Robot Kinematics & Dynamics
Chains of transforms, the Jacobian as a map between velocity spaces, and why inertia depends on posture.
A manipulator is a chain of rigid transforms with a few parameters left free. That sentence connects this topic to the pose algebra earlier, and it is worth taking literally: forward kinematics is composition, the Jacobian is the derivative of that composition, and singularities are where the derivative loses rank.
The figures use a planar two-link arm throughout. It is the smallest system in which every phenomenon — multiple IK solutions, singularities, manipulability, configuration-dependent inertia — is genuinely present rather than simplified away.
- 01 Forward kinematics as composition Foundation 1 interactive 8′ Each joint contributes one transform. The tip pose is their product, and the product is the whole model. Repairs Learning Denavit–Hartenberg parameters as a procedure, without seeing that they are one particular way of writing a product of poses.
- 02 The Jacobian Core 11′ A linear map from joint velocities to tip velocities, whose singular values are the arm’s instantaneous capability. Repairs Knowing that $\dot{x} = J\dot{q}$ without a picture of what the columns, the rank, or the singular values mean physically.
- 03 Dynamics: why control is harder than kinematics Advanced 10′ Inertia depends on configuration, and the coupling terms mean no joint moves alone. Repairs Assuming each joint can be controlled as an independent second-order system with a fixed inertia.