HCR - Chain of Joints

Let’s take a chain of joints:

The generic formula will be:$\left(\begin{array}{l}v\\ w\end{array}\right)=\overline{\stackrel{ˉ}{J}}\cdot \left(\begin{array}{l}{v}_{J1}\\ v_{J2}\\ v_{J3}\end{array}\right)$Where:

• $v$ and $w$ : velocity and angular velocity of the end-effector.
• $\overline{\stackrel{ˉ}{J}}$ : Jacobian matrix
• $v_{Ji}$ : velocity of joint $i$, more specifically:$v_{Ji}=\{\begin{array}{lc}{\dot{\Theta }}_i& \text{if joint i is an angular joint}\\ {\dot{d}}_i& \text{if joint i is a prismatic joint}\end{array}$Where:
  • $d_i$: distance of $i$-esim joint from $(i-1)$-esim joint.