I have another example, this time with the CD-850 transmission which is triple differential type. This is a diagram of the CD-850-5, I’ve highlighted the planetary gear set ups, including the two planetary differentials, for easier clarification.
In red are the two planetary differentials, green is the planetary gear set that provides the low and high gear ratios/ranges, and blue is the reverse planetary which provides the reverse gear ratio.
We can ignore the blue and green, and focus solely on the steering, which comes from the differential in the middle. Assuming 2800 input RPM, and that the torque converter is perfectly efficient, the differential receives 2791.79 RPM due to the initial gear ratios (22:34 and 31:20). This is a spur gear differential, and when the brakes aren’t applied on either side it rotates all together as a complete unit. On each side of the differential are 69 teeth gears, one for each side, which mesh with another 69 teeth gears, for 1:1. Finally we have a 109:31 gear ratio which goes into the sun gears of the planetary differentials, which gives us 793.99 RPM on those sun gears.
Whenever you apply one of the brakes on the steering differential (assume we apply them fully), one side is fully stopped, meaning the other side speeds up. This means the braked side receives 0 RPM on the planetary differential sun gear, and the other receives 1587.99.
No matter what gear is selected, those are the values you will get on those sun gears with the engine at 2800 RPM.
In effect, with triple differential steering, what happens is that instead of the steering input being connected and disconnected to the planetary differentials, it is always connected, and in straight line drive it provides equal RPM to both sides. When turning, RPM from one side is removed and added to the other side. But (assuming the brakes are fully applied), it is always the same RPM addition and subtraction. This means that at a specific engine speed, the difference between track speeds when turning is always the same. However because track speed is much higher when on a high gear, it means that the same difference makes up a smaller percentage, leading a wider turning radius on high gears than on low gears.