Small guide for rapid formation of manipulator (12): inverse kinematics analysis

Directory ：

Quick completion guide of mechanical arm （ zero ）： Main contents and analysis methods of the guide

Quick completion guide of mechanical arm （ One ）： The development of manipulator

Quick completion guide of mechanical arm （ Two ）： Application of mechanical arm

Quick completion guide of mechanical arm （ 3、 ... and ）： Mechanical structure of mechanical arm

Quick completion guide of mechanical arm （ Four ）： Reducer of key components of mechanical arm

Quick completion guide of mechanical arm （ 5、 ... and ）： End actuators

Quick completion guide of mechanical arm （ 6、 ... and ）： Stepper motor driver

Quick completion guide of mechanical arm （ 7、 ... and ）： Description method of robot arm posture

Quick completion guide of mechanical arm （ 8、 ... and ）： Kinematic modeling （ standard DH Law ）

Quick completion guide of mechanical arm （ Nine ）： Forward kinematics analysis

Quick completion guide of mechanical arm （ Ten ）： Reachable workspace

Quick completion guide of mechanical arm （ Twelve ）： Inverse kinematics analysis

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         stay guide （ Nine ） We have obtained the functional relationship between the joint variable and the position and orientation of the end effector , namely Forward kinematics equation . But in practice , We often need to solve the corresponding joint variables through the given pose of the end effector , To determine the rotation angle of each joint , Then control the manipulator to complete the movement in space .

One 、 The introduction

1. Number of inverse solutions of manipulator

         The solution of the inverse kinematics problem of the series manipulator is more complex than the forward kinematics problem . The degree of freedom of the manipulator studied in this paper is 6, In the analysis of inverse kinematics problems, it is necessary to align the sub transformation matrix Independent of each other 6 A nonlinear transcendental equation is solved . The equations contain 6 An unknown quantity , The existence of solutions and the number of solutions are determined by the kinematic structure of the arm 、 Whether the target point is located Flexible workspace And the restrictions on joints in the program . When there is a feasible solution , Its number is usually not unique , For a six degree of freedom manipulator , At most 16 Two different solutions , Get the number and DH In the parameter a（Oi And Oi' Distance between ） The relationship is shown in the following table .

ai	Number of inverse solutions
a1=a3=a5=0 ​​​​​​​	≤4
a3=a5=0 ​​​​​​​	≤8
a3=0 ​​​​​​​	≤16
All ai≠0 ​​​​​​​	≤16

 2.pieper The criterion

         obviously , It is of great significance to judge the existence of the solution before solving the inverse kinematics of the manipulator . Current research shows , All series type 6 All manipulators with degrees of freedom are solvable , But this kind of solution can only be obtained by numerical solution , It is difficult to calculate , High complexity . therefore , Before determining the inverse kinematics solution , It is necessary for us to explore whether the analytical solution of the inverse kinematics of the manipulator exists ,Pieper Criterion is a commonly used criterion in robot field to judge the existence of analytical solutions .

        Pieper In his paper, he pointed out ：6 The sufficient condition for the existence of an analytical solution of a DOF Manipulator is that the rotation axes of three adjacent joints intersect at a point . To meet this condition and reduce control costs , Most Manipulators on the market are designed according to this , Have a specific structure , Such as manipulator with spherical wrist , As shown in the figure below .

The joint position diagram of the manipulator used in this paper

Two 、 Solution method of inverse kinematics of manipulator

         Solution of inverse kinematics