Assume the resistance torque on the ball-valve is known as T, a constant, how much axial force F needed on the piston or two pins to drive the ball rotation?
As you can see I have done two calculations for the initial position in the middle section of attachment, I think the right one, method 2, is the correct one. My question is how to dissolve the for F? Method 1 or 2 or #3? In the pic, arm is the vertical distance for the torque.
But in the lower section of attachment, you can see I also run a solidworks motion study and found the curve for piston force which seems to indicate the method 1 is correct. If the method 2 is correct, then I should see a horizontal line with value at T.
You would be better off to draw your vector diagram showing radial and tangential components.  The torque value will be then directly proportional to the radial component. Both the radial and tangential components will change in magnitude and direction as the ball rotates.
The torque equation neglecting friction will be proportional to the radial force and the distance between the pin and the ball centers.  That distance will also change during the ball rotation explaining the slots to allow for the movement of the pins.

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