Software Lab 02: Modeling Wall Finder

The questions below are due on Tuesday February 12, 2019; 08:25:00 PM.
 

Files

This lab description is available as a PDF file here. There is no code distribution for today's lab.

1) Swann's Way

In last week's Design Lab, you programmed the robot to keep itself a fixed distance from a wall. One of the strategies we explored to accomplish this task was proportional control, wherein the robot's velocity is set to be its "error" (i.e., how far is the robot from its desired distance), scaled by a constant factor which we referred to as a gain.

In today's lab, we will use our intuition about the physical world and our experience with the robot to develop a model of how the wall-finding robot moves through the world.

Ultimately, this will be useful not only as an intellectual exercise, but to help us choose values of the gain that lead to the performance we would like from the robot. What we start here today will continue into the Design Lab, where we will develop a program to simulate arbitrary LTI systems and use that simulator to reason about and improve our model from today.

2) Signals and Systems

The brain/robot system with which you experimented in Design Lab 1 is an example of a control system with a single input (the desired distance from the wall, provided by the user) and a single output (the robot's actual distance from the wall, as reported by the sonar sensors). Thus, we can think about the wall finder system by the way it transforms the input signal into the output signal:

3) Breaking It Down

The wall-finder is a pretty complicated system, and so in order to think about it, we will try to break it down into smaller systems, about which we can think more easily. We will start by breaking the system down into two parts:

The controller represents our control scheme. In this application, it is, in some sense, the soar brain we wrote to control the robot. We will model the controller as taking in the error signal (E, which is the difference between the desired distance, D_d, and the distance from the wall as measured by the sonars, D_s) as input and yielding a commanded velocity, V, as output.

The plant represents the part of the system that we are controlling¹. In this application, the plant represents the robot's locomotion system. Unlike the controller, we do not have much say in how this system behaves; our job is to model, as accurately as possible, the actual physics of the plant we are given.

3.1) Controller

We will start by building a model of the controller. Notice that the controller takes in an error (the desired distance minus the sensed distance) and outputs a velocity.

Write a difference equation that describes the controller (i.e., that describes the relationship between the error e[\cdot] and the velocity v[\cdot]).

Note that a difference equation, in general, has the form:

Enter your difference equation into the box below by specifying the "c coefficients" and "d coefficients" from the form above. Enter each set of coefficients as a list of Python numbers representing [c_0, c_1, c_2, \ldots c_{k-1}] and [d_0, d_1, d_2, \ldots, d_j].

Note that the "d" coefficients are associated with the input (in this case, the error signal E) and the "c" coefficients are associated with the output (in this case, the velocity signal V). You may want to write your answer out as a difference equation first, and then to translate it to the form described above afterwards (and don't hesitate to ask for help if you're having trouble with that representation!).

Your answer can depend on K (the constant of proportionality) and/or T (the length of a timestep, in seconds).

Draw a block diagram for this system in the handout, including labels for its input and output signals.

3.2) Plant

Now we will model the plant.

The plant is, in itself, a fairly complicated system as well. It takes a commanded velocity from soar and transforms that into a distance as reported by the sonar sensors. We can make this more manageable by splitting the plant into a few subsystems. So rather than thinking about the following:

We can think about the plant itself as being made up of two subsystems:

1. \text{plant}_1 transforms the robot's velocity V into the robot's actual position in the physical world, which we will call P.
2. \text{plant}_2 transforms the robot's actual position P into a distance as reported by the sonars.

3.2.1) Coordinates

We will think about the robot's position P (meters) as being measured relative to the position of the wall. That is, the robot's position will be 0 when the robot is touching the wall, and will be negative when the wall is in front of the robot. Positive velocities (meters / second) cause the robot to move forward (toward the wall).

We will move forward by writing a model for \text{plant}_2, before moving on to \text{plant}_1.

3.3) Modeling plant2

Write a difference equation that describes plant2 (i.e., that describes the relationship between the position p[\cdot] and the sensed distance d_s[\cdot]).

Note that a difference equation, in general, has the form:

Enter your difference equation into the box below by specifying the "c coefficients" and "d coefficients" from the form above. Enter each set of coefficients as a list of Python numbers representing [c_0, c_1, c_2, \ldots c_{k-1}] and [d_0, d_1, d_2, \ldots, d_j].

Your answer can depend on K (the constant of proportionality) and/or T (the length of a timestep, in seconds).

Draw a block diagram for plant2 in the handout, including labels for its input and output signals. When building your model, you may assume that the sonars do not introduce any delay (they can instantaneously measure the exact distance to the wall).

3.4) Modeling plant1

Write a difference equation that describes plant1 (i.e., that describes the relationship between the velocity v[\cdot] and the position p[\cdot]).

Note that a difference equation, in general, has the form:

Enter your difference equation into the box below by specifying the "c coefficients" and "d coefficients" from the form above. Enter each set of coefficients as a list of Python numbers representing [c_0, c_1, c_2, \ldots c_{k-1}] and [d_0, d_1, d_2, \ldots, d_j].

Your answer can depend on K (the constant of proportionality) and/or T (the length of a timestep, in seconds).

Draw a block diagram for plant1 in the handout, including labels for its input and output signals.

3.5) Putting It Together

What is the difference equation for the entire combined system that relates D_d to D_s?

Hint: you may find it helpful to use operator notation in order to solve for this relationship.

Draw a block diagram for the entire wall finder system in the handout, labeling all of the signals discussed above (D_d, E, V, P, and D_s). If you need extra scratch paper, please ask a staff member.

 

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Footnotes

¹The term "plant" is a holdover from the early days of control theory. One of the first applications of this theory was in controlling processes in manufacturing plants, and the term has stuck around now to mean, more generally, the system we are controlling. It doesn't represent a leafy green thing in the system. (click to return to text)