Continuation is a numerical technique for computing discrete approximations of implicitly-defined manifolds, for example corresponding to parametrized families of dynamical-systems trajectories, given as the collection of solutions to a non-linear equation F(u)=0 in terms of a vector-valued function mapping F. Starting with a single chart, i.e., a point on the manifold together with a representation of the tangent space at this point, continuation employs a covering algorithm for computing nearby charts. The process is subsequently repeated for each of the nearby charts. The manifold through the initial point is called a branch or a family and the computed atlas of charts is a covering of this branch.
A number of computational tools for continuation and bifurcation analysis of characteristic classes of dynamical systems trajectories have been developed in the past. These include general algebraic and two-point boundary-value-problem solvers for ordinary differential equations, such as AUTO (and specialized drivers, such as HOMCONT, SLIDECONT, and TC-HAT), CONTENT, MATCONT, and SYMPERCON; boundary-value solvers for delay differential equations, such as DDE-BIFTOOL and KNUT; and tools for large-scale systems, such as LOCA.
The use of such computational tools, primarily applied to the bifurcation analysis of low-dimensional dynamical systems, as revolutionized the field of nonlinear dynamics and offers distinct advantages to brute-force forward-time simulation. Through a process of judicious exploration, such tools enable prediction of behavior and response without the need for a large collection of simulations based at distinct initial conditions. This methodology has extended the reach of methods of complex systems analysis far outside the confines of applied mathematics to a wide range of engineering, physics, chemical, and biochemistry applications.
With support from the National Science Foundation programs for Computational Mathematics and Dynamical Systems, I was the lead investigator on a research effort called "An Algorithm Suite for Computational Nonlinear Analysis of Power Systems," whose objectives were to scale continuation methods to complex networked systems with hybrid system trajectories and tens of thousands of states, by
- developing new multiscale, multisegment, trajectory-discretization algorithms based on asynchronous collocation methods;
- developing new mesh adaptation algorithms suitable for the asynchronous collocation methods that accommodate segment-specific discretization error bounds;
- constructing domain decomposition methods particular to the network topology and the asynchronous collocation formulation that enable efficient parallel execution.
This research project built on the Computational Continuation Core (informally known as COCO) recently documented in the extensive graduate-level textbook "Recipes for Continuation," published in May 2013 by the Society for Industrial and Applied Mathematics, and through a series of video tutorials.
With support from the National Science Foundation program for Integrated NSF Support Promoting Interdisciplinary Research and Education, I am the lead investigator on a research effort called "Asynchronous communication, self-organization, and differentiation in human and insect networks," whose objectives are the original development of a theory of parallel and asynchronous communication in complex human and insect networks and an unprecedented study of the implications of this framework for the robustness and resilience of such networks to external perturbations. The research explores the emergence of spatiotemporally correlated interaction patterns in uncoordinated, multi-agent networks and the formulation of a theoretical, experimental, and computational framework for characterizing the resultant parallel and asynchronous communication systems.
This highly interdisciplinary effort brings together experts in nonlinear dynamics, cognitive psychology, and entomology in a high-risk venture with high-reward implications for theories of human language and communication, self-organization and differentiation in complex communication systems, and social behavior in insect colonies.
Specific research activities include:
- high-resolution characterization of spatiotemporal interaction networks of bees in an experimental hive with emphasis on mature populations as well as the emergence of structure in artificially created single-cohort colonies;
- multimodal characterization of spatiotemporal interaction and coordination networks in small and large groups of humans engaging in virtual or actual multiplayer games with emphasis on phase transitions that distinguish between failed and successful coordination;
- development of computational algorithms and tools able to identify spatiotemporal differentiation of network agents and, in real time, long-range correlations that suggest opportunities for purposeful intervention; and
- construction of a paradigm of non-sequential grammars that addresses differentiation between content and function as well as modeling of the emergence of communication systems supporting such a grammar using dynamical systems methods.
Bioassistive device technology may be used to augment normal function, to prevent injury to function, and/or to rehabilitate injured function. Assistive augmentation may be introduced to directly enhance an operator's ability to perform a kinematic and/or kinetic task beyond the natural capabilities of the operator, for example, by injecting or dissipating kinetic energy at opportune moments during the task motion. When applied to the lower-limb musculoskeletal apparatus, exoskeletal devices may be designed to reintroduce or enhance propulsive joint actuation. In the context of injury, exoskeletal devices may support rehabilitation or sustained long-term function, e.g., travel by foot across greater distances or at enhanced speeds. Exoskeletal load-carrying devices may be designed to provide further support for extended ambulatory tasks. Assistive augmentation may also be introduced to compensate for environmental disturbances, thereby enabling the operator to perform tasks at a level commensurate with his or her natural capabilities in predictable environments. In irregular or adverse terrain, such exoskeletal devices may be used to enhance stance or gait stability, thereby augmenting the operator's ability to perform even normal musculoskeletal function.
With support from the NASA SBIR program through a collaboration with CU Aerospace, I was the lead university investigator on a research project called "Adaptive Bioassistive and Telerobotic Devices for Human-Robotic Systems," whose objectives were the design and control of adaptive bioassistive and telerobotic devices that augment normal musculoskeletal function in rapidly changing environments while providing a predictable dynamic response. We relied on principles of L1 adaptive control to decouple the task of adaptation to model and environmental uncertainty from the response perceived by the operator, thereby enabling the design of a range of apparently nonintrusive augmenting and telerobotic device technologies. The L1 paradigm significantly widens the domain of safe operation within which operator-induced instability can be eliminated without tuning.
With support from the USDA National Institute of Food and Agriculture, and in collaboration with Deere & Co, I am the principal investigator on a a research project titled "Cooperative Human-Robot Networks for Agricultural Applications." This project targets the theoretical development and experimental validation of control and planning architectures for cooperative networks of humans and robotic manipulators on mobile platforms in agricultural applications, using new insights from the field of adaptive control theory and scaled physical hardware realizations.
Emphasis is placed on two key applications in precision agriculture:
- semi-automated seeding operations with dedicated, robotic refilling vehicles and
- crop inspection and treatment operations in uncertain environments.
Versatile control designs are sought that sustain guaranteed and coordinated performance even in the presence of significant delays in network communication and actuation channels, and that do not depend on detailed model knowledge of individual nodes in the network. The proposed architectures provide a technology innovation in support of robust human-robot interfaces, as well as interchangeability of tools, platforms, and crops with successful operation across highly variable terrain.
Until recently, the field of nonlinear engineering dynamics has primarily focused on documenting and predicting phenomenology in existing systems and devices. Recent interest from the applied technology community in relying on system nonlinearities for improved performance brings both challenges and opportunities for the field. At a 2007 workshop on Applications of Non-Linear Dynamics in Nanomechanical Systems sponsored by the Microsystems Technology Office (MTO) of the Defense Advanced Research Projects Agency, the MTO program manager stated that the meeting sought to "[identify and assess]. . . unique and unexpected benefits that might accrue through effective exploitation of. . . nonlinear effects. . . " In particular, "rather than continuing to struggle to avoid nonlinearities. . . it is of interest to. . . effectively capitalize on these nonlinear phenomena."
The engineering of man-made devices to exhibit desirable nonlinearities has the potential to dramatically broaden the toolbox of the applied engineer and to change the performance characteristics in existing applications by orders of magnitude. Examples of such efforts are microoscillator mass sensors designed to exhibit hardening parametric resonance curves and smooth fold bifurcations; broadband, dissipative mechanisms realized experimentally in macroscopic devices through the purposeful introduction of essential nonlinearities; and radio-frequency microresonators that rely on high-velocity impacts for a sustained broadband response.
Contact interfaces and frictional/adhesive interactions, whether sustained or intermittent, are natural sources of nonsmoothness and nonlinearity in many natural and engineered mechanisms. With support from the National Science Foundation program for Dynamical Systems, I was the lead investigator on a research project called "Ultrafast and Robust, Resettable Threshold Sensors Based on Discontinuity-Induced Nonlinearities," whose objectives are to
- experimentally validate theoretical predictions of impact-induced instabilities in the threshold response of a macroscale, inductively-excited electromechanical circuit and demonstrate ultrafast switch rates due to the onset of low-velocity contact;
- experimentally validate theoretical predictions of contact-induced instabilities in the threshold response of a microscale, capacitively-excited electromechanical device and demonstrate the successful regulation of switch rates using active feedback control;
- formulate a fully nonlinear, large-deformation, continuum beam model for capacitively-excited microbeams and computationally investigate contact-induced instabilities and their regulation through passive design and active feedback.
A fascinating and unexpected theoretical discovery is the possibility of friction-induced reverse chatter to free flight, induced during slip transitions in sustained rubbing mechanical contact.