Motion Planning via Manifold Samples

We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably simpler sampling-based approaches that are appropriate for higher dimensions. In order to facilitate the transfer of advanced geometric algorithms into practical use, we suggest taking samples that are entire low-dimensional manifolds of the configuration space that capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms for analysis of low-dimensional manifolds then provide powerful primitive operations. The modular design of the framework enables independent optimization of each modular component. Indeed, we have developed, implemented and optimized a primitive operation for complete and exact combinatorial analysis of a certain set of manifolds, using arrangements of curves of rational functions and concepts of generic programming. This in turn enabled us to implement our framework for the concrete case of a polygonal robot translating and rotating amidst polygonal obstacles. We demonstrate that the integration of several carefully engineered components leads to significant speedup over the popular PRM sampling-based algorithm, which represents the more simplistic approach that is prevalent in practice. We foresee possible extensions of our framework to solving high-dimensional problems beyond motion planning.Comment: 18 page

The dynamics of linearized polynomials

Let F = GF(q). To any polynomial G element of F[x] there is associated a mapping on the set I_F of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of the mapping for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem. Assume that G is not of the form x^(q^l), where l>= 0 (in which event the dynamics is trivial). Then, for every integer n >= 1 and for every integer k >= 0, there exist infinitely many mü element of I_F having preperiod k and primitive period n with respect to the mapping. Previously, Morton, by somewhat different means, had studied the primitive periods of the mapping when G = x^q - ax, a a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials

Primitivity, freeness, norm and trace

Given the extension E/F of Galois fields, where F = GF(q) and E = GF(q^n), we prove that, for any primitive b element of F*, there exists a primitive element in E which is free over F and whose (E, F)-norm is equal to b. Furthermore, if (q,n) unequal (3,2), we prove that, for any nonzero b element of F, there exists an element in E which is free over F and whose (E,F)-norm is equal to b. A preliminary investigation of the question of determining whether, in searching for a primitive element in E that is free over F, both the (E,F)-norm and the (E,F)-trace can be prescribed is also made: this is so whenever n>=9

Actions of linearized polynomials on the algebraic closure of a finite field

Let g and h be monic polynomials in F[x], where F is the finite field of order q. We define a dynamical system by letting the q-linearized polynomial associated with g act on equivalence classes of a certain F-subspace of the algebraic closure of F in which related elements of the closure lie in the same orbit under the action of the q-linearized polynomial associated with h. When h = x, this is equivalent to the system in which the dynamic polynomial g acts on irreducible polynomials over F as discussed in [CH], where a conjecture of Morton [M] was proved as regards linearized polynomials. A generalization of that result is proved here. This states that when g and h are non-constant relatively prime polynomials, then there are infinitely many classes with prescribed preperiod and primitive period in the (g,h)-dynamical system

Primitive normal bases with prescribed trace

Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a element of F be nonzero. We prove the existence of an element w in E satisfying the following conditions: (1) W is primitive in E, i.e., W generates the multiplicative group of E (as a module over the ring of integers). (2) the set {w^g I g element of G} of conjugates of w under G forms a normal basis of E over F. (3) the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q <= 97 and n <= 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically