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Design, Modelling and Validation of a Novel Extra Slender Continuum Robot for In-situ Inspection and Repair in Aeroengine

Mingfeng Wang, Xin Dong, Weiming Ba, Abdelkhalick Mohammad, Dragos Axinte, Andy Norton

arXiv:1910.04572v1cs.ROeess.SY

TL;DR

Aeroengine maintenance through borescope ports requires a long, narrow, sufficiently stiff robot that can navigate confined combustor spaces and carry repair tools. The paper designs a 16-DoF dual-stage continuum robot and develops segment-based kinematic and Kirchhoff-rod static models; experiments validate its shape accuracy and payload capability.

  • Problem

    In-situ aeroengine repair requires a robot that can pass through small ports, reach deep combustor regions, perform multi-axis movements, and carry miniaturized end-effectors.

  • Method

    The paper combines a rigid-compliant dual-stage robot design with segment-based piecewise constant-curvature kinematics and Kirchhoff elastic-rod static modelling.

  • Results

    The robot achieved a C-c curve radius error of 1.02% and carried 136.2 g with deflections below 2.79% in the straight configuration and 4.45% in the C-c configuration.

  • Takeaways & Limitations

    The validated 12.7 mm-diameter, 715 mm-long robot provides the demonstrated reach, shape control, and payload capability for in-situ aeroengine maintenance.

Abstract

from arXiv · show

In-situ aeroengine maintenance works are highly beneficial as it can significantly reduce the current maintenance cycle which is extensive and costly due to the disassembly requirement of engines from aircrafts. However, navigating in/out via inspection ports and performing multi-axis movements with end-effectors in constrained environments (e.g. combustion chamber) are fairly challenging. A novel extra-slender (diameter-to-length ratio <0.02) dual-stage continuum robot (16 degree-of-freedom) is proposed to navigate in/out confined environments and perform required configuration shapes for further repair operations. Firstly, the robot design presents several innovative mechatronic solutions: (i) dual-stage tendon-driven structure with bevelled disks to perform required shapes and to provide selective stiffness for carrying high payloads; (ii) various rigid-compliant combined joints to enable different flexibility and stiffness in each stage; (iii) three commanding cables for each 2-DoF section to minimise the number of actuators with precise actuations. Secondly, a segment-scaled piecewise-constant-curvature-theory based kinematic model and a Kirchhoff-elastic-rod-theory based static model are established by considering the applied forces/moments (friction, actuation, gravity and external load), where the friction coefficient is modelled as a function of bending angle. Finally, experiments were carried out to validate the proposed static modelling and to evaluate the robot capabilities of performing the predefined shape and stiffness.

I. INTRODUCTION

The paper targets in-situ aeroengine inspection and repair with an extra-slender continuum robot that can traverse borescope ports while carrying miniaturized tools. It addresses coupled design and modelling challenges arising from the required small diameter, long reach, stiffness, and hybrid rigid-compliant structure.

  • Motivation: Extra-slender robots for deep aeroengine inspection and repair must combine a diameter below 15 mm, length above 500 mm, and sufficient stiffness for miniaturized end-effectors.These dimensions and payload demands make both robot design and modelling challenging.
  • Modelling challenge: Existing constant-curvature kinematic models can incur significant errors because friction, gravity, and hybrid rigid-compliant joints affect the actual arm shape.The paper identifies these effects as especially important as compliant-joint length increases.
  • Paper approach: The paper proposes a hybrid rigid-compliant continuum robot and segment-based PCCT and Kirchhoff-rod models to support design, control, and validation.The work also specifies requirements and validates the resulting mechatronic system experimentally.
  • Application: The proposed technology aims to deploy a miniature flame-spray repair tool through borescope ports into combustion chambers for in-situ coating repair.The robot must also deliver oxyacetylene gas and powder materials through its central working channel.
  • Technical requirements: The target robot requires at least 700 mm reach, no more than 15 mm diameter, at least 11 sections, and a 125 g tip payload.These requirements support access through inspection ports and use of inspection or repair end-effectors.

III. SOLUTIONS: DESIGN AND MODELLING

The proposed robot combines a two-stage, 16-DoF architecture with rigid-compliant joints and tendon actuation to navigate the combustor and perform multi-axis maintenance operations. Its design allocates flexibility and motion capability differently between the body and tip stages.

  • Design target: The design establishes a 12.7 mm-diameter, 715 mm-long, fully actuated arm with a diameter-to-length ratio below 0.02.The combination of compliant and rigid joints is intended to satisfy reach, slenderness, stiffness, and hyper-redundancy requirements.
  • Robot architecture: The robot has 13 sections and 16 DoFs, comprising ten 1-DoF body sections and three 2-DoF tip sections.The body navigates through the combustor, while the tip provides 6-DoF movement for maintenance.
  • Robot architecture: The body stage delivers the tip through small inspection ports, while the tip stage performs spatial maintenance movements in confined spaces.The two stages are assigned different roles for navigation and operation.
  • Mechanical design: Twin-pivot joints combine rigid disks with superelastic NiTi rods to provide section bending flexibility and enhanced torsional stability.Body sections use eight revolute joints constrained by NiTi rods, whereas tip sections use short NiTi rods.
  • Actuation: Tendon actuation uses cable pairs for body sections and three-cable groups for tip sections, with full actuation through 29 cable-motor units.Magnetic encoders and load cells provide cable position and force feedback.

B. Coordinate Systems and Modelling Assumptions

The modelling framework represents the robot through arc-based configuration variables and separates actuator-to-configuration and configuration-to-task mappings. It uses segment-level constant-curvature kinematics while retaining friction and other equilibrium effects through static modelling assumptions.

  • Coordinate systems: The robot is modelled as planar body and spatial tip continuum manipulators with coordinate frames for the world, sections, disks, and end-effector.The task space is represented by end-effector position and orientation.
  • Coordinate systems: The arc configuration uses curvature κ, rotational angle ϕ, and arc length ℓ to describe each continuum segment.The end-effector task space comprises its position and rotation matrix.
  • Modelling assumptions: The model assumes thin, inextensible planar elastic rods without shear or extension deformation in body backbones and tip segments.This is the principal backbone idealization used for the kinematic and static analyses.
  • Modelling assumptions: Cable-routing friction is retained, with its coefficient treated as a function of bending angle rather than a constant.The model also assumes fixed cable locations in disk cross-sections and static equilibrium during manipulation.
  • Kinematics analysis: Segment-based PCCT decomposes each section into curvature units whose bending angles are obtained from the static model before concatenation.The kinematics then maps actuator or cable lengths to configuration and configuration to end-effector pose.

1) Configuration-Cable Kinematics:

The configuration-cable model relates cable routing geometry and segment bending to cable-length changes across the body and tip sections. It accounts for distinct pitch-circle diameters, phase angles, and the different 1-DoF and 2-DoF segment structures.

  • Cable routing geometry: Cable routing uses distinct pitch-circle diameters and phase angles for body and tip actuation paths.Body cables use D1 or D2, while tip cables route through D2 and then D3.
  • Body-segment kinematics: The body sections are modeled as repeated 1-DoF segments containing two adjacent disks and a continuous compliant joint.Their through-disk cable lengths remain constant, so bending changes arise from gap-length variations.
  • Body-segment kinematics: Body cable-length changes are computed from bevel slopes, segment bending angle, routing geometry, and the accumulated contributions of all segments.The bevel slopes are denoted θ_s1 and θ_s2, while θ_i,j is the segment bending angle.
  • Tip-segment kinematics: The tip sections are modeled as repeated 2-DoF segments containing three adjacent disks and two twin-pivot joints.A group of three cables shapes each tip segment in the proposed structure.
  • Tip-segment kinematics: Total cable-length changes in tip sections are obtained by combining the gap changes between adjacent disks across the segments.The formulation uses the bending angles of neighboring segment joints and accounts for the disk count in each tip section.

2) Configuration-Task Kinematics:

The configuration-task model maps segment configurations to the end-effector pose by composing homogeneous transformations through the body and tip structures. It uses constant-curvature translations for body segments and sequential rotations for the twin-pivot tip segments.

  • Body-segment transformations: A body segment transforms one disk frame to the preceding frame through rotation about the local x-axis and translation along a constant-curvature arc.The translation depends on bending angle θ_i,j and segment arc length Δℓ.
  • Tip-segment transformations: A tip segment composes rotations about local y- and x-axes with translations to represent its two degrees of freedom.The two transformations are concatenated through the intermediate disk frame.
  • Whole-robot transformation: The entire robot transformation is obtained by multiplying the transformations of all segments from the thirteenth-section end disk to the first-section base disk.This composition connects the complete continuum structure to the base frame.
  • End-effector pose: The end-effector task pose is represented by its global position and orientation obtained from the composed transformation matrix.The position is O14 = [x14, y14, z14]^T and the orientation is the corresponding rotation matrix.
  • End-effector pose: The robot configuration angles required by the task and cable mappings are determined using the static model.Configuration-cable and configuration-task kinematics therefore depend on the preceding static analysis.

D. Static Modelling

The static model establishes equilibrium for the combined compliant-rigid robot by including actuation, gravity, backbone elasticity, and external loading. Its friction coefficient varies with bending angle, reflecting the changing cable-routing geometry.

  • Loading formulation: The static formulation includes four loading categories: cable actuation, disk gravity, backbone elasticity, and external loading.Cable actuation includes tension, friction, and contact force.
  • Actuation loading: Cable actuation forces and moments are formulated from the routing geometry between adjacent disks and transformed into local and base frames.The formulation uses cable tensions, contact forces, hole locations, and vector cross products for the generated moments.
  • Actuation loading: The friction coefficient μ_θi,j is modeled as a function of bending angle rather than as a constant.The model applies a Coulomb friction formulation to cable–routing-hole friction.
  • Gravity loading: Gravity loading is retained because the robot’s diameter-to-length ratio is below 0.02, making the supporting disks’ gravity effect non-negligible.Gravity is defined along the negative world-frame Z-axis and depends on each disk’s mass and gravitational acceleration.
  • Elasticity loading: Backbone elasticity is modeled with Kirchhoff elastic rod theory, approximating each NiTi-backbone segment as an arc undergoing planar bending.The bending moment depends on curvature, Young’s modulus, and moment of inertia.
  • External loading: External force and moment applied at the end-effector are transformed from the world frame into the local frame of each disk.The transformation matrix relates the end-effector load to the point about which local moments are calculated.

E. Solutions

The solution procedure recursively applies Newton–Euler equilibrium across repeated joint units. Given a desired configuration and mechanical properties, it computes the actuation inputs needed by the model.

  • Recursive solution procedure: Newton–Euler equations establish static equilibrium for the whole robot by recursively applying the derived equations to adjacent joint units.The recursion exploits the robot’s repeated segment structure.
  • Recursive solution procedure: Known configuration angles and mechanical properties provide the inputs for solving cable tensions and displacements.The listed properties include segment length, Young’s modulus, moment of inertia, disk mass, and μ(θ).

IV. EXPERIMENTAL VALIDATION

Validation tests assessed friction, single-section static-model accuracy, whole-arm configuration following, and stiffness under payloads.

  • The experiments measured variable friction and validated the static model with and without payload before evaluating whole-arm performance.The test program also assessed configuration-shape following and stiffness capability.

A. Static Model Validation and Analysis

Friction measurements and single-section bending tests evaluated the static model under gravity and payload, showing small prediction errors.

  • Friction test: The friction test varied cable angle and measured normal force, friction force, and friction coefficient for the model.Trials used standard weights from 0.8908 kg to 4.0908 kg at cable angles of 1.25°, 5°, 10.25°, and 20°.
  • Single-section validation: The single-section prototype comprised nine disks, with one actuated cable and coordinate measurements taken along the backbone.The prototype represented one body section and was tested in planar motion.
  • Single-section validation: The section was tested through 0° to 90° planar bending without payload and with a 90.8g payload.Experimental measurements were compared with static-model-calculated configurations.
  • Single-section validation: 0.43 mm average and 0.79 mm maximum errors occurred without payload, corresponding to 0.79% and 1.47% of the 54 mm section length.The prototype performed the predefined bending curve under gravity, while the model accurately predicted deflections.
  • Single-section validation: 0.94 mm average and 1.59 mm maximum errors occurred with the 90.8g payload, corresponding to 1.74% and 2.94% of section length.The simulated shape coincided well with the loaded prototype shape.
  • Conclusion: The proposed kinematic and static models predicted the novel robot structure under different payloads with small errors.These models were subsequently used when designing the mechanical structure.

B. C-c curve Following Validation

Whole-arm tests compared model-based and VICON-measured positions in straight and full C-c configurations, demonstrating close static-shape agreement and identifying a coupling-related limitation.

  • C-c configuration test: The full robot was tested in the required C-c configuration, with section 1 forming the c-shape for port navigation and other sections forming the C-shape.A VICON optical motion-capture system tracked marker plates on the robot.
  • C-c configuration test: Model-based positions and VICON measurements were compared for the initial straight shape and the full C-c configuration.Calculated results used red solid lines with circles, while measured data used blue dashed lines with stars.
  • Straight shape: 3.54 mm average and 5.88 mm maximum errors in the straight shape represented 0.49% and 0.82% of the 715 mm arm length.The reported assessment considered marker-plate assembly errors and cable-hole clearance.
  • Full C-c shape: 558.37 mm measured radius differed by 1.02% from the 564.14 mm model-based radius in the full C-c shape.The measured arm positions slightly undershot the simulation, mainly because cable tension reduced through the cable guide.
  • Conclusion: The static tests showed that the system was sufficient for the desired final C-c shape and static positioning.Further work was identified for dynamic positioning during navigation.

C. Stiffness Capability Test

Stiffness tests applied payloads in straight and C-c configurations, finding limited deflection below 90.8g and larger but bounded deflection at 136.2g.

  • Test setup: Payloads from 0g to 136.2g, in 45.4g increments, were tested in straight and C-c configurations.Deflections were measured from snapshots and plotted for both configurations.
  • Deflection results: Payloads below 90.8g produced deflections under 8 mm, or under 1.12% of the 715 mm arm length.This result applied to the tested straight and C-c configurations.
  • Deflection results: At 136.2g, deflection remained under 20 mm in the straight shape and 32 mm in the C-c shape, corresponding to 2.79% and 4.45% of arm length.The robot was considered capable of matching the 125g end-effector payload requirement.

V. CONCLUSIONS

The paper develops and validates an extra-slender, fully actuated continuum robot for in-situ aeroengine combustor maintenance through borescope ports. Its segment-scaled models and experiments support accurate shape prediction, C-c configuration following, and payload carrying.

  • The 12.7-mm-diameter, 715-mm-long robot has a diameter-to-length ratio below 0.02 and carries miniaturised end-effectors weighing at least 125 g.
  • Its two-stage, 13-section architecture combines ten 1-DoF body sections, three 2-DoF tip sections, bevelled disks, and rigid-compliant joints.
  • Segment-scaled piecewise constant-curvature kinematics and Kirchhoff elastic rod statics incorporate a variable friction coefficient and support configuration-cable solving for desired C-c shapes.
  • 0.79% and 1.74% average errors were obtained for single-section in-plane bending without and with a 90.8-g payload, respectively.
  • The robot followed the desired C-c shape with less than 1.02% average curve-radius deviation and carried 136.2 g with deflections of 2.79% straight and 4.45% in the C-c shape.
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