Control Systems for Robots

Prof. Robert MarmelsteinCPSC 527 – Robotics

Spring 2010

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

Robot State Description

Physical States: Position Orientation (pose) Velocity Acceleration Sensor states Actuator States

Internal States: Plans Tasks Behaviors

Assumed vs. Actual

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

Example: Forced Spring

A forced spring can be modeled by a linear differential equation

k

b

m

x

kxdt

dxb

dt

xdmtF

2

2

)(

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:

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

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

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

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)

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

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

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?

Example of a Closed Loop Controller

Computation(Brain)

Actuator Control(Auto Pedals)

Velocity(Engine Power)

Sensing(Eyes)

-+

ErrorInput Output

Desired Position

Actual Position

Velocity

Time

Velo

cit

y

Velocity decreases as the car gets closer to the desired position

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

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

OFF ON

temp < target

temp > target

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

Leveraging Hysteresis

Single Threshold System(no hysteresis)

Use of temperature dead zone to induce hysteresis

Closed Loop Controller Issues

Low Gain – Sluggish High Gain – Unstable

JLJ Text – Fig 2.5

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

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

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.

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

PID Controller (cont.)

• KP – Proportional Gain• KI – Integral Gain• KD – Derivative Gain

PID Controller Response Curve

Settling time

Overshoot

Controlledvariable

Time

Reference

%

Steady StateTransient State

Steady state error

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.

Effect of Increasing PID Factors

KP KI KD

NT: No trend

KP = 20

KP = 200 KP = 500

Control Performance – Proportional Control Source: CUNY – Dr. Jizhong Xiao

KP = 50

KI = 200

Control Performance – Integral Control(KP = 100)

KI = 50

Source: CUNY – Dr. Jizhong Xiao

KD = 5

KD = 20 KD = 10

KD = 2

Control Performance – Derivative Control(KP = 100, KI =200) Source: CUNY – Dr. Jizhong Xiao

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

Zeigler-Nichols Tuning Rule for PID Controllers

Yields ~25% overshoot and good settling time

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

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.

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

Non-Linearity Issues (cont.)

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

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.

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

Self-Balancing Robot (Pics)

Light Sensor

[Simplified] PID Controller Algorithm

// Get Balance Reading for Light SensormidVal = Sensor_Read(LS_Port);

// Now Compute Power (PID) Value for Balancing Robotwhile (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);}

Other Material

PID Controller for Dead Reckoning CMU PID Tutorial DePaul Self-Balancing Robots – Trials and

Tribulations

Questions?