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In point C.8 of this page of the book Robotics and Control it is stated that "the mass of a link used in a robot can be assumed as a point mass acting at the distal end of the link".

However, I cannot see why this assumption is true. For example, I would expect for a 2-link planar manipulator as the one shown below that, in the case of considering the mass of the links as point masses, their location should be the center of mass of the link, $\frac{a_1}{2}$ for link 1 and $\frac{a_2}{2}$ for link 2 and not the distal end of the link.

Figure

Furthermore, in Example 6.7 of the book Introduction to Robotics: Mechanics and Control the same assumption is made about the links present in the Figure.

Here is an extract of the example I'm referring to (I have wiped out some non relevant parts):

enter image description here

How is the stated assumption valid?

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If C8 applies to links with any distribution of mass, then the result in C8 seems to contradict the result in C7.

I think that C8 is not making a general statement which is true for any kind of link (with any distribution of mass). I think it is saying that this is a question about one kind of link (with mass concentrated at the distal end). Such a link could consist of a massless rigid rod with a point mass attached to the end. C8 is then asking you to find the inertia tensor for this case only. In comparison, C7 asks for the inertia tensor if the mass is assumed to be concentrated at the midpoint of a link.

Possibly in Robotics it is a common approximation that the mass is concentrated at the joints. That is where the "heavy machinery" is required - universal joints to swivel around multiple axes, and high-torque motors to power the joint. The link itself has no movement and could be a relatively light but strong rod.

So I think your question is really about the interpretation of the problem. It does not require an explanation using concepts of physics.

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  • $\begingroup$ sammy gerbil, this could cerainly be true, as no information about the link is given in that example. However, taking a look at the other example (from the book Introduction to Robotics: Mechanics and Control) I have just added to the question to me it seems that it is indeed an assumption which involves some physics concepts that I am not able to fully understand. I may be wrong, in which case I apologize for taking your time. $\endgroup$ – Ioannes Oct 25 '16 at 21:10
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    $\begingroup$ How do explain the difference between C7 and C8? The inertia tensor is different, but the frame of reference is the same. The difference is where the CM is located.... Your example clearly states that an assumption is being made that the mass is concentrated at the end of the link. This assumption is not necessary. The result C8 only applies when this assumption (mass at end) is made.... The assumption does not involve any physics concepts. It is made either for convenience, or because it is a good approximation to reality, but not because it is required because of physics concepts. $\endgroup$ – sammy gerbil Oct 25 '16 at 21:26
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Because one end is stationary, both an integration of the momentum, and an integration of the energy along the link can be simplified to a point mass at a given point along the link (except, of course, exactly at the stationary end). So the text says just put it at the distal end for simplicity. Consider that a tangential force at some point along the arm will result in some acceleration of that arm. You can certainly imagine some mass at some distance to provide that acceleration from that force.

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