Robot mathematics foundation 3D space position representation space position

3D There are many ways to express posture . Common ones are Euler angle 、 Four yuan number 、 Rotating vector 、 Rotation matrix, etc .

One 、 Euler Angle

Euler angle can express 3D Any rotating in space 3 It's worth . Altogether 3 Kind of Euler angle ： Pitch angle (Pitch)、 Yaw angle (Yaw) And roll angle (Roll).

  The pitch angle is the angle that looks up or down , The first picture . The second picture shows the yaw angle , The yaw angle indicates how far we look left and right . The roll angle represents how we roll the camera , Usually used in spacecraft Cameras . Each Euler angle has a value to represent , Combining the three angles, we can calculate 3D Any rotation vector in space .

At the same time, about these three rotation angles , We have two ways to select the rotation axis , Thus, two different Euler angles are derived ：

1. Static Euler angle , As is often said in graphics World coordinate system Benchmarking , The coordinate system is stationary , So it is called static .
2. Dynamic Euler angle , With object Self coordinate system Benchmarking , The coordinate system is stationary relative to the object itself , But after each rotation , Will change relative to the world coordinate system .

  Four yuan number

Quaternions are simple Super complex number . The plural By The set of real Numbers add An imaginary unit i form , among i²= -1. Similarly , Quaternions are real numbers plus three imaginary units i、j and k form , And they have the following relationship ： i² = j² = k² = -1, iº = jº = kº = 1 , Every quaternion is 1、i、j and k The linear combination of , That is, quaternions can generally be expressed as a + bi+ cj + dk, among a、b、c 、d Is the set of real Numbers .

about i、j and k The geometric meaning of itself can be understood as a kind of rotation , among i Rotation represents Z Shaft with Y In the plane where the axes intersect Z Axial forward direction Y Positive rotation of the shaft ,j Rotation represents X Shaft with Z In the plane where the axes intersect X Axial forward direction Z Positive rotation of the shaft ,k Rotation represents Y Shaft with X In the plane where the axes intersect Y Axial forward direction X Positive rotation of the shaft ,-i、-j、-k Represent the i、j、k Reverse rotation of rotation .

Rotating vector

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 In mathematics , vector （ Also known as Euclidean vectors 、 Geometric vectors 、 vector ）, Of having size （magnitude） And the amount of direction .
 It can be visualized as a line with an arrow . The arrow points to ： Represents the direction of the vector ; segment length ： Represents the size of the vector .
 The quantity corresponding to a vector is called quantity （ It's called scalar in physics ）, Number （ Or scalar ） Only in size , No direction .
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                     A*R = B

This formula says , Yes A、B Two coordinate systems , from B Coordinate to A The rotation transformation of coordinates can be determined by this R The matrix represents ; among ,R Each column of the matrix is B Coordinate x, y, z Axis in A Representation in coordinates ; Because the transpose of the rotation matrix is its inverse ,R Each row of the matrix is A Coordinate x, y, z Axis in B Representation in coordinates .

 

for instance , The above two coordinate systems , We can put R Write out ： First column , because XB Shaft with ZA Axis coincidence , So it is [0; 0; 1]; Second column ,YB Shaft with YA The axis direction is exactly the opposite , So it is [0; -1; 0]; Empathy , The third column is [1; 0; 0]. If you write according to each line , Then the first line is with ZB Coincident XA Axis [0,0,1], The second line is with YB The axis is in the opposite direction YA Axis [0, -1, 0], The third line is with XB Axis coincident ZA Axis [1, 0, 0].

If we want to add displacement , What shall I do? ; It is the following rotation matrix ;

Rotation matrix

3D Transformation matrix ： translation 、 The zoom 、 rotate .

 

3D The transformation matrix is a 4x4 Matrix , by 16 A two-dimensional array of real numbers , In 3D space , Any linear transformation can be represented by a transformation matrix . This paper introduces how to extract translation matrix from transformation matrix 、 The zoom 、 The method of rotating a vector , The complexity of the extraction formula is “ translation ;

The last column is the translation vector ; front 3*3 For rotation , The zoom ;

The following picture can be seen intuitively , translation 、 The zoom 、 The positional relationship of rotation in the transformation matrix