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【Main line】

$Kinematics\color{red}Kinematics$

$\color{red}kinetics (to be added)$

【Supplementary explanation】

foreword

The content of this chapter mainly introduces the problems encountered in solving the inverse solution of robot kinematics. After summarizing the content of the textbook, the understanding of the following three types of problems is obtained.

The related problems of kinematics inverse solution are mainly the existence, uniqueness and singularity of the solution on the Cartesian path (whether the path points are in the work space or not). The following is the content of the main text of this article, including three types of questions that are counter-answered.

text

Three Types of Problems Reversed

1. The middle point is outside the working space

Although the starting point and the ending point of the path planned in Cartesian space are inside the workspace, some points on the planned path may exceed the workspace, causing errors during the robot’s movement, while planning in joint space will not cause such problems.

2. The joint velocity increases sharply near the singularity point

A six-axis robot has two kinds of singularities:

Singular shape of the workspace boundary
Take the planar 2R manipulator as an example, when $sin θ_2=0$ 时， $θ_2=0°$ (extended) and $θ_2=180°$ (retracted) is at the boundary of the workspace, at which point there is only one degree of freedom operable. But this singularity is easier to avoid.
Internal singularity in the working space
The internal singularity is usually caused by the coincidence of two joint axes or multiple joint axes, and the movement of each joint of the manipulator arm cancels out, so no operation movement occurs. For example, of PUMA 560 θ_3=0 $i_{3} = 0$ at $- Near 90 °$ , the arm is in a straight shape, and at this time it is in a boundary singular state; and when $θ_5=0$ , the axes of joint4and joint6coincide, and the six-axis robot loses one degree of freedom at this time, and is in an internal singular state at this time.

When the robot is in a singular pose, the joint velocity corresponding to the operating velocity (Cartesian space velocity) may not exist (or be infinite). If the robot moves along a straight path in Cartesian space to the vicinity of the singularity, some joint velocities will tend to infinity. But in fact, the allowable joint speed of the robot is limited, which will cause the manipulator to deviate from the expected trajectory.

3. There are multiple solutions for the starting point and the target point

Although there are multiple solutions for the starting point and the target point, the number of feasible solutions is reduced due to the constraints of joint variables or obstacles. The problem is that if the starting point and the target point do not use the same inverse solution, this is the constraint of the joint variables, which will cause problems. Although each node on the path point can be reached by prime hi, it is not reached by the same solution. Therefore, before controlling the movement of the robot, the planning system of the robot should have a checking function and an error display function, so that it is feasible to deal with the reverse solution.

Summarize

Because of these problems in trajectory planning in Cartesian space, most existing industrial robot control systems have trajectory generation methods in joint space and Cartesian space. userUsually the joint space method is used, and only when necessary, the Cartesian space method is used。

References

"Basics of Robotics" - Xiong Youlun

[Matlab six-degree-of-freedom robot] Questions about the inverse solution of robot kinematics