IMO, stability (i.e., roll axis while on ground) will be influenced by several factors, some more readily apparent than others:
- Center of Gravity
- Track
- Main gear camber and toe
- Suspension compliance, both compression and rebound
- Roll Center (the point about which the gyro suspension wants to rotate irrespective of CG)
Everyone talks about the first 4 items but very little is said about the last item. If we use an automobile analogy, the roll center (RC) is typically designed to be near the CG. The RC can be approximated by the intersection of the A-arms. The lower the RC relative to CG, the more the car wants to roll in a turn which adds 'bite' to the outside wheel. This roll tendency is reduced by the use of anti-sway (aka anti-roll) bars. In a performance automobile, the RC may be designed to be near the road surface. If too low, the car tends to want to "go-kart" with ineffective handling.
If we look at a gyro application, we see two basic extremes. Rigid suspension ala a go-kart with both the advantages and disadvantages of driving a kart plus the impact of landing mitigated only by rather extreme low tire pressures.
At the other end of the spectrum is systems such as the Dominator. Relatively compliant suspension, minimal damping in the air shocks, and relatively long travel. Locating the RC of the Dom is very simple - the point of intersection between the A-arms. Put a piece of tape there. Then compare it with the CG which can be approximated by the thrust line just for the sake of discussion as the Dom is a CLT machine. On my Dom, the RC is some 10" inches ABOVE the CG. This is a good design in that the lower CG with respect to RC tends to add stability to ground handling. This can be readily verified by taxiing the Dom with the rotor tied in place. With the differential braking and castering nose wheel, the Dom is surprisingly stable.
So why do the Doms have a reputation as being "tipsy"? The problem, IMO, is the forces exerted by the disc on the rolling chassis. It is, in effect, a big "hand" pressing on the end of a long moment arm (i.e., the mast). The Dom is unusually tall with respect of rotor to ground so it has a long moment arm. If pilot landing technique is improper, that force will unsettle the gyro regardless of suspension.
To mitigate this phenomena, most pilots choose to improve the damping of the stock air shocks by using springs and dampers. The net effect is not unlike the use of an anti-roll bar on a car with the improved compression and rebound characteristics slowing down the action and giving the pilot an opportunity to get the disk into a more proper attitude.
Proper technique requires that the landing be as square as possible - tail wheel touching first to insure proper yaw attitude by pulling the craft straight, but the pilot needs to work roll so that both mains touch together. IMO, a slight amount of wheel camber (top in) and toe-out will help settle the aircraft as it lands. And never, never, touch the nose gear first.
My thoughts reflect suspension engineering basics, not practical experience. Your mileage may vary.