81,012 research outputs found
From Uncertainty Data to Robust Policies for Temporal Logic Planning
We consider the problem of synthesizing robust disturbance feedback policies
for systems performing complex tasks. We formulate the tasks as linear temporal
logic specifications and encode them into an optimization framework via
mixed-integer constraints. Both the system dynamics and the specifications are
known but affected by uncertainty. The distribution of the uncertainty is
unknown, however realizations can be obtained. We introduce a data-driven
approach where the constraints are fulfilled for a set of realizations and
provide probabilistic generalization guarantees as a function of the number of
considered realizations. We use separate chance constraints for the
satisfaction of the specification and operational constraints. This allows us
to quantify their violation probabilities independently. We compute disturbance
feedback policies as solutions of mixed-integer linear or quadratic
optimization problems. By using feedback we can exploit information of past
realizations and provide feasibility for a wider range of situations compared
to static input sequences. We demonstrate the proposed method on two robust
motion-planning case studies for autonomous driving
Efficient Humanoid Contact Planning using Learned Centroidal Dynamics Prediction
Humanoid robots dynamically navigate an environment by interacting with it
via contact wrenches exerted at intermittent contact poses. Therefore, it is
important to consider dynamics when planning a contact sequence. Traditional
contact planning approaches assume a quasi-static balance criterion to reduce
the computational challenges of selecting a contact sequence over a rough
terrain. This however limits the applicability of the approach when dynamic
motions are required, such as when walking down a steep slope or crossing a
wide gap. Recent methods overcome this limitation with the help of efficient
mixed integer convex programming solvers capable of synthesizing dynamic
contact sequences. Nevertheless, its exponential-time complexity limits its
applicability to short time horizon contact sequences within small
environments. In this paper, we go beyond current approaches by learning a
prediction of the dynamic evolution of the robot centroidal momenta, which can
then be used for quickly generating dynamically robust contact sequences for
robots with arms and legs using a search-based contact planner. We demonstrate
the efficiency and quality of the results of the proposed approach in a set of
dynamically challenging scenarios
Heron triangles with two rational medians and Somos-5 sequences
Triangles with integer length sides and integer area are known as Heron
triangles. Taking rescaling freedom into account, one can apply the same name
when all sides and the area are rational numbers. A perfect triangle is a Heron
triangle with all three medians being rational, and it is a longstanding
conjecture that no such triangle exists. However, Buchholz and Rathbun showed
that there are infinitely many Heron triangles with two rational medians, an
infinite subset of which are associated with rational points on an elliptic
curve with Mordell-Weil group , and they observed a connection with a pair of Somos-5
sequences. Here we make the latter connection more precise by providing
explicit formulae for the integer side lengths, the two rational medians, and
the area in this infinite family of Heron triangles. The proof uses a combined
approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT)
maps in the plane, from several different viewpoints: complex analysis, real
dynamics, and reduction modulo a prime.Comment: Minor typos and one entry in Table 9 correcte
Indices of fixed points not accumulated by periodic points
We prove that for every integer sequence satisfying Dold relations there
exists a map , , such that
, where denotes the origin, and
.Comment: 11 pages, 2 figures. Final version to appear in Topol. Methods
Nonlinear Ana
Fast Non-Parametric Learning to Accelerate Mixed-Integer Programming for Online Hybrid Model Predictive Control
Today's fast linear algebra and numerical optimization tools have pushed the
frontier of model predictive control (MPC) forward, to the efficient control of
highly nonlinear and hybrid systems. The field of hybrid MPC has demonstrated
that exact optimal control law can be computed, e.g., by mixed-integer
programming (MIP) under piecewise-affine (PWA) system models. Despite the
elegant theory, online solving hybrid MPC is still out of reach for many
applications. We aim to speed up MIP by combining geometric insights from
hybrid MPC, a simple-yet-effective learning algorithm, and MIP warm start
techniques. Following a line of work in approximate explicit MPC, the proposed
learning-control algorithm, LNMS, gains computational advantage over MIP at
little cost and is straightforward for practitioners to implement
Szemeredi's theorem, frequent hypercyclicity and multiple recurrence
Let T be a bounded linear operator acting on a complex Banach space X and
(\lambda_n) a sequence of complex numbers. Our main result is that if
|\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is
frequently universal then T is topologically multiply recurrent. To achieve
such a result one has to carefully apply Szemer\'edi's theorem in arithmetic
progressions. We show that the previous assumption on the sequence (\lambda_n)
is optimal among sequences such that |\lambda_n|/|\lambda_{n+1}| converges in
[0,+\infty]. In the case of bilateral weighted shifts and adjoints of
multiplication operators we provide characterizations of topological multiple
recurrence in terms of the weight sequence and the symbol of the multiplication
operator respectively.Comment: 18 pages; to appear in Math. Scand., this second version of the paper
is significantly revised to deal with the more general case of a sequence of
operators (\lambda_n T^n). The hypothesis of the theorem has been weakened.
The numbering has changed, the main theorem now being Th. 3.8 (in place of
Proposition 3.3). The changes incorporate the suggestions and corrections of
the anonymous refere
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