# Velocity Relation for Parallel Robots

In my course of "Advanced Robotics" with "Fundamental of Robotic Mechanical Systems" as the reference book I saw the following equation as the velocity relation for parallel manipulators such as the one depicted on the pic.

$$\mathbf{J}\mathbf{\dot{\theta}}=\mathbf{K}\mathbf{t}$$

$\mathbf{\dot{\theta}}$ is the joint rate vector and $\mathbf{t}$ is the twist.

where $\theta_{J1}$ for $1\leq J\leq 3$ is the actuator.

It was indicated there that usually the matrix $\mathbf{J}$ is a diagonal matrix referring to the borders of the work space. But the proof for this velocity relation was just given case by case. In another word there was no general proof of this equation.

So here are my questions

1. Is there a way to derive a general velocity formula for parallel manipulators ? (a formula that would show the relation between joint rates and twists).

2. Are there cases in which $\mathbf{J}$ is not a diagonal matrix ?