CS 282

Any question about…
◦ SVN
 Permissions?
 General Usage?
◦ Doxygen
 Remember that Project 1 will require it
 However, Assignment 2 is good practice place
◦ Assignment 2
 Is posted on the class website

Go over methods of integration
◦ Euler’s method
◦ Analytical
◦ Approximation

Examine the boat problem

Program Runge-Kutta
◦ Compare against analytical and euler methods
◦ Plot position vs. time (due next week’s lab)


We use integration to move our system
forward in time given a force/control
If you remember the falling ball lab, we used
an Euler method (kind of like cheating) to
integrate
◦ velocity = old_velocity + delta_time * acceleration
◦ position = old_position + delta_time * velocity

There exists better methods, however, for
integration.

Let’s say we are trying to solve the equation
x–5=0

Analytical methods solve for the exact answer
◦ In this case it’s easy to come up the analytical answer
x=5

Let’s assume we weren’t so smart. Then we could
develop an algorithm to numerically solve this
problem on a computer
◦ The algorithm could have the computer “guess” the
answer by testing different values of x. We could specify
our “error tolerance” to speed this process up.
Forces on the boat:
B = Buoyancy
W = Weight
T = Thrust
R = Resistance
If B = W, we can ignore the movement in y

new_velocity = old_velocity + acceleration * dt
new_position = old_position + new_velocity * dt

Pros:

◦ Easy to compute
◦ Easy to understand
◦ This means it’s also easy to program

Cons:
◦ Highly erroneous
◦ Not stable (i.e. tends to not converge to the true solution)


Open up this week’s framework
Examine rigid_body.cpp
◦ Notice that it is conveniently modeled like a boat
◦ Take a look at the function “euler_update_state”
 Make sure you understand each line in this function
 This function is the Euler method of updating the
boat’s state

Compile the code
Run the executable

Now let’s examine the analytical solution



Here we see the equations for the velocity, distance
traveled, and acceleration of the boat
Pro
◦ The one, true solution

Cons:
◦ Often hard to derive
◦ For complex systems, infeasible to implement


Why not take advantage of the fact that we’re
computer scientists?
Pros
◦ More realistic compared to Euler’s method
◦ Utilizes computational power
◦ We do not have to analytically solve (which for some
systems is infeasible)

Con
◦ Not as accurate since its an approximation
◦ Power depends on skill of programmer and the
computational capacity of the computer
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

The numerical method we will be using today
is called “Runge-Kutta”
Essentially, Runge-Kutta uses Euler’s Method
and splits up the integration into four parts. It
then takes the average of these parts, and
computes the new integration.
The reason this works better is because after
each split, it uses a newly computed control
to re-integrate.





First, we compute k1 using Euler’s method of
integration.
After this, we integrate while adding half of k1 to our
velocity (or state). This allows us to solve for k2.
Next we integrate while adding half of k2, and solve for
k 3.
Finally, we integrate by completely adding k3 and solve
for k4.
After averaging together all kx, we now have a new
velocity (or state).
1
k1  2k2  2k3  k4 
6
k1  h f  x, y 
f 
1
1 
 
k 2  h  f  x  h, y  k1 
2
2 
 
1
1 
 
k 3  h  f  x  h , y  k 2  
2
2 
 
k 4  h f  x  h, y  k3 
k2
h  step _ size
k4
k3
k1
x
i
Credit to:
Dr. E.W. Sandt
Civil Engineering Department
Texas A&M University
f
xi +
h/2
xi + h
Consider
Exact Solution
dy
2
 yx
dx
The initial condition is:
The step size is:
Credit to:
Dr. E.W. Sandt
Civil Engineering Department
Texas A&M University
y  2  2x  x  e
2
y0  1
h  0.1
x
The example of a single step:


k1  h f  x, y   0.1 f 0,1  0.1 1  0 2  0.1
 
1
1 
k 2  h  f  x  h, y  k1   0.1 f 0.05,1.05  0.10475
2
2 
 
 
1
1 
k3  h  f  x  h, y  k 2   0.1 f 0.05,1.  k 2 / 2   0.104988
2
2 
 
k 4  h f x  h, y  k3   0.1 f 0.1,1.104988  0.109499
1
yn 1  yn  k1  2k 2  2k3  k 4   1.104829
6
Credit to:
Dr. E.W. Sandt
Civil Engineering Department
Texas A&M University

The goal for today is to implement RungeKutta for simulating the movement of a boat.
◦ And if you finish early, to start working on the
graph due next week.

At the beginning of next week’s lab, you
must turn in a graph comparing the position
of the boat over time using
◦ the Euler Method (already implemented)
◦ the Analytical Method (plug into the equations)
◦ Runge-Kutta

Let’s do the first step of Runge-Kutta
◦ Follow the Euler method and compute k1
◦ This will involve you computing the Force and
Acceleration
 This gives us k1 = delta_time * acceleration

Now that we have k1, we can use this to
compute the halves (k2 and k3)

Step two starts off nearly the same
◦ However, when computing force, be sure to add
half of k1 to your velocity
 F = thrust - (drag * (velocity + k1 /2) )
◦ Solve for acceleration
◦ Compute k2 like before

Step three is exactly the same as step two,
except use k2 now.

Step four (surprise!) is exactly the same
◦ Except, since we are integrating over the entire
period, we do not halve our duration
◦ Solve for k4 now

We need to average together our kx’s
◦ k2 and k3 account for double because we integrated
only half of them.
◦ So, we have 6 parts in total to average.

Finally, add the average to your velocity, and
compute position like normal.
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Go to lab5.cpp, and look in the DrawGLScene
◦ Comment out the euler method, and add the
appropriate call to the runge-kutta method
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
Examine Runge-Kutta running
It will be hard to visually notice a difference.
Instead, run the Euler method for a short
period, and compare the numbers.

For next week, you will be turning in a graph
comparing the three methods.
◦ Euler Method (already implemented), Analytical
Method (plug into the equations), Runge-Kutta
◦ Output the position at each time step to a file
◦ Copy those numbers into excel (or the software of
your choice) and plot a graph over time comparing
the positions for each method.
◦ For the analytical method, you can
 a) Program the equations and run it
 or b) Plug in time, solve them, and put the numbers
into the graph