Geometric mechanics and control

Differential Geometry provides a common formalism for analytical mechanics and nonlinear control theory. Problems concerning the control of mechanical systems with dynamics that hinge on conservation laws derived from symmetries, on velocity constraints that break symmetries, or on the interplay of the two can be framed elegantly and analyzed systematically using geometric constructs. Models for the locomotion of robotic vehicles can often be framed, for instance, in terms of connections in principal fiber bundles, and the related notion of geometric phase can provide the basis for algorithmic motion planning. Some basic manifestations of this idea are described in [1]. A canonical class of driftless systems admitting this formalism comprises jointed aquatic vehicles that operate at low Reynolds number or — like the three-link swimmer simulated here and realized in styrofoam here — that shed negligible vorticity. A geometric approach can also clarify the mechanics of phenomena like dissipation-induced self-recovery, seen here to limit the ability of another styrofoam robot to execute a turn by counter-spinning a momentum wheel, as analyzed in [2].

Aquatic propulsion via vortex shedding

Efficient macroscopic locomotion in water frequently entails vortex shedding. This is a fundamentally viscous phenomenon, but idealized models for propulsive vortex shedding can be realized in terms of inviscid hydrodynamics. A model for planar fish-like swimming is developed in [3], for instance, by amending a Hamiltonian model from [4] for the interaction of a free solid body with a distribution of discrete planar vortices to accommodate body deformation and a Kutta condition. Here's a simulation based on this model in which a single shape parameter serves as a control input. The compatibility of this model with control design is illustrated in [5]. An analogous model for jellyfish-like locomotion is developed in [6] from a Hamiltonian model in [7, 8] for the interaction of a free solid body with a distribution of closed vortex filaments. Substantial model reduction can be achieved by modeling the forces on a body resulting from vortex shedding without explicitly accounting for the corresponding fluid dynamics. The almost Hamiltonian structure and chaotic dynamics of a model of this kind are examined in [9] and [10].

Minimally actuated robotic locomotion

Simplicity can be a source of robustness in mechanical design, and nonlinear dynamics can afford significant leverage to a small number of inputs to a mechanical system. Students of nonlinear control are usually familiar with the problem of reorienting a spacecraft using only two internal rotors, exploiting the Lie bracket of two vector fields to achieve motion along a third, but even a single input can interact with nonholonomic constraints in dynamically rich ways. The system shown here, for instance, comprises a cart supported by two casters and a wheel — colloquially a Chaplygin sleigh — with a horizontal rotor on top. It's shown in [11] that such a cart can be induced to translate asymptotically in any direction at any speed by spinning the rotor appropriately, and that the same control principle — arguably an example of energy shaping — can be used to control the swimming of a single-input fish-like robot. A physical realization of the terrestrial robot (with simple hovercraft in place of casters) is shown here, propelling itself from rest as a consequence of underdamped heading control. Single-input systems provide a rarefied context in which to study issues like the comparative efficiency of impulsive actuation for mechanical propulsion. Here two versions of the cart described above accelerate very differently from rest, the rotor atop one spinning steadily and the rotor atop the other — dynamically related to the first through the variation of the Zeeman catastrophe machine — spinning fitfully. The superior performance of the latter mimics that of many biomechanical systems that exhibit the gradual accumulation and intermittent release of elastic energy. Here and here the single-input swimmner from [3] executes efficient maneuvers from rest involving fitful changes in camber.

Coordination of multi-vehicle systems

The problem of coordinating the behavior of several mobile robots is typically framed in terms of an underlying graph structure representing the topology of information flow within the system. An individual robot may receive commands from a centralized controller or from other robots and may respond to sensor-based assesments of its nearest neighbors' behavior. Decentralizing the control authority in such a system can provide robustness and adaptability. An approach to the decentralized control of robot vehicles moving in formation is developed in [12, 13, 14] from the notion that partial difference equations (PdEs) on graphs exhibit solutions analogous to solutions of partial differential equations (PDEs) in continuum mechanics. Command authority is limited to certain robots whose movements represent analogues of boundary conditions for PDEs like the heat equation or wave equation; the remaining robots respond according to solutions to analogous PdEs.

Mutualistic energy harvesting in schools and flocks

Free solid bodies that are near to each other in a fluid can influence each other's dynamics significantly. This principle is exploited by birds that glide in formation to reduce their collective drag, and animals or robots that swim in formation can harvest energy from each other's unsteady wakes similarly for collective benefit. This is apparent from the experiment shown here, in which three rubber paddles flap sinusoidally to propel a carriage across the top of a pool. Adjusting the spacing between the paddles — without altering the actuation of any individual paddle — can cause the average speed of the carriage to vary by a factor greater than three. At the level of an individual swimmer, participation in a mutualistic school can be viewed as a problem in feedback control. The movie here depicts a fish-like swimmer, initially at rest, varying a single shape parameter according to an elementary principle described in [15] to extract propulsive energy from an inverse Kármán vortex street modeling the wake of a preceding swimmer. When a system of mutualistic swimmers is considered collectively, the optimization of propulsive efficiency may be viewed as a problem in distributed control (akin to problems of coordination in the absence of hydrodynamic coupling) or in network synchronization (with the view that energetically preferable states are locally stable). The pairwise synchronization of swimmers as a result of energy transfer through their shared medium is akin to the synchronization of pendulum clocks observed by Huygens in the 17th century; the related entrainment of a following swimmer by a leader is akin to the entrainment of one wheeled robot by another that may occur when the two operate on a common compliant substrate [16].

Boundary control of material transport in fluids

If a fluid's boundary is deformed periodically in time, individual fluid particles generally won't describe periodic trajectories. Inertial bodies suspended in the fluid, furthermore, will be displaced neither periodically nor in the manner of fluid particles, but instead in a manner depending on their sizes, shapes, and densities. The local excitement of a fluid's boundary can therefore enable the controlled transport of fluid or the contact-free manipulation of objects suspended therein. This is the principle allowing sessile microorganisms like Vorticella to gather food by beating cilia near their mouths, and artificial cilia can be employed in engineered settings to mediate processes ranging from the high-throughput sorting of biological cells for disease diagnosis to the precision machining of brittle surfaces using abrasive slurries. At the extremes of Reynolds number, the manipulation of a fluid using an isolated vibrating cylinder admits a treatment in terms of geometric phase as in [17]. The dynamics of inertial micro-particles driven by the streaming flow near a vibrating cylinder at low but finite Reynolds number are characterized using a combination of analysis and numerics in [18], with additional computational studies of particle capture within multi-cylinder systems presented in [19]. A complementary physical experiment is described in [20] along with a discussion of reduced-order modeling and simple control; a streaming flow generated by the experimental apparatus depicted therein is visible here. Models for the dynamics of micro-particles advected by flows under boundary control can also describe the dynamics of sparingly actuated micro-vehicles advected by biophysical flows or (on a larger scale) robotic drifters advected by atmospheric or oceanic flows.

Nonlinear systems in physiology and medicine

Nonlinear dynamics arise in problems relevant to medicine from the biochemical scale to the scale of human populations. Mathematical models in these contexts often represent significant simplifications of very complex systems, but enable the model-based design of potentially robust strategies for human intervention. Models at the scale of disease physiology enable the systematic development of protocols for measurement-based clinical decision-making or even for automated drug delivery. A model of this kind for the physiology underpinning retinopathy of prematurity, a leading cause of blindness in children, is described in [21] along with a computational study of potential clinical mediations. Models at the epidemiological scale can inform feedback control though the dynamic allocation of financial and clinical resources to disease management. A model of this kind for the spread of HIV is described in [22].

References

Current courses

I'm scheduled to teach the following courses in Spring 2025:


MEGR 3122: Dynamic Systems II

MEGR 2240: Computational Methods For Engineers

Previous courses

In previous semesters at UNC Charlotte, I've taught these courses:

MEGR 2240: Computational Methods for Engineers
MEGR 3121: Dynamic Systems I
MEGR 3122: Dynamic Systems II
MEGR 3114: Fluid Mechanics
MEGR 3890: Reinforcement Learning for Robotics
MEGR 3890: Design and Dynamics of Vacuum Tube–Based Guitar Amplifiers
MEGR 7090/8090: Linear Vector Spaces and Tensors
MEGR 7090/8090: Geometric Methods in Mechanics, Dynamics, and Control I
MEGR 7090/8090: Geometric Methods in Mechanics, Dynamics, and Control II
MEGR 7090/8090: Advanced Engineering Mathematics I
MEGR 7090/8090: Advanced Engineering Mathematics II
MEGR 7090/8090: Nonlinear Dynamics and Chaos I
MEGR 7090/8090: Nonlinear Dynamics and Chaos II
MEGR 7090/8090/3890: Geometric Methods in Nonlinear Control and Robotics
MEGR 7090/8090: Nonlinear Control
MEGR 7090/8090: Introduction to Stochastic Processes
MEGR 7145/8145/3890: Advanced Topics in Dynamics
MEGR 7223/8223/3890: Mathematical Concepts for Dynamics and Control
MEGR 7224/8224/3890: Analytical Mechanics
MATH 8184: Differential Geometry I
MATH 8185: Differential Geometry II

I taught a subset of the same courses at the University of Illinois at Urbana-Champaign. When I was a graduate student, I also taught SAT prep courses and courses in high school algebra, geometry, calculus, and English composition for an educational consulting company.

Teaching philosophy

Maintaining transparent dialogue in the classroom

Even the most lucid and thorough of lecturers has experienced the disappointment — perhaps while grading an exam — of discovering that a student has failed to internalize a key point. The larger the classroom, the less feasible it is for an instructor to monitor individual students' degrees of engagement with course material, but students are often reluctant to assume responsibility for alerting the instructor when an important idea hasn't come across, even when only the briefest rephrasing or elaboration is needed. Students may be embarrassed to admit in front of their peers that they're lost — a possibility amplified in larger classes — or may be uncertain that the instructor will respond sympathetically and usefully to a plea for help.

I believe it's imperative to establish complete trust in the classroom, and I consciously dedicate time at the beginning of every new course to cultivating an atmosphere of informal dialogue, regardless of class size. Every student should recognize me as an unconditional ally to whom any question or comment, no matter how small or nonlinear, can be made without fear of awkwardness. It's obvious to every professor that there's mutual benefit in a student's interrupting a lecture for clarification, or that any question a student might ask is likely to be in the minds of additional students as well, but these facts aren't obvious to students unless they're pointed out. I think my success in this endeavor is apparent in the regularity with which students approach me with questions concerning courses other than my own, or even seek my counsel regarding topics unrelated to academics.

Emphasizing fundamentals and abstraction

Working example problems at the blackboard can be one of the most efficient ways to illustrate the practical application of engineering concepts. When I poll students for their opinions of my teaching methods, I consistently receive both praise for my frequent use of examples and entreaties for more examples still. There's a danger in emphasizing examples in engineering classes, however, in that students may come to believe that a course is about the examples themselves and not about the deeper concepts they're intended to illustrate. The potential for this misperception is increased when classroom examples, homework problems, and exam problems too closely duplicate one another. It's easiest for the student and for the instructor when only the numbers change and the figures do not (so to speak), but it's a disservice to students to release them into the workplace expecting to solve problems that look just like those they've solved before.

In developing a suite of examples to support my presentation of a concept, I seek to illustrate the diversity of circumstances under which that concept obtains. The problems my students confront in the course of homework assignments and exams — all of which I construct from scratch to complement one another — can't be solved by matching patterns of technique, but require abstract insight to identify appropriate techniques a priori. I'm explicit in class about my expectation that students should understand what they're doing and not just how to do it, and students come to recognize this added challenge as the path to a lasting reward.

Cultivating translational skills

One of the strengths of an engineering education is its scope. Students survey mathematics and the fundamental sciences, then advance to treatments of disparate sub-disciplines within their chosen fields, preparing to enter a professional community of increasing breadth. The sequencing of their courses is deliberate, designed to reinforce and build upon connections among different subjects, yet students are frequently apprehensive about applying knowledge from one course to solving problems in another without being told explicitly to do so. Differential equations arise as models for dynamic phenomena in a variety of engineering courses, for instance, yet students are loath to believe that the methods they've learned in mathematics classes to solve such equations constitute the "right" approach when attacking problems with physical significance. Indeed, students seldom complete a course with the feeling that they're responsible for retaining course material for future semesters, and it can be difficult for an instructor to hold students accountable for material taught in prerequisite courses without seeming unfair.

I believe that the improved coordination of course content across disciplinary lines should be a programmatic priority. I've sought to address this issue both by volunteering for inter-departmental liaison positions and — more directly — by volunteering to teach two-semester sequences of courses that would traditionally be divided among faculty with different disciplinary affiliations. At the undergraduate level, for instance, I've repeatedly taught the junior-level sequence comprising introductory dynamics and introductory fluid mechanics. During the second of these courses, I deliberately present problems that hinge on concepts from the first. One illustration: A popular lecture of mine teaches students about the interplay of aerodynamic lift, wing-wake interactions, and gyroscopic effects responsible for the twisted flight of a boomerang. Students invariably elect to participate in the optional building and throwing workshop I offer following the lecture.

My academic ancestry includes a number of luminaries, and some of my research interests can be traced directly to work done by some of them more than a century ago. My PhD advisor was Richard Murray, whose advisor was Shankar Sastry, whose advisor was Charles Desoer, whose advisor was Robert Fano, whose advisor was Ernst Guillemin, whose advisor was Arnold Sommerfeld, whose advisor was Ferdinand von Lindemann, whose advisor was Felix Klein, whose advisors were Julius Plücker and Rudolf Lipschitz. Plücker’s advisor was Christian Gerling, whose advisor was Carl Friedrich Gauss, whose advisor was Johann Friedrich Pfaff, one of whose advisors was Johann Bode. One of Lipschitz’s advisors was Lejeune Dirichlet, whose advisors were Simeon Poisson and Jean-Baptiste Fourier. Fourier’s advisor was Joseph-Louis Lagrange and Poisson’s advisors were Lagrange and Pierre-Simon Laplace. Laplace’s advisor was Jean Le Rond d’Alembert. Lagrange’s advisor was Leonhard Euler, whose advisor was Johann Bernoulli, one of whose advisors was Jacob Bernoulli (does that seem fair?), whose advisor was Nicolas Malebranche, whose advisor was Gottfried Leibniz, whose advisors included Christiaan Huygens and Erhard Weigel. One of Huygens’s two academic grandparents was Marin Mersenne. One of Weigel’s academic great-great-great-grandparents was Nicolaus Copernicus. My postdoctoral advisor Jerry Marsden’s PhD advisor, meanwhile, was Arthur Wightman, whose advisor was John Archibald Wheeler, whose advisor was Karl Herzfeld, one of whose advisors was Sommerfeld. One of Herzfeld’s other advisor’s academic grandparents was August Kundt, whose advisor was Gustav Magnus.

Somewhat relatedly, my Erdös number is currently four. I wrote a paper with Jerry Marsden, who wrote a book with Michael Hoffman, who wrote a book with Allen Shields, who wrote a paper with Paul Erdös.

Additionally…

I held a visiting faculty position in the department of Systems and Control Engineering at IIT Bombay for the 2014-15 academic year.

At UNC Charlotte, I served as director of my department’s graduate programs in 2010 and 2011 and oversaw this seminar series.

I belong to several professional societies, including (consistently) SIAM, AMS, and MAA and (off and on) ASME, APS, IEEE, and AAAS. I've served as Secretary/Treasurer (2009-10) and President (2010-11) of SIAM-SEAS.

I've given more than eighty different presentations at scientific meetings and more than fifty invited seminars at different academic institutions.

I also belong to the Prometheus Society and I was Time magazine's Person of the Year in 2006, but spinors still confuse me.