4,426 research outputs found
Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
Variational integrators are well-suited for simulation of mechanical systems
because they preserve mechanical quantities about a system such as momentum, or
its change if external forcing is involved, and holonomic constraints. While
they are not energy-preserving they do exhibit long-time stable energy
behavior. However, variational integrators often simulate mechanical system
dynamics by solving an implicit difference equation at each time step, one that
is moreover expressed purely in terms of configurations at different time
steps. This paper formulates the first- and second-order linearizations of a
variational integrator in a manner that is amenable to control analysis and
synthesis, creating a bridge between existing analysis and optimal control
tools for discrete dynamic systems and variational integrators for mechanical
systems in generalized coordinates with forcing and holonomic constraints. The
forced pendulum is used to illustrate the technique. A second example solves
the discrete LQR problem to find a locally stabilizing controller for a 40 DOF
system with 6 constraints.Comment: 13 page
Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
Variational integrators are well-suited for simulation of mechanical systems
because they preserve mechanical quantities about a system such as momentum, or
its change if external forcing is involved, and holonomic constraints. While
they are not energy-preserving they do exhibit long-time stable energy
behavior. However, variational integrators often simulate mechanical system
dynamics by solving an implicit difference equation at each time step, one that
is moreover expressed purely in terms of configurations at different time
steps. This paper formulates the first- and second-order linearizations of a
variational integrator in a manner that is amenable to control analysis and
synthesis, creating a bridge between existing analysis and optimal control
tools for discrete dynamic systems and variational integrators for mechanical
systems in generalized coordinates with forcing and holonomic constraints. The
forced pendulum is used to illustrate the technique. A second example solves
the discrete LQR problem to find a locally stabilizing controller for a 40 DOF
system with 6 constraints.Comment: 13 page
SAT Modulo Linear Arithmetic for Solving Polynomial
Polynomial constraint solving plays a prominent role in several areas of
hardware and software analysis and verification, e.g., termination proving, program
invariant generation and hybrid system verification, to name a few. In this paper we
propose a new method for solving non-linear constraints based on encoding the problem
into an SMT problem considering only linear arithmetic. Unlike other existing methods,
our method focuses on proving satisfiability of the constraints rather than on proving
unsatisfiability, which is more relevant in several applications as we illustrate with
several examples. Nevertheless, we also present new techniques based on the analysis
of unsatisfiable cores that allow one to efficiently prove unsatisfiability too for a broad
class of problems. The power of our approach is demonstrated by means of extensive
experiments comparing our prototype with state-of-the-art tools on benchmarks taken
both from the academic and the industrial world
Polyhedral Approximation of Multivariate Polynomials using Handelman's Theorem
International audienceConvex polyhedra are commonly used in the static analysis of programs to represent over-approximations of sets of reachable states of numerical program variables. When the analyzed programs contain nonlinear instructions, they do not directly map to standard polyhedral operations: some kind of linearization is needed. Convex polyhe-dra are also used in satisfiability modulo theory solvers which combine a propositional satisfiability solver with a fast emptiness check for polyhedra. Existing decision procedures become expensive when nonlinear constraints are involved: a fast procedure to ensure emptiness of systems of nonlinear constraints is needed. We present a new linearization algorithm based on Handelman's representation of positive polynomials. Given a polyhedron and a polynomial (in)equality, we compute a polyhedron enclosing their intersection as the solution of a parametric linear programming problem. To get a scalable algorithm, we provide several heuristics that guide the construction of the Handelman's representation. To ensure the correctness of our polyhedral approximation , our Ocaml implementation generates certificates verified by a checker certified in Coq
Suspended Load Path Tracking Control Using a Tilt-rotor UAV Based on Zonotopic State Estimation
This work addresses the problem of path tracking control of a suspended load
using a tilt-rotor UAV. The main challenge in controlling this kind of system
arises from the dynamic behavior imposed by the load, which is usually coupled
to the UAV by means of a rope, adding unactuated degrees of freedom to the
whole system. Furthermore, to perform the load transportation it is often
needed the knowledge of the load position to accomplish the task. Since
available sensors are commonly embedded in the mobile platform, information on
the load position may not be directly available. To solve this problem in this
work, initially, the kinematics of the multi-body mechanical system are
formulated from the load's perspective, from which a detailed dynamic model is
derived using the Euler-Lagrange approach, yielding a highly coupled, nonlinear
state-space representation of the system, affine in the inputs, with the load's
position and orientation directly represented by state variables. A zonotopic
state estimator is proposed to solve the problem of estimating the load
position and orientation, which is formulated based on sensors located at the
aircraft, with different sampling times, and unknown-but-bounded measurement
noise. To solve the path tracking problem, a discrete-time mixed
controller with pole-placement constraints
is designed with guaranteed time-response properties and robust to unmodeled
dynamics, parametric uncertainties, and external disturbances. Results from
numerical experiments, performed in a platform based on the Gazebo simulator
and on a Computer Aided Design (CAD) model of the system, are presented to
corroborate the performance of the zonotopic state estimator along with the
designed controller
Keyframe-based visual–inertial odometry using nonlinear optimization
Combining visual and inertial measurements has become popular in mobile robotics, since the two sensing modalities offer complementary characteristics that make them the ideal choice for accurate visual–inertial odometry or simultaneous localization and mapping (SLAM). While historically the problem has been addressed with filtering, advancements in visual estimation suggest that nonlinear optimization offers superior accuracy, while still tractable in complexity thanks to the sparsity of the underlying problem. Taking inspiration from these findings, we formulate a rigorously probabilistic cost function that combines reprojection errors of landmarks and inertial terms. The problem is kept tractable and thus ensuring real-time operation by limiting the optimization to a bounded window of keyframes through marginalization. Keyframes may be spaced in time by arbitrary intervals, while still related by linearized inertial terms. We present evaluation results on complementary datasets recorded with our custom-built stereo visual–inertial hardware that accurately synchronizes accelerometer and gyroscope measurements with imagery. A comparison of both a stereo and monocular version of our algorithm with and without online extrinsics estimation is shown with respect to ground truth. Furthermore, we compare the performance to an implementation of a state-of-the-art stochastic cloning sliding-window filter. This competitive reference implementation performs tightly coupled filtering-based visual–inertial odometry. While our approach declaredly demands more computation, we show its superior performance in terms of accuracy
On processing development for fabrication of fiber reinforced composite, part 2
Fiber-reinforced composite laminates are used in many aerospace and automobile applications. The magnitudes and durations of the cure temperature and the cure pressure applied during the curing process have significant consequences for the performance of the finished product. The objective of this study is to exploit the potential of applying the optimization technique to the cure cycle design. Using the compression molding of a filled polyester sheet molding compound (SMC) as an example, a unified Computer Aided Design (CAD) methodology, consisting of three uncoupled modules, (i.e., optimization, analysis and sensitivity calculations), is developed to systematically generate optimal cure cycle designs. Various optimization formulations for the cure cycle design are investigated. The uniformities in the distributions of the temperature and the degree with those resulting from conventional isothermal processing conditions with pre-warmed platens. Recommendations with regards to further research in the computerization of the cure cycle design are also addressed
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