Trunk lean

Stand up straight and your pelvis and chest are close to vertical. Lean back and both tip backwards. Arch your back and they disagree: the hips push forward while the chest tips back, and the lower spine bends between them.

To measure it we take a body part’s “up” axis $\hat z$ (a unit vector fixed to the pelvis or to the chest) and the direction the hips face on the floor, $\hat h$. The lean is the angle of $\hat z$ along $\hat h$:

$$\theta = \arcsin\!\big(\hat z_{xy} \cdot \hat h\big)$$

with $\theta > 0$ leaning forward and $\theta < 0$ leaning back. Measuring along the hips’ heading means a turn of the whole body doesn’t count as a lean.

For a fighter, leaning back matters twice. The centre of mass moves towards the back edge of the feet (the support polygon), so a small push can finish the job. And the head stays where it was while the weight goes back, so the fighter can’t move or answer for a moment.

In this project

The rear knee teacher kneeR3 leaned its chest and pelvis 12.4° (in the air) and 9.7° (at the bag) further back than its reference after the knee landed: the arch the coach saw on video. A reward term on the lean past the reference brought it to 1.3° and 0.9° in kneeR4. The full story is in Correcting the knee.