Underactuated robotics

Watched the first lecture of underactuated robotics by Prof Tedrake. It was great. His lecture note/book is available online. And the example code is directly available at colab.

So what is underactuated robotics? Consider a standard manipulator equation with state $latex q$

$latex M(q) \dot{q}+C(q,\dot{q}) \dot{q} = \tau_g(q) + B(q) u,$

where L.H.S. are the force terms, R.H.S. are the “Ma” terms, $M(q)$ is mass/inertia matrix and positive definite, $latex u$ is the control input, and $latex B(q)$ maps the control input to $latex q$.

We can rearrange the above to

$latex \ddot{q}= M(q)^{-1} [ \tau_g(q) + B(q) u – C(q,\dot{q} )\dot{q}] =\underset{f_1(q,\dot{q})}{\underbrace{M(q)^{-1}[ \tau_g(q)  – C(q,\dot{q} )\dot{q}]}} +\underset{f_2(q,\dot{q})}{\underbrace{M(q)^{-1} B(q) }}u .$

Note that if $latex f_2(q,\dot{q})$ has full row rank (or simply $latex B(q)$ has full row rank since $latex M(q)$ is positive definite and hence full-rank), then for any desired $latex \ddot{q}^d$, we can achieve that by picking $latex u$ as

$latex u = f_2^{\dagger} (q,\dot{q}) (\ddot{q}^{d} – f_1(q,\dot{q})),$ where $latex f_2^{\dagger}$ is the pseudo-inverse of $latex f_2$. We say such robotic system is fully actuated.

On the other hand, if $latex f_2(q,\dot{q})$ does not have full row rank, the above trivial controller will not work. We then have a much more challenging and interesting scenario. And we say the robotic system is underactuated.



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