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Minimum Energy Performance


Experiments have shown humans to walk in an energy efficient manner. In [10], a detailed study is presented of energy expenditure in actual human walking where researchers have explored the relationships which exist between real energy data and human walking motion. A simple yet fairly accurate relationship is one in which the energy expended is quadratic in the forward velocity,


Here v is the average forward velocity in m/min and tex2html_wrap_inline756 is the energy requirement in cal/kg/min for the average human subject. An often more desirable set of units for walking energy is to measure it per distance traveled (m = meters) rather than per time elapsed (min = minutes) since this conveys more the notion of energy economy. We denote this form of energy as tex2html_wrap_inline756 . Its units are cal/kg/m, and it is related to tex2html_wrap_inline760 and the previous relationship by


This function will now have a hyperbolic shape. This and most other functional relationships such as (gif) indicate a minimum energy motion of an 80 m/min walking velocity with an energy expenditure of 0.8 cal/kg/m. The experiments also show average cadences of 105 steps/min and an average step length of 0.75 m for an adult male.

In our study, we shall minimize a quantity proportional to the injected energy into the system, the integral of the applied torques,


where tex2html_wrap_inline762 is the time at the end of the first phase (swing phase), and T is the time at the end of the second phase (double-contact phase). Dividing by the step length, the distance between successive heel strikes, gives the expended energy per meter traveled. If an impulsive force is added, then this control parameter will also be added into the performance,


This general form of minimal energy performance was also used in [11].

The performance J is not a measure of the mechanical work performed on the system, and we are unable to determine the change in energy of the body from J. In fact, for our biped model, we are minimizing a quantity proportional to the energy required for a motion. In humans, this is analogous to the difference between mechanical energy and metabolic energy. As no system, not even a human, is perfectly efficient, these quantities will differ and their relationship in humans still remains a very difficult and unanswered problem [10]. In robotics, we do not have metabolic energy, but for a simple actuation model, our approach amounts to minimizing the energy required for direct drive motors at the joints to produce the required torques. This approach provides a more numerically tractable way of reaching our performance objectives.

next up previous
Next: Numerical Optimal Control Up: Obtaining Minimum Energy Biped Previous: Nonlinear Inequality Contraints

Michael W. Hardt
Mon Oct 11 17:19:43 MET DST 1999