I have a question regarding the dynamic model of a robotic manipulator. It is commonly written as follows: $$ \tau = M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + J^\intercal F_{ext} $$ From what I have seen, the equation $\tau_{ext}=J^\intercal F_{ext}$ only holds true at very low velocities, so authors usually remove the velocity and acceleration terms and write: $$ \tau = G(q) + J^\intercal F_{ext} $$ So my question is, what happens when our task is highly dynamic with high velocities and accelerations ?
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1 Answer
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The first equation is always true. So at high velocities and accelerations, you use the first equation. See this
At low velocities $\dot{q}<<$ and low accelerations $\ddot{q} <<$, the first two terms are negligible, and thus you use the last equation.
$\tau_{ext} = J^T F_{ext}$ is also always true. It is the contribution of external forces. See this
There are control laws, when you want to exert a desired force $F_d$ at the environment, when the end effector does not move (thus $\dot{q}=0$ and $\ddot{q}=0$): $$\tau_{control} = J^T F_{d}$$ and this can be even more precise if you add the gravity term. This leads to your last equation. See this
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1$\begingroup$ So if I understand correctly, (3) is always true even at high velocities, but it is the (4) that assumes a quasistatic condition, right ? $\endgroup$ Jul 14 at 14:36
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$\begingroup$ Yes, but (4) is a control law, it doesn't describe the dynamics. The dynamics are again : $M\ddot{q} + C \dot{q} + G + J^T F_{ext}= \tau_{control} = J^T F_d $ . In quasistatic conditions, $\dot{q}$ and $\ddot{q}$ are negligible, and thus $J^TF_{ext} =\tau_{control}-G$ $\endgroup$ Jul 17 at 8:25