Aerospace and Mechanical Engineering

Homework 1

B. Goodwine
Spring, 2003 |
Issued: January 17, 2002 Due: January 22, 2002 |

Consider the robot arm illustrated in the following figure, with
a mass *m* and moment of inertia *I*.

- 1.
- Derive the equation of motion for the arm and linearize the equation about (we did this in class).
- 2.
- Via numerical simulation of the linearized equations, (pick your own values for
*I*,*m*,*l*and , and use any programming language and/or environment you want) verify the following ``rules of thumb'' for PID control in light of the definitions of rise time, peak time, overshoot, settling time and steady state error from the following figure.Submit plots indicating the response of the system as well as the gain values used. - (a)
- For proportional control,
*i.e.*,*k*_{p}> 0,*k*_{d}= 0 and*k*_{I}=0, the solutions are oscillatory, and increasing*k*_{p}increases the frequency of oscillation (which decreases the rise time and peak time) but decreases the mean steady state error. The settling time is infinite. - (b)
- If derivative control is added to the proportional controller,
*i.e.*,*k*_{p}> 0,*k*_{d}> 0 and*k*_{I}=0, then- i.
- for small
*k*_{d}the solutions are decaying oscillations; - ii.
- increasing
*k*_{d}decreases the settling time; - iii.
- increasing
*k*_{d}sufficiently eliminates the oscillatory behavior completely, resulting in an solution which exponentially decays to the final, steady state value; - iv.
- increasing
*k*_{p}decreases the final steady state error; - v.
- increasing
*k*_{p}decreases the rise time.

- (c)
- Adding integral control (PID control)
- i.
- eliminates the steady state error, even for small values of
*k*_{p}, - ii.
- increasing
*k*_{I}generally increases the overshoot and settling time; - iii.
- increasing
*k*_{p}decreases rise time, but may increase overshoot; - iv.
- increasing
*k*_{d}increases damping and stability.

*Last updated: January 17, 2003.*

B. Goodwine (*goodwine@controls.ame.nd.edu*)