Fase de laboratorio

The bending performance of the models have been highlighted in Figure 5.3. Here, the Con- stant Curvature Model is plotted next to the Modified Model, for an angle of 90 degrees. These models are compared to the physical representation of the bending module, represented in the left image of the figure. This approximation is based on the centerline selection of the bent module. In the right image, the Modified Model is adjusted to approach the physical response, which is done by scaling the constant curvature section of the Modified Model. The length of the Constant Curvature Model in the right image is scaled as well.

The results of implementing both models in the simulation algorithm have been compared in the next chapter, in for example Figure 6.13 and Figure 6.19.

5.4 Discussion

As the currently addressed models are mere approximations of the endoscope module beha- viour, some remarks must be taken into account when considering its implementation. 5.4.1 Physical Module vs. Modified Model

When regarding the left image of Figure 5.3, the shape of the physical response seems to be more similar with respect to the Modified Model compared to the Constant Curvature Model. The approximation of the physical module response also contains rather straight sections at the base and the tip, similar to the Modified Model.

0 1 2 3 4 X (cm) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Y (cm)

Physical Module vs. Models ( is 90 )

Constant Curvature Assumption Modified Model

Physical Module Approximation

0 1 2 3 4 X (cm) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Y (cm)

Modified Model: Scaled Constant Curvature Length

Physical Module Approximation Modified Model (Scaled) Constant Curvature Model (Scaled)

Figure 5.3: Comparison between model(s) and physical module response. The left illustration shows the responses of the physical module, the Constant Curvature Model, and the Modified Model. On the right, the physical approximation is compared to the Modified Model response with an increased size of the constant curvature section (from 3 cm to 4.5 cm), as well as to a scaled version of the Constant Curvature Model (from 5.5 cm to 6.5 cm).

It must be mentioned that the approximation of the physical response is constructed from manual selection of the centerline of the module during bending, which is subject to errors in representation. Due to a failed attempt using image processing, in which the module skel- eton was deformed, and the lack of time available, this method for choosing the correspond- ing centerline is chosen. Therefore, the comparison serves merely as an indication of physical module growth and an increased similarity with respect to the Modified Model, as little can be said about the actual differences.

Furthermore, it must be taken into account that this comparison is solely based on one scen- ario, which is the 90 degrees bending. More scenarios should be examined to be able to confid- ently state the claim of approximating the behaviour rather closely, together with defining the lengthening behaviour throughout the bending.

5.4.2 Lengthening Principle

As could be seen in Figure 5.3, there seems to be a misfit between the models and the actual module response, mainly regarding the size. Arguing from a physical point of view, module bending occurs due to chamber expansion, which is not accounted for in the models itself. The right image of Figure 5.3 is showing the Modified Model response when the length of the constant curvature section would be increased from 3 cm to 4.5 cm, an increase of 50 %. The scaled version of the Constant Curvature Model has a different length compared to the Modi- fied Model, as a bigger length would result in a worse approximation of the physical response. The Modified Model seems to approximate the physical response better compared to the Con- stant Curvature Model.

The phenomenon of lengthening of the module during bending is thus far excluded from the bending approximation, although it is affecting the response quite a lot. Due to lack of time, this was not researched in detail. Using the data from the EM tracker, the module (tip) po- sition could be mapped and compared the (tip) position of Modified Model, to examine the difference. By means of adjusting the length of the constant curvature section, which is the principal section for expansion, an estimate can be defined for lengthening per bending angle. This would hold under the condition that the remaining part of the model represents the be- haviour rather well. These results could be integrated in the current model as well. During

CHAPTER 5. KINEMATIC MODEL: BENDING MODULE 31

experimentation, it was found that the module increased in size more when bending with two chambers compared to one. This also has to be taken into account.

The lengthening has not been incorporated in the implemented model for the rest of the study, and remains a recommendation for future research.

5.5 Conclusion

Regarding the model of the bending endoscope module, two approaches have been con- sidered; the Constant Curvature Model and the Modified Model. The Modified Model con- tains a constant curvature section with the length of a module chamber, surrounded by two rigid blocks that do not attribute to the further bending. The Modified Model seems to have a more similar shape with respect to the physical bending scenario than the Constant Curvature Model. Regarding physical inspection, the Modified Model is also deemed a better approxima- tion. Lengthening of the modules seems to affect the bending response, which should be focus for future research. The Modified Model is eventually taken into account in the Endoscope Kinematic Planner, as well as integrated into the final experiments regarding path planned in- sertion.

6 Path Planning Algorithms

The knowledge obtained from the fabricating and characterizing the modules (Stage A1 and A2), in combination with the defined model of the endoscopic module bending behaviour (Stage B), serve as a basis for the development of algorithms that define the path for the endo- scope to follow, as well as determine and store the necessary bending angles to so. The angles, together with the discovered pressure-bending relationships, will define the necessary pres- sures for the physical endoscope response.

The following sections describe the algorithms developed to create a path throughout the en- vironment and to simulate the endoscope movement throughout the scenario when following this path. These two separate tasks, defining a path and simulating the endoscope movement when following it, are represented by the Path Planning algorithm and the Endoscope Kin- ematic Planning algorithm (also referred to as Path Tracker algorithm), which are stages C1 and C2, respectively. This chapter covers the answer to sub-question 4.

6.1 Path Planner

To determine the path the soft robotic endoscope has to follow, path planning is used. This section will treat the working principle of the currently used Path Planner algorithm, which starts with the working principle and is ending with simulations of the utilized scenarios. As has been addressed before, the RRT-algorithm is chosen for planning the path throughout the pre-defined scenarios.

6.1.1 Elaboration of the Principle

To eventually let an endoscope follow a path in simulation, first a path needs to be acquired. Knowing the environment and the dimensions of the endoscopic modules, a path can be de- termined. The goal is to let an algorithm define this path automatically, while also considering several criteria that come with the control of the soft robotic endoscope inside an environment. Some general requirements need to be fulfilled for the algorithm:

• A path should be defined between the starting point and the target point. • This path should be collision-free.

In general, path planning algorithms also take care of the mapping of the obstacles inside the given environment, as well as include obstacle margins, such that the dimensions of the mov- ing object are taken into consideration. Several criteria concerning the current study applica- tion should be met as well:

• The dimensions of the endoscope should be taken into account in the obstacle margins. • The endoscope dimensions should fit in the environment.

• The path should not contain corners which the endoscope should not be able to make. To realize this, the RRT principle is utilized, which has been described in Section 2.2.1. The 2D path planning algorithm that has been used throughout this study is inspired by the work of Sai Vemprala (Vemprala, 2017), who based his work on the work of LaValle (LaValle, 1998) and Karaman et al. (Karaman and Frazzoli, 2011). The code consists of an RRT* algorithm, which is a variation of RRT and has been discussed earlier. The work of Sai Vemprala (Vemprala, 2017) has been adjusted to fit the conditions of the current study.

CHAPTER 6. PATH PLANNING ALGORITHMS 33

More details regarding the separate sections of the algorithm can be found in Chapter H of the Appendix.

Modifications to the Algorithm

Several scenarios have been predefined, representing parts of the endoscopic environment. These scenarios are shaped using obstacle coordinates, which limit the growth of the tree in certain directions. The starting point and the target point also have been defined beforehand. To increase speed of computation, the points that spawned inside the obstacles were disreg- arded, just like the approach in the study of Noreen et al. (Noreen et al., 2016b). Furthermore, a target region was selected around the target point, which also is mentioned in the study of Alterovitz et al. (Alterovitz et al., 2011). In this way, the goal is reached earlier. To limit the angles found inside the created tree, and thus the eventual angles inside the path, an algorithm is used to define the angle and quit when the potential angle surpasses the threshold.

To determine the angle, the following formula was utilized:

θtree=cos−1 µ (A·B) ||A||||B|| ¶ (6.1) WhereAand Bare respectively the vectors from theqnear to the new point and the qnear to the previous point. The lengths of these connections are determined by taking the norm. A multiplication of these vector lengths serves as the denominator, whereas the numerator is the dot product of the two vectors. A representation of the angle determination can be found in Figure 6.1.

6.1.2 Simulations

To test the applicability of the path planner, two scenarios have been developed. These scen- arios are basic representations of the segments inside the human colon and can be seen as the two main alternative bending scenarios. The shape of the endoscope environments is kept rather basic; in reality, the colon consists of flexible curved corners and does not really contain

𝒒𝒏𝒆𝒂𝒓 𝒒𝒏𝒆𝒘 A B 𝜃′ 𝜃 𝜃= 180 −𝜃′ = 180 − cos−1 𝑨∙𝑩 𝑨 𝑩

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Figure 6.2:The RRT* algorithm performing the search for a path in Scenario 1. Obstacles are indicated in green, the target region in red. The coordinates of the path are transferred to the Endoscope Kinematic Planner.

any straight segments. Furthermore, the environment of the colon is rather dynamic, meaning that the consistency also shifts over time. It was attempted to find the dimensions of the colon in literature and use them as a reference for the sizes of these scenarios. Several sources were found, investigating the dimensions of the colon (Witmer, 2007) (Sadahiro et al., 1992)(Houn- nou et al., 2002). Based on Witmer (Witmer, 2007), one could define the average colon diameter to be between 4 cm and 8 cm, which is why an average diameter of 6 cm is chosen.

Scenario 1 consists of a double-cornered bend, which requires bending in only one direction. The second scenario, Scenario 2, includes bending towards both sides. Simulations of both scenarios will be addressed below.

Scenario 1

A representation of Scenario 1 can be seen in Figure 6.2. The Path Planner algorithm is given the dimensions of the obstacles, together with information regarding the module width and length, as well as the potential bending capacity. Furthermore, the position of the starting point as well as the target has been given. Based on this information, the algorithm finds a suitable path. In the figure, the dark green indicates the obstacles, whereas the lighter green is representing the module dimensions. At this moment in the image, the target is still searched for. When the target is reached, the path is traced back towards the base. The black lines indicate the tree growth, whereas the green lines show the RRT* property, attempting to reduce the distance cost.

The resulting path with respect to the centerline, which is created by the Path Planning al- gorithm, is shown in Figure 6.3. What stands out from Figure 6.3 is the deviations of the planned path with respect to the corners of the centerline. Due to the angle limitations set in the Path Planner, these corners are avoided. This benefits the eventual behaviour of the endoscope dur- ing simulation, as this promotes more gradual transitions.

CHAPTER 6. PATH PLANNING ALGORITHMS 35 4 6 8 10 12 14 16 18 X (cm) 0 1 2 3 4 5 6 7 8 9 10 Z (cm)

Planned Path and Centerline: Scenario 1

Centerline Planned Path

Figure 6.3:The planned path with respect to the centerline of Scenario 1. As can be seen, the planned path avoids the sharp corners of the centerline.

Scenario 2

Scenario 2 is rather similar to Scenario 1, but it has the second corner in the opposite direction. Figure 6.4 shows the building of the path inside Scenario 2. Similar settings were used.

The results of this scenario can also be converted in a comparison between centerline and planned path, as is depicted in Figure 6.3.

Duration

The simulations are performed using MATLAB on a Dell Latitude E6540 with 8 GB RAM and 2.7 GHz CPU. Regarding Scenario 1, it takes the algorithm on average around 5.9±1.2 seconds to compute a path for this scenario (N=10). For Scenario 2, the average duration is about 10.2±1.6 seconds to realize a path (N=10). Simple ways to improve speed of computation are by building the algorithm using faster platforms such as Python and C++, by using a faster computer or by removing visual computation during simulation.

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