Inverse kinematics: redundancy and numerical methods

Inverse kinematics answers: given a desired tool pose, what joint angles achieve it? This lesson covers existence, uniqueness, redundancy, and practical numerical approaches.

Figure

Many configurations, one target

A 2-link planar arm can reach almost every point in its workspace two different ways (elbow up / down). 7-DOF arms offer infinite continuous solutions.

Learning objectives

Explain why IK can have no solution, one solution, or many solutions.
State the role of the Jacobian in differential IK.
Describe null-space motion and one use for it.

Prerequisites

Forward kinematics lesson.
Partial derivatives at a “conceptual calculus” level.

Step 1 — IK as solving nonlinear equations

Let $\mathbf{f}(\mathbf{q})$ be the tool pose (or position) as a function of joints. IK seeks $\mathbf{q}$ such that

\mathbf{f}(\mathbf{q}) = \mathbf{g}^\ast

for a commanded goal $\mathbf{g}$ in task space.

Nonlinear + periodic joints ⇒ complicated solution sets.

Checkpoint: For a redundant arm, why is “pick any solution” underspecified?

Step 2 — Analytic IK vs numerical IK

Analytic: closed-form solutions exist for some architectures (e.g. many 6R arms with spherical wrists) — fast and predictable.

Numerical: iterate with gradients/Jacobians — general but can get stuck, oscillate, or violate joint limits without care.

Exercise: When would you prefer numerical IK even if analytic IK exists?

Step 3 — The Jacobian relates joint rates to tool velocity

Let $\mathbf{x}$ be a task-space representation (e.g. tool position in $\mathbb{R}^3$ or a minimal orientation parameterization).

\dot{\mathbf{x}} \approx \mathbf{J}(\mathbf{q})\, \dot{\mathbf{q}}

where $\mathbf{J}$ is the Jacobian (matrix of partial derivatives).

For small steps, differential IK uses:

\Delta \mathbf{q} \approx \mathbf{J}^{\dagger}\, \Delta \mathbf{x}

with $\mathbf{J}^{\dagger}$ a pseudoinverse (Moore–Penrose or damped).

Checkpoint: What does a singular configuration imply about $\mathbf{J}$ ?

Step 4 — Damped least squares (avoiding blow-ups)

Near singularities, naive pseudoinverses can produce enormous joint velocities. Damping trades off task error vs joint velocity magnitude:

\Delta \mathbf{q} = \mathbf{J}^\top (\mathbf{J} \mathbf{J}^\top + \lambda^2 \mathbf{I})^{-1} \Delta \mathbf{x}

You do not need to memorize the formula — you need to know why damping appears in real controllers.

Step 5 — Redundancy and null-space projections

If $\mathbf{J}$ has more columns than rows (redundant), there exists a null space of joint velocities that do not change the task to first order:

\mathbf{J}\, \dot{\mathbf{q}}_{\text{null}} \approx \mathbf{0}

You can use null-space motion to:

avoid joint limits,
minimize energy,
maintain manipulability.

Figure

Wiggle without moving the tool

Several elbow positions reach exactly the same end-effector pose. The leftover degrees of freedom are the null space — useful for dodging joints limits or obstacles.

Exercise: Describe a secondary objective you might optimize in null space while keeping the tool pose fixed.

Check your understanding

Why can IK have infinitely many solutions for a 7-DOF arm reaching a point in space?
What is a kinematic singularity in one sentence?
Why is IK harder than FK computationally in general?

Lab-style stretch goal (optional)

Implement Jacobian-transpose IK for a planar 3-link arm to follow a circle in the plane; log joint angles and task error per iteration.