Key Engineering Takeaways
- •The Geometric Jacobian maps joint velocities to end-effector twists: V = [v, w]ᵀ = J(q) · q̇.
- •By principle of virtual work, static joint torques required to exert end-effector wrench F are: τ = J(q)ᵀ · F.
- •At singular configurations, the Jacobian loses rank (det(J)=0), meaning the robot loses the ability to move in one or more Cartesian directions.
Prerequisites
- • Partial derivatives
- • Forward Kinematics
Formulating the Geometric Jacobian J(q)
The Jacobian is a $6 \ imes n$ matrix where the top 3 rows map to linear velocity $\mathbf{v}$ and the bottom 3 rows map to angular velocity $\oldsymbol{\omega}$.
Tags:#Jacobian#Singularities#Manipulability#Statics#Velocities#Robotic Arm