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 
where L.H.S. are the force terms, R.H.S. are the “Ma” terms, $M(q)$ is mass/inertia matrix and positive definite, 
is the control input, and 
maps the control input to .
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 .](//s0.wp.com/latex.php?latex=%5Cddot%7Bq%7D%3D+M%28q%29%5E%7B-1%7D+%5B+%5Ctau_g%28q%29+%2B+B%28q%29+u+-+C%28q%2C%5Cdot%7Bq%7D+%29%5Cdot%7Bq%7D%5D+%3D%5Cunderset%7Bf_1%28q%2C%5Cdot%7Bq%7D%29%7D%7B%5Cunderbrace%7BM%28q%29%5E%7B-1%7D%5B+%5Ctau_g%28q%29%C2%A0+-+C%28q%2C%5Cdot%7Bq%7D+%29%5Cdot%7Bq%7D%5D%7D%7D+%2B%5Cunderset%7Bf_2%28q%2C%5Cdot%7Bq%7D%29%7D%7B%5Cunderbrace%7BM%28q%29%5E%7B-1%7D+B%28q%29+%7D%7Du+.&bg=ffffff&fg=000&s=0 "\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 
has full row rank (or simply has full row rank since 
is positive definite and hence full-rank), then for any desired 
, we can achieve that by picking as
where 
is the pseudo-inverse of 
. We say such robotic system is fully actuated.
On the other hand, if 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.
![\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 .](http://s0.wp.com/latex.php?latex=%5Cddot%7Bq%7D%3D+M%28q%29%5E%7B-1%7D+%5B+%5Ctau_g%28q%29+%2B+B%28q%29+u+-+C%28q%2C%5Cdot%7Bq%7D+%29%5Cdot%7Bq%7D%5D+%3D%5Cunderset%7Bf_1%28q%2C%5Cdot%7Bq%7D%29%7D%7B%5Cunderbrace%7BM%28q%29%5E%7B-1%7D%5B+%5Ctau_g%28q%29%C2%A0+-+C%28q%2C%5Cdot%7Bq%7D+%29%5Cdot%7Bq%7D%5D%7D%7D+%2B%5Cunderset%7Bf_2%28q%2C%5Cdot%7Bq%7D%29%7D%7B%5Cunderbrace%7BM%28q%29%5E%7B-1%7D+B%28q%29+%7D%7Du+.&bg=ffffff&fg=000&s=0 "\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 .")
