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It's interesting to see all the talk about linear servos. But some time ago I did the required calculations and realized that linear servos are just a mute point with today’s radios. I would like to make something VERY clear. I'm not recommending people to setup expos or any other mixes in the radio to make the movement linear because in applications where the circular arm drives another circular movement such as a bellcrank there is nothing more linear than a circular movement driving another circular movement. Given that both have the same radius of rotation the bellcrank "copies" the servo exactly. Any introduction of expo will actually make it less linear. This analysis applies only to a case where the circular servo drives a straight line movement. First of all in some cases such as say the Freya's elevator, you have a circular servo arm movement controlling a circular "X arm" controlling the swashplate. In this case 10 degrees anywhere in the travel will be 10 degrees in the elevator swashplate rotation. You can't get more linear than that and that's why they say for FAI the mechanical mixing is the best. Nevertheless with today's radios whenever you need linear from a circular to straight movement you can make it almost perfectly linear with the mixing options/expos in your radio. As a matter of fact just applying expo makes it so close to linear that's just a waste of time to try to make it better. To demonstrate the point and for those of you who like the reasoning behind an opinion here's an example with the calculations. 
Above is an example of a typical non-linear servo installation.
Let’s study the case of a servo where as usual you hopefully made the pushrod 90 degrees with the servo arm at the center of the travel.What we are interested is to find a relationship between servo angular travel and effective linear travel. To do that lets name the following quantities: Ed = Effective distance Edi = Effective distance after servo moved i degrees T = Total effective travel = Ed - Edi In this example we have four known quantities: Rs = Servo arm Radius = Known quantity Lp = Length of pushrod = Known quantity s = angle of movement at servo arm = Known quantity b = 90 degrees in this example Lets now calculate the rest: Looking at the figure to the left: From Pythagoras we know that: Ed^2 = Lp^2 + Rs^2 so Ed = sqrt( Lp^2 + Rs^2 ) By the way "^2" is used here to mean "squared" or “raised to the power of two”. Therefore "Ed" is now a known quantity. (Known so far: RS, Lp, s, Ed, b ) b = 90 degrees Being the "b" angle 90 degrees we can assume that the tangent of "c" is Rs/Lp Tan(c) = Rs/Lp then c = arcTan ( Rs / Lp ) now "c" is also a know quantity (Known so far: RS, Lp, s, Ed, b, c ) From trigonometry we know that the sum of all angles of any triangle totals 180 degrees.(1) a + b + c = 180 a = 180 - b - c = 180 - 90 - c = 90 - c Since we know "c" then a = 90 - c So "a" is now a know quantity.(Known so far: RS, Lp, s, Ed, b, c, a ) Going to the figure on the right: s = servo arm travel angle The new angle "ai" is the sum of the old angle "a" plus the servo arm angle "s" ai = a + s Since a is known and s is also know then a1 is now known too.(Known so far: RS, Lp, s, Ed, b, c, a, ai ) From trigonometry using the “Law of the Sines” we know that the ratio of the side divided by the opposing angle is the same for every side/opposing angle pair so; Edi / Sin(bi) = Lp / Sin(ai) = Rs / Sin(ci) Using the bi angle and the ai angle pairs and since we are looking for "Edi" we have; Edi = ( Lp / Sin(ai) ) * Sin(bi) Lp = Known ai = Known bi = ? To get the bi angle we need the ci angle so lets find the ci angle; Lp / Sin(ai) = Rs / Sin(ci) ci = arcSin( (Rs * Sin(ai) ) / Lp) Since Rs, ai and Lp ar know then ci is also known now.(Known so far: RS, Lp, s, Ed, b, c, a, ai, ci) From (1) again; ai + bi + ci = 180 bi = 180 - ai - ci Since we know a1 and c1 now bi is also known. (Known so far: RS, Lp, s, Ed, b, c, a, ai, ci, bi) Now that we have ai and bi determined and since Lp is also known we can now calculate Edi using the previous formula: Edi = ( Lp / Sin(ai) ) * Sin(bi) (Known: RS, Lp, s, Ed, b, c, a, ai, ci, bi, Edi) Now to calculate the actual effective travel "T" we only need to subtract Ed minus Edi to get "T" T = Ed – Edi Following is a list of the results and a chart:  Above is a list of the calculated values using a servo radius of 1 inch and a pushrod length of 10 inches.  The blue one is the actual travel without any compensation. Notice that the actual linear travel per degree close to the center of travel is higher than at the ends. The ideal travel (in green) is the one that gives a linear travel across the whole travel. If we take the difference between the actual and the ideal and subtract if from the ideal we can see what a correcting curve should look like in Red. Guess what it looks like? Yes, it looks like an expo travel compensation so there you have your linear traveling servo. Actually you always had it in your radio. If you want to get picky you can use the formulas above and using your actual measured lengths calculate the exact expo required for your application. Augusto
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