### Summary

In this paper, the authors analyze certain properties of interpolated human motion for physical correctness. More specifically, they analyze the flight phase of a motion paying particularl attention to linear and angular momentum of the center of mass. The authors observe that the Z component of the center of mass's trajectory should be equal to gravity, yet rarely is. In terms of angular momentum, the authors determine angular momentum is preserved in the interpolated motion if the principal axis of both rotations is the same.

They also analyze the contact phase of a motion, e.g. when feet are in contact with the floor. They assert that the feet of the character should not slide, that its center of mass should be in the support polygon formed by the feet, and that the contact forces should not require an unreasonable amount of friction. 

### Problem

Interpolation of physically correct motions does not necessarily result in a physically correct motion.

### Solution/Methods

The authors use simple laws of physics and observations to construct rules for verifying the physical correctness of an interpolated motion as well as provide modifications to interpolation methods in an attempt to preserve physical correctness.

### Contributions

1. By interpolating the center of mass rather than the root node, linear momentum of the center of mass is preserved, given that the input motions exhibited this property.

2. Angular moment of the system should be constant during flight. This is proved to be true if the principal axis of rotation in the two motions are the same (or there is no rotation in either motion).

3. Feet should not slide during contact with the floor. The authors propose interpolating only non-redundant degrees of freedom, i.e. those degrees of freedom not inhibited by constraints, as well as interpolating the constraints on each motion.

4. The authors prove that if the interpolated motions result in the projection of the COM on the ground is in the support polygon of the feet for the input motions, the same is true for the interpolated motions. Similarly, they prove that friction forces are physically correct in an interpolated motion if the same holds true for the input motions.

### Questions and Comments

The authors mention interpolating constraints, but I am curious how "correct" this is. In general, it makes sense, e.g. if the left heel should be at a certain position over a certain period of time in both motions, the interpolation point should probably make sense. It seems that in the case of orientations being interpolated, though, this may not always be the case though I can't come up with an example.

Also, I am curious how the authors deal with the case where there is a constraint on one motion but not the other.