[Turn] attitude calculation of four-rotor aircraft small knowledge points

1, Inertial measurement unit IMU (inertialmeasurement unit)

Attitude Heading Reference System ahrs (attitude and Heading Reference systems)

Geomagnetic angular velocity gravity marg (magnetic, Angular rate, and Gravity)

MEMS (Micro electrical Mechanical systems)

DOF Dimension DOF (Dimension of Freedom)

Unmanned aerial Vehicle UAV (unmanned aerial Vehicle)

Extended Kalman Filter EKF (Extended Kalman Filter)

Lossless Kalman Filter UKF (unscented Kalman Filter)

Inertial navigation systems INS (inertial Navigation System)

Global navigation Satellite System GNSS (Navigation satellite System)

Astronomical navigation systems CNS (celestial Navigation System)

Vertical Landing VTOL (Vertical Take-off and Landing)

2, the common navigation system: inertial navigation, astronomy navigation, satellite navigation, road signs navigation, radio navigation, calculated navigation, combined navigation.

3, there are two basic coordinate system:"geographic " coordinate system and " carrier" coordinate system. The "geographic" coordinate system refers to the "Northeast Day (ENU)" coordinate system on Earth, and the "carrier" coordinate system values the four axes of their own coordinate system.

4, in the "geographical" coordinate system, the value of gravity is always (0,0,1g), the value of geomagnetic is always (0,1,x). These values are measured by the sensors placed on the four axes.

5. The "geographic" and "vector" coordinate systems are two different coordinate systems and need to be transformed. The conversion method is the coordinate system conversion, there are currently three ways: four Yuan (q0123), Euler angle (yaw (Z axis)/pitch (Y axis)/roll (x axis) belongs to one of the rotation sequence Z-y-xà aviation order Euler angle), A direction cosine matrix (9 coefficients).

6, the so-called posture , is the formula + coefficient . For example: Euler angle formula and Euler angle coefficients ( tumbling, tilt, yaw )

7, the attitude of the data source has 5: gravity, geomagnetic, gyroscope, accelerometer, electronic compass . The first two are from the "geographic" coordinate system, and the last three from the "vector" coordinate system.

8, the basic principle of navigation is to ensure the correct conversion of two basic coordinate system, no error . Only by implementing this principle can a carrier complete a series of actions in its own coordinate system and be converted into a geographic coordinate system that looks correct. In order to achieve this goal, two coordinate systems need to be calibrated and corrected in real time. Because the coordinate system has three axes, the yaw yaw correction is obtained by the electronic Compass (carrier-based), geomagnetic (geo-based) contrast correction error compensation. Corrections on tilt pitch and roll rolls are obtained from the accelerometer (carrier-based), Gravity (geo-based) contrast correction error. After completing the basic principle, that is to ensure the correct transformation of two coordinate system, using the gyroscope based on the carrier to perform integral operation, the attitude data based on the carrier coordinate system is obtained, and after a series of PID control, the control amount is given, and the stability control based on the carrier coordinate system is completed. Reaction to the stability of the geographical coordinate system control, so as to achieve our observed high, yaw, tumble, tilt and other actions.

As can be seen from the above discussion, the navigation attitude in theory only with the gyroscope can accomplish the task . However, due to the error accumulation in the integration process , the addition of white noise and poor temperature will result in the solution of navigation attitude gradually increasing with the passage of time. Therefore, it is necessary to use the accelerometer in the horizontal face of gravity for comparison and compensation to correct the vertical error of the gyroscope. But for the rotation on the vertical axis, the accelerometer is powerless, and the electronic compass is used at this time. He can also measure the geomagnetic direction in the horizontal plane to correct the horizontal error of the gyroscope. The correction compensation of the two devices makes the gyroscope work more stably and reliably.

9, the accelerometer on the Earth is the acceleration of gravity, if the carrier rotates along the z axis, the accelerometer is unable to perceive his movement; Similarly, the electronic compass measures the direction of the Earth's magnetic field, and if the carrier rotates along the y axis, the electronic compass is also unable to perceive his movement. To sum up, the accelerometer and the electronic compass can only get 2-dimensional angle relationship , through some way of fusion, you can get the correct three-dimensional attitude information.

10, here to find out a problem, the previous 8th said about the geographic coordinate system and the vector coordinate system between the mutual transformation. In this way, there are two conversion directions: one is to convert the B-system (carrier) to N-Series (geography), and the other is to transfer the N-series to the B-series. When we are in real control, we are obviously concerned with the change in the carrier coordinate system relative to the geographic coordinate system, so the rotation matrix we usually use is the one that transfers N to the B system (the relationship between the two is a transpose relationship). For example, in the use of the accelerometer to calculate the attitude error, you can use the last four-dollar attitude in the N-series of three axes of the vertical components to the vertical component in the B series to calculate the error.

The right side of the formula is n to the third column of the rotation matrix of the B series (which is exactly the value of gravity g in the B-series)

11, the displacement in the unit time is defined as the speed, speed wired speed and angular velocity of the points, respectively, corresponding to two sensors to measure the two different speed: linear speed sensor (accelerometer), angular velocity sensor (gyroscope). Therefore, the gyroscope is used to measure angular velocity, for the rotation of the coordinate system, that is, the navigation posture. The accelerometer can only measure line speed, the most typical example is the gravitational acceleration, if coupled with the acceleration on the horizontal coordinate system, the resultant force F produces a. Considering a missile, his flight speed is measured by an accelerometer and the rotational posture during the flight is measured by a gyroscope .

12, when we take the accelerometer in the hands of the random rotation, we see the gravitational acceleration on the three axis of the component value, can not be visualized to see the three axis of the acceleration is how much. To achieve this (see real acceleration on each axis), we need a rotation matrix that converts the accelerometer values placed in the vector coordinate system to the reference coordinate system, where the values on the three axes are always (0,0,1). So when we fix the accelerometer at any angle in space, regardless of the value of the three axes of the accelerometer, when the rotation matrix is transformed, the value that is output in the reference coordinate play is always (0,0,1)--and this indicates that in the reference coordinate system, The object has no acceleration on the x and Y axes, and only the gravitational acceleration exists on the z axis. But here again there is a problem, since the output of the z-axis is 1, which means that there is acceleration, the object should be moving up. But the object here is not moving. Why is the output 1? This involves the design of the accelerometer: the accelerometer measures acceleration by a specific force rather than by acceleration. you can understand it by imagining a small ball in a box. The accelerometer has an output of 0 only when it is free-falling.

13, easy to remember an example is how to from the castle Peak to the Yellow lake. For a person, from the castle Peak to the Yellow Lake, must meet two requirements: 1, you must have a map of Wuhan, and know the location of the yellow lake and the location of the castle Peak. 2. You must have a directional navigation system to update your current orientation in real time. Corresponding to the flight navigation above, the location of the yellow lake corresponds to the "geographic" coordinate system, and the location of the castle peak corresponds to the "carrier" coordinate system. Your goal is to have the two coordinate systems correctly transformed and calibrated. This part of the work is given to accelerometers and electronic compass processing. As far as you go, you can go by bike, by bus or by taxi, the gyroscope uses the integral function to determine its own dynamic posture, corresponding to the flight navigation.

14. The simplest example of single-axis fusion:

Here, the k= controls the cycle/sensor sampling period.

15, in the plural domain, two-dimensional coordinates through the complex number of subtraction operations can be quickly and easily expressed, especially rotation. Now consider the stretching and rotation of the complex vectors of the three-dimensional space, or the higher dimensions. Then need a complex domain coordinate system, easy to think of the form is H=A+BI+CJ, it turns out that in the two-dimensional complex domain simple addition of a j is not able to form three-dimensional complex space, in fact, four parameters to be able to build three-dimensional complex space ( two variables determine the direction of the axis, A variable determines the angle of rotation, a variable determines the scaling ratio , i.e. h=a+bi+cj+dk. This is the basic expression of the four-dollar number (where i2=j2=k2=-1). That is, four variables to express the position of three-dimensional space coordinates, which is the difference between the complex and real fields. However, such a definition is conditional, that is, the Exchange law of multiplication is sacrificed. For example, two $ four hp≠ph. As a result, the Q8 multiplication matrix table appears.

16, the further analysis of four yuan, found that four yuan can be written as a real number plus a three-dimensional vector sum, namely h=d+u (where D is a real, U is a three-dimensional vector). Make P=w+v, then

and

where real multiplication and inner product have multiplicative commutative laws, but the outer product of three-dimensional vectors is different, there is u x v =-V x u. So, hp-ph is twice times the outer product of two vectors. if the outer product of two vectors is 0, then the multiplication is exchangeable.

17, for four yuan multiplication PQ, is in the four-dimensional space F on a linear transformation, so there must be two perpendicular to each other two-dimensional invariant subspace, respectively (1,0,0,0) and u spanned two-dimensional plane (this plane in the four-dimensional space, we can not see the whole picture, only to see a line that intersects with us, That is, u) and a two-dimensional plane consisting of U1 and U2 (U1 and U2 are found in the three-dimensional space of U, a group of 22 perpendicular to the right-handed rule, which we can see). So the geometric meaning of the multiplication of the four-dollar number is the linear transformation of the telescopic rotation in the two-dimensional invariant subspace. Angle. Scaling Factor is | | p| | (from (1,0,0,0) to u rotation, from U1 to U2). if P is on the right, the first rotation is the same as the above direction, but the second direction of rotation is the opposite of the above. The contents of this article all take place in the four-dimensional space, remember, the four-dollar number cannot represent all the stretch rotation in the four-dimensional space, because he requires the same rotation angle on the two invariant subspace. But he can fully represent all the stretching rotations in three-dimensional space . If you want to discuss three-dimensional space, then the four-dollar number is fully capable.

18, in three-dimensional space with the application of four yuan multiplication to do linear transformation, there will be two rotations, one from (1,0,0,0) to u rotation, the second from U1 to U2 rotation. The former rotation occurs in the four-dimensional space, we do not see, only see u this line of intersection. But the second rotation takes place in three-dimensional space, which we can see.

19. Let's look at how it rotates in three-dimensional space. Given a three-dimensional vector P (0,x,y,z), this is represented by a four-dollar number. Then do the linear transformation in the four-dimensional space Rpq (q is the conjugate vector of R, and R is a unit of four yuan, that is, N (R) =1), the answer is (0,x ', y ', Z '). where r= (cos (THETA/2), Alpha*sin (THETA/2), Beta*sin (THETA/2), Gama*sin (THETA/2)), and alpha2+ beta2+gama2=1. This means: in three-dimensional space, the P-vector around the (Alpha,beta, GAMA) axis counterclockwise rotation theta angle, the length is unchanged. Why is THETA/2, because in the four-dimensional space actually only turned theta angle.

20, about the high-dimensional space knowledge. low-dimensional things cannot perceive things and actions that occur in high dimensions. For example, we will be a paper bag rotated in half and end connected to the side of the tape along the straight line has been drawn line, in two plane we always think we go straight, but in three-dimensional we are walking round, but the end-to-end, two-dimensional can not perceive, this is the three-dimensional dry things. and the low-dimensional object can only observe high-dimensional objects in the lower latitudes of the projection image, such as poker people see the human model is to use a piece of paper to cut our human body, for example, we see in the real life of the person's appearance is actually four-dimensional space in three dimensions of the projection just. The line seen on two-dimensional space is likely to be a circle in three-dimensional space, so the straight line seen on three-dimensional space is likely to be a circle in the thinking space. So when we use four of dollars to express three-dimensional space, what we see is actually a tangent in the four-dimensional space, and the straight line we see is likely to be a circle in the four-dimensional space.

21. Spherical polar projection is a good way to understand from low to high dimensions. such as the Earth's Map dome projection.

22, multiply a number by 1, equivalent to find the corresponding to the origin of the mirror opposite number, and then multiply-1 and then back to the original position. Such a -1x-1 process is equivalent to turning the number 360 degrees. That is, 1 means rotating the number 180 degrees. Now define a number, which only needs to be rotated 90 degrees, which appears . Special attention here, we operate on the horizontal axis is only one-dimensional real numbers, so that the definition will appear a number not on the horizontal axis, so that the need to expand the dimension, so that the definition I for rotation of 90 degrees, corresponding to the vertical horizontal axis to draw the ordinate, there is a complex plane. Since it is a two-dimensional plane, you need two numbers to represent the coordinates, just like the x and Y coordinates in our real plane. However, complex numbers require only a complex number to represent the stretching and rotation of a planar position.

23, four P=[w,u] (where w is scalar, U is vector). Describes a rotation axis and a rotation angle. If you multiply a vector by a number of four p, it means that the vector rotates at a specific angle on the axis of rotation.

24. There are many ways to represent rotation:axis/angle, Euler angle, direction cosine matrix, four-yuan number. compared to several other representations, the four-tuple has the gimbal lock problem that there is no Euler angle, only 4 coefficients instead of the 9 coefficients of the directional cosine matrix, two four-tuple is more easily interpolated, and two four-tuple multiplication represents rotation.

There are too many coefficients in the direction cosine matrix, which is difficult to interpolate.

Euler's angle is simple, but there is a gimbal lock problem (i.e. it may lose one degree of freedom)

The question of Axis/angle is like Euler's angle.

25, the use of four yuan direct rotation is very difficult, so we can use Euler angle to represent, but in the calculation of space rotation and interpolation, the Euler angle and the number of four is required to convert, because the direct calculation of Eulerian angle will encounter gimballock problems, However, there are no such problems with the four-dollar number in the four-dimensional space, and the interpolation is simple (because the interpolation in the thinking space is the shortest path problem in the three-dimensional spherical space, the personal understanding may be wrong). This is the pros and cons of complementarity: The use of Euler angle to represent the current vector posture, and in the specific calculation of the conversion to four yuan.

26, the fusion scheme is the accelerometer and the value of the magnetometer after the quest algorithm fusion to calculate the four-number ABCD, and then the gyroscope output (angular rate) after the Kalman filter to give an estimate of the object of four Yuan Q. The qurest algorithm can be replaced by a Gaussian algorithm (which requires a large number of matrix operations, may require a DSP) or a gradient descent algorithm (tradeoff algorithm).

27.

28, this diagram gives the concrete basis of the fusion theoretically. The middle vertical bar in the figure shows the Gaussian algorithm, the lower left corner of the four-dollar differential equation is very important, the equation will be four and the angle of the change rate to form a constant coefficient homogeneous linear differential equation. after the integration of two points and then normalization, we can get the expression of the object's attitude four-dollar number. Then the Euler angle transformation can be converted to the Roll,pitch,yaw we have known.

29, from one coordinate system to another coordinate system of conversion before the translation of a number of methods: Euler angle method, the direction of the cosine matrix method, four Yuan method. The core idea of Euler's Angle method is:a coordinate system can be expressed in three-space rotations of another reference coordinate system. There are two ways to rotate a coordinate system: one is to rotate three different axes in turn, and the other is to rotate different axes two times. The first method of rotation is calledTait–bryan Angles(X-y-z, Y-z-x, Z-x-y,x-z-y, Z-y-x, y-x-z are available in the order of choice); The second method of rotation is calledEuler Angles(Z-x-z, X-y-x, Y-z-y,z-y-z, X-z-x, y-x-y) are available in the order of choice. In addition there are two concepts, external rotation (extrinsic rotations) and inner rotation (intrinsic rotations). Our fixed reference coordinate system is XYZ, and the coordinate system that needs to be rotated is ABC. The initial state of the two coordinate values is fully coincident, and now the goal is to rotate the coordinate ABC to reach the specified position. The so-called external rotation refers to the rotation axis of each rotation in three rotations, which is an axis in the XYZ axis of the fixed reference system. For example: The xyz order of the Tait–bryan angles, then when rotating ABC, each rotation rotates the ABC coordinate system around an axis of the fixed reference system XYZ, while the inner rotation refers to the rotation of ABC, each rotation around the axis is the last ABC rotation of an axis. For example, it is like the mathematics of the sequence of questions, the topic is generally given is N and n-1 terms of the relationship expression, the value of n is derived from the previous item, based on the previous value, and the general formula can be directly through the expression of N to calculate the value of any nth item, For example, the value of the 10th item can be calculated directly by the expression of n, without the need to calculate 9th, 8th ... Until the first item is pushed back. External rotation is like a general formula, and each rotation is rotated by a fixed reference system of XYZ, regardless of the ABC state during rotation. and the inner rotation needs to be rotated on the basis of the rotating shaft after the last rotation, so the rotation axis is variable, like the N and n-1 of the sequence. About the relationship between inner rotation and external rotation,If you swap the first and third rotations of one of the rotations, they are equivalent. For inner rotation. (N and n-1 relationships of associative series formulas)

A rotation represented by Euler angles (α,β,γ) = (−60°, 30°, 45°), using z-x '-z″intrinsic rotations

For external rotation. (The general formula of the associative series formula)

The same rotation represented by (γ,β,α) = (45°, 30°,−60°), usingz-x-z extrinsic rotations

You can see that the final coordinate system has the same attitude.

Finally about Tait–bryan angles, because it is in the three reference coordinate system XYZ rotation, so just can use this property to navigate, the formation of roll, pitch, yaw and other concepts. But this is an outward rotation, and we draw often using inner rotation (because it is easy to remember, good to draw), so we need to take advantage of the relationship between internal rotation and external rotation: the position of the first rotation and the third rotation. already explained. And in some references (James, d.,representing attitude:euler angles, Unit Quaternions, and Rotation Vectors) Some illustrations of the Euler's conversion are involved, If a tait–bryan angle is present, such as 1-2-3 in order, but the inner rotation is used in the plot, the true rotation order is 3-2-1, which replaces the position of 1 and 3. Special attention!!!

Transferred from: http://demo.netfoucs.com/nemol1990/article/details/16924745

[Turn] attitude calculation of four-rotor aircraft small knowledge points