Details on Lectures and Exercises

The material of the course is divided into four parts:

The 1st part of the course (Lecture 1 and Exercise 1) is introductory. It is aimed at presenting:

• Classical concepts such as the Poincare first return map, the moving Poincare section, transverse dynamics;
• Methods and tools such as linearization of the Poincare map and transverse linearization, averaging, Lyapunov lemma and centre-focus problem.
  
  The methods are of common use for detecting periodic solutions in dynamical systems and for analysis of their properties. With the focus on mechanical systems, and aiming at a useful representation of periodic motions, we introduce an associated concept of virtual holonomic constraint and suggest a new approach for detecting cycles.  

In the 2nd part (Lecture 2 and Exercise 2), a procedure for computing transverse linearization for dynamics of mechanical system along its motion is presented. The procedure is generic and works for fully or over-actuated systems as well as for under-actuated or even passive ones. It allows  

• Analysis of properties of the motion such as orbital stability or instability and sensitivity to disturbances and structural uncertainties;
• Synthesis of feedback controllers to achieve orbital stabilization or redesign of a known stabilizing controller to increase robustness or convergence rate.

The development is illustrated in details on several benchmark examples. Namely, it is shown how  

• To shape stable oscillations of a pendulum on a cart around its unstable upright equilibrium;
• To organize a swing-up of the Furuta pendulum taking into account limitations on the travel of the first link;
• To shape stable oscillations of the Pendubot around any of its unstable equilibria;
• To synchronize stable oscillations of many (100) copies of pendulums on carts around their unstable equilibria.
   

In the 3rd part of the course (Lecture 3 and Exercise 3), we develop a procedure for computing hybrid transverse linearization of dynamics of impulsive mechanical systems along its hybrid motions. Here, we reformulate and improve the known algorithm for searching cycles of hybrid systems, and show how the concept of hybrid transverse linearization can be exploited for 

• Computing a linearization of the first return Poincare map analytically for a passive gait of a walking robot;
• Synthesizing a feedback controller to stabilize a gait for an under-actuated walking robot
• Analyzing sensitivity of stable gaits to uncertainties and disturbances in the model such as the slope angle
  
  The arguments for motion planning, motion representation and control design are illustrated on a model of a compass-gait biped and a model of a quadruped.
  
  

In the last part we suggest a discussion on how to use the proposed approach for analysis of human motions from the recorded data (Lecture 4) and present ideas for searching human-like motions for humanoid robots (e.g. pitching a ball, sit-down and stand up from a chair). The comprehensive descriptions of lab experiments that have been accomplished are given in Lecture 5.

Open problems are posted in the end of the course.