1. This paper analyzes the motions produced by linear interpolation for physical
 correctness and suggests small modifications to the standard interpolation technique 
that in some circumstances will produce significantly more natural looking motion.

2. Straightforward linear interpolation could well introduce visually apparent errors in 
the physics of the motion.

3. The interpolated motion is  analyzed in terms of a number of basic physical properties: 
(1) linear and angular momentum during flight; 
(2) foot contact, static balance and friction with the ground during stance; 
(3) continuity of position and velocity between phases.

4. Analysis of the flight phase
(1) linear interpolation of the root position and the joint angles of the character can 
result in a non-linear trajectory for the center of mass. A simple fix is to interpolate
 the center of mass trajectories instead of the root positions. The root position can then
 be computed from the center of mass.

(2) Interpolate the center of mass positions and the rotation angles, the angular momentum
 of the interpolated motion is constant during flight.

5. Analysis of the contact phase
(1) Center mass and root interpolation can not preserve Non-sliding Foot Contact
 constraint.  A solution that preserves physics is to interpolate only the non-redundant
 degrees of freedom of the character and the constraints.

(2) Static balance exists when the projection of the center of mass of the character onto
 the ground is within the support polygon of the feet. Statically balance can be preserved
 during interpolation. 

(3) Friction constraint can be preserved during interpolation.

6. Transition between phases
Motions with rotation during the flight phasemay have significant discontinuities at the 
transition between flight and stance phases. To reduce these problems, the subsequent 
motion of the root (or center of mass) in the original motions can be rotated to align 
them with the interpolated motion at the end of the flight phase.

7. Key contributions
(1) during flight, interpolate the center of mass positions instead of the root positions; 
(2) during ground contact, interpolate the positions of the feet, the center of mass 
positions and all non-redundant degrees of freedom to prevent the feet from sliding on the ground; 
(3) Recompute the timing of each phase.

8. Limitations
(1) The analysis presented here only looked at physical correctness.
(2) Our analysis cannot be directly applied to methods with dynamic time-warping techniques.
(3) Ingore the physical property that the center of pressure should fall within the support polygon of the feet.