Jacobian: matrix gear ratio, effect on force/torque multiplication

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 20:22

Summary:

This video discusses the role of the Jacobian matrix $J$ in force/torque transmission in a robot arm (or in a generic mechanism, of course). In previous videos, chain rule justified that $ṙ=J\dot{q}$, being $ṙ$ end effector speed and $\dot{q}$ the joint speeds. Here, we discuss that $\tau =J^{T}f$ is the joint torque needed to compensate end-effector force $f$ in order to keep the robot arm motionless.

We do it in two ways: a first one, sort of informal, via an energy (power) conservation argumentation, extracting intuition from a pinion-gear thought experiment; the second one is a more accurate account of what happens, deriving the same expression $\tau =J^{T}f$ through the Euler-Lagrange equations of motion.

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