# Have some nuggets

What? These are some random thoughts, including open problems, musings, puzzles, and curiosity about research communities and opportunities. Enjoy!

**It is hard to find a full-range spherical joint.**(27 Aug 12)

In robotics, we study models of how attached bodies can move with respect to each other. A common one is the spherical or ball joint. It allows three rotational degrees of freedom, but in physical implementations there always seem to be joint limits that prevent one body from achieving*any*orientation with respect to the other. In the city center of Oulu, Finland, I saw this fountain, in which you can walk up and easily roll a gigantic stone ball into any orientation. So, it is the perfect, complete spherical joint, fully unleashing the power of SO(3)! The design is generally called a Kugel ball. Perhaps there is one near you.**Want me to Skype to your class or group?**(16 Jan 12)

At our faculty retreat, Danny Digs said that while teaching, he invited textbook authors to skype to his class for a brief conversation. I think that would be really fun. If any of you are teaching a class on planning algorithms and want me to drop in and virtually visit your classroom, just send me an email with the time and date. Even if it is just a group meeting or a few students who want to ask questions, just let me know. Alternatively, I wonder whether I could have some sort of on-line office hour devoted to planning questions. It would be interesting to try! Free free to send me an email with any suggestions.**Fast sensor preimage computations?**(14 Oct 11)

Ideal sensors can be modeled as a mapping from a state space*X*to an observation space*Y*. Given an observation*y*in*Y*, what is the set of possible state in*X*that could have produced it. This subset of*X*is called a*preimage*and each sensor mapping induces a partition of*X*into preimages. This leads to many important, fundamental questions. For a given sensor mapping, characterize the set of all preimages. Are they all homeomorphic? For an example, consider a point mobile robot with translation and rotation in a polygonal environment. If the sensor measures only the distance to the boundary in the direction the robot is facing, then what are the preimages (subsets of the 3D configuration space that produce the same distance observation)? They are usually two dimensional sets, but can we characeterize them for various polygons, or better yet, develop algorithms that can compute them? This case seems similar to motion planning for a line segment robot (see Section 6.3.4 of this chapter).**Beautiful or useful?**(23 Sep 11)

Students and colleagues sometimes ask me what research results I find interesting, either for my own goals or for evaluating other works. My answer is that it should be*beautiful*or*useful*. Note that this is**or**; very few results are both beautiful**and**useful. (Shannon's information theory comes to mind as an example of both.) Beauty is subjective, of course; nevertheless, beautiful research results, in my opinion, have properties like elegance, simplicity, and insight. Too often people are distracted by a third criterion, which is how*impressive*or technically difficult the research appears to be. If it looks technically difficult but has no use or beauty, then I am not very interested!**Open-loop control or perfect time feedback?**(3 Sep 11)

The most common form of a control law (or policy) in control theory is a function that maps from a state space*X*into an input space*U*. This is considered as*perfect state feedback*because the current state needs to be measured and known to determine which input*u*in*U*to apply. It is also the most common example of a*closed loop*system: The loop is formed by measuring the state and feeding it around as input to the policy. This is contrast to an*open loop*policy, which is a function that maps from a time interval [0,t] into*U*. This seems an odd name to me. Doesn't it imply that time is perfectly measured rather than state? In this case, I would recommend calling it*perfect time feedback*and it seems to be yet another example of a "closed loop". So, in the truly "open loop" case, both time and state must remain uncertain, right? Magnus Egerstedt and I wrote a CDC paper about this in 2007. There is so much more to do on this topic, though! What can be controlled in the truly open-loop case? More generally, shouldn't there be interesting and useful policies that map from subsets of the Cartesian product of*X*and [0,t] into*U*?**An RRT in a big disc has 3 main branches. Why?**(17 Jul 11)

If you grow a simple RRT with fixed step size from the center of a "very large" disc, simulations show that it will more or less create three main branches that are roughly spaced 120 degrees apart. See this picture. If the disc is large enough, then all of the random samples are so far away that they fall outside of the convex hull of the RRT vertices. Therefore, you can imagine the RRT in this case as a growing convex polygon. So, why does the hull tend to be an equilateral triangle? In*d*dimensions, the hulls even appear (in simulation) to form a regular*d*simplex (the higher dimensional generalization of the equilateral triangle). I made these observations around the year 2000. Why is this happening? UPDATE: Check out this 2012 paper.