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Fig. 4. Predicting optimal behavior by finding the points where constraint lines
are tangent to cost contours. Cost contours are shown as black curves. Cost is
least in the region bounded by the central curve and greater for curves lying
outside each other. Constrained optimization predicts that for any given
constraint gait, parameters will be chosen such that cost of transport is
minimized. This occurs at the tangent of the constraint line and a cost
contour, because any other point on the constraint line lies outside the
contour and indicates a greater cost. This method is equivalent to predicting
optimal behavior by finding the points where one of the partial derivatives is
equal to zero and may be used to verify the optimal behavior predictions shown
in Fig. 5. Speed and frequency
constraints can be visualised as horizontal and vertical lines, respectively,
and step length constraints can be visualised as lines radiating from the
origin whose slopes are equal to the specified step lengths - i.e. lines whose
equations are of the form v=fd, where d=constant.
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