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 q

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, u is the control input, and B(q) maps the control input to q.

We can rearrange the above to

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

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

On the other hand, if 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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