Ep 3: Kinematics (Displacement Vectors)





Previously, when working with rotational motion, we dealt with rotation matrices. However, what about translational motion? Well, as stated in the previous post, we use something called, 'displacement vectors'.

Hence:

  • Rotational motion: Rotation matrices.
  • Translational motion: Displacement vectors.
In our case, when we consider the example of the 3 or 6-DOF robotic arm, what the displacement vector does is, give the 'coordinates' of the end defector, which really helps us determine the position of the robot.

However, unlike the rotation matrices, wherein we determined the rotation (with respect to the base frame) of the end defector, by multiplying the preceding rotational matrices from 0 (or base frame) to n-1 frame; where n is the end defector frame, one cannot get the displacement vector of the end defector (with respect to the base frame displacement vector) by plain multiplication. 

To compensate for that, we consider the case of Homogeneous matrices:
Fig 1: The above is a homogeneous matrix of frame 1 with respect to frame 0.
However, upon repeated multiplication, as we saw in the case of rotation matrices, we get
the homogeneous matrix of the end defector frame with respect to the base frame.
The benefit? We get the coordinates of the end defector, upon extraction 
of the displacement vector from the homogeneous matrix of the end defector frame with respect to the base frame. The figures below explain how the 'extraction' works.

Fig 3
Fig 2






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