Due in class at 10:00 on 11 July. To be completed by pairs of students. Pair assignments can be found on D2L.
Compile a brief report (2-3 pages) that provides answers and brief commentary on the questions given below. Attach a hard copy of your lua scripts to the report (copy it into an external text editor and bold the parts that you created or modified). You must also submit your scripts on D2L.
Introduction
Consider a system composed of three simple robots: a, b, and c. The system’s state vector is as follows:
There is no input to this system (i.e. no u(t)). Each robot’s velocity comes from averaging the vectors that point to the other two robots. So for example, the velocity for robot a is as follows:
The velocities for robot’s b and c are defined similarly. Overall, the behaviour of these robots will be to aggregate together.
- Write the two equations giving the velocities for robot’s b and c. Now form the A matrix of the state equation.
- Compute the eigenvalues of A. You can use Matlab or some other tool for this purpose. Characterize the internal stability of this system. It is insufficient to just say “stable” or “unstable”. You should justify your answer with respect to the eigenvalues. You should certainly indicate if the system is asymptotically stable or not.
- Download the V-REP sketch PA2.ttt. Enter the scripts for RobotA, RobotB, and RobotC. Look for “STUDENT” and make the appropriate changes to implement the aggregation behaviour. You only have to modify two lines for each robot and those lines correspond to the velocity equations for each robot. Execute the behaviour and observe the result.Now re-consider your answers to question 2. Interpret the stability of the system with respect to the observed behaviour. Explain what you would expect to observe for all of the following types of stability: unstable, stable, asymptotically stable, marginally stable.
- We will now alter the system to disperse the robots as opposed to aggregating them. Each robot should now move directly away from the average of the other two robots. Give the resulting velocity equations and the new A matrix. Compute the eigenvalues and comment on the system’s stability just as you did in question 2.
- Create a copy of your V-REP sketch called PA2_disperse.ttt. Modify it to use the equations from question 6. Execute the behaviour and observe the result. Now interpret the stability of this system in the light of your observations and your answers to question 4.