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You can calculate the inertia tensor for some primitive geometries of a linkage using known formula. Most of these are listed here. If you do not have a regular shape for which you can find the inertia tensor by a simple formula, you need to obtain this through integration (using the generalised formula). As this is not practicable by hand, most CAD programs ...


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To expand on my comment, the last line of your DH table gives the transform between END_EFFECTOR-1 and END_EFFECTOR frames, right? I'm saying instead of <θ2, 0, 0, 0> to have <180+θ2, 180, 0, 0>. There's no movement between your end effector and the object you're attaching to the end effector, so there's no point in keeping the end effector frame....


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You do not neceserily create an explicit C-Space representation which you sample. You have to choose, if you want to create your path in axis space or operational space (Cartesian space). This is a design choice and has some influence on the final shape of the trajectory. I am going to assume, that you plan in operational space (Cartesian-like). You sample ...


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The best (and cheapest) method would be to use AprilTags. I have added the python bindings for it here. For a great starting point, refer to this blog that explains the setup in C++. It can not only tell you the distance but also the orientation of the object I hope it helps.


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