Zone-Chang Lai and Chia-Hsiang Meng
IEEE Journal of Robotics and
Automation, Vol.4, No.1, February 1988
Summary
Many investigation have been one on the
determination of extreme position of the end-effector of a manipulator, this
paper is regarding instead studies on dexterous workspace with 6 joints axes,
which results not been investigated due to complexity in computations.
A dexterous space is defined as a space in which a manipulator’s hand
can rotate fully about al axes through any point.
In the manipulator’s
workspace, the sphere, called “service sphere” ( SS(P)) , is constructed around
a point P with the hand size of the manipulator as a the radius. When the
manipulator’s hand reached pint P, then the position of its six joint (J6)
must be located on the sphere. The so called “service point” is the point on
the sphere SS(P) which allows to access the position point P. The region
containing more service point is called “service region”, if the manipulator
can freely move in the region while keeping its hand in contact with P, then
this point is defined as “free service region”.
O4(H) is
defined as the set of hand orientations with joints 4 to 6 able to rotate
freely, so no boundary exists for a free service region corresponding to
appoint in the dexterous workspace. It has been proven that the boundary of a
free service region is described by either the boundary of W1(4),
which is the reachable space of joint 4 when joint from 1 to 3 are free to
rotate, or the boundary of O4(H).
In a dexterous space, the
robot’s wrist should be capable of generating a full range of orientations,
defining what is called “dexterous wrist”, so that when J6 is at the
boundary of a free service region, J4 is at that boundary of W1(4),
so that it could be considered that the boundary of dexterous workspace is
governed by the boundary of W1(4).
Before performing
computation hypothesis for the problem must be checked: it must be check is the
robot’s wrist is dexterous, the condition that J4 is not at the
boundary of W1(4) for all hand orientation must be determined. If a
wrist has unlimited revolute joints and its twist angle of joints 4 an 5 equal
±90°, then conditions for the wrist being dexterous are verified.
The boundary of W1(4)
can be described or by rotational limits of joint 1 to 3 or by the dimensional
constraint of the first three links. The equation which governs the boundary of
W1(4) can be obtained by solving det(G)=0, where G is the Jacobian
matrix of the robot’s first three links. G is defined: [dx4 dy4
dz4]T=G[dθ1
dθ2
dθ3]T,
where x4, y4 and
z4 are X, Y and Z coordinated of J4.
Solving det(G)=0 then the
function f1(θ1
θ2
θ3]=0
represents symbolically the W1(4)
boundary conditions.
The dexterous workspace
may then be obtained by finding the hand position such that: f1(θ1
θ2
θ3]≠0
and θi1< θ1< θi2,
where
i=1,2,3 are the joints.
For
6 d.o.f. robots this computation may be very difficult, the authors introduce 3
simple cases for commond practice: a PUMA type manipulator with an interesting orthogonal
wrist, manipulator with a roll pitch-yaw wrist (nonintersting orthogonal wrist)
and a manipulator with all unlimited joints and a roll pitch-yaw wrist and
limited revolute joints.
Key
Concepts
Dexterous Manipulators,
Workspace Boundaries
Key Results
The PUMA
type manipulator with an orthogonal wrist appears to have a spherical
workspace, as already shown with different computation by Ruth. The manipulator
with a roll pitch-yaw wrist (nonintersting orthogonal wrist) also appears to
have spherical workspace, computations for both manipulator are reported at
page 101 and 102. The manipulator with all unlimited joints and a roll
pitch-yaw wrist and limited revolute joints has the difference that has to be
described either by the dimensional constraint or by the rotating limit, since
joint aren’t unlimited.
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