Nascent Roboticist: Calculus of Variations

Thursday, June 13

Calculus of Variations, 2/2

In the first post of this series, the Barchistochrone problem was introduced and we looked at how the performance of a particular curve could be evaluated. Recall that the total transit may be expressed in this way:

$[;T = \int^{x_B}_{x_A}\frac{\sqrt{1+f'(x)^2}}{v}\,dx;]$

Also, because the bead is traveling through a constant gravitational field, the velocity may be expressed as a function of the change in height by conservation of energy.

$[; v = \sqrt{2g(y_A-y)};]$

$[; v = \sqrt{2g(f(x_A)-f(x))};]$

Therefore, a functional that evaluates the transit time for some curve f(x) in the brachistochrone problem is given by:

$[; T(f(x),f'(x),x) = \int^{x_B}_{x_A} \frac{\sqrt{1+f'(x)^2}}{\sqrt{2g(f(x_A)-f(x))}}\,dx ;]$

which may be generalized with a function L:

$[; T(f(x),f'(x),x) = \int^{x_B}_{x_A} L(f(x),f'(x),x)\,dx ;]$

where

$[; L(f(x),f'(x),x) = \frac{\sqrt{1+f'(x)^2}}{\sqrt{2g(f(x_A)-f(x))}} ;]$

Now that we have an expression for the total transit time of the bead for any continuous differentiable function between points A and B, imagine how changing that function might change the transit time of the bead. For the optimal curve, any perturbation in the curve will increase the transit time. If we can come up with a relationship between perturbations of a function and changes in the functional T, we can look at how infinitesimal perturbations of the function change the transit time. The motivation here is to use that relationship in order to find a function for which infinitesimal perturbations produce no change in the transit time - at which point the function can be said to minimize T.

Thursday, May 16

Calculus of Variations, 1/2

I've decided to post on this topic for two reasons. First, I want to solidify my own comprehension; I was introduced to variational calculus as a supplemental portion of a Finite Element Analysis course that I took during the third year of my BME. When my FEA professor lectured on it, I wasn't particularly comfortable with the method; I got the gist of what was being done but I wasn't able to reconstruct it on my own when I sat down to a piece of paper. Second, since that lecture, I have recognized it in a handful of applications in the realm of optimization, mechanics, control theory, and FEA; Those topics are pretty damned important for someone interested in robotics. For those reasons, although plenty of people have produced better summaries and course materials on the topic, I feel compelled to make a modest attempt for my own sense of completion.

To start, it is sometimes useful to learn about the history of a subject in order to understand the context in which it was developed. As far as mathematical methods go, there is usually some famous problem for which it was developed. In the case of variational calculus, that problem was finding the Brachistochrone Curve. The problem was posed by Johann Bernoulli as part of a correspondence between friends and contemporary mathematicians of his time. He asked his colleagues to determine the optimal curve that a bead would follow through a uniform gravitational field between two points in the shortest amount of time. As one might expect, the name is derived from the latin brachis meaning "shortest" and chron meaning "time".

The Brachistochrone Problem