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.)
  • 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,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 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)

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