“Introduction to Robotics ”
Lecture 3
Denavit and Hartenburg
Dr. Greg R. Luecke
Associate Professor
Mechanical Engineering
Iowa State University
Earth fixed axes-1st X, then Y, finally Z
IOWA STATE UNIVERSITY
©2008 Dr. Greg R.Luecke
More on rotation: Body fixed axes-1st X, then Y, finally Z
“Euler Axes”
Note the order!
IOWA STATE UNIVERSITY
©2008 Dr. Greg R.Luecke
Link descriptions, kinematics, standard frames
Two types of links: Revolute (R)
Prismatic(P)
Each has a single degree of freedom.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Robots are constructed with a number of links and joints to
allow the desired motion of the end effector.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Here is a general link connecting two joint axes:
The robot link connects
these two joint axes, and
motion is constrained by
the axis directions.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s assign reference frames according to the DenavitHartenburg conventions:
1) Identify the joint axes
with infinite lines.
2) At the point of
intersection, or the
point of intersection
of the common
perpendicular, assign
the origin of the ith
reference frame.
3) Assign the Zi-axis
pointing along the ith
axis.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s develop Denavit-Hartenburg parameters:
4) Assign Xi pointing
along the common
perpendicular, or if
the axes intersect,
assign Xi to be
perpendicular to both
axes.
5) Assign Yi according
to the right hand
rule.
6) Assign frame {0} to
be coincident with {1}
when the joint
parameter is zero,
and choose frame {N}
so as to make link
parameters zero.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Denavit-Hartenburg link parameters:
1) Link length, ai.
The distance
measured from Zi
to Zi+1 along Xi.
2) Link twist, ai. The
angle between Zi
and Zi+1 measured
about Xi.
3) Link offset, di.
The distance from
Xi-1 to Xi,
measured along Zi.
4) Joint Angle, qi.
The angle between
Xi-1 and Xi,
measured about
Zi.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Denavit-Hartenburg link parameters:
1) Link length, ai.
The distance
measured from Zi
to Zi+1 along Xi.
2) Link twist, ai. The
angle between Zi
and Zi+1 measured
about Xi.
3) Link offset, di.
The distance from
Xi-1 to Xi,
measured along Zi.
4) Joint Angle, qi.
The angle between
Xi-1 and Xi,
measured about
Zi.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s look at the transformation from frame {i-1} to {i}.
Note that this is
accomplished in steps:
First, the transform
from {i-1} to {P}, a
rotation about Xi-1 by
an angle ai-1 to align
Zi-1 with Zi.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s look at the transformation from frame {i-1} to {i}.
Next, the transform
from {P} to {Q}, a
translation along Xi-1
by a distance ai-1 to put
the origin of {Q} on
the axis, Zi.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s look at the transformation from frame {i-1} to {i}.
Third, the transform
from {Q} to {R}, a
rotation by an angle qi
to make XR parallel
with Xi+1.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s look at the transformation from frame {i-1} to {i}.
Finally, the transform
from {R} to {i}, a
translation by di to
align the origin of {R}
with the origin of {i}.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Let’s look at the transformation from frame {i} to {i+1}.
Let’s concatenate these
transforms:
di
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Here is a rotational transform:
And a translational transform:
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Expanding and multiplying each of these matrices:
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Now, for any given robot link, we can use this formula to
describe the motion of that link. Here is a simple robot, let’s
compute the robot kinematics.
1) Identify the joint axes
with infinite lines.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Now, for any given robot link, we can use this formula to
describe the motion of that link. Here is a simple robot, let’s
compute the robot kinematics.
2) At the point of
intersection, or the
point of intersection
of the common
perpendicular, assign
the origin of the ith
reference frame.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Now, for any given robot link, we can use this formula to
describe the motion of that link. Here is a simple robot, let’s
compute the robot kinematics.
3) Assign the Zi-axis
pointing along the ith
axis.
Z2
Z1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Now, for any given robot link, we can use this formula to
describe the motion of that link. Here is a simple robot, let’s
compute the robot kinematics.
4) Assign Xi pointing
along the common
perpendicular, or if
the axes intersect,
assign Xi to be
perpendicular to both
axes.
X1
Z2
Z1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Now, for any given robot link, we can use this formula to
describe the motion of that link. Here is a simple robot, let’s
compute the robot kinematics.
5) Assign Yi according
to the right hand
rule.
Y1
X1
Z2
Z1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Now, for any given robot link, we can use this formula to
describe the motion of that link. Here is a simple robot, let’s
compute the robot kinematics.
Y2
Y1
X1
X2
6) Assign frame {0} to
be coincident with {1}
when the joint
parameter is zero,
and choose frame {N}
so as to make link
Z2
parameters zero.
Z1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Make a table of D-H parameters:
Y2
X
2
Y1
X1
Z2
Z1
i
1
2
ai-1
0
L
ai-1
0
0
di
0
0
qi
q1
q2
1) Link length, ai.
The distance
measured from Zi
to Zi+1 along Xi.
2) Link twist, ai. The
angle between Zi
and Zi+1 measured
about Xi.
3) Link offset, di.
The distance from
Xi-1 to Xi,
measured along Zi.
4) Joint angle, qi.
The angle between
Xi-1 and Xi,
measured about
Zi.
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
Compute transform matrices.
X2
Y2
Y1
X1
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©2003 Dr. Greg R.Luecke
Concatenating the transform gives us the forward kinematics:
X2
Y2
Y1
X1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
To get the end effector position, we may want to add a final
reference frame:
{E}
X2
Y2
Y1
X1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke
“Forward Kinematics”—given the joint variables, find the
robot position.
{E}
X2
Y2
Y1
X1
IOWA STATE UNIVERSITY
©2003 Dr. Greg R.Luecke