IntermediateTopic 58 of 102Manufacturing, Robotics & Systems
Jacobian and singularities
Map joint velocities to tip velocity; singularities lose mobility.
In depth
The Jacobian J(q) maps joint velocities to end-effector velocity: v=J q̇; also static force mapping τ=Jᵀ F. Singularities occur where J loses rank (det J=0): the arm loses a direction of motion and joint velocities become unbounded. Manipulability measure √det(JJᵀ) quantifies dexterity.
Key points
- Velocity kinematics
- Fully extended arm is singular
- Damped inverse
ExampleA 2-link arm is singular when fully stretched or folded (θ₂=0 or π).
What is Jacobian and singularities?
Map joint velocities to tip velocity; singularities lose mobility.
Why is it important to understand jacobian and singularities?
The Jacobian relates joint speeds to tool speed, and singularities reveal positions where the robot loses the ability to move in some direction.
What is a singularity?
A configuration where the Jacobian loses rank, so the tool cannot move in one or more directions and joint speeds become very large.
How do I avoid singularities?
Plan paths that keep the arm away from fully stretched or aligned configurations, or use redundant joints to move around them.
How is Jacobian and singularities used in practice?
Typical applications include checking whether a robot can move its tool in a required direction at a given pose, avoiding a singular pose when planning a welding path and converting a desired tool velocity into joint speeds.
What should I learn before Jacobian and singularities?
Study "Inverse kinematics" first (topic 57 of 102 in Manufacturing, Robotics & Systems). It covers the groundwork this topic builds on.
What are the key points of Jacobian and singularities?
Velocity kinematics. Fully extended arm is singular. Damped inverse.
Can you give an example of Jacobian and singularities?
A 2-link arm is singular when fully stretched or folded (θ₂=0 or π).
What comes after Jacobian and singularities?
The next topic is "Robot dynamics", which builds directly on this one.
How difficult is Jacobian and singularities?
It is classed as Intermediate in Manufacturing, Robotics & Systems (topic 58 of 102). It is an intermediate topic. Work through the beginner topics in Manufacturing, Robotics & Systems first.