Melvin Leok’s Professional Blog

My first PhD student, Taeyoung Lee, successfully defended his PhD thesis entitled, “Computational Geometric Mechanics and Control of Rigid Bodies,” on May 8, 2008.

He was jointly co-advised by Prof. N. Harris McClamroch. The other members of the doctoral committee were Prof. Anthony Bloch, Prof. Jessy Grizzle, and Prof. Daniel Scheeres.

Amongst Taeyoung’s many honors, he recently received the Distinguished Achievement Award and the Ivor K. McIvor Award (outstanding research in applied mechanics) from the College of Engineering, University of Michigan.

Taeyoung will be starting an assistant professorship in the Department of Mechanical and Aerospace Engineering at the Florida Institute of Technology in Fall of 2008.

Diana Sosa Martín, who recently received her PhD in Mathematics from the University of La Laguna in Spain, will be joining Purdue University as a visiting assistant professor for the 2008/2009 academic year. Her research interests include geometric mechanics on Lie groupoids and algebroids.

Tomoki Ohsawa, a mathematics graduate student at the University of Michigan, will be visiting the group during the summer of 2008. His advisor at Michigan is Prof. Anthony Bloch.

Charles Roldan, an undergraduate mathematics major and physics minor, will be returning in his second summer as a NSF REU student.

Wooi-Chen Ng, an undergraduate computer science major, will be joining the group as a summer undergraduate research assistant.

For candidates interested in joining the Computational Geometric Mechanics @ Purdue group, research positions for postdoctoral scholars and graduate students in the broad area of geometric numerical methods in geometric mechanics and control remain available. Please contact Prof. Melvin Leok for further details.

[ PDF | arXiv:0803.1515 [math.DS] ]

This paper introduces a global uncertainty propagation scheme for rigid body dynamics, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, as it allows one to consider probability densities that are global, and are not supported on only a single coordinate chart on the manifold. The use of Lie group variational integrators, that are symplectic and stay on the Lie group, as the underlying numerical propagator ensures that the advected probability densities respect the geometric properties of uncertainty propagation in Hamiltonian systems, which arise as consequence of the Gromov nonsqueezing theorem from symplectic geometry. We also describe how the global uncertainty propagation scheme can be applied to the problem of global attitude estimation.

I have received a National Science Foundation Faculty Early Career Development (CAREER) Award in support of my work on “Computational Geometric Mechanics: Foundations, Computation, and Applications.” This is being funded by the Applied Mathematics Program of the Division of Mathematical Sciences.

NSF DMS-0747659
CAREER: Computational Geometric Mechanics: Foundations, Computation, and Applications
Melvin Leok, Purdue University

Abstract:

Symmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes. The proposed research will combine theoretical and computational tools arising from Dirac mechanics and geometry, noncommutative harmonic analysis, and uncertainty quantification to dramatically extend the applicability of computational geometric mechanics and geometric control to engineering problems that evolve intrinsically on nonlinear spaces, such as Lie groups and homogeneous spaces. This will provide insights into the canonical discretization of Dirac constraints, nonholonomic constraints, and interconnected systems. In addition, the study of uncertainty in the context of geometric control will improve the robustness and reliability of the resulting numerical and computational tools.

This research will improve our ability to control interconnected systems of autonomous vehicles in a robust and efficient fashion, by explicitly taking into account the uncertainty inherent in our knowledge of the surrounding environment. Our results will be applicable to the control of distributed sensor networks, consisting of an interconnected set of satellites, unmanned aerial vehicles and underwater vehicles. Such sensor networks are an exciting new development in the field of remote sensing that has the potential to dramatically increase the efficiency, coverage, and reliability of the information we obtain about our oceans, environment, and climate. More broadly, most complex engineering systems can be expressed as an interconnected system of more elementary components, and our mathematical framework will allow us to more readily understand complex systems in terms of the behavior of its component parts and the manner in which they are interconnected.

Synopsis of CAREER Program
The Faculty Early Career Development (CAREER) Program is a Foundation-wide activity that offers the National Science Foundation’s most prestigious awards in support of the early career-development activities of those teacher-scholars who most effectively integrate research and education within the context of the mission of their organization. Such activities should build a firm foundation for a lifetime of integrated contributions to research and education.

Research positions for postdoctoral scholars, graduate students, and undergraduates are available in the broad area of geometric numerical methods in geometric mechanics and control. A description of my current research may be found in my research statement and my bibliography.

These positions are funded in part by research grants from the National Science Foundation in Applied Mathematics and Computational Mathematics.

Please contact me by email if you are interested in any of these positions.

I will be giving a seminar during the Applied Mathematics pizza lunchtime seminar on Friday, November 16, 2007, from 11:30am to 12:15pm in REC 122. It will be geared towards first and second year graduate students. The title and abstract follows:

Lie Group and Homogeneous Variational Integrators and their Applications to Geometric Optimal Control Theory

Melvin Leok

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the optimal control of mechanical systems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton’s principle that preserves the Lie group structure of the configuration space. In contrast to traditional Lie group integrators, issues of equivariance and order- of-accuracy are independent of the choice of retraction in the variational formulation. The importance of simultaneously preserving the symplectic and Lie group properties is also demonstrated.

In addition, we will introduce a numerically robust shooting based optimization algorithm that relies on the conservation properties of geometric integrators to accurately compute sensitivity derivatives, thereby yielding an optimization algorithm for the control of mechanical systems that is exceptionally efficient. The role of geometric phases in these control algorithms will also be addressed.

Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometric exact rods, structures and mechanisms.

We will place recent work in the context of progress towards a coherent theory of computational geometric mechanics and computational geometric control theory, which is concerned with developing a self-consistent discrete theory of differential geometry, mechanics, and control.

The research has been supported in part by NSF grants DMS-0726263, DMS-0714223 and DMS-0504747.

I will be offering a graduate special topics course on geometric numerical integration in Spring 2008. Students who are interested are strongly encouraged to preregister for the course before the registration deadline of November 1, 2007, in order to ensure that the course has sufficient enrollment to be offered.

Course Description

MATH 692: Geometric Numerical Integration
Spring 2008, TTh 12pm-1:15pm

Target Audience:
This course is relevant to engineers, scientists, and mathematicians with an interest in long-time simulations of mechanical systems, including applications to robotic motion planning, astrodynamics, rigid-body, molecular and stellar dynamics. The application areas addressed will be tailored to the interests of the course participants.

Course Description:
Many differential equations of interest in the physical sciences and engineering exhibit geometric properties that are preserved by the dynamics. Recently, there has been a trend towards the construction of numerical schemes that preserve as many of these geometric invariants as possible.

Such methods are of particular interest when simulating mechanical systems that arise from Lagrangian or Hamiltonian mechanics, wherein the preservation of physical invariants such as the energy, momentum, and symplectic form can be important when simulating long-time dynamics of such systems.

In applications arising from astrodynamics and robotics, the dynamics evolve on nonlinear manifolds such as Lie groups, and in particular the rotation group, and the special Euclidean group. Numerical schemes that respect the underlying nonlinear manifold structure will also be discussed.

This course will begin with an overview of classical numerical integration schemes, and their analysis, followed by a more in depth discussion of the various geometric properties that are of importance in many practical applications, followed by a survey of the various geometric integration schemes that have been developed in recent years. Issues pertaining to the analysis and implementation of such schemes will also be addressed.

Background:
A strong undergraduate background in linear algebra, and differential equations; Some familiarity with numerical methods, classical mechanics and differential geometry would be helpful, but not essential; Programming experience in any language, e.g., C/C++, FORTRAN, MATLAB.

Instructor:
Prof. Melvin Leok, Department of Mathematics
Office: Math 430, phone: 49-63578, email: mleok@math.purdue.edu
http://www.math.purdue.edu/~mleok/

Textbook:
Simulating Hamiltonian Dynamics
Cambridge Monographs on Applied and Computational Mathematics),
Leimkuhler, Reich, Cambridge University Press, 2005. ISBN: 0521772907

Course Topics:
Numerical methods, Hamiltonian mechanics, Geometric integrators, Modified equations, Higher-order methods, Constrained mechanical systems, Rigid body dynamics, Adaptive geometric integrators, Highly oscillatory problems, Molecular dynamics, Hamiltonian PDEs.

Grading:
40 % — 6 Theoretical homework assignments (1 week assignments)
30 % — 3 Numerical implementation projects (2 week assignments)
30 % — 1 Final project involving a non-trivial application of geometric integrators.

[ PDF | arXiv:0709.2514 [math.OC] ]

A time optimal attitude control problem is studied for the dynamics of a rigid body. The objective is to minimize the time to rotate the rigid body to a desired attitude and angular velocity while subject to constraints on the control input. Necessary conditions for optimality are developed directly on the special orthogonal group using rotation matrices. They completely avoid singularities associated with local parameterizations such as Euler angles, and they are expressed as compact vector equations. In addition, a discrete control method based on a geometric numerical integrator, referred to as a Lie group variational integrator, is proposed to compute the optimal control input. The computational approach is geometrically exact and numerically efficient. The proposed method is demonstrated by a large-angle maneuver for an elliptic cylinder rigid body.

I was awarded a NSF Computational Mathematics grant as part of the focused topic area on Long Time Behavior (LTB) of Numerical Methods in Large Scale Scientific Computing. It is entitled Generalized Variational Integators for Large-Scale Scientific Computation, and involves combining variational integrator methods from geometric integration with techniques from numerical analysis, approximation theory, and high-performance computing.

[ PDF | arXiv:0707.0022 [math.NA] ]

Euler-Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two-spheres. The geometric structure of a product of two-spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler-Lagrange equations on two-spheres which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton’s principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems.

Animation

3-body problem on the sphere

Double spherical pendulum

Geometrically exact elastic rod model

Spherical pendula networks coupled with linear springs

2-dimensional array of magnets