Kinematics of Articulated Planar Linkages

Jing-Shan Zhao,Song-Tao Wei

doi:10.3389/fmech.2021.774814

Abstract

This paper proposes a kinematics algorithm in screw coordinates for articulated linkages. As the screw consists of velocity and position variables of a joint, the solutions of the forward and inverse velocities are the functions of position coordinates and their time derivatives. The most prominent merit of this kinematic algorithm is that we only need the first order numerical differential interpolation for computing the acceleration. To calculate the displacement, we also only need the first order numerical integral of the velocity. This benefit stems from the screw the coordinates of which are velocity components. Both the forward and the inverse kinematics have the similar calculation process in this method. Through examples of planar open-chain linkage, single closed-chain linkage and multiple closed-chain linkage, the kinematics algorithm is validated. It is particularly fit for developing numerical programmers for forward and inverse kinematics in the same procedures, including the velocity, displacement and acceleration which provide the fundamental information for dynamics of the linkage.

Highlights

Kinematics of a linkage aims at studying its motion without regard to forces (Norton, 2004) for the synthesis of a mechanism (Suh and Radcliffe, 1978) or to accomplish the desired motion (Shigley and Uicher, 1980) and determine its rigid-body dynamic behavior (Bottema and Roth, 1979; Waldron and Kinematics, 2004)
As the twist of an articulated rigid body includes the angular velocity and linear velocity, the corresponding displacements of all joints are obtained through one-order integration of the velocity solutions and the accelerations are represented by the first order numerical differential interpolation
Compared to the traditional methods in which the displacement parameters are the only variables that will surely lead to the second order differential interpolations for the accelerations, the advantages of this method is that both the forward and inverse kinematics of a mechanism can be expressed in a same way and only one-order differential interpolation is needed to get the acceleration and one-order integral is required to calculate the displacement

Summary

INTRODUCTION

Kinematics of a linkage aims at studying its motion without regard to forces (Norton, 2004) for the synthesis of a mechanism (Suh and Radcliffe, 1978) or to accomplish the desired motion (Shigley and Uicher, 1980) and determine its rigid-body dynamic behavior (Bottema and Roth, 1979; Waldron and Kinematics, 2004). From Equation 12, we know that the twist of the end effector with a marking point of the origin of the coordinate frame can be FIGURE 3 | A series kinematic chain. We let the angular velocity of joint O1 is 2 rad/s and with the structure parameters and initial conditions, we programmed the method in MATLAB and obtained the forward displacement, velocity and acceleration for each joint (see Figure 7) by numerical methods based on Equations 32, 27 for validating the method. We let the angular velocity of joint O1 is 2 rad/s and with the structure parameters and initial conditions, we programmed the method in MATLAB and gained the forward displacement, velocity and acceleration for each joint (see Figure 9) by numerical methods with Equation 35. We let the angular velocity of joint O1 is 3 rad/s, the angular velocity of joint O2 is 4 rad/s, and with the structure parameters in Table 5, the displacement, velocity and acceleration curves for each joint are illustrated in Figure 11 by numerical formulas of Eq 32, 27 in accordance to Equation 37

CONCLUSION

DATA AVAILABILITY STATEMENT