You have a jointed arm — a character's shoulder, elbow and wrist, or a robot's servos — and a target in space you want the hand to touch. Forward kinematics answers the easy question: given the joint angles \theta, where does the hand x end up? Inverse kinematics (IK) asks the hard one in reverse: what angles put the hand there? For anything past two or three joints there is no tidy formula — so we iterate. We nudge the joints a little, look at how much closer the hand got, nudge again, and repeat until the hand arrives.

The engine of that nudging is the Jacobian J — the matrix that says how a small change in the joint angles produces a small change in the hand's position:

That is a simple linear equation. IK is nothing more than the art of solving it backwards — finding the \Delta\theta that produces the \Delta x we want — over and over. This page walks the whole family of ways to do that, from the crudest (transpose) to the one every production rig actually ships (damped least squares).

The update loop

Every Jacobian IK solver is the same four-step loop, run for a handful of iterations per frame:

Because J is only a local, linear snapshot of a curved relationship, no single big step lands the hand on the goal — that is why we take small steps and rebuild J each time. Everything below is a different answer to the one question in step two: given e and J, what is \Delta\theta?

Method 1 — the Jacobian transpose

The cheapest trick imaginable: don't invert anything, just multiply the error by the transpose of the Jacobian and scale it down:

with a small positive step size \alpha. This looks like a wild guess, but there is a beautiful reason it works. Define the goal-distance cost C(\theta) = \tfrac12 \lVert e \rVert^2 = \tfrac12 \lVert x_{\text{target}} - x(\theta) \rVert^2. Its gradient with respect to the joints turns out to be exactly \nabla C = -J^{T} e. So the transpose update is \Delta\theta = -\alpha \nabla C — plain gradient descent rolling downhill on the squared error.

The catch is that gradient descent is slow and zig-zaggy. Because it only ever knows the downhill direction — not how far to go — it can shuffle back and forth across a narrow valley, taking many iterations to converge, and the choice of \alpha is fiddly (too big and it overshoots and oscillates; too small and it crawls). It is a fine fallback and a great way to understand IK, but rarely the production choice on its own.

Method 2 — the Moore–Penrose pseudo-inverse

If J were square and invertible we would just write \Delta\theta = J^{-1} e and land the hand in one step. But a useful arm is redundant: it has more joints than the task has dimensions (think of all the elbow positions that keep your fingertip on the same spot), so J is a wide, non-square matrix with infinitely many solutions. The Moore–Penrose pseudo-inverse J^{+} picks a canonical one:

Through the SVD J = U\Sigma V^{T}, the pseudo-inverse simply inverts each singular value: \sigma_i \mapsto 1/\sigma_i. That is elegant — and it is also the seed of disaster. Near a singularity (the arm stretched straight, or folded flat) one singular value \sigma shrinks toward zero, so 1/\sigma rockets toward infinity, and \Delta\theta explodes.

Worked example: the blow-up at a singularity

Reduce the picture to a single troublesome direction — the near-singular one — so the algebra is scalar. Suppose the relevant singular value is \sigma = 0.05 (we are close to a singularity) and we ask the hand to move e = 0.10 units along that stiff direction.

Pseudo-inverse. It inverts the singular value:

Two radians of joint rotation demanded in a single frame — the arm snaps violently, overshoots, and the next frame it flails back. This is the classic IK "pop" near full extension.

Damped least squares (next section) replaces 1/\sigma with the bounded factor \sigma/(\sigma^2 + \lambda^2). Take \lambda = 0.10:

Same Jacobian, same error — but a fifth of the joint motion, and crucially a move that stays bounded no matter how small \sigma gets. The hand approaches the target a touch more slowly, but the arm never snaps.

Method 3 — damped least squares (DLS / Levenberg–Marquardt)

The fix that industry converged on adds a damping term \lambda inside the inverse:

This is exactly the Levenberg–Marquardt update from nonlinear least squares. Instead of demanding the hand motion be met exactly, it minimises a blended cost \lVert J\Delta\theta - e\rVert^2 + \lambda^2 \lVert \Delta\theta \rVert^2: the first term wants to reach the target, the second penalises big joint moves. In SVD terms the damaging map \sigma \mapsto 1/\sigma becomes the tamed

So \lambda is a dial trading a little accuracy for a lot of stability: too small and singularities still bite; too large and the solver becomes sluggish and never quite reaches the target. A small non-zero \lambda (often adapted on the fly, larger near singularities) is the industry-default IK solver.

Watch the damping tame the blow-up

Below is the scalar model from the worked example with unit error (e = 1). The horizontal reference line is the pseudo-inverse step 1/\sigma — flat, because it ignores damping entirely. The curve is the DLS step \sigma/(\sigma^2 + \lambda^2) as you sweep the damping \lambda. Drag the \sigma slider toward zero to walk into a singularity: the pseudo-inverse line shoots off the top of the chart, while the DLS curve — which meets it at \lambda = 0 — bends down and stays bounded the moment you give it any damping at all.

The gap between the flat line and the curve at a given \lambda is precisely the joint velocity that damping removes — the snap it prevents. Notice that when \sigma is large (away from any singularity) the two are nearly on top of each other: damping costs you almost nothing where you don't need it.

Nullspace projection — a second goal for free

A redundant arm has spare freedom: many joint configurations put the hand in the same place. The pseudo-inverse spends none of it (least-norm), but we can use it for a secondary goal — steer toward a comfortable pose, dodge a joint limit, keep the elbow up — without disturbing the hand. The tool is the nullspace projector:

This is how a character's whole body can shift naturally while a hand stays glued to a prop, or how a robot avoids its joint stops while its gripper tracks a path. Primary task first, personality in the nullspace.