Supplementary Material: Linearization of Quadrotor Equations of Motion

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En provenance du cours de University of Pennsylvania

Robotics: Aerial Robotics

1922 notes

University of Pennsylvania

Robotics: Aerial Robotics

1922 notes

Cours 1 sur 6 dans Specialization Robotics

How can we create agile micro aerial vehicles that are able to operate autonomously in cluttered indoor and outdoor environments? You will gain an introduction to the mechanics of flight and the design of quadrotor flying robots and will be able to develop dynamic models, derive controllers, and synthesize planners for operating in three dimensional environments. You will be exposed to the challenges of using noisy sensors for localization and maneuvering in complex, three-dimensional environments. Finally, you will gain insights through seeing real world examples of the possible applications and challenges for the rapidly-growing drone industry. Mathematical prerequisites: Students taking this course are expected to have some familiarity with linear algebra, single variable calculus, and differential equations. Programming prerequisites: Some experience programming with MATLAB or Octave is recommended (we will use MATLAB in this course.) MATLAB will require the use of a 64-bit computer.

À partir de la leçon

Planning and Control

Welcome to Week 3! We have developed planar and three-dimensional dynamic models of the quadrotor. This week, you will learn more about how to develop linear controllers for these models. With this knowledge, you will be required to complete the second programming assignment of this course, which focuses on controlling the quadrotor in two dimensions. We encourage you to start working on the assignment soon. This week ends with a discussion on motion planning for quadrotors.

Time, Motion, and Trajectories9:08

Time, Motion, and Trajectories (continued)9:23

Motion Planning for Quadrotors12:04

Supplementary Material: Minimum Velocity Trajectories from the Euler-Lagrange Equations2:12

Supplementary Material: Solving for Coefficients of Minimum Jerk Trajectories5:57

Supplementary Material: Minimum Velocity Trajectories3:44

Supplementary Material: Linearization of Quadrotor Equations of Motion5:47

Rencontrer les enseignants

Vijay Kumar
Nemirovsky Family Dean of Penn Engineering and Professor of Mechanical Engineering and Applied Mechanics
School of Engineering and Applied Science

In lecture, we claimed that in the linearized equations of motion for

the quadrotor, the second derivative of position is proportional to U1,

and the fourth derivative of position is proportional to U2.

In this segment, we'll derive these relationships explicitly.

Recall the quadrotors equations of motion, which come from the linear and

angular momentum balances.

We can use U1 to represent the total thrust applied and

U2 to represent the moment vector.

We could then rewrite the equations of motion in terms of U1 and

the components of U2.

At the equilibrium hover configuration, the position and

yaw angle of the quadrotor can be at some arbitrary value r0 and sin0 respectively.

However, the angles theta and phi, as well as r dot, theta dot,

phi dot and sy dot are all zero.

We want to derive expressions for

the equations of motion when the quadrotor is near this equilibrium configuration.

First, let's consider the value of cosine theta

near the equilibrium configuration when theta equals zero.

Around the theta equals 0, the function cosine of theta can be approximated

using the Taylor expansion which is shown here.

We can consider all the terms after the first two terms in this series to be

negligibly small.

The value of cosine at theta equals 0 is 1.

The derivative of cosine is negative sin theta.

Since sin theta, a theta equals 0 is 0.

The approximation of cosine theta near theta equals 0 is simply 1.

Can you confirm this qualitatively?

By looking at the value of the cosine function, near theta equals 0.

We can plot the cosine function for angles negative 30 to 30 degrees.

We can see that the cosine values are indeed close to 1.

We can repeat this process to approximate the function sin theta near

theta equals 0.

Again, we use the Taylor series.

For sin theta we arrive at the following expression.

Since sin theta, a theta equals 0 is 0, and the derivative

of sin theta is cosine theta, we can simplify the expression as follows.

Since the value of cos(theta) at theta=0 is 1,

the value of sin(theta) near theta=0 is approximately theta.

This suggests that around theta = 0, we expect the sin function to look linear.

If we again plot the values of sin theta for angles from -30 to 30 degrees,

we see that the sin function indeed looks linear.

We will use these two approximations to linearize the equations of motion of

the quadrotor.

Again, we want to use this approximations

to help direct simplified versions of the equations of motion that

apply when the quadrotor is near the equilibrium hover configuration.

First, consider the linear momentum equation.

We can explicitly write the rotation matrix in terms of the Euler angles.

We can then perform the matrix multiplication to arrive at the following

second order differential, which at equilibrium,

the pitch angle theta and the roll angle phi are both approximately 0.

Therefore we can use the equation we found earlier to approximate the sins and

cosines of these angles.

Substituting in these approximations reduces the differential equations

to the ones shown here.

You can clearly see that the second derivative of position

is proportional to the input, u1.

Now consider the relationship between the angular velocity components p, q,

r and the first derivatives of the Euler's angles.

Again, we carry out the matrix multiplication to arrive at three

equations to relating p, q ad r to phi dot, theta dot and psi dot.

We can then substitute in the approximations for sins and

cosines of theta and phi.

This will resolves in the equation shown here.

Next, we can approximate all terms that are the product of an angle and

an angles derivative as 0.

Near Hover, theta and phi in all angular derivatives are close to 0.

The product of two terms near 0 will be very small.

And as a result you can approximate these terms as 0 itself.

Substituting these approximations into the equations, tells us that around hover,

the angular velocity components are approximately

the time derivatives of the Euler angles.

Finally consider the angular momentum equation.

First, we approximate the off diagonal inertia terms as close to 0.

This allows us to simplify the inertia matrix.

Performing the matrix multiplication gives us the following set of equations.

We saw earlier that around Hover, p, q and r are approximately phi dot,

theta dot and sin dot respectively and are therefore also close to 0.

The pattern of any two angular velocity components can then be approximated as 0.

This gives us the following set of equations.

Using again the approximation of the angular velocity components as the Euler

angle derivatives, we arrive at this set of second order differential equations.

Now let's go back to the linear momentum equation in the x direction.

We can differentiate this equation twice.

We can then substitute the approximations of the angular momentum equations of

motion into the expression for the fourth derivative of x.

Omitting terms that do not contain the second derivative of an Euler angle,

we arrive at the following expression for the fourth derivative of x.

Carrying out this procedure in y and z directions yields similar equations.

You can see that the fourth derivative of position is indeed proportional to u2.

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