The full series will include all of the following posts:. For a rigid body in a 3-D world we can describe the location of all points on the vehicle with 6 coordinates. Starting out, we will assume a flat-Earth model, meaning our world the vehicle can move around in is assumed to be above flat ground, in a rectilinear coordinate system, and the gravity is constant.

With constant gravity the center of mass will also be equivalent to the center of gravity cgso we will treat these terms as interchangeable. If we are in a 3-D world, how can there be 6 degrees of freedom? The idea of a rigid body is that every point on the vehicle has a fixed location relative to the cg.

The opposite of this would be a flexible body, where different points on the body are free to flex and vibrate and introduce additional dynamics to the system. The final 3 coordinates represent the orientation of the vehicle represented by three angles:. The 3 position vector components I think are easy enough to understand.

This coordinate frame could be aligned with North, East, and vertical height and the values of the components would be some distance in each of those directions. But 3 coordinates that describe the attitude of a vehicle is not as intuitive to grasp. There are multiple means of representing orientation, but the two most common are Euler angles and quaternions. However they do have a limitation as for certain orientations an ambiguity arises where the three coordinates do not define a unique orientation.

This singularity is referred to as gimbal-lock and can be mitigated by using quaternions instead. Often times simulations and controllers for vehicles will do their calculations in quaternions, but convert back to Euler angles for output because of their ease of use. I will discuss quaternions in more detail in a later post, but for now you can accept that Euler angles are sufficient to describe all the orientations of the vehicle we expect it to fly in.

Before we go into Euler angles we will first introduce the idea of multiple reference frames. For our model, there will be two reference frames coordinate systems we will care about:. Now that we have two reference frames, we need a way to switch between them.

If we can establish a general relationship that allows us to transform from one set of coordinates to the other, then we can do calculations in whichever is most convenient. Then we can transform that solution to the other reference frame when needed. This is the same concept of rectangular and polar coordinates you may have encountered in grade school algebra classes: you can describe the same location in both, and use mathematical relationships to switch back and forth between them.

But you learn to use both because each has different advantages for different situations. Our end goal is to use the two coordinates to separate out translational and rotational motion.

The translation representation is simply the vector distance of the body frame from the inertial. The idea of Euler angles is that one can represent any final orientation of a reference frame via a set of 3 sequential rotations about specific axes. To get used to the matrix notation we will use to describe the rotations, let us first start with the trivial case where both coordinate frames have the same orientation.

Here we are representing the coordinate frames with unit vectors [x, y, z] and [b1, b2, b3]. A convenient way to transform one vector to another is through matrix multiplication. When all 3 components are parallel to each other, it can be seen in the figure above that the transformation matrix is simply the identity matrix.

Now, what if we rotate the body frame about one of its axes by a specific amount? We can describe the other body frame axes with trigonometry as a function of the inertial components and the angle of rotation. These equations can be condensed again into a matrix multiplication expression as shown below, looking at each axis individually. Now imagine that we perform those 3 rotations sequentially in the order of rotating about b 3then b 2then b 1.

The body frame will end up looking something like this:. Notice how the second and third rotations are about axes which have already been rotated.

This is why the order of Euler angles must be consistent. There are many sequences you can choose from and different disciplines have different standards. However once a sequence of the three angle rotations is chosen you must always transform them in that order, otherwise you may not get the correct final orientation.

For our modeling we will follow the standard commonly used in the aircraft industry which uses Body angles, as illustrated above.The most popular representation of a rotation tensor is based on the use of three Euler angles. Early adopters include Lagrange, who used the newly defined angles in the late s to parameterize the rotations of spinning tops and the Moon [ 12 ], and Bryan, who used a set of Euler angles to parameterize the yaw, pitch, and roll of an airplane in the early s [ 3 ].

Today, Euler angles are widely used in vehicle dynamics and orthopaedic biomechanics. As discussed in [ 45 ], the Euler angle representation dates to works by Euler [ 67 ] that he first presented in Although the paper [ 7 ] dates to the 18th century, it was first published posthumously in One interpretation of the Euler angles involves a decomposition of a rotation tensor into a product of three fairly simple rotations:.

In representation 1denote the Euler angles, and the set of unit vectors is known as the Euler basis. In general, is a function of andand is a function of. Because there are three Euler angles, the parameterization of a rotation tensor by use of these angles is an example of a three-parameter representation of a rotation.

Furthermore, there are 12 possible choices of the Euler angles. For example, Figure 1 illustrates these angles for a set of Euler angles:. While the Euler basis is composed of a set of unit vectors, the basis is typically non-orthogonal and, in many cases, not right-handed.

This can be observed from the animation of the Euler angle sequence provided in Figure 2where the Euler basis is highlighted in cyan. Notice that the Euler basis vector is perpendicular to the plane formed by andi.

If we assume that the Euler basis vector is constant, then the angular velocity vector associated with the Euler angle representation of a rotation tensor can be established by using relative angular velocity vectors. In this case, there are two such vectors: and. For the first rotation, because its axis of rotation is fixed, the angular velocity vector is simply. The calculation of the relative angular velocity vectors is similar to the example considered in our discussion of relative rotations and their angular velocities.

Combining 45and 6we conclude that the angular velocity vector associated with is. The simplicity of this representation is remarkable. If the rotation tensor transforms the fixed vectors into the corotational vectorsthen it is possible to express the Euler basis in terms of either set of vectors. Alternative approaches to establishing 7 can be found in various textbooks. In one such approach, all three Euler angles are considered to be infinitesimal.

A good example of this approach can be found in Section 2. Another approach, which can be found in [ 910 ] and dates to Euler, features spherical geometry.

Finally, a third, but lengthy, approach involves differentiating the rotation tensor 1computing the angular velocity tensorand then calculating its corresponding axial vector. For the Euler angles to effectively parameterize all rotations, we must assume that we can find and such that 7 holds for any given angular velocity vector. For this to happen, it is necessary and sufficient that the Euler basis vectors, and span. When these vectors are not linearly independent, we say that the Euler angles have a singularity.

Thus, it is necessary to restrict this angle to avoid these singularities. For future developments, it is also convenient to define the dual Euler basis such thatwhere is the Kronecker delta.

The dual Euler basis vectors are analogous to the contravariant basis vectors in differential geometry and have similar uses in dynamics. We will use the dual Euler basis later when discussing particular sets of Euler angles; a rapid summary of this and other uses can be found in [ 1213 ].

One method of determining the dual Euler basis is to express the Euler basis vectors in terms of a right-handed basis, say. Then, for example, to determinewe could write and solve the three equations, and for the three unknowns, and.Compared to quaternions, Euler Angles are simple and intuitive and they lend themselves well to simple analysis and control.

On the other hand, Euler Angles are limited by a phenomenon called "Gimbal Lock," which we will investigate in more detail later.

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Euler angles provide a way to represent the 3D orientation of an object using a combination of three rotations about different axes. For convenience, we use multiple coordinate frames to describe the orientation of the sensor, including the "inertial frame," the "vehicle-1 frame," the "vehicle-2 frame," and the "body frame.

The vehicle-1 and vehicle-2 are intermediary frames used for convenience when illustrating the sequence of operations that take us from the inertial frame to the body frame of the sensor. It may seem unnecessarily complicated to use four different coordinate frames to describe the orientation of the sensor, but the motivation for doing so will become clear as we proceed. For clarity, this application note assumes that the sensor is mounted to an aircraft.

All examples and figures are given showing the changing orientation of the aircraft. The "inertial frame" is an Earth-fixed set of axes that is used as an unmoving reference. CH Robotics' sensors use a common aeronautical inertial frame where the x-axis points north, the y-axis points east, and the z-axis points down as shown below.

Note that because the z-axis points down, altitude above ground is actually a negative quantity. The sequence of rotations used to represent a given orientation is first yaw, then pitch, and finally roll. As shown in Figure 1, yaw represents rotation about the inertial-frame z-axis by an angle.

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The yaw rotation produces a new coordinate frame where the z-axis is aligned with the inertial frame and the x and y axes are rotated by the yaw angle. We call this new coordinate frame the vehicle-1 frame. The orientation of the vehicle-1 frame after yaw rotation is show in Figure 2.

The vehicle-1 frame axes are colored red, while the inertial frame axes are gray. Rotation of a vector from the Inertial Frame to the Vehicle-1 Frame can be performed by multiplying the vector by the rotation matrix. For clarity, the inertial-frame axes are not shown.

The vehicle-1 frame axes are shown in gray, and the vehicle-2 axes are shown in red. It is important to note that pitch is NOT rotation about the inertial-frame Y-axis. The rotation matrix for moving from the inertial frame to the vehicle-2 frame consists simply of the yaw matrix multiplied by the pitch matrix:.

The body frame is the coordinate system that is aligned with the body of the sensor. On an aircraft, the body frame x-axis typically points out the nose, the y-axis points out the right side of the fuselage, and the z-axis points out the bottom of the fuselage. For clarity, the inertial frame and vehicle-1 frame axes are not shown.

The vehicle-2 frame axes are shown in gray, while the body-frame axes are shown in red.

## Understanding Euler Angles

The rotation matrix for moving the opposite direction - from the body frame to the inertial frame - is given by. Performing the multiplication, the complete rotation from the body frame to the inertial frame is given by. Note that all this does is reverse the order of operations and reverse the direction of rotation. Gimbal lock occurs when the orientation of the sensor cannot be uniquely represented using Euler Angles.

The exact orientation at which gimbal lock occurs depends on the order of rotations used. On CH Robotics' sensors, the order of operations results in gimbal lock when the pitch angle is 90 degrees. Intuitively, the cause of gimbal lock is that when the pitch angle is 90 degrees, yaw and roll cause the sensor to move in exactly the same fashion. Consider Figure 5 for an illustration of the gimbal lock condition.

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By following the sequence of rotations discussed in this paper, it should be easy to see that the orientation in Figure 5 can be obtained by yawing and then pitching, OR by pitching and then rolling. An orientation sensor or AHRS that uses Euler Angles will always fail to produce reliable estimates when the pitch angle approaches 90 degrees.Documentation Help Center.

Implement Euler angle representation of six-degrees-of-freedom equations of motion. The 6DOF Euler Angles block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame X bY bZ b about a flat Earth reference frame X eY eZ e. For more information about these reference points, see Algorithms. The block assumes that the applied forces are acting at the center of gravity of the body, and that the mass and inertia are constant. Euler rotation angles [roll, pitch, yaw], returned as three-element vector, in radians. Coordinate transformation from flat Earth axes to body-fixed axes, returned as a 3-by-3 matrix.

Angular rates in body-fixed axes, returned as a three-element vector, in radians per second. Angular accelerations in body-fixed axes, returned as a three-element vector, in radians per second squared. Accelerations in body-fixed axes with respect to body frame, returned as a three-element vector. Accelerations in body-fixed axes with respect to inertial frame flat Earthreturned as a three-element vector.

You typically connect this signal to the accelerometer. This port appears only when the Include inertial acceleration check box is selected. The Simple Variable selection conforms to the previously described equations of motion. The Quaternion selection conforms to the previously described equations of motion. Initial location of the body in the flat Earth reference frame, specified as a three-element vector.

The three-element vector for the initial Euler orientation angles [roll, pitch, yaw], in radians. Assign unique name to each state. You can use state names instead of block paths during linearization. To assign a name to a single state, enter a unique name between quotes, for example, 'velocity'. Each name must be unique. You can specify fewer names than states, but you cannot specify more names than states.To browse Academia.

Consider the following sequence of rotations: about the x-axis, then about the y-axis, then about the z-axis, each rotation being applied about one of the world axes as opposed to one of the body axes. This can be written with, etc.

### The Euler angle parameterization

Now suppose we are given a matrix and are required to extract Euler angles corresponding to the above rotation sequence, i. We may note that this will also cause and to be very small or zero since will be nearmaking the extraction of equally problematic.

This matrix provides an example of the phenomenon of gimbal lock, in which the 1st and 3rd axes are brought into alignment by the 2nd rotation, effectively losing a degree of freedom because now and act in combination as though they were a single parameter. However, it can be dangerous to use in cases which fall just outside the threshold.

When the routine is passed a real-world single-precision matrix whose elements are subject to some typical rounding errors, can be expected to take on a fairly chaotic set of values. This is not surprising, as we expect near-gimbal-lock orientations to produce wild jumps in the angle values, and any numerical errors can accentuate these jumps.

Any wobble in the value of can be counteracted by a suitable anti-wobble in the value of. However, we have a couple of major expectations of the extracted angles. First, they should approximately reproduce the original matrix when passed to the inverse function.

Second, although we may not be able to extract an accurate or in isolation, we still expect their resultant angle to be correct: the extracted values should not be independent. In single-precision code these rounding errors can be very large, leading to completely erroneous results when the computed angles are used in an attempt to reconstruct the original matrix.

Sure enough, the result of passing the extracted angles into the inverse function is often a completely different matrix. We might try substantially increasing the threshold, to make the results less sensitive to rounding error, but the trouble with this approach is that the approximation used in the gimbal lock cases would no longer be valid.

Have we all simply lived with this problem? Fortunately, there seems to be an easy fix: compute the rotation generated by the first and second extracted angles, and work out the rotation needed in the third angle to match the target matrix.

This is easily derived by pre-multiplying the target matrix by the transpose of the reconstructed first-and- second-angle matrix. Reading across the middle row ofwe obtain and hence which requires that we compute the sine and cosine of the value we extracted for.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.

## Understanding Euler Angles

Well balanced mix of theory and practical applicability. Explanation of the material is also very good. The course is very good and is designed such that even beginners can get a good grasp on the content that is made available. The discussion forums are great and help in making life easier. Welcome to Week 2 of the Robotics: Aerial Robotics course!

We hope you are having a good time and learning a lot already!

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In this week, we will first focus on the kinematics of quadrotors. Then, you will learn how to derive the dynamic equations of motion for quadrotors. To build a better understanding on these notions, some essential mathematical tools are discussed in supplementary material lectures. In this week, you will also complete your first programming assignment on 1-D quadrotor control. If you have not done so already, please download, install, and learn about Matlab before starting the assignment.

Loupe Copy. Robotics: Aerial Robotics.

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Course 1 of 6 in the Robotics Specialization. Enroll for Free. From the lesson. Transformations Rotations Euler Angles Angular Velocity Taught By. Try the Course for Free. Explore our Catalog Join for free and get personalized recommendations, updates and offers. Get Started. All rights reserved.So far, our analysis of rotational motion has been of essentially one dimensional, or more precisely one angular parameter, motion: rotating about an axis, rolling, precessing, etc.

But this leaves out many interesting phenomena: the wobbling of a slowing down top, nutation, and so on. We need a well-defined set of parameters for the orientation of a rigid body in space to make further progress in analyzing the dynamics.

The standard set is Euler's Angles. What you actually observe as you watch a child's top beginning to wobble as it slows down is the changing direction of the axis -- this direction is given by the first two of Euler's angles: which are just the usual spherical coordinates, the angle measured from the vertical direction and the azimuthal angle about that vertical axis.

Euler's third angle, specifies the orientation of the top about its own axis, completing the description of the precise positioning of the top. To describe the motion of the wobbling top as we see it, we evidently need to cast the equations of motion in terms of these angles. The rotational motion of a rigid body is completely defined by tracking the set of body principal axeswith origin at the center of mass, as they turn relative to a set of fixed-in-space axes The principal axes can be completely defined relative to the fixed set by three angles: the two angles fix the direction of but that leaves the pair free to turn in the plane perpendicular to the angle fixes their orientation.

To see these angles, start with the fixed axes, draw a circle centered at the origin in the horizontal plane. Now draw a circle of the same size, also centered at the same origin, but in the principal axes plane. Landau calls the line of intersection of these circles or discs the line of nodes. It goes through the common origin, and is a diameter of both circles. The angle between these two planes, which is also the angle between since they're the perpendiculars to the planes is labeled.

The angle between this line of nodes and the axis is It should be clear that together fix the direction of then the other axes are fixed by giving the angle between and the line of nodes The direction of measurement of around are given by the right-hand or corkscrew rule.

Since the position is uniquely defined by Euler's angles, angular velocity is expressible in terms of these angles and their derivatives.

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The strategy here is to find the angular velocity components along the body axes of in turn. Once we have the angular velocity components along the principal axes, the kinetic energy is easy.

From the diagram, is along the line and therefore in the plane: notice it is at an angle with respect to Its components are therefore. Now is about the axis. The principal axis is at angle to the axis, so has component about and in the plane, that latter component along a line perpendicular toand therefore at angle from the axis.

The angular velocity is already along a principal axis. To summarize, the Euler angle angular velocities components along the body's principal axes are:. For a symmetric top, meaning the rotational kinetic energy is therefore. For this symmetrical case, as Landau points out, we could have taken the axis momentarily along the line of nodes giving.

As a warm up in using Euler's angles, we'll redo the free symmetric top covered in the last lecture. With no external torques acting the top will have constant angular momentum. We'll take in the fixed direction. The axis of the top is along. Taking the axis along the line of nodes in the figure on the previous page at the instant considered, the constant angular momentum. Remember, this new axis see diagram! Denoting the distance of the center of mass from the fixed bottom point as along the axis the moment of inertia about a line perpendicular to the axis at the base point is.

The Lagrangian is being the origin, in direction. Notice that the coordinates do not appear explicitly, so there are two constants of motion:. That is, the angular momentum about is conserved, because the two forces acting on the top, the gravitational pull at the center of mass and the floor reaction at the bottom point, both act along lines intersecting the axis, so never have torque about The angular momentum about is conserved because the gravitational torque acts perpendicular to this line.

We have two linear equations in with coefficients depending on and the two constants of motion The solution is straightforward, giving. Using the constants of motion to express in terms of and the constantsthen subtracting a -independent term to reduce clutter, we have.

The range of motion in is given by.