1. Control Systems for Robots
Prof. Robert Marmelstein
CPSC 527 – Robotics
Spring 2010

2. Introduction to Robot Control
We want robots to do things that might otherwise be done by intelligent, physical beings
Biological nervous systems are often thought of as control systems for living organisms
To control a robot in this manner, we need to:
Define the state of the robot
Sense the state of the robot
Compare the state of the robot to a desired state
Decide what actions need to be taken to achieve the desired state and issue the associated commands
Translate those command into physical action
Measure the effect of those physical actions

3. Robot State Description
Physical States:
Position
Orientation (pose)
Velocity
Acceleration
Sensor states
Actuator States
Internal States:
Plans
Tasks
Behaviors
Assumed vs. Actual

4. Control Engineering
Control engineering is the application of mathematical techniques to the design of algorithms and devices to control processes or pieces of machinery.
It almost always requires a model of the entity that is being controlled
If a system can be modeled by a set of linear differential equations there are well understood techniques for getting exact analytical solutions, and so designing controllers so that the output of the system is the required one

5. Example: Forced Spring
A forced spring can be modeled by a linear differential equation

6. Real-World Systems
Unfortunately, most real-world systems are non-linear in nature
Example: Pendulum
In these cases, the nonlinear system is often approximated by a linear system
For the Pendulum, assume sin() ≈ , which yields:

7. Control System Components
Target Value – The desired operating point of the overall system, which is speed.
Measured Value – The actual operating point of the system. It is affected by external factors such as hills, and internal factors such as the amount of fuel delivered to the engine.
Difference Value – This is the difference between the target value and the measured value. Translates into feedback.
Control Input – This is the main adjusting point of the control system. The amount of fuel delivered to the engine is the primary control input to the cruise control
Control Algorithm – Determines how to best regulate the control input to make the difference value as close to zero as possible. It does this by periodically looking at the difference value and adjusting the control input

8. Feedback Defined
Feedback: continuous monitoring of the sensors and reacting to their changes.
Feedback control = self-regulation
Two kinds of feedback: positive and negative
Negative feedback acts to regulate the state/output of the system
e.g., if too high, turn down, if too low, turn up
thermostats, toilets, bodies, robots...
Positive feedback acts to amplify the state/output of the system
e.g., the more there is, the more is added
lynch mobs, stock market, ant trails...
Often results in system instability

9. Open Loop Controller
The Open Loop Controller (OLC) is the simplest kind
The controller sends an input signal to the plant
It does not compensate for disturbances that occur after the control stage
Actual effects are assumed – not measured
No feedback to match actual to intended

10. Open Loop Controller (cont.)
The OLC is commonly used for behavior-based systems
If a trigger condition is met , then the behavior is activated
Behavior is performed until the condition is no longer met
If the condition is not met, the (assumedly) some other behavior is activated
You would likely use an OLC if you have no way of measuring your operating point (e.g., the value you are trying to control)

11. Problem with Open Loop Controllers
The effectiveness of OLCs are very context dependent
The amount of force that is applied has different effects dependent on the surface type

12. Closed Loop Controller
In Closed-loop control, the output is sensed and compared with the reference. The resulting error signal is fed back to the controller [feedback].
Components:
Reference – Desired State
Controller – Issues Commands
Plant – Actuator

13. Closed Loop Controller (cont.)
[Negative] Feedback keeps the operation of the system smooth and stable
Closed Loop Controller issues:
How quickly will the system respond to error?
How long will it take the system to reach equilibrium?
What, if any, residual error will remain?
Under what conditions will the system become unstable?

14. Example of a Closed Loop Controller
Computation
(Brain)
Actuator Control
(Auto Pedals)
Velocity
(Engine Power)
Sensing
(Eyes) - +
Error
Input
Output
Desired
Position
Actual
Time
Velocity
Velocity decreases as the
car gets closer to the
desired position
Velocity

15. Bang-Bang Controller (BBC)
The simplest type of closed-loop controller is the Bang-Bang controller. It typically consists of two states and a target value
The BBC typically monitors one item (quantity) of interest—its job is to keep that quantity at a certain target value
OFF ON
temp < target
temp > target
If the quantity is too high or low (vs. the target) the BBC compensates to change it
The system continually transitions between states, often abruptly

16. Taking the Edge off Bang-Bang Control with Hysteresis
Hysteresis provides a sort of "guard band" around the desired set point of the system.
In other words, when the temperature goes above (or below) the desired control point, there is a margin which needs to be exceeded before compensation is applied
The result a lag which causes the system to run much smoother, avoiding the jerkiness of a purely Bang-Bang controller

17. Leveraging Hysteresis
Single Threshold System
(no hysteresis)
Use of temperature dead zone to induce hysteresis

18. Closed Loop Controller Issues
JLJ Text – Fig 2.5
Low Gain – Sluggish
High Gain – Unstable

19. Proportional Integral Derivative (PID) Controller
Closed loop controllers that only use proportional control can easily become unstable if the gain is too high or sluggish if the gain is set too low
PID controllers help solve this problem
It use the measured error compute an input to the Plant based on three distinct controls: Proportional, Integral and Derivative (see right)
Proportional control – Computed based on the actual error (times a gain factor). Thus, The larger the error, the bigger correction the control will make
Serves to control response time to error
For high gains or large errors, tends to overshoot and oscillate the desired output
There is typically a steady state error that cannot be corrected

20. Proportional Integral Derivative (PID) Controller
Integral control – Reduces steady-state error by adding (integrating) the actual errors over time.
Once the error reaches a predetermined threshold, the controller will compensates
Too little can result in undershoot; too much can result in overshoot
Derivative (D) control – Computed based of difference between current and previous error. Thus, the output of this control is proportional to the change in error
Prevents oscillations due to overshoot
Reduced settling time by giving a better dynamical response
Generally, this control has a positive effect

21. PID Controller (cont.) The tunable factors are:
Kp – Proportional Gain factor
KI – Integral Gain Factor
KD – Differential Gain Factor
These factors are cross-coupled, so the performance of the system cannot be optimized by tuning each factor independently
Some systems can be engineered without all three PID components
The P components is always required, but P controllers alone can result in instability
PI is not accurate but converges quickly
PD converges relatively quickly reducing oscillations as it approaches the goal.
PID accurately maintains a position, but isn’t very fast.

22. Helpful PID Terms
Gain(s) --The parameter(s) that determine the magnitude of the system’s response.
Gain values determine whether or not the system stabilizes or fluctuates.
Finding effective gains is a trial and error process, requiring testing and recalibration.
Proportional Gain – When the value of the gain is proportional to the error.
Damping – The process of systematically decreasing a system’s fluctuations
A system is damped if it does not oscillate out of control.
Generally, the gains have to be adjusted for a system to be damped
Steady State Error – The amount of error that remains after the system has reached equilibrium

23. PID Controller (cont.) KP – Proportional Gain KI – Integral Gain
KD – Derivative Gain

24. PID Controller Response Curve
Controlled
variable
Overshoot
Steady state error
%
Reference
Transient State
Steady State
Time
Settling time

25. PID Response Curve (cont.)
Rise Time (Tr) – The time for the plant output y to rise beyond 90% of the desired level for the first time
Overshoot – How much the peak level is higher than the steady state, normalized against the steady state
The time required for the output to reach its maximum level is called the Peak Time (Tp)
Settling Time (Ts) – The time it takes for the system to converge to its steady state
Transient State – The period from the detection of error until its approximate correction, resulting in the steady state
Steady-state Error – The difference between the steady-state output and the desired output.

26. Effect of Increasing PID FactorsKP KI KD
NT: No trend

27. Control Performance – Proportional Control
Source: CUNY – Dr. Jizhong Xiao
KP = 20
KP = 50
KP = 200
KP = 500

28. Control Performance – Integral Control (KP = 100)
KI = 50
KI = 200
Source: CUNY – Dr. Jizhong Xiao

29. Control Performance – Derivative Control (KP = 100, KI =200)
Source: CUNY – Dr. Jizhong Xiao
KD = 2
KD = 5
KD = 10
KD = 20

30. Optimizing Performance
PID Tuning – By Hand
Boost Kp until it oscillates
Boost KD to stop oscillation, back off Kp by 10%
Dial in KI to Hold position or velocity smooth
Trial and error
PID tuning – By Design
Zeigler-Nichols Method (next slide)
Other:
Work to minimize environmental interference and sensor error (two are typically coupled)
Smart design helps too

31. Zeigler-Nichols Tuning Rule for PID Controllers
Yields ~25% overshoot and good settling time

32. Why Care about the PID Controller?
Because PID Controllers are everywhere!
Due to its simplicity and excellent if not optimal performance in many applications, PID controllers are used in more than 95% of closed-loop industrial processes.
It can be tuned by operators without extensive background in Controls, unlike many other modern controllers that are much more complex but often provide only marginal improvement.
In fact, most PID controllers are tuned on-site.
The lengthy calculations for an initial guess of PID parameters can often be circumvented if we know a few useful tuning rules. This is especially useful when the system is unknown

33. Non-Linear Control
Linear controllers are generally valid only over small operational ranges.
Hard non-linearity cannot even be approximated by linear systems.
Model uncertainties can sometime be tolerated by the addition of non-linear terms.
Non-linear systems often have multiple equilibrium points, plus periodic, or chaotic attractors.
In these systems, small disturbances (even noise) can induce radically different behaviors.

34. Control System Non-Linearity Issues
Saturation – Occurs when the input signal to a certain device exceeds the ability of the device to process it
Input – sensors
Output – motors
For output, saturation means that the required compensation can no longer be applied to the control system
In general, it is good practice to limit the signal to the saturation value in software
When an input reaches the saturation point, it no longer provides a reliable estimate of the real world

35. Non-Linearity Issues (cont.)
Dead Zone – Because the sensitivity of actuators is limited, not every non-zero input will result in action. The Dead Zone is the +/- region above a zero (0) input that will result in no actuator movement.
Backlash – Term describing actuator hesitation and overshoot caused by small gaps between motor gears
Can result in small, but unnecessary, oscillations of the actuator position

36. Self-Balancing Robot Lab
Problem – Create a PID controller that will keep an NXT robot balanced on two wheels
Use a light sensor to determine balance:
Light diminishes as it tilts backward
Light gets stronger as it tilts forward
Collect reference value when the robot is perfectly balanced
PID controller will use the light sensor data to compute the compensation for the motors (plant)
Move backward if tilting back
Move forward if tilting forward

37. Self-Balancing Robot (Pics)
Light Sensor

38. [Simplified] PID Controller Algorithm
// Get Balance Reading for Light Sensor
midVal = Sensor_Read(LS_Port);
// Now Compute Power (PID) Value for Balancing Robot
while (true) {
lightVal = Sensor_Read(LS_Port);
error = lightVal – midVal;
diffErr = error – prevError;
intErr = intErr + error;
pVal = kp * error;
iVal = (ki*intErr)*dampFactor;
dVal = kd*diffErr;
prevErr = error;
power = (pVal+iVal+dVal)*scale;
MotorFWD(motorPorts, power); }

39. Other Material PID Controller for Dead Reckoning CMU PID Tutorial
DePaul Self-Balancing Robots – Trials and Tribulations

40. Questions?