Inverse kinematics, four legs, and one very patient thesis advisor

Forward kinematics asks: given these joint angles, where does the foot end up? Easy. Inverse kinematics asks the useful question: I want the foot here — what angles get me there? That one has many answers, or none, and figuring out which is which took me most of a year.

Constraints come first

A leg cannot bend backwards. A knee has a range. The body has a mass that has to stay over the support polygon or the whole thing falls over. Before you solve anything you write down what is forbidden, and the forbidden list is what makes the solvable space small enough to reason about.

I write backends the same way now. Before a single endpoint, the constraints: what must never be true of this data? A booking cannot overlap itself. A balance cannot go negative. Put those in the schema, not in a validator someone will forget to call, and most of the bugs never get a chance to exist.