Homework and Reading guide

Homework guidelines: Homework is due at midnight Fri. in the box outside 122F Votey Hall. On every homework assignment please do the things listed below. After a few weeks of getting used to these you will have to follow all of them to get any credit on homework.

a) On the top right corner neatly print the following, making appropriate substitutions as appropriate:
    Sally Rogers
     HW 2 Due Oct. 5, 2007
     ME 111
b) STAPLE your homework in the top left corner.
c) Please: use only the front side of each sheet.
d) You are encouraged to work with others on homework; you will gain more by explaining to and learning from other students. At the top clearly acknowledge all help you got from TAs, faculty, students, or ANY other source (but for lecture, text, and office hours only). Examples could be "Mary Jones pointed out to me that I needed to draw the second FBD in problem 2." or "Nadia Chow showed me how to do problem 3 from start to finish." or "I basically copied this solution from a Tau Beta Sigma frat file." etc. If your instructor or grader thinks you are taking too much from other sources he/she will tell you. In the mean time, don't violate academic integrity rules by being being unclear about what of your presentation you worked out on your own.
e) Every use of force, moment, momentum, or angular momentum balance must be associated with a clear correct free body diagram . Every use of electrical laws must be accompanied by a clear correct circuit diagram, etc.
f) Your vector notation must be clear and correct.
g) Every line of every calculation must be dimensionally correct. (Carry your units.)
h) Your work should be laid out neatly enough to read. Part of your job as an engineer is not just to get the right answer, but to communicate its justification clearly. So, that is part of your job on the homework as well.
i) All computer code and output must be adequately documented and commented; your name should be clearly visible, as printed by the computer (e.g., title plots with your name, put your name in a comment in the first line of any *.m files, etc.) All computer plots must be titled and have axes labeled.
j) Write equations in symbolic form before plugging in numerical values and solving. It makes it easier for you and us to check your work. This is also a much desired practice for your exam solutions.

Hints : download the analogous example from class plus its MATLAB solutions here: Massless multiple spring-damper system example

(2) My First Robotics Problem: Balance the inverted Pendulum.

In class we studied the stability of the two equilibriums of the simple pendulum (no friction at hinge or air drag, point mass m, massless rod of length ell). Using a LINEARIZED STABILITY ANALYSIS, we found that theta=0 (inverted or upright pos.) is an UNSTABLE equilibrium and theta=pi (hanging) is also an UNSTABLE (or, at best 'NEUTRALLY STABLE) equilibrium, in response to small disturbances or PERTURBATIONS.

In this problem, we replace the pendulum hinge with a motor that can supply a torque T_m = T_M(theta, theta_dot); that is, the motor torque is determined by an instantaneous measurement of the angle theta and its time rate of change, theta_dot. The aim is to design the motor to provide a torque that can stabilize the unstable inverted pendulum; that is, by using sensors to measure angle and angle rate, the goal for the automatic motor measurement and control system to stabilize the pendulum near theta=0 by providing appropriate torques to 'dampen' small disturbances to the perfect system (the system at rest, at theta=0). To picture what is going on, imagine that a small handle is rigidly attached to the pendulum at its base at right angles to its plane of motion. While holding the pendulum by the handle, imagine trying to restore it to the vertical as it tips, by twisting the handle in response to the tipping, where your response is proportional to the 'tip' and 'rate of tip' as you 'see' and 'feel' those quantities. (Of course, in the idealized model, we are ignoring the significant effect of the twisting friction between your hand and the handle.)

In particular, I have suggested a combination of LINEAR PROPORTIONAL and DERIVATIVE control torques, T_m= b*theta + c*theta_dot; many other functions of angle and angle rate will work, too.

(A) To start, created a new FBD and re-derive the equations of motion using the same configuration from class, but this time with the motor torque acting on the end of the pendulum rod.

(B) Next, your job is to find constants b and c as functions of m, ell, and g so that the equilibrium q^* = {theta^*, theta_dot^*}={0,0} is stable; that is, so that perturbations to the system at this equilibrium die away exponentially. In mathematical terms, equilibriums are stable if the REAL PARTS OF ALL OF THE EIGENVALUES of the JACOBIAN (or 'derivative') MATRIX evaluated at q^*={0,0} ARE NEGATIVE. Use the linearized stability analysis procedure we employed in several examples in lecture.

(C) Then, once you have determined stabilizing values for b and c, test them on the FULL NONLINEAR equation of motion. That is, in the usual manner, do the followin:

(i) convert the new torque-driven equation of motion into a system of first order o.d.e's;

(ii) pick values for m and ell and find b and c in terms of those (and g); and,

(iii) integrate the system of equations using ode45.m and show that small initial disturbances to the system (i.e., small theta_0 and theta_dot_0), decay over time, thus demonstrating that the motor control torque can stabilize the pendulum ion the neighborhood of theta=0. Produce plots that show this behavior in the angle and angle rate.

(D) What is biggest disturbance that the systems is stable with respect to?