Templates, languages, and topological theories

The template-anchor paradigm is central to the formal approach to hierarchical planning.  Roughly speaking, one wants to decompose the difficult problem of controlling a high degree-of-freedom dynamical system (e.g., a locomoting legged robot) into two pieces: (i) finding a controller for a low degree-of-freedom template system (e.g., a limit cycle for a gait), and (ii) finding a formal relationship between the template system and a higher-dimensional anchor guaranteeing that the anchoring behavior will approximate that of the template.   At the highest level of abstraction, one can consider sequences of symbolic commands as templates for the dynamical systems that implement them.  Our aim is to formalize empirically useful template-anchor relationships using the tools of topology and category theory, thereby enabling modularity and reuse of controllers.  This involves defining classes of hybrid systems amenable to composition and hierarchy, on scales ranging from the physical to the symbolic.


Jared Culbertson, Paul Gustafson, Dan Koditchek, Peter Stiller, Matt Kvalheim, Mee Seong Im, Remy Kaldawy, Mikhail Khovanov, Zachary Lihn


  • J. Culbertson, P. Gustafson, D. E. Koditschek, P. F. Stiller, Formal composition of hybrid systems, Theory and Applications of Categories 35 (2020), no. 45, 1634-1682.
  • M. Kvalheim, P. Gustafson, D. E. Koditschek, Conley’s fundamental theorem for a class of hybrid systems, SIAM J. Appl. Dynam. Syst. 20 (2021), no. 2, 784-825.
  • P. Gustafson, M. S. Im, R. Kaldawy, M. Khovanov, and Z. Lihn. Automata and one-dimensional TQFTs with defects. arXiv preprint arXiv:2301.00700 (2023).


  • ONR N000141612817, a Vannevar Bush Faculty Fellowship held by Koditschek
  • UATL 10601110D8Z, a LUCI Fellowship held by Culbertson
  • W911NF1810327 under the SLICE Multidisciplinary University Research Initiatives (MURI) Program
  • AFRL grant FA865015D1845 (subcontract 669737-1)