Note: Descriptions are shown in the official language in which they were submitted.
CA 02385602 2002-04-25
FOUR DEGREE-OF-FREEDOM WIRE ACTUATED PARALLEL ROBOT
FIELD OF THE INVENTION
This invention relates to a four degree-of-freedom cable actuated parallel
robot.
BACKGROUND OF THE INVENTION
Designing robotic systems for space applications requires great attention
to light weight, compactness prior to deployment, reliability, failure
tolerance and,
recently, moderate cost. Traditional serial robots are typically optimized
with little
regard to the overall weight of the system; rather, weight is redistributed
and
optimized to minimize the moment produced about revolute joints and to reduce
the forces required by motors. For space applications, weight is a critical
factor,
and must be aggressively minimized. For terrestrial applications a lower
moving
mass reduces the overall system weight and cost through reduced power
requirements and lower weight for structural elements. For applications where
repeatability requirements are not exacting, wire actuation can produce light
weight and low cost manipulators capable of moving substantial payloads
compared to their own mass [1]. The architecture under consideration was
designed to be suitable for several tasks related to space exploration. These
areas include use as a dextrous grappling device during automated docking of
spacecraft and satellites and as a digging arm for soil sampling on other
planets
or the Moon.
Parallel robot systems are composed of closed kinematic chains that
produce rigid structures when joints are locked or held in place, provided the
robot is not at singularity. This rigidity allows the use of lightweight
elements as
the individual structural links do not have to be particularly stiff to
produce
satisfactory repeatability. Direct kinematics can be problematic with parallel
robots, as determining the end effector position from the joint variables can
be a
difficult problem. The design discussed here, however, has a sensed serial-
CA 02385602 2002-04-25
equivalent central linkage, making direct kinematics straightforward. Such a
passive sensed branch in parallel robots can simplify the solution to the
forward
displacement problem [2].
Conventional parallel robots typically have restricted workspaces relative
to their size and mass. This aspect of parallel systems works against their
use in
space applications. Linear actuators used in some parallel designs are also
problematic in a vacuum environment due to the increased difficulty in sealing
and preventing lubricant loss. Parallel robots do, however, have the advantage
of high stiffness relative to their weight, offering the potential for design
of
extremely lightweight systems that can be stowed in compact configurations.
This requires that the links comprising the robot be built with high strength
and
lightweight structure. These requirements and considerations lead to a
parallel
robot that has only a few solid links to minimize weight, and wire tension
members which also serve as actuators.
Previously wire actuators have been used as tendons to actuate serial
robots or externally framed wire actuated systems [3]. Earlier work also
includes
three and six degrees of freedom (DOF) Stewart platform type manipulators
actuated by cable tendons [4]. Additionally, substantial work has been done by
NIST on the RoboCrane [5], a wire actuated and supported robotic crane,
including one variant where a central spine allowed the robot to exert force
downwards (which is otherwise impossible in the RoboCrane as gravity keeps
the wires taut).
It would be very advantageous to provide a cable actuated robot which
overcomes the aforementioned problems associated with cable actuated robotic
structures.
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CA 02385602 2002-04-25
SUMMARY OF THE INVENTION
The present invention provides a cable actuated parallel robot having four
degree of freedom, comprising;
a) a structural base assembly,
b) a passively jointed central linkage connected to said structural base
assembly and descending therefrom, said passively jointed central linkage
including a pair of spaced upper linkage beams each having first and second
opposed ends, each of said pair of spaced upper linkage beams being linked at
said first ends to said structural base assembly, a tie linkage beam having
opposed ends and being linked at said opposed ends to the second ends of said
pair of spaced upper linkage beams, a medial linkage beam having opposed
ends and being linked at said opposed ends to the second ends of said pair of
spaced upper linkage beams, a first lower linkage beam having opposed ends
with one end linked to said second end of one of said pair of spaced upper
linkage beams and the other of said opposed ends being linked to an end
effector raft, a second lower linkage beam having opposed ends with one end
linked to said second end of the other of said pair of spaced upper linkage
beams
and the other of said opposed ends being linked to said end effector raft; and
c) first and second cables, said first cable being connected at one end
thereof to said end effector raft, said second cable being connected at one
end
thereof to said end effector raft, said first and second cables running from
said
end effector raft to said structural base assembly and being connected to
winch
means for extending and retracting said first and second cables, a third cable
attached at one end to a point on the first lower linkage beam and a fourth
cable
attached at one end to a point on the second lower linkage beam, said third
and
fourth cables running from said first and second lower linkage beams to said
structural base assembly and being connected to winch means for extending and
retracting said third and fourth cables, fifth and sixth cables running from
positions in close proximity to each other on the structural base assembly and
being attached in close proximity to each other on the central linkage, said
fifth
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CA 02385602 2002-04-25
and sixth cables being connected to winch means for extending and retracting
said fifth and sixth cables.
BRIEF DESCRIPTION OF THE DRAWINGS
The present invention will now be described by way of example only,
reference being had to the accompanying drawings in which:
Figure 1 is a perspective view of a four degree of freedom wire actuated
parallel robot in accordance with the present invention;
Figure 2 shows the world coordinate frame;
Figure 3 is a perspective view of the central linkage of the robot of Figure
1;
Figure 3.4 is a view of a portion of the robot showing the parameters R1,
R2, LU, LL and LS/2, as defined in Table 1;
Figure 4 shows the relationship between joint space and world space;
Figure 5 shows constant orientation workspace extents with zero pitch;
Figure 6 shows a workspace program flowchart; and
Figure 7 shows a plot of position error vs. end effector speed for 4501 time
steps of the spiral-on-cone test (60 s at high speed, 300 s at moderate speed
and 1500 s at low speed).
DETAILED DESCRIPTION OF THE INVENTION
Referring to Figure 1, a cable actuated truss - 4 DOF (CAT-4DOF)) robot
is shown generally at 10 and is a four DOF parallel robot. Robot 10 is
comprised
of a wishbone-shaped structural base assembly 12 at the top, which may be
designed to be collapsible for transit and forms the backbone of the
manipulator.
A passively jointed central linkage 14 descends from the centre of base
assembly 12, and is comprised of six rigid links (two upper linkage beams 20
and
22, two lower linkage beams 24 and 26, a medial beam 28 and a tie beam 30),
each of which may be a truss in order to minimize structural weight. The
central
linkage thus comprises the upper linkage defined by upper beams 20 and 22 and
lower linkages defined by beams 24 and 26. That is, the lower linkage is
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CA 02385602 2002-04-25
connected to the upper linkage through the medial beam 28. Beams 24 and 26
of the lower linkage are connected to an end effector raft 34.
Referring particularly to Figure 3, these six truss elements 20, 22, 24, 26,
28 and 30 that form the central linkage are connected with eighteen revolute
joints which are indicated by the letter j with a numeral subscript. A subset
of
these joints are sensed by position encoders or potentiometers to allow
determination of the end effector position and orientation. This jointed
central
linkage gives the end effector raft 34 attached to the lower linkage beams 24
and
26, the required three translational DOF and one rotational DOF (pitch angle).
Brakes may be used on a subset of the central linkage joints in order to
ensure
single-string failsafe operation.
Figure 3 shows the joints on the left side, labelled j,,...,js. In cases where
a distinction between joints on the left side and joints on the right side is
required,
the joints will be referred to as j~, (joint one on the left side) through j~s
(joint nine
on the left side) and, similarly, jR~ through jRS indicating the right side.
Where it is
unimportant which side is being referred to, the subscript indicating the side
is
omitted. Kinematic analysis requires joint data from only one side, hence
throughout, the left side is chosen for purposes of the analysis presented.
The
eighteen passive revolute joints j; can alternately be described as being four
revolute joints: J4L, J4R~ jm and j~R; four universal joints, formed by the
pairs of
revolute joints: (J5~ & js~), (J5R & jsR), (js~ & js~) and (j8R & jsR); and
two spherical
joints, formed by the sets of revolute joints: (j», j2~ & j3~) and (J1R, j2R &
J3R). In
order to determine the position of the end effector, the position of joints
j~, j2, j4
and js on one side must be sensed. Redundancy can be achieved by sensing
joints on both sides of the linkage, or by sensing joints j3 and j5 in
addition. It
should be noted that since the positian of joints j3 and j5 are each dependent
on j2
and j4, redundancy is achieved only for j2 and j4 by sensing j3 and j5.
Since the left and right sets top three joints of the upper linkage which
form a spherical joint group of three intersecting revolute joint axes (joints
1,2
and 3; also labeled as j1, j2, j3), the upper beams 20 and 22 of the upper
linkage
could move such that they be in a position not being parallel, i.e., the upper
CA 02385602 2002-04-25
linkage could be a non-planar (spatial) linkage. To constrain this motion, the
tie
beam 30 is used. Therefore, the tie beam 30 constrains the left and right
upper
beams 20 and 22 of the upper linkage to be/remain parallel during the motion
of
the end effector (while the upper linkage rotates about three orthogonal axes,
e.g., X-world, Y-world and Z-world axes).
Six actuated cables drive the manipulator, each with a motor for winching
the cable located on the base assembly to reduce moving mass. Two of these
wires (labeled cable 1 and cable 2) attach to distinct points on the end
effector
raft 34, and the remaining four (labeled cables 3, 4, 5, 6) attach to the
central
linkage. Cables 3 and 4 run between the base assembly and points on the lower
linkage beams on the left and right side respectively, while cable 5 and cable
6
both originate from approximately the same point on the base assembly and
terminate at approximately the same point on the central linkage, thus
appearing
redundant. The two cables 5 and 6 are preferred, however, to overcome
mechanical interference. Cable 1, which leads to the end effector raft 34 will
cross the line between the attachment point of cables 5 and 6 whenever the Y
axis position of the end effector 34 is zero (i.e. when the end effector
crosses the
XZ plane of the world coordinate frame). Since wire one crosses paths with
wires 5 and 6 during normal motion of the robot, it is necessary to run cables
5
and 6 to either side of wire 1 and slack the interfering cable (one of cable
five or
six).
Cables 5 and 6 are shown in Figure 1 connected to the medial beam 28.
Due to the symmetry of the central linkage, these two cables are attached half
way from the right and left sides. The cables should also have some clearance
between each other. That is, the attachment point will not be exactly the
middle
of the length of medial beam (L4). That is, because the central linkage is
symmetric by attaching cables 5 and 6 with equal distance from right and left
sides (e.g., cable 5 a distance (0.5L4 - 0.5c) from left end of the medial
beam
and cable 6 a distance (0.5L4 - 0.5c) from right, with c being the clearance
between cables 5 and 6) as long as the distance/clearance between cables 5
and 6 is not too large.
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CA 02385602 2002-04-25
The cables 5 and 6 are attached to the medial beam 28 assuming that the
beam has a thickness (diameter), i.e., the cables are attached to the
surface/periphery of the medial beam 28 and they should not be attached to the
center line (axis) of the medial beam 28 otherwise it will not function
properly
regarding the force capabilities of the end effector. The preferred attachment
point is on the medial beam, about half way between the ends of the medial
beam taking into account the requirement for clearance between cables 5 and 6.
Since the cables can only produce a force in a single direction, along their
length as a result of wire tension, it should be noted that the combined
action of
all the wires is required to maintain a pose or move in any direction.
However, in
general, the individual wires produce forces that are primarily responsible
for
motion in specific directions. Motion in the positive X direction is primarily
a
function of wires five and six, while motion in the negative X direction is
driven by
wires three and four. Motion in the positive and negative Y direction is
predominantly due to wires three and four respectively. The combined effect of
cables one through four allows motion in the positive Z direction, while
cables
five and six are capable of moving the end effector in the negative Z
direction.
Wire two allows the end effector to pitch up, while wire one can cause the end
effector to pitch down.
In simulation, both analytical and computational, cables 5 and 6 are
assumed to be a single cable able to cross through wire 1 without interference
however there is a small offset relative to the nominal attachment points
shown
for wires five and six to the left and right, respectively, at both the
central linkage
assembly 14 and tl~ base assembly 12. This offset is needed to provide
sufficient space for the two cables to attach to the central linkage 14
without
interfering with each other. The offset distance produces no effect on the
calculated length of cables 5 and 6 provided that the offset is the same at
all
attachment points for wires five and six. Ln is defined as the length of wire
n, in
the special case of cables 5 and 6, the notation L5/6 is used to denote the
length
of the active cable.
CA 02385602 2002-04-25
The manipulator uses joint position sensors on certain passive joints of the
central linkage, as opposed to sensing the extension of the actuating cables.
Due to the design of the central linkage, a serial chain of rigid links and
joints
(each of which is either sensed or dependent on the sensed joints) exists from
the base assembly to the end effectar. This serial chain contains three
revolute
joints with intersecting axes of rotation, therefore the forward kinematics
problem
is tractable, and an analytical solution can be found. In general, use of a
redundant branch in a parallel robot reduces workspace, and increases the
weight of the manipulator [2]. In the case of the CAT4 robot, however, the
utility
provided by the central linkage in constraining the motion of the robot
offsets
these disadvantages, and is, in fact, essential to the function of the robot.
The CAT4 robot disclosed herein has been evaluated both analytically,
and using computational simulation. To facilitate discussion of the robot
features,
a coordinate system is defined and Figure 2 shows the world coordinate system
used. Additionally, it should be noted that left and right refer to the
positive and
negative Y direction respectively. Table 1 below specifies the 16 parameters
required to determine a particular geometry of a CAT4 robot design, an
explanation of each term, and the value of each parameter for the specific
configuration and Figure 3.4 shows the parameters R,, R2, L~, L~ and LS,2, as
defined in Table 1.
Figure 1, in addition to being a representational layout of the CAT4
architecture, also is of the same proportions as the specific model used in
simulation. The following two reference frames are used: the origin of the
world
coordinate frame is defined as the midpoint of the line between the upper
spherical joints (joints one, two and three on the left and right). The origin
of the
end effector coordinate frame is defined as the midpoint of the line between
the
lower universal joints (joints five and six on the left and right).
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CA 02385602 2002-04-25
Table 1 - Parameters for the specific configuration of the CAT4 robot
ParameterDescription Value
The length of the upper linkage beams along
their long axis, as shown in Figure
3.4. This can also be expressed as the:
1'U distance between the midpoint of the line 1$00 111111
between the world reference frame origin
and the midpoint of the medial linkage
beam.
The length of the lower linkage beams along
their long axis, as shown in Figure
j_,L 3.4. This can also be expressed as the distance1$00 mm
between the midpoint of the
medial linkage beam and the origin of the
end effector coordinate frame.
The distance between the origin of the world
coordinate frame and the
intersection of either t:he left or the 4$0 mm
right upper spherical joints, as shown
in
Figure 3.4.
The X coordinate of the attachment point
btx of wire one to the base assembly in 400 mm
world coordinates.
The Z coordinate of the attachment point
btZ of wire one to the base assembly in 400 mm
'
world coordinates. The ~
coordinate is 0.
The X coordinate of the attachment point
b2X of wire two to the base assembly in 2$00 mm
world coordinates. Both the Y-axis and Z
coordinates are 0.
The X coordinate of the attachment point
b3x of wire three to the base assembly in 1$90 mm
world coordinates.
The Y coordinate of the attachment point
ba of wire three to the base assembly in 2120 mm
y world coordinates. The Z coordinate is 0.
The X coordinate of the attachment point
b4x of wire tur to the base assembly in 1$90 mm
world coordinates.
The Y coordinate of the attachment point
b4y of wire four to the base assembly in 2120 mm
world coordinates. The Z coordinate is 0.
The X coordinates of the attachment points
b5x of wires five and six to the base 1600 mm
assembly in world coordinates.
The Z coordinates of the attachment points
of wires five and six to the base
assembly in world coordinates. The Y coordinates
b5Z are small equal values plus 1$0 mm
and minus of zero, which has no impact on
the kinematics provided the same
offsets occur at the medial beam attachment
points.
The location of the attachment point of
wires three and four along the long axis
of the lower linkage beam of the central
linkage, as shown in Figure 3.4.
t
This parameter is expressed as the fi-action900 mm -
of the length of the lower linkage 3
bean, along the line between the point formed1$00 mm $
by the intersection of the axes of
rotation of joint tour and joint seven and
the point formed by the intersection of
the axes of rotation of j~:>int five and
joint six.
The location of the attachment point of 400 mm
wires three and four normal to the long 4
axis of the lower linkage beam of the central_ _
linkage, as shown in Figure 3.4.
RZ 1$00 mm 1$
As with the parameter Ri, the value is normalized
relative to the lower linkage
beam length, L~.
The distance from thc: origin of the end
effector reference frame, midway
(1t between the intersections of the axes of 4$0 rnrn
rotation of joints five and six on the
left
and right, to the attachment point of wire
one on the end effector raft.
The distance from the: origin of the end
effector reference frame, midway
(12 between the intersections of the axes of 4$0 mm
rotation of joints five and six on the
left
and right, to the attachment point of wire
two on the end effector raft.
CA 02385602 2002-04-25
The mechanism possesses symmetry about the XZ plane of the world
coordinate frame, and therefore the left and right sides are mirrors of each
other,
with nine revolute joints on each side, j~,...,j9. In order to determine the
position
of the end effector, the position of joints j~, j2, j4 and j6 on one side must
be
sensed. Redundancy can be achieved by sensing joints on both sides of the
linkage, or by sensing joints j3 and j5 in addition. It should be noted that
since the
position of joints j3 and j5 are each dependent on j2 and j4, redundancy is
achieved only for j2 and j4 by sensing j3 and j5. The relationship between
these
joints are given by equations (1) and (2).
Forward kinematic analysis provides a transformation from joint space to
world space, while the inverse kinematic analysis provides the reverse, the
transformation from world space to joint space, as shown in Figure 3.
Figure 1, in addition to being a representational layout of the CAT4
architecture, also is of the same proportions as the specific model used in
simulation. The following two reference frames are used: The origin of the
world
coordinate frame is defined as the midpoint of the line between the upper
spherical joints (joints one, two and three on the left and right). The origin
of the
end effector coordinate frame is defined as the midpoint of the line between
the
lower universal joints (joints five and six on the left and right).
In order Xo leverage the advantages of both serial and parallel robots in
the design of the CAT4 robot, the first, second, forth and sixth revolute
joints of
the central linkage are sensed. This allows the forward displacement
kinematics
to be found in the same manner as for serial robots, and an exact analytical
solution is given for the forward displacement analysis as equations (3)
through
(g).
CA 02385602 2002-04-25
63 =-arccos(c°s8= ~cosP, ~ (1l
sin 8_ ~ sin 8,
8s =+arcsin c°s0= ~~l
C sin A, )
"PX=oP% =ca(cn_c~-sis3~L~ -cisz(saL~ +Lu)
"Py =-°PZ =s_c3c,L~ +c_(s,L~ +Ly (4)
wPZ=°Py=-c,(s,c:c;+c,s,)L, +s,s=(syL, +L~,) (5)
ty-arctan cb(cs(c,(s,c=c,+c,s;)-s,sa,,~+s,(c,c3-
s,c2s,~)+s~(s4(s,c:c3+c,s,~+s,szc,) (6)
(Cfi(CS(Ca(C,C~C3 -S,S,)-C~S:S,,)-S,(S~C, +C:iC_$,~~+S6(Sq(C~C;C3 -
S,S,~+C,S:C4~)
Where "P%, "Py, "PZ denote the X, Y and Z position of the end effector,
respectively, in the world coordinate frame.
~y is the pitch of the end effector.
s" indicates the sine of 8", the angle of joint j". Similarly c" indicates the
cosine of 8".
8" is the displacement of joint n.
L~ is the length of the upper linkage beam an<l LL is the length of the lower
linkage beam.
A, = arctan (Yi + X ) ('7)
Where i is the imaginary number, ~ , and the terms X and Y are given by:
X= T:T,~P,Ts+~P.Lu (T<+T.T6 T~ T,' +aLr' T,-Lu' +4T,Ts +4T,Ts + T_'T,"p~Tb
L~,T,(T,,+T~T")
Y- T.T,"p"Ts+"p~L~, (Ts+T.T~ T_T,'+4L~-T,-Lu' +4T,T~, +4~;T5 + T_'T,"pxTb (9)
LaT,(T., +T:Tb)
T
62 = -arccos " PY ~ : ~ ( 10)
(Lu T, +2L~
T
NP, 4- ' 2Lr
L~'L~
A; _ +arccos - ~c ( I 1 )
(T~+~LL=) I- HP, T~
L~'~T, +2L~'~
A, _ +arcsinl 2LTIL ) ( 12)
l a
8s=+arcsin( 2~p,Lr-1 (13)
lT, +2L J~
T=wpx +~~Pz +~~P, -Lu -LL Ta=T,-"P, +Lu +L~.z=~~Px2+NP~
TZ =4Lu'L~' -T,' Ts =-4L1,'L~'T, =-4L~,rLr=("px-+,~p~z~ (14)
T - Ts + Lu-T, T~ = Li, -a p'z
Ts + T2 T6_
CA 02385602 2002-04-25
Inverse displacement equations are solved both for the requirement to
determine the wire lengths for any position and orientation of the end
effector,
and also for the inverse displacement analysis of the first through to the
fifth
joints of the central linkage, which are required to find the inverse
displacement
kinematics for the actuated wire lengths, and are used directly in determining
the
workspace of the robot. The inverse displacement equations for joints one
through five of the central linkage are given as equations (7) through (14).
The
inverse displacement equations for the actuated wires which drive the CAT4
have been found, however, these are omitted here due to their length.
The existence of exact solutions to the forward and inverse displacement
equations provides a benefit in comparison with many parallel manipulators
which require numerical methods to calculate a forward displacement solution.
Controller hardware can be less powerful and less expensive as solving the
closed form equations is computationally trivial, in comparison with iterative
methods that can be necessary 'where closed form solutions are not tractable.
The workspace of the manipulator can be characterized by the space
formed by the intersection of two volumes, defined by the two sets of
constraints
on the manipulator, the passive linkage joint limits, and the ability of the
wires to
operate in tension. The resulting workspace is a complex volume that is
determined by a computational method.
The passive joint limits are dependent on the detailed mechanical design
of a given CAT4 implementation, however, all CAT4 configurations must restrict
the displacement of joint five to be within some margin of ~90°, as the
central
linkage is in a singular configuration at these values. As joints five and
seven,
which are constrained to be equal, approach ~90°, the central linkage
ceases to
have the ability to effectively constrain the motion of the manipulator. The
practical effect of this is that when the robot is operated with these joints
very
close to t90°, the central linkage has significant interference
problems, as the
lower linkage beams are not prevented from colliding with each other in this
condition. This would result in high joint forces, mechanical interference of
the
two lower linkage beams, and possible mechanical failure. In order to prevent
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CA 02385602 2002-04-25
this from occurring, it is recommended that joints five and seven be limited
in
their range of motion to operate between approximately 85° and
+85°, which
reduces the available workspace, such that lateral motion in the direction of
the
Y-axis is restricted when the position of the end effector has small
displacements
in both X- and Z-axes.
The remaining joint limits will of course produce some limitation on
workspace to avoid mechanical interference of the links. These limits must be
determined by the detailed mechanical design of the robot structure, which is
beyond the scope of this research. As such, reasonable values for joint limits
have been assumed as follows:
0°<_9, s+Z~o°
-iso°_<8, so°
-so°__<9, s+8o° ~~5~
-ss° s85 < +ss°
In order to move the manipulator, the wires must be able to produce a
component of force in the desired direction through tension. In order for a
point
to be within the workspace, the wires must be able to exert a component of
force
in any arbitrary direction allowed by the degree of freedom of the
manipulator,
considering the restriction that wires can only generate forces through
tension.
In order to achieve this canstraint with the CAT4 manipulator, several
requirements are imposed on any point within the workspace. The constant
orientation workspace, for zero pitch, is shown in Figure 5. The workspace
visualization shown has contour lines to assist in the visualization of the
shape of
the workspace, with a spacing of 100mm in the X-, Y- and Z-directions.
Additionally, a transparent cylinder with a radius of 2.2 meters and a height
of 2.0
meters is superimposed on the CAT4 workspace, to indicate the comparison
between the actual workspace and the original goal.
The workspace is calculated by a computational method that tests points
for inclusion within the workspace based on certain criteria, grouped as
follows:
~ The point must lie below a specified upper clipping plane, which must be
below the world origin. The level of the upper clipping plane and the level of
a
ground plane are input into the workspace program at run time by the user.
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CA 02385602 2002-04-25
~ The point must lie within a reachable sphere of radius L~+L~ (the sum of the
lengths of the upper and lower linkage beams), which is the maximum
possible reach of the central linkage ignoring any joint limits.
~ Inverse joint displacement equations are used to calculate the joint
displacements for the workspace point and a check is performed to ensure
that the central linkage joint displacements are within the prescribed limits
given in equation (15).
~ The wires are tested to ensure they are able to remain in tension for the
specified workspace point. Due to the presence of the articulated central
linkage, with the majority of wires not directly connected to the end
effector, a
conservative approximate technique is used to ascertain whether the wires
are capable of exerting a force in tension to move the end effector in an
arbitrary direction. The approximation requires that the wire direction from
the
base attachment point be within limits chosen such that wire tension and
arbitrary motion is always possible.
The flowchart for the workspace computation program is shown in Figure 6.
The algorithm uses a recursive subdivision approach to determine whether
successively smaller boxes are included in, or excluded from the workspace.
When the program is first executed, parameters are accepted from the user to
define the upper clipping plane and the ground plane, the size of the initial
box
size before any refinement, the minimum box size to test, and the interval
between test points. The program then uses the maximum possible reachable
workspace size, given by L~+L~ and creates a three-dimensional grid of cube
shaped boxes with the maximum size that fully covers this possible workspace
from the upper clipping plane to the ground plane. Each of these major boxes
is
then passed to the box testing function, which will make a determination
whether
the box is completely within the workspace, completely outside of the
workspace,
or has points that are within and points that are outside the workspace.
The box testing part of the workspace determination algorithm is a subroutine
that accepts a cube by its lowest valued corner and the length of its sides.
The
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CA 02385602 2002-04-25
function operates by checking points on the surface of the box against the
criteria
given above for valid points within the workspace. Each of the six sides are
divided into grids and tested at an interval specified by the user. For
efficiency,
the points shared by more than one side are only tested once. The box testing
subroutine then returns a result that indicates one of three alternatives:
"Accept" All points passed the workspace test,
"Discard" All points failed the workspace test, or the box size was below the
minimum box size, or
"Refine" Some, but not all, points failed the workspace test.
Since the large number of points that must be checked for a complete box can
be
computationally expensive, the function will exit as soon as there is at least
one
point that fails and one point that passes giving a "refine" result.
On a "discard" result, the box tested is discarded, as it is either completely
outside the workspace, or a refinement of a cube that is too small to be
subdivided further, and therefore not included in the workspace at the
specified
resolution.
When the function returns a "refine" result, the cube is split into eight
cubes, each having a side length half that of the original, and each of these
is
tested in turn. This aspect of the program is therefore recursive. A minimum
box
size is established below which cubes are discarded rather than refined in
order
to establish an upper limit on the depth of recursion and ensure the program
completes.
On an "accept" result, the box tested is added to the list of boxes within
the valid workspace, assuming that all points within the box are within the
valid
workspace. The preferred form of the list of boxes within the workspace is
highly
dependent on the purpose for which the workspace data structure is intended.
In
the current program, the intended use is to produce a parsed text file that
can be
read by a graphics rendering program to produce visualizations of the
workspace. As such, the workspace list is kept in an unordered linked list (a
common type of data structure) for maximum memory efficiency, and periodically
dumped to the output text ale in a form readable by the graphics package
CA 02385602 2002-04-25
PovRay 3.1g. This raytracing tool is used once the evaluation is complete to
process the output files into an image of the workspace. Since, due to the
symmetry of the workspace of the CAT4 robot about the XZ plane at Y=0, the
workspace program only outputs the +Y half of the workspace. Some post-
processing of the PovRay input files is therefore required to obtain the full
workspace shown in Figure 5.
Mechanical interference is a key issue in parallel manipulators, where
multiple branches must operate within the same work volume and may interfere,
requiring a reduction in reachable workspace. Mechanical interference is
defined
as the interference of portions of the robot structure, such as actuators or
structural components, which would result in two parts of the mechanism
colliding during normal operation. Avoiding this interference in parallel
manipulators requires care in design and restriction of the workspace. The
CAT4
robot has relatively benign characteristics in this regard, however, attention
is
required in several aspects of the design discussed in this section in order
to
prevent mechanical interference.
Portions of the end effector raft 34 (Figure 3) extend forward and to the
rear to provide appropriate moment arms for cables 1 and 2 to control the end
effector pitch. It is therefore preferred to provide sufficient clearance
between the
end effector raft 34 and lower linkage beams 24 and 26 for the minimum acute
angle permitted by the joint limits in the lower part of the central linkage
14. This
has been achieved by designing the lower linkage beams 24 and 26 with
substantial flare at their lower ends (see Figure 1 ) so that they do not
interfere
with the attachment points of cables 1 and 2 to the end effector raft 34 shown
most clearly in Figure 3. This flare therefore is necessary to provide
sufficient
clearance for wires one and two in operation.
Additionally, since cables 5 and 6 and cable lone cross each other and
thereby interfere as the robot translates from the left side of the workspace
to the
right, it is necessary to use cables 5 and 6 as duplicates, with one operating
on
each side of cable 1 while the other is left slack. Cable 1 is then passed
between
cables 5 and 6 to eliminate this interference problem. Due to the symmetry of
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CA 02385602 2002-04-25
the CAT4 design the actuated wire switches between cable 5 and cable 6 as the
end effector moves through the XZ plane at Y=0, while the other wire is
allowed
to become slack. Cable 5 is defined as the left (+Y) wire, and cable 6 is
defined
as the right ( Y) cable, therefore, cable 5 is the active wire when the end
effector
has a negative Y coordinate, while cable 6 is active when the end effector has
a
positive Y coordinate. The cable that is not active at a given time is slack
and
free to extend as required such that it travels around wire one, although a
small
tension should be applied so that excess slack is taken back in, preventing
entanglement. As discussed previously, because this change in controlling
cable
will take a small but finite period of time, a small separation between cables
5
and six is necessary. Increasing the amount of separation decreases the
difficulty of the control changeover, and may ameliorate the friction between
the
slack inactive wire and wire one. Provided that the separation distance is not
overly large, and the separation distance from the centreline is the same at
the
base and linkage attachment points for cables 5 and 6, this does not affect
the
displacement kinematics.
Cable interference can also restrict the workspace of the robot for certain
poses of the end effector where the end effector is operating close to the
origin.
This can cause the central linkage to assume a position that results in cables
5
and 6 interfering with the end effector raft 34. The exact region where this
occurs is heavily dependent on the detailed design of the linkage and can be
ameliorated by making the lower linkage beams 24, 26 somewhat longer than the
upper linkage beams 20, 22. For the workspace shown in Figure 5, this
interference is ignored, since the maximum height of the end effector raft has
been specified such that it does not closely approach the origin of the world
coordinate frame, the region in which this issue becomes problematic.
The dynamic motion of the CAT4 manipulator has been simulated using
msc.VisuaINastran, in order to obtain an approximate evaluation of the
repeatability and potential difficulties with vibration, which are common
problems
with wire actuated manipulators. Additionally, these simulations verified the
DOF
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CA 02385602 2002-04-25
of the CAT4 manipulator, and provided numerical data with which the derived
forward and inverse displacement equations were verified.
A path that consists of the end effector moving in a spiral path down the
surface of a cone was chosen as the standard motion for the manipulator. The
path was chosen such that the end effector would be within the estimated
workspace of the robot yet approach the workspace boundaries closely during
the test. Wire damping, gravity and end effector speed were then adjusted to
observe their effects on the end effector positioning error. In these tests,
open
loop control was used, with the wire length controlled so that the end
effector raft
would follow an imaginary target point travelling along the test path, while
varying
end effector pitch.
Figure 7 shows a graph of the error for the test path with three different
speeds for the end effector. It can be seen that despite a 25-fold difference
between the slowest and fastest speed, error remains reasonable, and the
response is stable. There is, however, a significant amount of vibration in
the
system while in motion. It is expected that the position error can be reduced
substantially through calibration to account for the steady state error, which
is
between 3 and 6 mm, and is a function of end effector position. A closed loop
control system has the potential to significantly improve performance. The
large
amount of vibration that occurs between 1300 and 3000 time steps is due to the
end effector being driven very close to the boundaries of the estimated
workspace. This proximity to the workspace boundary causes large changes in
the position of the links of the central linkage for relatively small changes
in the
end effector position, which increases the error in dynamic motion.
Simulations
which examine the steady state error that occurs after the manipulator is
stopped
from the position of worst case error shown in Figure 7 result in an error of
less
than 6 mm.
As used herein, the terms "comprises" and "comprising" are to be
construed as being inclusive and open ended, and not exclusive. Specifically,
when used in this specification including claims, the terms "comprises" and
"comprising" and variations thereof mean the specified features, steps or
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CA 02385602 2002-04-25
components are included. These terms are not to be interpreted to exclude the
presence of other features, steps or components.
The foregoing description of the preferred embodiments of the invention
has been presented to illustrate the principles of the invention and not to
limit the
invention to the particular embodiment illustrated. It is intended that the
scope of
the invention be defined by all of the embodiments encompassed within the
following claims and their equivalents.
References
[1] Merlet, J.-P., Parallel Robots, KluwerAcademic Publishers, 2000.
[2] Notash, L., Podhorodeski, R., Forward Displacement Analysis and
Uncertainty Configurations of Parallel Manipulators with a Redundant
Branch, J. Robotic Systems, 13(9), pp. 587-601, 1996.
[3] H. A. Akeel: US Patent #5313854, Light Weight Robot Mechanism, May 24,
1994.
[4] S. E. Landsberger and T. B. Sheridan: US Patent #4666362, Parallel Link
Manipulators, May 19, 1987.
[5] R. Bostelman, J. Albus, N. Dagalakis and A. Jacoff: "RoboCrane Project: An
Advanced Concept for Large Scale Manufactureing", Proc. AUVSI Conf.,
paper 407, 1996.
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