Interpolation

If you know a value at a few points, interpolation estimates it in between. The simplest is linear: halfway between two known values, take their average. On a grid of two inputs, bilinear interpolation does this in both directions. For a point $(x, y)$ inside a cell with corners $f_{00}, f_{10}, f_{01}, f_{11}$ and fractions $s, t$ across it:

$$f(x, y) \approx (1-s)(1-t)\, f_{00} + s(1-t)\, f_{10} + (1-s)\,t\, f_{01} + s\,t\, f_{11}$$

Extrapolation uses the same idea past the last known point. It is far less safe: nothing checks the guess, and small trends in the data turn into large errors.

Blending whole poses this way has its own trap. Averaging two good poses joint by joint doesn’t always give a good pose: a knee can end up twisted, or a foot off the floor, in a pose neither original had.

In this project

Each technique’s reference motion is built as an aim table: the same strike re-solved for a grid of targets a little higher, lower, left and right (for the kicks ±10 cm each way, for the jab 5 heights × 5 sides × 5 depths), with inverse kinematics moving the striking limb. In training, the target is anywhere inside the grid, and the reference is interpolated between its nearest entries. The coach checks the grid’s corners and in-between poses on video before training, because that is where knees twist awkwardly. In the low and mid kicks the foot IK once flicked the knee in by about 40° over 4 frames, which had to be smoothed out before training.