This post is going to put a bunch of people off, but I was reminded of it when discussing something with my room-mate (and realizing how much I’d forgotten). Once upon a time I used to be an engineering student before going over to the dark side (save the world, burn less carbon). So…in memory of kinematics and dynamics, here’s what seemed to me to be a quite fascinating demonstration of the importance of variable selection in solving a dynamics problem. Note that this problem is theoretically quite uninteresting, completely understood, and utterly insoluble if you don’t make the right choices.

Consider the spacecraft in the figure with an attached robot arm, used perhaps to hurl hapless fools into the nether regions of frigid space. If the ship is stationary, external forces and torques are balanced and you have nine degrees of freedom corresponding to the three rigid parts of the arm with the spacecraft just sitting there. In order to figure out what torques (and thrusts) the driving motors should generate you need to write the differential equations governing motion. Some of these come from geometry (half to be precise) – thats the kinematics bit. The rest come from dynamics. 18 equations in total. The key is in how you write them.

Using Newtons Laws about the centre of mass is a complete mess, and you wouldn’t really try that. What might be worth trying is to use a Lagrangian formulation, using motion variables and their derivatives. Lagrange’s method has us define position variables as a set of angles. Three for the orientation of 1 in N, 2 for the twin motors at B (rotations about **by **and then a rotation of 2 about **bz**), 1 rotation about **cz** and finally three for the ball and socket joint connecting the grasper E to the arm. Thats 9 if you count, and well – bear with me!

So now Lagrange’s formulation leads to these angles and their derivatives as the position and motion variables. The kinematic equations in this case are trivial since they merely state that motion variables are defined as derivatives of position variables. The * shortest* dynamic equation is all of 300kms long in 11 point font. Modern dynamics (Kane dynamics) uses an alternative set of position and motion variables generated using a set of guidelines, and of the form of affine functions of positions and angles, and velocities and angular velocities respectively. Follow the right method and you get equations a few lines long, with the longest being a few pages.

The difference between a 300km long equation and a couple of lines is quite simply the difference between saying you can solve a Newtonian dynamics problem, and saying you understand the principle. Figuring out ways to come up with efficient formulations is less than a couple of decades old. Writing F=ma dates back to the 17th century.

I am completely taken aback as to why in hell would one write such a verbose engineering text about kane dynamics. You could have rather explained (and given a link) as to what kane dynamics really is.

Also about the 300 km long equation, that is out of newtonian mechanics and not via lagrangian mechanics which uses the variable T = KE-PE as its base.

4 February 2008at1pm@someone: Well its not really *that* verbose is it? Certainly not by the standards of an engineering text! Here’s a link to a fuller explanation though – http://ijr.sagepub.com/cgi/content/abstract/15/5/522. As to why I wrote it – well, partly to see whether it would produce a reaction of any kind (and it seems to have got one out of you : p), and partly because I do think its faintly interesting :)

That said, the energy expression you’ve given me is somewhat beside the point. That is just a quantity to be minimized and one way to derive the dynamics equations. The issue is how do you express KE or PE, i.e which variables do you choose? The purpose of this post was just to point out that its surprisingly important that you choose wisely. And yes, using classical generalized coordinates as position variables with their derivatives as motion variables does result in overly long equations.

4 February 2008at2pmSo basically your article tells us that choices matter? The awareness that the style of the solution matters goes further back than you give it credit for- and while this discussion will take us far afield from the wonder that is kane dynamics (really, why are you having these discussions with your roommates? Isn’t the superbowl sufficiently interesting for you?)- I felt this might be the right time to point them out. The question of what constitutes a solution to a posed question goes about as far back as mathematics, and we’ve been blessed with a few characters with strong opinions on the subject. Par example, we have Galileo, who believed that post solution, all intuition used to construct it should be stripped out, leaving the purest form- when asked why, he responded that no-one in their right minds asked architects to leave scaffolding around a completed building so one might study how it had been built. Or Borel, a man who didn’t believe that a proof by contradiction was a proof at all- he was of the ‘constructivist’ school of mathematics. One of the massive contributions of Feynman toward building the Bomb was a method of simplifying the differential equations involved (by ‘splitting them) so that the vaccuum tube computers of the era could solve it. The history of numerical math is a history of clever algorithms, conversions, tricks, what have you, to make possible calculations and computations that otherwise were not. Think Vedic mathematics, think steepest descent algorithms, think of any and every approximation to e^x, log x. If anything, our exposure to overly powerful computers and MATLAB leave us unwilling to put in the mental work necessary to come up with more efficient solutions. Simulations substitute proofs and solutions, bad simulations substitute good ones, and so on. The fact that you thought of kane mechanics is amazing. Lesser people would have fed F=Ma and the question at hand to matlab, and waited a few days while the computing cluster at their university figured it out.

4 February 2008at4pmMinus complexities please. Thank you.

5 February 2008at12amI liked it. Thansk.

9 July 2009at4am