*** Refer to the attached graph when reading this description ***


Let X be a joystick axis with value ranging from -1 to +1.

Plot y=x and you get a straight line with slope (gain) = 1


Now plot y = x^3 and you get a cubic output, also ranging from -1 to +1.

The gain of this cubic curve is less than 1 for all x values between -sqrt(1/3) and +sqrt(1/3), and greater than 1 outside that range.  At x=0 the gain is zero (essentially giving a small deadband), and at x=1 the gain is 3.


Now plot y = a(x^3) + (1-a)x, where "a" is a constant in the range 0<=a<=1.  You get a family of curves lying between the curve y=x and y=x^3.  When a=0 you get y=x, and when a=1 you get y=x^3. When 0<a<1, you get a "blend" in-between y=x and y=x^3.  The range of y is still -1 to +1 for all curves in the family.

The gain of each curve is given by:

dy/dx = 3a(x^2) + (1-a)

Set this gain equal to 1 and solve for x:

3a(x^2) + (1-a) = 1  =>  x = +/- sqrt(1/3) = +/- 0.577

... so the "gain crossover" between y=x and y=a(x^3)+(1-a)x always occurs at the same x value, namely 0.577.

The gain of y=a(x^3)+(1-a)x at x=0 is (1-a), and the gain at x=1 is 1+2a:

x=0		gain = 1-a
x=0.577		gain = 1
x=1		gain = 1+2a


You can set "a" equal to a hard-coded constant in your software, or you can use (for example) the throttle on a joystick to vary the value from 0 to 1 so the driver can select the desired sensitivity.