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.