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

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.

Key words： cable-driven parallel robots; parallel robots; redundant robots; evaluation index; force solution space

Nomenclature

Abbreviations
IFA	Internal force antagonism
PID	Proportion‒integral‒derivative
RCDPR	Redundant cable-driven parallel robot
RRPR	Redundant rigid parallel robot
Variables
Ai	ith connection point between the linkage and the base
ai	Position vector of Ai in O
Bi	ith connection point between the linkage and the moving platform
{}^{O^{\prime }}{\mathit{b}}_i	Position vector of B_i in O^{\prime }
Cλ	Initial value constants
E	Young’s modulus of the nylon material
f1, f2	Tensions of the first and second cables, respectively
fd	Dynamic antagonistic force
fs	Static antagonistic force
F	External force
G(s)	Transfer function
G	Gravity of the moving platform
i	Number of iterations
imax	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
kD	Differential parameters in PID control
kI	Integral parameters in PID control
kmax	Maximum number of selections
kp	Proportional parameters in PID control
L	Length of the cable
\dot{\mathit{l}}	Velocity vector of the linkages
li	Linkage vector from Bi to Ai
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
Nba	Element in the bth row and ath column of N
N	Basis vector for the general solution of Eq. (8)
Nj	jth column of N
O	Base coordinate system
O^{\prime }	Moving coordinate system
Om, On	m- and n-dimensional zero vectors, respectively
p, \dot{\mathit{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^{\prime } relative to O
s	Micro elements of the transfer function
S	Second matrix after singular value decomposition of J
t	Time
ti	Value of the ith joint force
ts1, ts2	First and second step time, respectively
Tsb	bth element of Ts
T	Value of the ith joint force
T(1)	Initial joint force
Tlast	Value of T for the last iteration
Ts	Special solution of the joint force
ui	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
xi	Initial position of the mass block
xm	Displacement of the mass block
xt	Target position of the mass block
xte1, xte2	First and second target positions with error, respectively
x	Pose of the moving platform
\dot{\mathit{x}}	Velocity vector of the moving platform
\varphi	Solution type
{\eta }_1	Index 1 (the maximum value of the Euclidean norm number of T)
{\eta }_2	Index 2 (the unit circle integral of {\Vert \mathit{T}\Vert }_{\text{max}} in the neighborhood of the coordinate \left[0,G\right])
\mathit{\lambda }	A random r-dimensional vector
{\lambda }_a^{(i)}	ath element of {\mathit{\lambda }}^{(i)}
\mathbf{\Theta }, \dot{\mathbf{\Theta }}	Angular displacement and velocity vectors of the moving platform, respectively
\tau	Update step
\mathrm{\Omega }	Output completion flag

Acknowledgements

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

Conflict of Interest

The authors declare that they have no conflict of interest.

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