Kinematic Modeling of Continuum Robots Using Artificial Intelligence Tools

Abstract

Continuum robots are progressively dwarfing traditional rigid robots in many areas, due to their outstanding flexibility and easy adaptation to almost any kind of trajectory, yet their kinematic modeling is presented as a drawback, in particular with multi-sections and variable curvature, which curbs the full benefit of these robots. In this context, this chapter presents the description of continuum robots and their kinematic modeling in detail. Firstly, the description of the behavior and forward kinematic models are presented for continuum robots with constant and variable curvatures. Then, to calculate the inverse kinematic model of continuum robots, artificial neural networks, and metaheuristic optimization techniques have been used to obtain the inverse kinematic model of single and multi-section continuum robots, respectively. To validate the utility of the proposed techniques, different trajectories are considered to be followed by considered robots. It is found that the developed techniques are powerful tools to deal with the high complexity of nonlinear equations, in particular when it comes to solving the inverse kinematic model of multi-section continuum robots with variable curvature. To have a closer look at the efficiency of the developed techniques during the follow-up of the proposed trajectories, 3D simulations have been carried out with different configurations.

1. Introduction

Robotics increasingly attract researchers’ interest over the past few decades, as it has become the solution to many problems in real world applications. Especially, when it comes to difficult and complex tasks that humans cannot handle or in industrial environments that may be hazardous to humans (for example near high temperatures and nuclear reactors or when dealing with chemical waste and sharp objects). As a consequence, the field of robotics has attracted more researchers to design more sophisticated robots to handle the required tasks. Initially, rigid robots were developed to perform specific tasks, for example in some military tasks, aerospace exploration and industrial operations because of their dexterity especially in repetitive tasks that require precision and rigidity.

Outside of industry, rigid robots have shown some drawbacks, in particular when it comes to surgery, rescue under buildings, inspection in hazardous environments and nuclear reactors which have labyrinth-like paths and small clustered environments. The deficiency of rigid robots is labeled to their rigid links and their inability to adapt with these kinds of paths. To this end, it would be tempting to find an alternative to these robots. Researchers have developed the so-called continuum bionic robots which can easily be adapted to any kind of paths thanks to their flexibility and high performance. Continuum robots are considered by a large number of researchers as an exciting and new technology because they have notable advantages such as flexibility, lightweight, inherent safety and so forth [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Continuum robots can also avoid obstacles, handle adjustable stiffness on irregular objects, and even adapt their body shape while maneuvering in an unknown and non-uniform environment. However, they still present an impediment to researchers, because of their complexity when it comes to modeling and control. Despite the fact that many researchers have come up with new methodologies to model them, yet they are not sufficiently reliable and further accuracy is required. Therefore, in this context, this chapter presents a contribution to the kinematics of continuum robots using artificial intelligence tools.

2. Forward kinematic modeling

The Forward Kinematic Model FKM is the set of relationships that allow finding the position of the robot end-effector according to the operating coordinates; unlike rigid robots, the variables that express the configuration of a flexible robot change with respect to the robot’s morphology and its type of actuation. For this reason, research on the modeling of flexible continuum robots investigates other means to develop exhaustive kinematic models taking into account maximum of specificities and mechanical properties of the studied robot. To this end, the main objective of this section is to establish the forward kinematic model of a flexible continuum robot with Variable Curvature (VC) and Constant Curvature (CC). First, a general description of the continuum robots is presented by defining the parameters and variables. After that, the forward kinematic modeling problem is formulated and explained. Finally, the workspace is graphically presented based on the established forward kinematic model of continuum robots with both (VC) and (CC).

2.1 Geometric description of cable-driven continuum robot

A cable-driven continuum robot is one of the most commonly used types. Cable-driven continuum robots consist of a flexible backbone with disks that are separated from each other by the same distance, and attached to the backbone.

In most of these robots, the cable is considered as a straight or smooth helical curve. The cables pass through holes that are located on the disks to guide the robot and make it bend to the desired position.

2.1.1 Geometry of the whole profile of the considered robot

Developing a thorough description of the behavior and the morphology of continuum robots are the first major step that is needed to derive precise kinematic models, which themselves help to simulate the movements and the behavior of these robots. Therefore, several models have been proposed by researchers for the geometric description of continuum robots assuming constant curvature [13, 14, 15, 16, 17, 18, 19, 20] and variable curvature [21, 22, 23, 24].

The kinematics description of the constant and variable curvature continuum robot depicted in Figure 1 are briefly presented, whose profile is similar to a backbone curve. The complex structure in each continuum robot’s has two sections. Each section k is mounted at the base of its first unit, while the remaining units are sequentially stacked on top. Each section has two degrees of freedom, a bending angle $\left({\theta }_{j,k}\right)$ and an orientation angle $\left({\phi }_{j,k}\right)$ actuated by three independent cables.

Figure 1.

(left) two-section constant curvature continuum robot with its coordinate frames for each section k; (right) two-section variable curvature continuum robot with its coordinate frames for each section k.

To have a simplified description of the continuum robot under consideration, the profile of the whole section is assimilated into a curve formed by a serial concatenation of circular arcs, (Figure 2) which represents the robot central axis [22, 23, 25].

Figure 2.

Frames placements on the central axis for each unit.

A detailed illustration of a cylindrical and a conical shape units showing arc parameters and geometric properties are discussed in the following.

2.1.2 The geometry of a unit

Grasping the geometry of a single unit paves the way to understanding the remaining units that constitute the backbone of the robot. Each unit consists of two platforms (Figure 3), which can be virtual as in the case of CBHA robot or real as in our case. In the general case, the base of the unit is fixed and the upper part is mobile. They are connected by three segments.

Figure 3.

Description of a conical and cylindrical unit and its cables.

For constant curvature continuum robot, each unit of the robot has a lower and an upper disk which have the same diameters (cylindrical shaped unit) and connected through the points $A_{i,j,k}$ and ${\stackrel{⌢}{B}}_{i,j,k}$. For variable curvature continuum robot, each unit has two disks with different diameters (conical shaped unit) and connected through $B_{i,j,k}$ and $A_{i,j,k}$ (i = 1, 2, 3). Interestingly, the differentiation of the disks’ diameters paves the way to establishing an equation which relates the units with each other as it is explained in detail in [22].

The length of cable that connects the disks of a cylindrical unit is denoted by ${\mathrm{\ell }}_{i,j,k}$ and the length of cable that connects the disks of a conical unit is denoted by ${\widehat{\mathrm{\ell }}}_{i,j,k}$. Each cylindrical unit is parameterized by its orientation angle ${\phi }_{j,k}$, its bending angle ${\theta }_{j,k}$, and its curvature ${\kappa }_{j,k}$.

2.1.3 The attached frames on the robot’s backbone

In this section, we define the different frames that must be attached to the general geometry of a flexible robot. First, the center of the upper platform of each section k is referenced by a frame named $R_k$. Then, a frame $R_O$ is fixed at the base of the first section as shown in Figure 4.

Figure 4.

Placement of frames on the structure of a flexible manipulator.

In order to calculate the FKM of each unit constituting the robot, an intermediate reference frame $R_{j,k}$ is assigned to the center of the platform of each unit (j, k).

The description of a flexible section from its base to its end is as follows: the indices of the units j increase from 1 to m, and for each intermediate platform (j, k), the distance between the central axis and the anchor points is noted $r_{j,k}$ (Figure 3). The unit (j, k), is modeled as an inextensible arc of a circle. This unit has one end fixed at the origin of the reference frame $O_{j-1,k}$, the other end $O_{j,k}$ is located at a point in the reachable workspace as shown in Figure 3.

2.2 Modeling assumptions

Before starting the modeling of the considered robot, it is necessary to point out the different used assumptions, on which the mathematical development throughout this work is based. The previously developed work to model the central axis of flexible continuum robots is based on the constant curvature [26], which considers the deformation of the robot sections as a circular arc. The researchers purposefully used it because it facilitates the calculation of kinematic models. However, the constant curvature does not show a better imitation of the central axis of the continuum robot. For this reason, recent research has opened new doors to modeling the central axis of the continuum robot, which describe in detail the deformation of each unit of the robot as an arc of a circle, typically called variable curvature [21].

The assumptions considered for modeling constant and variable curvature along this chapter can be summarized as follows [22]:

• The flexible continuum robot is described as an open kinematic chain of n sections.

• Each section is a set of equidistant units.

• Each unit is modeled as an inextensible circular arc having its individual parameters.

• The cable lengths are homogeneously fragmented along the robot.

• Robot deformations at sections and units are done without torsion (neglected torsion).

In the last section, a particular case is taken into account where the robot’s backbone can be extensible (Figure 5).

Figure 5.

Global view of the modeling.

2.3 Forward kinematic model

The forward kinematic model consists of calculating the position and the orientation of the flexible continuum robot’s end-effector as a function of the cable lengths. The FKM for the considered robot in this work can be derived in three steps:

• The FKM of a single unit.

• The FKM of a single section.

• The FKM of a multi-section flexible continuum robot.

2.3.1 FKM of a single unit

As shown in Figure 3, an elaborate description of the conical and cylindrical shape of the flexible continuum robot is described.

Overall, the FKM of a unit consists of representing the operational coordinates $X_{j,k}$ as a function of the length of the cables ${\stackrel{⌢}{Q}}_{j,k}$ through the arc parameters $K_{j,k}$, (See Figure 5), and which is obtained by two transformations, the first is the specific transformation and the second is the independent one. Both transformations will be discussed in detail in the rest of this chapter. Assuming that each unit deforms in the form of an arc. The geometric parameters of each unit can be illustrated in a three-dimensional frame as shown in Figure 6. The homogeneous transformation can be written by the following equation [18]:

$$T_{j,k}^{j-1,k}=\left[\begin{array}{cc}{R}_{j,k}^{j-1,k}& O_{j,k}^{j-1,k}\\ 0_{1\times 3}& 1\end{array}\right]$$E1

with $R_{j,k}^{j-1,k}$ and $O_{j,k}^{j-1,k}$ are respectively the (3 × 3) matrix and the (3 × 1) vector defining the orientation and the position of the reference frame origin $R_{j,k}$ in the reference frame $R_{j-1,k}$.

Based on the previously defined assumptions, the orientation matrix $R_{j,k}^{j-1,k}$ is given by the following equation (Figure 6) [19]:

Figure 6.

The geometric parameters of a unit’s central axis in 3D space.

$$\begin{array}{ll}{R}_{j,k}^{j-1,k}& =\mathit{rot}\left(Z_{j-1,k},{\phi }_{j,k}\right)\mathit{rot}\left(Y_{j-1,k},{\theta }_{j,k}\right)\mathit{rot}\left(Z_{j-1,k},-{\phi }_{j,k}\right)\\ & =\left[\begin{array}{ccc}{c}^2{\phi }_{j,k}c{\theta }_{j,k}+s^2{\phi }_{j,k}& c{\phi }_{j,k}c{\theta }_{j,k}s{\phi }_{j,k}-c{\phi }_{j,k}s{\phi }_{j,k}& c{\phi }_{k}s{\theta }_{j,k}\\ c{\phi }_{j,k}c{\theta }_{j,k}s{\phi }_{j,k}-c{\phi }_{j,k}s{\phi }_{j,k}& s^2{\phi }_{j,k}c{\theta }_{j,k}+c^2{\phi }_{j,k}& s{\phi }_{j,k}s{\theta }_{j,k}\\ -c{\phi }_{j,k}s{\theta }_{j,k}& -s{\phi }_{j,k}s{\theta }_{j,k}& c{\theta }_{j,k}\end{array}\right]\end{array}$$E2

with: $c.=cos\left(.\right)$ and $s.=sin\left(.\right)$.

The position vector is defined as follows:

$$O_{j,k}^{j-1,k}=\left\{\begin{array}{c}\frac{l_{j,k}}{{\theta }_{j,k}}\left(1-cos\left({\theta }_{j,k}\right)\right)cos\left({\phi }_{j,k}\right)\\ \begin{array}{l}\frac{l_{j,k}}{{\theta }_{j,k}}\left(1-cos\left({\theta }_{j,k}\right)\right)sin\left({\phi }_{j,k}\right)\\ \frac{l_{j,k}}{{\theta }_{j,k}}sin\left({\phi }_{j,k}\right)\end{array}\end{array}\right.$$E3

2.3.2 Specific transformation

In order to show the length of the cables in the orientation matrix, a specific transformation is established. It expresses the length of the cables ${\mathrm{\ell }}_{i,j,k}$, of each unit (j, k) as a function of the arc parameters which are the orientation angle ${\phi }_{j,k}$, the bending angle ${\theta }_{j,k}$ [rad] and its curvature ${\kappa }_{j,k}$ [1/mm]. First, the orientation angle ${\phi }_{j,k}$ must be calculated.

For the variable curvature continuum robot the diameter of each disk is calculated by a general equation that allows determining each of them as follows [22]:

$$r_{j,k}=r_{max,k}-\frac{j}{k}\left(r_{max,k}-r_{min,k}\right)$$E4

with $r_{max,k}$ and $r_{min,k}$ are, respectively, the maximum and the minimum radius of the section k. When the orientation angle is equal to 0, the central axis of the unit takes the shape of a circular arc with the center ${\left[d_{j,k}\mathrm{,0,0}\right]}^T$.

Similarly, the points $A_{i,j,k}$ and $B_{i,j,k}$ where the cables are attached takes a place in an arc of circle of center ${\left[d_{i,j,k}\mathrm{,0,0}\right]}^T$. Based on these results, we can obtain the following equation:

$$l_{j,k}\left[\mathit{mm}\right]={\theta }_{j,k}{d}_{j,k}=\frac{{\theta }_{j,k}}{{\kappa }_{j,k}}$$E5

The different curvature radii of the unit can be described by the Eq. (6) based on Figure 7, where $d_{j,k}$ is the radius of curvature of the unit (j, k) and $d_{i,j,k}$ is the radius of curvature for each cable. In addition, there is a relationship between the angle ${\phi }_{j,k}$ and ${\gamma }_{i,k}$ (i = 1, 2, 3).

Figure 7.

Definition of the different curvature radii of the unit (j, k).

with ${\gamma }_{1,k}=-{\phi }_{j,k},{\gamma }_{2,k}=\frac{2\pi }{3}-{\phi }_{j,k},{\gamma }_{3,k}=-\frac{2\pi }{3}-{\phi }_{j,k}$. Through the Figure 7, the radius of curvature can be expressed as follows:

$$\left\{\begin{array}{c}{d}_{1,j,k}=d_{j,k}-r_{j,k}cos\left({\gamma }_{1,k}\right)\\ d_{2,j,k}=d_{j,k}-r_{j,k}cos\left({\gamma }_{2,k}\right)\\ d_{3,j,k}=d_{j,k}-r_{j,k}cos\left({\gamma }_{3,k}\right)\end{array}\right.$$E6

Based the Eq. (5) and by multiply the three values of the Eq. (6) by ${\theta }_{j,k}$, we can obtain the following equation:

$$\left\{\begin{array}{c}{\mathrm{\ell }}_{1,j,k}=l_{j,k}-r_{j,k}{\theta }_{j,k}cos\left({\gamma }_{1,k}\right)\\ {\mathrm{\ell }}_{2,j,k}=l_{j,k}-r_{j,k}{\theta }_{j,k}cos\left({\gamma }_{2,k}\right)\\ {\mathrm{\ell }}_{3,j,k}=l_{j,k}-r_{j,k}{\theta }_{j,k}cos\left({\gamma }_{3,k}\right)\end{array}\right.$$E7

According to Eq. (7), one way to get ${\phi }_{j,k}$ out of this equation is to subtract member by member the second equation from the first equation and the third equation from the first equation as follows:

$$\left\{\begin{array}{c}{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}=r_{j,k}{\theta }_{j,k}\left(cos\left({\gamma }_{2,k}\right)-cos\left({\gamma }_{1,k}\right)\right)\\ {\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{3,j,k}=r_{j,k}{\theta }_{j,k}\left(cos\left({\gamma }_{3,k}\right)-cos\left({\gamma }_{1,k}\right)\right)\end{array}\right.$$E8

By dividing the equations member by member (8), we obtain the following equation:

$$\frac{{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}}{{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{3,j,k}}=\frac{\left(cos\left(\frac{2\pi }{3}-{\phi }_{j,k}\right)-cos\left(-{\phi }_{j,k}\right)\right)}{\left(cos\left(-\frac{2\pi }{3}-{\phi }_{j,k}\right)-cos\left(-{\phi }_{j,k}\right)\right)}$$E9

After applying the trigonometric properties to Eq. (9) we find the following equality

$$\frac{{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}}{{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{3,j,k}}=\frac{cos\left(\frac{2\pi }{3}\right)cos\left({\phi }_{j,k}\right)+sin\left(\frac{2\pi }{3}\right)sin\left({\phi }_{j,k}\right)-cos\left(-{\phi }_{j,k}\right)}{cos\left(-\frac{2\pi }{3}\right)cos\left({\phi }_{j,k}\right)+sin\left(-\frac{2\pi }{3}\right)sin\left({\phi }_{j,k}\right)-cos\left(-{\phi }_{j,k}\right)}$$E10

After simplification the Eq. (10) becomes:

$$\frac{{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}}{{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{3,j,k}}=\frac{-\frac{1}{2}cos\left({\phi }_{j,k}\right)+\frac{\sqrt{3}}{2}sin\left({\phi }_{j,k}\right)-cos\left(-{\phi }_{j,k}\right)}{-\frac{1}{2}cos\left({\phi }_{j,k}\right)-\frac{\sqrt{3}}{2}sin\left({\phi }_{j,k}\right)-cos\left(-{\phi }_{j,k}\right)}$$E11

In the following, a detailed procedure for obtaining the arc parameters is established.

1st simplification:

$$\begin{array}{l}\frac{1}{2}{\mathrm{\ell }}_{1,j,k}cos\left({\phi }_{j,k}\right)-\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{1,j,k}sin\left({\phi }_{j,k}\right)-{\mathrm{\ell }}_{1,j,k}cos\left({\phi }_{j,k}\right)+\frac{1}{2}{\mathrm{\ell }}_{1,j,k}cos\left({\phi }_{j,k}\right)\\ +\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{2,j,k}sin\left({\phi }_{j,k}\right)+{\mathrm{\ell }}_{2,j,k}cos\left({\phi }_{j,k}\right)=-\frac{1}{2}{\mathrm{\ell }}_{1,j,k}cos\left({\phi }_{j,k}\right)+\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{1,j,k}sin\left({\phi }_{j,k}\right)\\ -{\mathrm{\ell }}_{1,j,k}cos\left({\phi }_{j,k}\right)+\frac{1}{2}{\mathrm{\ell }}_{3,j,k}cos\left({\phi }_{j,k}\right)-\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{3,j,k}sin\left({\phi }_{j,k}\right)+{\mathrm{\ell }}_{3,j,k}cos\left({\phi }_{j,k}\right)\end{array}$$E12

₂nd simplification:

$$\begin{array}{l}-\sqrt{3}{\mathrm{\ell }}_{1,j,k}sin\left({\phi }_{j,k}\right)+\frac{1}{2}{\mathrm{\ell }}_{2,j,k}cos\left({\phi }_{j,k}\right)+\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{2,j,k}sin\left({\phi }_{j,k}\right)+{\mathrm{\ell }}_{2,j,k}cos\left({\phi }_{j,k}\right)\\ =\frac{1}{2}{\mathrm{\ell }}_{3,j,k}cos\left({\phi }_{j,k}\right)-\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{3,j,k}sin\left({\phi }_{j,k}\right)+{\mathrm{\ell }}_{3,j,k}cos\left({\phi }_{j,k}\right)\end{array}$$E13

₃rd simplification:

$$\begin{array}{l}-\sqrt{3}{\mathrm{\ell }}_{1,j,k}sin\left({\phi }_{j,k}\right)+\frac{3}{2}{\mathrm{\ell }}_{2,j,k}cos\left({\phi }_{j,k}\right)+\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{2,j,k}sin\left({\phi }_{j,k}\right)\\ =\frac{3}{2}{\mathrm{\ell }}_{3,j,k}cos\left({\phi }_{j,k}\right)-\frac{\sqrt{3}}{2}{\mathrm{\ell }}_{3,j,k}sin\left({\phi }_{j,k}\right)\end{array}$$E14

₄th simplification:

$$\frac{\sqrt{3}}{2}\left(-2{\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)sin\left({\phi }_{j,k}\right)=\frac{3}{2}\left({\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}\right)cos\left({\phi }_{j,k}\right)$$E15

₅th simplification:

$$\frac{sin\left({\phi }_{j,k}\right)}{cos\left({\phi }_{j,k}\right)}=\frac{\sqrt{3}\left({\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}\right)}{\left(-2{\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)}$$E16

Finally, we obtain following formula the orientation angle:

$${\phi }_{j,k}={tan}^{-1}\left(\frac{\sqrt{3}\left({\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}\right)}{2{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{3,j,k}}\right)$$E17

After having determined the orientation angle, we calculate the radius of curvature using the previously mentioned Eq. (7) by summing the three members:

$$3l_{j,k}={\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}+r_{j,k}{\theta }_{j,k}\left(cos\left({\gamma }_{1,k}\right)+cos\left({\gamma }_{2,k}\right)+cos\left({\gamma }_{3,k}\right)\right)$$E18

Through Figure 7, we can write:

$$\sum _{i=1}^{3}cos\left({\gamma }_{i,k}\right)=0$$E19

This makes it possible to write:

$$l_{j,k}=\frac{{\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}}{3}$$E20

Substituting the Eqs. (5) and (20) into the Eq. (6), and after trigonometric simplifications, the formula of the curvature’s radius can be expressed as follows:

$$d_{j,k}=\frac{1}{{\kappa }_{j,k}}=\frac{r_{j,k}\left({\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)}{2\cdot \sqrt{{\mathrm{\ell }}_{1,j,k}^2+{\mathrm{\ell }}_{2,j,k}^2+{\mathrm{\ell }}_{3,j,k}^2-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}{\mathrm{\ell }}_{3,j,k}}}$$E21

The last parameter of the arc is the bending angle ${\theta }_{j,k}$, which can be calculated by substituting the Eqs. (20) and (21) into (5), we then obtain:

$${\theta }_{j,k}=\frac{2\cdot \sqrt{{\mathrm{\ell }}_{1,j,k}^2+{\mathrm{\ell }}_{2,j,k}^2+{\mathrm{\ell }}_{3,j,k}^2-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}{\mathrm{\ell }}_{3,j,k}}}{3r_{j,k}}$$E22

Finally, the expressions the arc of a circle can be summarized in the Eq. (23):

$$\left\{\begin{array}{c}\begin{array}{l}{\phi }_{j,k}={tan}^{-1}\left(\frac{\sqrt{3}\left({\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}\right)}{2{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{3,j,k}}\right)\\ {\theta }_{j,k}=\frac{2\cdot \sqrt{{\mathrm{\ell }}_{1,j,k}^2+{\mathrm{\ell }}_{2,j,k}^2+{\mathrm{\ell }}_{3,j,k}^2-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}{\mathrm{\ell }}_{3,j,k}}}{3r_{j,k}}\end{array}\\ {\kappa }_{j,k}=\frac{2\cdot \sqrt{{\mathrm{\ell }}_{1,j,k}^2+{\mathrm{\ell }}_{2,j,k}^2+{\mathrm{\ell }}_{3,j,k}^2-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}{\mathrm{\ell }}_{3,j,k}}}{r_{j,k}\left({\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)}\end{array}\right.$$E23

Knowing that or the VC continuum robot each unit has a conical shape, the Eq. (23) must be expressed in terms of the cable lengths ${\stackrel{⌢}{\mathrm{\ell }}}_{i,j,k}$ instead of ${\mathrm{\ell }}_{i,j,k}$. Therefore, according to the assumption mentioned before; each conical shape unit is modeled as an inextensible arc with its individual arc parameters and deforms at small angles, the relationship between these two cable lengths is given by the law of cosines [21]:

$${{\stackrel{⌢}{\mathrm{\ell }}}^2}_{i,j,k}={\mathrm{\ell }}_{i,j,k}^2+{\left(r_{j-1,k}-r_{j,k}\right)}^2-2{\mathrm{\ell }}_{i,j,k}\left(r_{j-1,k}-r_{j,k}\right)cos\left({\beta }_{i,j,k}\right)$$E24

with:

$$cos\left({\beta }_{i,j,k}\right)=sin\left(\frac{{\kappa }_{j,k}{l}_{j,k}}{2}\right)cos\left(\frac{2}{3}\pi \left(k-1\right)-{\phi }_{j,k}\right)$$E25

After solving the Eq. (24), the cables length ${\mathrm{\ell }}_{i,j,k}$ can be expressed as follows:

$$\begin{array}{c}\begin{array}{l}{\mathrm{\ell }}_{i,j,k}=\sqrt{{\stackrel{⌢}{\mathrm{\ell }}}_{i,j,k^2}^{}-{\left(r_{j-1,k}-r_{j,k}\right)}^2+{\left(r_{j-1,k}-r_{j,k}\right)}^2{cos}^2\left({\beta }_{i,j,k}\right)}\\ +\left(r_{j-1,k}-r_{j,k}\right)cos\left({\beta }_{i,j,k}\right)\end{array}\end{array}$$E26

According to Eq. (26), the length of the cable ${\mathrm{\ell }}_{i,j,k}$ is in function of the cable lenghts ${\stackrel{⌢}{\mathrm{\ell }}}_{i,j,k}$, the disks’s radius $\left(r_{j-1,k}-r_{j,k}\right)$ and the angle ${\beta }_{i,j,k}$. To find an approximate analytical solution between the lengths of the conical unit cable ${\stackrel{⌢}{\mathrm{\ell }}}_{i,j,k}$ and the configuration state ${\kappa }_{j,k}$, the influence of ${\beta }_{i,j,k}$ on the lengths of the cylindrical unit cable according to Eq. (26) must be negligible. Since the unit’s conical shape cannot be changed, angles ${\beta }_{i,j,k}$ close to $\frac{\pi }{2}$ are required. This can be achieved by choosing a high number of units, because the bending angle ${\theta }_{j,k}$ of the unit decreases with an increasing number of units per section. In this case, the Eq. (26) simplifies to:

$${\mathrm{\ell }}_{i,j,k}=\sqrt{{\stackrel{⌢}{\mathrm{\ell }}}_{i,j,k^2}^{}-{\left(r_{j-1,k}-r_{j,k}\right)}^2}$$E27

2.3.3 Independent transformation

To express the cartesian coordinates $\left(x_{j,k},y_{j,k},z_{j,k}\right)$ in terms of the arc parameters, we substitute Eq. (23) into (3), the cartesian coordinates $\left(x_{j,k},y_{j,k},z_{j,k}\right)$ as a function of the cable lengths can be expressed as follows:

$$\left\{\begin{array}{c}{x}_{j,k}=\frac{r_{j,k}\left({\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)}{M_{j,k}}{sin}^2\left(\frac{M_{j,k}}{3r_{j,k}}\right)cos\left({tan}^{-1}\left(\frac{\sqrt{3}\left({\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}\right)}{2{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{3,j,k}}\right)\right)\\ \begin{array}{l}{y}_{j,k}=\frac{r_{j,k}\left({\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)}{M_{j,k}}{sin}^2\left(\frac{M_{j,k}}{3r_{j,k}}\right)sin\left({tan}^{-1}\left(\frac{\sqrt{3}\left({\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}\right)}{2{\mathrm{\ell }}_{1,j,k}-{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{3,j,k}}\right)\right)\\ z_{j,k}=\frac{r_{j,k}\left({\mathrm{\ell }}_{1,j,k}+{\mathrm{\ell }}_{2,j,k}+{\mathrm{\ell }}_{3,j,k}\right)}{2M_{j,k}}sin\left(\frac{2M_{j,k}}{3r_{j,k}}\right)\end{array}\end{array}\right.$$E28

with: $M_{j,k}=2\cdot \sqrt{{\mathrm{\ell }}_{1,j,k}^2+{\mathrm{\ell }}_{2,j,k}^2+{\mathrm{\ell }}_{3,j,k}^2-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{2,j,k}-{\mathrm{\ell }}_{1,j,k}{\mathrm{\ell }}_{3,j,k}-{\mathrm{\ell }}_{2,j,k}{\mathrm{\ell }}_{3,j,k}}$.

2.3.4 FKM of a single section

As mentioned before, a section k is a concatenation of m units. Moreover, each section deforms without torsional effect, so we can have the forward kinematic model of a section by the multiplication of transformation matrices, which can be expressed by the following relation:

$$T_k^{k-1}=T_{m,k}^{0,k}=\prod _{j=1}^m{T}_{j,k}^{j-1,k}$$E29

where $T_{j,k}^{j-1,k}$ is the (4 × 4) matrix defining the orientation and position of the origin of the reference frame $R_{j,k}$ in the frame $R_{j-1,k}$.

For a VC continuum robot, an approximate formula is developed in [22], which allows to express the bending angle of the robot’s units in function of the first bending angle of the first unit in the same section. The developed formula is expressed as follows [22]:

$${\theta }_{j,k}=\frac{r_{1,k}}{r_{j,k}}{\theta }_{1,k}$$E30

As the number of units increases, the accuracy of the solutions of the Eq. (30) also increases, that is, when the angle ${\beta }_i$ is closer to $\frac{\pi }{2}$ (Figure 3). Therefore, using Eq. (30), the forward kinematic model of a single section can only be expressed by two variables ${\theta }_{1,k}$ and ${\phi }_k$. This approximation leads to a remarkable simplification and reduction of the number of variables involved in the FKM.

2.3.5 FKM of the multi-section conical robot

Based on the Eq. (30), the identification of the robot’s end-effector posture becomes relatively simple because the different unit angles will be expressed in terms of the bending angle of the first unit. Thus the forward kinematic model for a multi-section continuum robot (see Figure 1) can be easily found by successive multiplication of the independent transformation matrices for each section k and the transformation matrix of the static frame of reference. The resulting matrix is given by the following equation:

$$T_n^{\mathit{\text{orig}}}=\left[\begin{array}{cc}{A}_n^{\mathit{\text{orig}}}& O_n^{\mathit{\text{orig}}}\\ 0_{1\times 3}& 1\end{array}\right]=T_0^{\mathit{\text{orig}}}\prod _{k=1}^n{T}_k^{k-1}$$E31

where the matrices $T_0^{\mathit{\text{orig}}}$ and $T_k^{k-1}$ represent the static transformation matrix and the independent transformation matrix of each flexible section respectively.

3. Workspace generation for a continuum robot

By definition, the workspace is the set of all positions accessible by the robot’s end effector.

Figure 8 shows the possible positions that can be reached by the first section of the continuum robot, as well as by the continuum robot with two sections in two-dimensional representations. These points were obtained using the FKM, by varying the first bending angle for each section in the interval ${\theta }_{1,k}\in \left[-\frac{\pi }{8},\frac{\pi }{8}\right]$.

Figure 8.

2D view of the continuum robot workspace at VC.

On Figure 8, we present some possible positions that the continuum robot’s end effector can reach, which are designated by the letter A (dotted curved lines in red) as well as the positions of the first section of the robot designated by the letter B (dotted curved lines in yellow).

3.1 Comparison between the workspace of a continuum robot with VC and CC

In this section, we compare the workspace of a flexible continuum robot with constant and variable curvature. Lets start with the comparison of a single section of the continuum robot, the characteristics of each section are shown on 1. The first bending angle and orientation of the section are varied in the interval $\left[0,\frac{\pi }{5}\right]$ and $\left[-\pi ,\pi \right]$ respectively. As it can be seen in Figures 9 and 10, the idea of variable curvature is able to produce bending angles with increasing curvature, namely, the end point of the section can reach much more curved positions with respect to the constant curvature.

Figure 9.

3D view of the workspace of a CC and VC section.

Figure 10.

View in the $\mathit{\Upsilon Z}$ plane of the workspace of a CC and VC section.

In the second example, we vary the first bending angle for each section in the same range ${\theta }_{j,k}\in \left[-\frac{\pi }{8},\frac{\pi }{8}\right]$.

Figures 11 and 12 show the workspace of a flexible continuum robot with CC and VC respectively. Comparatively, the end effector of a VC flexible continuum robot can reach much more curved positions than a CC continuum robot as shown in Figure 13.

Figure 11.

2D workspace view for a flexible continuum robot with constant curvature: $r_{\max }=r_{\min }=25\mathit{mm}$_.

Figure 12.

2D workspace view for a flexible continuum robot with variable curvature: $r_{\max }=25\mathit{mm},r_{\min }=10\mathit{mm}$.

Figure 13.

Comparison of workspace for a flexible continuum robot with CC and VC.

4. Inverse kinematic modeling

For any robotic system, the Inverse Kinematic Model IKM can be summarized as follows: for a given posture (position and orientation), we determine the necessary actuation variables that satisfy certain movement constraints. In the continuum robots’ world, the inverse kinematic models are a very nonlinear problem, which is inferred by calculating the inverse function of forward kinematic models. IKMs can easily orient robots by cartesian position equations rather than curvature and rotational equations. But obtaining IKMs remains a difficult task, especially when it comes to multi-section continuum robots. In almost cases IKMs cannot be analytically calculated and have several problems that lie in the following:

• Unavailability of solutions when the target points are outside the robot’s workspace.

• Several solutions or endless solutions when the target point can be reached by different configurations.

For the robots studied in this chapter, the inverse kinematic model consists in calculating the cable lengths and geometric parameters corresponding to the desired position of the robot’s end-effector. To this end, the artificial neural networks and meta-heuristic optimization will be used to solve the inverse kinematic model of a single section continuum robot and a multi-section continuum robot respectively. The development of the used approaches to solve the IKMs is done through the forward kinematic models.

4.1 ANN for solving the IKM of single section continuum robots with VC

Neural Networks have achieved a great success in many areas due to their learning and generalization capabilities as well as their parallelism. They have been successfully used in many applications, such as classification, noise filtering, system modeling and control…etc. One of the fields where NN has received an increasing interest is that of solving the robots’ IKMs.

Due to the complexity of continuum robots models, very few works which aim at solving their IKMs have been carried out, especially with varible curvature. To this end in this section, the Multi-Layer Perceptron (MLP) neural networks will be used and developed to find the IKMs of single section continuum robots with variable curvature. The hidden layers contains neurons with sigmoid activation function and a linear activation function in output layers. For the sake of avoiding the complexity of the neural networks a minimum number of neurons and hidden layers that gives the good learning are chosen.

The forward kinematic model is calculated for the sake of obtaining the $P_{x_i}$, $P_{y_i}$, and $P_{z_i}$ coordinates of the robot’s end tip which themselves are used to train the proposed MLP. the used objective function for training neural networks is given by the following Eq. (32):

$$F=\frac{1}{N}\sum _{i=1}^N\left({\left(P_{x_i}-X_{c_i}\right)}^2+{\left(P_{y_i}-{\Upsilon }_{c_i}\right)}^2+{\left(P_{z_i}-Z_{c_i}\right)}^2\right)$$E32

Where N represent the number of data. $X_{c_i}$, ${\Upsilon }_{c_i}$, and $Z_{c_i}$ represent the spatial coordinates of a located position on the prescribed trajectory. $P_{x_i}$, $P_{y_i}$, and $P_{z_i}$ represent the position of the robot’s end tip for each specific position of the prescribed trajectory, which are obtained from the FKM. Explicitly, they represent the three first components of the forth column of Eq. (31) which describe the FKM of the considered robots.

4.1.1 Outline and simulation

Two simulations examples are considered, for the first simulation; a single section continuum robot with variable curvature must follow up arc-shaped trajectory and for the second simulation the robot must follow up circular trajectories.

The characteristics of the considered robot are given in Table 1.

Parameter	value	Description
$m_k$	5 units	Number of units per section
$L_k$	300 mm	Total length of the section
$r_{min,k}$	17.5 mm	Radial cable distance
$r_{max,k}$	25 mm	Radial cable distance

Table 1.

Parameters of the single section flexible continuum robot with VC.

4.1.2 IKM of single section continuum robot

The FKM of a single section is first calculated using Eq. (31). After that, to find the IKM of the considered robot the orientation angle is calculated analytically using the FKM by dividing the position $P_y$ by the position $P_x$. Finally, the static MLP (Figure 14) with the following configuration is used to calculate the bending angles:

• The input layer contains four inputs $\left(x,y,z,\varphi \right)$

• One hidden layer with 20 neurons.

• One linear output neuron that gives the approximated $\theta$.

Figure 14.

Neural network structure for spacial single section continuum robot.

Learning rate: $\alpha =0.01$.

• Type of optimizer: Levenberg-Marquardt backpropagation.

A random dataset that covers all-inclusive possible positions which can be attained by the robot’s first section, namely its workspace, are generated using the FKM to train the developed model.

The training result error value to the bending angle is given by Figure 15. To evaluate the effectiveness of the obtained IKM. A second randomly dataset that cover all-inclusive possible positions in the robot’s workspace is generated using the FKM. Figure 16 gives the error test results of the developed neural model for the bending angle $\theta$. Figures 15 and 16 show the accuracy and the effectiveness of the obtained IKM. The values of the mean square error (MSE) of the achieved model are given in Figure 17.

Figure 15.

Training error results of the neural model for the bending angle $\theta$.

Figure 16.

Error test results of the neural model for the bending angle $\theta$.

Figure 17.

Training performance of the model.

After tested the effectiveness of the obtained model (IKM), the considered robot is tested on practically realistic-like trajectories (circular and arc-shaped trajectories) in order to assess the robot’s behavior via 3D simulations.

As it is shown in Figure 18, the considered single section continuum robot with VC can accurately follow the arc-shaped trajectory within its 3D workspace (yellow curved arcs).

Figure 18.

Two configurations of a single section continuum robot following an arc shaped trajectory within its 3D workspace.

To show the accuracy of the generated trajectory, euclidean errors are calculated. As it is shown in Figure 19, the maximum error between the generated and the desired trajectory is of the order of 0.002 mm (Figure 20). Similarly to the first simulation example, the continuum robot track a circular trajectory within its workspace (See Figure 20). Figure 21, shows the euclidean error of the trained NN.

Figure 19.

Error between the reference trajectory (arc-shaped) and the trajectory (arc-shaped) given by the NN.

Figure 20.

Representation of the continuum robot’s first section tracking the desired circular trajectory.

Figure 21.

Error between the reference circular trajectory and the circular trajectory given by the NN.

4.2 Methaheuristic optimization for solving the IKM of multi section continuum robots with VC

The main objective of adopting meta-heuristic optimization techniques unlike artificial neural networks to solve the inverse kinematic model of multi-section continuum robots is the ability to deal with the complexity of these robot’s models and their endless solutions, especially when including orientation angles. As well as their capabilities to make the robot avoid obstacles simply by taking the appropriate configuration to accomplish the required task. To this end in this section, a meta-heuristic optimization algorithm called teaching learning-based optimization (TLBO) is used to solve the inverse kinematic model of a multi-section continuum robot with the ability to make the robot dexterously avoid obstacles.

4.2.1 Teaching learning based optimization algorithm

TLBO Algorithm is originally inspired from the process’ effect of teaching a set of students. It was proposed by Rao et al. [27] and Rao and Savsani [28]. The algorithm consists of two phases, teacher and learners phases. In the process of this algorithm, a set of learners is considered as population, the whole population is provided with various topics, which are supposed to be the variables of the optimization problem. The learners’ results illustrate the fitness value of the optimization problem. The outstanding solution among the whole population is considered to be the teacher, while the variables are the objective function’s parameters and its best value is the best solution to the considered problem.

4.2.2 Teacher phase

In the first phase, the teacher thoroughly tries to improve the mean result of the whole class according to the taught subject. At each attempt (iteration) i, there are m topics and $n_p$ learners in a specific topic j. The best learner in the classroom is chosen as the teacher. The difference $d_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}$ between the existing mean results of each topic and the teacher result for each subject is given by:

$$d_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}=r\left(X_{\mathrm{j},{\mathrm{k}}_{\text{best}}}^{\mathrm{i}}-T_F{M}_{\mathrm{j}}^{\mathrm{i}}\right)$$E33

where

$$M_{\mathrm{j}}^{\mathrm{i}}=\frac{{\sum }_{\mathrm{k}=1}^n{X}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}}{n}$$E34

and $X_{\mathrm{j},{\mathrm{k}}_{\text{best}}}^{\mathrm{i}}$ is the best selected learner’s result in a topic j, $T_F$ is a factor which can be either 1 or 2, r is a random number in the range [0,1], $M_{\mathrm{j}}^{\mathrm{i}}$ is the mean result. Based on Eq. (33), the existing solution is updated in the teacher phase as follows:

$${\mathit{\text{Xnew}}}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}=X_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}+d_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}$$E35

where ${\mathit{\text{Xnew}}}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}$ is the updated value of $X_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}$ which is sent to the learner phase.

4.2.3 Learner phase

In this particular phase, the learners enhance their level of knowledge by a random interaction among themselves. The learner can improve his/her knowledge by interacting with a learner who has more knowledge than him/her. The whole process can be mathematically formulated as follows:

$$\left\{\begin{array}{c}{\mathit{\text{Xnew}}}_{\mathrm{j},A}^{\mathrm{i}}=X_{\mathrm{j},A}^{\mathrm{i}}+r\left(X_{\mathrm{j},A}^{\mathrm{i}}-X_{\mathrm{j},B}^{\mathrm{i}})\right.\mathit{\text{if}}{F}_A^{\mathrm{i}}<F_B^{\mathrm{i}}\\ {\mathit{\text{Xnew}}}_{\mathrm{j},A}^{\mathrm{i}}=X_{\mathrm{j},A}^{\mathrm{i}}+r\left(X_{\mathrm{j},B}^{\mathrm{i}}-X_{\mathrm{j},A}^{\mathrm{i}})\right.\mathit{\text{if}}{F}_B^{\mathrm{i}}<F_A^{\mathrm{i}}\end{array}\right.$$E36

where A and B are two randomly selected learners, $F_A^i$ and $F_B^i$ are the fitness values of $X_A$ and $X_B$ for A and B, respectively. It is noteworthy to say that the aforementioned Eqs. ((29), (30), (33)) are valid exclusively when it comes to minimization problems.

4.2.4 Comparison of TLBO with other optimization algorithms

Generally, almost all optimization methods require specific parameters that have a crucial role on the algorithm’s functionality as well as its performance. Table 2 shows the required parameters for some metaheuristic algorithms. Unlike other optimization techniques, TLBO does not require any specific parameters, which makes it a handy algorithm to deal with the complicated problems of engineering. TLBO uses the best solution at each iteration to shift to the best existing solution in the population, which paves the way to a fast convergence.

Algorithms	Required parameters
GA	Crossover probability, mutation rate, selection method
PSO	Acceleration constants, weight variation, maximum velocity
ABC	Onlooker bees number, employed bees number, food sources number
HS	Range of each variable, pich rate, number of improvisation

Table 2.

The required parameters for each algorithm.

4.2.5 Objective function and problem formulation

The resolution of the IKM can be addressed using an objective function, which describes the distance between the prescribed position and the robot’s end tip (Figure 22), where the lowest possible distance can be considered as a solution for a specifically prescribed position. The cost function contains the desired position coordinate and the robot’s end tip coordinates it can be mathematically expressed as follows:

Figure 22.

Global view of the governing objective function and the adopted strategy for the obstacle avoidance.

$$F=\sqrt{{\left(P_{X_i}-X_{c_i}\right)}^2+{\left(P_{{\Upsilon }_i}-{\Upsilon }_{c_i}\right)}^2+{\left(P_{Z_i}-Z_{c_i}\right)}^2}$$E37

Furthermore, $X_{c_i}$, ${\Upsilon }_{c_i}$, and $Z_{c_i}$ represent the spatial coordinates of a located position on the prescribed trajectory. $P_{X_i}$, $P_{{\Upsilon }_i}$, and $P_{Z_i}$ represent the position of the robot’s end tip obtained from the FKM. These positions represent the three first components of the fourth column of the matrix (31).

$$D_{\mathrm{j},\mathrm{k}}^2\le {\left(r_{\mathit{\text{obstacle}}}+r_{\mathrm{j},\mathrm{k}})\right.}^2$$E38

Where $r_{\mathit{\text{obstacle}}}$ is the radius of the spherical-like shape obstacle, and $r_{j,k}$ is the radius of the selected robot’s disks. The obstacle avoidance can be fulfilled if the distance between the robot’s disks radius summed up with the obstacle’s radius is bigger than $D_{j,k}$. The studied robot in this section is given by 3., which is consists of two sections $n=2$. Each section has five disks with different diameters; therefore, the robot’s 3D workspace is the zone where the operation of searching for a specific solution is performed by TLBO (Table 3).

	Section (k = 1)	Section (k = 2)	Description
$m_k$	5 units	5 units	Number of units per section
$L_k$	300 mm	300 mm	Total length of the section
$r_{min,k}$	17.5 mm	10 mm	Radial cable distance
$r_{max,k}$	25 mm	17.5 mm	Radial cable distance

Table 3.

Parameters of the two-section flexible continuum robot.

4.2.6 TLBO implementation to solve the IKM of VL continuum robot

Three Simulation examples through Matlab are carried out. For the former, a 3D simulation of a two-section continuum robot with variable length following two types of trajectories, namely a linear and spherical trajectory, is considered. The needed variables to be found by the TLBO are $\left({\theta }_1,{\phi }_1\right)$ for the robot’s first section and $\left({\theta }_2,{\phi }_2\right)$ for the robot’s second section. The second simulation addresses a free-collision example during the follow up of a circular trajectory. In the third simulation, a 3D simulation of a three-section continuum robot during the follow up of a linear trajectory is carried out.

4.2.7 Step of the optimization algorithm

• Step 0: initialization

  • Choose the population size $\left(n_p\right)$, the number of variables (m), the termination criterion $\left(\mathrm{ϵ}\right)$, the maximum number of iteration $\left({\mathrm{k}}_{\mathit{max}}\right)$, the admissible maximum and minimum values for each variable $\left(X_{{\mathrm{j}}_{\mathit{max}}},X_{{\mathrm{j}}_{\mathit{min}}}\right)$.

  • For each point of the reference trajectory do the following steps:

• Step 1: initial solution

For $\mathrm{k}=1:n_p$.

For $\mathrm{j}=1:m$

• Choose randomly the initial solution as follows: $x_{\mathrm{j},\mathrm{k}}^1=\mathit{rand}\left(0,1\right)\ast \left(X_{{\mathrm{j}}_{\mathit{max}}}-X_{{\mathrm{j}}_{\mathit{min}}}\right)+X_{{\mathrm{j}}_{\mathit{min}}}$ where: r is a random number between 0 and 1

  • End

    • End

    • Select the desired reference trajectory point.

Set $\mathrm{i}=1$

• Step 2: teacher phase

For $\mathrm{k}=1:n_p$

• Evaluate the corresponding objective function $\left(F_{\mathrm{k}}\right)$ using Eq. (37).

  • End

  • Select the student who gives the minimum value of the objective function $\left(F_{{\mathrm{k}}_{\mathit{\text{best}}}}\right)$ as a teacher.

For $\mathrm{j}=1:m$

• Calculate the mean result $\left(M_{\mathrm{j}}^{\mathrm{i}}\right)$ using Eq. (34).

  • End

For $\mathrm{j}=1:m$

• Calculate the difference mean $\left(d_{\mathrm{j}}^{\mathrm{i}}\right)$ using Eq. (33).

  • End

For $\mathrm{k}=1:n_p$.

For $\mathrm{j}=1:m$

• Calculate the new solutions $\left({\mathit{\text{Xnew}}}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}\right)$) using Eq. (35).

• End

  • End

For $\mathrm{k}=1:n_p$

• Evaluate the new corresponding objective function $\left({\mathit{\text{Fnew}}}_{\mathrm{k}}\right)$ using the new solutions $\left({\mathit{\text{Xnew}}}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}\right)$ according to Eq. (37).

  • End

  • Apply a greedy selection between the new and the old solutions.

• Step 3: learners phase

  • Choose randomly a q pairs of solutions.

For $\mathrm{k}=1:n_p$.

For $\mathrm{j}=1:m$

• Calculate the new solutions $\left({\mathit{\text{Xnew}}}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}\right)$ using Eq. (36)

• End

  • End

For $\mathrm{k}=1:n_p$

• Evaluate the new corresponding objective function $\left({\mathit{\text{Fnew}}}_{\mathrm{k}}\right)$ using the new solutions $\left({\mathit{\text{Xnew}}}_{\mathrm{j},\mathrm{k}}^{\mathrm{i}}\right)$ according to Eq. (37)

  • End

  • Apply a greedy selection between the new and the old solutions.

• Step 4: iterative process

  • If $F_{{\mathrm{k}}_{\mathit{\text{best}}}}<\mathrm{ϵ}$ or $\mathrm{i}>{\mathrm{k}}_{\mathit{max}}$

• Go to step 5

  • Else

$\mathrm{i}=\mathrm{i}+1$

• Go to step 2

  • End if

• Step 5: obtaining the final solution

  • Apply the computed best solution.

  • Go to step 1.

4.2.8 Simulation of a two-section continuum robot following a circular trajectory in the presence of an obstacle

In this application, a two-section continuum robot with variable curvature is considered to follow a circular trajectory (see Figure 23), which is defined by Eq. (39), in the presence of a static obstacle as previously explained in Figure 22. The robot has to follow the prescribed circular trajectory in the condition that it does not collide or smoothly touch the spherical trajectory.

Figure 23.

(left) VL continuum robot following a circular trajectory in the presence of a static obstacle; (right) different configuration of the central axis of the robot.

$$\left\{\begin{array}{c}X=30\mathit{cos}\left(\frac{\pi }{5}t\right)\\ \Upsilon =30\mathit{sin}\left(\frac{\pi }{5}t\right)\\ Z=440+30\mathit{sin}\left(\frac{\mathit{\pi t}}{5}\right)\end{array}\right.$$E39

with t = 0:0.25:10 As it can be seen from Figure 23, the variable curvature continuum robot follows the circular trajectory with a free collision. The whole operation is performed based on TLBO algorithm which generates the required angles allowing the robot to properly follow the circular trajectory with respect to the length variation. Remarkably, based on the Euclidean error between the generated trajectory by TLBO and the desired trajectory (Figure 24), TLBO can be considered as a powerful optimization algorithm for solving the inverse kinematic model of continuum robots.

Figure 24.

Error between the desired circular trajectory and the generated one by TLBO.

Figure 25 represents the cables length allowing the robot to follow the circular trajectory with free collision.

Figure 25.

The cables length generated by TLBO ensuring the free-collision during circular trajectory.

It is noteworthy to say that the calculation of the cables length is mainly based on angles which are calculated from the proposed algorithm and on the strategy of calculating the parallel robots’ IKMs, in other words, the three cables for each unit are considered as the three segments connecting the parallel robot’s platforms, namely two platforms are linked through three segments with prismatic articulation.

5. Conclusion

The aim of this chapter was to develop methods for modeling of continuum robots, which can provide satisfactory and effective performance. In fact, neural networks and optimization methods have been widely used for modeling and controlling nonlinear systems. The secret of neural networks’ success lies in the fact that any non-linear system can be modeled with a certain precision using a simple neural network having learning and generalization capabilities. On the other hand, the emergence of many optimization methods has made it possible to consider more complex optimization problems for which numerical methods cannot give acceptable solutions.

In this chapter, the different types of continuum robots and their modeling were properly studied. This study allowed us to examine the limitations and advantages of the adopted approaches and maked it easier for us to suggest several ways to improve the modeling and control methods based on artificial intelligence tools. The accomplished work can be summarized as follows:

• A recently developed meta-heuristic approach so-called teaching learning based optimization was adopted to solve the inverse kinematic model of a continuum robot with variable curvature. The TLBO algorithm is not fussy and does not require any specific parameters to be tuned up. In other words, unlike the other developed meta-heuristic algorithms which need specific parameters to perform correctly, TLBO can be considered as the best optimization method to tackle the complexity of continuum robots. To emphasize, based on the available state of art about optimization methods, it is concluded that the TLBO algorithm is way better accurate than the other optimization algorithms for solving the IKM.

• An artificial neural network is developed to solve the inverse kinematic model of single-section continuum robot with variable curvature. The development of the used artificial neural network is done through the forward kinematic model which is mainly used to train the ANN. At the training of the neural network is performed based on a random generated database from the FKM. The obtained neural network is tested on solving the IKM of continuum robots through various examples. The bending angles which allow the robot to the trained properly follow a desired trajectory can be easily and accurately generated by ANN.

Acknowledgments

This work is supported by the DGRSDT (Direction Générale de la Recherche Scientifique et du Développement Technologique), Algeria.

Parts of this chapter were previously published in the doctoral thesis by Abdelhamid GHOUL: Contribution to the modeling and control of continuum and deformable robots using artificial intelligence tools. 2022. UNIVERSITÉ DE BLIDA 1, Faculté de Technologie, Département d’Automatique et Électrotechnique. Available from: https://di.univ-blida.dz/jspui/handle/123456789/25562

Conflict of interest

The authors declare no conflict of interest.