numerical inverse kinematics

Numerical inverse kinematics solves the backward problem — what joint angles reach the target? — not with a single magic formula but by educated trial and improvement. You start from some guess of the joint angles, use forward kinematics to see where the hand actually lands, measure how far that is from the goal, then nudge the joints a little in the direction that shrinks the gap. Repeat, and the hand creeps closer and closer until it is close enough. It is like adjusting a shower's hot and cold taps: feel the water, turn the knobs a touch, feel again, and home in on the right temperature.

The key to choosing each nudge intelligently is the Jacobian, a quantity that says, near the current pose, how a small turn of each joint moves the hand. Inverting that relationship tells the solver which way and how much to adjust the joints to step toward the target. The big advantage of this approach is generality: it works on almost any robot, including odd shapes and arms with extra joints that no neat formula can crack. The trade-offs are that it can be slower, it needs a decent starting guess, it may stall near awkward configurations where the Jacobian misbehaves (the singularities), and it usually returns just one solution rather than listing them all — so software often runs it from several starting guesses to find good options.