So I empirically found that -(orth_speed^2)/(d) was preventing my pods from turning around checkpoints. And the taylor expand was a "proof" I found later on to convince myself that this was optimal. For example I had -orth_speed^2 for a while during the contest but looking at alot of replays it seemed like it needed a /d. For me the taylor expand thing made sense of my heuristic. Hoping it's clearer for you.