2.5 Robot Kinematics

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• 2.5 Robot Kinematics
• The key problem to be solved is how to set up
  coordinate systems on individual bodies. This is
  related to the structure of robot.
• 2.5.1 Link description
• Types of joint are shown in Fig.2-20.
• A link is composed of a line connecting two
  joints with

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their types being as shown in Fig. 2-20. Remark
1 Those types are also called one degree of
freedom joint. In the rare case, the joint type
has two degrees of freedom. Fig.2-21 shows a
typical layout of the link.
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Fig.2-20
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• The following features are concluded from Fig.
  2-21
• - The mutual perpendicular distance, a i-1
• - The link twist, i-1

Fig.2-22 shows two links connected. The following
Features or parameters are concluded from this
figure - d i link offset - joint
angle
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Fig.2-21
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Fig.2-22
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• Labeling of links
• The base link 0
• The last link n
• The length and link twist depend on joint axes i
  and i1. Hence a1 through a n-1 and 1 and
  n-1 are defined as discussed above, but this
  rule does not hold for the base link and the last
  link
• The idea is to set up the base frame and the
  last frame such that a0 a n0
  0

n
0
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- Regarding d1 (dn) and ?1 (?n), the rule is to
make them 0.0
a) If joint 1 is revolute, define the base frame
and the first frame such that d 1 0.0, and
?10.0 at the initial time
b) If joint 1 is prismatic, define the base
frame and the first frame such that d1 0.0 at
the initial time, and ?10.0.
c) If joint n is revolute, define the last frame
and the n-1 frame such that d n 0.0, and ?n0.0
at the initial time
d) If joint n is prismatic, define the last
frame and the n-1 frame such that dn 0.0 at the
initial time, and ?n0.0.
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Summary of the link parameters The previous
rule to set up the frame has produced four
parameters for each link. Each time there is
only one is time varying. This is rule is
called Denavit-Hartenberg (D-H) notation.
2.5.2 Convention for affixing frame w to links In
order to describe the location of each link
relative to its neighbors we define a frame
attached to each link. The link frames are named
by number according to the link to which they are
attached. That is, frame (i) is attached rigidly
to link
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Intermediate links in the chain (1) The Z-axis
of frame (i), Zi , is coincident with the joint
axis i. The origin of frame (i) is located where
the ai perpendicular intersects the joint i axis.
Xi points along ai in the direction from joint i
to joint i1. In the case of ai O, Xi is
normal to the plane of Zi and Zi1. We define ai
as being measured in the right-hand sense about
Xi, so we see that the freedom of choosing the
sign of a in this case two choices are
available.
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Yi is formed by the right-hand rule to complete
the i-th frame. Fig.2-23 shows the location of
frames i-1 and i for a general manipulator.
Fig. 2-23
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First and last links in the chain Attach a
frame to the base of the robot, or link O, called
frame (O). This frame can be treated as a
reference frame for measuring the position and
orientation of all other frames. Since frame (O)
is arbitrary, it always simplifies matters to
choose Z0 along axis 1 and to locate frame (O) so
that it coincides with frame (1) when joint
variable 1 is zero. Using this convention we
have ao 0.0, ao 0.0. Additionally, this
ensures that d1 0.0 if joint 1 is revolute, or
? 1 0.0 if joint 1 is prismatic.
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For joint n to be a revolute one, the direction
of XN is chosen so that it aligns with X N-1 when
? N 0.0, and the origin of frame (N) is chosen
so that dn 0.0. For joint n to be a prismatic
one, the direction of XN is chosen so that ?N
0.0, and the origin of frame (N) is chosen at the
intersection of X N-1 and joint axis n when dn
0.0. Summary of the link parameters in terms of
the link frames If the link frames have been
attached to the links according to our
convention, the following definitions of the link
parameters are valid ai the distance from Zi
to Zi1, measured along Xi
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di the distance from Xi_, to Xi-1 measured
along Zi and i, the angle between Zi, and
Z i1 measured about Xi. i the angle
between X i-1 and Xi, measured about Zi We
usually choose ai gt 0 since it corresponds to a
distance however, other three are signed
quantities. A final note on uniqueness is
warranted. The convention outlined above does not
result in a unique attachment of Frames to links.
First of all, when we first align the Zi axis
with joint axis i, there are two choices of
direction in which to point Zi. Furthermore, in
the case of intersecting joint axes (i.e., ai
O), there are two choices for the direction of Xi,
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corresponding to the choice of signs for the
normal to the plane containing Zi and Zi1,. When
axes i and i1 are parallel, the choice of origin
location for (i) is arbitrary (though generally
chosen in order to cause di to be zero).
Example
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Fig.2-24
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Fig.2-25
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