HYBRID DYNAMICS IN MECHANICAL SYSTEMS

This course is for undergraduate students in their last year of studies and graduate students. Course will have pre-recorded lectures.

The course deals with dynamics of mechanical and robotic systems under constraints, and specifically kinematic constraints of contact with friction. Motion of such systems is typically described by a hybrid dynamical system with non-smooth solutions due to transitions between different contact states and impact at the contacts. These systems are useful for motion analysis and control design of robotic systems for object grasping and manipulation, as well as legged mobile robots and wheeled or tracked vehicles. The class discusses basic topics in addition to phenomena which are unique to hybrid systems, as well as advanced research problems. The course involves four homework assignments and a final project, all involving analysis and numerical computation in Matlab

Topics: Lagrange’s equations, formulating the dynamics of mechanical systems with holonomic and non-holonomic constraints, with mechanical and/or kinematic inputs. Contact kinematics and contact forces. Friction, Coulomb’s model, planar multi-contact statics – graphical and computational analysis. Dynamics under different contact states, Painlevé paradox and dynamic jamming. Impact models with and without friction, Zeno phenomenon, simple models of dynamic walking, Poincaré map, stability of periodic orbits in hybrid systems.

Semester Start Date: October 29, 2025

Contact Hours: 3

Day & Time: Tuesday, 14:30-17:30 (Israeli Time which is one hour ahead of Germany, France, Denmark, Switzerland, Czech Republic, and the Netherlands; same time zone as Estonia)

Text Books: (some are available in electronic format online or via Technion’s library)

1. M. T. Mason, "Mechanics of robotic manipulation", MIT Press 2001 LINK to Mason’s class lecture notes at CMU
2. R. M. Murray, Z. Li, S. Sastry, "Mathematical Introduction to Robotic manipulation", CRC press 1994. LINK to author’s online printout.
3. B. Brogliato, “Nonsmooth Mechanics: models, dynamics, and control”, Springer 1999
4. E. R. Westervelt, J. W. Grizzle, C. Chevallereau, J. H. Choi, and B. Morris, "Feedback Control of Dynamic Bipedal Robot Locomotion", CRC Press 2007

At the end of the course the student will be capable of:

1. analysis of planar problems in statics with multiple unilateral frictional contacts by methods and by solving a linear programming.

2. formulating equations of motion for a mechanical system with contact constraints.

3. formulating conditions for occurrence of painleve's paradox in a system with a single frictional contact.

4. formulating impact laws for a single contact with and without friction in a system of rigid bodies.

5.formulating equations of motion of a mechanical system with intermittent contacts as a hybrid system.

6.conducting numerical simulations of mechanical systems with intermittent contact.

7. finding periodic solutions of a hybrid dynamical system and their stab

Hybrid lectures, both in class at Technion for Technion students, and with recordings for remote participants. Homework, oral defense exam via Zoom, as well as office hours via Zoom. The lectures will be given in English.

Exam Type:

Written assignments during the semester and one oral defense exam which will be held via Zoom No mandatory attendance or participation

Dynamics, Linear Systems, Introduction to Control or Mechanics of vibrations

Lectures, Homework & Final Project Assignment