Rigid body motion state description

rigid body in space

To describe the state of a rigid body in space, it generally requires 6 parameters, 3 translational parameters and 3 rotational parameters, which respectively correspond to the three axes X, Y, and Z of the world's rectangular coordinate system.

Integrated expression of the state of the rigid body: establish a coordinate system rigid system (body frame) on the rigid body , and the origin of the coordinate system is established on the center of mass of the rigid body. What needs to be noted here is that the coordinate axes of this coordinate system are not necessarily parallel to the coordinate axes of the world's rectangular coordinate system .

When the rigid body translates, it is determined by the origin position of the rigid system; when the rigid body rotates, it is determined by the attitude of the rigid system compared to the world coordinate axis.

Of course, such an expression can only express the state of the rigid body at a certain moment, and we record all these state parameters at each moment in the entire motion trajectory of the rigid body.

By using the time differential of the translation parameters at each moment, the displacement can be converted into motion states such as speed and acceleration; by using the time differentiation of the rotation parameters at each moment, the attitude can be converted into motion states such as angular velocity and angular acceleration.

Two ways for vectors to express spatial relationships

Express a position in space
2. The above formula expresses the position of the origin of a certain rigid system in the world coordinate system.
express a direction in space
2. The above figure expresses the various axes \(X_B, Y_B, Z_B\) of the rigid system . They can also be represented by vectors in the world coordinate system. They only represent the direction. When expressing the direction, a unit vector with modulo 1 is generally used.

direct cosines

Directional cosines refer to the three-directional cosines of a vector in analytical geometry, which are the cosines of the vector and the three axis directions of the space rectangular coordinate system.

Directional cosine matrix: A matrix formed by the direction cosines between two different sets of unit vectors of orthonormal basis. It can be used to express the relationship between a set of orthonormal bases and another set of orthonormal bases. It is also possible to express the direction cosine of a vector with respect to another set of orthonormal bases.

Quantitative expression

Translation: Use vectors to describe the state of the origin of the rigid system {B} relative to the world coordinate system.

Let us explain

We project the origin of {B} into the world coordinate system, and their values on the X, Y, and Z axes are 10, 3, and 3 respectively.

Rotation: Describes the attitude of the rigid system {B} with respect to the world coordinate system - Rotation Matrix

In the figure above, the direction pointed by each axis of the rigid system {B} can be expressed by the following formula

The above formula is a matrix that represents {B} represented by the world coordinate system {A}. Each column of the matrix is a 3-dimensional column vector, representing the direction of each axis of {B} in {A}. The entire matrix is A 3*3 pattern, which is a rotation matrix.

The three column vectors of R are the basis of the rigid system {B} in the world coordinate system {A} , and they are a set of orthogonal basis . (For the concept of basis, please refer to the basis of space in linear algebra arrangement and the orthogonal basis and standard orthonormal basis in linear algebra arrangement (2) ). We can use the direction cosine to describe the attitude of a rigid body.

We assume that the unit vectors of the three coordinate axes of the world coordinate system {A} are \(A_1,A_2,A_3\) , and the unit vectors of the three coordinate axes of the rigid system {B} are \(B_1,B_2,B_3\ ) . Define the direction cosine between the coordinate axes of the rigid system {B} and the world coordinate system {A} as

\(a_{ij}=cosθ_{ij}=A_i⋅B_j\)

Because \(A_i,B_j\) are all unit vectors with modulus 1, so the above formula holds. It can be expressed as the total arrangement of any axis of the two coordinate systems. Therefore

can be expressed as

Let us illustrate with the following example

In the above figure, the blue coordinate system is the world coordinate system {A}, and the red coordinate system is the rigid body coordinate system {B}. Now we require the posture of {B} relative to {A} .

First, the X'' axis of {B} is opposite to the Z axis of {A}, so

It represents the X'' axis of {B} in the direction of {A}.

The Y'' axis of {B} is in the same direction as the Y axis of {A}, so

The Z'' axis of {B} is in the same direction as the X axis of {A}, so

Therefore, the posture of {B} relative to {A} is:

Because the value of {A} itself is \((1,1,1)^T\) .

Let’s look at a more general situation.

In the figure above, the Z-axis of the world coordinate system {A} coincides with the Z'-axis of the rigid system {B}. So let's take a look at the top view of the XY plane.

We project \(X_B\) and \(Y_B\) of rigid system {B} to the X-axis and Y-axis of {A} respectively. Then the relative value obtained is

And because the Z axis coincides, then there is

Then the final posture of {B} relative to {A} is

rotation matrix

We knew the meaning of this formula before. Furthermore, for the point product, we can understand it as the projection of the X-axis of the rigid system on the X-axis of the world coordinate system, and the others are the same.

Because the dot product of vectors satisfies the commutative law, the above formula can be written as

From the row vector of this matrix , it can be seen as the projection of an axis of the world coordinate system on each coordinate axis of the rigid system. For example , it can be regarded as the projection of the X-axis of the world coordinate system on the X-axis of the rigid system, and everything else is the same. Because the vectors we usually talk about refer to column vectors, it will be written here , which refers to the transposition of the X-axis of the world coordinate system {A} using the rigid system {B}.

The above formula can also be written as

Therefore, it can be seen that the two coordinate systems represent each other, but the only difference is a transposition .

The transpose of the rotation matrix multiplied by itself will result in an identity matrix \(I_3\) , which in turn shows that they are mutually reversible. (For the content of this part, you can refer to the inverse of the matrix in linear algebra and the standard orthogonal matrix in linear algebra (2) ). This is very convenient when we convert different coordinate systems. We only need to convert Just set it without having to find the inverse of the matrix.

Three uses of rotation matrices

Describes the attitude of one coordinate system relative to another coordinate system.
Convert a point from the expression of a certain coordinate system to the expression of another coordinate system with only relative rotation to the coordinate system.
2. The expression of the above formula is that the position vector of point P in the B coordinate system can be obtained by multiplying the rotation matrix on the left to obtain its position vector in the A coordinate system.
Rotate a point in the same coordinate system.
3. The above formula is expressed as the position vector of point P left multiplied by the rotation matrix R(θ) to obtain the position vector of the other position to which it is rotated. This is done in the same coordinate system.

Rotation matrix and angle

Rotation in space has three parameters. We need to decompose the posture expressed by the rotation matrix into three rotation angles to deal with the three parameters.

Things to note when breaking down into 3 spins:

Rotation is different from translation. The order of translation can be ignored because a particle moves first in the X direction and then in the Y direction or first in the Y direction and then in the X direction. The effect is the same. This Properties are called commutable. However, the sequence of multiple rotations needs to be made clear , otherwise, the posture after rotation will be different.
The axis of rotation also needs to be clearly defined . Is it a rotation of the "fixed" axis, or a rotation of the "rotated coordinate system"?

Two ways of disassembly:

Rotating an axis with a "fixed" direction is called fixed angles .
Rotation in the direction of the rotation axis of a constantly rotating rigid system (the coordinate axis of the rigid system itself changes at any time) is called Euler angles .

The third use of the rotation matrix can further describe the rotation state of the object. The explanation is based on the rotation matrix.

1. Rotate around the Z axis

The blue spatial rectangular coordinate system in the above figure is the initial state. It rotates counterclockwise around the Z-axis by an angle θ, and the red spatial rectangular coordinate system of the final state is obtained. The Z-axis coincides with the initial state. The rotation matrix at this time is

This is simplified to

2. Rotate around the X axis

The rotation matrix is

3. Rotate around the Y axis

The rotation matrix is

Example: Rotate the axis \(30^∘\) , find .

First get the rotation matrix

Finally ask for

fixed angle

This is mainly for continuous rotation

After the spatial rectangular coordinate system {A} (blue dotted line) is defined in the above figure, the directions of its axes will no longer change, so it is called a fixed angle.

And our rigid body coordinate system {B} (red line part) initially coincides with {A}, then rotates counterclockwise along the X-axis by an angle γ, then rotates counterclockwise along the Y-axis by an angle β, and then rotates counterclockwise along the The axis is rotated counterclockwise by an angle α. The rotation matrix it finally gets is recorded as

The meaning of this formula is that the γ angle, β angle, and α angle are rotated around the put in front ). It should be noted here that matrix multiplication does not satisfy the commutative law of multiplication, so attention must be paid to the order of multiplication factors. The result of this formula is

For information on matrix multiplication, please refer to Matrices and Matrix Multiplication in Linear Algebra .

Example: Using a fixed angle, first rotate the X-axis by 60 degrees, and then rotate the Y-axis by 30 degrees; first rotate the Y-axis by 30 degrees, and then rotate the X-axis by 60 degrees. Find the respective values of these two rotations .

First rotate the X axis 60 degrees, then rotate the Y axis 30 degrees

First rotate the Y axis 30 degrees, then rotate the X axis 60 degrees

What needs to be noted here is that if you rotate the Y axis first and then the X axis, you cannot use the previous results, but you need to re-calculate the new calculation formula. What I can deduce here is

The above are all based on known angles to calculate the rotation matrix. Now if the rotation matrix is known, the angle can be calculated in reverse. Now we define the rotation matrix as follows

It can be obtained if \(β≠90^∘\)

\(β=Atan2(-r_{31},\sqrt{r_{11}^2+r_{21}^2})\)
\(α=Atan2({r_{21}\over cβ},{r_{11}\over cβ})\)
\(γ=Atan2({r_{32}\over cβ},{r_{33}\over cβ})\)

If \(β=90^∘\)

\(α=0^∘\)
\(γ=Atan2(r_{12},r_{22})\)

If \(β=-90^∘\)

\(α=0^∘\)
\(γ=-Atan2(r_{12},r_{22})\)

Robotic arm kinematics finishing

Rigid body motion state description

Guess you like