# How to compute associated (partial) Jacobian matrix for each joint of a serial manipulator?

Given DH parameters of a serial manipulator, the forward kinematics gives me

T0n = T01*T12*T23*...*Tn-1n , n is the DOF of manipulator.

I can calculate the Jacobian matrix J for the end-effector by taking first order partial directives.

Then I know the relation between joint speed and end-effector speed

eeSpeed = J*jointSpeed or jointSpeed = J_inv*eeSpeed

Now my question is how to calculate the Jacobian for joints 1 to n-1, and how to derive the relationship between joint speed and speed of any point on a robot link?

Well, the answer is quite straight forward. With the given equation $$T^n_0 = \ldots$$ you calculate the transformation from the $$n^\mathrm{th}$$ coordinate frame (attached to the end effector) to the inertial coordinate frame.
Just imagine you want to know the Jacobian from the joint $$n-1$$, you do the exact same procedure. Calculate $$T^{n-1}_0 = \ldots$$ and calculate the first partial derivative.
In other words, the Jacobian method is just the use of the chain rule from differentiation. Given the translational velocity $$v=\frac{\mathrm{d}}{\mathrm{d}t}p$$ of any point $$p$$. Any point $$p$$, expressed in the inertial frame depends on the joint configuration $$q(t)$$, so $$p(q(t))$$. Therefore you can apply the chain rule $$\frac{\mathrm{d}}{\mathrm{d}t}p = \frac{\partial}{\partial q} p \frac{\mathrm{d}}{\mathrm{d}t}q ~.$$ The partial derivative $$\frac{\partial}{\partial q} p$$ is called the Jacobian of the point $$p$$. If $$p$$ describes the end effector position, the Jacobian is called the manipulator jacobian.
Note: $$p$$, $$v$$ and $$q$$ are usually vectors, therefore the partial derivative yields a matrix.