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to reduce..

- the peak acceleration on the robot mechanism

- vibration in the structure of the robot

- the size of the motors required to drive the robot

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

path

a set of points in space 

trajectory

a set of points in space and a schedule for reaching each point

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

A singularity occurs when the middle rotation in the sequence makes the rotation axes of the first and third rotations parallel.

(A singularity in a three-angle sequence is sometimes referred to as gimbal lock.)

Think about the PUMA560 robot whose 4th joint and 6th joint are aligned. If 4th joint increases and 6th joint decreases with same rates at the same time, the orientation of end effector will not be changed.

So, we lose some degree of freedom!

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

For N-dimensional space the number of angles required is quadratic N(N-1)/2. The Wikipedia page for Euler angles has the explanation.

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

Extrinsic rotation (fixed angle)

• 12 representations

Intrinsic rotation

• Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)
• Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z).
  (cardan angles, roll-pitch-yaw, ...)

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

[2] http://en.wikipedia.org/wiki/Euler_angles

We can recognize the rotation matrix is the transposed matrix of the upper equation.

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn