Linearization of servos, and expo setting info...
Taking the mystery out of expo settings on radios --- finally!
Did you ever wonder what those functions actually were that set the "expo" for your radios? Well, wonder no longer!
These are actually CUBIC functions that are symmetrical about the y-axis (relative throw) and are designed to either reduce or enhance response up to full throws.
After some digging, here is the formula!!!
y = rx³ + (1-r)x
x = control throw (stick position) amount. -1<x<1 for most radios;
r = the exponential rate setting in decimal equivalent of %.
(so for example, 25% expo = 0.25, and x = ±1.00 for full stick movement)
I have verified this result with two sources: One on the net, and another with software - Real Flight expos do the same on expo rates on models.
So, what does this really mean? The higher the expo, (above zero), the more dead band we find in the center of the stick positions - this IS useful to linearize a model's deflection of control surfaces. Did you think that throws are linear? Nope, not quite. The SERVOS are, in angle versus input from the sticks, but the movement is actually the SINE of the angle. I have some data below that illustrates what the relative motions of the surfaces behave for three angles: ±45°, ±60°, and ±75°.
±60 degrees is the normal throws at 100% for most servos. 75° is a 125% setting on control throws, and 45° is at the 75% throw rate.
The reason this is important to consider is that the geometry of the deflection is linear, and the throws from the servos are directed in an angular fashion, both from the servo arms and from the control surface hinges. For those rusty, the sine of the angle is the ratio of the side opposite the angle of interest and the hypotenuse, or in this case, the control surface itself (length). Since this surface of the rudder, elevator, etc. during deflection is a constant, it is the linear throw distance which is the one we wish to linearize, or at least try to. One can also consider the arm of the servo as the hypotenuse, and the linear throw direction (which moves the control surface via the linkage) as well.
In that case, we are looking at the RELATIVE amounts of throw versus angular input signals (which are sent as pulses from the TX) which are not linear with respect to the throw distance!
For example, using 60° as the standard "100%" throw in angle off the servo gear, the linear throw will be sin(60) or 0.866... times the servo arm length. If we use no expo, then if I say go to 1/2 the stick input on my TX, the radio sees this to direct a 1/2 angular throw on the servo gear, making the 1/2 throw at 30°. Sin(30) = 0.5. The relative throw is now not 1/2, but 0.5/0.866 = ~0.577, or almost 58%! This is a 8% difference in throw from what we think we have on our stix to what is actuated by the servo. As we get to higher throws, the relative changes begin to diminish and the response on an uncompensated servo system is actually deader on the extremes, and more sensitive on the middle and near-center settings of the stix. Basically, an inherent negative expo response! by adding positive expos, we can re-linerize this relative servo throw response.
Below are some sine curves for the throws mentioned above, and the expo settings that should yield a close-to-linear throw response.
For 50-75% rates, a 15% expo is close to ideal. For full rate, 25% expo is close. For 125% rates, a 35% expo is close to linear. If one wishes more dead band in the center, then they should use higher expos than these to yield a softer middle response from their radios. Some radios use negative values for flattening the middle response, so then use this same value (but negative. ie: -25%) in those cases. Graphs below have been normalized to 100% for both axes.
Hopefully, this initial article here will help take some of the mystery out of expos!
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