Problem in Jacobian matrix

Question

I have a 5DOF robotic arm were 4 joints rotates around a Z axis and one around an x-axis. I am not that good with the Jacobian, but I think I have this. $$ J[v,\Omega]=[Z_{i-1}×(O_n-O_{i-1}), Z_{i-1} Z_{i-1}×(O_n-O_{i-1}), z_{i-1} ... Z_{n-1}×(O_n-O_{n-1})]. $$

I am a bit confused about Jacobian matrix on the part ($Z_{i-1}$), Eg say $Z_1=[0,0,1]$. whereby I know that "$Z$" is the axis of actuation. What about when one of the joint in my robotic arm rotates around an x-axis does it affect the matrix??

Is it correct to write, $X_{i-1}$, eg $X_1=[1,0,0]$?

Answer 1

Usually the axes of robots are determined using the so called Denavit Hartenberg convention. Based on this convention all actuated axes are Z axes (and the transformation matrices are defined in a way which enables this). This is why everybody assumes that the actuation axis is the Z axis (like in the formula above).

If you do not follow the DH Convention, you can have an actuation axis, which is not Z. In this case it is correct to use $X_i$ instead of $Z_i$. Your assumption, that is equals $[1, 0, 0]$ might be correct. If the $O$ matrices are $4x4$ in size, then $X$ would equal $[1, 0, 0, 1]$ (as in this case $Z$ would also be $[0, 0, 1, 1]$). Also, please make sure that the positive movement (rotation or translation) direction of the motion axis matches the positive direction of the geometrical axis.