Teaching > ECE1658
ECE1658 - Geometric Nonlinear Control of Robotic Systems (Winter 2025)
Instructor
M. Maggiore
Office: GB344
Email address: maggiore (at)
ece.utoronto.ca
Lectures
Course starts on January 8, 2025
| Mon 1PM-2:30PM | BA4164 |
| Wed 1PM-2:30PM | BA4164 |
Course Description
This course presents recent developments on control of underactuated robots, focusing on the notion of virtual constraint. Traditionally, motion control problems in robotics are partitioned in two parts: motion planning and trajectory tracking. The motion planning algorithm converts the motion specification into reference signals for the robot joints. The trajectory tracker uses feedback control to make the robot joints track the reference signals. There is an emerging consensus in the academic community that this approach is inadequate for sophisticated motion control problems, in that reference signals impose a timing on the control loop which is unnatural and inherently non robust. The virtual constraint technique does not rely on any reference signal, and does not impose any timing in the feedback loop. Motions are characterized implicitly through constraints that are enforced via feedback. Through judicious choice of the constraints, one may induce motions that are surprisingly natural and biologically plausible. For this reason, the virtual constraints technique has become a dominant paradigm in bipedal robot locomotion, and has the potential of becoming even more widespread in other area of robot locomotion. The virtual constraint approach is geometric in nature. This course presents the required mathematical tools from differential geometry and surveys the basic results in this emergent research area. Below is the lecture schedule:
- Introduction (1 lecture)
- Motivation for virtual constraints
- Virtual constraints versus motion planning
- Sample virtual constraints for the acrobot
- Unit 1: Robot modelling (4 lectures)
- Review of Euler-Lagrange robot models
- Modelling of impulsive impacts
- The falling stick example
- Models of planar bipedal robots
- The walking acrobot example
- Unit 2: Notions of geometric nonlinear control -- part 1 (3 lectures)
- Prelude to relative degree and feedback linearization
- Invariant and controlled invariant sets
- Prelude to zero dynamics manifold
- Initial definition of virtual holonomic constraint (VHC)
- Feedback controllers enforcing VHCs
- Unit 3: Notions of differential geometry (6 lectures)
- Smooth manifolds -- definition
- Topology of smooth manifolds
- Smooth maps between manifolds, diffeomorphisms, and smooth curves on manifolds
- Tangent spaces and tangent bundle
- The derivative map
- Smooth vector fields on manifolds
- Maximal integral curves of vector fields
- Inverse function theorem
- Integral curves of vector fields and flows of vector fields
- Lie derivatives and Lie brackets
- F-related vector fields
- Embedded submanifolds and the preimage theorem
- Unit 4: Notions of geometric nonlinear control -- part 2 (3 lectures)
- Geometric treatment of relative degree and feedback linearization
- Controlled invariant manifolds and zero dynamics of nonlinear control systems
- Normal form for input-output feedback linearization
- Dynamics on the zero dynamics manifold
- Example: bicycle moving along a circular path
- Unit 5: Virtual holonomic constraints (VHCs) (5 lectures)
- Geometric definition
- Regularity and enforcement of regular VHCs
- Constrained dynamics resulting from regular VHCs
- Conditions for Lagrangian constrained dynamics
- Qualitative properties of the constrained dynamics
- Unit 6: Virtual constraints for walking robots (2 lectures)
- Hybrid invariance
- Hybrid constrained dynamics
- Grizzle's conditions for existence and stability of hybrid limit cycles
Prerequisite:This course has no formal prerequisites, but assumes knowledge of vector calculus, linear algebra, and Lagrangian modelling of robots. Ideally, the student taking this course will have taken an introductory course on nonlinear control theory, such as ECE1647F in this institution, and be familiar with the Lagrangian modelling of robots from a course like ECE470 in this institution.
Course Deliverables
This course has two main components:
Assignments: There will be three homework assignments giving you an opportunity to practice the theoretical concepts presented in the lectures and testing your understanding of the course material. Assignments are a group activity: you will form groups of four students and submit your work as a group.
Course project: You will design a virtual holonomic constraint and associated controller making a robot walk with constant speed on flat ground. Then, in the creative component of the project, you will pursue your own inspirations building upon the basic walking controller. You will also give a conference-style presentation of your development and the main results you have obtained.
Composition of Final Mark
| Homework assignments | 40% |
| Final project | 40% |
| Final presentation | 20% |
References
- Books on differential geometry
- Loring W. Tu, An Introduction to Manifolds, 2nd edition, Spring. 2010
- J.M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer, 2013
- V.I. Arnold, Ordinary Differential Equations, MIT Press, 1978
- V. Guillemin, A. Pollack, Differential Topology, Prentice Hall, 1974
- W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., Academic Press, 1986
- Books on geometric nonlinear control
- Alberto Isidori, Nonlinear Control Systems, 3rd edition, Springer, 1995
- H. Nijmeijer, A.J. van der Schaft, Nonlinear Dynamical Systems, Springer, 1990
- F. Bullo, A. Lewis, Geometric control of mechanical systems, Springer, 2005
- V. Jurdjevic, Geometric Control Theory, Cambridge University Press, 1996
- A. Agrachev, Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer, 2004
- Book on robotic locomotion
- E. Westervelt, J. Grizzle, C. Chevallereau, J. Choi, B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion, CRC Press, 2007