Resolution-Exact Algorithms for Link Robots Yi-Jen Chiang 1 * Zhongdi Luo 2 † Chee Yap 2‡ 1 Polytechnic Institute of NYU 2 New York University Algorithmic motion planning has had a 30-year run in which various approaches and theories have competed. Divergent paths have been taken by practical robotics and theoretical motion planners. There are three main approaches to algorithmic motion planning: exact, sam- pling and subdivision approaches [9]. The exact ap- proaches have been developed by Computational Ge- ometers [6] and in computer algebra [2]. However, their correct implementation is highly non-trivial because of numerical errors. Although an approach known as Ex- act Geometric Computation (EGC) can lead to correct implementation [5], it is still highly complex to imple- ment and expensive to compute. According to [13], there is no known good implementation of exact planners for more than 3 degrees of freedom (DOF). The sampling approach includes the famous PRM [7] framework and its many variants [10]. This is currently the dominant paradigm among roboticists. The subdivision approach is one of the earliest approaches to motion planning [3]. Recently, we have revisited this approach from a theoret- ical standpoint [11, 12]. What is new is the introduction of soft predicates and resolution-exactness [11, 12]: taken together, we can completely avoid exact computation. They lead to new classes of practical and theoretically sound motion planning algorithms. They seem to recover all the practical advantages of the PRM framework, but provide much stronger theoretical guarantees. In this paper we continue this line of research. In the robotics community, an informal measure of the practical success of any method is whether it “can solve” motion planning for various canonical robots. Since we currently lack appropriate complexity analysis, what is meant by “can solve” is that these methods terminate in reasonable time on judiciously chosen input obstacle environments. This is a reasonable way to try to un- derstand the limits and applicability of these methods. Canonical robots are first classified by the dimension- ality of the physical space (i.e., planar or spatial), and then by their degrees of freedom (DOF). Choset et al. [4] pointed out that sampling methods “can solve” (in the above sense) robots with medium to high DOFs; these are out of reach for exact methods. Subdivision meth- ods are said to reach medium DOFs (say 4-10 DOFs). In * Email: yjc@poly.edu. † Email: zl562@nyu.edu. ‡ Email: yap@cs.nyu.edu. particular, they noted that a certain link robot [8] with 9 DOFs can only be solved by sampling methods. The three approaches (sampling, subdivision and ex- act) provide increasing strength in their algorithmic guarantees. So the above empirical observations about their relative abilities is not surprising. Barring other issues, we should try to use the strongest algorithmic method available to solve motion planning problems for a given robot. For instance, existing exact techniques can solve for planar disc robots very efficiently, even with the correct implementation of exact predicates. What we argue for subdivision methods 1 is that (1) it is better fit for the requirements of robotics than exact methods, and (2) it avoids the halting problem of sampling meth- ods [11]. Hence our interest in developing the subdivision methods. We believe such methods “can solve” consider- ably higher DOFs than is often suggested. This cannot be done in simplistic ways: certainly we cannot afford to use a tree whose size is exponential in the depth d (cf. [1]). Our adaptive framework avoids this. Moreover, it is critical that the rotational DOFs be given a different treatment from translational DOFs. This paper makes a contribution of new techniques to this end. We follow the general framework of subdivision motion planning in [11, 12], where we draw attention to the roles of soft predicates and global search strate- gies. The present paper focuses on the 2-link planar robots with 4 DOFs. We make three main contributions: (A) Soft predicates for 2-link robots. As envi- sioned in [11], soft predicates can exploit a wide variety of techniques that trade-off ease of implementation against efficiency. Here we introduce the notion of length-limited forbidden angles for link robots. (B) A “T/R Splitting” technique based on splitting translational and rotational DOFs in different phases. Since a 2-link robot has 4 DOFs, naive subdivision would split each box into 2 4 = 16 children. This would con- siderably slow down the algorithm. A natural idea [11] is to consider two regimes: boxes are originally in the “large regime” in which we only split the translational DOFs. When the boxes are sufficiently small, in the 1 Properly construed. E.g., we should not use subdivision meth- ods as another way to solve exact problems, thus relying on exact predicates.