The only thing new needed for 6x6 is a way to solve the Arc Centers (the X-Centers can be solved using the 4x4 sequences as shown here). The first step to solving the Arc Centers is to eliminate the ones that are are quarter-twisted by using the following mirror-image sequences. To use this sequence it is only necessary that there are two centers with quarter-twisted arrows in the locations shown on the top face. The direction of the arrows does not have to match the setup nor do the rest of the arrows on the face. The quarter-twisted pair will either be solved or half-twisted (as in no longer quarter-twisted) as a result of the sequence. The half-twisted pair on the front face can be ignored while doing this step. This is similar to the method used for the 4x4 Supercube, the eight move sequence is used twice (moves 2-9 and 12-19).

Another way to do it is with the quarter-twisted centers on the front face (instead of top) as shown below. The following sequences are the inverse mirror-images of the above. This alternate method is used in the Example Solve of Arc Centers.

When only half-twisted Arc Centers remain, use 5x5 sequences to fix centers where possible like the following example where two pairs on the front face and another two pairs on the top face can be fixed. This can also be done if one face has only a single twisted pair (like top face on above cube). In that case the single twisted pair will be swapped with the other pair that is not twisted, the net result will be that there is no change in the number of twisted centers on that face but the two pairs on the other face will both be fixed.

When no more Arc Centers can be fixed with 5x5 sequences use one of the following sequences (or mirror-image) to fix two twisted pairs. The first three fix pairs on the front and top faces, the fourth one fixes pairs on the front and back faces and the last one fixes two pairs on the front face. The sequence on the first cube below can also be used to quarter-twist centers with a slight modification as shown here.

You can try solving 6x6 Supercubes With Random Arc Centers.