CA2385602A1 - Four degree-of-freedom wire actuated parallel robot - Google Patents

Abstract

There is disclosed a cable actuated truss - 4 degrees of freedom (CAT4) robot. The robot is a novel, passively jointed, parallel robot utilizing six control cables for actuation. The robot utilizes a passive jointed linkage with 18 revolute joints to constrain the end effector motion and provide the desired structural stability, restricting the end effector to 3 translational degrees of freedom (DOF) and 1 DOF for end effector pitch. This central mechanism together with winched cable actuation gives a number of important benefits for applications where the advantages of a parallel robot are required in conjunction with light weight.
Six electric motor driven winches control the length of the cable actuators that extend from the top frame to points on the end effector raft and jointed linkage to create a stiff, but lightweight, actuated robot.

Description

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-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.

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 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 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 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 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 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.

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.

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).

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.

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).

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_ 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 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.

~ 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 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 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 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 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 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 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

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 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.

Claims (9)

1. 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;
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 and sixth cables being connected to winch means for extending and retracting said fifth and sixth cables.

2. The robot according to claim 1 wherein said fifth and sixth cables are connected in close proximity to said medial linkage beam.

3. The robot according to claims 1 or 2 wherein said fifth and sixth cables are connected substantially midway between the ends of said medial linkage beam.

4. The robot according to claims 1, 2 or 3 wherein said pair of upper linkage beams, said medial linkage beam, said tie linkage beam and said first and second lower linkage beams are linked together with revolute joints.

5. The robot according to claim 4 including position sensors on a selected number of said revolute joints on said passively jointed central linkage.

6. The robot according to claims 1, 2, 3, 4 or 5 wherein said structural base assembly is collapsible.

7. The robot according to claims 1, 2, 3, 4, 5 or 6 wherein said pair of upper linkage beams, said tie linkage beam, said medial linkage beam and said first and second lower linkage beams are trusses.

8. The robot according to claims 1, 2, 3, 4, 5, 6 or 7 wherein said first and second lower linkage beams are longer than said pair of upper linkage beams.

9. The robot according to claims 1, 2, 3, 4, 5, 6 or 7 wherein said first and second lower linkage beams are flared at the lower end portions thereof.