# Design of a flexible bio-inspired robot for inspection of pipelines

Swaminath Venkateswaran, Damien Chablat

► To cite this version:

Swaminath Venkateswaran, Damien Chablat. Design of a flexible bio-inspired robot for inspection of pipelines. Biologically Inspired Series-Parallel Hybrid Robots: Design, Analysis, and Control, Elsevier, pp.427-449, 2024, 978-0323884822. 10.1016/B978-0-32-388482-2.00030-5. hal-04861021

HAL Id: hal-04861021

https://hal.science/hal-04861021v1

Submitted on 10 Apr 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Design of a flexible bio-inspired robot for inspection of
pipelines

Swaminath Venkateswaran¹,*, Damien Chablat²

¹Léonard de Vinci Pôle Universitaire, Research Center, 92916 Paris La Défense, France

²Centre National de la Recherche Scientifique (CNRS), Laboratoire des Sciences du Numérique de Nantes (LS2N), UMR CNRS 6004, Nantes, 44321 France

**Abstract**

This article presents the design of a flexible piping inspection robot for the inspection of industrial pipelines. By imitating the locomotion of a caterpillar, a rigid bio-inspired piping inspection robot was initially developed. For working through pipe bends and junctions, a tensegrity mechanism is introduced into this robot to have a flexible design. An experimental test bench is then developed to understand the working limits of the tensegrity mechanism. In order to work inside a certain diameter range of pipelines, an optimization approach is then followed to determine the size of motors and leg mechanisms for the robot assembly. The static modeling of the robot assembly is then proposed for horizontal orientation where the assembly resembles a series-parallel hybrid manipulator.

**Keywords:** Bio-inspired, Piping inspection, Tensegrity, Optimization, Hybrid manipulator

**1. Introduction**

Piping inspection robots have created a major attraction in the scientific re-
search community. These robots can carry out an inspection or an intervention
on areas that are difficult to access by humans. They also reduce the chances
of hazards or risks to human life as these robots can potentially work within an
irradiated or polluted environment. With the automation of control systems,
inspection robots can accomplish the desired task within a short period as well
as with better accuracy. Generally, the locomotion principles of these robots can
employ either mechanical systems or inspire their motion from nature, otherwise
referred to as Bio-inspired techniques. Kassim et al.[1] proposed a distinction
between these two categories where the mechanical systems can use wheels and
pulleys [2], telescopic system [3], impact modules [4] or natural peristalsis [5]

*Corresponding author: swaminath.venkateswaran@devinci.fr
<figure><img src="image_2.png" /><figcaption>Figure 1: Possible profiles encountered by an inspection robot inside an industrial pipeline: (a) varying diameter; (b) curvature; (c) inclination; (d) branching; and (e) uneven surface walls</figcaption></figure>

to accomplish the locomotion. On the other hand, the bio-inspired systems can mimic their locomotions from animals such as earthworms [6], snakes [7], milipedes [8], lizards [9] or an octopus [10]. Currently, several piping inspection robots have been designed and developed by many researchers. These robots exploit either the mechanical [11] or bio-inspired approach [12, 13] for the locomotion. However, the design and development of an inspection robot for pipeline diameters less than 150 mm remains to be a big challenge. Park et al.[14] proposed different kinds of obstacles that a piping inspection can generally encounter and it is shown below in Figure 1.

This article will focus on arriving at the design of a piping inspection robot that can operate in a pipeline having a diameter of 150 mm or less and at the same time, a design which is capable of overcoming bends or intersections (Figures 1b to 1d). Also, the design will be capable of adapting its structure when it encounters a diameter change inside a pipeline (Figure 1a). The article is organized as follows. Section-2 focuses on the initial design of a rigid caterpillar-type piping inspection robot. Section-3 focuses on the design and prototyping of a tensegrity mechanism that will be incorporated onto the robot studied in Section-2. Section-4 deals with the identification of optimal design parameters of the flexible robot in order to work inside straight or curved pipe profiles. Followed by that the static modeling of the robot assembly inside a horizontal pipeline is proposed in Section-5. The article then ends with conclusions and future perspectives.

## 2. Rigid bio-inspired piping inspection robot-An overview

As a part of a project with AREVA, a novel bio-inspired piping inspection robot that mimics the locomotion of a caterpillar was proposed and developed by Henry et al.[15] and Chablat et al.[16]. This robot is capable of addressing the issue of the design of inspection robots for pipelines having diameters less than 150 mm. Using electrical actuators with leg mechanisms and spindle drive units, this robot accomplishes the motion of a caterpillar and can work inside 40–94 mm diameter pipelines. Three architectures were studied for optimization viz: slot-follower, four-bar, and six-bar mechanisms in [15] for the choice of leg mechanism for this robot. By following a heuristic optimization [17] technique in MATLAB, the slot-follower mechanism was chosen for the robot [15]. A detailed force analysis was then performed on the robot during clamping and locomotion conditions. The static force model was analysed using
<figure><img src="image_3.png" /><figcaption>Figure 2: Prototype of the caterpillar type piping inspection robot developed at LS2N, France</figcaption></figure>

Coulomb's law during the clamping phases of the leg modules with the pipeline walls [16]. For the dynamic force model, the forces on the central actuator were identified under locomotion using the Newton-Euler recursive algorithm [18]. Upon comparison, it was observed that the static forces were higher than the dynamic forces. However, in the numerical analysis, it was not possible to validate electromechanical factors and frictional coefficients. Using DC-motors, spindle drives, leg mechanisms, and standard fasteners, the entire robot was realized at LS2N, France. Experiments were then carried out on this prototype, where the peak results of the static force models were set as threshold limits for ensuring tight clamping with the pipeline walls. The real-time forces induced on the actuators under locomotion were then interpreted using a Force-control algorithm with the help of a BeagleBone (BB) black micro-computer inside a 74 mm diameter straight pipe. The robot was able to overcome the issue of working inside pipelines having diameters less than 150 mm that can have variable diameters. One major disadvantage of this prototype is that it was a rigid model and it cannot be employed for curved pipe profiles represented in Figures 1b to 1d. The spindle drive unit used in each module of the robot was also currently oversized as it has a screw length of 102 mm. This excess screw length restricted the robot to overcome a pipeline having bends or curvatures if the robot was made flexible. Thus, in the following section, a solution will be proposed which can assist this prototype to work inside curved pipe profiles. The prototype of the caterpillar-type rigid piping inspection robot is shown below in Figure 2.

### 3. Design, analysis and synthesis of a tensegrity mechanism

In order to overcome pipe bends and junctions, the design of the caterpillar-type inspection robot discussed in the previous section was modified. An articulation unit can be incorporated between the modules to overcome this issue. Generally, articulation units can be classified into active and passive systems. In the literature, there exist several robots that incorporate one of these solutions such as the robot THES-I [19] which proposed an actuated universal joint and the robot of Kwon et al.[13] which employed a passive compression spring. Most of these robots employed articulation units that are either passive or active, but not a combination of both. Two types of tensegrity mechanisms that employ a passive universal joint and either three or four tension springs
<figure><img src="image_4.png" /><figcaption>Figure 3: The 3-SPS-U(left) and 4-SPS-U(right) tensegrity mechanisms at their home-poses</figcaption></figure>

were proposed as articulation units for the rigid piping inspection robot. While encountering a pipe bend at 90°, both mechanisms can work in a passive mode. In the event of a junction or a T-union, cables that pass through the springs of each mechanism are actuated to tilt in a certain direction. These mechanisms were designed by taking into account three parameters namely [20]: Passive compliance (autonomous mode), active compliance (actuation), and tilt limits.

### 3.1. Identification of optimal design parameters

The tensegrity mechanisms were analyzed by correlating it to parallel manipulators of type 3-SPS-U and 4-SPS-U. Here S indicates the spherical joint, U indicates the universal joint, and P indicates the actuated prismatic joints or the springs. The representation of the tensegrity mechanisms at their home poses and the correlation to a parallel manipulator is shown below in Figure 3.

The fixed coordinate frame of the base is represented by $\Sigma_0$ with the origin at $B_0$. The spring mounting points on the fixed base are represented by $B_i$ ($i = 1, 2, 3$) and they form the imaginary equilateral triangle of the manipulator base whose median is $r_f$ for the 3-SPS-U mechanism. The base mounting points $B_i$ ($i = 1, 2, 3, 4$) forms an imaginary square for the 4-SPS-U mechanism. The diagonal length for the square is $2r_f$. The vector coordinates for the base mounting points are given by:

$$ \mathbf{b}_i = [r_f \cos(\frac{i2\pi}{j}), r_f \sin(\frac{i2\pi}{j}), -r_f h]^T \quad (1) $$

where $j = 3$, $i = 0$ to 2 for 3-SPS-U and $j = 4$, $i = 0$ to 3 for 4-SPS-U

The moving coordinate frame of the end-effector is represented by $\Sigma_1$ with its origin at $C_0$. The spring mounting points of the end-effector is represented by $C_i$ ($i = 1, 2, 3$) and $C_i$ ($i = 1, 2, 3, 4$) for the 3-SPS-U and 4-SPS-U mechanisms respectively. For estimating the vector coordinates of the end-effector mounting points, the XY Euler angle matrix is employed with respect to the central point A of the universal joint. The vector coordinates for the end-effector mounting
points are given by:

$$ \mathbf{c}_i = \mathbf{E}[r_f \cos\left(\frac{i2\pi}{j}\right), r_f \sin\left(\frac{i2\pi}{j}\right), r_f h]^T \quad \text{with} \quad \mathbf{E} = \mathbf{R}_x(\eta)\mathbf{R}_y(\phi) \quad (2) $$

where $j = 3$, $i = 0$ to 2 for 3-SPS-U and $j = 4$, $i = 0$ to 3 for 4-SPS-U

In Equation 2, $\mathbf{E} \in SE(3)$ represents the spatial transformation matrix obtained from the Euler angles of the universal joint, and it is used to identify the end-effector coordinates as indicated in Equation 2. The inverse kinematic problem (IKP) for the mechanism is simpler and it determines the length of springs between the base and end-effector at home pose and working conditions. The equation is given by:

$$ l_i = \sqrt{(b_{ix} - c_{ix})^2 + (b_{iy} - c_{iy})^2 + (b_{iz} - c_{iz})^2} \quad (3) $$

with $i = 1$ to 3 for 3-SPS-U & $i = 1$ to 4 for 4-SPS-U

In line with the flange units employed in the piping inspection robot, the value of $r_f$ was taken as 11 mm. However, the value of parameter $h$ plays an essential role in the stability of the mechanism under static modes. According to Lagrange, for a moving system, the equation of motion [21] is given by:

$$ \tau = \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\mathbf{q}}} \right) - \frac{\partial T}{\partial \mathbf{q}} + \frac{\partial U}{\partial \mathbf{q}} \quad \text{where} \quad \mathbf{q} = [\eta, \phi]^T \quad (4) $$

Under static conditions, the total potential energy of the tensegrity mechanisms is given by:

$$ U_{c1} = \sum_{i=1}^{3} F_i l_i \quad U_{s1} = \sum_{i=1}^{3} \frac{1}{2} k(l_i - l_0)^2 \quad \text{for 3-SPS-U} \quad (5) $$

$$ U_{c2} = \sum_{i=1}^{4} F_i l_i \quad U_{s2} = \sum_{i=1}^{4} \frac{1}{2} k(l_i - l_0)^2 \quad \text{for 4-SPS-U} \quad (6) $$

$$ U_{3-SPS-U} = U_{c1} + U_{s1}, \quad U_{4-SPS-U} = U_{c2} + U_{s2} \quad (7) $$

In Equations 5 and 6, $U_c$ and $U_s$ represents the potential energy contributed by the cables and the springs. The applied forces along the cables were considered as $F_1$ to $F_3$ and $F_1$ to $F_4$ for the 3-SPS-U and 4-SPS-U mechanisms respectively. The total potential energy for both mechanisms is given by $U_{3-SPS-U}$ and $U_{4-SPS-U}$ in Equation 7. For the stability analysis, the springs were considered to be massless and their free lengths $l_0$ were set as 0 mm for the analysis. A meta-heuristic optimization approach using genetic algorithm in MATLAB was followed to identify the parameter $h$ and spring stiffness $k$ for the two tensegrity mechanisms at their home poses where the tilt angles are zero. As the mechanism depends on the two pose variables $\eta$ and $\phi$, it was necessary to estimate the determinant of the Hessian matrix and the value of the second-order derivative of the potential energy with respect to one of the tilt angles. For having a stable design configuration, both these parameters must be greater
<figure><img src="image_5.png" /><figcaption>Figure 4: Solutions for optimization problem obtained by Genetic Algorithm for the (a) 3-SPS-U and (b) 4-SPS-U mechanisms</figcaption></figure>

than zero. As the tensegrity mechanism is proposed to be integrated with the piping inspection robot, a preloading of around 6 N exists due to the robot weight. Thus, a preloading of 2 N and 1.5 N along each spring for the 3-SPS-U and 4-SPS-U mechanisms were considered for the analysis. The optimization problem is stated as:

Maximize: $f_{11}(\mathbf{x})$
subject to constraints: $g_1$: $\det(\mathbf{H}) \ge 0$ , $g_2$: $f_{11}(\mathbf{x}) \ge 0$ , where $\mathbf{x} = [h, k]^T$

In the above definition, $\mathbf{H}$ is the Hessian matrix and $f_{11}(\mathbf{x})$ is the second-order derivative of the potential energy with respect to one of the tilt angles. Their equations are of the form:

$$ \mathbf{H}_i = \begin{bmatrix} \frac{\partial^2 U_j}{\partial \eta^2} & \frac{\partial^2 U_j}{\partial \eta \partial \phi} \\ \frac{\partial^2 U_j}{\partial \eta \partial \phi} & \frac{\partial^2 U_j}{\partial \phi^2} \end{bmatrix} = \begin{bmatrix} f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix}, \det(\mathbf{H}_i) = f_{11}f_{22} - f_{12}^2 \quad (8) $$

where for $i = [1, 2]$, $j = [3-SPS-U, 4-SPS-U]$

The objective function aims to maximize the derivative $f_{11}$ such that it remains positive along with the determinant of the Hessian matrix throughout the optimization process. The results of solutions obtained from the optimization problem are represented below in Figure 4.

The solutions obtained by the genetic algorithm in MATLAB are depicted as black-colored scatter points over the contour of the objective function in Figure 4. It could also be observed that optimum values of $h$ are distributed between 0.6 to 0.7. For each value of $h$, an optimum value for spring stiffness $k$ was obtained. Based on a spring stiffness of 0.75 N/mm that was available at LS2N, the corresponding value of $h$ was chosen from the results, which were around 0.649 and 0.663 for each mechanism. For ease of calculations and prototyping, a common value of $h = 0.6$ was taken for both mechanisms. The stability plot for both mechanisms with the optimized design parameters under the presence of a preload is represented below in Figure 5.
<figure><img src="image_6.png" /><figcaption>Figure 5: Plot of total potential energy versus the tilt angles $\eta$ and $\phi$ for $h = 0.6$ at the home-pose condition for the (a) 3-SPS-U and (b) 4-SPS-U mechanisms</figcaption></figure>

From Figure 5, a stable configuration could be observed for both mechanisms under the presence of a preload. These design parameters will be useful to analyze the singularity and tilt limits of both architectures.

## 3.2. Singularity and workspace analysis

With the determination of stable design parameters, the next step focuses on the analysis of singularities, which provides information on the tilt limits of both architectures. This analysis also enables us to identify the best architecture that can be accommodated for the piping inspection robot. For a parallel manipulator, the singularity equation is given by the well-known equation [22]:

$$At + B\dot{\rho} = 0 \qquad (9)$$

where $t$ represents the angular velocity vector and $\dot{\rho} = [\dot{l}_1, \dot{l}_2, \dot{l}_3]^T$ represents the joint velocity vector

Parallel singularities occurs when the determinant of the direct kinematics matrix $A$ of Equation 9 vanishes. This phenomenon can be observed when the end-effector platform aligns itself with one of the springs of the mechanism. There exist no serial singularities for the mechanism as the determinant of the inverse kinematics matrix $B$ does not vanish as the length of the prismatic springs cannot zero. Since the matrix $A$ does not correspond to a $n \times n$ square matrix, parallel singularities were analyzed by sub-dividing the manipulator into three and four sets of 2-SPS-U architectures for the 3-SPS-U and 4-SPS-U mechanisms [23]. By using the Groebner base elimination technique, the joint limits equations that provided solutions to the IKP were generated using the SIROPA library in Maple [24]. The singularity equations were generated for both mechanisms to determine their workspace limits [20]. The joint limits and singularity equations were then employed to determine the singularity-free workspace for the mechanisms. By using the Cylindrical Algebraic Decomposition (CAD) technique, the real solutions were generated for the problem [25]
using SIROPA library in Maple. For isolating the aspects around home-pose,
the singularity and joint limits equations were transformed as inequalities [26]
in Maple. For generating the traces of workspace and singularities, the joint
limits were set as [$l_{min}$, $l_{max}$] = [5, 30] mm for both mechanisms. The lower
limit $l_{min}$ contributes mainly to the boundaries of the aspects centered at $\eta =
\phi = 0$ radians.

3.2.1. *Analysis of the 3-SPS-U architecture*

The workspace and singularity curves for the 3-SPS-U mechanism are rep-
resented below in Figure 6a.

<figure><img src="image_7.png" /><figcaption>Figure 6: Representation of the (a) Feasible workspace with joint limits at $l_{min} = 5$ mm and (b) singularity-free workspace with joint limits at $l_{min} = 6.5$ mm for the 3-SPS-U mechanism</figcaption></figure>

The orange regions indicate the workspace for the mechanism with singu-
larity boundaries. The joint limits for the three springs are represented by red
lines and they appear superimposed on the plots. Parallel singularities can be
observed for the mechanisms at the corners of the orange regions especially at $\eta$
$= \phi = 0$ radians. The singularity-free workspace for the mechanism is bounded
by a triangular region which has to be extracted to determine the minimum
limit for the springs. One edge of triangle is extracted at [$\eta, \phi$] = [0, $-\pi/3$] ra-
dians. At this position, one or two of the springs reach their minimum position
with no singularities. A value of 6.5 mm is estimated for the lengths $l_2$ and $l_3$
at this position. This is the minimum limit for which the spring can go to avoid
parallel singularities. For determining the remaining edges of the triangle, the
minimum limits for the other length pairs ($l_1 - l_2$ & $l_1 - l_3$) are fixed as 13.5 mm.
The values of $\eta$ are taken as $\pm\pi/3$ radians from the workspace obtained in Fig-
ure 6a. A value of 0.67 radians is found for $\phi$ at these positions. With the
modified lower limit for the springs, the workspace for the 3-SPS-U tensegrity
mechanism is recalculated and plotted as shown in Figure 6b. It could be seen
that a singularity-free workspace in the form of a Reuleaux triangle is obtained.
The minimum square zone within this triangle is bounded by the limits [$\eta, \phi$] $\in$
$[-2\pi/15, 2\pi/15]$ radians. Thus, in order to avoid singularities during operating
conditions, the 3-SPS-U mechanism can tilt up to $\pm 2\pi/15$ radians.

3.2.2. *Analysis of the 4-SPS-U architecture*

The workspace and singularity surfaces were generated for the 4-SPS-U ar-
chitecture and it is represented below in Figure 7a.
<figure><img src="image_8.png" /><figcaption>Figure 7: Representation of the (a) Feasible workspace with joint limits at $l_{min} = 5$ mm and (b) singularity-free workspace with joint limits at $l_{min} = 5.3$ mm for the 4-SPS-U mechanism</figcaption></figure>

In Figure 7a, the joint limits are represented by the colored lines, and the workspace is represented by the brown-colored region. Compared to the 3-SPS-U architecture, a wider workspace with singularity regions is observed for the 4-SPS-U mechanism. Based on the analysis, parallel singularities occur when both $\eta$ and $\phi \in [\pm\pi/3, \pm\pi/3]$ radians. At these boundaries, two of the four legs reach their minimum length. At $\eta$ equal to 0 radians, $\phi$ attains maximum tilt values of $\pm\pi/4$ radians. Using the joint limits and the minimum spring length value at the singularity boundary, the maximum tilt limits with singularity-free workspace were estimated for $[\eta, \phi]$ at $[0, \pi/4]$ radians. A tilt limit of $\pm 5\pi/18$ radians was obtained and the minimum spring length was estimated to be 5.3 mm. The recalculated workspace was in the form of a square for the 4-SPS-U mechanism and it is represented in Figure 7b. The singularity-free workspace for this mechanism is superimposed on this square, where the tilt limits are given by $\pm 5\pi/18$ radians.

### 3.2.3. Discussions

From the workspace analysis, it could be observed that the 4-SPS-U mechanism generates a higher tilt limit when compared to the 3-SPS-U mechanism. The articulation unit for the bio-inspired robot must be able to overcome pipe bends at $\pi/2$ radians in a passive mode. With the 3-SPS-U mechanism, it will be difficult to address this problem as there are possibilities that the mechanism might reach singular poses within a narrow tilt limit range. On the other hand, the 4-SPS-U mechanism offers higher tilt limits with a singularity-free workspace. The possibilities of reaching singular positions by this mechanism are comparatively less than the 3-SPS-U mechanism. The tilt limit issues for the 3-SPS-U mechanism can be addressed by using stacked modules. Based on the results of the algorithm, it can be concluded that the 4-SPS-U architecture is suitable for addressing passive and active compliance issues. The narrow tilt range of 3-SPS-U architecture restricts its application for the issue of active compliance.

### 3.3. Prototyping and control of the tensegrity mechanism

In order to demonstrate the validation of the issue of active compliance, a prototype of the 3-SPS-U tensegrity mechanism was realized at LS2N us-
ing springs (stiffness of 0.75 N/mm), universal joint and ABS platforms. A scaled-up model was constructed with $r_f = 56.7$ mm. As the mechanism is similar to a parallel robot, using the IKP solutions, the Jacobian matrices were extracted from the direct-kinematics matrices. Using a BB black microcomputer, a closed-loop PID controller law was developed to tilt the mechanism. DC-motors coupled with a high reduction gear trains were employed to have a static model. Encoders were coupled with these motors to transmit angular position data to the user. A circular trajectory was demonstrated within the singularity-free workspace zone of the tensegrity mechanism (Figure 6b).

### 3.3.1. Trajectory planning and control

For the experiments, the IKP was solved, where the tilt angles $\eta$ and $\phi$ were passed as inputs to the control loop. The pose variables and the articular variables for the mechanism are given by $\mathbf{q} = [\eta, \phi]$ and $\rho = [l_1, l_2, l_3]$ [23]. The simpler form of the IKP is provided in Equation 3. The desired angular positions of the mechanism ($\theta_{di}$) can be calculated with respect to the IKP and the measured angular positions can be calculated as a function of the motor parameters (encoder data $E$, gear reduction ratio $G$ and counts per revolution $C$). The corresponding equations are given by:

$$ \theta_{di} = \frac{(l_i - l_{home})}{r}, \quad \theta_{mi} = \frac{E_i \pi}{2CG}, \quad \text{with } i = 1, 2, 3 \qquad (10) $$

In Equation 10, the value of $l_{home}$ is 68 mm for $r_f = 56.7$ mm and $h= 0.6$. During the change of tilt angles from home-pose to the desired position, the cartesian velocities of the prismatic joints can be calculated using the equation [27]:

$$ \begin{bmatrix} \dot{l}_1 \\ \dot{l}_2 \\ \dot{l}_3 \end{bmatrix} = \mathbf{J}_c \begin{bmatrix} \dot{\eta} \\ \dot{\phi} \end{bmatrix}, \quad \text{with } \mathbf{J}_c = \begin{bmatrix} \frac{\partial l_1}{\partial \eta} & \frac{\partial l_1}{\partial \phi} \\ \frac{\partial l_2}{\partial \eta} & \frac{\partial l_2}{\partial \phi} \\ \frac{\partial l_3}{\partial \eta} & \frac{\partial l_3}{\partial \phi} \end{bmatrix} \qquad (11) $$

In Equation 11, $\mathbf{J}_c$ represents the direct kinematics matrix ($\mathbf{A}$) of the 3-SPS-U tensegrity mechanism. For estimation of intermediate positions from the home-pose to the final pose, the fifth-degree polynomial equation of Khalil [28] is employed for displacement, velocity and acceleration profiles. The values of tilt angles to perform a circular trajectory are given by:

$$ \mathbf{P}(t) = \begin{bmatrix} \eta \\ \phi \end{bmatrix} = \begin{bmatrix} R \sin(r_0 + (r_1 - r_0)s(t)) \\ R \cos(r_0 + (r_1 - r_0)s(t)) \end{bmatrix} \qquad (12) $$

In Equation 12, $R$ corresponds to the radius of the circular trajectory which is equal to the tilt angle chosen within the singularity-free workspace. The values $r_0$ and $r_1$ maps the initial and final points of circle and it goes from 0 to $-2\pi$ to perform a counter-clockwise (CCW) circular trajectory. For the control of the tensegrity mechanism, a force control algorithm was employed to reach the
desired position for input tilt angles. Through the application of motor torques and current, the mechanism was made to tilt to reach the desired position. A closed-loop feedback system was employed which tries to minimize the errors between the desired and measured positions. The classical relation for the PID control scheme which provides the motor torque, $\Gamma$ for the experimental setup is given by the equation:

$$ \Gamma = J(\ddot{\theta}_i + K_P(\theta_{di} - \theta_{mi}) + K_D(\dot{\theta}_{di} - \dot{\theta}_{mi}) + K_I \int_0^t (\theta_{di} - \theta_{mi})) \quad , \quad i = 1, 2, 3 \qquad (13) $$

In Equation 13, $\Gamma$ is the regulated torque obtained at the gearbox output shaft after PID correction. For computing the torques induced on each motor, Equation 13 was multiplied by the gear reduction ratio $G$ of the planetary gearbox. The inertia of motor-gearbox assembly $J$ was taken as $4.1e-7 \text{ kg.m}^2$ from the catalog of motor. The values for the PID terms of Equation 13 are calculated using the motor dielectric constants and resistance. The constants are given by:

$$ \omega = \frac{k_t k_e}{R J} = 14 \text{ rad/s}, \quad K_P = 3\omega^2 = 588, \quad K_D = 3\omega = 42, \quad K_I = \omega^3 = 2744 \qquad (14) $$

An overview of the experimental setup and the PID controller block is shown below in Figure 8.

<figure><img src="image_9.png" /><figcaption>Figure 8: (a) Experimental setup of the tensegrity mechanism and (b) PID controller block employed in the program</figcaption></figure>

The force control algorithm was executed in C-program with the help of a BB black microcomputer.

### 3.3.2. Results of experiments

With the interfacing being done between the prototype test bench and microcontrollers, the experiments were carried out for two orientations of the tensegrity mechanism: Vertical and Horizontal. The vertical orientation is the natural pose of the mechanism as depicted in Figure 8a. This orientation is also inline with the vertical orientation of the piping inspection robot. The horizontal
orientation is the posture when the tensegrity mechanism is coupled with the bio-inspired robot and the assembly is moving inside a horizontal pipeline. For experiments, the sampling frequency of the control loop was set at 1000 Hz. Before the circular trajectory was performed, a linear tilt was initiated to tilt the mechanism along one of the springs. This operation was simpler when compared to the equations of circular trajectory and this can be understood in detail in [29]. For smoother operations, sleep routines were introduced in the control law. Thus, the total time to perform the experiments was around 124 s. The initial linear tilt was performed from 0 to $\pi/10$ radians. The circular trajectory was created from this tilt position in the counter-clockwise direction. During the experimental cycle, two of the three prismatic springs extended to a maximum length of 84.9 mm from the home position. The springs connected to each motor also reached a minimum length of 50 mm during the circular trajectory. From the obtained results, the error was calculated between the measured and desired angular positions of the pulley after gear reduction using the equation:

$$ Error = 180 \frac{(\theta_{mi} - \theta_{di})}{\pi} \qquad (15) $$

Due to the frequency issues associated with the BB black microcomputer, higher noises were observed in the plots. In order to reduce these errors, the frequency of the results was measured in MATLAB and the Savitzky-Golay filtering method [30] was applied to reduce the noise. This filtering technique uses the least-squares method for smoothing signals without distorting it. The plot for the error data after filtering for both orientations of the mechanism is represented below in Figure 9.

<figure><img src="image_10.png" /><figcaption>Figure 9: Joint position errors along the circular trajectory for the (a) vertical and (b) horizontal orientations of the mechanism</figcaption></figure>

It could be observed that the global error range lies between -0.01° to 0.02° for vertical orientation and -0.046° to 0.06° for horizontal orientation. The difference between measured and desired data was very much smaller and this proved the effectiveness of the PID control law. At each instance of the experiment, the torque induced on each motor to perform cable pull or push was also evaluated. The representation of motor torques after application of Savitzky-Golay filter for both orientations is provided below in Figure 10.
<figure><img src="image_11.png" /><figcaption>Figure 10: Motor torques under operation along the circular trajectory for the (a) vertical and (b) horizontal orientations of the mechanism</figcaption></figure>

For the vertical orientation of the mechanism, the torque values were between -0.0021 N.m to 0.0023 N.m. However, for the horizontal orientation of the mechanism, the torque values were between -0.0025 N.m to 0.0027 N.m. This is because, the self-weight of the mechanism created an influence on the motor torques. On a global scale, these effects were comparatively smaller. With the addition of an external loading, significant differences can be observed between both orientations. Also, it must be noted that the operating torques were around one-third of the nominal motor torques and these values were well within the operating range of the DC-Motor.

### 3.3.3. *Discussions*

The experiments with the circular trajectory allows to correlate with the theory of Tilt & Torsion [31]. Initially, the mechanism was made to tilt along a given direction. Followed by that, the azimuth angle was chosen from 0 to -2π radians for the given tilt position to have a complete circular trajectory. The corresponding X – Y Euler angle equations for this trajectory were determined and passed as inputs to the control loop to execute the circular path. Owing to the higher simulation time and incorporation of intermittent sleep periods of around 20 s in the control loop, the initiation time for the circular trajectory was much longer. This phenomenon could be observed in the plots of errors and torques between 25–50 s. During this period, the motor torques were also equal to zero. When the mechanism is coupled with the bio-inspired robot, the circular trajectory can be incorporated in the control law of the entire robot assembly. Based on the pipeline profile encountered, the circular trajectory can be performed to align the axis of the robot with that of the pipeline after overcoming a bend or a junction.

## 4. Optimal design of the bio-inspired robot

When the tensegrity mechanism proposed in the earlier section was assembled with the bio-inspired piping inspection robot, the assembly proved to be oversized for passing through a 90° pipe bend or a junction. The main factors
were associated with the sizing of the spindle drive screw and the slot-follower leg mechanism. In order to overcome this issue, two optimization problems were defined and solved. The first optimization problem aims to identify the size of motor units without the presence of leg mechanisms while the second problem aims to determine the size of the leg mechanism for the results of first optimization problem.

### 4.1. First optimization problem

The first optimization problem aims to determine the sizing of motor modules of the inspection robot inside the test bench without the presence of leg mechanisms. The optimization problem was tested for pipeline diameters ranging from 70 to 160 mm. From the results of optimization, a suitable motorspindle combination unit was identified from the catalogue of Maxon. Apart from the motor sizing, the existing bio-inspired robot had issues associated with velocities and cable management. Based on results obtained from the optimization problem, the Maxon motors were identified in such a way that the spindle drive has a lower gear reduction ratio to have higher velocity. The robot was modeled as multi-body planar blocks with a tensegrity mechanism between each module in MATLAB. Each module consisted a Maxon brushless DC-motor coupled with a spindle drive. The robot geometry considered for the optimization problem is represented below in Figure 11.

<figure><img src="image_12.png" /><figcaption>Figure 11: Representation of the robot assembly with various design parameters for the first optimization problem</figcaption></figure>

In Figure 11, $[l_{k1}, w_{k1}]$ represents the length and width of motor and spindle drive unit (without the screw) for module $k$ whereas $[l_{k2}, w_{k2}]$ represents the length and width of output screw drive of spindle unit for module $k$. Here $k$ indicates a module and it assumes values from 1 to 3. The spindle screw length was one of the main factors that hindered the existing bio-inspired robot to overcome a pipe bend. This is the reason why the screw unit and motorspindle unit were considered as separate design variables. The dimensions of the tensegrity mechanism were taken from the results of Section-3. The entire robot assembly depicted in Figure 11 was simulated inside a test pipe bench constructed in MATLAB through discretization equations. At each position of the robot, the constraints were evaluated which ensured that there exists no collision between each module and the pipeline walls [32]. The first optimization problem is then defined as follows:

$$ \text{maximize:} \quad \sum_{i=1}^{n} \sum_{k=1}^{3} (l_{k1}w_{k1} + l_{k2}w_{k2}) $$
<figure><img src="image_13.png" /><figcaption>Figure 12: Representation of the design parameters of the leg mechanism during (a) declamped and (b) clamped phases</figcaption></figure>

subject to the constraints: $g_1, g_2, g_3, g_4, g_5, g_6$

where $k$ indicates the module, $i = 1..n$ indicates the discretized positions

The six inequality constraints ensured that there exists no collision between inner and outer walls of the pipeline on each module. Using the fmincon function in MATLAB, the optimization problem was solved for 70 to 160 mm diameter pipelines. Finally, a motor-spindle drive combination with $[l_1, w_1, l_2, w_2] = [51.5, 16, 45, 5]$ mm was chosen [32, 33] based on dimension and velocity factors. This motor-spindle drive combination holds good for pipeline range of 100–120 mm diameters.

## 4.2. Second optimization problem

Using the optimized dimensions of the motor-spindle unit, the dimensions of the leg mechanism were then identified. Unlike in [15], a geometric approach was followed to determine the dimensions. The design parameters of the leg mechanism for the second optimization problem are represented below in Figure 12.

The second optimization problem which is used to estimate the size of the leg mechanism is defined by:

$$ \text{maximize} \quad \sum_{i=1}^{n} \sum_{k=1}^{2} (\|\mathbf{e}_{k1} - \mathbf{e}_{k3}\| \|\mathbf{e}_{k3} - \mathbf{e}_{k2}\|) $$

subject to constraints: $g_1, g_2, g_3, g_4, h_1, h_2$

where $k$ indicates front & rear modules, $i$ indicates the discretized positions

For the second optimization problem, the central module was not taken into account as it does not accommodate the leg mechanisms. Similar to the first optimization problem, the inequality constraints $g_1$ to $g_4$ ensured that there exists no collisions between leg mechanisms and pipeline walls. The two additional equality constraints $h_1$ and $h_2$ ensured that the stroke length $\rho_1$ remains at 32 mm and the parameter $l_{s1}$ remains constant during clamping and
de-clamping modes. Upon solving the problem, the dimensions of the leg mech-
anism were found to be $l_{s1} = 57$ mm and $l_{s2} = 9.7$ mm. By assembling the
tensegrity mechanism, the DC-motor with spindle drive unit identified from the
catalogue of Maxon, the optimized slot-follower leg mechanism and standard
fasteners such as circlips, bolts, etc., the entire robot was realized in a CAD
software. The flexible robot resembles an “Elephant trunk” during a passive
mode and it is demonstrated below in Figure 13.

<figure><img src="image_14.png" /><figcaption>Figure 13: The correlation of the flexible robot assembly to an “Elephant trunk”</figcaption></figure>

**5. Static force modeling of the flexible robot assembly**

The static force modeling goes back to the models proposed in [16], how-
ever with the presence of the tensegrity modules in the assembly. When one
set of leg mechanisms (left) are clamped with the pipeline walls, the self-weight
of the robot imposes forces and moments at the CG position of the assembly.
However, the tensegrity mechanism experiences a deflection due to the weight
of the assembly. While working inside a closed environment, the robot assem-
bly can assume any orientation about the z-axis. From previous studies and
analysis, the tensegrity mechanism undergoes the maximum deformation when
one of the spring is at a distance of $r_f$ from the origin as demonstrated below
in Figure 14a.

<figure><img src="image_15.png" /><figcaption>Figure 14: The tensegrity mechanism at (a) home-pose and (b) load of the robot assembly</figcaption></figure>

The tensegrity mechanism under deflection due to the robot assembly is
represented in Figure 14b. In Figure 14b, *w* is the weight of the entire robot
assembly, which can be extracted using the digital model of the prototype and
$\phi$ is the deflection of the mechanism. This analysis is currently under study and for ensuring a static stability, the optimal design parameters of the spring such as the stiffness, number of windings and coil diameter will be determined. This study will then be further extended by considering stacked tensegrity modules. The assembly sequence of this setup resembles to that of a series-parallel hybrid manipulator and the digital model of the robot that will be studied in the future is represented below in Figure 15.

<figure><img src="image_16.png" /><figcaption>Figure 15: Representation of the flexible robot assembly with stacked tensegrity modules</figcaption></figure>

## 6. Conclusions & future works

This article mainly focused on the design of a flexible bio-inspired piping inspection robot. The operational context was first defined wherein the robot must be capable of working inside pipelines having diameters less than 150 mm subject to varying diameters as well as profiles. In-line with the context, a rigid bio-inspired caterpillar-type robot proposed by Henry et al.[15] was taken into study. Numerical force models were created for this robot and experiments were carried out on the prototype using microcontrollers inside a test pipe. A comparison was then made between the simulations and the experiments. The robot was then converted into a flexible system by the incorporation of a tensegrity mechanism, which can potentially operate in passive and active modes. Two types of tensegrity mechanisms were studied. The optimal design parameters were identified based on stability and preloading parameters for these mechanisms. Followed by that, singularity and workspace analysis were performed and it was found that the 4-SPS-U architecture was an ideal architecture that can address both active and passive compliance issues. Experiments were then carried out using a simple closed-loop PID control algorithm on the 3-SPS-U tensegrity mechanism in vertical and horizontal orientations. For given input tilt angles, the IKP was solved for the mechanisms for a circular trajectory. An optimization approach was then carried out to identify the sizing of motors and the leg mechanisms for the flexible robot to work through 90° pipe bends. From the results of optimization, the digital model of the optimized robot assembly was realized in CAD software to demonstrate an “Elephant trunk” type model.

Currently, the static force modeling of the flexible robot assembly is being carried out. An overview of the analysis was discussed in Section-5 with the 3-SPS-U architecture. Further analysis will be carried out by considering stacked 3-SPS-U mechanisms. The proposal of the stacked two-stage 3-SPS-U
mechanism can offer similar tilt limits as the 4-SPS-U. Initially and to meet the space constraints, the tensegrity mechanisms will be passive. To operate all the modules, a change in control architecture is necessary. Controllers such as EtherCAT or CANopen is proposed to be employed as they allow greater flexibility in mounting and simplifies the wiring between the motors and the control unit. The results of the static force model will help in identifying the appropriate actuator for the tensegrity modules. At present, DC motors are employed for actuation, however alternative solutions such as the Shape Memory systems are also being considered.

## References

[1] I. Kassim, L. Phee, W. S. Ng, F. Gong, P. Dario, C. A. Mosse, Locomotion techniques for robotic colonoscopy, IEEE Engineering in Medicine and Biology Magazine 25 (3) (2006) 49–56.

[2] F. Cosentino, E. Tumino, G. R. Passoni, E. Morandi, A. Capria, Functional evaluation of the endotics system, a new disposable self-propelled robotic colonoscope: in vitro tests and clinical trial, The International journal of artificial organs 32 (8) (2009) 517–527.

[3] S. Masuda, Apparatus for feeding a flexible tube through a conduit, UK patent 1 (6) (1978).

[4] S. Kim, H. Min, H. Lim, Y. Lim, J. Park, B. Kim, Magnetic impact actuator for robotic endoscope, in: Intelligent Microsystem Symposium, 2001, pp. 45–49.

[5] G. J. Iddan, D. Sturlesi, In vivo video camera system, uS Patent 5,604,531 (Feb. 18 1997).

[6] M. Drapier, V. Steenbrugghe, B. Successeurs, Perfectionnements aux cathéters médicaux, France patent 1 (1961) 15.

[7] K. Ikuta, M. Tsukamoto, S. Hirose, Shape memory alloy servo actuator system with electric resistance feedback and application for active endoscope, in: Proceedings. 1988 IEEE International Conference on Robotics and Automation, Ieee, 1988, pp. 427–430.

[8] M. Utsugi, Tubular medical instrument having a flexible sheath driven by a plurality of cuffs, uS Patent 4,148,307 (Apr. 10 1979).

[9] M. R. Treat, W. S. Trimmer, Self-propelled endoscope using pressure driven linear actuators, uS Patent 5,595,565 (Jan. 21 1997).

[10] I. Ginsburgh, J. A. Carlson III, G. L. Taylor, H. Saghatchi, Method and apparatus for fluid propelled borescopes, uS Patent 4,735,501 (Apr. 5 1988).

[11] A. Nayak, S. Pradhan, Design of a new in-pipe inspection robot, Procedia Engineering 97 (2014) 2081–2091.
[12] Y. Zhang, M. Zhang, H. Sun, Q. Jia, Design and motion analysis of a flexible squirm pipe robot, in: 2010 International Conference on Intelligent System Design and Engineering Application, Vol. 1, IEEE, 2010, pp. 527–531.

[13] Y.-S. Kwon, H. Lim, E.-J. Jung, B.-J. Yi, Design and motion planning of a two-moduled indoor pipeline inspection robot, in: 2008 IEEE International Conference on Robotics and Automation, IEEE, 2008, pp. 3998–4004.

[14] J. Park, D. Hyun, W.-H. Cho, T.-H. Kim, H.-S. Yang, Normal-force control for an in-pipe robot according to the inclination of pipelines, IEEE transactions on Industrial Electronics 58 (12) (2010) 5304–5310.

[15] R. Henry, D. Chablat, M. Porez, F. Boyer, D. Kanaan, Multi-objective design optimization of the leg mechanism for a piping inspection robot, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 46360, American Society of Mechanical Engineers, 2014, p. V05AT08A001.

[16] D. Chablat, S. Venkateswaran, F. Boyer, Mechanical design optimization of a piping inspection robot, Procedia CIRP 70 (2018) 307–312.

[17] R. V. Rao, G. Waghmare, A new optimization algorithm for solving complex constrained design optimization problems, Engineering Optimization 49 (1) (2017) 60–83.

[18] D. Chablat, S. Venkateswaran, F. Boyer, Dynamic model of a bio-inspired robot for piping inspection, in: ROMANSY 22–Robot Design, Dynamics and Control, Springer, 2019, pp. 42–51.

[19] S. Hirose, H. Ohno, T. Mitsui, K. Suyama, Design of in-pipe inspection vehicles for $\phi$ 25, $\phi$ 50, $\phi$ 150 pipes, in: Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No. 99CH36288C), Vol. 3, IEEE, 1999, pp. 2309–2314.

[20] S. Venkateswaran, D. Chablat, A new inspection robot for pipelines with bends and junctions, in: IFTOMM World Congress on Mechanism and Machine Science, Springer, 2019, pp. 33–42.

[21] L. Meirovitch, Fundamentals of vibrations, Waveland Press, 2010.

[22] C. Gosselin, J. Angeles, et al., Singularity analysis of closed-loop kinematic chains, IEEE transactions on robotics and automation 6 (3) (1990) 281–290.

[23] S. Venkateswaran, D. Chablat, Singularity and workspace analysis of 3-sps-u and 4-sps-u tensegrity mechanisms, in: International Symposium on Advances in Robot Kinematics, Springer, 2020, pp. 226–233.

[24] R. Jha, D. Chablat, L. Baron, F. Rouillier, G. Moroz, Workspace, joint space and singularities of a family of delta-like robot, Mechanism and Machine Theory 127 (2018) 73–95.
[25] D. Chablat, E. Ottaviano, G. Moroz, A comparative study of 4-cable planar manipulators based on cylindrical algebraic decomposition, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 54839, 2011, pp. 1253-1262.

[26] D. Chablat, P. Wenger, Working modes and aspects in fully parallel manipulators, in: Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No. 98CH36146), Vol. 3, IEEE, 1998, pp. 1964–1969.

[27] S. Caro, D. Chablat, P. Lemoine, P. Wenger, Kinematic analysis and trajectory planning of the orthoglide 5-axis, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 57144, American Society of Mechanical Engineers, 2015, p. V05CT08A004.

[28] W. Khalil, E. Dombre, Modeling identification and control of robots, CRC Press, 2002.

[29] S. Venkateswaran, D. Chablat, Trajectory planning for a 3-sp su tensegrity mechanism, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 85451, American Society of Mechanical Engineers, 2021, p. V08BT08A004.

[30] Savitzky-golay filtering, https://fr.mathworks.com/, accessed: 2020-08-28.

[31] S. Venkateswaran, M. Furet, D. Chablat, P. Wenger, Design and analysis of a tensegrity mechanism for a bio-inspired robot, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 59230, American Society of Mechanical Engineers, 2019, p. V05AT07A026.

[32] S. Venkateswaran, D. Chablat, P. Hamon, Design of a piping inspection robot by optimization approach, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 83990, American Society of Mechanical Engineers, 2020, p. V010T10A018.

[33] S. Venkateswaran, D. Chablat, P. Hamon, An optimal design of a flexible piping inspection robot, Journal of Mechanisms and Robotics 13 (3) (2021).