Aerial Robotics Lecture 3B Supplemental_1 Supplementary Material - Minimum Velocity Trajectories From the Euler-Lagrange Equations

 Lecture 3B.Supplemental_1

Supplementary Material - Minimum Velocity Trajectories from the Euler-Lagrange Equations Presented by Lucy Tang, PhD student. In lecture, we saw that we can use the Euler-Lagrange equations to deduce the general form of a minimum velocity traectory for a first-order system. In this segment, we!ll go through the details of that calculation. In lecture, we considered "roblems where we had to find the function #$%t& that minimi'es the integral of a cost function L%T&( T 

 x

$

%t &  arg min  x % t &

 L% x, x, t &dt  )

*hen loo+ing for the minimum-velocity traectory, this cost function is  x  . The Euler-Lagrange equation gives the necessary condition that must be satisfied by the o"timal function #%t&.

   L    L )       x dt     x d 

Let!s evaluate the Euler-Lagrange equation for the "roblem of finding the minimum velocity traectory. gain, the cost function for this "roblem is  L% x , x, t &  x . irst let!s find the individual terms in the Euler-Lagrange equation. /ere, it is im"ortant to ta+e care to differentiate with res"ect to the "ro"er variable. The "artial derivative of L with res"ect to # is ). This is because the term # does not a""ear in L. The "artial derivative of L with res"ect to  x  is  x . 0ote that in this ste" we are not yet differentiating with res"ect to time. inally, we can evaluate the time-derivative of L1  x . 2sing the value of L1# found in the last ste", we see that we need to ta+e the time-derivative of  x  which is  .  x

 0ow we can substitute all these terms into the Euler-Lagrange equation. The equation  becomes(

  )  )  x which is equivalent to sim"ly(

   x

)

This is the condition we saw in the lecture for a minimum velocity traectory. *e can integrate this condition once with res"ect to t to get the velocity(

   c3  x /ere, c3 is an arbitrary constant. *e can integrate the velocity to get the "osition function for the minimum velocity traectory(  x %t &

 c3t   c)

/ere c) is another arbitrary constant. In the lecture, we discussed how c ) and c3, can  be found from the "roblem!s boundary boundary conditions.