Aerial Robotics Lecture 2C_4 Quadrotor Equations of Motion

 Lecture 2C.4

Quadrotor Equations of Motion  Now let’s return to the Quadrotor dynamics. These equations tell us the net force and the net moment:

 F 



 F 1



 F 2



 F 3



 F 4

mg a 3





 M   (r 1  F 1 )  (r 2  F 2 )  (r 3  F 3 )  (r 4  F 4 )   M 1   M 2   M 3  M 4

f we com!ine the net force and net moment with the Newton"#uler #quations we $et these two sets of equations: %  %           %   R % mr       mg   F 1   F 2   F 3   F 4   L( F 2   F 4 )  p     p   p        q   I  q   I  q   L( F 3   F 1 )          r    M 1   M 2   M 3   M 4   r   r 

&n the ri$ht side of the first equation' we hae the total thrust which is u1 (this thrust ector is nown in the !ody"fi*ed frame). The matri* + is rotatin$ this thrust ector to an inertial frame. ,t the !ottom you see the net moment' also nown in the !ody fi*ed frame. The equations as they-e !een written hae comonents in the inertial frame on the to' and in the !ody"fi*ed frame at the !ottom. These are the Newton"#uler equations' and these are the equations we-ll use to deelo controllers and lanners for our ehicles. , reasona!le question to as is: how do we actually calculate these arameters. &r in an online settin$' how do we estimate these arameters. n fact' the arameters we really need to now are those corresondin$ to the $eometry' such as the len$th' /' for e*amle' and those corresondin$ to the hysical  roerties' lie the mass' m and inertia' . These arameters aear linearly in these equations' so' if we hae a system that allows us to measured ositions' elocities' and accelerations' it-s actually not that hard to estimate the len$ths' masses and inertias. t is worth erifyin$ that it-s quite easy to calculate the an$ular elocity in the !ody" fi*ed frame. f you now the itch' roll' and yaw an$les' and also the rate"of"chan$e

of the itch' roll' and yaw an$les (on the ri$ht of the equation !elow)' a simle transformation yields the an$ular elocity comonents ' q' and r alon$ ! 1' !2' and !3.

 p  cos   %  cos  sin      q   %     1 sin         r   sin   % cos  cos       This model is quite comlicated. t inoles three comonents of osition' elocity' and acceleration' and three comonents of rotations' an$ular"elocities' and an$ular "accelerations. To $et a feel for the control ro!lem' let’s loo fist at the 0lanar ersion of the model.  e start !y looin$ at the equations of motion in the " lane. e will assume that the ro!ot cannot moe out of this lane or' in other words' that there is no motion in the *"direction. e will also assume that there are no yaw or itch motions:

n this confi$uration' we come u with the three equations that you see here:

 1    sin   %   y  %   m  u  1 1  z          g   cos   %        m  u 2    %    1      %   I  xx   These equations descri!e the rates of chan$e of elocity in the y and 4 directions' and the rate of chan$e of the an$ular"elocity in the roll direction. To descri!e these inds of systems it is useful to define a state ector. n the three" dimensional case' we hae si* aria!les that descri!e the confi$uration of a ro!ot' and a state ector that includes: the confi$uration and its deriatie:

 x   y     z  q  q   ' x    q          f Q is the si*"dimensional confi$uration ector' q and q  constitute a 12 dimensional descrition off the state. The sace of all such state ectors is called a state sace. f we loo at the equili!rium confi$uration' and that confi$uration is defined !y the  osition *%' y%' 4%' the confi$uration !y definition also has a 4ero roll an$le and a 4ero  itch an$le. t could' of course' hae any yaw yaw an$le. The equili!rium confi$uration must also corresond to 4ero elocity. f we write it down in terms of a state sace ector' we hae the equili!rium confi$uration q % and a 4ero"elocity ector:

 x%   y   %  z %  q e  q%   ' xe    % % %       This $ies a 12"dimensional ector' where the !ottom si* elements are all %. 5imilarly' for lanar quadrotors' we hae a three dimensional confi$uration sace' a si*"dimensional state"sace' and equili!rium state that can !e similarly defined: