# actuation

## **Design of Autonomous Systems**
### csci 6907/4907-Section 86
### Prof. **Sibin Mohan**

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a controller → generates a _control signal_

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a controller → generates a _control signal_

**actuate** a physical component → make it move
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## actuator

part of device/machine → achieve physical movements

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## actuator

part of device/machine → achieve physical movements

converting **energy** → mechanical force

Note:
- They're like muscles on a human body -- converting energy to **physical** action

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## actuators

present everywhere!

- mobile phones
- vehicles
- industrial devices
- robots

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## actuators | examples

- electric motors
- stepper motors
- jackscrews
- muscular stimulators in robots

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## actuator

definition:

> a device that converts energy, which may be electric, hydraulic, pneumatic, etc., to mechanical in such a way that it can be **controlled**

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**quantity** and **nature** of input depend on:

- kind of energy to be converted 
- function of actuator

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an example of a simple electric actuator

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an example of a simple electric actuator

converts **electric current** or **voltage** → mechanical motion

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Note:
- They are responsible for ensuring a device such as a robotic arm is able to move when electric input is provided. 
- A car uses actuators in the engine control system to regulate air flaps for torque and optimization of power, idle speed and fuel management for ideal combustion.

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an actuator requires,

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an actuator requires,

- a control device → provides control signal

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actuator requires,

- a control device → provides control signal
- a source of energy

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actuator based on → **displacement**

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actuator based on → **displacement**

- linear (motors)
- rotary (motors)

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### another broad classification

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### another broad classification

1. **continuous**-drive actuators

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### another broad classification

1. **continuous**-drive actuators
2. **incremental**-drive actuators

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### another broad classification

1. **continuous**-drive actuators → [brushed motors](#brushed-dc-motors)
2. **incremental**-drive actuators → [brushless motors](#stepper-motors)

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### **Brushed** DC motors

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### **Brushed** DC motors

move **continuously** → DC voltage applied to terminals

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### **Brushed** DC motors

move **continuously** → DC voltage applied to terminals

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### **stepper** motors

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### **stepper** motors

**brushless** motors → rotate in series of small/discrete angular steps

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### **stepper** motors

**brushless** motors → rotate in series of small/discrete angular steps

Note:
Animation of a simplified stepper motor turned on, attracting the nearest teeth of the gear-shaped iron rotor
- with the teeth aligned to electromagnet 1, they will be slightly offset from right electromagnet (2)
- Frame 2: The top electromagnet (1) is turned off, and the right electromagnet (2) is energized, pulling the teeth into alignment with it. This results in a rotation of 3.6° in this example.
- Frame 3: The bottom electromagnet (3) is energized; another 3.6° rotation occurs.
- Frame 4: The left electromagnet (4) is energized, rotating again by 3.6°.

When the top electromagnet (1) is again enabled, the rotor will have rotated by one tooth position; since there are 25 teeth, it will take 100 steps to make a full rotation in this example.

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### **stepper** motors

**brushless** motors → rotate in series of small/discrete angular steps

can be set to any given step position → **no need for position sensor**

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### **stepper** motors

can be set to any given step position → **no need for position sensor**

step position → rapidly increased or decreased → continuous rotation

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### **stepper** motors

converting **train of square wave pulses → precise increment**

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|||
|-----|-----|
|digital signals | <scb>0</scb> and <scb>1</scb>|
|analog signals| **continuous** range of values |
||

---

---

---

remember **adc**?

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remember **adc**?

need opposite → convert **digital** to **analog**

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## digital-to-analog convertor (DAC)

- many devices have internal DACs
- DAC is **expensive**
- takes up a lot of silicon on chip

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better method?

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## Pulse Width Modulation (PWM)

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## Pulse Width Modulation (PWM)

method to **control analog devices** → using **digital** signals

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## Pulse Width Modulation (PWM)

method to **control analog devices** → using **digital** signals

- output "**analog-like signal**" from microcontroller

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## Pulse Width Modulation (PWM)

method to **control analog devices** → using **digital** signals

- output "**analog-like signal**" from microcontroller 
- control motors, lights, actuators

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### "**analog-like signal**"

- **not** a true analog signal
- digital one modified to **behave** like one

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a **rectangular wave** with varying "duty cycles" and periods

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a **rectangular wave** with varying "duty cycles" and periods

Note:
-an idealized inductor is driven by a set of voltage pulses (in <font color="blue">blue</font>) that result in a sine-wave-like current (in <font color="red">red</font>)

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## Pulse Width Modulation (PWM)

controlling **average** power/amplitude delivered by → electrical signal

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## Pulse Width Modulation (PWM)

controlling **average** power/amplitude delivered by → electrical signal

- controlled by switching supply <scb>0%</scb> ↔ <scb>100%</scb>

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## Pulse Width Modulation (PWM)

controlling **average** power/amplitude delivered by → electrical signal

- controlled by switching supply <scb>0%</scb> ↔ <scb>100%</scb>
- **faster** than → time for load to change

---

example | consider a fan

---

example | consider a fan | **two** options

<div>

</div>

<div>

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>
</div>

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example | consider a fan | **two** options

<div>

</div>

<div>

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>
</div>

I want the fan to operate at <scb>50%</scb> → possible?

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example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch

</div>

---

example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch
- wait → fan  (close to) <scb>50%</scb>

</div>

---

example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch
- wait → fan  (close to) <scb>50%</scb>
- turn it <scb>off</scb>

</div>

---

example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch
- wait → fan  (close to) <scb>50%</scb>
- turn it <scb>off</scb>
- slows down → turn <scb>on</scb> again

</div>

---

example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch
- wait → fan  (close to) <scb>50%</scb>
- turn it <scb>off</scb>
- slows down → turn <scb>on</scb> again
- repeat **fast** → close to <scb>50%</scb>

</div>

</div>
---

example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch
- wait → fan  (close to) <scb>50%</scb>
- turn it <scb>off</scb>
- slows down → turn <scb>on</scb> again
- repeat **fast** → close to <scb>50%</scb>

</div>

<br>

the faster you do this → closer to desired value of <scb>50%</scb>

---

example | consider a fan | **two** options

I want the fan to operate at <scb>50%</scb> → possible?

|switch state| fan **moves** to|
|------------|-----------------|
| <scb>on</scb> | <scb>100%</scb> speed|
| <scb>off</scb> | <scb>0%</scb> speed|
||

</div>

<div>

- turn <scb>on</scb> switch
- wait → fan  (close to) <scb>50%</scb>
- turn it <scb>off</scb>
- slows down → turn <scb>on</scb> again
- repeat **fast** → close to <scb>50%</scb>

</div>

<br>

the faster you do this → closer to desired value of <scb>50%</scb> (aka **setpoint**)

---

I don't recommend doing this...

---

### PWM wave

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### PWM wave

|||
|------|------|
|<scb>on</scb>-<scb>off</scb> switching| **pulse**|

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### PWM wave

|||
|------|------|
|<scb>on</scb>-<scb>off</scb> switching| **pulse**|
|duration → pulse at **high** state| **pulse width**|

---

### PWM wave

|||
|------|------|
|<scb>on</scb>-<scb>off</scb> switching| **pulse**|
|duration → pulse at **high** state| **pulse width**|
| $T$ | **period** |
||

---

### PWM wave | **properties**

---

### PWM wave | **properties**

two important properties

1. [duty cycle](#duty-cycle)
2. [period](#pwm-period)

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## Duty Cycle

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## Duty Cycle

recall that logic **high** → <scb>on</scb>

Note:
- or `off` depending on the system, but pick one for consistency

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## Duty Cycle

recall that logic **high** → <scb>on</scb>

duty cycle → represents <scb>on</scb> time

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## Duty Cycle

> duty cycle describes the proportion of 'on' time to the regular interval or 'period' of time

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## Duty Cycle

> duty cycle describes the proportion of 'on' time to the regular interval or 'period' of time

represented as **percentages**

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## Duty Cycle | **examples**

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## Duty Cycle | **calculations**

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## Duty Cycle | **calculations**

$$D = \frac{T_{on}}{T} * 100$$

---

## Duty Cycle | **calculations**

$$D = \frac{T_{on}}{T} * 100$$

</div>

<div>

|||
|----|----|
| **D** | duty cycle (percentage)|
| $T_{on}$ | duration → signal is <scb>on<scb>|
| $T$ | total period|
||
</div>

</div>

---

## Duty Cycle | **calculations**

consider → periodic pulse wave, $f(t)$,

---

## Duty Cycle | **calculations**

consider → periodic pulse wave, $f(t)$,

</div>

<div>

|||
|----|----|
|low value | $y_\text{min}$ |
|high value | $y_\text{max}$ |
| duty cycle | $D$ |
||
</div>

</div>

---

## Duty Cycle | **calculations**

consider → periodic pulse wave, $f(t)$,

</div>

<div>

**average** value of wave is,

$$\bar{y} = \frac{1}{T}\int^T_0f(t)\ dt$$

</div>
</div>

---

## Duty Cycle | **calculations**

consider → periodic pulse wave, $f(t)$,

</div>

<div>

$f(t)$ → **pulse** wave

|||
|-----|-----|
| $y_{max}$ | $0<t<D.T$|
| $y_{min}$ | $D.T<t<T$ |
||

</dv>

</div>

---

## Duty Cycle | **calculations**

we can expand,

$$\bar{y} = \frac{1}{T}\int^T_0f(t)\ dt$$

---

## Duty Cycle | **calculations**

$$
  \bar{y} = \frac{1}{T}\int^T_0f(t)\ dt
$$

$$
       = \frac{1}{T} \left(\int_0^{DT} y_\text{max}\ dt + \int_{DT}^T y_\text{min}\ dt\right)
$$

---

## Duty Cycle | **calculations**

$$
  \bar{y} = \frac{1}{T}\int^T_0f(t)\ dt
$$

$$
       = \frac{1}{T} \left(\int_0^{DT} y_\text{max}\ dt + \int_{DT}^T y_\text{min}\ dt\right)
$$

$$
 \bar{y} = \frac{1}{T} \left(D \cdot T \cdot y_\text{max} + T\left(1 - D\right) y_\text{min}\right)
$$

---

## Duty Cycle | **calculations**

$$\bar{y} = D\cdot y_\text{max} + \left(1 - D\right) y_\text{min}$$

---

## Duty Cycle | **calculations**

$$\bar{y} = D\cdot y_\text{max} + \left(1 - D\right) y_\text{min}$$

can now compute → how long signal at ($y_{max}$, $y_{min}$)

---

### PWM Period

determines → number of times signal **repeats per second**

depends on **application**

---

### PWM Sampling Theorem

---

### PWM Sampling Theorem

> number of pulses in the waveform is equal to the number of Nyquist samples

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### PWM Sampling Theorem

> number of pulses in the waveform is equal to the number of Nyquist samples

Nyquist rate → **twice** the highest frequency

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### PWM Sampling Theorem

> number of pulses in the waveform is equal to the number of Nyquist samples

Nyquist rate → **twice** the highest frequency

directly related to → [Nyquist-Shannon Sampling Theorem](https://autonomy-course.github.io/textbook/autonomy-textbook.html#adc-sampling-rate)

---

### PWM example | servo motor

---

### PWM example | servo motor control

servos/RC servos → small, cheap servomotors/actuators

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### PWM example | servo motor control

servos/RC servos → small, cheap servomotors/actuators

used for radio control and small-scale robotics

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### PWM example | servo motor control

- controlled by → PWM signal

---

### PWM example | servo motor control

- controlled by → PWM signal
- width/duty-cycle of pulse → position of motor

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### PWM example | servo motor control

---

### PWM Generation in Microcontrollers

- **built-in PWM components**
- timer ICs

---

### PWM Generation in Microcontrollers | _e.g.,_ **Arduino**

generating PWM → **simple lines of code**!

```
analogWrite(pin, value)
```

Note:
- that **not all pins** of an Arduino can generate a PWM signal. In the case of Arduino Uno, there are only 6 I/O pins (3,5,6,9,10,11) that support PWM generation and they are marked with a tilde (~) in front of their pin number on the board.

---

### PWM Generation in Microcontrollers | _e.g.,_ **Arduino**

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### PWM Generation in Microcontrollers | _e.g.,_ **Arduino**

various duty cycle examples

```
analogWrite(PWM_PIN, 64);   // 25% Duty Cycle or 25% of max speed
analogWrite(PWM_PIN, 127);  // 50% Duty Cycle or 50% of max speed
analogWrite(PWM_PIN, 191);  // 75% Duty Cycle or 75% of max speed
analogWrite(PWM_PIN, 255);  // 100% Duty Cycle or full speed
```

---

### PWM Generation in Microcontrollers | _e.g.,_ Raspberry Pi

---

### PWM Generation in Microcontrollers | _e.g.,_ Raspberry Pi

multiple **GPIO pins** → for generating PWM signals

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### PWM Generation in Microcontrollers | _e.g.,_ Raspberry Pi

consider this [code](https://circuitdigest.com/microcontroller-projects/raspberry-pi-pwm-tutorial) → controlling LED brightness

```python[0|1|4-5|6|9-11|12-14]
import RPi.GPIO as IO       #calling header file which helps us use GPIO’s of PI
import time                 #calling time to provide delays in program
IO.setwarnings(False)       #do not show any warnings
IO.setmode (IO.BCM)         #we are programming the GPIO by BCM pin numbers. (PIN35 as ‘GPIO19’)
IO.setup(19,IO.OUT)         # initialize GPIO19 as an output.
p = IO.PWM(19,100)          #GPIO19 as PWM output, with 100Hz frequency
p.start(0)                  #generate PWM signal with 0% duty cycle
while 1:                    #execute loop forever
    for x in range (50):    #execute loop for 50 times, x being incremented from 0 to 49.
        p.ChangeDutyCycle(x) #change duty cycle for varying the brightness of LED.
        time.sleep(0.1)      #sleep for 100m second
    for x in range (50):     #execute loop for 50 times, x being incremented from 0 to 49.
        p.ChangeDutyCycle(50-x)  #change duty cycle for changing the brightness of LED.
        time.sleep(0.1)          #sleep for 100m second
```