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Abstract

The article deals with development and application of snake robot for inspection pipes. The first step involves the introduction of a design of mechanical and electrical parts of the snake robot. Next, the analysis of the robot locomotion is introduced. For the curved pipe, potential field method is used. By this method, the system is able to generate path for the head and rear robot, linking the environment with obstacles, which are represented by the walls of the pipe. Subsequently, the solution of potential field method is used in inverse kinematic model, which respects tasks as obstacle avoidance, joint limit avoidance, and singularity avoidance. Mentioned approach is then tested on snake robot in provisional pipe with rectangular cross section. For this research, software Matlab (2013b) is used as the control system in cooperation with the control system of robot, which is based on microcontrollers. By experiments, it is shown that designed robot is able to pass through straight and also curved pipe.

Keywords

Inverse kinematics microcontroller obstacle avoidance pipe potential field snake robot

Introduction

A pipe inspection task is an often performed operation, which is done using a broad spectrum of robotic systems based on wheeled chassis, tracked chassis, bristled robots, or snake robots. Development of snake robots started in 1970s by Prof. Shiego Hirose. Idea of designing snake robots comes from nature and mainly because of the high-level of flexibility of snakes to their environment. High degrees of freedom (DOF) enables this kind of mechanism to be more adaptable to its rough terrain and environment.¹

A considerable amount of snake robots were developed until now, with lateral undulation locomotion using passive wheels, which represent or simulate repulsion of robot from an obstacle.^2,3 Our study concerns snake robot without any wheels, wherein its forward motion is enabled using contact forces with walls of the pipe.

Up to now, several snake robots were developed for the purpose of pipe inspection. Wakimoto et al.⁴ studied the inspection of pipe using micro snake robot. The aim of the article was to develop snake robot that would be able to pass through the pipe by overcoming the changing diameter of the pipe. The principle of locomotion is based on generating sinusoidal curve by snake robot. This research also deals with stabilization of camera mounted on the head link. Fjerdingen et al.⁵ developed snake robot with active wheels with which it is able to pass through the horizontal and vertical pipes. Robot serves in inspection, maintenance, and repair of a diverse set of pipes and pipe-like structures. A two DOF snake robot that is able to pass through straight and curved pipe is developed in this work by Brunete et al.⁶ However, this work doesn’t offer mathematical background of the proposed locomotion patterns. Maneewarn and Maneechai⁷ studied the parameters that contribute to the crawling performance of the snake robot inside an inclined pipe. The authors tested several types of curves for locomotion in the pipe. The work⁸ presents the influence of fixation curves by which snake robot can be fixed in the pipe using concertina locomotion. Everist and Shen⁹ deal with mapping of a robot environment in the pipe. Authors proposed techniques for (1) pose estimation of a snake robot using a self-posture motion model, (2) correcting error in pose estimation using only the snake’s internal configuration overtime, and (3) building environmental features using only self-occupancy and contact detection. Concertina locomotion was also used in the study by Trebuňa et al.¹⁰ The developed snake robot, LocoSnake, has unique kinematic configuration which uses one revolute and one prismatic for each link.

Based on the study of previous works on snake robots moving in the pipe as well as based on our own research in this area, our research team continue with the development of new snake robot that would able to take the nonnature kind of locomotion and also able to overcome both straight and curved pipes. The article is organized as follows.

At first, snake robot SnIPE is introduced. Next section offers view on locomotion modes in straight and curved pipes. To perform locomotion through the pipe, the designed methodology used potential field method implemented as inverse kinematics solver. The designed approach is then demonstrated on a developed snake robot.

Design of snake robot SnIPE

The research of the Department of Mechatronics, Faculty of Mechanical Engineering at Technical University of Košice (Slovakia) led to the development of snake robot named SnIPE, for pipe inspection purposes. The robot has the ability to pass through the pipe with rectangular cross section in order to perform pipe inspection. The next requirement is the ability of robot to transport material (e.g. cables) through the pipe. This robot should be able to pass through not only straight pipes but also curved pipes. The robot consists of 11 identical segments (links) with configuration, which allows motion in two-dimensional. The number of snake robot links depends on the application to which the robot is put. By designing robot as a modular system, the links of robot can be added or reduced according to the requirements of application. Head link has three-dimensional stereo camera for inspection purposes.

Design of mechanical parts

The kinematic configuration of the robot is designed so that it is able to perform planar motion. Revolute motion is ensured by servo motors Hitec HS-8380 TH, Hitec with catalogue torque 3.43 Nm, which is satisfactory for our purposes. The range of motion of servo motor is ±90°, but the mechanical design of robot decreases this angle to ±45°. Skeleton of robot is made from the aluminum ENAW5754 using waterjet cutting technology. Cover of robot link (CoRL) is made from acrylonitrile butadiene styren (ABS) using rapid prototyping technology. CoRL has a spherical shape and it ensures point contact with walls of the pipe.

Figure 1 shows the model of snake robot link. The design of CoRL involves an important element in designing mechanical parts. As was mentioned, robot is designed for motion in a pipe, and robot links will always be in contact with the walls of the pipe. It appears from this that contact force will always be acting on CoRL. Therefore, the inside of CoRL is adapted to skeleton in order to lock together and to prevent the rotation of CoRL. Composed snake robot SnIPE is shown in Figure 2.

Figure 1.

Link of snake robot.

Figure 2.

Snake robot SnIPE.

The diameter of snake robot link is L = 102 mm and weight of link is w = 406 g. Assuming all 11 robot links and also camera on head link, the whole robot length is 1045 mm and the weight of robot is 4480 g.

Design of electronic parts

Individual links are connected together through the communication interface Serial Peripheral Interface (SPI). Each robot link is controlled by the microcontroller, ATmega8-16PU, Atmel, with a clock frequency 16 MHz. This microcontroller controls the positioning of servo motor concerning the required position based on the requirements of master robot link. So, all of other links are in slave position. Master microcontroller is in rear robot link and using SPI communication it communicates with the main control system—PC through universal asynchronous receiver/transmitter (UART). Slave microcontrollers also detect the actual position of servo motor by signal from servo motor potentiometer.

Each robot link has independently powered servo motor of 7.4 V and independently control unit of 5 V from power supply by cables connected to the rear link. All robot links also have a three-axis accelerometer MMA7361LC, Freescale Semiconductor for measuring the kinematic variables during an experimental analyses. The scheme of robot control systems is shown in Figure 3. The inside view of SnIPE can be seen in Figure 4.

Figure 3.

Control system of snake robot SnIPE.

Figure 4.

View to inside of snake robot link, inside of CoRL.

Design of locomotion pattern

For pipe inspection tasks, several kinds of mechanisms such as wheel-based or track-based robots can be applied, as was mentioned earlier. From the view of control, these robots are not difficult and therefore they are often used for inspections. Next kind of inspection mechanisms are bristled mechanisms that use friction difference between bristles and walls of the pipe. However, their load capacity is very low and thus they are not suitable for more complex applications. All the mentioned mechanisms are designed mainly for straight pipes, eventually for pipes whose diameter is high enough in order to possibly pass through the curved pipes (wheeled and tracked robots).

The use of snake robots is very suitable for these applications. Their kinematic configuration allows them adapt to the roughness of terrain and so they are appropriate for complex environment, where conventional mechanisms should fail.

The core of this research lies on applying the snake robot for inspection of provisional pipe with rectangular cross section. Robot should be able to pass through straight and curved pipe as well. Within this research, the type of locomotion, the so-called traveling wave (for straight pipe) is investigated. Using this type of locomotion, robot uses interaction of its body with walls of the pipe. By gradual undulated motion with the aid of continual contact with the walls of the pipe at minimum two points, robot will perform forward motion. The following figure shows the sequence of motion of snake robot moving by traveling wave.

Of course, there are also certain restrictions to this locomotion. For example, while passing through the pipe, there have to be some contact points of snake robot links and the walls of pipe. The robot has to have at least two contact points during its straight pipe locomotion.

According to Figure 5, with respect to the geometric aspects of Figure 6, the distance travelled can be expressed during one locomotion cycle:

Figure 5.

Motion of snake robot by traveling wave.

Figure 6.

Relation between snake robot and pipe.

δ = 4 L {1 - \cos [arcsin (\frac{D - L}{2 L})]}

From equation (1), the distance travelled c δ is a function of pipe diameter D and diameter of robot link L

d = D - L

Equations (1) and (2) are correct when robot link has a spherical shape. The angle of curve, which robot describes by its body, depends on the pipe diameter D and diameter of robot link L

α = arcsin (\frac{D - L}{2 L})

According to Figure 6, the curve of snake robot presents the discrete function of sin. On one hand, within this research, narrow pipe is considered for demonstration of motion in straight pipe and also wider pipe by, which will be demonstrated, passing through the curved pipe.

Robot passing through the curved pipe is shown in Figure 7. According to Figure 7, robot reached the end of the pipe (phase 1) and now it needs to continue so that it changes its direction about 90°. During phase 2,coordinates of the point need to be determined, where robot should move its head link. By this way, it can be ensured that links 1–5 will be fixed and robot can move other links 6–11 (phase 3). During phase 3, robot bends links 6–11 so that the rear link can be moved to the corner of the pipe. In the last phase, robot reaches the initial position as in phase 1 but with the direction turned about 90°.

Figure 7.

Snake robot passing through the curved pipe.

To generate path for head and rear link during particular phases, can be snake robot in Figure 7 considered as manipulator. Manipulator has typically its base fixed to the ground, while its links are movable. The same view can be applied on the snake robot. It is important to ensure during all phases (curved pipe) that robot links form such fixed base and other links can move according to the requirements. Based on this idea, approach will be described for path generating and subsequently this generated path will be implemented in inverse kinematics solver. The methodology is demonstrated in the section “Experimental verification of designed control system with snake robot” of the article.

Mathematical background of control system

The aim of this section is introduction of basic principles for generating path for snake robot links by considering the walls of the pipe. The output of this solution is used as input to the inverse kinematic model with consideration of joint limit avoidance, obstacle avoidance, and singularity avoidance. Based on these aspects, a control system can be designed. For generating links’ path, potential field method is used. This approach will be demonstrated for motion in curved pipe.

Generating of path for head and rear link

In context of this research, path generation is understood as generating path for head and rear link with consideration of the sequence of motion according to Figure 7. Of course, the walls of pipe represent geometrical constraints which particular robot links have to avoid. Now the task is to model the walls of the pipe as a set of obstacles using the potential field method.

Potential field method

Path planning for the snake robot by potential field method is based on a simple idea. The main idea is to attract the navigated robot to goal position at the same time as navigated robot is repulsed out of obstacles. This method defines potential functions of goal position and obstacles position and generates the direction of motion.¹¹ In general, a potential field is a vector field obtained by applying a gradient operator as a function. The reference position of robot is evaluated using the higher value of potential as the goal value, which is also the global minimum of potential. The positions of obstacles are evaluated by higher potential as free space around the robot. So, the finding of suitable path for robot actually means finding of global minimum. Disadvantage of this method can be stopping robot in local minimum.¹²

Potential fields can be divided into two fields, namely, attractive and repulsive field. Total field is expressed as the sum of both¹³

U_{total} (s) = U_{att} (s) + U_{rep} (s)

where s represents position $s = {[x, y]}^{T}$ with consideration of two dimensional workspace. Output of generated path is matrix $G \in ℝ^{p x m}$ , where p is the amount of geometrical points between reference position (e.g. initial position of head link) and goal position.

Attractive potential function

An attractive potential function is a function of relative distance between head (rear) link and goal, to which the link tends. In general, attractive field can be described as¹⁴

U_{att} (s) = \frac{1}{2} ξ ∥ s_{target} - s_{robot} ∥^{k}

where ξ is the positive scaling factor, s_target is the goal position, s_robot is the position of robot, and k is the positive number higher than 0. If k = 1, then it is a linear function and if k = 2, then it is a quadratic function. Corresponding virtual attractive force can be expressed as negative gradient of attractive potential

F_{att} (s) = - \nabla U_{att} (s) = - \frac{\partial U_{att} (s)}{\partial s}

An example of phase 2 is demonstrated in Figure 8. The head link has to move into goal position, which is indicated as dark blue. Dimensions in graph are in scale.

Figure 8.

Attractive forces with focus on goal point.

Repulsive potential function

The role of repulsive potential field is to generate high potential in an area of obstacles so that gradient flow tends out of the obstacles. This ensures avoidance of collision between obstacle and robot. Repulsive function can be expressed by the following equation¹⁴

U_{rep} (s) = {\begin{matrix} η {(\frac{1}{d} - \frac{1}{d_{0}})}^{2}, d \leq d_{0} \\ 0, d > d_{0} \end{matrix}

where $d = ∥ s_{obstacle} - s_{robot} ∥$ is the distance between robot and obstacle, d₀ represents force influence distance, and η is an adjustable parameter. Corresponding gradient to this function is:

\nabla U_{rep} (s) = {\begin{matrix} η {(\frac{1}{d} - \frac{1}{d_{0}})}^{2} \frac{(s - s_{0})}{d^{3}}, d \leq d_{0} \\ 0, d > d_{0} \end{matrix}

where s is the actual position of robot and s ₀ is the distance to the obstacle. These relations consider obstacles in the form of circle. Using the above mentioned relations, environment can be described, in which the robot will be applied. For modeling of robot environment, the set of adjoining circles is used, while the distance between two adjoining centers of these circles equals to its radius. An example of a pipe modeled in the shape of L is shown in Figure 9.

Figure 9.

Repulsive forces shaping the pipe.

Graph is shown in a scale again. As can be seen from Figure 9, by suitable positioning of particular circles, it is able to generate repulsive forces that shape continuous objects. Set of obstacles shaping the pipe can be defined as $o \in ℝ^{h x m}$ , where h represents the number of circles, by which all objects are defined representing an environment of the robot. Particular parameters from the equations concerning the repulsive forces can be chosen by estimation. The pipe shown in Figure 9 was cut in the direction of axis z because of too high values. Cutting of too high values of course should be such that it doesn’t influence the results . The appearance of obstacles can be changed by corresponding parameters from equation (7).

The resultant potential field can be obtained by addition of attractive and repulsive field. Subsequently, field vector F can be determined and computed as negative gradient of potential field.

The curved pipe and initial position of head link (brown color) at the beginning of phase 2 are shown in Figure 10. The goal position into which the head link has to be moved is also shown (red color). It can be seen that the positions of obstacles act as repulsive forces. The path of head link during phase 2 is indicated as green. By analogy path can be generated for head or rear link, for particular phases of locomotion. While head link has to follow the generated path, other links have to keep such generalized variables $q \in ℝ^{n}$ , which are the outputs of inverse kinematic solver and at the same time they are input to control system of the snake robot.

Figure 10.

Resultant potential field.

Inverse kinematics task

The task of inverse kinematics is computing and time-consuming considerably more exacting task as task of direct kinematics. This is true mainly in the cases like snake robots, which can be considered as redundant or hyper-redundant mechanisms, difficult from the view of control. The problem of inverse kinematics can be defined as calculation of corresponding variables (position, velocity, and acceleration) from work space to the joint space.

Let’s have a task space trajectory x(t) and the goal is to find suitable trajectories of particular joints q(t). The differential kinematic equation is¹⁵

ẋ = J (q) \dot{q}

for q $\in ℝ^{n}$ , x $\in ℝ^{m}$ , where n is the dimension of the joint space (n = 11) and m is the dimension of the task space (m = 2). By assuming the snake robot, we get redundant mechanism and matrix J is so nonrectangular. From this reason, it is necessary to determine the so-called pseudo-inverse Jacobian

\dot{q} = J^{†} (q) ẋ

where

J^{†} = J^{T} {(J J^{T})}^{- 1}

provided that n > m (n = 11, m = 2).

The task of inverse kinematics is to position particular snake robot links to required positions. Within this research, we also assume that robot moves in environment with the obstacles (walls of pipe) and therefore there is necessary to include the obstacles avoidance task into the inverse kinematics solver. Moreover, based on the section “Design of snake robot SnIPE,” snake robot SnIPE has restriction in the form of joint limitation ±45°. This fact has to be also included into inverse kinematics solver as joint limit avoidance task.

The inverse kinematics task can be divided into several tasks. In our study, the main task is positioning of head or rear link into required position during motion in curved pipe. The additional tasks are joint limit avoidance as well as obstacle avoidance. The last task is singularity avoidance.

Joint limit avoidance task

Joint limit avoidance task is performed only at certain time. In other words, this task is activated or deactivated based on whether particular robot link is in the area of its geometrical limitation. This additional task is activated or deactivated by suitable choice of weight matrix W _l . In the area around the center of joint motion is usually zero for W _l . When the joint reaches the area with certain leakage (close area of joint limit) ρ_i , then weight matrix W _l increases from zero to maximum at the upper limit q_i _max or lower limit q_i _min. Diagonal values of this matrix are determined by¹⁶

W_{l i} = {\begin{matrix} W_{w} \leftarrow q_{i} < q_{i min} \\ \begin{matrix} \frac{W_{w}}{2} {1 + \cos [π (\frac{q_{i} - q_{i min}}{ρ_{i}})]} \leftarrow q_{i min} \leq q_{i} \leq q_{i min} + ρ_{i} \\ 0 \leftarrow q_{i min} + ρ_{i} < q_{i} < q_{i max} - ρ_{i} \end{matrix} \\ \begin{matrix} \frac{W_{w}}{2} {1 + \cos [π (\frac{q_{i max} - q_{i}}{ρ_{i}})]} \leftarrow q_{i max} - ρ_{i} \leq q_{i} \leq q_{i max} \\ W_{w} \leftarrow q_{i} > q_{i max} \end{matrix} \end{matrix}

W_w is the constant weight. This value is usually higher than the value chosen for main task and singularity avoidance task. Next, it is necessary to define Jacobian J _l for this additional task

J_{l} = \frac{\partial e}{\partial q}

where e = q, e is the additional task. From equation (13) results that Jacobian Jl is unit matrix in the case, when joint limit has to be assumed in each robot links. Jacobian is defined as $J_{l} \in ℝ^{n_{l} x n_{l}}$ where n_l is the number of robot links with limits in their motion range.

Obstacle avoidance task

In obstacle avoidance task, the solver periodically evaluates the presence of obstacle in the area of moving robot link. In our case, solver will periodically evaluate the distance between particular robot links and walls of the pipe (defined in previous section). Links of robot will be modeled as segment from the joint i at the Cartesian coordinates x_i to the joint i + 1 at the Cartesian coordinates x_{i +} ₁. Unit vector determining direction of link with length l_i is expressed as¹⁶

e_{i} = \frac{x_{i + 1} + x_{i}}{l_{i}}

Considering Figure 11, to each obstacle (also obstacle with unsymmetrical shape) the so-called obstacle active area (OAA) or influence of obstacle can be assigned, which is circular shape with radius r ₀. Let the center of the assuming obstacle has the coordinates x ₀. Then the projection of a line from joint i to the center of the OAA on the robot link i is

Figure 11.

Snake robot with obstacle.

p_{i} = e_{i}^{T} (x_{0} - x_{i})

Subsequently, the coordinates of potential point of robot collision with obstacle can be expressed as

x_{c i} = x_{i} + p_{i} e_{i}

The coordinates of potential point of robot collision are used for expression of mutual distance between this point and center of OAA

d_{c i} = ∥ x_{c i} - x_{0} ∥

Next important issue within modeling of obstacle avoidance task is to determine Jacobian J _a . In the case of snake robot, it is necessary to assume that each robot link has its own state of collision with obstacles and therefore each row of Jacobian matrix of this additional task is computed independently on other robot links. Consequently, ith row of Jacobian J _ai is

J_{a i} = - u_{i}^{T} J_{x_{a i}}

where u represents unit vector from OAA to center of obstacle and it is expressed as

u_{i} = \frac{x_{a i} - x_{0}}{d_{c i}}

Jacobian $J_{x_{a i}}$ is

J_{x_{a i}} = \frac{\partial x_{a i}}{\partial q}

Jacobian J _a includes submatrices J _ai and it is defined as $J_{a} \in ℝ^{n_{d} x n_{d}}$ , where n_d is the number of moving robot links, which can come into collision with the obstacle.

Solution of inverse kinematics

The basic idea of inverse kinematics solution in our case is to minimize the cost function. The main task is defined as $ẋ = J_{e} (q) \dot{q}$ and singularity avoidance as $\dot{s} = J_{s} (q) \dot{q}$ . Velocity of joints $\dot{q}$ is determined so that errors of particular tasks are minimalized, while high joint velocities are penalized. Based on this consideration, it is possible to write cost function as

c = {(J_{e} \dot{q} - ẋ^{d})}^{T} W_{e} (J_{e} \dot{q} - ẋ^{d}) + {(J_{l} \dot{q} - ė^{d})}^{T} W_{l} (J_{l} \dot{q} - ė^{d}) + {(J_{a} \dot{q} - {\dot{a}}^{d})}^{T} W_{a} (J_{a} \dot{q} - {\dot{a}}^{d}) + {\dot{q}}^{T} W_{s} \dot{q}

where $ẋ^{d}$ is the required main task velocity and $ė^{d}$ and ${\dot{a}}^{d}$ are required additional tasks velocities. Dimensions of particular weight matrices are $W_{e}$ (m × m), $W_{l}$ (l × l), $W_{a}$ (n × n), and $W_{s}$ (n × n). These matrices are diagonally unit matrices multiplied by coefficient of priority. This coefficient considers level of priority of main and additional tasks. For example, if the main task has 10 times more higher priority as additional task, matrix W _e will be multiplied by coefficient of priority 10, while matrix W _l will be multiplied by coefficient of priority 1.¹⁶ Jacobian J _e is defined as $J_{e} \in ℝ^{m x n}$ .

Cost function can be solved as

\frac{\partial c}{\partial {\dot{q}}^{T}} = 0

By substitution of equation (21) into equation (22), one obtains

\dot{q} = {(J_{e}^{T} W_{e} J_{e} + J_{l}^{T} W_{l} J_{l} + J_{a}^{T} W_{a} J_{a} + W_{s})}^{- 1} (J_{e}^{T} W_{e} ẋ^{d} + J_{l}^{T} W_{l} ė^{d} + J_{a}^{T} W_{a} {\dot{a}}^{d})

Now, equation (23) includes one main task that has to ensure precise positioning of robot links, two additional tasks consisting of joint limit avoidance as well as obstacle avoidance. The last task is singularity avoidance. Let’s consider $J_{e}^{T} W_{e} ẋ^{d} + J_{l}^{T} W_{l} ė^{d} + J_{a}^{T} W_{a} {\dot{a}}^{d}$ . Joint velocities $ẋ^{d}$ are nonzero, because the goal is to move robot links into required positions q $\in ℝ^{n}$ . The joint velocities have to be eliminated in this case, when particular joints reach their limits, thus term $ė^{d}$ has to be zero. Next, if some robot link comes near to the defined obstacle (walls of pipe), the solver has to avoid the collision between these objects. From this reason, the term ${\dot{a}}^{d}$ has to be also zero. The solution of inverse kinematics now can be implemented into control system and tested using snake robot SnIPE.

Experimental verification of designed control system with snake robot

Designed methodology of control system can now be demonstrated with snake robot SnIPE. Snake robot SnIPE was tested within experiments with designed methodology using straight as well as curved (shape L) provisional pipe built from modular profiles. The aim of the experiments was not to measure precise path following robot links but test methodology with mentioned locomotion modes using approaches mentioned earlier.

Control system

The algorithm of control system was implemented using software Matlab. Although having a remote control of snake robot would be a more convenient solution¹⁷ because of closed environment (pipe), the robot works on cables. The inputs into control system are basic parameters given by user, parameters such as diameter of pipe (constant along the pipe), diameter of robot link, and type of pipe (straight and curved). The scheme of the control system is shown in Figure 12.

Figure 12.

Control system.

Considering straightforward locomotion, the output of Matlab script involves a sequence of joint angles $q \in ℝ^{n}$ , changing according to particular phase of motion. PC communicates with robot through USART, so the output information from Matlab script are given to the rear robot link, which is a master link. Then master link sends necessary information to other slave links. Subsequently, each robot link controls position of servo motor to required position.

Snake robot is equipped by the sensor of obstacles, Sharp GP2Y0A41SK0F, ©SHARP Corporation, and also by stereo camera DUO3D, DUO3D™ Division. Through the PC, the user can watch information from sensor as well as record from camera. The stereo camera will serve in navigation tasks in the future.

Control algorithms

Particular algorithms used for this experimental study differ from each other, that is, according to how the robot moves through straight or curved pipe. Straightforward motion of snake robot uses the following algorithm:

Algorithm 1 Motion in Straight Pipe

Definition of basic parameters

Determination of snake robot curvature and required angles for particular phases → α

function Angles (D, L, α)

for phase = 1 to n

for i = 0 to α

send q(i) through USART: PC → Master microcontroller

read sensors data

end for

end function

Parameter phase depends on the curvature that the snake robot describes during its motion and also on the number of snake robot links of which robot consists of. Angles vector α represents the final values of generalized variables q, which are continually changed.

More difficult algorithm is used in the case of curved pipe. Now, navigation method and also inverse kinematics have to be implemented into control algorithm.

Algorithm 2 Motion in Curved Pipe

Definition of basic parameters

Determination of goal points for head and rear links $G = {[x y]}^{T}$ using potential field method

function Phase_1

for i = 1 to length of g // number of navigation points

x_required = g(i)

while $x_{r e a l} \neq x_{r e q u i r e d}$

Compute joint limits, relations for obstacle avoidance, Jacobians

$q = q_{previous} + {(M)}^{- 1} [J^{T} W_{e} (x_{required} - x_{r e a l})]$

$q_{p r e v i o u s} = q$

end while

send q (i) USART PC→ Master microcontroller

read sensors data

end for

end function

In equation (23), matrix M represents $J_{e}^{T} W_{e} J_{e} + J_{l}^{T} W_{l} J_{l} + J_{a}^{T} W_{a} J_{a} + W_{s}$ . Algorithm 2 is applied for phase 2 of passing through the curved pipe (Figure 7). Other phases work by analogous algorithms.

Algorithm for master microcontroller lies on reading the information through USART from Matlab and then sending these information about required the position of particular links to slave microcontrollers. Slave microcontrollers constantly read information from master microcontroller and also control servo motors.

Motion in straight pipe

Testing of snake robot SnIPE was performed in provisional pipe built from modular profiles. The pipe width is D = 170 mm and diameter of robot link is L = 102 mm. Straightforward motion of snake robot was applied using locomotion according to Figure 5, which consists of six phases. During testing, control algorithm works with infinite loop. The distance traveled by snake robot can be seen in Figure 13.

Figure 13.

Traveled distance in straight pipe.

The course of traveled distance changes quasi by steps, which is caused by fall time of snake robot wave from rear link to head link (see Figure 5). Each robot link includes three-axis accelerometer for purposes of determination of kinematic variables during experiments.

The record of robot motion in straight pipe is shown in Figure 14. Every 2 s, a sequence of images are obtained. It is important to say that robot moves on plastic smooth surface. When the surface would be replaced by rougher surface, which means higher coefficient of friction, this kind of locomotion should be slower and in the case of too high friction coefficient almost impossible. Nevertheless, more detailed analysis of this theme is out of range of this article.

Figure 14.

Locomotion of snake robot in straight pipe.

Head link of snake robot SnIPE includes stereo camera DUO3D MLX, DUO3D™ Division with built-in Infrared (IR) filters, whereby it can monitor the pipe. Shots from stereo camera are shown in Figure 15.

Figure 15.

Shots from stereo camera during locomotion.

DUO3D offers output in several forms. During our application, camera was used only for video recording.

Motion in curved pipe

During next experiment, algorithm 2 was tested and designed, which serves for passing through the curved pipe (shape L). The provisional pipe was built from modular profiles again. Diameter of the pipe is D = 390 mm. Input parameters like diameter of the pipe, required points for path generation $G \in ℝ^{p x m}$ as well as the definition of obstacles in environment (walls of pipe) $o \in ℝ^{h x m}$ were defined by the user in Matlab. Shots from record are shown in Figure 16.

Figure 16.

Locomotion of snake robot in curved pipe.

The experiment started by phase 2, when robot reaches the end of the pipe (phase 1 in Figure 7). In time 0–12 s, the robot used its rear links for fixation purposes, while the head link with other fore links can move to the required position of head link. After 12 s, robot starts fixing its body by fore links and its rear links move in to position parallel with the wall of pipe. Subsequently, rear link moved into pipe corner. From 28 s robot started push from pipe corner and snake robot body obtains shape that it had at the beginning of the phase 2 (Figure 7) but turned about 90°. From this moment, robot can continue with its locomotion by algorithm 1.

Conclusion and future work

The article shows application of snake robot for inspection purposes in straight and curved provisional pipe. At first, the snake robot SnIPE is introduced, which was developed for experimental analysis of pipe inspection tasks. In this article, the way of locomotion through straight and curved pipe is established. For these purposes, there was shown potential field method with modeling of the walls of pipe as robot obstacles. Output of this method (generated robot path) is subsequently used by solver of inverse kinematics. Inverse kinematics in this research includes obstacle avoidance task, joint limit avoidance task, as well singularity avoidance task. Introduced model was demonstrated on snake robot moving in the pipe. The goal of the work was verification and testing of designed methodology. According to the results obtained, designed methodology can be used for snake robot motion in pipe. The main contribution of this article is new methodology for overcoming elbow of pipe by snake robot using potential field method with cooperation of inverse kinematic model.

Within this research, robot works in cooperation with user by defining the required positions in the pipe, which were used for generating path for head and rear links. In the future, our research team would like to extend this application to be autonomous which means that robot based on its sensors equipment should consider geometric aspects between robot and pipe and be able to compute required points for path following. Robot would be fully autonomous during its locomotion and inspection of the pipe.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: A contribution to the research of measuring strategy in coordinate measuring machines. This contribution is also the result of the project implementation: Centre for research of control of technical, environmental and human risks for permanent development of production and products in mechanical engineering (ITMS: 26220120060) supported by the Research & Development Operational Programme funded by the ERDF. This article is also contribution is also result of grant project VEGA 1/0872/16 and VEGA 1/0751/16.

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An inspection of pipe by snake robot