Cálculo de ángulos de Euler en un robot de 6 ejes

¿Cómo se representa la orientación en el espacio con los ángulos de Euler?

Si utiliza un brazo robótico de seis ejes, como el Meca500 de Mecademic utilizado en este tutorial como ejemplo, lo más probable es que esté interesado en posicionar su herramienta (efector final ) en varias orientaciones. En otras palabras, debe poder programar su robot para mover su efector final tanto a la posición deseada como a la orientación deseada (es decir, a la postura deseada). ). Por supuesto, siempre puede mover el efector final de su robot o guiarlo manualmente hasta aproximadamente la pose deseada, pero este llamado método de programación en línea es tedioso y muy impreciso. Es mucho más eficiente calcular y definir la pose deseada fuera de línea. Además, para definir el marco de referencia de la herramienta asociado con su efector final (como en la imagen que se muestra aquí), deberá calcular la posición del marco de referencia de esa herramienta con respecto al marco de referencia de la brida .
En el espacio 3D, necesita un mínimo de seis parámetros para definir una pose. Por ejemplo, la posición del efector final del robot, o más precisamente del TCP (punto central de la herramienta ), normalmente se define como x , y y z coordenadas del origen del marco de referencia de la herramienta con respecto al marco de referencia universal . Pero, ¿cómo se define entonces la orientación en el espacio?

La representación de la orientación en el espacio es un tema complejo. Teorema de rotación de Euler establece que, en el espacio (3D), cualquier desplazamiento de un cuerpo rígido de tal manera que un punto en el cuerpo rígido permanezca fijo es equivalente a una sola rotación alrededor de un eje que pasa por el punto fijo. En consecuencia, dicha rotación puede describirse mediante tres parámetros independientes:dos para describir el eje y uno para el ángulo de rotación. Sin embargo, la orientación en el espacio se puede representar de varias otras formas, cada una con sus propias ventajas y desventajas. Algunas de estas representaciones utilizan más del mínimo necesario de tres parámetros.

La forma más común de transformar las coordenadas de posición de un marco de referencia cartesiano (3D), F , a otro, F' , es la matriz de rotación . Por lo tanto, esta matriz de 3×3 se puede utilizar para representar la orientación del marco de referencia F’ con respecto al marco de referencia F . Sin embargo, esta representación, aunque a menudo es necesaria como veremos más adelante, no es una forma compacta e intuitiva de definir la orientación.
Otra forma mucho más compacta de definir la orientación es el cuaternión . Esta forma de representación consiste en un vector normalizado de cuatro escalares. El cuaternión se usa generalmente en controladores de robots, ya que no solo es más compacto que la matriz de rotación, sino que también es menos susceptible a errores de aproximación. Además, durante una interpolación entre dos orientaciones diferentes, los elementos del cuaternión cambian continuamente, evitando las discontinuidades inherentes a las parametrizaciones tridimensionales como los ángulos de Euler. Sin embargo, el cuaternión rara vez se usa como medio de comunicación entre un usuario y el controlador del robot porque no es intuitivo.

Definición detallada de los ángulos de Euler

Con mucho, la forma más común de comunicar una orientación en el espacio a un usuario, o de permitir que un usuario defina una orientación, en un software CAD o en un controlador de robot, es el uso de ángulos de Euler . Debido a que el término ángulos de Euler a menudo se usa incorrectamente, hemos preparado este tutorial interactivo.

Los ángulos de Euler son un conjunto (o más bien una secuencia) de tres ángulos, que se pueden denotar, por ejemplo, con α , β , y γ . (A menudo, los ángulos de Euler se denotan por rodar , tono y guiñada .) Los ángulos de Euler se definen de la siguiente manera:Considere dos marcos de referencia 3D cartesianos diestros, uno de los cuales se denominará arbitrariamente fijo marco y el otro se denominará móvil cuadro. Los dos marcos de referencia coinciden inicialmente. Para definir la orientación de un tercer marco (los tres marcos comparten el mismo origen), el marco móvil se hace coincidir, en el orden que se muestra a continuación, con el tercer marco girando el marco móvil

1. sobre la x , y , o z eje del marco fijo o la x’ , y' , o z' del marco móvil, por α grados,
2. entonces sobre la x , y , o z eje del marco fijo o la x’ , y' , o z' del marco móvil, por β grados,
3. y finalmente sobre la x , y , o z eje del marco fijo o la x’ , y' , o z' del marco móvil, por γ grados.

El orden en que se realizan las tres rotaciones es importante. Así, tenemos un total de 216 (6 ³ ) secuencias posibles: x →y →z , y →y →z , z →y →z , x’→y →z , y’→y →z , z’→y →z , Etcétera. Sin embargo, una secuencia de tres rotaciones en las que dos rotaciones consecutivas son sobre el mismo eje (por ejemplo, y →y →z ) no puede describir una orientación general. Además, antes de la primera rotación, x coincide con x ', y coincide con y ', y z coincide con z '. En consecuencia, de todas estas 216 combinaciones, solo existen doce secuencias de rotaciones ordenadas significativas únicas, o doce convenciones de ángulo de Euler :XYX, XYZ, XZX, XZY, YXY, YXZ, YZX, YZY, ZXY, ZXZ, ZYX, ZYZ.
Dicho esto, cada una de las doce combinaciones es equivalente a otras tres secuencias. En otras palabras, cada convención de ángulo de Euler se puede describir de cuatro maneras diferentes. Por ejemplo, el ZYX la convención es equivalente a las secuencias z →y →x , x '→y '→z ', y →z '→x y y →x →z '. Afortunadamente, nadie describe los ángulos de Euler con secuencias en las que algunas rotaciones son sobre los ejes del marco móvil y otras sobre los ejes fijos (por ejemplo, secuencias como y →z '→x y y →x →z ‘).
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 x →y →z  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, 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 bellow the arrow. Finally, you can drag your mouse over the reference frame to change the viewpoint.

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α  :

β  :

γ  :

R  = R x (0°) R y (0°) R z (0°) =

1.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 1.000

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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:

R x (α ) =

1 0 0 0 cos(α ) −sin(α ) 0 sin(α ) cos(α )  ,

R y (β ) =

cos(β ) 0 sin(β ) 0 1 0 −sin(β ) 0 cos(β )  ,

R z (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 R _x (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  = R _x (α )R _y (β )R _z (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 the F , which will act as a mobile frame, then rotate this frame about its z ‘ axis at θ  degrees, 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:

R desired  = R z (θ )R y (φ ) =

cos(θ )cos(φ ) −sin(θ ) cos(θ )sin(φ ) sin(θ )cos(φ ) cos(θ ) sin(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 the x , 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 R _desired  = 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:

R desired  =

r 1,1 r 1,2 r 1,3 r 2,1 r 2,2 r 2,3 r 3,1 r 3,2 r 3,3  .

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:  r _1,3  ≠ ±1 (i.e., the z’  axis of frame F’  is not parallel to the x  axis of frame F ).

β  = asin(r _1,3 ),   γ  = atan2(−r _1,2 , r _1,1 ),   α  = atan2(−r _2,3 , r _3,3 ).

Case 2:  r _1,3  = ±1 (i.e., the z’  axis of frame F’  is parallel to the x  axis of frame F ).

β  = r _1,3 90°,   γ  = atan2(r _2,1 , r _2,2 ),   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. This is particularly useful with the mobile XYZ Euler angle convention, since the redutant 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).

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  = R _z (45°)R _y (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?

Representational singularities and orientation errors

In the case of the mobile XYZ Euler angle convention, if the z’  axis of frame F’  is parallel to the x  axis of frame F , there are infinite pairs of α  and γ  that will define the same orientation. Obviously, you only need one to define your desired orientation, so we have arbitrarily set α  to be equal to zero. More specifically, if β  = 90°, then any combination of α  and γ , such than α + γ = φ , where φ is any value, will correspond to the same orientation, and be output by Mecademic’s controller as {0,90°,φ}. Similarly, if β  = −90°, then any combination of α  and γ , such that α − γ = φ , where φ is any value, will correspond to the same orientation, and be output by Mecademic’s controller as {0,−90°,−φ}. Note, however, that if you try to represent the orientation of a frame F’  with respect to a frame F  and the z’  axis of frame F’  is almost parallel to the x  axis of frame F  (i.e., β is very close to ±90°), the Euler angles will be very sensitive to numerical errors. In such a case, you should enter as many digits after the decimal point as possible when defining the orientation using Euler angles.
Consider the following situation which has caused worries to several users of our Meca500. You set the orientation of the tool reference frame with respect to the world reference frame to {0°, 90°, 0°}, which is a representational singularity. Then you keep this orientation and move the end-effector in space to several positions. At some positions, because of numerical noise, the controller does not detect the condition r _1,3  = ±1 (Case 2, as mentioned above) and calculates the Euler angles as if the orientation did not correspond to a representational singularity. Thus, the controller returns something like {41.345°, 90.001°, −41.345°}, which seems totally wrong and very far away from {0°, 90°, 0°}. Well it’s not.
Unlike position errors, which are measured as √(Δx ²  + Δy ²  + Δz ² ), orientation errors are not directly related with the variations in the Euler angles, especially close to representational singularities. To better understand this so-called non-Euclidean nature of Euler angles, consider the spherical coordinates used to represent a location on Earth. At the North Pole, the latitude is 90° (North), but what is the longitude? Longitude is not defined at the North Pole, or it can be any value. Now imagine that we move only 1 mm away from the North Pole in the direction of Greenwich. In this case, the latitude will be 89.99999999°, but the longitude will now have the value of 0°. Imagine once again that you return to the North Pole and move 1 mm in the direction of Tokyo. The new longitude will be approximately 140°. Between your two locations, the error in longitude is 140°! However, the real angular error will be approximately 0.00000002°.
The situation described above is similar in all other Euler angles conventions. Depending on the Euler angle convention, the correspoding representation singularity occurs when a specific axis of frame F’  is parallel to another specific axis of frame F . In such a representation singularity, the first and third rotation become dependant.
In conclusion, unless you master Euler angles (or use sophisticated offline programming software), and more specifically the convention used for programming your robot, you will hardly be able to program anything but simple pick and place operations. Because robotics is not simple, we do our best to help you understand the basics.

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