## Brachistochrone Optimal Control

## Apps.BrachistochroneProblem History

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Deleted lines 46-63:

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Changed lines 19-23 from:

{$\quad \frac{dx(t)}{dt} = v sin(u)$}

{$\quad \frac{dy(t)}{dt} = v cos(u)$}

{$\quad \frac{dv(t)}{dt} = g cos(u)$}

{$\quad \frac{dy(t)}{dt} = v cos(u)$}

{$\quad \frac{dv(t)}{dt} = g cos(u)$}

to:

{$\quad \frac{dx(t)}{dt} = v \; \sin(u)$}

{$\quad \frac{dy(t)}{dt} = v \; \cos(u)$}

{$\quad \frac{dv(t)}{dt} = g \; \cos(u)$}

{$\quad \frac{dy(t)}{dt} = v \; \cos(u)$}

{$\quad \frac{dv(t)}{dt} = g \; \cos(u)$}

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{$\mathrm{subject~~/~~;to}$}

to:

{$\mathrm{subject\;to}$}

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{$\mathrm{subject~~ ~~to}$}

to:

{$\mathrm{subject/;to}$}

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{$subject to$}

to:

{$\mathrm{subject to}$}

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{$~~"~~subject to~~"~~$}

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{$subject to$}

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{$\quad x(t_f)=2, \quad y(t_f)=2, \quad v(t_f)=~~"~~Free~~"~~$}

to:

{$\quad x(t_f)=2, \quad y(t_f)=2, \quad v(t_f)=Free$}

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<iframe width="560" height="315" src="https://www.youtube.com/embed/pgJ0jbfFBUE" frameborder="0" allowfullscreen></iframe>

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Attach:download.png [[Attach:brachistochrone_xend_study.zip|Download Brachistochrone Parametric Solution in Python]]

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Attach:download.png [[Attach:brachistochrone_xend_study.zip|Download Brachistochrone Parametric Solution in Python]]

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%width=~~300px~~%Attach:brachistochrone_solution.png

to:

%width=350px%Attach:brachistochrone_solution.png

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The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position while ''v'' is the velocity. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

to:

The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position in the down direction while ''v'' is the velocity. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

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Attach:brachistochrone_parameteric_study.png

to:

%width=500px%Attach:brachistochrone_parameteric_study.png

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An interesting extension to this problem is to compare the solutions to when the value of x is varied from 1.0 to 10.0. The final value of y is fixed at 2.0 below the starting point.

Attach:brachistochrone_parameteric_study.png

Attach:download.png [[Attach:brachistochrone_xend_study.zip|Download Brachistochrone Parametric Solution in Python]]

Thanks to Trent Okeson for providing a solution to this problem.

Attach:brachistochrone_parameteric_study.png

Attach:download.png [[Attach:brachistochrone_xend_study.zip|Download Brachistochrone Parametric Solution in Python]]

Thanks to Trent Okeson for providing a solution to this problem.

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Attach:brachistochrone_solution.png

to:

%width=300px%Attach:brachistochrone_solution.png

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!!!! Solution

Attach:download.png [[Attach:brachistochrone_solution.zip|Download Brachistochrone Solution in Python]]

Attach:brachistochrone_solution.png

Thanks to Trent Okeson for providing a solution to this problem.

Changed lines 9-10 from:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. ~~The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position while ''v'' is the velocity. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').~~

to:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

Added lines 12-13:

The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position while ''v'' is the velocity. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

Changed lines 23-25 from:

{$\quad x(0)=0, y(0)=0, v(0)=0$}

{$\quad x(t_f)=2, y(t_f)=2, v(t_f)="Free"$}

{$\quad x(t_f)=2, y(t_f)=2, v(t_f)="Free"$}

to:

{$\quad x(0)=0, \quad y(0)=0, \quad v(0)=0$}

{$\quad x(t_f)=2, \quad y(t_f)=2, \quad v(t_f)="Free"$}

{$\quad x(t_f)=2, \quad y(t_f)=2, \quad v(t_f)="Free"$}

Changed line 25 from:

{$\quad x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

to:

{$\quad x(t_f)=2, y(t_f)=2, v(t_f)="Free"$}

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{$\~~mathrm{subject~~ to~~}~~$}

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{$\mathtt subject to$}

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{$subject to$}

to:

{$\tt{subject to}$}

Changed lines 23-25 from:

{$ x(0)=0, y(0)=0, v(0)=0$}

{$ x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

{$ x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

to:

{$\quad x(0)=0, y(0)=0, v(0)=0$}

{$\quad x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

{$\quad x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

Changed lines 13-28 from:

subject

initial boundary conditions

x

v

final boundary conditions

x

y(tf) = 2

v(tf) = Free

to:

{$\min t_f$}

{$subject to$}

{$\quad \frac{dx(t)}{dt} = v sin(u)$}

{$\quad \frac{dy(t)}{dt} = v cos(u)$}

{$\quad \frac{dv(t)}{dt} = g cos(u)$}

{$ x(0)=0, y(0)=0, v(0)=0$}

{$ x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

{$subject to$}

{$\quad \frac{dx(t)}{dt} = v sin(u)$}

{$\quad \frac{dy(t)}{dt} = v cos(u)$}

{$\quad \frac{dv(t)}{dt} = g cos(u)$}

{$ x(0)=0, y(0)=0, v(0)=0$}

{$ x(t_f)=2, y(t_f)=2, v(t_f)=Free$}

Changed line 9 from:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

to:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position while ''v'' is the velocity. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

Changed line 9 from:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''~~tf~~''. The variable ''x'' is the horizontal position and ''y'' is the vertical position. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

to:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''t'_f_'''. The variable ''x'' is the horizontal position and ''y'' is the vertical position. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

Changed line 9 from:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''tf''. The variable ''x'' is the horizontal position and ''y'' is the vertical position. The parameter ''g'' is the gravitational constant (assume 9.81 m/s^2).

to:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''tf''. The variable ''x'' is the horizontal position and ''y'' is the vertical position. The parameter ''g'' is the gravitational constant (assume 9.81 m/s'^2^').

Changed line 9 from:

This is a classic problem that has been solved with calculus of variations~~. This~~ particular ~~demonstration~~ is ~~a numerical method to solve this same~~ problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

to:

This is a classic problem that has been solved with calculus of variations but this particular task is to use a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction. The adjustable parameter ''u'' is the slope and can be adjusted over the minimized time horizon ''tf''. The variable ''x'' is the horizontal position and ''y'' is the vertical position. The parameter ''g'' is the gravitational constant (assume 9.81 m/s^2).

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%height=~~70px~~%Attach:brachistochrone.gif

to:

%height=100px%Attach:brachistochrone.gif

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A classic optimal control problem is to compute the [[https://en.wikipedia.org/wiki/Brachistochrone_curve|brachistochrone curve]] of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time.

Attach:brachistochrone.gif

Attach

to:

A classic optimal control problem is to compute the [[https://en.wikipedia.org/wiki/Brachistochrone_curve|brachistochrone curve]] of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. The following animation (source: Wikipedia) shows different trial solutions (blue) and the optimal solution (red) for a particular starting and end point.

%height=70px%Attach:brachistochrone.gif

%height=70px%Attach:brachistochrone.gif

Changed lines 5-9 from:

A classic optimal control problem is to compute the [[https://en.wikipedia.org/wiki/Brachistochrone_curve|brachistochrone curve]] of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time.~~ ~~This is a classic problem that has been solved with calculus of variations. This particular demonstration is a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

to:

A classic optimal control problem is to compute the [[https://en.wikipedia.org/wiki/Brachistochrone_curve|brachistochrone curve]] of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time.

Attach:brachistochrone.gif

This is a classic problem that has been solved with calculus of variations. This particular demonstration is a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

Attach:brachistochrone.gif

This is a classic problem that has been solved with calculus of variations. This particular demonstration is a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

Changed line 5 from:

A classic optimal control problem is to compute the ~~brachistochrone curve of fastest descent~~. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. This is a classic problem that has been solved with calculus of variations. This particular demonstration is a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

to:

A classic optimal control problem is to compute the [[https://en.wikipedia.org/wiki/Brachistochrone_curve|brachistochrone curve]] of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. This is a classic problem that has been solved with calculus of variations. This particular demonstration is a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

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(:title Brachistochrone ~~Solution with MATLAB and Python~~:)

to:

(:title Brachistochrone Optimal Control:)

Changed line 5 from:

A classic optimal control problem is to compute the brachistochrone curve of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. ~~The problem has been solved with calculus of variations. This is a demonstration of~~ a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

to:

A classic optimal control problem is to compute the brachistochrone curve of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. This is a classic problem that has been solved with calculus of variations. This particular demonstration is a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

Added lines 1-45:

(:title Brachistochrone Solution with MATLAB and Python:)

(:keywords Python, MATLAB, nonlinear control, Brachistochrone, dynamic programming, optimal control:)

(:description Minimize final time on which a bead slides without friction under the influence of a uniform gravitational field to a given end point in the shortest time.:)

A classic optimal control problem is to compute the brachistochrone curve of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. The problem has been solved with calculus of variations. This is a demonstration of a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

!!!! Problem Statement

minimize tf

subject to

dx/dt = v * sin(u)

dy/dt = v * cos(u)

dv/dt = g * cos(u)

initial boundary conditions

x(0) = 0

x(0) = 0

v(0) = 0

final boundary conditions

x(tf) = 2

y(tf) = 2

v(tf) = Free

!!!! Solution

----

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(:keywords Python, MATLAB, nonlinear control, Brachistochrone, dynamic programming, optimal control:)

(:description Minimize final time on which a bead slides without friction under the influence of a uniform gravitational field to a given end point in the shortest time.:)

A classic optimal control problem is to compute the brachistochrone curve of fastest descent. A point mass must slide without friction and with constant gravitational force to an fixed end point in the shortest time. The problem has been solved with calculus of variations. This is a demonstration of a numerical method to solve this same problem. The solution curve is independent of both gravitational force and the mass of the object. The solution is different if there is an initial velocity or if there is friction.

!!!! Problem Statement

minimize tf

subject to

dx/dt = v * sin(u)

dy/dt = v * cos(u)

dv/dt = g * cos(u)

initial boundary conditions

x(0) = 0

x(0) = 0

v(0) = 0

final boundary conditions

x(tf) = 2

y(tf) = 2

v(tf) = Free

!!!! Solution

----

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