Comprehensive kinetostatic modeling and morphology characterization of cable-driven continuum robots for in-situ aero-engine maintenance

Abstract

In-situ maintenance is of great significance for improving the efficiency and ensuring the safety of aero-engines. The cable-driven continuum robot (CDCR) with twin-pivot compliant mechanisms, which is enabled with flexible deformation capability and confined space accessibility, has emerged as a novel tool that aims to promote the development of intelligence and efficiency for in-situ aero-engine maintenance. The high-fidelity model that describes the kinematic and morphology of CDCR lays the foundation for the accurate operation and control for in-situ maintenance. However, this model was not well addressed in previous literature. In this study, a general kinetostatic modeling and morphology characterization methodology that comprehensively contains the effects of cable-hole friction, gravity, and payloads is proposed for the CDCR with twin-pivot compliant mechanisms. First, a novel cable-hole friction model with the variable friction coefficient and adaptive friction direction criterion is proposed through structure optimization and kinematic parameter analysis. Second, the cable-hole friction, all-component gravities, deflection-induced center-of-gravity shift of compliant joints, and payloads are all considered to deduce a comprehensive kinetostatic model enabled with the capacity of accurate morphology characterization for CDCR. Finally, a compact continuum robot system is integrated to experimentally validate the proposed kinetostatic model and the concept of in-situ aero-engine maintenance. Results indicate that the proposed model precisely predicts the morphology of CDCR and outperforms conventional models. The compact continuum robot system could be considered a novel solution to perform in-situ maintenance tasks of aero-engines in an invasive manner.

Graphical abstract

Keywords

kinetostatic modeling / morphology characterization / variable friction / continuum robots / in-situ maintenance

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Nomenclature

Abbreviations
CDCR	Cable-driven continuum robot
D‒H	Denavit‒hartenberg
DC	Direct current
DOF	Degree of freedom
ESC	Electronic speed controller
FEM	Finite element method
LPC	Low-pressure compressor
MAE	Mean absolute error
MAPE	Mean absolute percentage error
PC	Personal computer
PCC	Piecewise constant curvature
Variables
Ci	Cable number
Di	Disc number
E	Young’s modulus of Ni‒Ti rod
f_{{\mathrm{D}}_i{\mathrm{C}}_j}	Friction generated by the jth cable on the ith disc
F_{{\mathrm{J}}_i{\mathrm{C}}_j}	Value of the jth cable tension in the ith joint
F_{\mathrm{S}{\mathrm{C}}_j}	Value of the jth cable tension on the force sensor
{\mathit{F}}_{{\mathrm{D}}_i\mathrm{C}}^{O_{i-1}}	Lumped force of actuating forces {\mathit{F}}_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}} on the ith disc expressed in frame \left\{O_{i-1}\right\}
{\mathit{F}}_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}}	Actuating force vector applied by the jth cable to the ith disc expressed in frame \left\{O_{i-1}\right\}
{\mathit{F}}_{{\mathrm{D}}_i}^{O_{i-1}}	Lumped forces on the ith disc expressed in frame \left\{O_{i-1}\right\}
FEX	Matrix of {\mathit{F}}_{\mathrm{E}{\mathrm{X}}_i}^{O_{\mathrm{G}}}
{\mathit{F}}_{\mathrm{E}{\mathrm{X}}_i}^{O_{\mathrm{G}}}	External force applied to the ith disc expressed in frame \left\{O_{\mathrm{G}}\right\}
{\mathit{F}}_{{\mathrm{J}}_i{\mathrm{C}}_j}^{O_p}	jth cable tension in the ith joint expressed in frame \left\{O_p\right\}
{\mathit{F}}_{\mathrm{S}\mathrm{C}}	Matrix of F_{\mathrm{S}{\mathrm{C}}_j}
{\mathit{G}}_{\mathrm{C}\mathrm{L}{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{\mathrm{G}}}	Gravity of the jth cable-locking device on the ith disc expressed in frame \left\{O_{\mathrm{G}}\right\}
{\mathit{G}}_{{\mathrm{D}}_i}^{O_{\mathrm{G}}}	Gravity of the ith disc expressed in frame \left\{O_{\mathrm{G}}\right\}
{\mathit{G}}_{{\mathrm{J}}_i{\mathrm{C}}_j}^{O_{\mathrm{G}}}	Gravity of the jth cable of the ith joint expressed in frame \left\{O_{\mathrm{G}}\right\}
{\mathit{G}}_{\mathrm{N}\mathrm{i}\mathrm{T}{\mathrm{i}}_i}^{O_{\mathrm{G}}}	Gravity of the Ni‒Ti rod of the ith joint expressed in frame \left\{O_{\mathrm{G}}\right\}
g	Gravitational acceleration
h	Thickness of disc
h_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{\mathrm{G}}}, H_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{\mathrm{G}}}	jth cable holes on the ith disc
Iz	Moment of inertia of Ni‒Ti rod
K	Number of sections
l_{{\mathrm{D}}_{2i-1}{\mathrm{C}}_j}, l_{{\mathrm{D}}_{2i}{\mathrm{C}}_j}	jth cable length in the ith segment
\mathrm{\Delta }{l}_{{\mathrm{D}}_{2i-1}{\mathrm{C}}_j}, \mathrm{\Delta }{l}_{{\mathrm{D}}_{2i}{\mathrm{C}}_j}	jth cable variations in the ith segment
L	Length of Ni‒Ti rod
\mathrm{\Delta }{L}_{{\mathrm{D}}_i{\mathrm{C}}_j}	Sum of the jth cable variation from the ith joint to the {\mathrm{D}}_Kth joint
m_{\mathrm{C}\mathrm{L}{\mathrm{D}}_i{\mathrm{C}}_j}^{}	Mass of the jth cable-locking device on the ith disc
m_{{\mathrm{D}}_i}^{}	Mass of the ith disc
m_{{\mathrm{J}}_i{\mathrm{C}}_j}^{}	Mass of the jth cable of the ith joint
m_{\mathrm{N}\mathrm{i}\mathrm{T}{\mathrm{i}}_i}^{}	Mass of the ith compliant backbone
{\mathit{M}}_{\mathrm{C}\mathrm{L}{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}}	Moment of the jth cable-locking device gravity {\mathit{G}}_{\mathrm{C}\mathrm{L}{\mathrm{D}}_i{\mathrm{C}}_j} relative to the point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{{\mathrm{D}}_i}^{O_{i-1}}	Lumped moments relative to the point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{{\mathrm{D}}_i\mathrm{C}}^{O_{i-1}}	Lumped moment of {\mathit{M}}_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}} relative to point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}}	Moment of actuating force {\mathit{F}}_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}} relative to point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{\mathrm{E}\mathrm{X}}	Matrix of {\mathit{M}}_{\mathrm{E}{\mathrm{X}}_i}^{O_{\mathrm{G}}}
{\mathit{M}}_{\mathrm{E}{\mathrm{X}}_i}^{O_{\mathrm{G}}}	External moment applied to the ith disc expressed in frame \left\{O_{\mathrm{G}}\right\}
{\mathit{M}}_{{\mathit{F}}_{{\mathrm{D}}_i}}^{O_{i-1}}	Moment of the lumped force {\mathit{F}}_{{\mathrm{D}}_i}^{O_{i-1}} relative to point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{F_{\mathrm{E}{\mathrm{X}}_i}}^{O_{i-1}}	Moment of the external force {\mathit{F}}_{\mathrm{E}{\mathrm{X}}_i}^{O_{\mathrm{G}}} relative to point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{{\mathit{G}}_{{\mathrm{D}}_i}}^{O_{i-1}}	Moment of the ith disc gravity {\mathit{G}}_{{\mathrm{D}}_i} relative to the point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{{\mathrm{J}}_i{\mathrm{C}}_j}^{O_{i-1}}	Moment of the jth cable gravity {\mathit{G}}_{{\mathrm{J}}_i{\mathrm{C}}_j} relative to the point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{{\mathrm{J}}_i}^{O_{i-1}}	Bending moment of the ith joint expressed in frame \left\{O_{i-1}\right\}
{\mathit{M}}_{\mathrm{N}\mathrm{i}\mathrm{T}{\mathrm{i}}_i}^{O_{i-1}}	Moment of Ni‒Ti rod gravity {\mathit{G}}_{\mathrm{N}\mathrm{i}\mathrm{T}{\mathrm{i}}_i} relative to the point O_{i-1} expressed in frame \left\{O_{i-1}\right\}
N	Number of segments
{\mathit{N}}_{{\mathrm{D}}_i{\mathrm{C}}_j}^{O_i}	Pressure generated by the jth cable on the ith disc expressed in frame \left\{O_i\right\}
{\mathit{n}}_{X_i}^{O_i}	Normal unit vector of the Y_i{O}_i{X}_i plane, expressed in frame \left\{O_i\right\}
O_{\mathrm{C}\mathrm{L}{\mathrm{D}}_i{\mathrm{C}}_j}^{O_{i-1}}	Gravity center of the jth cable-locking device on the ith disc, expressed in frame \left\{O_{i-1}\right\}
O_{{\mathrm{D}}_i}^{O_{i-1}}	Gravity center of the ith disc, expressed in frame \left\{O_{i-1}\right\}
O_i^{O_p}	Point O_i expressed in frame \left\{O_p\right\}
O_{{\mathrm{J}}_i{\mathrm{C}}_j}^{O_{i-1}}	Gravity center of the jth cable of the (2i − 1)th joint, expressed in frame \left\{O_{i-1}\right\}
O_{\mathrm{N}\mathrm{i}\mathrm{T}{\mathrm{i}}_i}^{O_{i-1}}	Gravity center of the ith compliant backbone, expressed in frame \left\{O_{i-1}\right\}
\left\{o_{2i}\right\}:o-x_{2i}{y}_{2i}{\mathit{\text{z}}}_{2i}	Revolute joint frame with origin o_{2i} at the axial intersection point of the (2i ‒ 1)th disc and the 2ith disc
\left\{O_{\mathrm{G}}\right\}:O-X_{\mathrm{G}}{Y}_{\mathrm{G}}{Z}_{\mathrm{G}}	World frame and YG-axis is considered to be along the gravity direction
\left\{O_i\right\}:O-X_i{Y}_i{Z}_i	ith disc frame with origin O_i at the center of the ith disc
rj	Distance of the center of disc and the jth cable hole
\mathbf{R}\mathbf{o}\mathbf{t}(x_i,\alpha )	Rotation matrix (around the xi-axis and the bending angle is \alpha)
{}_{}^{}i+1i\mathit{T}	Homogeneous transformation matrix from \left\{Q_i\right\} to \left\{O_{i-1}\right\}
\mathbf{T}\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{s}(x,y,\mathit{\text{z}})	Translation matrix
{\rho }_{\mathrm{c}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}}	Linear density of cables
{\beta }_{i,1}, {\beta }_{i,2}	Joint angles of the ith segment
\mathit{\beta }	Results of the bending angle matrix
{\mathit{\beta }}^{\ast }	Bending angle matrix during solving the kinetostatic equations
{\mathit{\beta }}_{2N\times 1}	Matrix of {\beta }_{i,1} and {\beta }_{i,2}
{\varphi }_j	Angle of the jth cable hole and Yi-axis
{\mu }_{2i,1}, {\mu }_{2i,2}	Friction coefficient of the 2ith disc
\theta	Cable-hole angle
{\theta }_{i,1}, {\theta }_{i,2}	Angel between the 2ith disc and cables
{\gamma }_{i,1}	Degree of the deviation of the center of the gravity of the Ni‒Ti rod in the ith segment

Acknowledgements

This work was sponsored by the National Natural Science Foundation of China (Grant Nos. 52105117, 52375125, and 52105118).

Conflict of Interest

The authors declare that they have no conflict of interest.

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