.73Β° results in a
nearly spherical momentum envelope outer surface (Yoon & Tsiotas, 2002).
For larger satellite systems on orbital paths, a non-circular envelope may be employed to
maximise the potential angular momentum in a certain axis at the expense of another. For the
purposes of this project a spherical momentum envelope has been deemed the most relevant
when considering the practical applications of a spacesuit mission. Figure 11 depicts the
momentum envelope of the conceptual system.
Mathematically singularity states can be defined by the βCMG gainβ of the system in the same
manner as Paradiso, 1991. Using the Jacobian matrix of the system as defined in Equation 5,
the CMG gain is defined as;
π = β|π¨π¨π|
where π½ = π΄ β΅ πΌπ΅ = π΄π = 0 for the simplified conceptual system case. As π approaches 0 the
system approaches a singular state and loses control in one or more dimensions.
(20)
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Figure 11 - Momentum envelope of conceptual system (Source: Yoon, 2004)
5.4. Elliptical and Parabolic Singularities
Before the steering laws can be addressed further clarification on the required control of the
system must be made. The singularity states can be categorised into two definitions;
βhyperbolicβ and βparabolicβ. Using Figures 12 through 16 as reference, hyperbolic states are
escapable through null motion, on the momentum envelope they are represented by all
internal surfaces. Parabolic states are inescapable and are represented by the outer surfaces.
They occur when all CMGs have aligned angular momentum vectors. If the system enters a
parabolic singularity state then it can only be escaped by an external torque being applied.
The CMG steering law problem is complex due to the different behaviours required for
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hyperbolic and parabolic states. Depending on the initial gimbal angles at the start of an
attitude manoeuvre, the system may need to transit through a hyperbolic state less the
system be restricted in operation for certain attitudes. In contrast, parabolic states must be
avoided at all costs.
Figure 12 - Hyperbolic singularity momentum envelope surface (Source: Yoon, 2004)
Figure 13 - Hyperbolic singularity CMG arrangement
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Figure 14 β Parabolic singularity momentum envelope surface (Source: Yoon, 2004)
Figure 15 - Parabolic singularity CMG arrangement
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Figure 16 - Momentum envelope of pyramid configuration CMG system (Source: Yoon, 2004)
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6. Steering Laws
There are many proposed steering laws that can be found in the literature that have potential
to control a spacesuit CMG system and avoid the described singularity states. Viewing the
steering laws in a broad sense there are two distinct types; singularity-avoidance laws and
singularity-robust laws (Jones, Zeledon, & Peck, 2012). The latter allows the system to enter a
hyperbolic singularity state (escapable and denoted by the internal surface on the momentum
envelope), in some circumstances where the transit through the singularity can be controlled.
Singularity-robust laws are the more modern method of control of the two types. The former,
singularity-avoidance laws, do not allow the system to enter a singularity state in any
circumstance. Neither of the steering law types allow the system to enter a parabolic,
inescapable singularity state. The only way to restore control to a saturated system in this
state is to apply an external torque which, for obvious, reasons is outside the control of any
potential steering law (Gui & Vukovich, 2016).
Further to this most modern steering laws can be placed into one of six categories that
differentiate between distinctive design mentalities (Kurokawa, 2007);
1. Moore-Penrose Pseudoinverse
2. Offline Planning
3. Preferred Gimbal Angle/Gimbal Reorientation
4. Linearly Constrained
5. Gradient Weighting/Null Motion
6. Singularity Robust Inverse
No singularity control
Singularity-avoidance
Singularity-robust
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The listed design mentality categories have different advantages and characteristics. Although
each may be broken down into further subcategories these are mostly trivial and do not
change the major design characteristics of the various solutions (Jones, Zeledon, & Peck,
2012). The special case of the Moore-Penrose Pseudoinverse has no inherent protocol to avoid
or deal with singularities but it has been used in scenarios where the potential momentum
envelope is significantly greater than the required envelope for a given mission/application. It
is worth noting that most steering laws use the Moore-Penrose solution as a basis and apply
various scaling factors to control singularities (NASA TM, 1972). For this reason, although it is
not suitable to use this law with a spacesuit system, it has been analysed to gain further insight
into the other steering laws.
The investigated control laws have been assessed in several areas as they pertain to use with
a spacesuit. The points of qualitative assessment are;
β’ Singularity handling method,
β’ Induced error and perturbations,
β’ System efficiency and the full use of the hardware momentum envelope,
β’ Convergence to singularity arrangements, and
β’ Potential for generalization to other CMG arrangements.
The last assessment criterium has been included as it is a good indication of how the steering
law will cope with partial system failure. If for reasons outside the control of the CMG
controller one or more CMGs fail, the controller must be adaptable to configurations other
than the nominally designed case.
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6.1. No Singularity Avoidance
6.1.1. Moore-Penrose Pseudoinverse
As previously discussed in Section 5.2, the Moore=Penrose Pseudoinverse (MPI) solution is
the basic form of the steering logic and is the solution to Equation 18 (Jung & Tsiotras, 2008);
οΏ½ΜοΏ½π = π¨π(π¨π¨π)β1π
The MPI solution is one that minimizes the required system input, gimbal angle rate (οΏ½ΜοΏ½π), to
achieve the desired system output, total CMG torqueing (π). Jung & Tsiotras (2008) define the
MPI solution as the one that minimizes the input gimbal rates;
minοΏ½ΜοΏ½
||οΏ½ΜοΏ½||2
subject to π¨οΏ½ΜοΏ½ = π
If a system employs an MPI steering controller then the system will eventually converge to a
singular state without fail (Jung & Tsiotras, 2008).
Gui & Vukovich (2016) conducted an analysis on potential steering laws for a spacecraft with
two parallel CMGs. The authors found that a controller employing an unmodified MPI solution
results in satisfactory performance up until a saturation limit was reached. At this point the
system was unable to escape the singularity. Further work by the George C. Marshall Flight
Center, NASA (1972) found that to make an unmodified MPI controller viable, the mission had
to be restricted to 50-60% of the potential momentum envelope cap abilities of the CMG
system. The authors also noted that the unmodified MPI controller can control the system in
the instance of a single CMG failure for a 4-CMG system. At the point of failure, the system
was still capable of calculating the required gimbal rates to achieve the desired attitude
change (NASA TM, 1972).
(21)
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6.2. Singularity-Avoidance
6.2.1. Offline Planning
Offline planning is an all-encompassing term and relates to any steering law logic which uses
predefined data sets to control the CMG system for a given scenario. For instance, a system
may be modelled for a range of potential attitude changes. If any particular manoeuvres
introduce singularity states to the system then these can be adjusted. For any similar
manoeuvers in the future the system can use memory based logic to avoid the singularity
state.
Significant research has been done in this area by Paradiso (1991). The author investigated
the use of heuristic optimization of pre-planned gimbal angle inputs for a range of torque-
producing trajectories. After the database of planned manoeuvres has been established then
a search protocol must be implemented. The search protocol is responsible for analysing the
system state and then comparing to cases stored in the database. Although this methodology
has the potential for optimization it does not allow for adaption to unplanned system changes
or states. The ISS has employed offline planning for the control of its 6 CMG control system
but this is highly dependent on the fact that the station only carried out schedule attitude
changes (Gurrisi, et al., 2010).
For use with a spacesuit, offline planning steering control is likely unfeasible. Unlike the ISS, a
user driven spacesuit must be subjectable to range to a range of attitude manoeuvers. Nor is
the steering law capable of dealing with system changes or component failure. Although it is
possible to model such a system accurately given enough computational power, such
resources are likely unavailable onboard a spacesuit system.
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6.2.2. Preferred Gimbal Angle/Gimbal Reorientation
An extension to offline planning is the preferred gimbal angle method. First proposed by
Vadali, Oh & Walker (1990), the method uses the MPI solution to find the most efficient initial
gimbal angles for a set of required attitude manoeuvres. The system is modelled for initial
states near saturation (the outer momentum envelope surface) and back integration using the
MPI solution is used to determine all attitude changes that are possible without the system
entering a singularity state, either hyperbolic or parabolic. The results are then compared to
the potential mission profile and then the most appropriate initial gimbal angle position is
determined (Jones, Zeledon, & Peck, 2012).
The preferred gimbal angle controller has the same advantages and characteristic as the
offline planning method. Vadali, Oh & Walker (1990) conducted numerical simulations of their
proposed steering logic on the Soviet space station MIR, and Space Station Freedom (the
original concept mission that later became the ISS mission).
The authors found that optimizing the initial gimbal angles allowed the system to utilize
regions of the momentum envelope closer to singularity states than the offline planning
method at the cost of full use of the rest of the envelope (Vadali, Oh, & Walker, 1990). Like
offline planning, the preferred gimbal angle method is still highly dependent on the foresight
or potential future attitude manoeuvers and is susceptible to unforeseen system changes and
failure.
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6.2.3. Linearly Constrained
The final singularity-avoidance law is the linearly constrained method. The steering law first
proposed by Kurokawa (1997), is a simple geometric restraint placed on the relative gimbal
angles. The author present the simple case for a 4 CMG pyramid configuration which is very
close to the conceptual system proposed in this report. Noting that internal singularity states
occur at any point where the angular momentum vector of one CMG aligns or negates that of
another, ie;
ππ = Β±ππ
By restricting the relative angles between the gimbals then singularity states along one or
more specific axes can be avoided. Using the case presented by Kurokawa (1997) the system
can be made non-singular by applying the constraint function;
π1 β π2 + π3 β π4 = 0
A geometric representation of the constraint equation for a system of 3 CMGs can be seen in
Figure 17. A draw back of the linearly constrained steering law is significant reduction in the
useable momentum envelope. Figure 18 shows the restricted momentum envelope
workspace for a 4 CMG pyramid system that prevents singularities in the π§-axis, note the
shaded region is the lost momentum envelope as a result of the linearly constrained steering
law.
(22)
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Figure 17 β Constrained steering law applied to system of 3 CMGs (Source: Jones, et al., 2012)
Figure 18 - Effects of constrained steering law on momentum envelope (Source: Kurokawa, 1997)
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6.3. Singularity-Robust
6.3.1. Gradient/Null Motion Weighted Matrix
The first type of singularity-robust steering law allows for the system to transit through
internal singularities by utilizing null motion. Null motion is the process of actuating gimbals
in such a way as to produce a net zero torque on the overall system, but which results in a
reorientation of the gimballed CMGs (Bedrossian, Paradiso, & Bergmann, 1990).
Mathematically the null motion method uses the CMG gain value, π = βπ¨π¨π, and weighting
matrix, π[π°], to determine the degree of introduced null motion in the system. Referring to
general MPI solution of Equation 18;
minοΏ½ΜοΏ½
||οΏ½ΜοΏ½||2
subject to π¨οΏ½ΜοΏ½ = π
The weight null motion solution has the form;
οΏ½ΜοΏ½ = οΏ½ΜοΏ½MPI + π[π°] β οΏ½ΜοΏ½NM
where, οΏ½ΜοΏ½MPI is the general gimbal angle solution in line with the MPI method and οΏ½ΜοΏ½NM is the
introduced null motion factor. There is an inversely proportional relation between CMG gain,
π, and the weighted matrix, π[π°];
π = βπ¨π¨π β π[π°]β1
Other sources from the literature detail alternative implementations of the null motion
methodology. Systems can employ step functions to introduce null motion into the gimbal
rate solutions (Yoshikawa, 1977), pair the null motion with scissoring limit laws that link
partner CMGs (Bedrossian, Paradiso, & Bergmann, 1990), or use null motion principles to
create oscillations about a singularity state to maximise the useable work envelope (Hefner &
McKenzie, 1983).
(23)
(24)
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Other works by Jin & Hwang (2011) and Zhang & Fang (2013) investigate the potential of back-
stepping control methods for use with double-gimbal VSCMG systems. The authors present a
dual-loop control system comprising of an attitude loop and an angular speed loop. Although
the authors investigate the control of a variable speed CMG system which is out of scope for
this project, the null motion principles are applicable to a constant speed CMG system. Zhang
& Fang (2013) present the control law using Euler notation, Jin & Hwang (2011) develop the
controller using quaternion notation however both follow the same logic (Jin & Hwang, 2011;
Zhang & Fang, 2013). As the system approaches the singularity state a small random
perturbation in null motion is introduced into the attitude control loop to allow the spacecraft
to reach the desired attitude while maintaining a non-singular gyroscope orientation (Zhang
& Fang, 2013).
Compared to singularity-avoidance laws, the useable work envelope available for a system
using a null motion control is greater. Although some methods introduced random null motion
perturbations rather than strictly controlled rates, both methods allow the system to reach
near singularity points, and oscillate about and through them. For a smaller system on a
spacesuit where hardware sizing is limited, the full utilization of the work envelope is crucial.
6.3.2. Singularity-Robust Inverse
The final steering law method, and the most modern, is an extension on the null motion
principles. Most null motion controls intentionally add error to the solution to oscillate the
system around and/or through singularities. Although easily implemented while still creating
useable torque from respective CMGs near singularities, the system torque is only accurate in
a small region of the momentum envelope. The closer to a singularity the system gets, the
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greater the attitude error. There is trade-off between the useable work envelope and accuracy
of attitude at different points in the envelope.
If random perturbations are introduced into the system then there is also a risk of control
gimbal rates that exceed hardware limitations. There is significant research on the topic of
singularity-robust inverse (SRI) steering laws that can be found in the literature.
Wie, Bailey & Heiberg (2001) present the generalized SRI solution. In comparison to the null
motion solution, the SRI solution introduces a new term into Equation 21 in an attempt to
calculate gimbal rates that produce an accurate, non-singular result. Taking Equation 21 and
adding the additional term, ππ, the general SRI solution is;
οΏ½ΜοΏ½π = π¨π(π¨π¨π)β1π + ππ
where π is the 1Γ4 matrix of form [πΆ1, πΆ2, πΆ3πΆ4] containing the optimized singularity measure
values that determine the amount of null motion from each CMG, and π is the step function;
π = { π6 for π β₯ 1πβ6 for π < 1
Note, π is the CMG gain of the system as defined previously.
The benefit of this solution is that null motion introduced into the system is proportional to
the state of the system and can be accounted for. This method is superior to the random
perturbations introduced with other methods and reduces inherent error with attitude
changes near singularities.
Jung & Tsiotras (2008) investigated the potential to add further terms to Equation 25. The
authors tested various SRI solutions on a 3 CMG system and concluded that the induced
torque error was reduced and that the energy efficiency of the system was greater when
compared to an MPI solution with randomly induced torqueing when near singularities.
(25)
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6.4. Qualitative Summary
Table 2 summarises the properties of the detailed steering laws against the metrics listed in
Section 6.0.
Table 2 - Qualitative comparison summary
Although each steering law methodology has various advantages and characteristics which
are applicable to a spacesuit system, the singularity-robust inverse method is the best fit for
purpose. The full use of the momentum envelope with a smaller package system like a
spacesuit is crucial. It is also essential that the system can adapt to any hardware changes or
error, system failure with a user outside of the parent spacecraft is unacceptable. The SRI
solution provides the user with redundancy without the need for excessive computational
Singularity Handling Method
Induced Error
Momentum Envelope Efficiency
Convergence to Singularity Arrangements
Generalization Capabilities for System Changes
Moore-Penrose Pseudoinverse
None None Full use Possible Fully adaptable
Offline Planning
Optimized pre-modelling None Slightly reduced Impossible Requires extensive computational power
Preferred Gimbal Angle
Optimized initial gimbal orientations
None Slightly reduced Impossible Requires extensive computational power and system adjustment
Linearly Constrained
Restricted relative gimbal angles
None Reduced dependent on initial gimbal angles
Impossible Fully adaptable
Null Motion Randomly induced torque scaling with proximity to singularities
Large near singularity
Full use but with large induced error in segments
Possible Fully adaptable
Singularity-Robust Inverse
Optimized null motion scaling factor from system Jacobian inertial matrix
Controlled near singularity
Full use Possible Fully adaptable
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power and the full use of the momentum envelope available unlike singularity-avoidance
methods.
Although the weighted null motion solution is also feasible, the reduced error with the SRI
solution is beneficial. If missions call for finer motor control of instruments for maintenance
or scientific processes, the reduced error results in greater attitude stabilization.
7. Simulation and Results
The SRI solution was tested against the MPI solution for the conceptual system. Figure 10
details the geometry of the system. The simulation was run using a time-step interval of 0.001
seconds. The work of Yime-Rodriguez et al. (2014) has been used heavily in the construction
of the computational model. Several simplifications have been made; the centre of mass of
the system has been placed at the base of the pyramid as indicated. The initial gimbal angles
have been set to 0Β° for all gimbals, and therefore the initial angular momentum vector of the
system is also 0 such that ππ = [π, π, π]π». The pyramid skew angle has been set at 53.47Β° to
ensure a nearly spherical momentum envelope. The angular momentum magnitude for all
flywheels has been set to a nominal value of 1 ππ β π/π 2 to ensure uniformity and to simplify
the model response. A gimbal rate limit of Β±25Β°/π and a gimbal angle limit of Β±100Β° was used
as per consensus from the literature on feasible hardware limits (Jung & Tsiotras, 2008).
The model requests a constant torque demand from the system in the π₯-axis (roll-axis) such
that οΏ½ΜοΏ½ β‘ π = [π, π, π]π». Applying the Euler method, at each time step the model calculates
the gimbal rates using the MPI or SRI steering law solution for each case respectively. An
estimate of the angular momentum vector, π, is then found. The singularity state of the CMG
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configuration can then be visualized by plotting the CMG gain against the magnitude of
angular momentum in the roll-axis, ie π versus |βπ₯|.
For the MPI test, the minimum gimbal rate solution to the identity equation οΏ½ΜοΏ½ = π¨π(π¨π¨π)β1π
was found for the final system π΄ matrix;
π΄ = [βcos π½ cos πΎπ
βsin πΎ1
sin π½ cos πΎ1
sin πΎ2
β cos π½ cos πΎ2
sin π½ cos πΎ2
cos π½ cos πΎ3
sin πΎ3
sin π½ cos πΎ3
β sin πΎ4
cos π½ cos πΎ4
sin π½ cos πΎ4
]
For the SRI test the same torque demand was used. The additional null motion component
was added to the minimum solution equation such that οΏ½ΜοΏ½ = π¨π(π¨π¨π + ππ°)β1π. The scaling
factor, π, was set using the values found from experimental optimization by Jung & Tsiotras
(2008) such that;
IF π > πcritical;
π = 0
ELSE
IF π0
π< πMAX;
π =π0
π
ELSE
π = πMAX
where πcritical = 1.0, π0 = 0.1, and πMAX = 0.2. This ensures that the introduced null motion
in the SRI solution is low when the system is far from a singular state, and increases as the
system approaches a singular state. Note, the system is not singular at π = πcritical. The
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induced torque error of the SRI system can be calculated by computing the difference
between the input angular momentum command, οΏ½ΜοΏ½ β‘ π = [π, π, π]π»Μ , and the angular
momentum that would result from the calculated gimbal rates at each time step, ie;
οΏ½ΜοΏ½error = |οΏ½ΜοΏ½input β (π¨ β οΏ½ΜοΏ½calculated)|
Figures 19 and 20 show the results. The raw data points are given in Appendix B.
Figure 19 - Comparison of MPI and SRI Solutions
Figure 20 - SRI Solution Induced Torque Error from Input Command
(26)
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8. Discussion
As expected the MPI system rapidly approaches a singular state at ππ₯ β 1.15 corresponding
to the angular momentum alignment of CMGs 1 and 3 at πΈ = [βππ, π, ππ, π]π» (Β°). This
position corresponds to an inescapable singularity state (Figures 14 and 15). The parabolic
response of the CMG gain near the singularity can be seen in Figure 19. At this point, CMGs 2
and 4 are not capable of producing a change in angular momentum in the roll-axis and
therefore the system rapidly approaches the singularity state.
Using the SRI system, it is shown that once the critical gain limit is reached then the control
law introduces a disturbing factor to avoid the singularity. As shown in Figure 20, the system
shows a deflection in the roll (π₯), pitch (π¦) and yaw (π§) axes near the singularity state. Although
it is still not possible to pass directly through the singularity with no introduced error, the
system can successfully reach an angular momentum vector very-near-to while maintaining a
higher CMG gain.
The results of the conceptual system model support the findings from the qualitative
comparison of the steering laws. The singularity-robust inverse solution avoided the
singularity state by inducing a small, controlled torqueing error when near a singularity state.
For implementation with a spacesuit the SRI solution should be used over an MPI solution. It
does not require excessive computational power like other control methods and it effectively
avoids singularity states. The trade-off of angular momentum error is deemed acceptable
when compared to the confidence that the system will avoid a singularity. In a worst-case
scenario for an astronaut during an EVA, if a singularity state is entered and there is no foreign
body with induce an outside torque on the system, then the astronaut may be stuck.
Further consideration should be made before implementing a CMG system into a spacesuit.
Although current computational power precludes aforementioned steering methods at the
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current time, faster computers or quantum computers may allow such methods to be used in
the future. The essential factor as to when machine learning and search algorithm dependent
laws become available is the ability to recalculate nominal gimbal rates in the event of a
system change or component failure in a timely manner.
Alternatively, different hardware configurations other than that detailed in this report can be
investigated. Different configurations other than a pyramid can be used to alter the
momentum envelope of the spacesuit. Double-gimbal CMG systems are possible where a
single flywheel can be actuated as to produce a torque vector in any direction in 3D-space. In
theory, a single double-gimbal CMG could be used to control a spacesuit but there is significant
strain placed on the gimbal mounting structure as a result of transverse bending during gimbal
actuation (Zhang & Fang, 2013).
Additionally, there is significant knowledge available in the literature regarding variable-speed
CMGs which allow for the flywheel speed of each CMG to be altered independently. Using the
example from the simulation in Section XXX, the singularity state could be avoided by altering
the speed of flywheels 1 and 3 in such a way that would allow the system to reach an angular
momentum vector of π = [1.15, 0, 0]π with a CMG gain, π, > 0 (Jin & Hwang, 2011). A new
area of research emerging is the potential to use a variable-speed CMG system for attitude
control and power storage. High inertia flywheels have been used on satellite systems as a
form of energy storage for some time. Yoon (2004) provides an overview of spacecraft control
and power management using such a system. Although feasible and potentially viable the
author notes the additional electrical motor and linkages required to enable the braking and
acceleration of the flywheel. Although conceivable for a satellite or spacecraft sized vehicle
the additional mass of the electrical motor as a percentage of total mass is much higher for a
spacesuit.
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9. Conclusion
The project has met the initial aim of conducting a qualitative and quantitative comparison of
control moment gyroscope system control laws for use with a spacesuit sized vehicle. Control
moment gyroscopes are a viable technology for use with a spacesuit. Utilizing momentum
exchange principles, a spacesuit could conserve resources and energy as compared to the
modern day manned-manoeuvring units. Providing astronauts with a means of attitude
control and stabilization will lead to the reduction of EVA related injuries of the hands and
forearms. It will also provide a solid foundation from which astronauts will be able to conduct
maintenance and scientific work.
Singularity-avoidance and singularity-robust control laws which are currently used with
satellite and larger-scale spacecraft can potentially be adapted for use with spacesuit. Broadly
speaking any given control law either induces a controlled torqueing error when near
singularities or restricts the available momentum envelope to avoid singularities altogether.
The singularity-robust inverse control law is deemed the most viable option. As shown in the
simulation results, a proportional torqueing error is introduced when the CMG gain of the
system approaches 0. The control law successfully navigates around singularities. This method
allows for the full use of the potential momentum envelope for the given hardware. Compared
to a larger spacecraft the mass of the CMGs as a percentage of total mass is much higher for
a spacesuit sized system. Maintaining full use of the momentum envelope is crucial. The
control law is also capable of accounting for system changes and/or component failure
without the need for excessive computational modelling or hardware adjustment.
Implementing a CMG control system, utilizing the singularity-robust inverse control law, into
a spacesuit will allow future astronauts to navigate space more efficiently than what is
possible today.
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10. References
Bedrossian, N., Paradiso, J., & Bergmann, E. (1990). Steering Law Design for Redundant Single-
Gimbal Control Moment Gyroscopes. Journal of Guidance, Control and Dynamics,
1083-1089.
Brown, D., & Peck, M. (2009). Energetics of Control Moment Gyroscopes as Joint Actuators.
Journal of Guidance, Control, and Dynamics, 1871-1883.
Carpenter, M., Jackson, K., Cohanim, B., Duda, K. R., Rize, J., Dopart, C., . . . Zumbado, J. R.
(2013). Next Generation Maneuvering System with CMGs for EVA Near Low-Gravity
Objects. Houston: American Institude of Aeronautics and Astronautics.
Cheng, J. P. (2010). Dynamic Model of an Astronaut Equipped with a Manned Maneuvering
Unit in Virtual Reality. Journal of Aerospace Engineering, 139-145.
Danaher, P., Tanaka, K., & Hargens, A. R. (2005, April 4). Mechanical Counter-Pressure vs. Gas-
Pressurized Spacesuit Gloves: Grip and Sensitivity. Aviation, Space, and Environmental
Medicine, pp. 381-384.
Dawson, R. (2013). Orbits and Locked Gimbals. New York: Springer Science + Business Media.
Gui, H., & Vukovich, G. (2016). Attitude Stabilization of a Spacecraft with Two Parallel Control
Moment Gyroscopes. Journal of Guidance, Control and Dynamics, 724-731.
Gurrisi, C., Seidel, R., Dickerson, S., Didziulis, S., Frantz, P., & Ferguson, K. (2010). Space Station
Control Moment Gyroscope Lessons Learned. Houston: NASA.
Harland, D. M. (2010). NASA's Moon Program - Paving the Way for Apollo 11. Dordrecht:
Springer.
Hefner, R., & McKenzie, C. (1983). A Technique for Maximizing the Torque Capability of
Control Moment Gyroscopes. Proceedings of the AAS/AIAA Astrodynamics Conference
(pp. 905-920). Lake Placid: AAS.
Jennings, R., & Bagian, J. (1996). Musculoskeletal injuries review in the U.S. space program.
Aviation, Space, and Environmental Medicine, pp. 762-766.
Jin, J., & Hwang, I. (2011). Attitude Control of a Spacecraft with Single Variable-Speed Control
Moment Gyroscope. Journal of Guidance, Control and Dynamics, 1920-1924.
Jones, L. L., Zeledon, R. A., & Peck, M. A. (2012). Generalized Framework for Linearly
Constrained Control Moment Fyro Steering. Journal of Guidance, Control and
Dynamics, 1094-1103.
Jung, D., & Tsiotras, P. (2008). An Experimental Comparison of CMG Steering Control Laws.
Atlanta: Georgia Institute of Technology.
N McNamara
47
Kim, D. (2011). A Novel Integrated Spacecraft Attitude Control System Using Variable Speed
Control Moment Gyroscopes: A Lyuapunov-Based Approach. University of Florida.
Kurokawa, H. (2007). Survey of Theory and Steering Laws of Single-Gimbal Control Moment
Gyroscopes. Journal of Guidance, Control and Dynamics, 1331-1340.
Leve, F. A., Hamilton, B. J., & Peck, M. A. (2015). Spacecraft Momentum Control Systems. New
York: Springer International.
MacKunis, W., Dupree, K., Fitz-Coy, N., & Dixon, W. (2008). Adaptive Satellite Attitude Control
in the Presence of Inertia and CMG Gimbal Friction Uncertainties. The Journal of
Astronautical Sciences, 121-134.
McMahon, J., & Schaub, H. (2009). Simplified Singularity Avoidance Using Variable-Speed
Control Moment Gyroscope Null Motion. Boulder: Journal of Guidance, Control and
Dynamics.
NASA TM. (1972). A Comparison of CMG Steering Laws for High Energy Astronomy
Observatories. NASA Marshall Space Flight Center.
Opperman, R. A., Waldie, J. M., Natapoff, A., Newman, D. J., & Jones, J. A. (2010, October).
Probability of Spacesuit-Induced Fingernail Trauma Is Associated with Hand
Circumference. Aviation, Space, and Environmental Medicine, pp. 907-913.
Paradiso, J. (1991). Global Steering of Signle GImballed cOntrol Moment Gyrosopes Using a
Directed Search. AIAA Guidance and Control Conference. New Orleans: The Charles
Stark Draper Laboratory.
Russel, S. P., Spencer, V., & Metrocavage, K. (2008). On-Orbit Propulsion and Methods of
Momentum Management for the International Space Station. AIAA/ASME/SAE/ASEE
Joint Propulsion Conference & Exhibit. Hartford: JPL.
Tewari, A. (2011). Automatic Control of Atmospheric and Space Vehicles. London: Springer.
Thomas, K. S., & McMann, H. J. (2006). US Spacesuits. Chichester: Springer Science and
Business Media.
Vadali, S., Oh, H., & Walker, S. (1990). Preferred Gimbal Angles for Single Gimbal Control
Moment Gyroscope. Journal of Guidance, Control, and Dynamics, 1090-1095.
Votel, R., & Sinclair, D. (2013). Comparison of Control Moment Gyros and Reaction Wheels for
Small Earth-Observing Satellites. Annual AIAA/USU Conference on Small Satellites.
Utah: Utah State University.
West, W., & Witt, V. (2010). Post-Shuttle EVA Operations on ISS. Houston, Texas: American
Institute of Aeronautics and Astronautics.
Wie, B. (2001). Singularity Robust Steering Logic for Redundant Single-Gimbal Control
Moment Gyroscopes. Journal of Guidance, Control and Dynamics, 865-872.
N McNamara
48
Yime-Rodriguez, E., Pena-Cortes, C., & Rojas-Contreras, M. (2014). The Dynamic Model of a
Four Control Moment Gyroscope System. Colombia: SciELO.
Yoon, H., & Tsiotas, P. (2002). Spacecraft Adaptive Attitude and Power Tracking with Variable
Speed Control Moment Gyroscopes. Journal of Guidance, Control and Dynamics, 1081-
1090.
Yoshikawa, T. (1977). Steering Law for Roof Type Configuration Control Moment Gyro System.
Automatica, 359-368.
Zhang, H., & Fang, J. (2013). Robust Backstepping Control for Agile Satellite Using Double-
Gimbal Variable-Speed Control Moment Gyroscope. Journal of Guidance, Control and
Dynamics, 1356-1367.
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11. Appendices
11.1. Appendix A β CMG Torque Derivation from First Principles
Brown & Peck (2009) outline the derivation of torque applied by a control moment gyroscope.
The angular momentum of a CMG about the centre of mass is the sum of the momentum of the
flywheel and the gimbal structure;
βCMG = πΌπ β ππΊπ + πΌπ β π
π π
= πΌπ β (ππΊπ΅ + π
π΅π) + πΌπ β (π
π πΊ + π
πΊπ΅ + π
π΅π)
where πΌπ and πΌπ is the moment of inertia of the gimbal and rotor respectively, and ππ
π is the angular
velocity of in the π frame relative to the π frame for; rotor frame, π , gimbal frame, πΊ, body frame, π΅,
and origin frame, π.
Letting;
πΌπ β ππ πΊ = βπ
β΄ βCMG = πΌCMG β (ππΊπ΅ + π
π΅π) + βπ
where, πΌπ + πΌπ = πΌCMG. Taking the time derivative in the origin frame;
πCMG = ββCMGπ β ππ
where;
βCMGπ = πΌCMG β (οΏ½ΜοΏ½πΓπ
π΅(π΅π
)β οΏ½ΜοΏ½πΓπ
π΅π) + (οΏ½ΜοΏ½π + π
π΅π) Γ (πΌCMG β (οΏ½ΜοΏ½π + π
π΅π) + βπ)
where;
οΏ½ΜοΏ½π = ππΊπ΅
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For the case where a CMG has a spherical momentum envelope (as is the case with most constant
speed flywheel systems), [(οΏ½ΜοΏ½π + ππ΅
π) Γ (πΌCMG β (οΏ½ΜοΏ½π + ππ΅
π) + βπ)] can be eliminated (Brown & Peck,
2009).
β΄ βCMGπ = πΌCMG β (οΏ½ΜοΏ½πΓπ
π΅(π΅π
)β οΏ½ΜοΏ½πΓπ
π΅π)
This equation can then be split into two separate equations representing the torques applied by the
gimbal structure and torque transferred to the parent system.
Neglecting friction, electro-magnetic forces, and flexible body effects;
πCMG = βοΏ½ΜοΏ½πΓβπ
ππ = πΌCMGοΏ½ΜοΏ½ + πΌCMGππ΅(
π΅π
)β π + (π
π΅πΓβπ) β π
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11.2. Appendix B β Simulation Data
The data in this section is represented with an angular momentum interval of 0.01 from 0 to 2.25.
MPI SRI
ππ π π Ξππ₯ Ξππ¦ Ξππ§
0.00 0.0000 0.0000 0 0 0
0.01 1.0800 1.0800 0 0 0
0.02 1.0790 1.0790 0 0 0
0.03 1.0765 1.0765 0 0 0
0.04 1.0733 1.0733 0 0 0
0.05 1.0700 1.0700 0 0 0
0.06 1.0667 1.0667 0 0 0
0.07 1.0629 1.0629 0 0 0
0.08 1.0588 1.0588 0 0 0
0.09 1.0544 1.0544 0 0 0
0.10 1.0500 1.0500 0 0 0
0.11 1.0452 1.0452 0 0 0
0.12 1.0398 1.0398 0 0 0
0.13 1.0344 1.0344 0 0 0
0.14 1.0293 1.0293 0 0 0
0.15 1.0250 1.0250 0 0 0
0.16 1.0213 1.0213 0 0 0
0.17 1.0177 1.0177 0 0 0
0.18 1.0146 1.0146 0 0 0
0.19 1.0119 1.0119 0 0 0
0.20 1.0100 1.0100 0 0 0
0.21 1.0085 1.0085 0 0 0
0.22 1.0072 1.0072 0 0 0
0.23 1.0060 1.0060 0 0 0
0.24 1.0053 1.0053 0 0 0
0.25 1.0050 1.0050 0 0 0
0.26 1.0053 1.0053 0 0 0
0.27 1.0061 1.0061 0 0 0
0.28 1.0072 1.0072 0 0 0
0.29 1.0086 1.0086 0 0 0
0.30 1.0100 1.0100 0 0 0
0.31 1.0118 1.0118 0 0 0
0.32 1.0140 1.0140 0 0 0
0.33 1.0167 1.0167 0 0 0
0.34 1.0194 1.0194 0 0 0
0.35 1.0220 1.0220 0 0 0
0.36 1.0245 1.0245 0 0 0
0.37 1.0270 1.0270 0 0 0
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0.38 1.0296 1.0296 0 0 0
0.39 1.0323 1.0323 0 0 0
0.40 1.0350 1.0350 0 0 0
0.41 1.0377 1.0377 0 0 0
0.42 1.0404 1.0404 0 0 0
0.43 1.0432 1.0432 0 0 0
0.44 1.0463 1.0463 0 0 0
0.45 1.0500 1.0500 0 0 0
0.46 1.0546 1.0546 0 0 0
0.47 1.0601 1.0601 0 0 0
0.48 1.0664 1.0664 0 0 0
0.49 1.0731 1.0731 0 0 0
0.50 1.0800 1.0800 0 0 0
0.51 1.0871 1.0871 0 0 0
0.52 1.0948 1.0948 0 0 0
0.53 1.1028 1.1028 0 0 0
0.54 1.1112 1.1112 0 0 0
0.55 1.1200 1.1200 0 0 0
0.56 1.1293 1.1293 0 0 0
0.57 1.1392 1.1392 0 0 0
0.58 1.1495 1.1495 0 0 0
0.59 1.1598 1.1598 0 0 0
0.60 1.1700 1.1700 0 0 0
0.61 1.1807 1.1807 0 0 0
0.62 1.1921 1.1921 0 0 0
0.63 1.2031 1.2031 0 0 0
0.64 1.2127 1.2127 0 0 0
0.65 1.2200 1.2200 0 0 0
0.66 1.2257 1.2257 0 0 0
0.67 1.2312 1.2312 0 0 0
0.68 1.2357 1.2357 0 0 0
0.69 1.2388 1.2388 0 0 0
0.70 1.2400 1.2400 0 0 0
0.71 1.2388 1.2388 0 0 0
0.72 1.2356 1.2356 0 0 0
0.73 1.2310 1.2310 0 0 0
0.74 1.2256 1.2256 0 0 0
0.75 1.2200 1.2200 0 0 0
0.76 1.2135 1.2135 0 0 0
0.77 1.2053 1.2053 0 0 0
0.78 1.1958 1.1958 0 0 0
0.79 1.1856 1.1856 0 0 0
0.80 1.1750 1.1750 0 0 0
0.81 1.1638 1.1638 0 0 0
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0.82 1.1515 1.1515 0 0 0
0.83 1.1383 1.1383 0 0 0
0.84 1.1244 1.1244 0 0 0
0.85 1.1100 1.1100 0 0 0
0.86 1.0954 1.0954 0 0 0
0.87 1.0805 1.0805 0 0 0
0.88 1.0649 1.0649 0 0 0
0.89 1.0484 1.0484 0 0 0
0.90 1.0304 1.0304 0 0 0
0.91 1.0107 1.0000 0 0 0
0.92 0.9886 1.1938 0.1826 0.0270 -0.0958
0.93 0.9633 1.1866 0.1883 0.0308 -0.1041
0.94 0.9347 1.1789 0.1941 0.0341 -0.1093
0.95 0.9031 1.1715 0.1999 0.0369 -0.1132
0.96 0.8684 1.1650 0.2054 0.0392 -0.1173
0.97 0.8305 1.1593 0.2105 0.0407 -0.1230
0.98 0.7870 1.1540 0.2153 0.0419 -0.1290
0.99 0.7384 1.1490 0.2199 0.0428 -0.1350
1.00 0.6856 1.1443 0.2242 0.0437 -0.1410
1.01 0.6292 1.1400 0.2282 0.0445 -0.1470
1.02 0.5682 1.1360 0.2319 0.0455 -0.1531
1.03 0.4943 1.1322 0.2359 0.0467 -0.1599
1.04 0.4137 1.1286 0.2397 0.0479 -0.1669
1.05 0.3351 1.1252 0.2429 0.0490 -0.1732
1.06 0.2672 1.1220 0.2447 0.0498 -0.1782
1.07 0.2160 1.1189 0.2449 0.0502 -0.1815
1.08 0.1711 1.1159 0.2442 0.0505 -0.1844
1.09 0.1309 1.1131 0.2429 0.0507 -0.1869
1.10 0.0959 1.1104 0.2412 0.0509 -0.1888
1.11 0.0670 1.1080 0.2391 0.0510 -0.1899
1.12 0.0438 1.1057 0.2366 0.0510 -0.1899
1.13 0.0232 1.1036 0.2323 0.0509 -0.1894
1.14 0.0086 1.1020 0.2265 0.0507 -0.1885
1.15 0.0031 1.1007 0.2199 0.0505 -0.1873
1.16 0.0004 1.1000 0.2132 0.0502 -0.1858
1.17 0.0004 1.1007 0.2068 0.0498 -0.1839
1.18 0.0031 1.1020 0.2001 0.0491 -0.1802
1.19 0.0086 1.1036 0.1931 0.0481 -0.1751
1.20 0.0232 1.1057 0.1859 0.0470 -0.1692
1.21 0.0438 1.1080 0.1786 0.0457 -0.1630
1.22 0.0670 1.1104 0.1713 0.0443 -0.1569
1.23 0.0959 1.1131 0.1636 0.0424 -0.1500
1.24 0.1309 1.1159 0.1558 0.0402 -0.1425
1.25 0.1711 1.1189 0.1482 0.0380 -0.1350
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1.26 0.2160 1.1220 0.1412 0.0359 -0.1281
1.27 0.2672 1.1252 0.1350 0.0342 -0.1221
1.28 0.3351 1.1286 0.1293 0.0327 -0.1167
1.29 0.4137 1.1322 0.1240 0.0313 -0.1116
1.30 0.4943 1.1360 0.1190 0.0299 -0.1068
1.31 0.5682 1.1400 0.1143 0.0286 -0.1022
1.32 0.6292 1.1443 0.1098 0.0273 -0.0978
1.33 0.6856 1.1490 0.1054 0.0259 -0.0932
1.34 0.7384 1.1540 0.1012 0.0245 -0.0888
1.35 0.7870 1.1593 0.0973 0.0232 -0.0848
1.36 0.8305 1.1650 0.0937 0.0223 -0.0814
1.37 0.8687 1.1715 0.0905 0.0218 -0.0787
1.38 0.9049 1.1789 0.0881 0.0216 -0.0760
1.39 0.9384 1.1866 0.0860 0.0215 -0.0736
1.40 0.9676 1.1938 0.0836 0.0213 -0.0717
1.41 0.9910 1.0000 0.0802 0.0209 -0.0706
1.42 1.0075 1.0075 0 0 0
1.43 1.0197 1.0197 0 0 0
1.44 1.0299 1.0299 0 0 0
1.45 1.0400 1.0400 0 0 0
1.46 1.0502 1.0502 0 0 0
1.47 1.0597 1.0597 0 0 0
1.48 1.0691 1.0691 0 0 0
1.49 1.0790 1.0790 0 0 0
1.50 1.0900 1.0900 0 0 0
1.51 1.1031 1.1031 0 0 0
1.52 1.1182 1.1182 0 0 0
1.53 1.1336 1.1336 0 0 0
1.54 1.1481 1.1481 0 0 0
1.55 1.1600 1.1600 0 0 0
1.56 1.1702 1.1702 0 0 0
1.57 1.1799 1.1799 0 0 0
1.58 1.1885 1.1885 0 0 0
1.59 1.1954 1.1954 0 0 0
1.60 1.2000 1.2000 0 0 0
1.61 1.2031 1.2031 0 0 0
1.62 1.2058 1.2058 0 0 0
1.63 1.2080 1.2080 0 0 0
1.64 1.2095 1.2095 0 0 0
1.65 1.2100 1.2100 0 0 0
1.66 1.2097 1.2097 0 0 0
1.67 1.2089 1.2089 0 0 0
1.68 1.2077 1.2077 0 0 0
1.69 1.2064 1.2064 0 0 0
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1.70 1.2050 1.2050 0 0 0
1.71 1.2033 1.2033 0 0 0
1.72 1.2012 1.2012 0 0 0
1.73 1.1988 1.1988 0 0 0
1.74 1.1967 1.1967 0 0 0
1.75 1.1950 1.1950 0 0 0
1.76 1.1936 1.1936 0 0 0
1.77 1.1923 1.1923 0 0 0
1.78 1.1911 1.1911 0 0 0
1.79 1.1903 1.1903 0 0 0
1.80 1.1900 1.1900 0 0 0
1.81 1.1905 1.1905 0 0 0
1.82 1.1919 1.1919 0 0 0
1.83 1.1941 1.1941 0 0 0
1.84 1.1968 1.1968 0 0 0
1.85 1.2000 1.2000 0 0 0
1.86 1.2055 1.2055 0 0 0
1.87 1.2144 1.2144 0 0 0
1.88 1.2256 1.2256 0 0 0
1.89 1.2379 1.2379 0 0 0
1.90 1.2500 1.2500 0 0 0
1.91 1.2627 1.2627 0 0 0
1.92 1.2768 1.2768 0 0 0
1.93 1.2917 1.2917 0 0 0
1.94 1.3063 1.3063 0 0 0
1.95 1.3200 1.3200 0 0 0
1.96 1.3336 1.3336 0 0 0
1.97 1.3475 1.3475 0 0 0
1.98 1.3608 1.3608 0 0 0
1.99 1.3720 1.3720 0 0 0
2.00 1.3800 1.3800 0 0 0
2.01 1.3859 1.3859 0 0 0
2.02 1.3914 1.3914 0 0 0
2.03 1.3958 1.3958 0 0 0
2.04 1.3989 1.3989 0 0 0
2.05 1.4000 1.4000 0 0 0
2.06 1.3989 1.3989 0 0 0
2.07 1.3959 1.3959 0 0 0
2.08 1.3915 1.3915 0 0 0
2.09 1.3861 1.3861 0 0 0
2.10 1.3800 1.3800 0 0 0
2.11 1.3714 1.3714 0 0 0
2.12 1.3588 1.3588 0 0 0
2.13 1.3435 1.3435 0 0 0
N McNamara
56
2.14 1.3268 1.3268 0 0 0
2.15 1.3100 1.3100 0 0 0
2.16 1.2917 1.2917 0 0 0
2.17 1.2708 1.2708 0 0 0
2.18 1.2492 1.2492 0 0 0
2.19 1.2283 1.2283 0 0 0
2.20 1.2100 1.2100 0 0 0
2.21 1.1939 1.1939 0 0 0
2.22 1.1788 1.1788 0 0 0
2.23 1.1647 1.1647 0 0 0
2.24 1.1517 1.1517 0 0 0
2.25 1.1400 1.1400 0 0 0