Ep 4: Denavit-Hartenberg method of finding homogeneous matrices.

 

In the last post, I mentioned Homogeneous matrices. To be honest, they are a lifesaver. However, as we will learn at the end of this blog, the process of finding them is pretty trivial. In other words, we have a shorter process for finding these matrices, known as "The Denavit Hartenberg method." 

In this process, we don't calculate the rotation matrices and displacement vectors for each of the individual frames, and later calculate the homogeneous matrices from the base frame to the end effector by repetitive multiplications. Rather, we make a parameter table to calculate the values (elements) for the respective homogeneous matrices.

Points to be noted:-

• The number of columns in the parameter table is 4. The reason? The number of parameters being considered is 4. Namely, 'theta', 'alpha', 'r', and 'd'. Where theta and alpha are rotation parameters, while r and t are linear parameters.
• The number of rows is equal to n-1, where n is the number of frames being considered. (Frame 0, or the base frame is not considered individually in the parameter table).

The parameters:-

• Theta: Represents the rotation between individual joints, at the same time one needs to consider the rotation (if required) to bring the X-axis of the previous frame to align with the X-axis of the consecutive frame.
• Alpha: Represents the rotation (if required) to bring the Z-axis of the previous frame to align with the Z-axis of the consecutive frame.
• r: Distance between the centre of the previous and next frame, along the X-axis of the next frame.
• d: Distance between the centre of the previous and next frame, along the Z-axis of the previous frame.
  
  
  
• Fig 1: Finding the homogeneous matrix of the next frame with respect to the previous frame.

Fig 2: Multiplying each of the homogeneous matrices of frames with respect to the previous frame, gives the homogeneous matrix of the end defector frame with respect to the base frame.