Three Dimensional
Manipulators and Denavit–
Hartenberg Parameters
Prof. Matthew Spenko
MMAE 540: Introduction to Robotics
Illinois Institute of Technology
Review from Undergraduate Dynamics Class –
Rotation Matrices
• Describe a point defined in one basis (or reference frame) in some
other basis.
• Right-Handed Unitary Orthogonal Bases
• To find out how two bases are related to each other, we use rotation
matrices.
RHOUBS
These are basis vectors, typically, ax, ay, az
• Given two RHOUBS, a and b
• aRb is a 3x3 rotation matrix:
The ith and jth element of rotation
matrix between the a and b frames
The angle between the
ai and bj unit vectors
The dot product between the orthogonal unit vectors
• where i,j = x, y, z
• A list of the cosines of the angles between all combinations of the
base’s vectors.
• Method to encapsulate the difference between to RHOUBS.
RHOUBS properties
• It is common to write out a rotation matrix in a rotation table:
• Use the rotation matrix to express a vector currently in the a-basis in
the b-basis (or vice versa).
• Example:
• More generally:
RHOUBs Properties
• Rotation Matrices have some special properties.
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•
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•
•
RT=R-1
each column and row are mutually orthogonal.
each column and row is a unit vector.
det R = 1
Another useful property that we will use to our advantage is that:
• But, they do not commute! The order is important
Constructing a Rotation Matrix
• If you every get confused, just remember how it is defined with dot
products and you will always be right.
A Few Other Comments
• Using the convention I have here:
• Using the convention in the book:
Rotation Matrices Summary
• Comprised of cosine between unit basis vectors
• Used (in this class) to write vectors expressed in one basis into
another basis
• Chain (multiply) rotation matrices together to move through more
than one vector basis
Rotation Matrix Mini-Quiz
• Given two rotation matrices, A and B, does AB=BA?
• What are the implications of this?
• Compute the rotation table between the n-basis and b-basis shown
below, where theta is the angle between bx and nx
• Compare your answers to your neighbor’s
Rotation Matrices and Reference Frames
• Reference frames have no
origin
• Rotation matrices can always
be used to translate vectors
between two bases.
• We want to develop the
forward kinematics of a
manipulator
• Rotation matrices alone cannot
do this because they do not
account for translations
Homogeneous Transformation Matrices
• Homogeneous Transformation Matrices
Rotation Matrix
Translation Vector
Scale Factor
Example: Homogeneous Transformation
Matrix
• A set of basic homogeneous transformations is given by
Example: Constructing a Homogeneous
Transformation Matrix
• A HTM can be constructed the same way as any rotation matrix.
• rotate a about x axis, then…
• translate a distance c along z axis, then…
• rotate angle b along y axis
Homogeneous Transformation Matrices
Summary
• Homogeneous transformation matrices are comprised of:
• A rotation matrix
• A translation matrix
• A scaling factor (always 1 for our purposes)
• Homogeneous transformation matrices:
• Can be multiplied together (in the proper order) to create a map that relates
a vector described in one basis to another basis.
Homogeneous Transformation Matrices MiniQuiz
• Describe the components of a HTM
• Compare your answer to your neighbor’s
Three Dimensional Manipulators
and Denavit–Hartenberg
Parameters
Kinematic Chains
Kinematic Chains
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•
•
•
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A robot manipulator with n joints has n+1 links
Joints are numbered 1 to n
Links are numbered from 0 to n
Therefore joint i connects link i-1 to link I (when joint i actuates, link i moves!)
A coordinate frame is attached rigidly to each link
• oixiyizi is the coordinate frame of link i
Rotary Joint 2
Prismatic Joint 3
Rotary Joint 1
SCARA Manipulator Example
Three Dimensional Manipulators
and Denavit–Hartenberg
Parameters
The Denavit-Hartenberg Parameters
Why Do We Need The DH Parameters?
• 6 DOF = 6 Homogeneous Transformation Matrices needed
• 3 displacements
• 3 rotations
• Very cumbersome
• Denavit Hartenberg reduces this to 4
• 2 translations
• 2 rotations
A Denavit-Hartenberg HTM
Always these HTMs and always in this order
• ai is the “link length”
• ai is the “link twist”
• di is the “link offset”
• qi is the “joint angle”
How is this possible?
• Two assumptions
• x1 is orthogonal to z0
• x1 is intersects to z0
Assigning Coordinate Frames
• More than one way to assign link frames such that the 2 DH
conditions are met
• End results, assuming the coordinate frames for the inertial and end
effector coordinate frames are the same, will always be the same
• Establish a base frame
• Assign zi to be the axis of actuation for joint i+1
• Example: make sure z0 is pointing in the direction of actuation of the first actuator
• Axis of rotation for revolute joint
• Axis of translation for prismatic joint
• Set axis xi so that it is perpendicular and intersects zi-1
• Work from the base frame toward the end effector
Rule 1 Example
• Set up the 0 frame
• z0 is the axis of actuation of
joint 1
• Remember, the z0 axis
points in the direction of
the joint that connects the
0 frame with the 1 frame
• Frame 0 is fixed to link 0
(the base frame)
Image from QUT robotics course
Suggested convention
• When the rules allow (i.e. just a suggestion)
• zn is the approach direction of gripper
• yn is the slide direction of gripper
• xn is by right hand rule
Image from QUT robotics course
Special Cases
• zi-1 and zi are not coplanar
• There exists a unique shortest line between zi-1 and zi that is perpendicular to
both
• This line segment is assigned to xi
• zi-1 and zi are parallel
• commonly choose xi to be directed from oi toward zi-1 or opposite of it
• zi-1 and zi intersect
• i should match i-1 when theta = 0
Conventions
• ai is the “link length” from zi-1 to zi in the xi direction
About i
frame!
• ai is the “link twist,” the angle from zi-1 to zi about the xi axis.
• di is the “link offset,” the distance from xi-1 to xi along zi-1
About i-1
• qi is the “joint angle,” the angle from xi-1 to xi about the zi-1 axis.
frame!
Example 1
• ai is the “link length” from
zi-1 to zi in the xi direction
• ai is the “link twist,” the
angle from zi-1 to zi about
the xi axis.
• di is the “link offset,” the
distance from xi-1 to xi
along zi-1
• qi is the “joint angle,” the
angle from xi-1 to xi about
the zi-1 axis.
Example 1 Continued
• Based on the DH parameters
• Write the DH Matrices
Example 1 Continued
• Based on the DH Matrices:
• Write the Homogeneous Transformation Matrix:
Orientation of the end effector frame
relative to the base frame
The x and y components of the
end effector in the base frame
Denavit-Hartenberg Mini-Quiz
• Why do we need to use the DH method?
• What homogeneous transformation matrices comprise the DH
matrix?
• Does the order of the HTMs matter? Why or why not?
• What two assumptions are critical for the DH method to work?
• Compare your answer’s with your neighbor’s
Example 2
x2
• Assign z axes in direction of
next joint axis of rotation,
starting at the base frame
• Assign x axis to be
perpendicular and intersecting
with previous z axis
y3
• z1 and z0 intersect
• Therefore, x1 must be orthogonal
to both
• z2 and z1 intersect…
y2
z1
x1, z2
x3
y1
z3
z0
• Assign y to follow right hand
rule
x0
y0
Example 2 Continued
x2
• ai is the “link length” from
zi-1 to zi in the xi direction
• ai is the “link twist,” the
angle from zi-1 to zi about
the xi axis.
• di is the “link offset,” the
distance from xi-1 to xi along
zi-1
• qi is the “joint angle,” the
angle from xi-1 to xi about
the zi-1 axis.
y2
z1
X1, z2
x3
y1
y3
z3
z0
x0
y0
DH Matrices
And then do a bunch of matrix multiplication…
Example 3
it makes it easy if we
keep x2 in the same
direction as x0 and x0
• ai is the distance from zi-1 to zi
in the xi direction
• ai is the angle from zi-1 to zi
about the xi axis.
• di is the the distance from xi-1
to xi along zi-1
• qi is the angle from xi-1 to xi
about the zi-1 axis
z0 points in the direction
of actuation of JOINT 1
It’s simple to keep the end
effector the same as the previous
end effector location
z1 and z0 intersect,
Therefore we should
make x1 and y1
match x0 and y0
Example 4
• ai is the distance from zi1 to zi in the xi direction
• ai is the angle from zi-1
to zi about the xi axis.
• di is the the distance
from xi-1 to xi along zi-1
• qi is the angle from xi-1
to xi about the zi-1 axis
Example 5
Patent application 20110137464