aerial robotics lecture 3b supplemental_3 supplementary material - minimum velocity trajectories

7/25/2019 Aerial Robotics Lecture 3B Supplemental_3 Supplementary Material - Minimum Velocity Trajectories

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Lecture 3B.Supplemental_3

Supplementary Material - Minimum VelocityTrajectories

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

tx

dtxtx

%

&

'(

) minarg'(

*e were able to use the +uler-Lagrange equation to find that the general form of the

equation is$

%'( ctctx

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$

&&dxdtds

*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,dt

dxx $

dtxds &


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To find the total length of the curve we #ust integrate d/ along the entire curve.

T

dtx

%

&dscurveofLength

*e can now mathematically represent the problem of finding the minimum-distance

tra#ectory in the familiar form$

T

tx

dtxtx

%

&

'(

) minarg'(

1inding the function, /)(t' that minimi2es the integral of a cost-function with respect

to t. In this case, the cost function, L, is$

dtxtxxL

&

',,(

3gain, the necessary condition for the optimal tra#ectory is given by the +uler-

Lagrange 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$

%x

L

The partial derivative of L with respect to x is$

& x

x

x

L

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 L4x .

5owever, it turns out that we don!t need to e/plicitly calculate this. 6ubstituting theterms we7ve found into the +uler-Langrange +quation, we get the following

e/pression for the necessary condition for the minimum-distance tra#ectory$

% &

x

x

dt

d

*e can directly integrate this e/pression with respect to time to get an e/pression for

the velocity of the tra#ectory$

K

x

x &

&

&

c

K

Kx


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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 cgives the

position function of the minimum-distance tra#ectory$

%'( ctctx

cand c%are arbitrary constants. *e see that this is the equation for the minimum-

velocity curve. Thus, given the same set of boundary-conditions, the minimum-

velocity tra#ectory will be the same as the minimum-distance tra#ectory.

aerial robotics lecture 3b supplemental_3 supplementary material - minimum velocity trajectories

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