Kiru Lab / Track

Embodied Control: Kinematics and the Inverse Jacobian

Where the derivative stops being an abstraction and starts moving a physical arm.

A robot arm is a chain of joints. Forward kinematics — joint angles to end-effector pose — is easy and has one answer. Inverse kinematics — a desired pose back to joint angles — is the hard direction, and the standard solution is to differentiate the forward map, get the Jacobian, and invert it iteratively. This is the same gradient machinery that trains a neural network, applied to geometry instead of loss. It is also, in outline, what the cerebellum appears to do when you reach for a cup.

The question this track answers

Given a place I want my hand to be, how do I decide what each joint should do?

By the end you can

Compose transforms and compute forward kinematics for a serial chain
Derive and interpret the manipulator Jacobian
Solve inverse kinematics with the pseudoinverse and with damped least squares
Explain singularities in terms of rank and conditioning
Use null-space projection to satisfy a secondary objective on a redundant arm

Come here from

The Mathematics of Learning Deep Learning

Scope

7 lessons, roughly 20 hours.

Module 1

Frames and Forward Kinematics

Coordinate frames, homogeneous transforms, and the easy direction of the problem.

Coordinate Frames and Homogeneous Transforms

Every rigid body carries a frame. A 4x4 matrix moves you between frames, and the algebra is matrix multiplication.

Forward Kinematics: The Easy Direction

Joint angles in, hand position out. One answer, always, computed by walking the chain.

Code Walkthrough · 35m

Module 2

The Jacobian

Differentiate the forward map and you get the matrix that relates joint velocity to hand velocity.

The Velocity Jacobian

The same object you met in track two, now with physical units: how fast the hand moves per unit of joint rate.

Code Walkthrough · 40m

Singularities and Conditioning

When the Jacobian loses rank, an entire direction of motion becomes unreachable and naive inversion explodes.

Module 3

Inverse Kinematics

The inverse Jacobian, damped least squares, null-space redundancy, and learned controllers.

Inverse Jacobian Methods

The centerpiece: solve the hard direction by iterating the easy one, exactly like gradient descent.

Hands-On Lab · 55m

Redundancy and the Null Space

With more joints than task dimensions there are infinitely many answers. Choose among them on purpose.

Learned Controllers and the Gap to Reality

When you learn the inverse map with a network instead of solving it, you inherit every failure mode from earlier tracks.

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