The most understandable explanation of coordinate transformation (push to + diagram)

Catalog

Detailed explanation of coordinate transformation

The function of coordinate transformation

Data required for coordinate transformation

Postures

The essence of rotation in coordinate transformation

The essence of translation in coordinate transformation

How to calculate the coordinate system B Each coordinate axis is in the coordinate system A The projection on the ？（ Multi coordinate transformation ）

How to realize coordinate transformation ？

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Detailed explanation of coordinate transformation

The function of coordinate transformation

In a robot system , Each measuring element measures the same object to get different information , And the reason is that “ The data measured by each measuring element is based on different coordinate systems ”, for example ：

  There is a laser radar in this car M And lidar W, The data measured by these two radars are quite different , But the position of the car relative to the measured object is unique , This means that “ The physical meaning represented by data measured by radars at different locations （ That is to say, they all represent the relative position between the vehicle and the object to be measured ） It's the same ”. Since the coordinates of the measured object in different coordinate systems are different but have the same physical meaning , This involves the mutual transformation of coordinates in different coordinate systems .

Data required for coordinate transformation

We usually use the vector starting from and ending at the coordinate point in the coordinate system to represent the position of the coordinate point in the coordinate system relative to the coordinate origin （ distance + bearing ）. The mutual transformation of coordinate systems can only be realized by taking the earth coordinate system as the medium , That is, the mutual transformation of coordinate systems must be known “ Each coordinate axis in any coordinate system is located in world Coordinates in a coordinate system ”：

  We wonder ：world What is the coordinate system ？

In space there will be n+1 Coordinate system , Only one coordinate system is used for calibration , in other words “ other n All three coordinate systems find their own position in space based on this coordinate system ”. Only everyone knows their specific position in the space , Coordinate conversion can proceed smoothly .

Postures

be based on O1 World coordinate system based on O2 The coordinate system of is as follows ：

  When describing robot motion , We often mention “ Postures ”, In fact, posture is a compound word , We can disassemble it into “ Location + Posture ”. Position means “ A moving joint of a robot / Measure the specific position of the sensor in the world coordinate system , Posture is ” Coordinate system based on this point Compare with World coordinate system Rotation performed “, As shown below ：

 

The essence of rotation in coordinate transformation

The essence of coordinate transformation is “ Projection ”. First , Let's explain how a vector is transformed into coordinates ：

 

Understanding the origin of vector coordinates is very important to understand the essence of coordinate transformation ！ Next, let's consider the projection of the unit vector in the coordinate system ：

 

The projection of the unit vector in the coordinate system is exactly the vector P Cosine value of the included angle with each coordinate axis .

 

We will use the coordinate system A As a reference coordinate system （world Coordinate system ）, Based on coordinate system A The coordinate system B And unites each vector , So we get a rotation matrix , The meanings of the elements of the rotation matrix are as follows ：

 

We mentioned earlier , The calculation of vector coordinates is nothing more than projection , So the vector coordinates from the coordinate system B Convert to coordinate system A It's just two projections ：

 

The practical meaning of coordinate transformation is nothing more than to transform a vector P In the coordinate system A The projections of each axis in the are added up to form a new coordinate . That's the question , How does accumulation work ？ This involves the rotation matrix and matrix multiplication .

 

Actually , The multiplication and convolution of this matrix are similar . The properties of rotation matrix ：

 

The essence of translation in coordinate transformation

The vector can move freely in the coordinate system , As long as you don't change the direction and size of the vector , The properties of the vector do not change . But we are studying the coordinate system B A coordinate point in the coordinate system A Mapping in , So we need to consider the coordinate system B The origin of O2 Compared to the coordinate system A origin O1 The distance of translation .

 

Each is based on a coordinate system B Vector progressiveness rotation transformation , Re and vector O1O2 Sum to get the vector P Then coordinate system A Actual mapping in .

How to calculate the coordinate system B Each coordinate axis is in the coordinate system A The projection on the ？（ Multi coordinate transformation ）

First , We need to know the world coordinate system A/ Coordinate system B The coordinate axes of are in the world coordinate system （ Reference coordinate system ） Coordinates of ：

 

We can get the coordinate system by using the property of rotation matrix B Transform to coordinate system A The rotation matrix of ：

 

The coordinate transformation process is as follows ：

 

How to realize coordinate transformation ？

We will P The point is in the coordinate system B The coordinates in are converted to P The point is in the coordinate system A Corresponding coordinates in ：

 

Coordinate transformation = rotate + translation , Therefore, the coordinate transformation expression is as follows ：

 

This is not conducive to matrix operation , We can rewrite it as follows ：

 

among O1O2 It's from O1 Point to O2 Vector .