The six-bar (like Dave Weagle's DW6) uses six articulated bars instead of four. What for? To adjust the leverage ratio, anti-squat, anti-rise and the axle path almost separately, without sacrificing one to improve another. The price: more weight, more bearings, more complexity. Used by Atherton (DW6) and high-end designs. It is the most complex of the systems.
If a four-bar is an instrument with four knobs, the six-bar has six: more control, but more things to tune and maintain.
To adjust LR, anti-squat, anti-rise and axle path more independently, without sacrificing one for another. It costs weight and complexity.
Only if the design needs to solve conflicting requirements (e-bikes, difficult packaging). For many bikes, a good four-bar is enough.
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