# 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 ### 2.1.1 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.)
• 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 groupSpecial 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 ,always Established
• Convert the cross product to a dot product: • beg  • For a fixed coordinate system {s}: • For the dynamic coordinate system {b}: ### 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 matrix And position vector composition
• 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}): ### 3.2.1 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 explainsRepresentation 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)   ### programmerwiki

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