## Gait Mapping with Vertical Leg Orientation

### A general system state for gait model simulations

In walking and running, the motion of the centre of mass can be understood as a kind of oscillation, however, this oscillation is sometimes hardly comparable with a sinus function. For investigations of gaits, a special kind of mapping, so called Poincaré maps are used. In Poincaré maps a comparison between the systems state at the beginning and at the end of an oscillation is conducted which removes the time from the dataset. In the case, the systems state variables are equal, a periodic motion is revealed. Poincaré originally used the period duration for fixing start and end events (in order to analyse the motion of planets). When simulating gaits with legged models, the period duration is not known prior the simulation start.

In gait simulations, the scientists apply physical system states to define start and end event of a simulation while two conditions are very popular in the biomechanics field, i.e. the touch down of a leg and the apex of the centre of mass. The touch down event is often used for simulating Passive Dynamic Walkers with rigid legs. In models with compliant legs, the apex is mostly used to describe start and end of a gait cycle. When the touch down event is used, the conditions of the counter leg need to be fixed a priori. Is the counter leg at the ground? Does it lift off at the same moment or does it swing without ground contact?

Similar a priori definitions are required when using the apex. Is the leg lifted at the instant of apex, the gait of running is selected.  In the walking gait, the active leg must have ground contact during apex. By definition, the apex is the highest point of the centre of mass during locomotion. However, the stop condition of a simulation is identified when one maximum is reached. That could be one maximum of possibly several peaks. In some simulations with a walking model, there were truly more than one maximum identified. Hence, the apex ore the maximum is not necessarily a unique event in walking and the apex return map is incomplete or maybe incorrect.

The instant of Vertical Leg Orientaion (VLO) with a simulated walking model.

At the Locomotion Lab in Jena another system state event for Poincaré maps was established, which allows to investigate both gaits, walking and running, with the same simulation and mapping. This event is called Vertical Leg Orientation (VLO). The definition is, that the active leg has ground contact and is oriented vertically or the hip joint is vertically above the foot point. While the active leg is on the ground, the counter leg is lifted. This system state exists in both gaits equally.

Well defined motion events for Poincaré maps ensure for a reduction of independent system variables in order to simplify the analysis. The system variables of the spring-mass model are the positions $x$ and $y$ of the centre of mass, which is located at the hip, and the velocities $v_x$ and $v_y$ of the centre of mass. At the instant of Vertical Leg Orientation, the system state can be reduced to two independent variables, i.e. the height $y$ and the angle of the velocity $\theta = \arctan(v_y/v_x)$.

Parameter map where symmetric walking patterns were found.

The novel method for gait analysis using VLO was applied for walking simulations first. There is a clear distinction between symmetric and asymmetric gait patterns. In symmetric walking, the velocity angle $\theta$ is always zero, which means, the first half of the stance phase is symmetrically identical to the second half. In asymmetric walking patterns is $\theta \neq 0$. Asymmetric walking patterns are not worse than symmetric patterns as there exist also self-stable solutions. A surprising finding is that in symmetric walking solutions, the centre of mass $y$ is always lifted at the event of VLO compared to the height at touch down $y_{TD}$.

Why is the novel event for Poincaré maps called VLO instead of “mid-stance”? The term mid-stance is already used in scientific literature but addresses various conditions. Mid-stance is for instance the time based centre of the stance phase, a period of time during the stance phase, or the event when the ground reaction force is perpendicular to the ground. In order to clearly define the event of vertical orientation and differentiate from a less explicit term, the name of Vertical Leg Orientation was established.

Simulation results on Walking and Running combined in a single map using VLO will be presented in another article.

#### Featured Paper

J. Rummel, Y. Blum, H.M. Maus, C. Rode, A. Seyfarth.
Stable and robust walking with compliant legs.
IEEE International Conference on Robotics and Automation, May 3-8, Anchorage, Alaska: 5250-5255, 2010.
DOI: 10.1109/ROBOT.2010.5509500
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