Abstract
In-situ maintenance is of great significance for improving the efficiency and ensuring the safety of aeroengines. 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.
![](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs11465-023-0756-0/MediaObjects/11465_2023_756_Fig1_HTML.jpg)
Article PDF
Avoid common mistakes on your manuscript.
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
- C i :
-
Cable number
- D i :
-
Disc number
- E :
-
Young’s modulus of Ni−Ti rod
- \({f_{{{\rm{D}}_i}{{\rm{C}}_j}}}\) :
-
Friction generated by the jth cable on the ith disc
- \({F_{{{\rm{J}}_i}{{\rm{C}}_j}}}\) :
-
Value of the jth cable tension in the ith joint
- \({F_{{\rm{S}}{{\rm{C}}_j}}}\) :
-
Value of the jth cable tension on the force sensor
- \({\boldsymbol{F}}_{_{{{\rm{D}}_i}{\rm{C}}}}^{{O_{i - 1}}}\) :
-
Lumped force of actuating forces \({\boldsymbol{F}}_{_{{{\rm{D}}_i}{{\rm{C}}_j}}}^{{O_{i - 1}}}\) on the ith disc expressed in frame {Oi−1}
- \({\boldsymbol{F}}_{_{{{\rm{D}}_i}{{\rm{C}}_j}}}^{{O_{i - 1}}}\) :
-
Actuating force vector applied by the jth cable to the ith disc expressed in frame {Oi−1}
- \({\boldsymbol{F}}_{_{{{\rm{D}}_i}}}^{{O_{i - 1}}}\) :
-
Lumped forces on the ith disc expressed in frame {Oi−1}
- F EX :
-
Matrix of \({\boldsymbol{F}}_{_{{\rm{E}}{{\rm{X}}_i}}}^{{O_{\rm{G}}}}\)
- \({\boldsymbol{F}}_{_{{\rm{E}}{{\rm{X}}_i}}}^{{O_{\rm{G}}}}\) :
-
External force applied to the ith disc expressed in frame {OG}
- \({\boldsymbol{F}}_{_{{{\rm{J}}_i}{{\rm{C}}_j}}}^{{O_p}}\) :
-
jth cable tension in the ith joint expressed in frame {Op}
- F SC :
-
Matrix of \({F_{{\rm{S}}{{\rm{C}}_j}}}\)
- \({\boldsymbol{G}}_{{\rm{CL}}{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{\rm{G}}}}\) :
-
Gravity of the jth cable-locking device on the ith disc expressed in frame {OG}
- \({\boldsymbol{G}}_{{{\rm{D}}_i}}^{{O_{\rm{G}}}}\) :
-
Gravity of the jth disc expressed in frame {OG}
- \({\boldsymbol{G}}_{{{\rm{J}}_i}{{\rm{C}}_j}}^{{O_{\rm{G}}}}\) :
-
Gravity of the jth cable of the ith joint expressed in frame {OG}
- \({\boldsymbol{G}}_{{\rm{NiT}}{{\rm{i}}_i}}^{{O_{\rm{G}}}}\) :
-
Gravity of the Ni−Ti rod of the ith joint expressed in frame {OG}
- g :
-
Gravitational acceleration
- h :
-
Thickness of disc
- \(h_{{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{\rm{G}}}},\,H_{{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{\rm{G}}}}\) :
-
jth cable holes on the ith disc
- I z :
-
Moment of inertia of Ni−Ti rod
- K :
-
Number of sections
- \({l_{{{\rm{D}}_{2i - 1}}{{\rm{C}}_j}}},\,{l_{{{\rm{D}}_{2i}}{{\rm{C}}_j}}}\) :
-
jth cable length in the ith segment
- \(\Delta {l_{{{\rm{D}}_{2i - 1}}{{\rm{C}}_j}}},\,\Delta {l_{{{\rm{D}}_{2i}}{{\rm{C}}_j}}}\) :
-
jth cable variations in the ith segment
- L :
-
Length of Ni−Ti rod
- \(\Delta {L_{{{\rm{D}}_i}{{\rm{C}}_j}}}\) :
-
Sum of the jth cable variation from the ith joint to the DKth joint
- \({m_{{\rm{CL}}{{\rm{D}}_i}{{\rm{C}}_j}}}\) :
-
Mass of the jth cable-locking device on the ith disc
- \({m_{{{\rm{D}}_i}}}\) :
-
Mass of the ith disc
- \({m_{{{\rm{J}}_i}{{\rm{C}}_j}}}\) :
-
Mass of the jth cable of the ith joint
- \({m_{{\rm{NiT}}{{\rm{i}}_i}}}\) :
-
Mass of the ith compliant backbone
- \({\boldsymbol{M}}_{{\rm{CL}}{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) :
-
Moment of the jth cable-locking device gravity \({{\boldsymbol{G}}_{{\rm{CL}}{{\rm{D}}_i}{{\rm{C}}_j}}}\) relative to the point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\rm{D}}_i}}^{{O_{i - 1}}}\) :
-
Lumped moments relative to the point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\rm{D}}_i}{\rm{C}}}^{{O_{i - 1}}}\) :
-
Lumped moment of \({\boldsymbol{M}}_{{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) relative to point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) :
-
Moment of actuating force \({\boldsymbol{F}}_{{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) relative to point Oi−1 expressed in frame {Oi−1}
- M EX :
-
Matrix of \({\boldsymbol{M}}_{{\rm{E}}{{\rm{X}}_i}}^{{O_{\rm{G}}}}\)
- \({\boldsymbol{M}}_{{\rm{E}}{{\rm{X}}_i}}^{{O_{\rm{G}}}}\) :
-
External moment applied to the ith disc expressed in frame {OG}
- \({\boldsymbol{M}}_{{{\boldsymbol{F}}_{{{\rm{D}}_i}}}}^{{O_{i - 1}}}\) :
-
Moment of the lumped force \({\boldsymbol{F}}_{{{\rm{D}}_i}}^{{O_{i - 1}}}\) relative to point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\boldsymbol{F}}_{{\rm{E}}{{\rm{X}}_i}}}}^{{O_{i - 1}}}\) :
-
Moment of the external force \({\boldsymbol{F}}_{{\rm{E}}{{\rm{X}}_i}}^{{O_{\rm{G}}}}\) relative to point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\boldsymbol{G}}_{{{\rm{D}}_i}}}}^{{O_{i - 1}}}\) :
-
Moment of the ith disc gravity \({{\boldsymbol{G}}_{{{\rm{D}}_i}}}\) relative to the point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\rm{J}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) :
-
Moment of the jth cable gravity \({{\boldsymbol{G}}_{{{\rm{J}}_i}{{\rm{C}}_j}}}\). relative to the point Oi−1 expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{{\rm{J}}_i}}^{{O_{i - 1}}}\) :
-
Bending moment of the ith joint expressed in frame {Oi−1}
- \({\boldsymbol{M}}_{{\rm{NiT}}{{\rm{i}}_j}}^{{O_{i - 1}}}\) :
-
Moment of Ni−Ti rod gravity \({{\boldsymbol{G}}_{{\rm{NiT}}{{\rm{i}}_i}}}\). relative to the point Oi−1 expressed in frame {Oi−1}
- N :
-
Number of segments
- \({\boldsymbol{N}}_{{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_i}}\) :
-
Pressure generated by the jth cable on the ith disc expressed in frame {Oi}
- \({\boldsymbol{n}}_{{X_i}}^{{O_i}}\) :
-
Normal unit vector of the YiOiXi plane, expressed in frame {Oi}
- \(O_{{\rm{CL}}{{\rm{D}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) :
-
Gravity center of the jth cable-locking device on the ith disc, expressed in frame {Oi−1}
- \(O_{{{\rm{D}}_i}}^{{O_{i - 1}}}\) :
-
Gravity center of the ith disc, expressed in frame {Oi−1}
- \(O_i^{{O_p}}\) :
-
Point Oi expressed in frame {Op}
- \(O_{{{\rm{J}}_i}{{\rm{C}}_j}}^{{O_{i - 1}}}\) :
-
Gravity center of the jth cable of the (2i − 1)th joint, expressed in frame{Oi−1}
- \(O_{{\rm{NiT}}{{\rm{i}}_i}}^{{O_{i - 1}}}\) :
-
Gravity center of the ith compliant backbone, expressed in frame {Oi−1}
- {O 2i}: O−X 2i Y 2i Z 2i :
-
Revolute joint frame with origin \({o_{2i}}\) at the axial intersection point of the (2i − 1)th disc and the 2ith disc
- {O G}: O−X G Y G Z G :
-
World frame and YG-axis is considered to be along the gravity direction
- {O i}: O−X i Y i Z i :
-
ith disc frame with origin Oi at the center of the ith disc
- r j :
-
Distance of the center of disc and the jth cable hole
- Rot(x i, α):
-
Rotation matrix (around the xi-axis and the bending angle is α)
- \(_{i + 1}^{\,\,\,\,\,i}{\boldsymbol{T}}\) :
-
Homogeneous transformation matrix from {Qi} to {Oi−1}
- Trans(x, y, z):
-
Translation matrix
- ρ cable :
-
Linear density of cables
- β i,1, β i,2 :
-
Joint angles of the ith segment
- β :
-
Results of the bending angle matrix
- β * :
-
Bending angle matrix during solving the kinetostatic equations
- β 2N×1 :
-
Matrix of βi,1 and βi,2
- ϕ j :
-
Angle of the jth cable hole and Yi-axis
- μ 2i,1, μ 2i,2 :
-
Friction coefficient of the 2ith disc
- θ :
-
Cable-hole angle
- θ i,1, θ i,2 :
-
Angel between the 2ith disc and cables
- γ i,1 :
-
Degree of the deviation of the center of the gravity of the Ni−Ti rod in the ith segment
References
Wang M F, Dong X, Ba W M, Mohammad A, Axinte D, Norton A. Design, modelling and validation of a novel extra slender continuum robot for in-situ inspection and repair in aeroengine. Robotics and Computer-integrated Manufacturing, 2021, 67: 102054
Dong X, Axinte D, Palmer D, Cobos S, Raffles M, Rabani A, Kell J. Development of a slender continuum robotic system for on-wing inspection/repair of gas turbine engines. Robotics and Computer-integrated Manufacturing, 2017, 44: 218–229
Dong X, Wang M F, Mohammad A, Ba W M, Russo M, Norton A, Kell J, Axinte D. Continuum robots collaborate for safe manipulation of high-temperature flame to enable repairs in challenging environments. IEEE/ASME Transactions on Mechatronics, 2022, 27(5): 4217–4220
Russo M, Raimondi L, Dong X, Axinte D, Kell J. Task-oriented optimal dimensional synthesis of robotic manipulators with limited mobility. Robotics and Computer-integrated Manufacturing, 2021, 69: 102096
Dong X, Palmer D, Axinte D, Kell J. In-situ repair/maintenance with a continuum robotic machine tool in confined space. Journal of Manufacturing Processes, 2019, 38: 313–318
Dong X, Raffles M, Cobos-Guzman S, Axinte D, Kell J. A novel continuum robot using twin-pivot compliant joints: design, modeling, and validation. Journal of Mechanisms and Robotics, 2016, 8(2): 021010
Dong X, Raffles M, Cobos Guzman S, Axinte D, Kell J. Design and analysis of a family of snake arm robots connected by compliant joints. Mechanism and Machine Theory, 2014, 77: 73–91
Li S Y, Hao G B. Current trends and prospects in compliant continuum robots: a survey. Actuators, 2021, 10(7): 145
Garriga-Casanovas A, Rodriguez Y Baena F. Kinematics of continuum robots with constant curvature bending and extension capabilities. Journal of Mechanisms and Robotics, 2019, 11(1): 011010
Webster R JIII, Jones B A. Design and kinematic modeling of constant curvature continuum robots: a review. The International Journal of Robotics Research, 2010, 29(13): 1661–1683
Rao P, Peyron Q, Lilge S, Burgner-Kahrs J. How to model tendon-driven continuum robots and benchmark modelling performance. Frontiers in Robotics and AI, 2021, 7: 630245
Guo H, Ju F, Cao Y F, Qi F, Bai D M, Wang Y Y, Chen B. Continuum robot shape estimation using permanent magnets and magnetic sensors. Sensors and Actuators A: Physical, 2019, 285: 519–530
Yang Z S, Yang L H, Xu L, Chen X F, Guo Y J, Liu J X, Sun Y. A continuum robot with twin-pivot structure: the kinematics and shape estimation. In: Liu X J, Nie Z G, Yu J J, Xie F G, Song R, eds. Intelligent Robotics and Applications. Cham: Springer, 2021, 466–475
Chen X Q, Zhang X, Huang Y Y, Cao L, Liu J G. A review of soft manipulator research, applications, and opportunities. Journal of Field Robotics, 2022, 39(3): 281–311
Yuan H, Zhou L L, Xu W F. A comprehensive static model of cable-driven multi-section continuum robots considering friction effect. Mechanism and Machine Theory, 2019, 135: 130–149
Chen Y Y, Wu B B, Jin J B, Xu K. A variable curvature model for multi-backbone continuum robots to account for inter-segment coupling and external disturbance. IEEE Robotics and Automation Letters, 2021, 6(2): 1590–1597
Godage I S, Medrano-Cerda G A, Branson D T, Guglielmino E, Caldwell D G. Modal kinematics for multisection continuum arms. Bioinspiration & Biomimetics, 2015, 10(3): 035002
Yuan H, Chiu P W Y, Li Z. Shape-reconstruction-based force sensing method for continuum surgical robots with large deformation. IEEE Robotics and Automation Letters, 2017, 2(4): 1972–1979
Gonthina P S, Kapadia A D, Godage I S, Walker I D. Modeling variable curvature parallel continuum robots using euler curves. In: Proceedings of 2019 International Conference on Robotics and Automation. Montreal: IEEE, 2019, 1679–1685
Singh I, Amara Y, Melingui A, Mani Pathak P, Merzouki R. Modeling of continuum manipulators using pythagorean hodograph curves. Soft Robotics, 2018, 5(4): 425–442
Mbakop S, Tagne G, Drakunov S V, Merzouki R. Parametric PH curves model based kinematic control of the shape of mobile soft manipulators in unstructured environment. IEEE Transactions on Industrial Electronics, 2022, 69(10): 10292–10300
Huang X J, Zou J, Gu G Y. Kinematic modeling and control of variable curvature soft continuum robots. IEEE/ASME Transactions on Mechatronics, 2021, 26(6): 3175–3185
Dong X. Design of a continuum robot for in-situ repair of aero engine. Dissertation for the Doctoral Degree. Nottingham: University of Nottingham, 2016
Morales Bieze T, Kruszewski A, Carrez B, Duriez C. Design, implementation, and control of a deformable manipulator robot based on a compliant spine. The International Journal of Robotics Research, 2020, 39(14): 1604–1619
Li S J, Kruszewski A, Guerra T M, Nguyen A T. Equivalent-input-disturbance-based dynamic tracking control for soft robots via reduced-order finite-element models. IEEE/ASME Transactions on Mechatronics, 2022, 27(5): 4078–4089
Zhang J Y, Fang Q, Xiang P Y, Sun D Y, Xue Y N, Jin R, Qiu K, Xiong R, Wang Y, Lu H J. A survey on design, actuation, modeling, and control of continuum robot. Cyborg and Bionic Systems, 2022, 2022: 9754697
Janabi-Sharifi F, Jalali A, Walker I D. Cosserat rod-based dynamic modeling of tendon-driven continuum robots: a tutorial. IEEE Access: Practical Innovations, Open Solutions, 2021, 9: 68703–68719
Ghafoori M, Keymasi Khalaji A. Modeling and experimental analysis of a multi-rod parallel continuum robot using the cosserat theory. Robotics and Autonomous Systems, 2020, 134: 103650
Rucker D C, Jones B A, Webster R J III. A geometrically exact model for externally loaded concentric-tube continuum robots. IEEE Transactions on Robotics, 2010, 26(5): 769–780
Bretl T, McCarthy Z. Quasi-static manipulation of a kirchhoff elastic rod based on a geometric analysis of equilibrium configurations. The International Journal of Robotics Research, 2014, 33(1): 48–68
Awtar S, Sen S. A generalized constraint model for two-dimensional beam flexures: nonlinear strain energy formulation. Journal of Mechanical Design, 2010, 132(8): 081009
Chen G M, Ma F L, Hao G B, Zhu W D. Modeling large deflections of initially curved beams in compliant mechanisms using chained beam constraint model. Journal of Mechanisms and Robotics, 2019, 11(1): 011002
Ma F L, Chen G M. Kinetostatic modeling and characterization of compliant mechanisms containing flexible beams of variable effective length. Mechanism and Machine Theory, 2020, 147: 103770
Qi F, Ju F, Bai D M, Chen B. Kinematics optimization and static analysis of a modular continuum robot used for minimally invasive surgery. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 2018, 232(2): 135–148
Rucker D C, Webster R JIII. Statics and dynamics of continuum robots with general tendon routing and external loading. IEEE Transactions on Robotics, 2011, 27(6): 1033–1044
Roy R, Wang L, Simaan N. Modeling and estimation of friction, extension, and coupling effects in multisegment continuum robots. IEEE/ASME Transactions on Mechatronics, 2017, 22(2): 909–920
Mohammad A, Russo M, Fang Y H, Dong X, Axinte D, Kell J. An efficient follow-the-leader strategy for continuum robot navigation and coiling. IEEE Robotics and Automation Letters, 2021, 6(4): 7493–7500
Virgala I, Kelemen M, Božek P, Bobovský Z, Hagara M, Prada E, Oščádal P, Varga M. Investigation of snake robot locomotion possibilities in a pipe. Symmetry, 2020, 12(6): 939
Nikitin Y, Božek P, Peterka J. Logical-linguistic model of diagnostics of electric drives with sensors support. Sensors, 2020, 20(16): 4429
Abramov I V, Abramov A I, Nikitin Y R, Sosnovich E, Božek P, Stollmann V. Diagnostics of electrical drives. In: Proceedings of 2015 International Conference on Electrical Drives and Power Electronics. Tatranska Lomnica: IEEE, 2015, 364–367
Acknowledgements
This work was sponsored by the National Natural Science Foundation of China (Grant Nos. 52105117, 52375125, and 52105118).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare that they have no conflict of interest.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution, and reproduction in any medium or format as long as appropriate credit is given to the original author(s) and source, a link to the Creative Commons license is provided, and the changes made are indicated.
The images or other third-party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Visit http://creativecommons.org/licenses/by/4.0/ to view a copy of this license.
About this article
Cite this article
Yang, Z., Yang, L., Sun, Y. et al. Comprehensive kinetostatic modeling and morphology characterization of cable-driven continuum robots for in-situ aero-engine maintenance. Front. Mech. Eng. 18, 40 (2023). https://doi.org/10.1007/s11465-023-0756-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11465-023-0756-0