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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.

enter image description here

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 ?

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  1. If you can write the forward kinematics equations of a parallel robot in an explicit form, you can derivate those equations and you get the formula for the velocities. This is generally valid approach for all robots, but the formulae obtained are only valid for the specific structure.

  2. If the jacobi matrix is diagonal it means that the motions of the robot are decoupled. i.e. if only one joint moves, only one component of the 6 (or 3 the planar case) velocities of the end effector will change. The ISOGlide type parallel structures have a diagonal jacobi matrix, as published here. Portal or Cartesian type structures, like this, also have a diagonal jacobi matrix. Delta robots, Stewart platfroms, Scara robots and 6 axis puma robots do not have a diagonal jacobi matrix.

(By looking at the schematic you attached, and not doing the math, I am not convinced that the robot you described has a diagonal jacobi matrix)

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  • $\begingroup$ Thank you so much for the answer. In the case of number one I'm more interested in a solution with geometrical interpretation (Like the solution that one would find on "Geometric Fundamentals of Robotics" about the serial manipulator's velocity relation). $\endgroup$
    – Arvin Rasoulzadeh
    Dec 18, 2016 at 20:55
  • $\begingroup$ and in the case of the robot depicted above, the Jacobian matrix J in fact is diagonal ... you can find the proof on page 407 of Angeles "Fundamental of Robotic Mechanical Systems". $\endgroup$
    – Arvin Rasoulzadeh
    Dec 18, 2016 at 20:57
  • $\begingroup$ Which edition of the mentioned book? $\endgroup$
    – 50k4
    Dec 18, 2016 at 20:59
  • $\begingroup$ third edition if I'm not wrong $\endgroup$
    – Arvin Rasoulzadeh
    Dec 18, 2016 at 21:01
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    $\begingroup$ This is a rather subjective advantage... $\endgroup$
    – 50k4
    Dec 19, 2016 at 4:21

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