12,980 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
Handling convexity-like constraints in variational problems
We provide a general framework to construct finite dimensional approximations
of the space of convex functions, which also applies to the space of c-convex
functions and to the space of support functions of convex bodies. We give
estimates of the distance between the approximation space and the admissible
set. This framework applies to the approximation of convex functions by
piecewise linear functions on a mesh of the domain and by other
finite-dimensional spaces such as tensor-product splines. We show how these
discretizations are well suited for the numerical solution of problems of
calculus of variations under convexity constraints. Our implementation relies
on proximal algorithms, and can be easily parallelized, thus making it
applicable to large scale problems in dimension two and three. We illustrate
the versatility and the efficiency of our approach on the numerical solution of
three problems in calculus of variation : 3D denoising, the principal agent
problem, and optimization within the class of convex bodies.Comment: 23 page
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