Design Lab 03: Staggering Proportions
Files
This lab description is available as a PDF file here. The code distribution may be downloaded
(in zip format) here, or by running athrun
6.01 getFiles from an Athena machine or a 6.01 lab laptop.
1) Proportional Wall Follower
In today's lab, we will explore a new problem which is closely related to the "wall finder" task we have been working on in the last two weeks: today, we wish to drive the robot at a constant forward velocity of V=0.1 m/s along a wall.
To maintain a constant distance between the robot and wall, we will steer left if the robot is too close to the wall on its right and steer right otherwise. We will think about this problem as a control problem in which we set the rotational velocity \omega [n] (not shown) to reduce the error between the desired distance d_i[n]=0.5 m and the actual distance d_ o[n]. Then if the error e[n] = d_ i[n]-d_ o[n] is positive (indicating that the robot is too close to the wall), we will set \omega [n] to be positive, which will steer left (which increases \theta [n], and ultimately increases d_ o[n]); if e[n] is negative (the robot is too far from the wall), we will set \omega [n] to be negative, causing the robot to turn back toward the wall. This process is depicted in the figure below:
2) Simulated Robot
Edit the file wallFollowerBrain.py to implement a proportional
controller. We have provided a procedure called get_distance_right to
calculate the perpendicular distance to the wall from inp.sonars, as
follows. If both sonars 6 and 7 hit the wall (as indicated by values less than
1.5 meters), then these hits establish a straight line, from which the
perpendicular distance to the center of the robot is calculated. If only one
of sonars 6 and 7 hit the wall, then the distance from the center of the robot
to the sonar hit point is returned (since we cannot triangulate the position of
the wall). If neither of sonars 6 and 7 hits, then None is returned.
Test your code by using soar in wallFollowerWorld.py.
Notice that the brain has code to plot the distance to the wall as a function
of discrete time (where one step is 0.1 seconds) after the brain has been
stopped. Experiment with a few values of the gain of your proportional
controller to determine how gain affects the resulting behavior. For each
oscillatory response, determine the period of the oscillation, and enter the period in the table in the
handout. If a gain does not cause the robot to oscillate, enter None in that row. ** Save plots for each of the values of gain below** to illustrate
the trends that you found.
Be sure to Reload Brain and World in soar after changing the
gain in your brain file in between trials.
| K | Period from Simulated Robot |
| 1 | |
| 2 | |
| 5 | |
| 10 | |
| 50 | |
| 100 |
What are the units of the periods in the table in the handout?
Show your plots to a staff member. What quantities (and units) are represented on each axis? Describe how the gain
K affects the
behavior of the wall-follower system. What value of K is best, and
why? Compare the effect of K in this lab and in last week's lab
("wall finder").
3) Modeling Behavior
We can think of the wall follower system as a control system with the following form (notice that, although we are modeling a different system from last week, it has a similar form). This week, we will consider the output of the system to be the robot's actual distance from the wall, rather than the distance reported by any particular sonar sensor.
3.1) Controller model
Assume that the controller can instantly set the rotational velocity \omega [n] to be proportional to the error e[n], with proportionality constant K. Express this relation as a system functional \frac{\Omega}{E}.
Enter your system functional for the controller below.
Enter your answer by entering the coefficient lists associated with the numerator and denominator polynomials of the system functional (in {\cal R}),
with the zeroth-order coefficient first.
You may use the variables
K, T, and V. Note that, like any other Python
variables, these are case-sensitive!
System Functional:
Numerator coeffs (in {\cal R}):
Denominator coeffs (in {\cal R}):
3.2) Plant Model
As in the wall finder system, we can break the plant into smaller systems in order to help us think about the plant itself. This week, we will break the plant into two pieces: the first part (a "rotator") determines how the angular position of the robot \Theta (in radians) depends on angular velocity \Omega (in radians per second), and the second part (a "translator") determines how distance to the wall D_ o (in meters) depends on angular position \Theta.
When the robot receives a new commanded angular velocity, we assume that the robot ** immediately** changes its angular velocity and then holds the new angular velocity constant until it receives the next command (i.e., the robot's angular acceleration is large enough that we can ignore the acceleration time).
3.2.1) Rotator
Determine the system functional of the first part of the plant, \frac{\Theta}{\Omega}, where \Theta is the robot's angular orientation with respect to the wall, which depends on its rotational velocity \Omega. Assume that the rotational velocity at time n-1 is \omega [n-1], and that this rotational velocity is constant until time n. Also assume that the time between steps is T (in seconds).
Enter your system functional for the "rotator" system below.
Enter your answer by entering the coefficient lists associated with the numerator and denominator Polynomial instances.
In each box, enter a Python list containing one or more numbers, with the coefficient associated
with the lowest power of (\mathcal{R}) first.
You may use the variables
K, T, and V. Note that, like any other Python
variables, these are case-sensitive!
System Functional:
Numerator coeffs (in {\cal R}):
Denominator coeffs (in {\cal R}):
This system (the rotator) should look familiar. What operation is it performing? Note that this is a general system that always applies this operation!
3.2.2) Translator
Determine the system functional of the second part of the plant, \frac{D_o}{\Theta}, where D_o is the robot's perpendicular distance from the wall, which depends on its angular position \Theta.
Assume that the angular displacement at time n-1 is \theta [n-1], and that this angular displacement is constant until time n. Assume that the velocity in the direction that the robot is traveling is constant, at V (meters/second).
Linearize the resulting difference equation using the small angle approximation (i.e., if \theta is small, then \sin \theta \approx \theta ). This approximation makes our model linear, allowing us to analyze it easily. It may be useful, however, to think about the consequences of this approximation when trying to account for behaviors in subsequent sections.
Enter your system functional for the "translator" system below.
Enter your answer by entering the coefficient lists associated with the numerator and denominator Polynomial instances.
You may use the variables
K, T, and V. Note that, like any other Python
variables, these are case-sensitive!
System Functional:
Numerator coeffs (in {\cal R}):
Denominator coeffs (in {\cal R}):
3.3) Block Diagram
Draw a block diagram for the entire system (which transforms D_i into D_o) in the space provided in the handout.
You will need this block diagram for the checkoff.
3.4) System Functional
Calculate the system functional for the entire system (which transforms D_i into D_o). Note that we have included scratch paper if you need extra space for algebra.
Enter the system functional associated with the complete system by entering the
coefficient lists associated with the numerator and denominator Polynomial instances.
You may make use of the following variables: T, V, K. Note that,
like any other Python variables, these are case-sensitive.
System Functional:
Numerator coeffs (in {\cal R}):
Denominator coeffs (in {\cal R}):
3.5) Poles
Find an algebraic expression for the poles of the system.
Enter the algebraic expression for one pole.
You should enter this using normal Python syntax. If you click "Check Syntax",
you will be shown a representation of how your answer was parsed. If you have
not entered a valid Python expression in the box, you will see ERROR.
You may make use of the following variables: T, V, K. Note that,
like any other Python variables, these are case-sensitive.
The function sqrt is available for you.
We won't have a way of specifying \pm, so enter either one of the poles in the box below.
Note that, for positive K, these poles are complex-valued. Thus, we will need to understand a few things about complex numbers in order to use the poles to reason about the expected behvaior of the system. If you do not have experience working with complex numbers, or if you need a refresher, please read section 1.5.2.3 of the Signals and Systems course notes and/or this set of supplemental notes.
We can express these poles in polar form as r\cdot e^{\pm j \phi}, where r is the magnitude of the poles, and \phi is an angle in the complex plane (one pole has angle positive \phi, and the other has angle -\phi).
Enter expressions for r and \phi below, in terms
of the variables T, V, K. Note that, like any other
Python variables, these are case-sensitive.
The functions sqrt and atan2 (as described in the Python
documentation for the math module) are available for you.
Determine the period of the response in terms of r and \phi.
Use `pi` to represent $\pi$, and `r` and `phi` to represent $r$ and $\phi$.
What are the units of the period?
3.6) Fundamental Modes
Choose one of the poles $p$, and make a sketch of the real part of $(p^ n)$ versus $n$, when $K=1$, $T=0.1$ second, and $V=0.1$ m/s. Also, sketch the imaginary part of $(p^ n)$ versus $n$. **Fully label all axes.** What do these plots tell you about the response of the system?
3.7) Root locus
On the complex plane below, indicate (with \times's) the pole locations associated with gains K=1, K=10, and K=100. Draw a line (or lines) on the complex plane to indicate the pole locations that result for all possible gains K.
Describe the poles of the wall follower system. How does the gain K affect the poles? What are the implications of the poles for system behavior?
4) Get Real
Test your code from checkoff 1 on the real robot (not in soar simulation):
- Locate a robot (there should be one near your table).
- Set up two long white bubble-wrapped boards to form a long, straight wall for the robot to follow.
- Start the robot with its center 0.3 meters from the wall, facing at an angle of approximately \pi / 6 radians away from the wall.
- Run the wall follower for each of the gains you used in checkoff 1. For each oscillatory response, estimate the period of the oscillation from the generated plot, and enter the gain and period in the table in the handout. Save the plot for each gain to illustrate the trends that you found.
Be sure to Reload Brain and World in soar after changing the
gain in your brain file in between trials.
| K | Period from Model | Period from Simulated Robot | Period from Real Robot |
| 1 | |||
| 2 | |||
| 5 | |||
| 10 | |||
| 50 | |||
| 100 |
predicted_periods to
contain a list of the periods predicted by your LTI model for each of the
gains above (in increasing order of gain), and a variable called
realrobot_periods to be a list of the periods you estimated from the
graphs generated by soar for those agains (again, in increasing order of
gain). Enter all periods in units of timesteps, not seconds.
In addition to checking your predicted periods, this will generate a graph of period versus gain. Note that this will only check your predicted values, which must be entered accurate to within 10% of actual period predicted by the pole. It will not check your estimated results; those will be discussed in the checkoff. Be sure to click "Show/Hide Detailed Results" to view this graph.
Show your plots to a staff member. Compare the results from the mathematical model to the results from the
soar simulation and the
real robot.