661 research outputs found
The Boltzmann–Hamel equations for the optimal control of mechanical systems with nonholonomic constraints
In this paper, we generalize the Boltzmann–Hamel equations for nonholonomic mechanics to a form suited for the kinematic or dynamic optimal control of mechanical systems subject to nonholonomic constraints. In solving these equations one is able to eliminate the controls and compute the optimal trajectory from a set of coupled first-order differential equations with boundary values. By using an appropriate choice of quasi-velocities, one is able to reduce the required number of differential equations by m and 3 m for the kinematic and dynamic optimal control problems, respectively, where m is the number of nonholonomic constraints. In particular we derive a set of differential equations that yields the optimal reorientation path of a free rigid body. In the special case of a sphere, we show that the optimal trajectory coincides with the cubic splines on SO (3). Copyright © 2010 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/79427/1/1598_ftp.pd
Some applications of quasi-velocities in optimal control
In this paper we study optimal control problems for nonholonomic systems
defined on Lie algebroids by using quasi-velocities. We consider both
kinematic, i.e. systems whose cost functional depends only on position and
velocities, and dynamic optimal control problems, i.e. systems whose cost
functional depends also on accelerations. The formulation of the problem
directly at the level of Lie algebroids turns out to be the correct framework
to explain in detail similar results appeared recently (Maruskin and Bloch,
2007). We also provide several examples to illustrate our construction.Comment: Revtex 4.1, 20 pages. To appear in Int. J. Geom. Meth. Modern Physic
On the Dynamical Propagation of Subvolumes and on the Geometry and Variational Principles of Nonholonomic Systems.
Their are two main themes of this thesis. The first is the theory and application of the propagation of subvolumes in dynamical systems. We discuss the integral invariants of Poincare-Cartan and introduce a new and closely related set of integral invariants, those of Wirtinger type, and relate these new invariants to a minimum obtainable symplectic volume. We will then consider the application of this approach to the orbit determination and correlation problem for tracking particles of space debris. The second theme is on the geometry of nonholonomic systems. In particular we will focus on the precise geometric understanding of quasi-velocity techniques and its relation to the formulation of variational principles for these systems. We will relate the Euler-Poincar'e equations for Lie groups to the Boltzmann-Hamel equations and further extend both these equations to a higher order form that is applicable to optimal dynamical control problems on manifolds.Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58444/1/jmaruski_1.pd
Analytical investigation of laminar flow through expanding or contracting gaps with porous walls
AbstractLaminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions is investigated analytically using optimal homotopy asymptotic method (OHAM). OHAM is a powerful method for solving nonlinear problems without depending to the small parameter. The concept of this method is briefly introduced, and it׳s application for this problem is studied. Then, the results are compared with numerical results and the validity of these methods is shown. After this verification, we analyze the effects of some physical applicable parameters to show the efficiency of OHAM for this type of problems. Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate (α) and permeation Reynolds number (Re) on the velocity, normal pressure distribution and wall shear stress. The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fluids through contracting or expanding vessels
Jacobi multipliers and Hamel''s formalism
In this work we establish the relation between the Jacobi last multiplier, which is a geometrical tool in the solution of problems in mechanics and that provides Lagrangian descriptions and constants of motion for second-order ordinary differential equations, and nonholonomic Lagrangian mechanics where the dynamics is determined by Hamel''s equations. © 2021 IOP Publishing Ltd
Network Plasticity as Bayesian Inference
General results from statistical learning theory suggest to understand not
only brain computations, but also brain plasticity as probabilistic inference.
But a model for that has been missing. We propose that inherently stochastic
features of synaptic plasticity and spine motility enable cortical networks of
neurons to carry out probabilistic inference by sampling from a posterior
distribution of network configurations. This model provides a viable
alternative to existing models that propose convergence of parameters to
maximum likelihood values. It explains how priors on weight distributions and
connection probabilities can be merged optimally with learned experience, how
cortical networks can generalize learned information so well to novel
experiences, and how they can compensate continuously for unforeseen
disturbances of the network. The resulting new theory of network plasticity
explains from a functional perspective a number of experimental data on
stochastic aspects of synaptic plasticity that previously appeared to be quite
puzzling.Comment: 33 pages, 5 figures, the supplement is available on the author's web
page http://www.igi.tugraz.at/kappe
Dynamical formulations and control of an automatic retargeting system
The Poincare equations, also known as Lagrange's equations in quasi coordinates,
are revisited with special attention focused on a diagonal form. The diagonal
form stems from a special choice of quasi velocities that were first introduced by Georg
Hamel nearly a century ago. The form has been largely ignored because the quasi
velocities create so-called Hamel coefficients that appear in the governing equations
and are based on the partial derivative of the mass matrix factorization. Consequently,
closed-form expressions for the Hamel coefficients can be difficult to obtain
and relying on finite-dimensional, numerical methods are unattractive. In this thesis
we use a newly developed operator overloading technique to automatically generate
the Hamel coefficients through exact partial differentiation together with numerical
evaluation. The equations can then be numerically integrated for system simulation.
These special Poincare equations are called the Hamel Form and their usefulness in
dynamic modeling and control is investigated.
Coordinated control algorithms for an automatic retargeting system are developed
in an attempt to protect an area against direct assaults. The scenario is for
a few weapon systems to suddenly be faced with many hostile targets appearing together.
The weapon systems must decide which weapon system will attack which
target and in whatever order deemed sufficient to defend the protected area. This
must be performed in a real-time environment, where every second is crucial. Four different control methods in this thesis are developed. They are tested against each
other in computer simulations to determine the survivability and thought process of
the control algorithms. An auction based control algorithm finding targets of opportunity
achieved the best results
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