1. CS 282

2.  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

3.  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)

4.  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.

5.  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.

6. Forces on the boat: B = BuoyancyT = Thrust W = WeightR = Resistance If B = W, we can ignore the movement in y

7.  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)

8.  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

9.  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

10.  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

11.  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.

12.  First, we compute k 1 using Euler’s method of integration.  After this, we integrate while adding half of k 1 to our velocity (or state). This allows us to solve for k 2.  Next we integrate while adding half of k 2, and solve for k 3.  Finally, we integrate by completely adding k 3 and solve for k 4.  After averaging together all k x, we now have a new velocity (or state).

13. xixi x i + h/2 x i + h k1k1 k2k2 k3k3 k4k4 Credit to: Dr. E.W. Sandt Civil Engineering Department Texas A&M University

14. Consider Exact Solution The initial condition is: The step size is: Credit to: Dr. E.W. Sandt Civil Engineering Department Texas A&M University

15. The example of a single step: Credit to: Dr. E.W. Sandt Civil Engineering Department Texas A&M University

16.  The goal for today is to implement Runge- Kutta 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

17.  Let’s do the first step of Runge-Kutta ◦ Follow the Euler method and compute k 1 ◦ This will involve you computing the Force and Acceleration  This gives us k 1 = delta_time * acceleration  Now that we have k 1, we can use this to compute the halves (k 2 and k 3 )

18.  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 + k 1 /2) ) ◦ Solve for acceleration ◦ Compute k 2 like before  Step three is exactly the same as step two, except use k 2 now.

19.  Step four (surprise!) is exactly the same ◦ Except, since we are integrating over the entire period, we do not halve our duration ◦ Solve for k 4 now  We need to average together our k x ’s ◦ k 2 and k 3 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.

20.  Go to lab5.cpp, and look in the DrawGLScene ◦ Comment out the euler method, and add the appropriate call to the runge-kutta method  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.

21.  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