1. Homogeneous Transformation Matrices
3x3 Rotation Matrix
3x1 Displacement Vector
4x4 Homogeneous Matrix
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

2. Homogeneous Transformation Matrices
What does a pure translation look like
What does a pure rotation look like
Multiply to move from one coordinate to another.
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

3. Robotics 1 Copyright Martin P. Aalund, Ph.D..
D-H Parameters
Denavit-Hartenberg (D-H) are used to describe a robot link.
One Coordinate System is created for each link.
Each Axes is orthogonal
Use the right hand rule
Link are assumed to be rigid
Four Parameters are used
dn Distance Between Links
an Common Normal Distance
qn Joint Angle
an Offset Angle
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

4. Robotics 1 Copyright Martin P. Aalund, Ph.D..an
Common Normal Distance usually a positive value.
Is the distance from axes n to axes n+1 along the common perpendicular
Common perpendicular is shortest distance between two lines in space.
Does not necessarily lie in the physical link.
If axes intersect then it is zero.
Not defined for Prismatic, set to zero z1 z2 a1
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

5. Robotics 1 Copyright Martin P. Aalund, Ph.D..dn
Distance between Links
Measured from xn-1 to xn along zn
Joint variable for Prismatic joint y2 z2 z1 x2 d1 x1 y1
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

6. Robotics 1 Copyright Martin P. Aalund, Ph.D..qn
Joint Angle
The angle between xn-1 and xn
Measured around zn z2 z1 x2 q2 x2 x1 y1
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

7. Robotics 1 Copyright Martin P. Aalund, Ph.D..an
The Offset Angle
Measured from zn to zn+1 around xn
Usually a multiple of 90 degrees z2 z1 a1
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

8. Assigning Coordinate Frames
The base frame (0) is fixed and does not move. It is usually picked to be coincident with frame 1
First place the Z axes along the joint axes of rotation, or translation.
Next place the X axes along the common perpendicular between axes n and n+1
Use the right hand rule to define the Y axes.
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

9. Robotics 1 Copyright Martin P. Aalund, Ph.D..
Special Cases
If an=0 (the Z axes intersect) then pick Xn to be normal to the plane defined by Zn and Zn-1. This leads to two possible Definitions of X and thus Y.
If Zn and Zn+1 are parallel then dn is 0
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

10. Robotics 1 Copyright Martin P. Aalund, Ph.D..
DH Parameters
Make a Matrix of the DH parameters for all the links in the robot.
This will make defining the transformation from one axes to another much easier.
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

11. DH Based Transformation Matrices
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

12. Robotics 1 Copyright Martin P. Aalund, Ph.D..
Forward Kinematics
Used to mover from Joint coordinates to end-effector coordinates
Just Multiply the individual T matrices.
For a robot with N joints we will get N Transformations
If we are interested in a point beyond the last joint such as an end-effector or tool. We define a coordinate system for the Tool point of interest and create a transform from the last joint to the point.
We will find it helpful to define other points in the robot work space with coordinate systems relative to the robot base system.
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..

13. Standard Frames include
Wrist Frame {W} Same as the last robot frame I.e. Frame {N}
Base frame {B} Located at the base of the manipulator same as frame {0}
Station frame {S} Usually used as the base point for robot motions. Calculated relative to the Base Frame
Tool Frame {T} Could be tip of a torch, point between finger of a gripper or the lens of a camera. specified relative to the {W} Frame
Goal Frame {G} Location where the robot is desired to move. At the end of motion the {G} and {T} should coincide.
So to get from the Station frame to the toll frame we simply calculate: Where
January 24, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D..