Lecture 3B.Supplemental_3
Supplementary Material - Minimum Velocity Trajectories Presented by Lucy Tang, PhD student. In lecture we posed the question of why is the minimum-velocity curve also the shortest-distance curve In this segment, we!ll discuss the answer to this question. "ecall that to find the minimum-velocity curve, we solved for the tra#ectory$ T
)
x (t '
arg min x & dt x ( t '
%
*e were able to use the +uler-Lagrange equation to find that the general form of the equation is$ x (t '
ct c%
ow, let!s find the minimum-distance tra#ectory. iven two points, a and b, we want to find the tra#ectory that is the shortest in total length. *e can find the length of a tra#ectory between a and b by integrating infinitesimal segments ds along the curve$
+ach infinitesimal segments ds has a corresponding change in dt and a change in d/. *e can find the length of the segment ds using the distance function$ ds
&
dt
dx &
*e can rewrite this function by factoring-out a factor of dt from under the square root, and then ma0ing use of the fact that, by definition, x ds
x &dt
dx dt
$
To find the total length of the curve we #ust integrate d/ along the entire curve. T
Length of curve ds
x & dt
%
*e can now mathematically represent the problem of finding the minimum-distance tra#ectory in the familiar form$ T
x
)
(t ' arg min x ( t '
x & dt
%
1inding the function, /)(t' that minimi2es the integral of a cost-function with respect to t. In this case, the cost function, L, is$
, x, t ' L( x
x &dt
3gain, the necessary condition for the optimal tra#ectory is given by the +ulerLagrange equation. To find this condition, we need to evaluate the +uler-Lagrange +quation for our cost-function, L. *e start by evaluating each term in the equation. The partial-derivative of L with respect to / is %, because / does not appear in the cost-function$ L x
%
The partial derivative of L with respect to x is$ L
x
x x &
ote that this is the partial-derivative of L with respect to x . *e are not ta0ing any derivatives with respect to time yet. To use the +uler-Lagrange equation we need to find the time-derivative of L4 x . 5owever, it turns out that we don!t need to e/plicitly calculate this. 6ubstituting the terms we7ve found into the +uler-Langrange +quation, we get the following e/pression for the necessary condition for the minimum-distance tra#ectory$
x % & dt x d
*e can directly integrate this e/pression with respect to time to get an e/pression for the velocity of the tra#ectory$
x x &
K
x
&
K
K &
c
5ere, 0 is an arbitrary constant. *e can solve this equation for x . *e see that x is a function of only the constant 8. Therefore, x is a constant and doesn7t vary with time. *e can re-label this constant as c. Integrating the e/pression x 9 c gives the position function of the minimum-distance minimum-distance tra#ectory$ x (t '
ct c%
c and c% are arbitrary constants. *e see that this is the equation for the minimumvelocity curve. Thus, given the same set of boundary-conditions, the minimumvelocity tra#ectory will be the same as the minimum-distance tra#ectory.
