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During the time of me learning the concept of ZMP, there are some questions arised. In the following I will discuss my current thoughts on some of those questions.
As we know, in order to compute torque we need to find distance vectors and in order to find distance vectors I need to know which axis or point we will rotate around. For a hinge joint it is obvious that our rotation axis is the hinge joint itself. On the other hand, when there is no obvious constraint that can be used as a rotation axis we simple choose center of mass. The questions is, why we don't always choose center of mass if we can always find the center of mass for a rigid body. The answer is that, center of mass does always work in torque/moment analysis regardless there is a constraint or not. We can think of a rigid body is consist of multiple particles with different weights. Center of mass indicates mass distribution, heavier particles are more reluctant to change its states, which results rotation. Thus it's reasonable to analyze rotation based on center of mass. The following is a toy example to demonstrate how center of mass will work when there is also a constraint presented. In a physics engine both an object's orientation and angular velocity can have an axis angle representation (a 3D vector) because both of them are rotations. In fact the difference between orientation and angular velocity is that, angular velocity is a rotation with a context of unit time and orientation is simply a rotation. When we pile two rotations, we can't simply sum the two vectors that represents the corresponding rotations simply because vector math is commutative, but composition of rotations is not, and we have to use matrix or quaternion to do so. One the other hand, we can pile two angular velocities by adding them together, and the proof in the following excerpt is straightforward.
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