Drone Control and the Complementary Filter

Summary

This video lecture introduces the concept of a complementary filter, a simple yet effective tool for blending data from different sensors, particularly useful in drone control systems. It explains how to estimate the roll angle using an IMU's gyro and accelerometer, highlighting the strengths and limitations of each method. The video demonstrates how integrating gyro data over time can accumulate errors, while accelerometer readings offer long-term stability despite short-term noise. It then illustrates the implementation of a complementary filter that combines the short-term accuracy of the gyro with the long-term stability of the accelerometer, providing a practical solution for more accurate roll angle estimation.

Takeaways

• 📚 The video introduces the concept of a complementary filter, a simple yet effective method for blending measurements from two different sensors.
• 🛠 The context of the explanation is a drone trying to estimate its roll angle using an IMU (Inertial Measurement Unit), which includes a gyroscope and an accelerometer.
• 🔄 The gyroscope measures angular rates and can be used to estimate the roll angle by integrating these rates over time, a method known as dead reckoning.
• 📉 Dead reckoning is susceptible to cumulative errors due to the integration of noise and bias in the gyroscope measurements, leading to inaccuracies over time.
• 🌐 The accelerometer measures acceleration, including gravity, and can be used to determine the roll angle by calculating the arctangent of the acceleration in the Y and Z directions.
• 🔍 While accelerometer measurements are stable long-term due to the constant gravity vector, they are noisy and can be affected by any acceleration, not just gravity.
• 🤝 The complementary filter combines the short-term accuracy of the gyroscope with the long-term stability of the accelerometer to produce a more reliable roll angle estimate.
• 🧩 The implementation of the filter involves using a low-pass filter on the accelerometer measurements and a high-pass filter on the gyroscope measurements, or a simplified approach by scaling and combining the two measurements.
• ⚙️ The filter can be tuned by adjusting the time constant for the low-pass filter, which affects the balance between short-term agility and long-term stability.
• 📉 The video demonstrates the effectiveness of the complementary filter through a JavaScript program, showing how it maintains a closer estimate to the true roll angle compared to using either sensor alone.
• 🔧 The video also discusses the discrete implementation of the filter in software, which involves scaling the gyroscope and accelerometer measurements and applying a low-pass filter to the sum.

Q & A

What is a complimentary filter in the context of control systems?

A complimentary filter is a simple yet effective method used to blend measurements from two different sensors, such as an IMU's gyro and accelerometer, to estimate a variable like the roll angle of a drone more accurately than either sensor could alone.

Why is the roll angle estimation using only a gyroscope prone to error over time?

Estimating the roll angle using only a gyroscope is prone to error over time because the integration of the angular rate measurements accumulates any bias or noise in the gyro readings, leading to a growing discrepancy between the estimated and true roll angles.

How does an accelerometer help in estimating the roll angle of a drone?

An accelerometer measures the linear acceleration, including the acceleration due to gravity. By knowing the direction of gravity relative to the drone's reference frame, the roll angle can be determined, although this method is less precise in the short term due to noise and other accelerations.

What is the main advantage of using a complementary filter over relying solely on either the gyroscope or the accelerometer?

The main advantage of using a complementary filter is that it combines the short-term accuracy of the gyroscope with the long-term stability of the accelerometer, resulting in a more reliable and accurate estimation of the roll angle.

How does the script describe the behavior of the gyroscope and accelerometer measurements over time?

The script describes the gyroscope measurements as being smooth and less jittery in the short term due to integration, but slowly wandering over time. In contrast, the accelerometer measurements are more stable long term because the gravity vector doesn't change, but are noisy and less reliable in the short term.

What is the purpose of the low-pass filter in the context of the complementary filter?

The purpose of the low-pass filter in the complementary filter is to attenuate high-frequency noise and emphasize the long-term stability provided by the accelerometer, while still allowing some of the short-term agility from the gyroscope measurements.

How is the high-pass filter related to the low-pass filter in the complementary filter?

In the complementary filter, the high-pass filter is the complement of the low-pass filter, meaning that if the low-pass filter is represented by G(s), the high-pass filter is represented by 1 - G(s). When combined, they sum up to 1, ensuring the integrity of the original measurements.

What is the significance of the time constant 'tau' in the low-pass filter of the complementary filter?

The time constant 'tau' in the low-pass filter determines the cutoff frequency of the filter. A lower 'tau' allows more high-frequency content from the gyroscope, while a higher 'tau' emphasizes more low-frequency content from the accelerometer.

Can you explain the discrete implementation of the complementary filter as described in the script?

The discrete implementation of the complementary filter involves scaling the gyroscope and accelerometer measurements, summing them, and then applying a low-pass filter. This is done at each time step, updating the roll angle estimate by taking a fixed fraction of the gyroscope measurement and a complementary fraction of the accelerometer measurement.

How does the script demonstrate the effectiveness of the complementary filter through a JavaScript program?

The script uses a JavaScript program to simulate the behavior of the complementary filter, showing how it holds close to the true down direction, unlike the gyroscope which wanders off, and the accelerometer which is noisy. The program visually demonstrates the filter's ability to balance short-term agility and long-term stability.

Outlines

00:00

🤖 Introduction to Complementary Filters in Control Systems

This paragraph introduces the concept of a complementary filter within the context of control systems, specifically for a drone's roll angle estimation using an Inertial Measurement Unit (IMU). The IMU consists of a gyroscope for angular rate sensing and an accelerometer for linear acceleration. The paragraph explains the limitations of using either sensor alone due to noise and bias in gyro measurements or inaccuracies in accelerometer measurements during motion. It sets up the problem of estimating the roll angle and hints at the solution involving a blend of both sensor inputs.

05:01

🔍 Analyzing Roll Angle Estimation Methods

This section delves deeper into the two methods for estimating the roll angle: using gyro measurements through dead reckoning and using accelerometer measurements based on gravity. It discusses the advantages and disadvantages of each method, highlighting the short-term accuracy of the gyro and the long-term stability of the accelerometer. The paragraph also introduces a JavaScript program to visually demonstrate the noise and stability issues of both methods, leading to the necessity of a complementary filter that combines the strengths of both sensors.

10:03

🛠 Implementing the Complementary Filter

The paragraph explains the implementation of a complementary filter that blends the short-term accuracy of the gyro with the long-term stability of the accelerometer. It describes the process of using low-pass and high-pass filters to create a filter that enhances the strengths of each sensor while compensating for their weaknesses. The explanation includes a mathematical breakdown of the filter's operation in both continuous and discrete domains, illustrating how the filter can be adjusted for optimal performance and how it is applied in practice through a software algorithm.

📚 Conclusion and Further Resources

In conclusion, the paragraph summarizes the benefits of using complementary filters for control problems, emphasizing their simplicity, computational efficiency, and ease of implementation and tuning. It invites viewers to ask questions or leave comments and promotes further learning through related videos on drone control and simulation available on the MATLAB channel. The paragraph also acknowledges Patreon supporters and offers a way for viewers to support the creator's work, including the option to download a digital copy of a book on control theory fundamentals.

Keywords

💡Complimentary Filter

A complimentary filter is a type of sensor fusion algorithm that combines measurements from two different sensors to produce a more accurate and reliable output than either sensor could alone. In the context of the video, it's used to estimate the roll angle of a drone by blending data from an IMU's gyroscope and accelerometer. The filter is designed to leverage the short-term accuracy of the gyroscope and the long-term stability of the accelerometer, resulting in a more robust estimation of the roll angle.

💡Control System

A control system is a device or set of devices that manage, command, direct, or regulate the behavior of other devices or systems. In the video, the control system is discussed in the context of drone stabilization, where the complimentary filter is a part of the control system that helps in maintaining the drone's orientation by estimating its roll angle.

💡IMU (Inertial Measurement Unit)

IMU stands for Inertial Measurement Unit, which is a device that provides orientation and angular velocity feedback based on the sensor data from a three-dimensional accelerometer and a three-dimensional gyroscope. In the script, the IMU is used on a drone to estimate the roll angle, highlighting its importance in the field of robotics and aerospace for navigation and control.

💡Gyroscope

A gyroscope is a device used for measuring or maintaining orientation and angular velocity. In the video, the gyroscope within the IMU is used to sense the angular rate of the drone, which is then integrated over time to estimate the roll angle. The script emphasizes the importance of gyroscopes in short-term motion tracking but also mentions their susceptibility to drift over time.

💡Accelerometer

An accelerometer is a sensor that measures acceleration forces. It is used in the video to sense linear acceleration, including the acceleration due to gravity. By knowing the direction of gravity, the accelerometer can help determine the roll angle of the drone. The script explains that while accelerometers are noisy in the short term, they provide a stable long-term measurement of the gravity vector.

💡Roll Angle

The roll angle is the rotation of an object around its longitudinal axis. In the video, estimating the roll angle is the central problem being addressed. The script discusses how both the gyroscope and accelerometer can be used to estimate this angle, with the complimentary filter combining their measurements to achieve a more accurate estimation.

💡Dead Reckoning

Dead reckoning is a method of navigation that involves estimating one's current position based on a previously known position and factors such as speed, heading, and the time elapsed since the last known position. In the context of the video, dead reckoning is used to describe the process of integrating gyroscope measurements to estimate the roll angle over time. The script points out that while effective for short durations, cumulative errors can lead to significant inaccuracies in the estimated roll angle.

💡Low-Pass Filter

A low-pass filter is a type of filter that allows low-frequency signals to pass while attenuating high-frequency signals. In the video, a low-pass filter is used to process the accelerometer measurements, smoothing out the noise and providing a stable long-term estimate of the roll angle. The script explains how the low-pass filter is part of the complimentary filter's design to balance the short-term accuracy of the gyroscope with the long-term stability of the accelerometer.

💡High-Pass Filter

A high-pass filter is the opposite of a low-pass filter; it allows high-frequency signals to pass while reducing the amplitude of low-frequency signals. In the script, the high-pass filter is conceptually used in the complimentary filter to emphasize the short-term changes in the gyroscope measurements, complementing the low-pass filtered accelerometer data.

💡Sensor Fusion

Sensor fusion is the process of intelligently combining data from multiple sensors to achieve more accurate and reliable information than could be obtained from any individual sensor alone. The video script discusses the complimentary filter as an example of sensor fusion, where data from the IMU's gyroscope and accelerometer are combined to estimate the drone's roll angle more effectively.

💡Time Constant

In the context of filters, the time constant is a parameter that determines the filter's response time. A lower time constant results in a faster response, allowing more high-frequency content to pass through the filter. In the video, the time constant of the low-pass filter within the complimentary filter is discussed as a tuning parameter that affects the balance between the short-term accuracy of the gyroscope and the long-term stability of the accelerometer.

Highlights

Introduction to the complimentary filter, a simple yet effective method for blending measurements from two different sensors.

The practical application of the complimentary filter in estimating the roll angle of a drone using its IMU.

Explanation of how to measure the roll angle using gyro and accelerometer readings from an IMU.

Dead reckoning method for estimating roll angle by integrating angular rate measurements over time.

Limitations of dead reckoning due to cumulative errors from noise and bias in gyro measurements.

Utilizing the accelerometer to measure the roll angle based on the gravity vector pointing downwards.

Challenges in using accelerometer for roll angle estimation due to noise and external accelerations.

The stability of the accelerometer for long-term roll angle estimation as gravity vector remains constant.

Visual demonstration of the complimentary filter's effectiveness using a JavaScript program with noisy sensors.

The concept of combining short-term accuracy of the gyro with the long-term stability of the accelerometer.

Technical explanation of the complimentary filter implementation using low-pass and high-pass filters.

Adjusting the time constant tau to tune the filter's balance between short-term and long-term accuracy.

Discrete implementation of the complimentary filter in software for digital control algorithms.

Algorithm explanation for combining gyro and accelerometer measurements to estimate the roll angle.

Demonstration of the complimentary filter's performance in maintaining the true down direction compared to individual sensors.

Mathematical breakdown of the discrete complimentary filter implementation and its similarity to the continuous domain approach.

The simplicity and computational efficiency of using the complimentary filter for control system applications.

Invitation for questions, comments, and further exploration of drone control and simulation topics.

Acknowledgment of Patreon supporters and offer to download a digital copy of the book on control theory fundamentals.

Transcripts

welcome back to control system lectures

in this video we're gonna talk about the

complimentary filter it's such a dead

simple filter which is a good reason to

learn it but it's also practical because

it produces nice results when blending

measurements from two different sensors

to understand how a complimentary filter

works and the situations where it's

useful we should set up a problem let's

imagine we have a drone and we're trying

to estimate the roll angle using its

onboard inertial measurement unit or IMU

the IMU has both a gyro that senses

angular rate and an accelerometer that

senses linear acceleration so the

question is how can we measure the roll

angle with the IMU now neither of these

sensors can measure the roll angle

directly however we can use either of

them to estimate the roll angle let's

look at how we would do that using just

the gyro measurements if we assume that

the roll angle is 0 degrees when the

drone is sitting on the ground then

after it takes off we can use the

measured roll rates to calculate how the

roll angle changes over time that is we

can determine how far the drone has

rotated in one time step and then add

that value to the current estimate of

roll angle for example if we read the

gyro ten times a second and at the first

time step the roll rate measures three

degrees per second then after that time

step or 0.1 seconds later we can

estimate that the drone has rotated

point three degrees we can then add that

Delta angle to the current roll estimate

then at the next time step we do it

again and then again and again basically

we are integrating the angular rate to

get angles that's pretty straightforward

right well this type of approach is

called dead reckoning and it's very good

for keeping track of motion over short

periods of time

how short well it depends on the noise

and error characteristics of the gyro

when we integrate the rate measurements

any uncorrect at bias in the gyro or

even just random high frequency noise

gets summed as well and with these

cumulative errors eventually there's

going to be a large difference between

the true roll angle and the angle we

estimate dead reckoning is a relative

measurement and there's no absolute

measurement that will correct the roll

over time

now let's look at how to estimate role

angle using the accelerometer gravity is

always pointing down and near the

surface of the earth that causes an

acceleration of 1g let's imagine that

the drone is sitting stationary on the

ground and since there's no other

accelerations acting on the drone the

accelerometer will only be measuring the

acceleration due to gravity now the

drone knows which direction is down and

knowing down relative to the drone

reference frame it can determine the

roll angle a simple way to do this is to

just take the arctangent of the

acceleration in the Y and z direction to

get angle this measurement however isn't

very precise in the short term because

the accelerometer is also noisy but more

importantly than that it measures all

accelerations any time the drone

accelerates in any direction the

measurement is no longer just from

gravity now we could ignore

accelerations that are outside of some

boundary but even really small

accelerations can throw off the

estimation of the down direction and

then therefore the roll angle even

rolling the drone will induce a linear

acceleration if the IMU isn't located

precisely at the center of rotation for

these reasons it's hard to rely solely

on the accelerometer for a short

duration very quick roll measurements

but the accelerometer is very stable

long term because the gravity vector

doesn't change over time the measurement

of the gravity vector isn't wandering

off like the rate-based roll angle did

we may not want to trust it much at any

given moment but at least we have an

absolute understanding of which way down

is over really long time scales so we

have two different ways of determining

the roll angle integrating the gyro

which is more accurate over the short

term and measuring the acceleration

which is more accurate over the long

term to let you visualize these two

methods I created a simple JavaScript

program that uses a noisy accelerometer

and a noisy gyro to show you the

direction the drone thinks is down with

either method

notice how the acceleration method is

bouncing around more than the gyro both

sensors have the same amount of noise

but the gyro is integrated which

attenuates high frequencies and makes it

less jittery than the acceleration

method so that smoothness makes it

better over the short-term like

we talked about but the gyro is slowly

wandering around at times further than

the accelerometer direction with these

two estimates of roll angle from two

different sensors we can now combine

them using a complementary filter let me

reset the values from the gyro and

accelerometer sensors and add a third

drone that uses a complimentary filter

to estimate roll angle this filter needs

to run for a little while for you to see

the benefits so I'm just going to

minimize this and go back to explaining

what the complimentary filter is doing

while it runs in the background

complimentary in this sense means that

we combine the two measurements in a way

that complete each other or in other

words we take some part of one

measurement and add it to the

complementary part of the other so that

the sum of the two parts is still one

whole measurement in our case we'd want

to keep the short-term benefits of the

gyro and add them to the long-term

benefits of the accelerometer making

these two parts complementary is easier

than you might think if we pass the

accel measurement through a low-pass

filter G of s then the filter that we

passed the gyro through is the high-pass

filter 1 minus G of s since adding these

two filters together equals 1 then

they're complimentary of each other this

is the basic complementary filter and G

of s can be any filter you want like a

notch filter or something and 1 minus G

of s will be its complement however more

often than not they are just the

standard low-pass and high-pass filter

now in our case with a rate sensor and

accelerometer this block diagram isn't

actually the way we would implement this

filter because it would result in a lot

of extra steps like integrating and

filtering multiple signals that really

just cancel out if we reduce this block

diagram I'll show you what I mean with

an example we'll use a first-order

low-pass filter for the acceleration 1

over tau s plus 1 this makes the high

pass filter tau s over tau s plus 1 and

since the integral is 1 over s we can

cancel out those s's and then factor the

low-pass filter 2 after the summing

Junction and look at this we're no

longer converting the rate measurements

into angles at all we're just scaling it

and then adding it to the acceleration

angle

and low-pass filtering the result to

tune this filter we simply need to pick

the time constant tau for the low-pass

filter a lower tau raises the cutoff

frequency and lets more Excel

frequencies in and fewer gyro

frequencies and a higher tau does the

opposite now picking the ideal cutoff

frequency depends on the noise

characteristics of the two sensors and

can be adjusted during simulation or

tests all right no the continuous domain

is a perfectly fine way to implement a

complementary filter but since most

control algorithms run on digital

computers let's talk about a simple and

intuitive way to implement a discreet

filter that we can code in software

let's start with the gyro measurement

which were reading every time step we

multiply this measurement by the length

of time for one time step to get the

angle traveled during that period now we

can add this angle to the existing

estimated roll angle I'm gonna explain

where we get this in just a second at

this point we have an updated roll

estimate for the current time step from

just the gyro but we also have another

roll estimate from the accelerometer

that we're also reading every time step

we can combine these two angles by

taking a fixed fraction of one and the

complementary fraction of the other so

for example we may take 98% of the gyro

measurement and add that to 2% of the

accel measurement so that ever so

slightly were nudging the gyro angle in

the acceleration direction this new

angle is the estimated roll angle that

we feedback at the next time step by

believing the gyro more we're allowing

the short-term speed and agility to make

it through but we're nudging it back

towards the absolute down direction over

time to keep the angle from wandering

off this is how we're getting the

long-term stability of the accelerometer

this is the algorithm that I implemented

in my JavaScript example in fact let's

check back in on that example and see

how it's doing first off check out how

well the complementary filter is holding

close to the true down direction whereas

the gyro has wandered off by 30 degrees

or so and if you watch the motion of the

complementary filter it's still moving

back and forth in the same

direction as the gyro getting those high

frequencies passed through but it's just

being pulled back by the excel which is

hanging around the true down position

this is the beauty of the complimentary

filter

all right so the question might be why

does this implementation that we just

walked through appear to work when there

doesn't seem to be an obvious low pass

or high pass filter well let's work

through the math a bit and reduce this

block diagram to see what we get

now I'm sipping through this pretty

quickly but if you pause the video you

can follow along with the algebraic

steps but the important part is that the

first half of the resulting function is

the summation of the scaled

accelerometer and scaled rate

measurements and the second half is a

discrete low-pass filter so we're

scaling the inputs summing them together

and applying a low-pass filter and that

is very similar to what we saw with the

continuous domain implementation so

that's pretty cool

two different ways to implement a

complimentary filter in one version we

tuned it by adjusting the cutoff

frequency and in the other we tuned it

using the ratio of one signal versus the

other now I hope this video helped you

understand a bit more about

complementary filters and has encouraged

you to see if they will work for the

control problems that you're working on

it's nice when you can get away with

simple filters like this because they're

cheap from a computational perspective

and they're easy to understand implement

and tune if you have any questions or

comments on this video please leave them

below and I'll try to answer them if I

can

if you're interested in learning more

about drone control and simulation check

out my other videos on that topic on the

MATLAB channel the link is in the

description below and at the end of this

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