This article was submitted to Robotic Control Systems, a section of the journal Frontiers in Robotics and AI

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Continuum robots are a type of robotic device that are characterized by their flexibility and dexterity, thus making them ideal for an active endoscope. Instead of articulated joints they have flexible backbones that can be manipulated remotely, usually through tendons secured onto structures attached to the backbone. This structure makes them lightweight and ideal to be miniaturized for endoscopic applications. However, their flexibility poses technical challenges in the modeling and control of these devices, especially when closed-loop control is needed, as is the case in medical applications. There are two main approaches in the modeling of continuum robots, the first is to theoretically model the behavior of the backbone and the interaction with the tendons, while the second is to collect experimental observations and retrospectively apply a model that can approximate their apparent behavior. Both approaches are affected by the complexity of continuum robots through either model accuracy/computational time (theoretical method) or missing complex system interactions and lacking expandability (experimental method). In this work, theoretical and experimental descriptions of an endoscopic continuum robot are merged. A simplified yet representative mathematical model of a continuum robot is developed, in which the backbone model is based on Cosserat rod theory and is coupled to the tendon tensions. A robust numerical technique is formulated that has low computational costs. A bespoke experimental facility with precise automated motion of the backbone via the precise control of tendon tension, leads to a robust and detailed description of the system behavior provided through a contactless sensor. The resulting facility achieves a real-world mean positioning error of 3.95% of the backbone length for the examined range of tendon tensions which performs favourably to existing approaches. Moreover, it incorporates hysteresis behavior that could not be predicted by the theoretical modeling alone, reinforcing the benefits of the hybrid approach. The proposed workflow is theoretically grounded and experimentally validated allowing precise prediction of the continuum robot behavior, adhering to realistic observations. Based on this accurate estimation and the fact it is geometrically agnostic enables the proposed model to be scaled for various robotic endoscopes.

Continuum robots are inspired by nature and enable positioning of an end effector via a backbone that bends continuously along its length

This category of robots are characterized by their high manoeuvrability in unstructured and confined environments

Nonetheless, this class of robots pose a significant challenge in terms of control. The issue lies in the manner of which such a flexible structure can be described. In classical robotics the mechanisms are rigid and have well defined shapes enabling an analytical description of their kinematics. For continuum robots alternative theoretical methods have been proposed. One of the most prominent approaches is a constant curvature approach, as used by

A variable curvature approach, which is also based on a geometric description, has also been applied to a continuum robot by

Additionally the continuum robot and the actuation method needs to be coupled together efficiently to give reliable results. For tendon driven continuum robots,

This work focuses on investigating a direct method of solving a continuum robot model based on Cosserat theory, chosen as it is independent of any specific discretization scheme and so is able to predict the behavior of the backbone under large deformation more accurately. A simplified approach for coupling the tendon tensions to the backbone is utilized to reduce the necessary numerical computations. A robust numerical technique is developed and bespoke code is formulated. This approach gives a compromise between the computation time and the model accuracy, however it is sufficiently precise for practical implementation with predictions representing the backbone curvature and position well. Additionally, the bespoke experimental facility is novel as it can be controlled automatically by regulating the tendon tension through motors and strain gauges to allows the backbone to be moved in a smooth and orderly motion. The model is built about the assumption that by adjusting the boundary condition within the classical Cosserat rod model to suitably account for the tendon loads which act on the continuum robots, as well as external loads. The research presented here is investigating this premise for a 3D continuum robot, with the model developed in

A continuum robot backbone model is developed, based on the classical nonlinear Cosserat rod theory. The backbone can be described effectively by an elastic rod in three dimensions, which accounts for the nonlinearity of rod bending [see

The centroid position along the backbone arc length

Backbone showing the backbone centroid at an arc length

The backbone internal force and moments in global coordinates are given by

Therefore, the set of governing equations for the global coordinates, formed from

The experimental set-up for evaluating the model can be seen in

Constructed continuum robot rig with backbone, actuated by tendons running through support disks by stepper motor and lead screws. Cantilever load-cells measure the tendon tension and OptiTrack markers are included for backbone position detection.

Acrylic circular support discs were used to minimize friction between the tendon and support discs; additionally Teflon spray was used to further reduce friction. A total of 13 support discs of 20 mm diameter and 1.5 mm thickness were used and spaced 20 mm apart along the backbone, such that an optimum ratio of support disc spacing to tendon offset from backbone of 2.5 mm was achieved as proposed by

Each tendon is actuated independently, through an assembly comprising of a lead screw mechanism driven by a stepper motor. The stepper motors have a resolution of

A cantilever load cell is fixed to the carriage, which in turn is connected to the tendon, causing it to move through a set of forces and a moment offering a loading feedback for each of the tendons. A 1 kg load cell (Phidgets, Canada) was chosen to identify the tension as a compromise between the maximum expected tendon tension and resolution.

The position of the backbone was recorded using a contactless sensor, namely OptiTrack (Target3D, United Kingdom) motion capture system; this eliminated any error that may be associated with contact sensors. Reflective markers of 6 mm diameter were incorporated on the face of every second support disc. Seven cameras were used, to ensure uninterrupted tracking of the markers, with a maximum root mean square positional error of 0.105 mm achieved. The initial position of the backbone, before any tendon loading, was identified once a stable tension reading of the desired value

The backbone model in

Single tendon continuum robot actuation (

When the backbone is undeformed, it is assumed moments

Based on the boundary conditions in

Schematic representation of the closed-loop control for tendon tensioning. The demand is calculated from the model and transmitted to the Arduino Uno and hold via a ZOH. The Control Loop runs at 100 Hz and runs a PID controller based on the error between the demand and the Force Feedback. The control signal are pulses for the Motor driver that generates the current to run the stepper motor. The lead-screw mechanism applies the tension to the tendon via the cantilever load cell. The measurements from the latter are being amplified and sampled by from the ADC at the rate of the Control Loop.

A PID controller architecture was selected given the overall behavior of the system. Although PD is the preferred method in continuum robots

Initially, the tuning procedure for the PID controller followed the classic Ziegler-Nichols method based on a step response of a single tendon being tensioned from 0 to 3 N. This procedure resulted in zero steady state error and minimal oscillations with the calculated PID gains for the proportional, integral and derivative terms of

Continuum robot control curves for two PID tuning techniques showing the tendon tension against time.

Therefore, to reduce the system rise time to 25 s, a two-stage proportional controller was implemented. This is similar to a divide and conquer gain-scheduling controller

A set of 15 tension combinations are used for all experimental evaluations. As tendons 1 and 3 act in the horizontal plane and tendons 2 and 4 act in the vertical plane, tendon 3 remained slack always and combinations of tendon 1 with either tendon 2 or 4 were used. This was to examine the effect of gravity, as tendons 2 and 4 acting against or with gravity, respectively. To minimize the backbone stress and enable reliable data for comparison with the model, only one tendon from each plane was tensioned. It is assumed that if tendon 1 remained slack, but combinations of tendon 3 with either tendon 2 or 4 were examined, solutions would be the same but mirrored along the central vertical position.

Each tendon could be tensioned with a value of 0, 1.5 or 3 N. The combinations were divided into three groups, a) In-Plane, b) Semi-Out-of-Plane and c) Out-of-Plane which can be seen in

Overlay of experimental tendon tension combinations. Three different regions are defined; In-Plane, Semi-Out-of-Plane and Out-of-Plane data points.

When conducting the experiments, the error, defined as the distance between the predicted and realized tip location, is quoted as a percentage of the backbone length, enabling comparison of the model accuracy with those approaches already available in literature.

In order to evaluate the validity of the results produced by the experimental set-up, the effect of tendon-disk interface friction on the system repeatability was investigated. The tension in tendons 1 and 3 were kept at 0 N while the tension in tendons 2 and 4 were changed, but the tension in only one tendon was non-zero at any given time, to a value of either 1.5 N or 3 N. Experiments were carried out where these values were achieved by increasing the tension from 0 N as well as by over tensioning the tendon to 5 N before decreasing it to the desired value which allowed for the removal of tendon slack from within the support discs. The relative distance of the backbone tip from the origin was recorded and analyzed for each repeat.

The results of the relative distance of the backbone tip from the origin for different tensions in the tendons in the vertical plane are shown in

Tip displacement from the global origin for tensions of 0, 1.5 or 3 N in tendon two or 4 (only one tendon is tensioned at a time) in the vertical plane when the desired value is achieved by increasing the tension from 0 N (red) or decreasing from 5 N (blue).

Standard deviation of the backbone tip location for different tensions when increasing from 0 N or decreasing from 5 N.

Vertical tendon tension (N) | |||||
---|---|---|---|---|---|

Tension (N) | Tendon 4 | Tendon 2 | |||

3 | 1.5 | 0 | 1.5 | 3 | |

Standard deviation when increasing from 0 N (mm) | 2.94 | 2.22 | 2.60 | 2.18 | 2.59 |

Standard deviation when decreasing from 5 N (mm) | 7.77 | 3.48 | 3.29 | 1.32 | 9.43 |

The impact of the support disks and the OptiTracker markers on the modulus of elasticity of the rod is investigated, to enable model validation. Based on these measurements the updated modulus is used to perform the calculation for the model and used subsequently. To identify the modified Young’s Modulus, values of

Total positional tip error between the theoretical prediction and experimental result for increasing Young’s Modulus.

In this fist set of experiments the accuracy of the predictions from the derived model is investigated with no end effector. The updated Young’s Modulus from

Predicted and realized backbone position for the

Mean positional error between the OptiTrack observations and model predictions for the three tension groups. The average result is included for overall comparison.

Mean positional tip location error together with the standard deviation for the three different regions.

Region | Error (mm) | Error (% of length) | Standard deviation (mm) |
---|---|---|---|

In-plane | 7.14 | 2.98 | 3.21 |

Semi-out-of-plane | 10.9 | 4.56 | 7.73 |

Out-of-Plane | 12.1 | 5.05 | 3.30 |

All | 9.48 | 3.95 | 4.99 |

In practice a continuum robot will have an end effector attached at the backbone tip location, therefore, investigations are carried out in which a vertical end load is acting on the tip of backbone. In this case the load was attached to the end support disk, where tendon 4 passes through it. The examined tensions in the different tendons are similar to those previously studied but with a reduce number of combinations of only 0 and 3 N for tendons 1, 2 and 4 with tendon 3 remaining slack. Four different loads were applied to the backbone tip of

Predicted and realized backbone position for the

Mean positional error for different OptiTrack sensor backbone locations for increasing end load; 0, 96.14, 189.92, 293.32 N.

Mean positional tip location error together with the standard deviation for increasing end load.

End load (N) | Error (mm) | Error (% of length) | Standard deviation (mm) |
---|---|---|---|

0 | 7.88 | 3.28 | 3.72 |

96.14 | 25.28 | 10.53 | 15.38 |

189.92 | 35.75 | 14.89 | 31.16 |

293.32 | 100.36 | 41.81 | 132.86 |

All | 42.32 | 17.63 | 73.33 |

The mean tip error and standard deviation across all datasets with zero added mass was found to be 3.95% and 4.99 mm respectively, which compares favourably with similar continuum robot validation studies carried out by

When investigating the effects of increasing or decreasing the tension to the desired tendon tension value, it was found that decreasing to the desired tension results in a significantly increased standard deviation of 7.77 and 9.43 mm for a tension of 3 N in tendon 4 and 2, respectively, whereas in the case of increasing the tension to the desired value, the maximum standard deviation is 2.94 and 2.59 mm, respectively. It is proposed that this situation arises because although the slack is removed in the case of over tensioning, the stiction/friction may cause a higher than desired tension remaining within the backbone past the base support disc. This higher energy state that the tendon has, is more likely to overcome the stiction/friction limit of the support disc and therefore more likely to randomly find a state of equilibrium. Therefore, the Teflon coated tendons used within the experimental facility still have significant friction effects. It would be advantageous to account for the effects of stiction/friction and slack within the model, as this would improve the repeatability of the continuum robot control as the desired tip position will most commonly be reached from an undeformed backbone position. The friction occurring between the tendons and the routing holes could be modeled as Coulomb/dry friction, and will effect the moment applied to the backbone tip

Good agreement between the predicated and realized positions is achieved when there is a 0 N end load. However, for the cases when the end load is non-zero, there is an error between the predicted and realized results, which increases with end load. For tensions in only one plane, the errors are still relatively small, however when tendons in two planes are tensioned the error becomes larger. This discrepancy could be due to the backbone bending away from a single plane and causing the end load to produce an additional moment which is not replicated within the model. Therefore, it is important to know the center of mass of any tools to be applied to the end of a surgical robot as the tip position results can be significantly effected. Nevertheless, the mass of a surgical tool is likely to be small and as such the model would still be a reasonable representation of the real system.

In this work theoretical and experimental descriptions of an endoscopic continuum robot are examined. The representative model of the continuum robot configuration was based on Cosserat rod theory including gravitational effects, end loads and end moments produced by the tendons. A robust and efficient numerical solver was implement and validation was provided through the bespoke experimental facility. The experimental system actuates four tendons via motorized lead screw mechanisms that are instrumented load cells. To minimize the rise time of tendon tension, while avoiding steady state oscillations, a switching PID control methodology is implemented. An OptiTrack system (a contactless system) was used to determine the backbone deformation.

The tendon tension range, end load range together with the repeatability of the experimental facility were examined. A model parameter calibration gave an optimized backbone modulus of elasticity of 168 GPa, and model predictions gave the backbone tip position to within 9.48 mm of the experimental data, or 3.95% of the backbone length over the full tendon tension range examined. The In-Plane tendon tensions gave the theoretical and realized position being closer than the Out-of-Plane tendon tensions with calculated tip position errors of 2.98 and 5.05%, respectively. Despite the increased tendon slack introduced within the rig, increasing to the desired tension was significantly more repeatable than the decreasing method for high tendon tension cases with maximum recorded standard deviations of 2.94 and 9.43 mm, respectively. The unloaded model was observed to predict very similar backbone deformations to those from the experimental data. Results show that the model is sufficiently precise for practical implementations with predictions representing the backbone curvature and position well, showing a good compromise between computationally time and accuracy.

This investigation has provided an overview of the effect of the frictional effects of the tendons in the body of a continuum robot. Moreover, the proposed modeling and numerical approach in describing the kinematics of the robot have been validated and follow the experimental observations. The next step is to translate these findings into a smaller scale system, applying these outcomes to an endoscopic task together with evaluating the proposed approach with a system and corresponding controller that has a higher dynamic response. By taking into account the effect and behavior of the actuation tendons the precision and accuracy of the final system can be greatly improved. Moreover, the proposed methodology is applicable to multi-segmented continuum robots, since the model can be applied in a cascade fashion between segments with appropriate boundary conditions and minimum computational load.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

In this work AI and IG designed the experimental system. AI constructed the experimental system and conducted the experiments. AI and NB developed and implemented the model described here. AI produced early technical reports and IG and NB have written the current manuscript.

This work was funded by a University of Bath small internal experimental grant.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at: