My team and I would like to make the arm our claw is going to be on to be telescoping, but we're not sure how to go about doing so. Suggestions?
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There are two problems to solve: how to mount two parts so they telescope, and how to make them move.

The best way to mount two parts that you want to slide but not rotate with respect to each other is to use the IGUS track and carts.  There is a trap to be aware of: the carts need to be almost perfectly aligned with each other or they will jam in the track.  One way to get them aligned and keep them aligned is to mount each cart with only one machine screw, and don't tighten the screws until the carts are in the tracks.  That sounds counterintuitive (my students don't want anything to wobble!), but it is a lot easier than trying to drill 4 perfectly co-linear holes in wood or PVC (unless you have a milling machine).

I can think of several ways to get sliding parts to move with respect to each other:

1) If you have access to a CNC machine and can teach your students how to use it, build a rack and pinion system.  Mount the rack on one piece, and mount a motor-driven pinion gear on the other piece.  If you don't have a CNC machine and have a student who really wants to try this with a jigsaw or a bandsaw, I have heard that there are gear templates online that you can print onto paper.  If your student wants to try doing this without a gear template, he or she will learn the importance of using the right tool for the job.

2) Mount a servo on one part and connect the servo horn to the other.  This will work only if you don't need to use much force and you don't need to move very far. If you look up the specs for the servo, you can figure out how much force you can apply over that distance.  For the kind of reach and force you probably want for this competition, this solution is a non-starter.

3) Use a threaded rod and a captured nut.  Mount a nut on one of the sliding parts, insert a threaded rod into the nut, mount a motor to the other part, and have the motor turn the threaded rod.  You can get a lot of force this way, but the motion is slow.  For example, if you use a 1/4-20 nut and rod, the nut will move 1 inch for every 20 turns of the motor.  I believe that the small motors have a maximum speed of 90RPM, and the toothed belt-pulley system has a mechanical advantage of 7:1, so you could combine these to get a maximum speed of (90RPM*7)/(20 Rev/inch)=31.5 inches/minute.

4) Use a motor-driven spool and strings.  Mount the spool on one of the sliding parts, attach a string to the spool, and tie the string to one end of the second sliding part.  This will allow you to pull the second part in one direction. To move in the other direction, you can use gravity (if the motion is mostly vertical), rubber bands, or a second string attached to the other end of the second sliding part.  The maximum force and speed of this approach is determined by your choice of motor and by the diameter of the spool.  My team has used this approach successfully in several competitions.
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This is a resource for the FIRST robotics competition, but the information can be applied here.
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In 2013, my team had nested PVC pipes where the inner pipe had cuts in it that mated with a gear.  That gear was mounted at the end of the outer pipe and to keep the weight on the arm low, the motor was mounted at the base of the arm.  The motor drove the gear via string and pulleys.  Here is one picture (not a great shot), but it shows the pulleys and gear.
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