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.




It is kind of mysterious that this works without using negative samples for self learning. See video and paper

  • The main idea is to train a representation network and a classifier so that the latter will predict the representation of an augmented data input.
  • The representation network for the augmented data has moving average parameter of the current representation. Similar tricks have been used in deep reinforcement learning
  • It is indeed quite surprising that this works without negative samples. Because there is nothing in the above model that avoids converging to trivial solution (everything maps to a constant)
  • Experimental results look good. But also may not be accounted for too much. Their implementation for some older approaches have way higher prediction performance. And they pulled numbers from papers (reasonable tho) for comparison. Approach is probably on par and without negative samples, they can train with a smaller batch size
  • They are using 512 TPUs for training for 7 hours…


video and paper.


  • Project embedding to lower dimension to save computational complexity and space
  • Some gain in speed but doesn’t look too significant. Tradeoff in performance seems larger than claimed
  • Theorem 1 based on JL-lemma did not used properties of attention itself. It seems that the same argument can be used to anywhere (besides attention). The theorem itself seems to be a bit a stretch
  • With the same goal of speeding up transformer, the “kernelized transformer” appears to be a better work
Copyright OU-Tulsa Lab of Image and Information Processing 2021
Tech Nerd theme designed by Siteturner