Actually, there's not much on a floating caliper that can be repaired. If grooves have formed between the piston and the cylinder, or corrosion has appeared through moisture in the brake fluid, the complete unit must be replaced anyhow. Of course, this is also valid if the piston can no longer be moved.
In the floating caliper, the piston has a special relevance. Not only must it transfer the pressure to the respective brake pad, it must also retract itself again when no pressure exists. This is achieved through a particularly elastic sealing ring (made from silicone material), which provides for an always constant amount of brake-release movement. In any event, it must always be able to move freely. In the above picture actually, a floating caliper with an integrated hand brake is installed. Obviously the surface of the of the piston has been slightly worked on, the sealing ring changed or the release mechanism of the hand brake has been treated to ensure a smooth working. In any event, here, the lubricated piston, without being tilted, is pressed into the cylinder with the sealing ring. The second figure shows the problem of disc brake adjustment a little more exactly. If the piston is simply pressed in, the post adjustment mechanism must turn (figure 1), or the piston is turned while being pressed in by means of a special tool (figure 2). Now, finally, it is also clear, why the floating caliper has a round hollow up front. The dismantling of the piston is only really worthwhile, if the sealing ring has to be replaced. Possible damage through tilting can only rarely be repaired. In this case, the best recommendation is probably the replacement of the whole caliper. When working on a disc brake, it is a good idea, apart from checking the dust protection collar and the that the seals are intact, to have a look at the brake disc. Even a casual look can show cracks or grooves of more than a half a millimeter in depth. In this case, the brake discs must be replaced. For further checking however, one must examine the components a lot closer and/or make use of measuring instruments. Because of the enormous heat development, the brake discs can become warped. A large amount of the heat is led off through the wheel hub. The heat transfer here is often complicated by a smaller cross-section (clearance groove), to keep the inside- and outside temperature of the disc from becoming excessive. One can recognise a warped disc by laying a ruler along the diameter. For measuring the thickness however, a special slide-gauge is necessary.