q-state vector used to define the transformation matrix? how?

Question

How can it be used to determine the transformation matrix?

an example could be at computing the inverse kinematics for small displacements: J(q)$\Delta$q = $\Delta$u

$\Delta$U is a vector defining the difference between current and desired position. The desires position can always be computed, but if keep solving this in such manner that every time you solve $$J(q)\Delta q = \Delta u$$

you do this

• q:= q + $\Delta$q
• Compute $T_{base}^{tool}(q)$
• Compute the difference between $[T^{tool}_{base}]_{desired~position} $ and $T_{base}^{tool}(q)$.
• If change is less than 10^-5 finish and output Q, if not resolve.

How would you compute The transformation matrix based on q state vector.

Answer 1

The "state vector" $\vec q$ is the vector of joint angles. Use Craig's, or Paul's, book for understanding how to compute the transformation matrix that relates the end effector position and orientation, $\vec u$, to $\vec q$. You can also use 3x3 rotation matrices and a vector loop. If you don't have those references, just search for Denavit-Hartenburg.

After you have that transformation matrix, you can find $J$. $J$ is the partial derivative of your transformation matrix with respect to the individual joint angles. That should be enough to feed the algorithm you describe.

Answer 2

No, you cannot determine the forward kinematics transformation matrix from the Jacobian. You can however go the other direction (kind of). The analytical Jacobian is simply the partial derivatives of the forward kinematics equation. Which makes sense because the forward kinematics equation gives you position, and the Jacobian (the derivative) gives you velocity. But this is in general different than the geometric Jacobian. Which is typically what people use.

The geometric and analytical Jacobians differ in how they handle rotational velocity. The geometric Jacobian uses angular velocity $\omega$, which is a more physically intuitive representation. Whereas the analytical Jacobian uses the time derivative of Euler angles $\dot\phi$, which suffers from a representational singularity. Furthermore, you can integrate $\dot\phi$ over time to get $\phi$, however this cannot be done with $\omega$.

So if you have the forward kinematics equation, you can differentiate to get the analytical Jacobian, which you can then integrate to get back the forward kinematics equation, (but why?). But if you start from the geometric Jacobian, you cannot integrate to get the proper forward kinematics equation.

See Robotics: Modelling, Planning and Control, by Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, Giuseppe Oriolo. Chapter 3.6 for full discussion.