U.S. patent number RE37,374 [Application Number 09/452,682] was granted by the patent office on 2001-09-18 for gyro-stabilized platforms for force-feedback applications.
This patent grant is currently assigned to Cybernet Haptic Systems Corporation. Invention is credited to Charles J. Jacobus, Gerald P. Roston.
United States Patent |
RE37,374 |
Roston , et al. |
September 18, 2001 |
**Please see images for:
( Certificate of Correction ) ** |
Gyro-stabilized platforms for force-feedback applications
Abstract
Force feedback in large, immersive environments is provided by
device which a gyro-stabilization to generate a fixed point of
leverage for the requisite forces and/or torques. In one
embodiment, one or more orthogonally oriented rotating gyroscopes
are used to provide a stable platform to which a force-reflecting
device can be mounted, thereby coupling reaction forces to a user
without the need for connection to a fixed frame. In one physical
realization, a rigid handle or joystick is directly connected to
the three-axis stabilized platform and using an inventive control
scheme to modulate motor torques so that only the desired forces
are felt. In an alternative embodiment, a reaction sphere is used
to produce the requisite inertial stabilization. Since the sphere
is capable of providing controlled torques about three arbitrary,
linearly independent axes, it can be used in place of three
reaction wheels to provide three-axis stabilization for a variety
of space-based and terrestrial applications.
Inventors: |
Roston; Gerald P. (Erie,
PA), Jacobus; Charles J. (Ann Arbor, MI) |
Assignee: |
Cybernet Haptic Systems
Corporation (San Jose, CA)
|
Family
ID: |
26674855 |
Appl.
No.: |
09/452,682 |
Filed: |
November 30, 1999 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
Reissue of: |
736016 |
Oct 22, 1996 |
05754023 |
May 19, 1998 |
|
|
Current U.S.
Class: |
318/561;
318/568.11; 318/649 |
Current CPC
Class: |
B25J
9/1689 (20130101); F16F 15/00 (20130101); G05B
5/01 (20130101); G05B 13/042 (20130101); G05G
5/03 (20130101); G06F 3/016 (20130101); G05B
2219/37164 (20130101); G05B 2219/37174 (20130101); G05B
2219/40122 (20130101); G05B 2219/41274 (20130101); G06F
2203/013 (20130101); G06F 2203/015 (20130101) |
Current International
Class: |
F16F
15/00 (20060101); G05B 13/04 (20060101); G05B
5/00 (20060101); G05B 5/01 (20060101); G05G
5/00 (20060101); G05G 5/03 (20060101); G05B
013/02 (); B25J 009/00 () |
Field of
Search: |
;318/561,566,567,568.1,568.11,628,648,649 ;414/4,5 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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0265011A1 |
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Apr 1988 |
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EP |
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0607580A1 |
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Jul 1994 |
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EP |
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0626634A2 |
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Nov 1994 |
|
EP |
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WO 92/00559 |
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Jan 1992 |
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WO |
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WO 95/12188 |
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May 1995 |
|
WO |
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WO 97/31333 |
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Aug 1997 |
|
WO |
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|
Primary Examiner: Ro; Bentsu
Attorney, Agent or Firm: Riegel; James R. Tucker; Guy V.
Parent Case Text
REFERENCES TO RELATED APPLICATIONS
This invention claims priority of U.S. Provisional Application
Serial No. 60/005,861, filed Oct. 26, 1995, the entire contents of
which are incorporated herein by reference.
Claims
What is claimed is:
1. A spatially unrestricted force-feedback device, comprising:
a body;
gyroscopic means connected to the body to provide an inertial
reference to stabilize the body in at least one spatial
dimension;
a user-interactable member connected to the body; and
force-feedback means coupled to the member, enabling a user of the
device to experience the feedback of forces relative to the
gyroscopically stabilized body.
2. The device of claim 1, wherein the user-interactable member is a
joystick.
3. The device of claim 1, wherein the user-interactable member
includes a handle.
4. The device of claim 1, wherein the user-interactable member is a
steering wheel.
5. The device of claim 1, wherein the user-interactable member is a
device associated with the simulation of a sport.
6. The device of claim 1, further including:
a computer system modeling a virtual environment including one or
more virtual objects; and wherein
the user-interactable member is in electrical communication with
the computer system to generate forces on the member as a function
of an activity involving an object within the virtual
environment.
7. The device of claim 1, wherein the gyroscopic means includes a
momentum wheel, and wherein a torque is produced on the member
through accelerating and decelerating the angular velocity of the
wheel.
8. The device of claim 7, including three momentum wheels to
stabilize the body in three dimensions.
9. The device of claim 1, wherein the gyroscopic means takes the
form of a reaction sphere operative to produce arbitrary reaction
torques about three linearly independent axes of the body.
10. The device of claim 1, further including an angular position
measuring device to determine the state of the body in space.
11. The device of claim 10, wherein the angular position measuring
device is a potentiometer.
12. The device of claim 10, wherein the angular position measuring
device as an encoder.
13. The device of claim 1, further including an angular velocity
measuring device to determine the state of the gyroscopic
means.
14. The device of claim 13, wherein the angular velocity measuring
device is a tachometer.
15. The device of claim 13, wherein the state of the gyroscopic
means is determined by numerically differentiating the angular
position of the body.
16. The device of claim 1, further including an active control
system to provide device stability.
17. A spatially unrestricted force-feedback device, comprising:
a body;
an active control system to stabilize the body in space;
three rotatable reaction wheels coupled to the body;
means for determining the angular velocity of each wheel;
an angular position measuring device to determine the state of the
body in the space;
a user-interactable member connected to the body; and
force-feedback means using the angular velocity and position of the
body as inputs to produce torque on the member about three
arbitrary axes through the coordinated acceleration and
deceleration of the angular velocity of each wheel.
18. The device of claim 17, wherein the angular position measuring
device is an inertial measuring unit.
19. The device of claim 17, wherein the angular velocity measuring
device uses numerical differentiation to determine the angular
position of the body.
20. A method of generating a spatially unrestricted haptic
environment, comprising the steps of:
providing a body in space having a user-interactable force-feedback
device;
geo-stabilizing the body in one or more dimensions;
simulating a virtual environment modeling one or more virtual
objects; and
interfacing the user-interactable force-feedback device to the
virtual environment, enabling the user to experience a force
representative of an activity within the virtual environment
involving one or more of the objects.
21. The method of claim 20, further including the step of:
slowly and continually removing angular momentum from the body so
as to minimize the effect on a user.
22. The method of claim 20, further including the steps of:
receiving an input disturbance on the body;
stabilizing the body through a pole placement, with the location of
the poles being determined through optimal control theory; and
cancelling out the disturbance inputs to produce a desired torque
output immune to the input disturbance.
23. The method of claim 20, further including the step of:
receiving an external force generated through a remote physical
device; and
producing a scaled representation of the force received relative to
a point on the physical device.
24. The method of claim 23, wherein the scaled representation is
such that the maximum force applicable to the physical device is
mapped into the maximum force which the device is capable of
producing..Iadd.
25. A spatially unrestricted force-feedback device, comprising:
a body;
a plurality of motors, each of said motors capable of spinning a
momentum mass about an associated axis of rotation and each of said
motors connected to said body to provide computer controllable
inertial forces on said body about said associated axis;
a user-interactable member connected to said body, wherein said
user-interactable member is in electrical communication with a host
computer system modeling a simulated environment including one or
more simulated objects, said host computer system commanding said
inertial forces on said body as a function of a simulated activity
involving at least one object within said simulated environment;
and
a computer mediated controller electrically connected to said
motors and in electrical communication with said host computer
system, said controller receiving signals from said host computer
system and simultaneously controlling each of said motors in
response such that said motors produce said inertial forces about
said axes..Iaddend..Iadd.
26. A spatially unrestricted force-feedback device as described in
claim 25, wherein said computer mediated controller sends position
data to said host computer system..Iaddend..Iadd.
27. A spatially unrestricted force-feedback device as described in
claim 25, wherein said computer mediated controller decodes
commands received from said host computer
system..Iaddend..Iadd.
28. A spatially unrestricted force-feedback device as described in
claim 25, wherein said computer mediated controller decodes
commands received on a serial communication bus..Iaddend..Iadd.
29. A spatially unrestricted force-feedback device as described in
claim 25, wherein said user-interactable member is a
joystick..Iaddend..Iadd.
30. A spatially unrestricted force-feedback device as described in
claim 25, wherein said user-interactable member is a steering
wheel..Iaddend..Iadd.
31. A spatially unrestricted force-feedback device as described in
claim 25, wherein said user-interactable member is associated with
the simulation of a sport..Iaddend..Iadd.
32. A spatially unrestricted force-feedback device as described in
claim 25, wherein said computer mediated controller includes a
processor that runs motor control code stored in Read-Only
memory..Iaddend..Iadd.
33. A spatially unrestricted force-feedback device as described in
claim 25, wherein at least a portion of said computer controllable
inertial forces stabilize said body in at least one spatial
dimension to counteract undesired torques produced by at least one
of said motors..Iaddend..Iadd.
34. A spatially unrestricted force-feedback device as described in
claim 25, wherein said computer controllable inertial forces
stabilize said body in at least one spatial dimension..Iaddend.
Description
FIELD OF THE INVENTION
The present invention relates generally to force feedback and, more
particularly, to the use of gyroscopic stabilization to provide an
inertial frame against which a force-reflecting device react.
BACKGROUND OF THE INVENTION
Force-feedback technology and related devices may be divided into
four broad application areas: medical, entertainment,
teleoperations, and virtual reality. Teleoperations, the research
of which provided the foundation for the development of
force-feedback devices, is the process of locally controlling a
remote device. The primary difference between virtual reality and
teleoperations is in the objects which they control. With
teleoperations, actual physical robots are manipulated in the real
world, whereas virtual reality involves simulated devices in
synthetic worlds. Force-feedback for telerobotics has evolved large
and bulky mechanical arms to more joystick-like designs. In
general, these devices are designed for six degree-of-freedom
(6DOF) force feedback, and have the capability to provide high
levels of force. More recently, finger-operated devices have also
been introduced for use in teleoperations applications.
The use of force feedback in medical training, simulation, and
teleoperations is also increasing, with the primary application
being minimally invasive surgical techniques which use laparscopic
tools to perform intricate tasks when inserted into body cavities
through small incisions. To realistically simulate laparoscopic
tool forces, special-purpose force-feedback devices are currently
under development.
The entertainment field is very difficult to address with
force-feedback technology, since the applications demand both
higher performance and lower costs. There are three primary markets
for force feedback devices in entertainment: location-based
entertainment (LBE), arcades, and home entertainment. LBE demands
the highest performance while home entertainment demands the lowest
cost. Despite the conflicting demands, progress is being made in
each of these fields.
It may be argued that each of the application domains just
described has its roots in virtual reality, which is becoming
dominant in all immersive applications. As a consequence, on-going
research in immersive applications is often termed "virtual
reality," whereas, when the research is completed, the application
is given a specific name, such as a surgical simulator. Overall,
virtual reality is becoming increasingly popular as a preferred
means of interacting with many scientific and engineering
applications. To cite two of many examples, molecular modeling and
automobile design are moving from standard graphics, carried out on
conventional graphics terminals, to more interactive environments
utilizing 3-D stereo graphics, head-mounted displays and force
feedback.
As visualization is a very important aspects of these applications,
interesting and useful technologies are being developed, including
graphical object representations and large working volumes (CAVES).
Concurrently, haptic interfaces are being perfected, which enable
manual interactions with virtual environments or teleoperated
remote systems. The haptic system is a unique sensory system in
that is can both sense the environment and allow a user to react
accordingly. As a result, haptic devices not only stimulate the
user with realistic sensor input (forces, tactile sensations, heat,
slip, etc.), but also sense the user's actions so that realistic
sensory inputs can be generated. Haptic devices are divided into
two classes, depending upon the type of sensory information being
simulated. The first, tactile, refers to the sense of contact with
the object. The second, kinesthetic, refers to the sense of
position and motion of a user's limbs along with associated
forces.
Broadly, these approaches point toward the same goal: to immerse a
person in a seemingly visual reality, complete with haptic
feedback. However, a major deficiency with all existing
force-generating devices is the requirement that they be connected
to a fixed frame, thus forcing immobility on the user.
State-of-the-art force-feedback devices, for example, are table
mounted, requiring the device to be mounted to an immobile object
in order to generate a fixed point of leverage for forces and/or
torques. Consequently, no existing force feedback device allows for
easy mobility and force generation. This problem is fundamental,
since many virtual reality applications require large working
volumes and the ability to move freely within these volumes, to
provide realistic visual and audio feedback during walk-through
scenarios, for example.
In summary, large, immersive environments such as CAVES currently
lack haptic feedback, primarily because the existing technology
will not support unrestricted motion. This leads to one conclusion
that force-feedback devices must migrate as visual technologies
have, that is, from the desktop to large-volume, immersive
environments. However, the design of a hand-held, spatially
unrestricted force-feedback device is fundamentally different from
existing devices, which typically use primarily electromechanical
or pneumatic actuators operating against fixed supports to achieve
active force feedback. Nor is the realization of such a device
intuitively obvious. To construct an n-axis joystick, requiring 1,
2, 3 to n+3 motors, presents significant challenges, for example,
since the additional motors may significantly increase the cost
and/or weight of the device.
SUMMARY OF THE INVENTION
The present invention addresses the need for force feedback in
large, immersive environmentally by providing a device that uses a
gyro-stabilization to generate a fixed point of leverage for the
requisite forces and/or torques. In one embodiment, one or more
orthogonally oriented rotating gyroscopes are used to provide a
stable body or platform to which a force-reflecting device can be
mounted, thereby coupling reaction forces to the user without the
need for connection to a fixed frame. In one embodiment, a
user-interactable member is physically coupled to a stabilized
body, with the control structure used for stabilization and that
used to mitigate force-feedback being substantially independent of
one another, enabling different stabilization mechanisms as
described herein to be used with existing force-feedback
capabilities. In alternative embodiments, inventive apparatus and
methods are used which take into account both the movements
associated with the gyroscopic stabilization, a user's movements,
and the application of torques and forces to realize a spatially
unrestricted force-feedback device requiring fewer motors and
structural elements. Specifically, an inventive control scheme is
used in these cases to accelerate and decelerate the motor(s)
associated with providing the gyroscopic stabilization such that
only the desired tactile feedback is experienced by the user. All
of the various approaches are applicable to single and multiple
degrees of freedom.
A three-axis implementation includes a set of three, mutually
perpendicular momentum wheels which form the gyro-stabilized
platform, an attitude measuring device, and a control system. The
attitude measuring device is employed to detect disturbances to the
gyro-stabilized platform, including reaction torques due to a
user's interactions with the device. The control system varies the
speed the momentum wheels in order to maintain the gyro-stabilized
platform in a fixed position. In an alternative embodiment, a
reaction sphere is used to produce the requisite inertial
stabilization. Since the sphere is capable of providing controlled
torques about three arbitrary, linearly independent axes, it can be
used in place of three reaction wheels to provide three-axis
stabilization for a variety of space-based and terrestrial
applications.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a drawing of a one-dimensional space gyroscopic model, as
seen from an oblique perspective;
FIG. 2 is a drawing of a three-axis stabilized system model, as
seen from an oblique perspective;
FIG. 3 is a drawing used to illustrate torque generation with
respect to a momentum sphere;
FIG. 4 is a block diagram illustrating a closed-loop control
system;
FIG. 5 is a block diagram illustrating a closed-loop control system
with disturbance;
FIG. 6 is a block diagram depicting plant feedback with optimal
feedback for linear regulations;
FIG. 7 is a representation of a mathematical model of a 1-D model
plant;
FIG. 8 is a state diagram used to illustrate position regulation of
a 1-D satellite plant using pole placement;
FIG. 9 is a state diagram used to illustrate a final design of a
1-D satellite model controller;
FIG. 10 is a skeletal representation of momentum sphere
housing;
FIG. 11 is a simplified drawing of an aspect of a momentum sphere
depicted infrared emitters and detectors;
FIG. 12 is a simplified drawing showing a great circle band of
reflective material around a momentum sphere;
FIG. 13 is a drawing, seen from an oblique perspective,
illustrating a different aspect of a momentum sphere;
FIG. 14 is a cross-sectional view of a momentum sphere illustrating
how a control subsystem may interact with optical emitters and a
reflective band;
FIG. 15 is a block diagram used to describe a momentum sphere
control environment;
FIG. 16 is a drawing, as seen from an oblique perspective, of a
spacecraft including a pitch momentum wheel;
FIG. 17 is a simplified drawing used to illustrate the
stabilization of a gimbal sensor platform;
FIG. 18 is a block diagram of a single-axis momentum wheel for
terrestrial applications;
FIG. 19 is a drawing of a momentum wheel viewed from a top-down
perspective, before the application of motor current;
FIG. 20 is a drawing of a momentum wheel after the application of
motor current;
FIG. 21 is a root-locus plot;
FIG. 22 is a time-response plot of a one-dimensional motor
application according to the invention;
FIG. 23 is a graph used to illustrate the control effort of a 1-D
motor;
FIG. 24 is a drawing, as seen from an oblique perspective, of a
hand-held force-feedback controller utilizing three momentum wheels
to provide inertial stabilization in three space:.
FIG. 25 is a drawing of a block digram of a spatially unrestricted
force feedback controller utilizing three momentum wheels to
provide inertial stabilization in three space.
DETAILED DESCRIPTION OF THE
PREFERRED EMBODIMENTS According to the invention, programmed
amounts of rotary force are used for motion compensating and/or the
stabilization of free-flying platforms, or to provide force/torque
outputs from platforms to attached frames. Specific embodiments are
disclosed with respect to spacecraft stabilization, as well as to
the application of forces and/or torques to hand-held force
generating devices, including joysticks, steering wheels, and
implements of arbitrary shape for specific applications, such as
sports simulations.
By way of introduction, reaction wheels use the inertia of one or
more (typically up to three) rotating flywheels to generate
torques. These wheels are typically accelerated using electric
motors which can be controlled to increase or decrease rotary
speed, thus changing rotational momentum. When the wheel on a
particular axis is accelerated through increased motor torque, an
equal and opposite reaction torque is generated and applied to the
base upon which the wheel is mounted.
Reaction wheels are the most precise type of attitude control
mechanism. However, when called upon to provide non-cyclic torques,
they must be periodically unloaded by other means (i.e. when the
motors have accelerated to maximum RPM in any direction, no
additional acceleration can be realized in that direction unless
the motors are slowed, generating torques in the opposite
direction). Moreover, to provide arbitrary torques, three wheel
axes must be provided.
This application describes how reaction wheels as currently only
applied only to spacecraft can be extended into several other
related terrestrial applications, including gyro-stabilized bodies
and tethered, force-generating/reflective input devices.
Preliminarily, the following description will demonstrate and how
three axes of reaction wheel can be reduced into a single, reaction
sphere, useful either in the space-based or terrestrial
applications. A description of reaction wheels and spheres will
first be presented, followed by a discussion of the extensions to
such technology made possible by the invention.
Single Plane of Torque Action
The single plane model for a torque consists of a spinning wheel
attached to a frame. There is a reference frame B, embedded in the
frame and a fixed reference frame, A, in the world. Reference frame
B is aligned with the axis of the spinning wheel. The system is
shown in FIG. 1.
The reference frame B has two degrees of freedom with respect to
reference frame A. These degrees of freedom are described by
generalized coordinates q.sub.1, q.sub.4, where q.sub.1 represents
the angular degree of freedom about unit vector a.sub.1 and q.sub.4
represents the linear degree of freedom along unit vector a.sub.1.
With this model, the unit vectors in frames A and B are related
by
The center of mass of the frame is located at the origin of the B
reference frame. The frame is assumed to be a cube with a mass of M
and height of R.
The location of the center of mass of the wheel, d, is given by the
vector p.sup.d,
where l is the offset from the center of the frame (in meters) and
is a run-time parameter. The notation .sup.A v.sup.B denotes some
vector v in reference frame B with respect to reference frame A.
The mass of the (assuming without loss of generality a solid
cylindrical disk) wheel is given by
where .rho. is the density of the material of the wheel, r is the
radius of the disk and h is the height of the wheel; and have the
units of kg/m.sup.3, meters and meters respectively.
The central inertia dyadic of the wheel is given by
where ##EQU1##
The orientation of the wheel with respect to the frame is given by
the generalized coordinates q.sub.3. This generalized coordinate is
about the b.sub.1 axis.
The central inertia dyadic of the frame is given by
where (assuming without loss of generality that the frame is cubic)
##EQU2##
Since there are two rigid bodies in this model (the frame and the
wheel), the angular velocities and accelerations for both must be
developed.
The reference frame B is said to have a simple angular velocity in
the reference frame A because there exists for all time a unit
vector whose orientation in both the reference frame B and
reference frame A is independent of time. This allows writing the
angular velocity of reference frame B as the magnitude of its
angular velocity times the fixed unit vector
To make the equations of motion concise, a generalized velocity
will be defined as
Using the definition in Equation (8), .sup.A.omega..sup.B can be
rewritten as
The wheel is said to have a simple angular velocity in the
reference frame B because there exists for all time a unit vector
whose orientation in both the wheel reference frame and reference
frame B is independent of time. This allows writing the angular
velocity of the wheel as the magnitude of its angular velocity
times the fixed unit vector
The angular velocity of the wheel in reference frame A is given
by
Defining another generalized velocity,
allows expressing Equation (12) in terms of generalized velocities
only as
The angular acceleration of reference frame B is found to be
and the angular acceleration of the wheel can be written as
The location of reference frame B is given by
The velocity .sup.A v.sup.B and acceleration .sup.A a.sup.B of this
frame are found to be
since the unit vectors a.sub.i are fixed in reference frame A.
Defining a generalized velocity
allows rewriting Equation (18) as
By defining the disturbance forces acting at the origin of
reference frame B as
the disturbance torque acting on the frame as
and the motor torque, applied to the wheel, as
the equations of motion are found to be:
Rewriting Equation (24) in matrix form yields ##EQU3##
To control this system, an expression for .tau..sub.d that allows
the system to move from any value of {q.sub.1,u.sub.1 } to any
other value of {q.sub.1,u.sub.1 } in the presence of disturbance
torques .tau..sub.x must be developed (see Section below). To gain
a understanding of the system, first set .tau..sub.x =0. Equation
(25) can now be written as
From control theory, it is known that this equation is not stable
since the poles lie on the imaginary axis. Thus, the form of
.tau..sub.d required to satisfy stability criteria must meet the
following two criteria:
1. It must move the poles of Equation (26) into the left-half
plane.
2. It should utilize values of {q.sub.1, u.sub.1, q.sub.7, u.sub.7
} to control the system as these state variables can be
measured.
If the disturbance torque is not set equal to zero, then Equation
(26) is rewritten as
and a third requirement for the control torque is added.
3. It must be robust for a specified set of disturbance torque
values and functional forms.
Some simple relationships are also developed to suggest appropriate
motor parameter values and sizes for the momentum wheels. For real
world application, it is important to be able to specify certain
aspects of the problem, such as force produced, the period of time
for which it is produced and the mass of the device. Also, to stay
firmly rooted in reality, it is important to specify the power
output of the motor.
Equation (28) shows the basic equations
where I is the moment of inertia of the momentum wheel (assuming
that it is a thin hoop; for a solid disk, I=mr.sup.2 /2 and, in
reality, the actual value will fall some place in between), m is
the mass of the momentum wheel, r is the radius of the momentum
wheel, .tau. is the torque applied to the operator (which is the
same as the torque produce by the motor), (.tau. is the angular
acceleration of the momentum wheel, .omega. is the angular velocity
of the momentum wheel, t is the period of time for which the torque
is felt and P is the power output of the motor.
To feel a torque produced by a motor that is not attached to some
fixed structure, the motor rotor must be accelerating. The rotor
will continue to accelerate until the motor reaches its maximum
angular velocity, a value that is determined by motor parameters
(but the calculation of which is not important for this analysis).
To increase the amount of time during which the torque can be felt,
it is necessary to slow down the angular acceleration of the motor
by increasing the moment of inertia of the rotor.
Equation (28) has four equations and eight parameters. Of these
parameters, an equation is formed that relates m,r,t,.tau. and P
because these are the parameters that can be controlled during the
design of the device. One such form of this equation is
##EQU4##
Arbitrary Torque Generation From Wheels
To generate arbitrary torques, a 3D platform consisting of three
spinning wheels attached to three non-coplanar axes of a frame is
required. (For simplicity, and without loss of generality, this
work assumes that the axes are mutually perpendicular.) There is a
reference frame, B, embedded in the frame and a fixed reference
frame, A, in the world. Reference frame B is aligned with the axes
of the three spinning wheels, thus defining a set of mutually
perpendicular unit vectors. The system is shown in FIG. 2.
The reference frame B has six degrees of freedom with respect to
reference frame A. These degrees of freedom are described by
generalized coordinates q.sub.1, . . . , q.sub.6, where q.sub.1, .
. . q.sub.8 represent the angular degrees of freedom about unit
vectors a.sub.1,a.sub.2,a.sub.3 respectively and q.sub.4, . . . ,
q.sub.6 represent the linear degrees of freedom along unit vectors
a.sub.1,a.sub.2,a.sub.3 respectively. The orientation of reference
frame B with respect to reference frame A is described using a Body
3: 1-2-3 representation. Table 1 shows the relationship between the
unit vectors a.sub.1, a.sub.2, a.sub.3 and b.sub.1, b.sub.2,
b.sub.3.
TABLE 1 Direction cosines b.sub.1 b.sub.2 b.sub.3 a.sub.1 c.sub.2
c.sub.3 -c.sub.2 s.sub.3 s.sub.2 a.sub.2 s.sub.1 S.sub.2 c.sub.3 +
s.sub.2 c.sub.1 -s.sub.1 s.sub.2 s.sub.3 + c.sub.3 c.sub.1 -s.sub.1
c.sub.2 a.sub.3 -c.sub.1 s.sub.2 c.sub.3 + s.sub.3 s.sub.1 c.sub.1
s.sub.2 s.sub.3 + c.sub.3 s.sub.1 c.sub.1 c.sub.2
The terms c.sub.i,s.sub.i are defined as cos (q.sub.i) and sin
(q.sub.i) respectively.
To simplify some expressions, the following terms are defined:
##EQU5##
Since the equations of motion will be developed using the unit
vectors in reference frame B, the unit vectors in reference frame A
are explicitly presented using the terms Z.sub.t defined in
Equation (30).
For simplicity, and without loss of generality, the center of mass
of the frame is located at the origin of the B reference frame and
the frame is assumed to be cubical with a mass of M and height of
R.
The control inertia dyadics of the frame is given by
where ##EQU6##
The locations of the center of mass of the wheels, d.sub.i, are
given by the vectors p.sup.d.sup..sub.1 , where
where l is the offset from the center of the frame (in meters). The
mass of each wheel (assuming without loss of generality that each
wheel is a solid cylinder) is given by
where .rho. is the density of the material of the wheel, r is the
radius of the fisk and h is the height of the wheel and have the
units of kg/m.sup.3, meters and meters respectively.
The central inertia dyadics of the wheels are given by
where ##EQU7##
The orientation of the wheels with respect to the frame are given
by the generalized coordinates q.sub.7, . . . , q.sub.9. These
generalized coordinates are about the b.sub.1, b.sub.2, b.sub.3
axes respectively.
Since there are four rigid bodies in this model (the frame and the
three wheels), the angular velocities and accelerations for all
four must be developed.
The angular velocity of the frame, .sup.A.omega..sup.B is found to
be
To make the equations of motion concise, three generalized
velocities will be defined as
Using the definition in Equation (39) .sup.A.omega..sup.B can be
rewritten as
The wheels are said to have a simple angular velocity in the
reference frame B because there exists for all time a unit vector
whose orientation is both the wheel reference frames and reference
frame B is independent of time. This allows writing the angular
velocities of the wheels as the magnitude of their angular velocity
times the fixed unit vector
The angular velocities of the wheels in reference frame A are given
by
Defining three more generalized velocities,
allows expressing Equation (42) in terms of generalized velocities
only as
The angular acceleration of reference frame B is found to be
The angular accelerations of the wheels can be written as
The following terms are defined to simplify the equations
##EQU8##
thus allowing Equation (46) to be rewritten as
There are four points of interest in this problem: the location of
reference frame B and the locations of the centers of mass for each
of the wheels. The location of reference frame B is given by
and its velocity .sup.A v.sup.B and acceleration .sup.A a.sup.B are
found to be
since the unit vector a.sub.i are fixed in reference frame A.
Defining generalized velocities
allows rewriting Equation (50) as
The velocities of the centers of masses of the wheels are found to
be
and the accelerations of the centers of masses of the wheels are
found to be
These are three sets of forces acting on this system: the applied
disturbances forces and torques applied to reference frame B that
represent, the gravity forces acting on the wheel and frame masses
and the motor torques applied to the wheels.
The disturbance forces acting at the origin of reference frame B is
defined as
the gravity force on the frame is defined as
or equivalently as
and three gravity forces, which act at the center of the wheels,
are defined as
or equivalently as
The disturbance torque acting on the frame is defined as
and the three motor torques are defined as
with the positive sense of the torque being applied to the wheel.
However, Newton's second law demands that there be an equal and
opposite torque applied to the frame, body B. Thus, the resultant
acting on body B is given by
The definitions of the generalized inertia forces was facilitated
by defining the following terms: ##EQU9##
The equations of motion are found to be:
Since there are nine generalized coordinates, there are 18
equations of motion, nine kinematics and nine dynamic. To solve
these equations numerically, they must be written in the form.
where the state vector y has the form y={q.sub.1, . . . , q.sub.n,
u.sub.1, . . . , u.sub.n }. This necessitates rewriting Equation
(39) to solve for the q.sub.i in terms of the u.sub.i. The nine
kinematic equations of motion can now be written as ##EQU10##
To write the dynamical equations of motion in the same manner,
Equation (64) must be solved for the u.sub.i. Since several of the
Z.sub.i include u.sub.i, these terms will need to be expanded. As a
first step, the following terms are defined
Next, Equation (64) is rewritten in the form Au.sub.i =K.sub.i,
thus providing a means for solving for the u.sub.i. K.sub.i is
given in Equation (68) and A is defined as ##EQU11##
where I.sub.4 =2I.sub.1 +I.sub.2 +I.sub.6.
A discussion of the control system is presented in below. To
simplify the equations of motion to facilitate control development,
those terms and equations that deal with the linear position/force
are eliminated because a gyro-stabilized platform can only
counteract torques, not forces. Rewriting Equation (69) as
indicated yields ##EQU12##
This equation can also be rewritten to explicitly express u.sub.i
##EQU13##
where I.sub.5 =(I.sub.4 +2ml.sup.2)-I.sub.2.
Arbitrary Torque Generation From a Sphere
The equations of motion for the sphere, see FIG. 3, can be derived
from those for the three wheel device by noting these two salient
differences between the systems: the inertia of the sphere is equal
in all directions and is unchanged with orientations; and the
center of mass of the sphere is located at the origin of reference
frame B. The equations of motion for the sphere are given by:
Because the cross-coupling through the linear velocity terms does
not exist for this device, controlling a system that employs this
device for stabilization is easier than controlling a system that
employs three reactions wheels for stabilization.
Control Issues
Control theory is defined as a division of engineering mathematics
that attempts, through modeling, to analyze and to command a system
in a desired manner. Of particular interest are closed-loop system.
In a closed-loop system, the forcing signals of the system (calling
inputs) are determined (at least partially) by the responses (or
outputs) of the system. In FIG. 4, a generic closed-loop control
system is shown. In order to explain the contents of this diagram,
the following example is used:
The object is to control the temperature of a room. In this case,
the sensor is the thermostat. The system input is set by selecting
a temperature. Through either some mechanical or electrical means,
the difference between the desired and actual temperature is
calculated, resulting in an error. If the actual temperature is
below the desired, the compensator sends out a control signal to
the furnace (or plant). If the control signal says heat on
(actually, the electromechanical equivalent), the furnace outputs
heat. This process continues until the compensator determines it is
not necessary to heat the room, and the control signal is changed
to a heat off signal.
Control theory can be classified in two categories: classical and
modern. Classical control theory is generally a trial-and-error
system in which various type of analyses are used iteratively to
force a electromechanical system to behave in an acceptable manner.
In classical control design, the performance of a system is
measured by such elements as settling time, overshoot and
bandwidth. However, for highly complex, multi-input/multi-output
(MIMO) systems entirely different methods of control system design
should be implemented to meet the demands of modern technology.
Modern control has seen wide-spread usage within the last fifteen
years or so. Advancements in technology, such as faster computers,
cheaper and more reliable sensors and the integration of control
considerations in product design, have made it possible to extend
the practical applications of automatic control to systems that
were impossible to deal with in the past using classical
approaches. Modern control theories are capable of dealing with
issues such as performance and robustness. The
spatially-unrestricted force-feedback system makes use of two
modern control design methods: disturbance rejection and optimal
control.
In the design of electromechanical systems, one can consider that
the system will be exposed to disturbances. A disturbance may be
defined as any unwanted input. In FIG. 5, the disturbance, w(t), is
shown as a second input to the plant. The effect of the disturbance
is added to the output of the plant.
Disturbance rejection design can be used to create a compensator
which is able to ignore the disturbance and cause the desired plant
output. In this section, the basic method of disturbance rejection
design is presented using a MIMO model. For this model, notation
must be established to designate the various elements of the
control design; let:
[A, B, C, D] be a state-space representation of the plant (with
state x), assuming (A,B) is completely controllable,
The lumped MIMO linear, time-invariant (LTI) system, may be
expressed as:
The model for the input (Equations (76)-(77)) and the noise
(Equations (78)-(79)) are:
The objectives in the design of the feedback system in FIG. 5 are
as follows:
Closed-loop system must be exponentially stable,
Achieve asymptotic tracking and disturbance rejection for all
initial states
Robustness
If this is true, then for all initial states of the system,
e(t).fwdarw.0.epsilon.A.sup.n.sup..sub.0 as t.fwdarw..infin..
Given the system [A,B,C,D], suppose it is minimal. Let the
compensator be given by
where
with ##EQU14##
Since A.sub.w and A.sub.r are known, [83] can be derived from the
equation
which is the least-common multiple of the characteristic equations
of A.sub.w and A.sub.r.
Under these conditions, if ##EQU15##
(which guarantees that the system is still completely controllable
with the addition of the compensator) then
The composite system is completely controllable
Asymptotic tracking and disturbance rejection holds
Asymptotic tracking and disturbance rejection are robust
The discussion contained here is establishes a mathematical basis
for the invention. Control of a gyro-stabilized force feedback
device is based on its ability to respond robustly to a control
signal and to respond correctly despite system noise. For the
single-input/single-output (SISO) case, this theorem reduces to the
classical control case where an integrator is required for robust
performance. This result is used in the design of the 1D experiment
which is similar to the classical satellite control problem.
Optimal control theory can be used to design compensators which are
able to take into account the cost of performing a particular
action. A classical example of optimal control is the use of fuel
to maneuver a satellite in orbit above the earth. Two extreme
scenarios are possible: movement taking minimum time or movement
taking minimum fuel. In the following section, discussion will
focus on the fundamental principles of optimal-control design.
The optimal control problem is to find a control u*(t) which causes
the system x(t)=a[x(t), u(t), t] to follow a desired trajectory x*
that minimizes the performance measure ##EQU16##
Other names for J include cost function, penalty function, and
performance index. Assume that the admissible state and control
regions are not bounded. (This removes all mechanical constraints;
these can be included in later development) Let the initial states,
x(t.sub.o)=x.sub.o, of the system and initial time, t.sub.o, be
known. Also, let x.epsilon.A.sup.n and u.epsilon.A.sup.m. The goal
now is to establish tote necessary conditions for optimality:
Assuming that h is differentiable and that initial conditions are
fixed and do not affect minimization, [b 87] can be expressed as
##EQU17##
For generality, apply the chain rule and include differential
equation constraints to form an augmented cost function:
##EQU18##
using Lagrange multipliers p.sub.1 (t), . . . , p.sub.n (t). To
simplify the notation, rewrite .quadrature. as follows:
##EQU19##
The necessary conditions for optimal control can be derived using
calculus of variations. Specifically, take the variations of the
functional J.sub.a (n) by .delta.x, .delta.x, .delta.u, .delta.p
and .delta.t.sub.f. (Increment of the functional J is defined as:
.DELTA.J(x, .delta.x)=.delta.J(x, .delta.x)+g(x,
.delta.x).multidot..parallel..delta.x.parallel.; .delta.J is linear
with respect to .delta.x; .delta.x is called the variation of the
function x.) From this, the necessary conditions may be derived:
##EQU20##
for all t.epsilon.[t.sub.o, t.sub.f ], and ##EQU21##
where
The principles of calculus of variations are applied to the design
of a linear regulator. The linear regulator is used in the control
of the motors used to spin the inertial masses to change the
attitude of the satellite system. The regulator design is
particularly useful in controlling unstable systems through optimal
pole placement. First, recall the state equation of a linear,
time-varying plant:
The cost function to be used is ##EQU22##
where t.sub.f is fixed, H and Q are real, positive-semi-definite
matrices, and R is a real, positive-definite matrix. The purpose of
the regulator is to maintain the state of the system as close to a
desired set of parameters as possible without excessive control
effort. The necessary conditions for optimality to be used are:
where the Hamiltonian is defined as
Equation (100) is easily solved for the optimal input for the
regulator, yielding
It is now possible to form an augmented, closed-loop state-space
equation of the regulated system: ##EQU23##
These 2n differential equations have a solution of the form:
##EQU24##
Note: .phi.(t.sub.f,t) is called the transition matrix, define by
d/dt.phi.(t.sub.f, t)=A(t).phi.(t.sub.f,t) with the initial
condition of .phi.(t.sub.o,t.sub.o)=1 and is solved through
numerical integration. By partitioning the state transition matrix,
.phi.(t.sub.f,t), the following solution for p*(t) can be
reached:
Therefore, the optimal control law is
the next step is to define a method of solving for K. This is
achieved using a Riccati-type differential equation:
which involves solving n(n+1)/2 first-order differential equations.
Fortunately, the motor system involved in the hand controller
control system can be considered time invariant. This simplifies
the previous equations, which can be summarized as: ##EQU25##
and the optimal control law is
As long as Q is positive definite, the closed-loop system is
guaranteed to be stable and the controller may be used for pole
placement design of the system, as shown in FIG. 6.
The design of the controller system for the 1D model is now
presented. The first segment of the design is a optimal
pole-placement. This is needed because a the 1D model of the
spatially unrestricted force feedback device (which is a simplified
version of the actual 3D version), which can be considered a
second-order system, is inherently unstable. Definitions of
"stable" vary; here, "stable" is considered any plant which has
only poles and zeros to the left of the imagery axis in the complex
plane (i.e., left-hand poles and zeros). Using previously
established results, the poles of the system are placed optimally
based on the inertia of a second-order linear model. Lastly,
disturbance rejection is augmented to the control system the
robustness.
the plant for a single DOF hand controller, FIG. 7, has the form
##EQU26##
where .alpha.(t) is the angular acceleration, I.sub.6 is the
inertial mass, and .tau.(t) is the torque.
Since the stability of this system is (at best) marginal, a pole
placement is performed. Further, optimal methods are employed for
placing these poles at the best locations. The new plant will
follow the model in FIG. 8. The optimal design will give the "best"
values to use for K.sub.1 and K.sub.2.
The first step is to choose the cost function to minimize, set
initial conditions, and select the necessary conditions and
boundary conditions which apply to this problem. Let the initial
states of the satellite be zero: x(0)=0;x(0)=0. The cost function
for minimal control effort is ##EQU27##
such that the amount of acceleration of the system, whether it is
positive or negative, for all time is minimal. This is frequently
used for satellites because the amount of acceleration is the
magnitude of the control input, or for satellite, the amount of
fuel, which is a limited resource. For this system, the following
parameters are known: ##EQU28##
with a state defined by ##EQU29##
and choose ##EQU30##
and R=1.
For the LTI Ricatti equation, [110], K has four solutions, but the
only positive-definite solution is ##EQU31##
which results in a regulator, F, of the following form:
##EQU32##
The two terms of the F vector are the position feedback and
velocity feedback required for optimal tracking, as in FIG. 8.
The final step is to include an integrator which provides the SISO
case with robustness. The final controller design is shown in FIG.
9.
There are some control issues that are specific to the momentum
wheel concept. These issues are those that deal with determining
the state of the sphere, which must be known to calculate the
sphere's angular momentum vector. Since the nature of a spherical
object allows it to be at any orientation relative to it's cavity,
a method that can detect the sphere's exact orientation relative to
the three fixed orthogonal axis of the sphere housing is used. This
is illustrated in FIG. 10.
Each of the three sphere housing axis is outfitted with a band of
optical infra-red emitters to detect the relative position of the
sphere. Each emitter will be placed between two (or more) infra-red
detectors as shown in FIG. 11. This technique will enable fine
position sensing and simultaneously minimize power requirements
since a single emitter will service two (or more) detectors.
The sphere is equipped with a single great circle band of
reflective material as illustrated in FIG. 12. As shown in FIG. 13,
each sensor band on the sphere housing covers one half of the great
circle band on each sphere housing axis. Consequently the
reflective band is always within range of at least three optical
emitter/detector pairs regardless of sphere orientation.
The IR emitter/detector sensors are located directly on the cavity
face to simplify construction of the sphere housing. Each emitter
and detector is directly interfaced to the housing cavity by a
fiber optic cable that ends at a lens mounted on the cavity face as
shown in FIG. 14. Using a lens permits the use of lower power
infra-red emitters.
As shown in FIG. 15, the infra-red emitters are driven by an output
bit from the Sphere Control Computer. Address decode logic and
latch bits contained in the Sphere Control Subsystem decode emitter
data from the control computer and turn the appropriate IR emitter
on. The control computer reads the associated IR receiver, via the
same decode multiplexor logic in the Sphere Control Subsystem.
Conventional Applications to Spacecraft
There are two inter-related branches of mechanics that are used to
spacecraft control: celestial mechanics and attitude mechanics. The
former deals with the position and velocity of the center of mass
of the spacecraft as it travels through space, whereas the latter
deals with the motion of the spacecraft about its center of mass,
see FIG. 16.
Attitude mechanics is divided into three components: determination,
prediction and control. Attitude determination is the process of
computing the current orientation of the spacecraft with respect to
some specified inertial frame. Attitude prediction is the process
of computing the future attitude of the spacecraft based on its
current state and motion. Attitude control is the process of
applying torques to the spacecraft to reorient it into some desired
future state. The devices mentioned in this patent deal primarily
with the control aspect of attitude mechanics.
For most modern spacecraft applications, three-axis control is
required. This method of control allows mission planners to specify
the orientation of the spacecraft at all times during the course of
a mission. Missions that employ this type of control include all
communications satellites, the space shuttle and earth-orbiting
scientific satellites.
To function properly, three axis stabilized spacecraft employ
sensing devices that identify the spacecraft's attitude by
determining two mutually perpendicular orientation vectors. Some
typical examples include two-axis sun sensors and magnetic field
sensors. Once the spacecraft's attitude is determined, the mission
profile determines the control requirements. Certain scientific
satellites require extremely precise attitude control (arc-seconds)
for the purpose of data collection. Others, such as C-band
television satellite, require less precise control (arc-minutes).
Since all satellites are subject to disturbances, some method of
maintaining proper orientation is required.
There are three primary means for controlling a satellites
attitude: gas jets, electromagnets and reaction wheels. Reaction
jets operate by expelling gas through an orifice to impart a moment
on the spacecraft. These devices can produce large (but imprecise)
torques, but since they expend fuel, there on-station operating
time is limited. Electromagnets operate by creating magnet fields
that interact with the magnetic field of a nearby body to produce a
torque on the satellite. Although these systems do not expend fuel,
they only function near bodies with large magnetic fields. Reaction
wheels operate by way of Newton's third law by accelerating a wheel
to absorb torque that is applied to the satellite. If the applied
disturbances are cyclic, these systems can operate indefinitely
since there is not net gain/loss of energy. For real-world systems,
reaction wheels typically operate in conjunction with gas jets,
which are used to bleed off excess momentum as the wheels approach
their operating condition boundaries. Reaction wheels provide a
very fine degree of attitude control.
Applications for Platform Stabilization
What differentiates space-based applications from other
applications is not the lack of gravity but rather the fact that
gravity is the same in all directions. Similar situations can occur
on the Earth: system with neutral buoyancy in a liquid and systems
that are fixed in the direction of gravity operate under similar
principals as space-based systems, see FIG. 17.
For example, consider the case where a sensor platform is to
collect data from a lake over a period of time. If this platform is
required to maintain a particular attitude, a gyroscopic system can
be used for stabilization. Similarly, a sensor platform mounted on
a research balloon may be required to maintain two-axis attitude
control for the duration of the mission. Again, a gyroscopic system
can be used to stabilize the two rotational degrees of freedom of
this system.
EXAMPLES
Two sets of experiments were carried out with the single degree of
freedom device. The first experiment was intended to validate
Equation (119). A second experiment was intended to demonstrate a
control system for a three DOF system.
To carry out these tests, a test stand was developed, as shown in
FIG. 18.
This test setup consists of the following components:
A turntable with an attached motor. The position of the turntable
is instrumented with an incremental encoder attached directly to
the turntable (not used in this experiment). The position of the
motor shaft was not instrumented, however, its angular velocity is
instrumented. The motor employed is a Hathaway model 1500, attached
to the turntable by means of an adapter block.
A momentum wheel attached to the motor shaft. This momentum wheel
is manufactured from a piece of stock, 2 inch diameter, cast iron
shaft.
The motor is attached to a CyberImpact.RTM. Intelligent Motor
Controller (IMC) system, a standard Cybernet product and is used
with all of our force feedback devices, which provides an interface
to a PC based controller that allows for a wide range of motion
commands to be programmed.
The IMC is attached to a PC. In this example, a simple, previously
developed interface to start and stop the motor was employed. This
interface presents the user with an input screen for directly
controlling the motor current. By setting the current to its
maximum allowable value, the maximum obtainable torque is observed.
By setting the current to zero, the motor comes to a stop.
A torque measuring system consisting of a spring and a camera.
Applied torque was measured by the displacement of a known spring
and the time for this to happen by counting video frames.
The position, velocity, and/or acceleration on a user-interactable
member is sensed and transmitted as a command to a computer model
or simulation which implements a virtual reality force field. In
turn, the force field value for the given position, velocity,
and/or acceleration is sent back to the member, which generates a
force command, thereby providing the user with direct kinesthetic
feedback from the virtual environment traversed. Although
applicable to controlling a virtual or simulated environment, the
technology is also well suited to the control of a remote or
physical device. Further, the present invention is suited for
application to any number of axes.
The operation of the IMC system and PC interface will be best
understood by referring to commonly assigned U.S. Pat. Nos.
5,389,865 and 5,459,382, and pending applications Ser. Nos.
08/513,488 and 08/543,606, the contents of each of which are
incorporated herein in their entirety by reference. These patents
and co-pending applications describe systems and methods for
presenting forces to an operator of a remote device or to a user
interacting with a virtual environment in multiple axes
simultaneously mediated through a computer controlled interface
system which provides a position, velocity, and/or acceleration (to
be referred to generally as "force") to a user interface which, in
turn, generates an electrical signal for each of a plurality of
degrees of freedom. These electrical signals are fed to a virtual
reality force field generator which calculates force field values
for a selected force field. These force field values are fed to the
force signal generator which generates a force signal for each of
the plurality of degrees of freedom of the user as a function of
the generated force field. These motion commands are fed back to
actuators of the user interface which provide force to the user
interface and, thus, to the user in contact with the interface
device.
Before discussing these applications in further detail, a
background will be provided with respect to inertial stabilization
as its relates to reaction wheels and space-based applications, as
certain principles of spacecraft platform stabilization have, for
the first time according to this invention, been applied to
spatially unrestricted terrestrial control.
Experimental Data for 1D Device Implementation
Since a known momentum wheel was used, the form of Equation (29) is
not quite right for this experiment. Instead, this equation is
rewritten as ##EQU33##
where the factor of two is used because a solid disk, not a thin
hoop, was used. The mass of the momentum wheel is 0.277 kg
(measured) and the inertia of the motor rotor is ignored.
Using the motor electrical parameters and the electrical
characteristics of the IMC chassis, the maximum torque that can be
applied by the motor is known to be 0.18 Nm. Inserting these values
into Equation (119) yields a time of 0.09 seconds.
To measure the torque, a spring with a spring constant 110 N/m was
attached to the adaptor block by way of a bolt, at a distance of
0.050 m from the center of rotation. Since F=kx and .tau.=Fd, these
terms can be related in the following manner ##EQU34##
Of course, the equations used are very primitive and do not account
for many of the real-world affects. The affects, which are
primarily frictive in nature, should tend to make the displacement
less than predicted and the time greater than predicted. The
results of these experiments are shown in FIG. 19 and FIG. 20. The
picture on the left shows the system just before current is applied
to the motor. The picture on the right shows the system at maximum
spring extension, which occurred five video frames, at 30 frames
per second later. The results show a displacement of 1.2 inches
(0.030 m) and a time of 0.16 seconds. Given the experimental setup,
these results are well within the range of experimental error, thus
giving credence to the model.
Experiments were also performed to control the position of the
turntable, in the face of disturbances, by controlling the speed of
the momentum wheel. The equations and methods used to develop this
control scheme were discussed previously. For this experiment, the
same setup was used as for the previous experiment with several
small modifications:
The instrumented readings from the turntable and the motor shaft
were used by the controller.
The spring was removed from the experimental setup.
A control program was written that interfaces directly with the IMC
system.
Using MATLAB, which is a PC based mathematical tool designed to aid
engineers in the development of complex mathematical systems, the
controller and plant were simulated. Since the amount of control
input is not a particular concern, optimal control parameters were
selected to produce a system that responds quickly. In the
following experiments, the values q.sub.1 =q.sub.2 =10 and R=1 were
selected. To select an appropriate value for the disturbance
rejection gain, a root locus plot of the system, FIG. 21, was
developed. From this diagram, the gain of the system, which is
selected to produce fast response time, has value of approximately
1.33.times.10.sup.-5. The response of the system of a unit step
disturbance is shown in FIG. 22 (plot generated from MatLab). These
parameters were then used in testing a real model of the
system.
The control parameters determined using the optimal control
techniques and the root-locus method were applied to the system
shown in FIG. 18 (without the spring). Since the control equations
require the moment of inertia of the platform, CAD tools were used
to calculate the moment of inertia of the motor, the adapter plate
and the bolt. One item that was not modeled in the simulation, or
the calculations for determining control parameters, was the
friction in the system.
In this particular device, there was a great deal of Coulomb
friction in the base bearing. The components mounted to the base
would not complete a single rotation before coming to a halt after
an initial spin. This has the effect of adding instability to the
system. In particular, what tends to happen is that the system will
stay at some point for some period of time while the integrator
error (the disturbance rejection) adds up. At some point, there is
sufficient energy to overcome the static friction, which is less
than the dynamic friction. Once moving, the system will tend to
overshoot the desired point and try to compensate, but the same
sequence of events occurs.
FIG. 23 shows actual data from an experiment to control the
physical device. Despite the friction problem, the results from
this test are as expected. The system does oscillate about the
control point, though it is quite noisy.
An experiment was also performed to determine if the forces
generated were noticeable by a human. To perform this experiment,
three motors with momentum wheels were mounted onto the adapter
block used in the previous experiments.
The motors were spun up to a speed of 5000 RPM. Individuals were
asked to handle the device and to make subjective evaluations of
the torques felt as the device was moved about. In all cases, the
subjects reported feeling appreciable forces that were deemed to be
sufficient for carrying out meaningful tasks. A picture of the
device is shown in FIG. 24.
The torques felt were generated because the control system had been
commanded to maintain the momentum wheels at a constant angular
velocity. By moving the device about, the angular momentum vectors
were changed, thus causing a torque. The control system compensated
for these motions by adjusting the output to the motors. Since the
motors were already spinning at high speed, the period of time for
which a torque could be applied was far more limited than for the
case where the motor is initially at rest.
Having demonstrated that forces can be generated in any direction,
the final task is to control the motors in an appropriate manner so
as to provide haptic feedback to the user. This task requires a
sophisticated control algorithm for two reasons: first, the
platform will be grossly displaced from its nominal operating
orientation, and second, for any motion of the platform (for
simplicity consider just rotations about the world coordinate axes
with which the device is initially oriented), some subset of the
motors will produce torques (due to changes in the orientation of
the angular momentum vectors) that are undesired. To counteract
these undesired torques, some subset of the motors will need to be
accelerated to produce counter torques. The control system must
model the full, non-linear dynamics of the system, have a high
speed attitude sensor and possibly a control to smoothly generate
the prescribed forces. A block diagram of the system is shown in
FIG. 25.
Applications
As discussed above, one family of applications for the devices
described above utilizes inputs received from a virtual
environment. For this type of application, the virtual environment
models some set of objects, and hand controller or other
force-reflection device produces forces that are representative of
some activity within the virtual environment. Since it is not
required that the forces produced correlate to any specific
activity, the only restriction placed on the commands sent to the
gyro-stabilized device is that the output forces be within the
range of forces that the device can produce. An alternative family
of applications for these devices produces forces in accordance
with inputs received from a (possibly remote) physical device. For
this type of application, the forces produced are typically a
scaled representation of the actual forces produced at some point
on the actual physical device. To provide the widest range of
haptic input, the scaling is typically designed such that the
maximum force that can be applied to the physical device is mapped
into the maximum force that the haptic device can produce.
To the first order, the devices described are marginally stable at
best. To control these devices to produce desired torque outputs in
the face of input disturbances, a two step controller is preferably
utilized. The first step stabilizes the controller by doing a pole
placement. The location of the poles can be determined using any
applicable method although optimal control is preferred. The second
step creates a robust controller by canceling out disturbance
inputs. Robust control theory is applied for this task.
With specific regard to platform stabilization, the desired input
is typically a zero input, i.e., that the system should not change
state. For these applications, sensor are employed to determine
when the system changes state due to disturbances and the
controller acts to return the system to the zero state.
Unloading
The human operator who controls the haptic device is, from the
perspective of the momentum device, equivalent to group. Although
any amount of angular momentum can be removed from the device when
it is coupled to ground, since this is a haptic device, the
strategy is to slowly and continually remove angular momentum so as
to have as minimal affect on the user as possible. In particular,
the momentum sphere has a maximum speed at which it can operate due
to the materials and construction techniques employed. When the
sphere approaches this maximum velocity, momentum must be unloaded
from the sphere for it to continue to function. To do this requires
the application of an external torque that will cause the angular
momentum vector to be diminished. This can be accomplished in three
ways: reaction jets, magnetic field torquers and/or spacecraft
reorientation. The first two methods work by applying a torque to
the spacecraft that diminished the angular momentum of the sphere.
The third method works if the following two conditions are met: the
disturbances to the spacecraft are primarily applied in the same
direction and the spacecraft can continue to operate at different
attitudes. If these conditions are met, the spacecraft can be
reoriented such that the disturbance torque act to cancel the
sphere's angular momentum. It may also be feasible to rigidly
couple the platform to ground for a brief period of time. While
coupled to ground, any amount of angular momentum can be removed
from the stabilized platform.
* * * * *