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It is known that with the help of the Jacobi matrix it is possible to transform the angular velocities of the drive links $\boldsymbol{\omega}$ into the angular velocity of the platform $\boldsymbol{\Omega}$ through the Jacobi matrix.

$\boldsymbol{\Omega}=\boldsymbol{J}\boldsymbol{\omega}$

where $\boldsymbol{J}$ - Jacobi matrix.

What matrix is used to transform the corresponding angular positions?

$\boldsymbol{\Theta}=?\boldsymbol{\theta}$

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You use Forward Kinematics to calculate the end-effector pose of your robot/platform, given the joint angles.

There are many different methods to calculate the Forward Kinematics, I remember using the Product of Exponentials method, which works well if you have a good visual model/description of your platform. You may use Geometric Methods and functions (trigonometric) if you have a fairly simple platform, such as a SCARA Robot. Also, I have never used it but I believe that the most common approach to Forward Kinematics is something called "Denavit–Hartenberg Parameters". You can make a search on these ones according to your platform and needs.

If you want to know about this because of a position-following application; for example if you want your platform to follow a position trajectory, and desire to calculate the joint angles; you use an iterative approach to achieve that. If that is the case you can have a look one of my previous answers, regarding a similar case by clicking here:

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  • $\begingroup$ This transformation is difficult to represent in the form of any matrix. $\endgroup$
    – dtn
    Oct 6 at 3:17
  • $\begingroup$ Why such a matrix does not exist for rotation angles? If $\dot{\theta}=J\dot{\omega}$, then $\theta=\int J\dot{\omega} dt$? Is it possible to find out what kind of formula (I mean, without integrals) lies behind $\int J\dot{\omega} dt$. $\endgroup$
    – dtn
    Oct 6 at 4:07
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    $\begingroup$ You wish to take the integral of the dot product of two matrices, right? Whether it is Jacobian, Inverse Jacobian, linear or angular velocities; they are matrices. When you wish to integrate a product of matrices, you do that by a some kind of "Product of Exponentials" formula. I actually never tried to integrate this expression, however, I strongly believe that it will point out to the relation between the Forward Kinematics via. Product of Exponentials and the Jacobian. I believe that this answer may show some guidance for you: math.stackexchange.com/a/537571/789117 $\endgroup$
    – kucar
    Oct 6 at 13:59
  • $\begingroup$ Well. I'll try to extract something useful. Thank you very much :) $\endgroup$
    – dtn
    Oct 6 at 14:15

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