M.Sc Thesis

M.Sc StudentEvstrachin Alexandrina
SubjectLocal Stability Analysis of Hybrid Dynamical Systems
via Saltation Matrix with Applications to Simple
Walking and Jumping Robots
DepartmentDepartment of Mechanical Engineering
Supervisor PROF. Miriam Zacksenhouse
Full Thesis textFull thesis text - English Version


Systems with hybrid dynamics, such as walking robots, are characterized by regions of smooth flow (e.g. continuous dynamics during the swing phase) divided by discontinuity surfaces (e.g. impact with the ground) at which the flow undergoes non-smooth or even discontinuous changes.

Discontinuity surfaces transverse to a limit cycle provide natural choices for defining a Poincare section and the associated Poincare map in order to investigate orbital stability. Orbital stability can be deduced from the eigenvalues of the linearized Poincare map around the fixed point at which the limit cycle intersects the Poincare section, however, their analytic computation is challenging. Hence, eigenvalues are usually computed numerically. The method is based on numerical integration of the smooth flow from the perturbed fixed point until the next intersection with the Poincare section, and computation of the states around first return map.

The numerical method is prone to numerical errors when perturbations are small, and is inaccurate due to non-linearities when perturbations are large. Furthermore, it does not provide insights into the effects of system or control parameters on the gait stability.

This research investigates an analytical approach for obtaining linearized Poincare map for hybrid dynamical systems with discontinuities in both flow and state based on the Saltation matrix, which describes the linear part of the Discontinuity mapping.

This method is applied to obtain the linearized Poincare map and to investigate the stability of simple active running and walking models: one dimensional (1D) hopper and two-dimensional (2D) compass biped.