129,201 research outputs found
Motion Planning of Legged Robots
We study the problem of computing the free space F of a simple legged robot
called the spider robot. The body of this robot is a single point and the legs
are attached to the body. The robot is subject to two constraints: each leg has
a maximal extension R (accessibility constraint) and the body of the robot must
lie above the convex hull of its feet (stability constraint). Moreover, the
robot can only put its feet on some regions, called the foothold regions. The
free space F is the set of positions of the body of the robot such that there
exists a set of accessible footholds for which the robot is stable. We present
an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n))
space for n discrete point footholds where alpha(n) is an extremely slowly
growing function (alpha(n) <= 3 for any practical value of n). We also present
an algorithm for computing F when the foothold regions are pairwise disjoint
polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log
n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly
growing function). These results are close to optimal since Omega(n2) is a
lower bound for the size of F.Comment: 29 pages, 22 figures, prelininar results presented at WAFR94 and IEEE
Robotics & Automation 9
Topology of quasiperiodic functions on the plane
The article describes a topological theory of quasiperiodic functions on the
plane. The development of this theory was started (in different terminology) by
the Moscow topology group in early 1980s. It was motivated by the needs of
solid state physics, as a partial (nongeneric) case of Hamiltonian foliations
of Fermi surfaces with multivalued Hamiltonian function. The unexpected
discoveries of their topological properties that were made in 1980s and 1990s
have finally led to nontrivial physical conclusions along the lines of the
so-called geometric strong magnetic field limit. A very fruitful new point of
view comes from the reformulation of that problem in terms of quasiperiodic
functions and an extension to higher dimensions made in 1999. One may say that,
for single crystal normal metals put in a magnetic field, the semiclassical
trajectories of electrons in the space of quasimomenta are exactly the level
lines of the quasiperiodic function with three quasiperiods that is the
dispersion relation restricted to a plane orthogonal to the magnetic field.
General studies of the topological properties of levels of quasiperiodic
functions on the plane with any number of quasiperiods were started in 1999
when certain ideas were formulated for the case of four quasiperiods. The last
section of this work contains a complete proof of these results. Some new
physical applications of the general problem were found recently.Comment: latex2e, 27 pages, 7 figure
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