1. Robotics 1 Copyright Martin P. Aalund, Ph.D.
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

2. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Kinematics
Kinematics is the science of motion without regard to forces.
We study the position, velocity, acceleration, jerk etc of objects
Concerned with the location of Objects
We will define coordinate systems or frames to define there location
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

3. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Definitions
Velocity: The derivative of position with respect to time.
Acceleration: The derivative of velocity with respect to time.
Jerk: The derivative of acceleration with respect to time.
Link: Nearly rigid structure between joints.
Joint: Allow relative motion between links.
Joint Angle: Measurement of the relative position of two links
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

4. Definitions Continued
Joint Space: Relative coordinates that are referenced to coordinate frames at the robot joints.
Cartesian Space or Task Space. Global or base coordinate frame
Jacobian: Specifies a mapping of Velocities in joint space to velocity in Cartesian or Task Space.
Singularity: Region or point at which the Jacobian is singular.
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

5. Mathematical Background
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

6. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Forward Kinematics
Lets look at a simple link (1DOF)
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

7. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Forward Kinematics
Want to know the end point of link in terms of X and Y
We have R and Theta
From Geometry we can determine the position: and then the velocity
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

8. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Two Link Example x Y y q2 q1 Q2 R2 R1
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

9. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Two Link Example X Y x y q2 q1 Q2
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

10. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Coordinate Frames X1 Y1 Y0 R X0
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

11. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Coordinate Frames
We need to express R which we no in Frame 1 in Frame 0
Equation is independent of values Y0 X1 Y1 R q q1 X0
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

12. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Coordinate Frames
Now lets put it in matrix form
So what if we want to map the other way?
What is the inverse of T? Why?
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

13. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Coordinate Frames
If we look at the columns and rows of T we see that they have a norm of one.
Also if we take the dot product of the columns we find they are orthogonal to each other.
So T is an ortho-normal Matrices. Thus its transpose is its inverse.
This was a simple 2DOF example what about 3.
If we project a Z axes out the plane generated by the X and Y axes, then a rotation around the Z axes will not affect the Z position of the vector R.
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

14. Robotics 1 Copyright Martin P. Aalund, Ph.D.
Coordinate Frames
The 3D transformation axes about the Z axes is:
Similarly for rotations around the X or Y axes we get
Shorthand notation is often used cos(x)=c x sin(x)=s x
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

15. Coordinate Transformations
For multiple transformation we simply multiply by more matrices.
If we have multiple rotations about the same axes we can just add the angles of the matrices. Does this make sense Substituting we get
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

16. Coordinate Transformation
So now we can rotate a vector to and from an arbitrary angle in space.
What if we want to translate it
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

17. Translating Coordinate FramesX0 Y0 R X1 Y1
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.

18. Translating Coordinate Frames
Now we can translate and rotate. Y1 Y0 R X1 X0
January 17, 2000
Robotics 1 Copyright Martin P. Aalund, Ph.D.