Fundamentals
Power is the rate at which work is done:
\[P = \frac{W}{T}\]Conservation of Energy
(Ignoring losses)
Power at the joints = Power at the tip.
Tip Velocity
\[v_{\text{tip}} = J(\theta) \, \dot{\theta}\]Power at Individual Joints
The power contribution from each joint is:
\[P_i = \tau_i \, \dot{\theta}_i\]Summing across all joints gives:
\[P = \tau^T \dot{\theta}\]Power at the End-Effector
At the end-effector, power is expressed in terms of force and velocity:
\[P_{\text{tip}} = f_{\text{tip}}^T \, v_{\text{tip}}\]Relating Torques and Forces
Substituting \(v_{\text{tip}} = J(\theta)\dot{\theta}\) into the conservation of power relation:
\[P_{\text{tip}} = f_{\text{tip}}^T \, J(\theta) \, \dot{\theta}\]Equating joint and tip power:
\[\tau^T \dot{\theta} = f_{\text{tip}}^T \, J(\theta) \, \dot{\theta}\]Since this must hold for all \(\dot{\theta}\):
\[\tau^T = f_{\text{tip}}^T \, J(\theta)\]Taking the transpose for conventional notation:
\[\tau = J(\theta)^T f_{\text{tip}}\]