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Comparison of internal force antagonism between redundant cable-driven parallel robots and redundant rigid parallel robots

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Abstract

The internal force antagonism (IFA) problem is one of the most important issues limiting the applications and popularization of redundant parallel robots in industry. Redundant cable-driven parallel robots (RCDPRs) and redundant rigid parallel robots (RRPRs) behave very differently in this problem. To clarify the essence of IFA, this study first analyzes the causes and influencing factors of IFA. Next, an evaluation index for IFA is proposed, and its calculating algorithm is developed. Then, three graphical analysis methods based on this index are proposed. Finally, the performance of RCDPRs and RRPRs in IFA under three configurations are analyzed. Results show that RRPRs produce IFA in nearly all the areas of the workspace, whereas RCDPRs produce IFA in only some areas of the workspace, and the IFA in RCDPRs is milder than that RRPRs. Thus, RCDPRs more fault-tolerant and easier to control and thus more conducive for industrial application and popularization than RRPRs. Furthermore, the proposed analysis methods can be used for the configuration optimization design of RCDPRs.

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Abbreviations

IFA:

Internal force antagonism

PID:

Proportion–integral–derivative

RCDPR:

Redundant cable-driven parallel robot

RRPR:

Redundant rigid parallel robot

A i :

ith connection point between the linkage and the base

a i :

Position vector of Ai in O

B i :

ith connection point between the linkage and the moving platform

O′ b i :

Position vector of Bi in O′

C λ :

Initial value constants

E :

Young’s modulus of the nylon material

f 1, f 2 :

Tensions of the first and second cables, respectively

f d :

Dynamic antagonistic force

f s :

Static antagonistic force

F :

External force

G(s):

Transfer function

G :

Gravity of the moving platform

i :

Number of iterations

i max :

Maximum number of iterations

J :

Jacobian matrix of the parallel robots

J − :

Inverse matrix of J

J + :

Moore–Penrose pseudo-inverse matrix of J

k 1, k 2 :

Stiffnesses of the first and second cables, respectively

k D :

Differential parameters in PID control

k I :

Integral parameters in PID control

k max :

Maximum number of selections

k p :

Proportional parameters in PID control

L :

Length of the cable

\({{\boldsymbol{\dot l}}}\) :

Velocity vector of the linkages

l i :

Linkage vector from Bi to Ai

m :

Number the linkages

m 1, m 2 :

Masses of the first and second cables, respectively

M :

Mass of the mass block

n :

Degrees of freedom of the robot’s motion

N ba :

Element in the bth row and ath column of N

N :

Basis vector for the general solution of Eq. (8)

N j :

jth column of N

O :

Base coordinate system

O′:

Moving coordinate system

O m, O n :

m- and n-dimensional zero vectors, respectively

p, ṗ :

Position and velocity vectors of the moving platform, respectively

r :

Degree of redundancy of parallel robots

R :

Radius of the cable

R :

Rotation matrix of O′ relative to O

s :

Micro elements of the transfer function

S :

Second matrix after singular value decomposition of J

t :

Time

t i :

Value of the ith joint force

t s1, t s2 :

First and second step time, respectively

T sb :

bth element of Ts

T :

Value of the ith joint force

T (1) :

Initial joint force

T last :

Value of T for the last iteration

T s :

Special solution of the joint force

u i :

Linkage unit vector

U, V :

First and third matrices after singular value decomposition of J, respectively

x 1, x 2 :

Displacements of the first and second cables, respectively

x i :

Initial position of the mass block

x m :

Displacement of the mass block

x t :

Target position of the mass block

x te1, x te2 :

First and second target positions with error, respectively

x :

Pose of the moving platform

ẋ :

Velocity vector of the moving platform

ϕ :

Solution type

η 1 :

Index 1 (the maximum value of the Euclidean norm number of T)

η 2 :

Index 2 (the unit circle integral of ∥T∥max in the neighborhood of the coordinate [0, G])

λ :

A random r-dimensional vector

\(\lambda _a^{(i)}\) :

ath element of λ(i)

\({\boldsymbol{\Theta }},\,{\boldsymbol{\dot \Theta }}\) :

Angular displacement and velocity vectors of the moving platform, respectively

τ :

Update step

Ω:

Output completion flag

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Acknowledgements

The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 51975307).

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Authors and Affiliations

  1. State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing, 100084, China

    Yuheng Wang & Xiaoqiang Tang

  2. Beijing Key Laboratory of Precision/Ultra-Precision Manufacturing Equipment and Control, Tsinghua University, Beijing, 100084, China

    Yuheng Wang & Xiaoqiang Tang

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  1. Yuheng Wang
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Correspondence to Xiaoqiang Tang.

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Wang, Y., Tang, X. Comparison of internal force antagonism between redundant cable-driven parallel robots and redundant rigid parallel robots. Front. Mech. Eng. 18, 51 (2023). https://doi.org/10.1007/s11465-023-0767-x

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  • DOI: https://doi.org/10.1007/s11465-023-0767-x

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