# Dynamic model of a robotic arm

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 ?

2. At low velocities $$\dot{q}<<$$ and low accelerations $$\ddot{q} <<$$, the first two terms are negligible, and thus you use the last equation.
3. $$\tau_{ext} = J^T F_{ext}$$ is also always true. It is the contribution of external forces. See this
4. 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
• 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 $J^TF_{ext} =\tau_{control}-G$ Jul 17 at 8:25