Poincaré return map

Compare one cycle with the next#

Choose a surface in state space that the motion crosses once per cycle, such as the state immediately before each foot impact. Starting from one crossing, simulate or measure the system until it returns to that surface. The Poincaré return map assigns the next crossing state to the current one.

This removes the phase along the cycle and focuses on step-to-step change. In the hybrid zero dynamics treatment, the map follows a biped through the impact update and continuous swing dynamics until the next crossing of the selected section.

A fixed point represents a periodic gait#

A state that the map sends back to itself is a fixed point. It corresponds to a periodic orbit of the original continuous or hybrid system. Linearizing the map around that point gives a local test of orbital stability: perturbations contract from cycle to cycle when the relevant eigenvalues lie inside the unit circle.

This is different from checking whether the robot holds a static pose. Walking repeats an orbit rather than settling at one state. The Grizzle and Chevallereau chapter uses restricted Poincaré maps to analyse periodic motions inside hybrid zero dynamics and discusses numerical analysis when the reduced map is not available in closed form.

The conclusion is local and model-dependent#

The map depends on the section, event definition, controller, contact model, and state coordinates. A missed contact or an extra crossing can make the numerical map discontinuous or undefined. Estimation noise is especially important when velocity is sampled at an impact event.

Eigenvalues from a linearization describe small perturbations near one nominal orbit. They do not prove recovery from large pushes, foot slip, actuator saturation, or a different contact sequence. Hardware tests should therefore complement the model-based return map and state the disturbance range over which recovery was observed.