# Kinematic Modelling for Hyper-Redundant Robots—A Structured Guide

^{*}

## Abstract

**:**

## 1. Introduction, Motivation and Definitions

#### 1.1. Definitions

**Nonredundant robot:**Any robot whose degrees of freedom are fewer than or equal to $DO{F}_{{\mathbb{R}}^{m}}$, that is, the minimal number of degrees of freedom to completely define the position and orientation of the robot endpoint.**Redundant robot**: A robot with more than $DO{F}_{{\mathbb{R}}^{m}}$ degrees of freedom. They are ensured to be capable of reaching a certain state through different joint configurations since there are infinite solutions for each state.**Hyper-redundant robots**: A robot with more than twice the minimal degrees of freedom required to completely define the endpoint state ($DOF\ge 2\xb7DO{F}_{{\mathbb{R}}^{m}}$).

#### 1.2. Classification of CDHR

- (1)
**Discreet Hyper-Redundant Robots:**The classical set of hyper-redundant robots and the most used in the origins of this field. Its main characteristic is that discreet robots are composed of a succession of rigid sections, joined together by, normally, one or two degree-of-freedom joints. The rigidity of the sections makes it possible to use traditional robotics techniques to obtain their kinematic model. MACH-I [14] (see Figure 1a) or Series II, X125 System from OC Robotics [18] are two examples of discreet hyper-redundant robots.- (2)
**Constant Curvature Continuous Robots**: in this group, we include any robot whose sections’ backbone can be mathematically modelled as a constant curvature segment. Therefore, actual continuous robots (soft robots, for example) might be included in this group, but also robots with discreet joints and deformable sections that form a curve. They are mathematically more challenging, but due to the assumption of constant curvature, their equations can be analytically obtained. Examples of this group of robots are the Elephant Trunk by Hannan and Walker [19] or Pylori-I (see Figure 1b), a pivotal discs CDHR [20].- (3)
**Other Continuous Robots**: In some cases, due to an excess of forces either in the backbone of the robot (excessive distributed weight) or in the endpoint, the constant curvature hypothesis cannot be applied. These cases are more difficult to manage, and they often imply using numerical methods to solve the constitutive equations of the robot. However, since this is a dynamic condition, they are morphologically equivalent to constant curvature robots (either with joints and deformable sections or soft robots). Most soft robots (such as Ruan [21], see Figure 1c, or Kyma [22]), specially those made by polymeric materials, need this kind of kinematic model.

## 2. Kinematic Modelling Stages for CDHR

#### 2.1. Definitions and Nomenclature for the Kinematic Problem

- The definition of the whole robot’s structure (the position of each of its points in space) is named the
**robot configuration.** - The discs (or equivalent physical structures) of the robot to which the cables are fixed are called
**active discs.** - The discs with no cables attached but that are crossed by them are called
**passive discs**. - Except as otherwise specified, the
**section**of the robot is considered as the portion of the robot between two consecutive active discs. - The determination of the position of each one of the infinite points of a certain section is named the
**section’s state**. It can be represented in different ways depending on the type of robot or the chosen model (a vector, a matrix, etc.). - The
**endpoint of a section**is the theoretical point that represents the ending of a certain section of the robot. Although it could be arbitrarily defined, in this work the usual convention of the geometric centre of the ending disc is chosen. The**endpoint of the robot**would then be the last section’s endpoint.

- The total number of sections of the robot is named n.
- The index k is used when referring to any particular section.
- The total number of cables or tendons the robot uses is named f.
- The endpoint of the robot is indicated with a subindex e.

#### 2.2. Direct Kinematics

#### 2.3. Inverse Kinematics

## 3. Discreet Hypothesis: Denavit–Hartenberg Method

#### 3.1. Applying the Hypothesis

#### 3.2. Method Explanation

## 4. PCC Hypothesis—Piecewise Constant Curvature

#### 4.1. Direct Kinematics

#### 4.1.1. Independent Transformation—Geometric Method

#### 4.1.2. Alternatives to Geometric Method

- (1)
- The first transformation is used to transform the problem in a two-dimensional curve, using the rotation $\varphi $.
- (2)
- The second transformation represents a rotation of ${\theta}_{2}=\frac{1}{2}\kappa s$ degrees, which is used to obtain a reference system pointing to the section’s endpoint.
- (3)
- The third transformation introduces the translation from the origin to the endpoint of the section curve ${d}_{3}=\frac{2}{\kappa}sin\frac{\kappa s}{2}$.
- (4)
- The fourth transformation rotates again ${\theta}_{4}=\frac{1}{2}\kappa s$ degrees to return the tangency to the curve in the endpoint.
- (5)
- Finally, the fifth transformation undoes the first transformation, returning to a three-dimensional problem.

#### 4.1.3. Dependent Transformation

#### 4.2. Inverse Kinematics

**$\varphi $**: Vertical axis z corresponds to the set of points in which $x=0$ and $y=0$, so that in those cases when the robot’s endpoint is in the z axis, any value of $\varphi $ can be set.**$\kappa $**: In this case, two possibilities must be taken into account. Whenever $z\ne 0$, then $\kappa =0$ and $l=z$ can be used, as long as the robot’s length can be varied and z has a positive value (other cases should be studied individually). On the other hand, if $z=0$, then the robot’s endpoint should be located precisely in the origin, thus forming a perfect circle. There could be many mathematical solutions to do so, choosing, as an example, $\kappa =0$, $\varphi =0$ and $\theta =2\pi $. It should be noted, though, that this situation is, in most cases, mechanically impossible to reach.

#### 4.3. Differential Kinematics

## 5. LSK—Linearised Segment Kinematics

## 6. Kinetic Robot Modelling Using Elastic Properties

#### 6.1. Pseudorigid Body

#### 6.2. Cosserat Rod Theory

## 7. Optimization Algorithm—CCD and Natural CCD Algorithms

- To tackle the possibility of a singularity, two viable alternatives can be used. In some cases, it would be enough to change to another joint (the movement of one of the joints might exit by itself the singularity that the first joint caused). However, in some cases where the singularity affects the whole robot (as when the robot is perfectly colinear and pointing to the goal), the Natural-CCD algorithm assigns a random d and $\theta $ that exit the position and allow it to continue.
- Collision management with itself is solved by designing an angle ${\theta}_{max}$ that represents a maximum bound for the possible range of the joint. This way, the angle can be defined so the segment can never collide with its predecessor. To do so, Martín et al. in [26] define ${\theta}_{max}$ as the supplementary angle to the interior angle of an N-sided polygon, N being the number of the robot’s joints.
- Finally, to minimise the effect of abrupt movements to the joints, Natural-CCD limits by a coefficient k the value of ${\theta}_{i}$ for the rotation. Particularly, the inverse of the distance between the robot’s endpoint and the desired position is suggested as a possible coefficient. However, k must be superiorly bounded by 1 in order for the algorithm to converge.$$k=\frac{1}{|\overrightarrow{{p}_{e}{p}_{f}}|}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}0>k\ge 1\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\theta}^{*}=k\theta $$

## 8. Tutorial for Hyper-Redundant Robot Modelling

#### 8.1. Determination of the Type of Robot

#### 8.2. Example of Discreet Robot: MACH-I

#### 8.3. Constant Curvature Hypothesis

#### 8.4. Example of Non-PCC Robot: Ruan

#### 8.5. Inside the PCC Model

#### 8.6. Example of PCC Robot: Pilory

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDHR | Cable-Driven Hyper-Redundant Robot |

DOF | Degrees Of Freedom |

HTM | Homogeneous Transformation Matrix |

PCC | Piecewise Constant Curvature |

LSK | Linearised Segment Kinematics |

PRB | Pseudorigid Body |

CRT | Cosserat Rod Theory |

CCD | Cyclic Coordinate Descent |

## References

- Moran, M.E. Evolution of robotic arms. J. Robot. Surg.
**2007**, 1, 103–111. [Google Scholar] [CrossRef] [PubMed] - Hirose, S. Biologically Inspired Robots: Snake-Like Locomotors and Manipulators; Oxford University Press: Oxford, UK, 1993; p. 282. [Google Scholar]
- Burdick, J.W. A Modal Approach to Hyper-Redundant Manipulator Kinematics. IEEE Trans. Robot. Autom.
**1994**, 10, 343–354. [Google Scholar] [CrossRef] - Hannan, M.W.; Walker, I.D. Novel Kinematics for Continuum Robots. In Advances in Robot Kinematics; Springer: Berlin/Heidelberg, Germany, 2000; pp. 227–238. [Google Scholar] [CrossRef]
- Jones, B.A.; Walker, I.D. Kinematics for multisection continuum robots. IEEE Trans. Robot.
**2006**, 22, 43–55. [Google Scholar] [CrossRef] - Gravagne, I.A.; Rahn, C.D.; Walker, I.D. Large Deflection Dynamics and Control for Planar Continuum Robots. IEEE/ASME Trans. Mechatronics
**2003**, 8, 299–307. [Google Scholar] [CrossRef] - Webster, R.J.; Jones, B.A. Design and kinematic modeling of constant curvature continuum robots: A review. Int. J. Robot. Res.
**2010**, 29, 1661–1683. [Google Scholar] [CrossRef] - Martín Barrio, A. Design, Modelling, Control and Teleoperation of Hyper- Redundant Robots. Ph.D. Thesis, Universidad Politécnica de Madrid, Madrid, Spain, 2020. [Google Scholar] [CrossRef]
- Kim, S.; Xu, W.; Ren, H. Inverse kinematics with a geometrical approximation for multi-segment flexible curvilinear robots. Robotics
**2019**, 8, 48. [Google Scholar] [CrossRef] - Xanthidis, M.; Kyriakopoulos, K.J.; Rekleitis, I. Dynamically efficient kinematics for hyper-redundant manipulators. In Proceedings of the 24th Mediterranean Conference on Control and Automation, MED 2016, Athens, Greece, 21–24 June 2016; pp. 207–213. [Google Scholar] [CrossRef]
- Gallardo-Alvarado, J.; Tinajero-Campo, J.H.; Sánchez-Rodríguez, Á. Kinematics of a configurable manipulator using screw theory. Rev. Iberoam. Automática E Inform. Ind.
**2020**, 18, 58. [Google Scholar] [CrossRef] - Zaplana, I.; Hadfield, H.; Lasenby, J. Closed-form solutions for the inverse kinematics of serial robots using conformal geometric algebra. Mech. Mach. Theory
**2022**, 173, 104835. [Google Scholar] [CrossRef] - Kane, S.N.; Mishra, A.; Dutta, A.K. Towards extending Forward Kinematic Models on Hyper-Redundant Manipulator to Cooperative Bionic Arms. J. Phys. Conf. Ser.
**2016**, 755, 011001. [Google Scholar] [CrossRef] - Martín-Barrio, A.; Roldán-Gómez, J.J.; Rodríguez, I.; Del Cerro, J.; Barrientos, A. Design of a hyper-redundant robot and teleoperation using mixed reality for inspection tasks. Sensors
**2020**, 20, 2181. [Google Scholar] [CrossRef] [PubMed] - Rao, P.; Peyron, Q.; Lilge, S.; Burgner-Kahrs, J. How to Model Tendon-Driven Continuum Robots and Benchmark Modelling Performance. Front. Robot. AI
**2021**, 7, 1–20. [Google Scholar] [CrossRef] [PubMed] - Xu, F.; Wang, H. Soft Robotics: Morphology and Morphology-inspired Motion Strategy. IEEE/CAA J. Autom. Sin.
**2021**, 8, 1500–1522. [Google Scholar] [CrossRef] - Chirikjian, G.S. Theory and Applications of Hyper-Redundant Robotic Manipulators. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, USA, 1992. Available online: https://resolver.caltech.edu/CaltechETD:etd-11082006-132210 (accessed on 30 June 2022).
- Robotics, O. Series II, X125 System. 2015. Available online: https://www.ocrobotics.com/technology-/series-ii-x125-system/ (accessed on 15 May 2022).
- Hannan, M.W. Theory and Experimentation with an ‘Elephant’s Trunk’ Robotic Manipulator; Oxford University Press: Oxford, UK, 2002; p. 649. [Google Scholar]
- Muñoz Sánchez, E. Construccion y Control de un Robot Continuo de Cables con Discos Pivotantes. Bachelor’s Thesis, Universidad Politécnica de Madrid, Madrid, Spain, 2022. Available online: https://oa.upm.es/69811/ (accessed on 2 July 2022).
- Terrile, S. Soft Robotics: Applications, Desing and Control. Ph.D. Thesis, Universidad Politécnica de Madrid, Madrid, Spain, 2021. [Google Scholar] [CrossRef]
- Martin-Barrio, A.; Terrile, S.; Diaz-Carrasco, M.; del Cerro, J.; Barrientos, A. Modelling the Soft Robot Kyma Based on Real-Time Finite Element Method. Comput. Graph. Forum
**2020**, 39, 289–302. [Google Scholar] [CrossRef] - Paez-Grandos, D.; Gualdron, O.E.; Valencia Ramón, J.L. Aprendizaje de la cinemática en robots redundantes utilizando mapas de bézier. Rev. Tecnol. J. Technol.
**2015**, 14, 23–32. [Google Scholar] [CrossRef] - Dehghani, M.; Moosavian, S.A.A. Modeling and control of a planar continuum robot. In Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, AIM, Budapest, Hungary, 3–7 July 2011; pp. 966–971. [Google Scholar] [CrossRef]
- Tommy Wang, L.C.; Cheng Chen, C. A Combined Optimization Method for Solving the Inverse Kinematics Problem of Mechanical Manipulators. IEEE Trans. Robot.
**1991**, 7, 489–499. [Google Scholar] [CrossRef] - Martín, A.; Barrientos, A.; Del Cerro, J. The natural-CCD algorithm, a novel method to solve the inverse kinematics of hyper-redundant and soft robots. Soft Robot.
**2018**, 5, 242–257. [Google Scholar] [CrossRef] - Denavit, J.; Hartenberg, R. A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices. ASME J. Appl. Mech.
**1955**, 22, 215–221. [Google Scholar] [CrossRef] - Barrientos Cruz, A.; Peñín, L.F.; Balaguer, C.; Aracil, R. Fundamentos de Robótica; McGraw-Hill: New York, NY, USA, 2007. [Google Scholar]
- Simaan, N.; Taylor, R.; Flint, P. A dexterous system for laryngeal surgery. In Proceedings of the IEEE International Conference on Robotics and Automation, ICRA ’04, New Orleans, LA, USA, 26 April–1 May 2004; Volume 2004, pp. 351–357. [Google Scholar] [CrossRef]
- Neppalli, S.; Csencsits, M.A.; Jones, B.A.; Walker, I.D. Closed-form inverse kinematics for continuum manipulators. Adv. Robot.
**2009**, 23, 2077–2091. [Google Scholar] [CrossRef] - Hannan, M.W.; Walker, I.D. Kinematics and the implementation of an elephant’s trunk manipulator and other continuum style robots. J. Robot. Syst.
**2003**, 20, 45–63. [Google Scholar] [CrossRef] [PubMed] - Jones, B.A.; McMahan, W.; Walker, I.D. Practical kinematics for real-time implementation of continuum robots. In Proceedings of the 2006 IEEE International Conference on Robotics and Automation, ICRA, Orlando, FL, USA, 15–19 May 2006; Volume 2006, pp. 1840–1847. [Google Scholar] [CrossRef]
- Camarillo, D.B.; Carlson, C.R.; Salisbury, J.K. Configuration Tracking for Continuum Manipulators with Coupled Tendon Drive. Springer Tracts Adv. Robot.
**2009**, 54, 271–280. [Google Scholar] [CrossRef] - Orin, D.E.; Schrader, W.W. Efficient Computation of the Jacobian for Robot Manipulators. Int. J. Robot. Res.
**1984**, 3, 66–75. [Google Scholar] [CrossRef] - Chembrammel, P.; Kesavadas, T. A new implementation for online calculation of manipulator Jacobian. PLoS ONE
**2019**, 14, 1–16. [Google Scholar] [CrossRef] [PubMed] - Webster, R.J.; Swensen, J.P.; Romano, J.M.; Cowan, N.J. Closed-Form Differential Kinematics for Concentric-Tube Continuum Robots with Application to Visual Servoing. Springer Tracts Adv. Robot.
**2009**, 54, 485–494. [Google Scholar] [CrossRef] - Barrientos-Diez, J.; Dong, X.; Axinte, D.; Kell, J. Real-Time Kinematics of Continuum Robots: Modelling and Validation. Robot. Comput.-Integr. Manuf.
**2021**, 67, 102019. [Google Scholar] [CrossRef] - Su, H.J. A pseudorigid-body 3r model for determining large deflection of cantilever beams subject to tip loads. J. Mech. Robot.
**2009**, 1, 1–9. [Google Scholar] [CrossRef] - Chen, G.; Xiong, B.; Huang, X. Finding the optimal characteristic parameters for 3R pseudo-rigid-body model using an improved particle swarm optimizer. Precis. Eng.
**2011**, 35, 505–511. [Google Scholar] [CrossRef] - Huang, S.; Meng, D.; Wang, X.; Liang, B.; Lu, W. A 3D Static Modeling Method and Experimental Verification of Continuum Robots Based on Pseudo-Rigid Body Theory. In Proceedings of the IEEE International Conference on Intelligent Robots and Systems, Macau, China, 4–8 November 2019; pp. 4672–4677. [Google Scholar] [CrossRef]
- Yuan, H.; Zhou, L.; Xu, W. A comprehensive static model of cable-driven multi-section continuum robots considering friction effect. Mech. Mach. Theory
**2019**, 135, 130–149. [Google Scholar] [CrossRef] - Altenbach, H.; Eremeyev, V.A.; Morozov, N.F. On the Influence of Residual Surface Stresses on the Properties of Structures at the Nanoscale. In Surface Effects in Solid Mechanics 2013; Springer: Berlin/Heidelberg, Germany, 2013; pp. 21–32. [Google Scholar] [CrossRef]
- Jones, B.A.; Gray, R.L.; Turlapati, K. Three dimensional statics for continuum robotics. In Proceedings of the 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2009, St. Louis, MO, USA, 10–12 October 2009; pp. 2659–2664. [Google Scholar] [CrossRef]
- Renda, F.; Laschi, C. A general mechanical model for tendon-driven continuum manipulators. In Proceedings of the IEEE International Conference on Robotics and Automation, St Paul, MN, USA, 14–18 May 2012; pp. 3813–3818. [Google Scholar] [CrossRef]
- Martín-Barrio, A.; del Cerro, J.; Barrientos, A.; Hauser, H. Emerging behaviours from cyclical, incremental and uniform movements of hyper-redundant and growing robots. Mech. Mach. Theory
**2021**, 158, 104198. [Google Scholar] [CrossRef] - Merlet, J.P. Parallel robots. In Part of Series: Solid Mechanics and Its Applications; Springer: Berlin/Heidelberg, Germany, 2006; Volume 128, pp. 1–413. ISBN 978-1-4020-4132-7. [Google Scholar] [CrossRef]

**Figure 1.**Three examples of CDHR exemplifying each category. (

**a**) MACH-I: a discrete CDHR. (

**b**) Pilory: a constant curvature CDHR. (

**c**) Ruan: a soft CDHR.

**Figure 6.**Example of Denavit–Hartenberg application for a two 2-DOF joint robot. (

**a**) Schematic figure for the application of the Denavit–Hartenberg algorithm to a simple kinematic joint chain. (

**b**) Denavit–Hartenberg parameters for such robot.

**Figure 10.**Application of Denavit–Hartenberg’s method to a robotic section of constant curvature. (

**a**) ${T}_{1}$. (

**b**) ${T}_{2}$. (

**c**) ${T}_{3}$. (

**d**) ${T}_{4}$. (

**e**) ${T}_{5}$. (

**f**) Section’s parameter for Denavit–Hartenberg’s method.

**Figure 11.**Auxiliary figures for the cables’ kinematic relationships. (

**a**) Three-dimensional image of the proposed robot. (

**d**) Geometric hypothesis to obtain the kinematic equations for the cables. (

**c**) Bird’s-eye view of the robot’s section, including the curvature plane projection.

**Figure 19.**Defining virtual sections for inverse kinematic modelling. (

**a**) Differences between work spaces. (

**b**) Discreet sections for inverse kinematic modelling.

**Figure 21.**Graphical summary for the Natural-CCD algorithm, produced based on [26].

**Figure 23.**Details of MACH-I that suggest the discreet CDHR classification. (

**a**) Detail of joints and straight cables between sections. (

**b**) Detail of sections and cables with curved robot.

**Figure 24.**DH method for MACH-I. (

**a**) Schematic figure for the application of the Denavit–Hartenberg algorithm to MACH-I. (

**b**) Denavit–Hartenberg parameters for such a robot.

**Figure 29.**Equations for the Pilory robot, as deduced in Section 4.

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**MDPI and ACS Style**

Cerrillo, D.; Barrientos, A.; Del Cerro, J.
Kinematic Modelling for Hyper-Redundant Robots—A Structured Guide. *Mathematics* **2022**, *10*, 2891.
https://doi.org/10.3390/math10162891

**AMA Style**

Cerrillo D, Barrientos A, Del Cerro J.
Kinematic Modelling for Hyper-Redundant Robots—A Structured Guide. *Mathematics*. 2022; 10(16):2891.
https://doi.org/10.3390/math10162891

**Chicago/Turabian Style**

Cerrillo, Diego, Antonio Barrientos, and Jaime Del Cerro.
2022. "Kinematic Modelling for Hyper-Redundant Robots—A Structured Guide" *Mathematics* 10, no. 16: 2891.
https://doi.org/10.3390/math10162891