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:

 

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.

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

which may be generalized with a function L:

where

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.

Underactuated Robotics

I recently came across materials from a class on MIT's Open Courseware (OCW) that has me pretty excited: Underactuated Robotics. I ran into one of the lecture notes during my investigation into inverted pendulum control schemes (a project which sort of branched out of my review of the calculus of variations). I was originally interested in modeling the dynamics of a pendulum in a variety of force fields and decided it would be more interesting to investigate controlling that system in ordinary conditions. I'm planning on implementing this system physically once I have a sense of what I would like to accomplish with it.

 The first lecture opened my ears to a fundamental bifurcation in the application of mechanical and control system design. Primarily, fully actuated systems which possess the same number of actuators as they do degrees of freedom and underactuated systems which have no direct influence over the unactuated degrees of freedom. More succinctly, fully actuated systems are capable of producing accelerations along arbitrary directions in the state space. For systems where the dynamics may have a linear relationship with the actuation vector:

$\ddot{q} = f_1(q,\dot{q},t) + f_2(q,\dot{q},t)u $

Where $q$ is a vector in the state space, $u$ is a vector of all actuators, and $f_2$ maps the actuators to some dimension in the state space. A fully actuated system will possess the following property:

$rank[f_2(q,\dot{q},t)] = dim[q]$

Whereas underactuated systems will not map the actuation space to all dimensions of the state space.

$rank[f_2(q,\dot{q},t)] < dim[q]$

In order to reach a particular position in the state space, an underactuated system must make use of the indirect and dynamic characteristics of the system that relate the actuated features of the state vector to the desired unactuated feature. For instance, a cart and pole inverted pendulum is underactuated. It may be modeled as an actuated prismatic joint coupled with an unactuated revolute joint with some mass at the end of the free member. This is apparent because there are two degrees of freedom in the system and only one actuator. It is generally possible to reach any point in the state space, but not necessarily possible to describe any path through the space. The available paths that may be taken through the state space are defined by the dynamics of the system.