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I paste this algorithm from textbook Modern Robotics P294 in chapter 8:

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This is coordinate invariance of Newton-Euler equations. In the textbook, they use the frame is centers of mass so the inertial matrix is simple and beautiful, but the result torque $\tau$ is applied in the centers of mass and NOT in the joint frame. so, when we use this frame, \tau maybe need some transformation from centers of mass to the joint frame?

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maybe need some transformation from centers of mass to the joint frame?

Isn't that what $A_i$ is? I don't have the book with me, but from your excerpt:

Let $A_i$ be the screw axis of joint $i$ defined in $\{i\}$

where it says at the top,

frames $\{1\}$ to $\{N\}$ [are attached] to the centers of mass of links $\{1\}$ to $\{N\}$

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  • $\begingroup$ Thank you! Looks like this. now I just confused the equations $\tau_i = F_i^TA_i$. the textbook says: since joint $i$ has only one degree of freedom, five dimensions of the six-vector $F_i$ are provided for free by the structure of the joint(it is clear when $A_i, F_i$ in the same frame), But how to explain $\tau_i = F_i^TA_i$ from physics, where $A_i$ in the joint frame, and $F_i$ in cm frames. $\endgroup$ – Ben Feb 26 at 3:35
  • $\begingroup$ I'm sorry to make some mistake in my above comment, $A_i$ is the screw axis of joint $i$ expressed in the ith center of mass frame, NOT in the joint frame. so $F_i^TA_i$ still in the center mass frame, and the $\tau$ $\endgroup$ – Ben Feb 26 at 3:54

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