## How is Orientation in Space Represented with Euler Angles?

By Ilian Bonev, Ph.D., Eng.

If you use a six-axis robot arm, like Mecademic’s Meca500 used in this tutorial as an example, you are most probably interested in positioning its tool (end-effector) in various orientations. In other words, you need to be able to program your robot to move its end-effector both to a desired position AND a desired orientation (i.e., to a desired pose). Of course, you can always jog your robot’s end-effector or hand-guide it to approximately the desired pose, but this so-called online programming method is tedious and very imprecise. It is much more efficient to calculate and define your desired pose offline. Besides, in order to define the tool reference frame associated with your end-effector (as in the figure shown below), you would need to calculate the pose of that tool reference frame with respect to the flange reference frame. In 3D space, you need a minimum of six parameters to define a pose. For example, the position of the robot’s end-effector, or more precisely of the TCP (tool center point), is typically defined as the x, y and z coordinates of the origin of the tool reference frame with respect to the world reference frame. But how do you then define orientation in space?
“The representation of orientation in space is a complex issue.”
The representation of orientation in space is a complex issue. Euler’s rotation theorem states that, in (3D) space, any displacement of a rigid body in such way that a point on the rigid body remains fixed is equivalent to a single rotation about an axis that passes through the fixed point. Accordingly, such rotation can be described by three independent parameters: two for describing the axis and one for the rotation angle. Orientation in space, however, can be represented in several other ways, each with its own advantages and disadvantages. Some of these representations use more than the necessary minimum of three parameters. The most common way of transforming position coordinates from one Cartesian (3D) reference frame, F, to another, F’, is the rotation matrix. This 3×3 matrix can therefore be used to represent the orientation of reference frame F’  with respect to reference frame F. However, this representation, while often necessary as we will discover later, is not a compact and intuitive way to define orientation. Another much more compact way of defining orientation is the quaternion. This form of representation consists of a normalized vector of four scalars. The quaternion is generally used in robot controllers, as it is not only more compact than the rotation matrix, but also less susceptible to approximation errors. Moreover, during an interpolation between two different orientations, the elements of the quaternion continuously change, avoiding the discontinuities inherent in three-dimensional parameterizations such as Euler angles. Nevertheless, the quaternion is rarely used as a means of communication between a user and the controller of the robot because it is unintuitive. (For more information, see our tutorial on quaternions.)

#### Detailed definition of Euler angles

By far the most common way to communicate an orientation in space to a user, or to allow a user to define an orientation, in a CAD software or in a robot controller, is the use of Euler angles. Because the term Euler angles is often misused, we have prepared this interactive tutorial.
“[…] the term Euler angles is often misused […]”
Euler angles are a set (or rather a sequence) of three angles, which can be denoted for example by α, β, and γ. (Often, Euler angles are denoted by roll, pitch, and yaw.) Euler angles are defined as follows: Consider two Cartesian right-handed 3D reference frames, of which one will be arbitrarily called the fixed frame and the other will be referred to as the mobile frame. The two reference frames coincide initially. To define the orientation of a third frame (all three frames share the same origin), the mobile frame is brought, in the order shown below, to coincide with the third frame by rotating the mobile frame
1. about the x, y, or z axis of the fixed frame or the x’, y’, or z’  of the mobile frame, by α degrees,
2. then about the x, y, or z axis of the fixed frame or the x’, y’, or z’  of the mobile frame, by β degrees,
3. and finally about the x, y, or z axis of the fixed frame or the x’, y’, or z’  of the mobile frame, by γ degrees.
The order in which the three rotations is done is important. Thus, we have a total of 216 (63) possible sequences: xyz, yyz, zyz, x’yz, y’yz, z’yz, and so forth. However, a sequence of three rotations in which two consecutive rotations are about the same axis (e.g., yyz) cannot describe a general orientation. In addition, prior to the first rotation, x coincides with x’, y coincides with y’, and z coincides with z’. Consequently, of all these 216 combinations, there exist only twelve unique meaningful ordered sequences of rotations, or twelve Euler angle conventions: XYX, XYZ, XZX, XZY, YXY, YXZ, YZX, YZY, ZXY, ZXZ, ZYX, ZYZ. That said, each of the twelve combinations is equivalent to three other sequences. In other words, each Euler angle convention can be described in four different ways. For example, the ZYX convention is equivalent to the sequences zyx, x’y’z’, yz’x and yxz’. Fortunately, no one describes Euler angles with sequences in which some rotations are about the mobile frame axes and others are about the fixed axes (e.g., sequences like yz’x and yxz’). Thus, while there are twelve different Euler angle conventions, each is typically described in two different ways: either as a sequence of rotations about the axes of the fixed frame or as a sequence of rotations about the axes of the mobile frame. Therefore, it can be convenient to talk about fixed and mobile conventions, although they are equivalent. For example, the fixed XYZ Euler angle convention is described by the xyz sequence, while the mobile ZYX Euler angle convention is described by the z’y’x’  sequence, but both are equivalent, as we will see later. In robotics, FANUC and KUKA use the fixed XYZ Euler angle convention, while ABB uses the mobile ZYX Euler angle convention. Furthermore, Kawasaki, Omron Adept Technologies and Stäubli use the mobile ZYZ Euler angle convention. Finally, the Euler angles used in CATIA and SolidWorks are described by the mobile ZYZ Euler angle convention.
“At Mecademic, we use the mobile XYZ Euler angle convention.”
At Mecademic, we use the mobile XYZ Euler angle convention, and therefore describe Euler angles as the sequence x’y’z’. Why be different? The reason is that we used to offer a mechanical gripper for handling axisymmetric workpieces (see video), which was actuated by the motor of joint 6. A six-axis robot equipped with such a gripper can only control two rotational degrees of freedom, or more specifically the direction of the axis of joint 6, that is to say the direction of the axis of symmetry of the workpiece. In the chosen Euler angle convention, angles α and β define this direction, while angle γ is ignored because it corresponds to a parasitic rotation that is uncontrollable. Our applet below will help you understand Euler angles. You can select one of the twelve possible Euler angle conventions by clicking on the x, y, and z boxes of the first, second and third rotation. (The default Euler angle sequence is the one used by Mecademic.) To switch between rotations about the axes of the fixed or mobile frames, you need to double-click on any of these nine boxes. The axes of the fixed frame are drawn in gray while the axes of the mobile frame are in black. Axes x and x’  are drawn in red, y and y’  in green, and z and z’  in blue. Gliding along any of the three blue horizontal arrows with your mouse changes the corresponding Euler angle. Alternatively, you can directly set the Euler angle value (in degrees) in the corresponding textbox below the arrow. Finally, you can drag your mouse over the reference frame to change the viewpoint.
 α : β : γ :
R = Rx(0°) Rx(0°) Rx(0°) =
 n/a n/a n/a n/a n/a n/a n/a n/a n/a

#### Calculating Euler angles via rotation matrices

With the above applet, you will see the orientation of the mobile frame with respect to the fixed frame, for a given set of Euler angles, in the far right subfigure. Unfortunately, however, in practice, the situation is usually the opposite. You frequently have two reference frames, and you want to find the Euler angles that describe the orientation of one frame with respect to the other. For orientations in which at least two axes are parallel, you could attempt to guess the Euler angles by trial and error. For example, look back at the image at the beginning of this tutorial and try to find the Euler angles used by Mecademic that define the orientation of the tool reference frame associated with the gripper, with respect to the flange reference frame. The answer is α = −90°, β = 0°, γ = −90°. Not so easy to get, is it? To be more efficient therefore, you must learn about rotation matrices after all. As we have already mentioned, any orientation in space can be represented with a 3×3 rotation matrix. For example, a rotation of α about the axis x, a rotation of β about the axis y, and a rotation of γ about the axis z, respectively correspond to the following three rotation matrices:
Rx(α) =
 1 0 0 0 cos(α) −sin(α) 0 sin(α) cos(α)
,
Ry(β) =
 cos(β) 0 sin(β) 0 1 0 −sin(β) 0 cos(β)
,
Rz(γ) =
 cos(γ) −sin(γ) 0 sin(γ) cos(γ) 0 0 0 1
.
We will refer to the above matrices as basic rotation matrices. To obtain the product of basic rotation matrices that corresponds to a sequence of rotations, start by writing the basic rotation matrix corresponding to the first rotation. For example, if the first rotation is about the x (or x’ ) axis, then write Rx(ψ), where ψ  is the angle of rotation. For every subsequent rotation, post-multiply (right multiply) the current result with the next rotation matrix, if the rotation is about an axis of the mobile reference frame, or pre-multiply (left multiply) the current result with the next rotation matrix, if the rotation is about an axis of the fixed reference frame. Use our applet to see the resulting product of basic rotation matrices. For example, the rotation sequence x’y’z’  corresponds to the product R = Rx(α)Ry(β)Rz(γ). Thus, the rotation matrix that corresponds to the Euler angles used by Mecademic is:
R(α, β, γ) =
 cos(β)cos(γ) −cos(β)sin(γ) sin(β) cos(α)sin(γ) + sin(α)sin(β)cos(γ) cos(α)cos(γ) − sin(α)sin(β)sin(γ) −sin(α)cos(β) sin(α)sin(γ) − cos(α)sin(β)cos(γ) sin(α)cos(γ) + cos(α)sin(β)sin(γ) cos(α)cos(β)
.
Therefore, for a given orientation, you will need to do two things: First, you need to find the rotation matrix that corresponds to your orientation. Second, you need to extract the Euler angles using a couple of simple equations. Let us first show you two ways to find your rotation matrix. Consider the example shown in the figure below where we need to find the rotation matrix representing the orientation of frame F’  with respect to frame F. (Recall that we always represent the x axis in red, the y axis in green, and the z axis in blue.) Here, it is easy to see that if we align a third reference frame with F, which will act as a mobile frame, then rotate this frame about its z’  axis at θ − 90°, and then rotate it about its y’  axis at φ degrees, we will obtain the orientation of F’. Thus, the rotation matrix we are looking for is:
Rdesired = Rz(θ − 90°)Ry(φ) =
 sin(θ)cos(φ) cos(θ) sin(θ)sin(φ) −cos(θ)cos(φ) sin(θ) −cos(θ)sin(φ) −sin(φ) 0 cos(φ)
.
Alternatively, we can obtain the above rotation matrix directly. Its first, second and third columns represent the coordinates of the unit vectors along thex, y and z axis, respectively, of frame F’, with respect to frame F. Now that you have the rotation matrix that represents your desired orientation, you simply need to solve the system of nine scalar trigonometric equations Rdesired = R(α,β,γ), for α, β, and γ. Fortunately, this problem has a generic solution and we’ll simply give you the equations to use. Let the desired orientation of a frame F’  with respect to a frame F be represented by the following rotation matrix:
Rdesired =
 r1,1 r1,2 r1,3 r2,1 r2,2 r2,3 r3,1 r3,2 r3,3
.
The Euler angles (in degrees), in keeping with the mobile XYZ convention used by Mecademic, are then obtained according to the following two cases: Case 1: r1,3 ≠ ±1 (i.e., the z’  axis of frame F’  is not parallel to the x axis of frame F).

β = asin(r1,3), γ = atan2(−r1,2, r1,1), α = atan2(−r2,3, r3,3).

Case 2: r1,3 = ±1 (i.e., the z’  axis of frame F’  is parallel to the x axis of frame F).

β = r1,390°, γ = atan2(r2,1, r2,2), α = 0°.

In the general Case 1, we actually have two sets of solutions where all angles are in the half-open range (−180°, 180°]. However, it is useless to calculate both sets of solutions, so only the first is presented, in which −90° < β < 90°. Also, note that we use the function atan2(y, x) in our solution. Beware that in some programming languages, in some scientific calculators and in most spreadsheet software, the arguments of this function are inverted. Finally, note that Case 2 corresponds to a so-called representation singularity. This singularity is present in any three-parameter representation of orientation in 3D space (not only in Mecademic’s choice of Euler angles). It is similar to the problem of representing points on a sphere by only two parameters. For example, longitude is not defined on Earth at the South and North Poles on the Earth. In other words, this singularity has nothing to do the singularities of mechanisms (e.g., the so-called gimbal lock), which correspond to actual physical problems (e.g., the loss of a degree of freedom).

#### Exercice

Consider the following real-life situation that occured to us. We wanted to attach a FISNAR dispensing valve to the end-effector of our Meca500 robot arm. Naturally, the engineer who designed and machined the adapter didn’t care about Euler angles and was only concerned with machinability and reachability. In his design, there were essentially two rotations of 45°. Firstly, he used two diametrically oposite threaded holes on the robot flange to attach the adapter, which caused the first rotation of 45°. Secondly, the angle between the flange interface plane and the axis of the dispenser was 45°. The figure above shows the actual installation (left) and the tool frame (right) that needed to be defined. Note that when using axi-symmetric tools, it is a common practice to allign the tool z-axis with the axis of the tool, but this may be wrong. This practice is useful with the mobile XYZ Euler angle convention, since the redundant rotation about the axi-symmetric tool corresponds to the third Euler angle, γ. Thus, the first two Euler angles define the axis of the tool, while the third one can be used to choose the optimal configuration of the robot (i.e., far from singularities). Thus, if your robot is using the mobile ZYX Euler angle convention, you should align the tool x-axis with the axis of the tool.
“When using axi-symmetric tools, it is a common practice to allign the tool z-axis with the axis of the tool, but this may be wrong.”
Returning to our example, we will show now that it is impossible to come up with the Euler angles according to the mobile XYZ convention by trial and error. Indeed, for this choice of tool reference frame, we can represent the final orientation as a sequence of the following two rotations: R = Rz(45°)Ry(45°). From here, we can extract the Euler angles according to the mobile XYZ convention using the equations previously described and obtain: α = −35.264°, β = 30.000°, γ = 54.735°. Are you convinced now that you do need to master Euler angles for situations like this?