Inverse and Forward Kinematics

Packaging machine designers are confronted with a bewildering array of technologies with which to implement motions - whether the operation be pushing, pulling, picking, folding or placing.

 We can choose ball screws, belt drives, gearboxes, chains or even ( if you have plenty of time )  a mechanism with links and things.  If the transforming mechanism between motor/cam and end effector is linear then the difference between inverse and forward kinematics becomes trivial.  If I need high speed, reciprocating motion along a linear path then I can choose a ball screw safe in the knowledge that my following error will probably be better than +/- 50 microns running whilst my load is batting up and down 60 times a minute.  Also the input/output gearing of a ball screw is linear.  If my motor turns 10 revs, my load will move 50 mm; if it turns 20 then by load moves 100 mm.  This simplifies my design massively doesn't it?  Ball screws are great at going up and down with precision, however maybe they limit the path options open to me. 

Suppose I need to push my load in a straight line, but after pushing I need to get back double quick to ensure the next pack has time to get in front my pusher before it pushes again. Ah great I think, the catalog says that my ball screw/ belt drive will get back easily, but hang on, I have got to get my next pack in place in 50 milliseconds. Oh dear. I wonder how many broken light bulbs before the customer wants his money back?

At this point I groan, because I know in my heart of hearts I need a proper mechanism to generate a path so that on my return stroke I go back over the top of the incoming pack.  This means that I need either two axes, too expensive, or a path generating mechanism and single servo/cam to control it. I find the right mechanism but now I need to consider the following question. If I design my motion at the pusher because i do not want an impact on contact with the pack, how do i find the servo-motor motion driving the rocker of the four bar that will generate the designed end effector motion? Big question in more ways than one. My four bar is effectively a function generator .

y = F(x)      where y is my output parameter along the path and x is servo-motor position

Trouble is I don't know 'F' unless i dust off my Theory of Machines text book ( no, no, no...) and do some serious very speculative mathematics ( as if...). Even if I found 'F' I would then need its inverse since I know y ( my motion ) and I want x.  Looks like back to the super high acceleration ball shaking screw option.  It'll look great until the customer sees the speed of that pusher on its return stroke and hears the load bang as the next pack flies in like a newly liberated champagne cork.

Fortunately I have MechDesigner so not only can I identify the most appropriate four bar for my path generation, but because forward and inverse kinematics are built in, also I immediately get the servo motor motion table. Success. This mechanism will work very quickly indeed and the replacing pack has a lot more time to be positioned and I might be able to pedal it and have two up and over pushers working in tandem at 180 degrees apart in machine time.  Oh no, can't have that, the customer would only need one machine not the two ordered !