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.
Introduction
Kinematics of a linkage aims at studying its motion without regard to forces () for the synthesis of a mechanism () or to accomplish the desired motion () and determine its rigid-body dynamic behavior (; ). Kinematic geometry (), geometric design (), theoretical kinematics of robotic mechanisms (; ; ; ), and the analysis, synthesis and optimization of spatial kinematic chains () are elaborated in previous works in the past decades. Kinematic analysis of a linkage requires the algebraic equations that might be iterated numerically. Computational kinematic analysis plays a vital role in the study of a mechanical system (). It is the most straightforward procedure for kinematics and dynamics to select the absolute coordinates of the reference point as the variables (). This selection has the advantage of normally leading to a facilitated expression for the constraints and Jacobian matrices of a multi-body system (). The forward kinematics of a serial linkage is easier than its inverse kinematics while the forward kinematics of a parallel linkage is more complex than its inverse kinematics (; ; ; ; ; ; ).
In the past half century, parallel manipulators witnessed very quick development. Parallel manipulators have been attracted a great attention ever since the industry application of Gough’s tire testing machine and Stewart’s platform for its superior performance over its serial counterparts in terms of loading capacity, rigidity and accuracy (; ). However, the forward kinematics usually contains a group of nonlinear algebraic equations that are complexly coupled and there are no general methods to solve them analytically (; ). It has been recognized as the general purpose of a software that standardized procedures should be proposed for reducing the kinematic analysis of mechanism to simplify the problem (). Different procedures were developed to establish the restriction equations to solve their time derivatives of first and second orders (). Actually, such procedures can be accomplished through vector-loop processing for planar linkages (; ). The forward kinematics are either established in the functions of structure parameters and input variables numerically () or presented in algebraic coordinates (; ). The inverse kinematic problem consists of finding the joint variables to achieve a desired configuration of a mechanism (; ).
To understand the kinematic performance of a linkage, many scholars have been proposing different theories (; ; ; ; ; ; ), methods (; ; ; ; ), algorithms (; ; ) and software (; ; ; ). This paper focuses on an algorithm in the screw coordinates to solve the velocity of articulated planar linkage and investigates the displacement and acceleration. A screw is a line vector accompanied by a secondary vector attached with a pitch. As the geometrical element, a screw with six components plays a vital role in kinematics and mechanics of a mechanism (). The paper develops an algorithm to analyze the displacement, velocity and acceleration of articulated planar mechanisms in twist coordinates of each joint. This is the first try to use screw coordinates to completely study the displacement, velocity and acceleration of linkages in a general systematic way. Because the kinematics analysis starts from the velocity, the solutions of forward and inverse kinematics of a mechanism have the same form in expression which facilitates the programming and calculation. The discussion is not restricted to the kinematics of revolute jointed planar linkages, and the similar principles also apply to spatial mechanisms.
Instantaneous Twist of the End Effector of a Series Mechanism
Definition of Twist
Table 1 is the definition of parameters in this paper.
TABLE 1
| Parameters | Definition |
|---|---|
| The relative angular velocity of joint | |
| The velocity of any point attached to the rigid link | |
| The vector direction of joint | |
| The screw of joint | |
| The unit screw of joint | |
| The screw of end effector | |
| The screw matrix of end effector | |
| The vector of relative angular velocity |
Definition of parameters.
Figure 1A shows an articulated link that is rotating around a fixed revolute joint with angular velocity of around the -axis. The velocity of any point attached to the rigid link can be expressed by . As a result, the velocity of point on the extended rigid body of link that is overlapped with the origin of the coordinate frame is where which is illustrated in Figure 1B.
FIGURE 1
Also, there is which is illustrated in Figure 1C. The dual 3-dimensional vectors and can fully determine the rotation of link . So, a dual vector can be defined aswhere is a screw that expresses the twist of link that is rotating around joint with a marking point superimposing with the origin of the coordinate system. Supposing that , Equation 1 can be rewritten aswhere is the angular speed of the rotation about joint . Letwhere is called unit screw because the norm of is (35].
In Equation 3, the first three components indicate the unit direction of a rotation and the last three components present implicitly the position of the axis of rotation with respect to the origin of the coordinate system (). Therefore, Equation 2 can be denoted by
Twist Matrix of a Series Pivoted Kinematic Chain
When a second link is jointed with at (see Figure 2A), the relative twist of with respect to can be analyzed by fixing with the ground which is indicated by Figure 2B.
FIGURE 2
With the similar procedure mentioned above, we getwhere is the relative speed of link with respect to rotating around the revolute joint , and
According to the principle of linear superposition, the absolute angular velocity of link is ()and the absolute velocity of link with the marking point that is superimposed with the origin at the moment iswhere indicates the velocity of a point on link that is at this instant superimposed with the origin illustrated in Figure 2C.
As a result, the twist of link with respect to the coordinate system is
Equation (9) can be rewritten aswhich can be expressed in matrix multiplication form:where and .
Similarly, the twist of the end effector, denoted by, of a kinematic chain in series illustrated by Figure 3 can be expressed as.whereand
FIGURE 3
Eq. 13 is called the unit twist matrix of a serial linkage while Equation 14 presents a vector including all relative angular speeds of each joint relative to its previous neighbor in the kinematic chain. Screw matrix 13) is made up of the geometry parameters of the mechanism. It can be programmed in the computer software. This procedure offers an explicit inference of kinematic attributes for velocities in the mechanisms that have the same topology.
Kinematics of a Revolute Jointed Mechanism With Serial Open Chain
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 directly obtained when are all prescribed. Denote the twist by
We know that is the absolute angular velocity of the end effector and is the velocity of the point attached with the end effector that is superimposed with the origin of the coordinate system at this moment. The velocity of the geometry center of the end effector is therefore denoted bywhere is the position vector of the geometry center of the end effector in the absolute ground coordinate system. As a result, we get the twist of the end effector at its central coordinate frame whose axes are parallel with the corresponding absolute ones at this instant:where is the absolute angular velocity of the end effector and is the absolute linear velocity of the center of the end effector. Eq. 17 is the forward kinematics of a serial linkage in screw form which provides all necessary parameters for developing the dynamics of the linkage.
After knowing the twist of the end effector with a marking point of the origin of the coordinate frame, we can left multiply at both sides of Equation 15:where is the transpose of matrix .
When , the serial linkage is either redundantly actuated or in its singularity configuration. Otherwise, we get:where is called the pseudo inverse of the unit twist matrix . Eq. 19 represents the inverse kinematics for the serial mechanism.
Figure 4 shows a planar linkage in series. In the absolute coordinate system, we gain the twist of the end effector:whereand
FIGURE 4
Supposing the length of the links are , and , the coordinates of each revolute joint can be expressed by , , , , , , respectively. So the unit twist matrix 18) is
In accordance to Equation 17, we know that the twist of the end effector with the marking point on its center iswhere .
By programming with the numerical algorithms, we obtain the forward kinematics of the linkage. With the initial conditions of , and , and , and , we get the first set of parameters, from equation (22). And then we gain the first twist of the end effector from equation (20) and from equation (23) and from equation (24)
Then we get the successive parameters of from Equation 23 and from Equation 24 by updating the data:where and is a finite small time increment. The absolute angular displacement of the end effector is
The angular accelerations of each joint can be numerically calculated bywhere represents the jth joint.
In forward kinematic, we let the angular velocity of screw joint be , the angular velocity of screw joint be , the angular velocity of screw joint be .with the structure parameters and initial conditions in Table 2, we programmed the above process in MATLAB and drew the forward displacement, velocity and acceleration for each joint (see Figure 5) by numerical methods based on Equations 25–27.
TABLE 2
| 200 | 210 | 300 |
Structure parameters and initial conditions.
FIGURE 5
Kinematics of Pivoted Linkages of Closed Chain
Kinematics of the 4-bar Linkage
Figure 6 1) shows a planar 4-bar linkage in series and 2) a 4-bar mechanism with closed loop. The twist of the end effector of the 4-bar linkage in series can be gained from Equation 12:where , and , , , .
FIGURE 6
Eq. 12 indicates that the kinematic chain forms a closed loop when the end effector is fixed with the frame (see Figure 6B). And therefore, there must be
Eq. 28 is called the loop equation of the mechanism which can be used to solve all angular velocities by taking other known conditions into account. In the coordinate frame shown in Figure 6, , , , , , , and , we get thatwhere , , .
Rearranging equation (29) presents
Therefore, we gain the forward kinematics of the closed-chain 4-bar linkage (Figure 6B):where
When the output of is known, we can also get that
Consequently, the inverse kinematics of the 4 bar mechanism is now rewritten aswhere , and .
In this regard, the forward velocity and inverse velocity have the same form in mathematical expressions which is one of the advantages of this algorithm. Then, we get the successive parameters of from Equation 30 or from Equation 31 by updating the data of , and with the interactions below:where represents the ith iteration and is a finite small time increment.
Compared with the Denavit-Hartenberg notation for a closed loop (), the kinematics algorithm in screw form here only requires to implement one numerical integration 32) for displacement and one numerical differential 27) for acceleration in the absolute coordinate frame. This provides a more convergent algorithm to develop computational kinematics of a linkage. We let the angular velocity of joint is 2 rad/s and with the structure parameters and initial conditions in Table 3, 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.
TABLE 3
| 2 | 100 | 150 | 200 | 210 | 0 |
Structure parameters and initial conditions.
FIGURE 7
Kinematics of an Articulated Linkage of 1 degree of Freedom With Multiple Closed Chains
Figure 8 illustrates a planar multi-closed-chain 6-bar linkage 1) in which there are two independent closed chains (b). For the first closed chain of a 4-bar linkage (Figure 8 (c)), the loop equation can be found from Equation 27:where is the input. Rearranging this loop equation yields:
FIGURE 8
Similarly, we get the loop equation of the second coupled 5-bar closed chain linkage (Figure 8 (d)) from Equation 28:
Rearranging this loop equation presents:
So the forward kinematics of the multiple closed chain linkage can be obtained by associating these double-loop Equations 33, 34:where , and .
We let the angular velocity of joint is 2 rad/s and with the structure parameters and initial conditions in Table 4, 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.
TABLE 4
| 2 | 100 | 150 | 200 | 210 | 300 | 400 | 0 | 150 | −100 |
Structure parameters and initial conditions.
FIGURE 9
Kinematics of a Planar Mechanism of More Degrees of Freedom With Single Closed Chain
Figure 10 illustrates 1) a planar 5-bar linkage in series and 2) a 5-bar linkage of closed loop. The twist of the end effector of the 5-bar linkage in series can be gained by Equation 12:where , and , , , , .
FIGURE 10
From Equation 12, we know that the kinematic chain forms a closed loop when the end effector is fixed with the frame (Figure 10B). Therefore, there must be . In the coordinate frame shown in Figure 10, , , , , , , , , and , and , we get that
Then there is
Therefore, we get the forward kinematics of the closed-chain 5-bar linkage (Figure 10B):where and .
Eq. 37 represents the forward velocity of the planar 5-bar mechanism of 2 degrees of freedom. In accordance to the numerical Eq. 32, 27, we obtain the forward displacement and acceleration of the mechanism with a single closed chain. We let the angular velocity of joint is 3 rad/s, the angular velocity of joint 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.
TABLE 5
| 3 | 4 | 100 | 300 | 300 | 100 | 200 | 400 | 0 |
Initial conditions and structure parameters.
FIGURE 11
Conclusion
This paper proposed a method to investigate the displacement, velocity and acceleration of a mechanism in screw coordinates in a general systematic way. 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. This method is validated by planar mechanisms in series, single closed loop and multiple closed loops. This method is particularly suitable for programming the computational software for forward and inverse kinematics of a mechanism, covering the velocity, displacement and acceleration. Although this paper discusses the kinematics of revolute jointed planar mechanisms, the same principles may apply to any spatial linkages.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
methodology, J-SZ; software, S-TW; validation, J-SZ, S-TW; formal analysis, J-SZ; investigation, J-SZ, S-TW; resources, J-SZ; data curation, S-TW; writing—original draft preparation, J-SZ; writing—review and editing, J-SZ, S-TW; visualization, J-SZ, S-TW; supervision, J-SZ; project administration, J-SZ; funding acquisition, J-SZ All authors have read and agreed to the published version of the manuscript.
Funding
This research was supported by the Natural Science Foundation of China under Grant 51575291.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmech.2021.774814/full#supplementary-material
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Summary
Keywords
kinematics, numerical algorithm, planar open-chain linkage, closed-chain linkage, articulated joint
Citation
Zhao J-S and Wei S-T (2021) Kinematics of Articulated Planar Linkages. Front. Mech. Eng 7:774814. doi: 10.3389/fmech.2021.774814
Received
13 September 2021
Accepted
04 November 2021
Published
14 December 2021
Volume
7 - 2021
Edited by
Hamid M. Sedighi, Shahid Chamran University of Ahvaz, Iran
Updates
Copyright
© 2021 Zhao and Wei.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jing-Shan Zhao, jingshanzhao@mail.tsinghua.edu.cn
This article was submitted to Solid and Structural Mechanics, a section of the journal Frontiers in Mechanical Engineering
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.