Core Concepts
Extending the Control Contraction Metrics approach to Lie groups provides a framework for analyzing control systems with manifold structures.
Abstract
The paper extends the Control Contraction Metrics (CCM) approach to Lie groups, viewing manifolds as constrained sets. It formulates conditions for CCM existence and controller design, demonstrating applicability in various domains. The content is structured into sections covering general results, convexified conditions, CCM application on Lie groups, and treatment of abstract manifolds. Detailed mathematical derivations and examples illustrate the theoretical concepts presented.
I. INTRODUCTION
- Contraction analysis overview.
- Application of CCM in nonlinear control synthesis.
- Recent developments in contraction analysis.
II. PRELIMINARIES
- Variational dynamics calculation.
- Design of tracking controller using CCM approach.
- Formulation of sufficient conditions for CCM existence.
III. CONTROL CONTRACTION METRICS ON LIE GROUPS
A. General results
- Formulation of CCM on embedded submanifolds.
- Characterization of CCM search by convex conditions.
B. Convexified conditions
- Modification of conditions for numerical implementation ease.
- Application to systems evolved on Lie groups.
C. CCM on Lie groups
- Illustrative case study with O(2) × R as an example.
D. More abstract manifolds
- Intrinsic treatment of CCM on abstract manifolds.
- Convexification via musical isomorphism.
IV. CONCLUDING REMARKS
- Summary of extending CCM to Lie groups.
- Future directions for practical system trajectory tracking.
Stats
"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 834142 (ScalableControl)."
"Dongjun Wu is with Department of Automatic Control, Lund University, Box 118, SE-221 00 Lund, Sweden dongjun.wu@control.lth.se."
"B. Yi is with Department of Electrical Engineering, Polytechnique Montreal, QC H3T 1J4, Canada bowen.yi@polymtl.ca."
"I.R.Manchester is with Australian Centre for Robotics and School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Australia ian.manchester@sydney.edu.au."