# What is an oblique symmetric matrix

## 1 Movement of rigid bodies in space

### MODERN ROBOTIC MECHANICS Chapter 3 Rigid body movements

tags: Robotic MODERN ROBOTICS Rigid-Body Motions

**This chapter is very important and contains a lot of content that forms the basis for subsequent chapters**

- The speed of the rigid body is determined by Twist: 3 angular speeds, 3 angular speeds
- Rigid body force is caused by wrenches: 3 moments, 3 forces
**free vector ：**Only the length and direction vector, such as the linear velocity vector, is the free vector**space frame {s}:**Can be understood as the coordinate system based on a fixed system, also called a fixed frame**boby frame {b}:**Can be understood as the coordinate system of the dynamic effector of the end effector- Cartesian coordinate system conventions:

- Express {b} in the {s} coordinate system.
**Note withThe vector of symbols is the unit vector**- place
- angle

### 2.1 rotation matrix

### 2.1.1 The type of rotation matrix

- The type of rotation matrix R:
- Consists of 3 columns with 3 elements in each column
- Each column is a unit vector
- Any two columns are perpendicular to each other
- Knowing linear algebra, we can see that the matrix satisfying the above three points is a standard three-dimensional orthogonal matrix

- The concept of the group
- Don't feel big
- As long as the following four properties are met, they can be called groups. Because A and B belong to a set (for example: A, B belong to the set of orthogonal standard matrices)
- Closure: AB is still in this collection,
**(The product of two standard orthogonal arrays is still a standard orthogonal array.)** - Combination law:
**(Matrix multiplication fulfills this law)** - Unit unit exists:
**(For standard orthogonal matrix sets, the identity matrix is the identity element.)** - There is an inverse element:
**(For the set of standard orthogonal matrices, the transpose of the matrix is the inverse of the matrix.)**

- Closure: AB is still in this collection,

- With the nature of the rotation matrix and the concept of groups, it is clear that the set of rotation matrices is what we call a group
**Special orthogonal group, abbreviated SO (3)**

### 2.1.2 Using the rotation matrix

- More general rotation formula

### 2.2 angular velocity

- Oblique symmetric matrix: For a given vector, the oblique symmetric matrix is called as follows (3) (Lie algebra)Then the obliquely symmetric matrix is defined as:

Application of an obliquely symmetric matrix

- For each,alwaysEstablished
- Convert the cross product to a dot product:

- beg：

- For a fixed coordinate system {s}:
- For the dynamic coordinate system {b}:

### 2.3 Representation of rotating exponential coordinates

### 2.3.1 Solving linear differential equations

- Differential equation

- When converting a in the above differential equation to matrix A.：

- The exponential equation of the matrix has the following properties:
- If the matrix A can be expressed as(Reading the evidence process)
in the case ,then

### 2.3.2 Representation of rotating exponential coordinates

- The index of the rotation matrix indicates how to calculate:

- The index of the rotation matrix is expressed as:

### 2.3.3 Rotated protocol matrix

### 3.1 Homogeneous transformation matrix

- Homogeneous Transformation Matrix: This term shouldn't be viewed as particularly peculiar as we often say by the rotation matrixAnd position vectorcomposition
- Special Euler group (
**Special Euclidean Group**): As mentioned earlier, the matrix set with the shape of a homogeneous transformation matrix is called a special Euler group as long as the closure, associativity, the presence of unit elements and the presence of zero elements can be grouped

### 3.1.1 Properties of the homogeneous transformation matrix

- The inverse of a homogeneous transformation matrix is also a homogeneous transformation matrix, and there is a simple method of inversion

- Standard Euclidean interior product

### 3.1.2 Use of a homogeneous transformation matrix

- Main purpose:
- The position and direction of the rigid body indicate:
- Change the reference coordinate system:
- Move (rotate and move) a vector or coordinate system:

- Coordinate transformation (relative to the fixed coordinate system {s}):
- Coordinate transformation (relative to the dynamic coordinate system {b}):

### 3.2 Twists

### 3.2.1 Understand phrases

- Adjoint matrix:begging adjunct matrix

- Adjoint mapping of T (
**T is used for position transformation and T for speed transformation**)：

- The type of adjoint matrix:

### 3.2.2 Explanation of the rotation

### 3.3 Exponential coordinate representation of the rigid body motion

### 3.3.1 Index representation of rigid body operation

### 3.3.2 Logarithmic representation of rigid body operation

- The previous section mainly dealt with the representation of kinematics-related physical quantities in exponential space. This section explains
**Representation of the physical quantities related to the dynamics in the exponential space** - Define a 6-dimensional spatial force in the coordinate systemThe representation in is also known as a wrench, which is the symbol(
**It's actually a combination of torque and power**)

- The performance has nothing to do with the choice of the coordinate system:

- Conversion between fixed coordinate system {s} and dynamic end effector coordinate system {b}

**Finally, a summary is given: (From "Introduction to the Mechanics and Control of Robotics", written by (US) John J. Craig)**

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