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

doi:10.1007/s11465-023-0767-x

Front. Mech. Eng.

2023, Vol. 18

Issue (4) : 51 https://doi.org/10.1007/s11465-023-0767-x

RESEARCH ARTICLE

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

Yuheng WANG^1,², Xiaoqiang TANG^1,²(

)

¹. State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
². Beijing Key Laboratory of Precision/Ultra-Precision Manufacturing Equipment and Control, Tsinghua University, Beijing 100084, China

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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.

Keywords cable-driven parallel robots parallel robots redundant robots evaluation index force solution space

Corresponding Author(s): Xiaoqiang TANG

About author: Peng Lei and Charity Ngina Mwangi contributed equally to this work.

Just Accepted Date: 28 August 2023 Issue Date: 17 November 2023

Cite this article:

Yuheng WANG,Xiaoqiang TANG. Comparison of internal force antagonism between redundant cable-driven parallel robots and redundant rigid parallel robots[J]. Front. Mech. Eng., 2023, 18(4): 51.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-023-0767-x
https://academic.hep.com.cn/fme/EN/Y2023/V18/I4/51

Fig.1 Redundant cable-driven parallel robot with two cables: (a) configuration diagram and (b) mechanical model.

Fig.2 Control block diagram of the simulation example. PID: proportion?integral?derivative.

Fig.3 Simulation results: response of (a) moving platform position and (b) cable tension.

Fig.4 Demonstration of target position errors caused by position sensors during actual control.

Fig.5 Simulation results of internal force antagonism for eight configuration cases.

Fig.6 General simplified model of parallel robots with

m

linkages (

m ? 2

Fig.7 Possible cases of the joint force solution space: (a) case 1, (b) case 2, (c) case 3, and (d) case 4.

Fig.8 Planar three-cable-driven redundant parallel robot in a particular configuration.

Fig.9 Joint force solution space graph of the planar three-cable-driven redundant parallel robot.

Fig.10 Some of the possible directions of external forces applied to the robot.

Fig.11 Plotting method of external force space graph.

Fig.12 External force space graph of the planar three-cable-driven redundant parallel robot. IFA: internal force antagonism.

Fig.13 Position space graph of the planar three-cable-driven redundant parallel robot. IFA: internal force antagonism.

Fig.14 Proof and analysis flow diagram. IFA: internal force antagonism.

Fig.15 Diagram of the three-linkage planar configuration using (a) cable linkages and (b) rigid linkages.

Fig.16 Comparison of the space graphs for the three-linkage planar configuration: (a) joint force solution space graph of the redundant cable-driven parallel robot (RCDPR), (b) external force space graph of the RCDPR, (c) position space graph of the RCDPR, (d) joint force solution space graph of the redundant rigid parallel robot (RRPR), (e) external force space graph of the RRPR, and (f) position space graph of the RRPR.

Parameter	Value
Size of the base	$2 m × 2 m × 2 m$
Radius of the static platform	$1 m$
Radius of the moving platform	$0.5 m$
Distance between two platforms	$1 m$
Included angle of the hinge points	$20 ∘$

Tab.1 Parameters of the seven-linkage spatial configuration

Fig.17 Diagram of the seven-linkage spatial configuration using (a) cable linkages and (b) rigid linkages.

Fig.18 Comparison of the space graphs for the seven-linkage spatial configuration: (a) joint force solution space graph of the redundant cable-driven parallel robot (RCDPR), (b) external force space graph of the RCDPR, (c) position space graph of the RCDPR, (d) joint force solution space graph of the redundant rigid parallel robot (RRPR), (e) external force space graph of the RRPR, and (f) position space graph of the RRPR.

Parameter	Value
Size of the base	$2 m × 2 m × 2 m$
Size of the moving platform	$0 .5 m × 0 .5 m × 0 .25 m$
$x$ -direction distance of the linkage connection points under the moving platform	$0.25 m$
$x$ -direction distance of the linkage connection points above the moving platform	$0 .5 m$
$z$ -direction distance of the linkage connection points on the base	$0.5 m$

Tab.2 Parameters of the eight-linkage spatial configuration

Fig.19 Diagram of the eight-linkage spatial configuration using (a) cable linkages and (b) rigid linkages.

Fig.20 Comparison of the space graphs for the eight-linkage spatial configuration: (a) joint force solution space graph of the redundant cable-driven parallel robot (RCDPR), (b) external force space graph of the RCDPR, (c) position space graph of the RCDPR, (d) joint force solution space graph of the redundant rigid parallel robot (RRPR), (e) external force space graph of the RRPR, and (f) position space graph of the RRPR.

Abbreviations
IFA	Internal force antagonism
PID	Proportion?integral?derivative
RCDPR	Redundant cable-driven parallel robot
RRPR	Redundant rigid parallel robot
Variables
A_i	ith connection point between the linkage and the base
a_i	Position vector of A_i in O
B_i	ith connection point between the linkage and the moving platform
$O ′ b i$	Position vector of $B i$ in $O ′$
C_λ	Initial value constants
E	Young’s modulus of the nylon material
f₁, f₂	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
k₁, k₂	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
$l ˙$	Velocity vector of the linkages
l_i	Linkage vector from B_i to A_i
m	Number the linkages
m₁, m₂	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, $p ˙$	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 T_s
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
x₁, x₂	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
$x ˙$	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
$λ a (i)$	ath element of $λ (i)$
$Θ$ , $Θ ˙$	Angular displacement and velocity vectors of the moving platform, respectively
$τ$	Update step
$Ω$	Output completion flag

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