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 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 . Its units are
*cal/kg/m*, and it is related to and the previous
relationship by

This function will now have a hyperbolic shape.
This and most other functional relationships such as ()
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 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.

Mon Oct 11 17:19:43 MET DST 1999