Stability and tolerance of compliant legs
In usual simulation studies focusing on walking and running, two equal legs are assumed. However, in biology as also in technical systems the existence of absolutely equal legs would almost never present. In humans, the leg length measured between ground and trochanter mayor of the hip is often slightly different. The difference can be a few millimetres. The force the leg will generate during stance is also rarely or never equal compared to the counter leg. Possible causes are variations in the shaping of muscles and the activation of them. Very impressive are the disparities of both legs in humans wearing a prosthesis. Another issue concerns the leg control before touch down of the foot, which is also almost never equal to the counter leg. One might expect, the technical legs could be manufactured identically, but small variations in joint friction or motor and sensor devices would lead to unequal leg behaviour.
Is the stability affected in walking when both legs are not identical?
This question was tried to be answered in the Lauflabor Locomotion Lab in Jena. Here, the bipedal spring mass model was used to apply walking with varying leg properties. The study was focused on the stability of walking patterns, which were identified and analysed in a previous study. The spring mass model offers the the options for varying three basic parameters, i.e. the leg length, the leg stiffness, and the leg’s angle of attack at touch down. As an additional parameter, the energy was considered, which partially represents the velocity. In the presented study, periodic and stable walking patterns were selected and the leg parameters were varied for the left and right leg in opposite direction.
One could expect that walking with slightly different legs would lead to loose gait stability. This, however, is not the case because of the compliant legs, represented by springs, which offer a tolerance regarding stability. In the proposed study, the leg parameters were varied separately between the two legs. In the case of a relatively soft leg, the leg stiffnesses $k_L$ and $k_R$ can differ about $30 \%$ without loosing stability. This shows a remarkably high tolerance of compliant legs with respect to different muscle or motor forces in the two legs. A lower tolerance is observed when considering differences in leg length. The difference should be not more than 8mm when the original leg length is 1m. However, until a difference about 4 or 5 mm the stability in walking is not endangered. An unexpected result of the study concerns the control parameter of leg angle at touch down or angle of attack $\alpha$. When the difference of the angle of attack increases a little bit until 3 deg, the range of stable gait patterns even extends.
In conclusion, when walking with compliant legs where the legs act like springs, not just self-stability exists, there is also a remarkable robustness against different properties of the two legs. While the tolerance is somewhat lower regarding leg length differences, the robustness against controller and actuator properties like leg stiffness and leg angle are remarkable. In order to apply natural walking a leg control with low performance seems to be adequate.