3.1. Adaptable pipeline diameter range of the compound drive mechanism

The compound drive mechanism, a critical component for motion switching in the robot, requires rigorous analysis of its adaptability to pipeline diameters. Let the angle between the line connecting points I and H and the positive X-axis be denoted as α.

The motion trajectory of point E is confined to a linear segment along the Y-axis. When the electric linear actuator II is fully extended (maximum stroke of 30 mm), the linear distance from point E to the origin O reaches 178 mm. Defining t as the vertical coordinate of point E, the permissible range of t is constrained by:

(1) \begin{equation} E_{y}=t\!\left\{\left.t\right| \right.\left.-178\leq t\leq -148\right\} \end{equation}

The motion trajectory of point Q is also confined to a linear segment along the Y-axis. Points E and Q are fixed relative to each other, with a vertical separation of 74 mm along the coordinate axis. Consequently, the vertical coordinate of point Q satisfies:

(2) \begin{equation} Q_{y}=t+74 \end{equation}

The motion trajectory of point F is confined to a linear segment along the linear guide. Since the linear guide is parallel to the X-axis and its vertical distance from the origin O is 46 mm, the vertical coordinate of point F remains constant:

(3) \begin{equation} F_{y}=-46 \end{equation}

The motion trajectories of points Q and F combined with the Y-axis can be geometrically represented as a right triangle, where the first link serves as the hypotenuse. Given that the motion trajectory of point F lies to the right of the Y-axis and the length of the first link is l ₁ = 58 mm, the horizontal coordinate of point F can be expressed in terms of t as:

(4) \begin{equation} F_{x}=\sqrt{58^{2}-\left| t+74+46\right| ^{2}} \end{equation}

Since the motion trajectory of point I is a circular arc centered at the fixed point H, and the distance between points H and I is l_HI = 100 mm, the coordinates of point I can be expressed as:

(5) \begin{equation} \left\{\begin{array}{l} I_{x}=-74-100\cos \alpha \\ I_{y}=-82-100\sin \alpha \end{array}\right. \end{equation}

The motion trajectory of point J is also a circular arc centered at the fixed point H. Given the distance between points H and J as l_HJ = 32 mm and the angles ∠IHJ = ∠I $^{\prime}$ H $^{\prime}$ J $^{\prime}$ = 55°, the coordinates of point J satisfy:

(6) \begin{equation} \left\{\begin{array}{l} J_{x}=-74+32\cos \left(180^{\circ}-\alpha -55^{\circ }\right)\\[3pt] J_{y}=-82-32\sin \left(180^{\circ}-\alpha -55^{\circ }\right) \end{array}\right. \end{equation}

The motion trajectories of points J and F, combined with the linear guide, form a right triangle where the second link serves as the hypotenuse. Given the length of the second link as l ₂ = 68 mm, the geometric relationship satisfies:

(7) \begin{equation} {l_{2}}^{2}=\left(F_{x}-J_{x}\right)^{2}+\left(F_{y}-J_{y}\right)^{2}=68^{2} \end{equation}

The horizontal coordinate of point F can be expressed in terms of the angle as:

(8) \begin{equation} F_{x}=\sqrt{{l_{2}}^{2}-\left(F_{y}-J_{y}\right)^{2}}+J_{x} \end{equation}

Through the representation of the x-coordinate of point F in Eqs. (4) and (8), a deterministic relationship between variables t and α in the form of an implicit function was derived. This implicit function was then input into MATLAB to generate a scatter plot, and curve fitting was subsequently performed using the fitting tool, as illustrated in Figure 4 Within reasonable error margins, the fitting results indicated a linear relationship between variables t and α. The fitted relationship can be expressed as:

(9) \begin{equation} \alpha =2.021t+440.3 \end{equation}

Figure 4.

Relationship between t and α.

The coordinates of point I can be expressed in terms of t:

(10) \begin{equation} \left\{\begin{array}{l} I_{x}=74+100\cos \left(2.021t+440.3\right)\\[3pt] I_{y}=-82-100\sin \left(2.021t+440.3\right) \end{array}\right. \end{equation}

To calculate the adaptable pipe diameter range of the compound drive mechanism, it is necessary to determine the diameter of the circumcircle defined by the three points projected onto the Cartesian coordinate system OXY – specifically, the axial drive wheel and the two circumferential drive wheels – when the electric linear actuator II is retracted to its shortest length and extended to its longest length. Let the pipe diameter be d. Based on the positional relationships described above and the coordinates of point I expressed in terms of t in Eq. (10), the following equation can be established:

(11) \begin{equation} d=2\sqrt{I_{x}^{2}+I_{y}^{2}}-50 \end{equation}

The electric linear actuator II has a stroke of 30 mm. However, due to geometric constraints imposed by the lengths and angles of link I, link II, and the L-shaped link in the compound drive mechanism, the vertical coordinate of the axial drive wheel center (point E) becomes t = −162.5 mm when the electric linear actuator II is fully retracted. At this configuration, the circumcircle diameter is calculated as d = 74 mm, corresponding to a minimum adaptable pipe diameter of 74 mm for the compound drive mechanism. Conversely, when the electric linear actuator II is fully extended, the vertical coordinate of point E shifts to t = −178 mm, resulting in a circumcircle diameter of d = 304 mm, which defines the maximum adaptable pipe diameter. Therefore, the adaptable pipe diameter range of the compound drive mechanism is:

(12) \begin{equation} d\in \left[74,304\right] \end{equation}

3.2. Adaptable pipe diameter range of the clamping mechanism

To ensure effective pipe clamping and obstacle-crossing capabilities, the clamping mechanism must satisfy the following geometric condition: when the two clamping arms grip the pipe, the projected angles between the lines connecting the pipe center to the contact points of the universal wheels (fixed at the arm ends) and the Cartesian coordinate system OXY are required to maintain 120°.

Given that the angle between the line connecting points D and C and the negative horizontal axis of the coordinate system is defined as θ, with point C fixed at coordinates (−137, −253), the motion trajectories of points B and D are arc trajectories with fixed point C as the center. The distance between points C and B is specified as l_CB = 102.5 mm, and between points C and D as l_CD = 104 mm. Consequently, the coordinates of points D and B satisfy the following relationships:

(13) \begin{align}& \left\{\begin{array}{l} B_{x}=-137-102.5\cos \left(173^{\circ }-\theta \right)\\[3pt] B_{y}=-253+102.5\sin \left(173^{\circ }-\theta \right) \end{array}\right. \end{align}

(14) \begin{align}&\qquad \left\{\begin{array}{l} D_{x}=-137-104\cos \theta \\[3pt] D_{y}=-253-104\sin \theta \end{array}\right. \end{align}

Let B ₁ be the extreme position of the electric linear actuator I when its stroke L ₁ = 0 mm, with its coordinates measured as (−185.9, −163.17). Substituting the coordinates of B ₁ into Eq. (13), the following result is obtained:

(15) \begin{equation} \theta _{1}\approx 111.5^{\circ} \end{equation}

Therefore, under this extreme position, the coordinates of point D ₁ and the distance between the centers of the two universal wheels are determined as follows:

(16) \begin{align}& \left\{\begin{array}{l} D_{1x}=-137-104\cos \theta _{1}\approx -98.9\\[3pt] D_{1y}=-253-104\sin \theta _{1}\approx -349.8 \end{array}\right. \end{align}

(17) \begin{align}&\qquad\,\,\, l_{DD_{1}^{}}=2\times 98.9=197.8 \end{align}

Based on the geometric configuration of triangle MDDʹ, the maximum adaptable pipe diameter d of the clamping mechanism is determined as:

(18) \begin{equation} d_{\max }=2\times \left(\frac{\frac{l_{DD_{1}^{\prime}}}{2}}{\sin \frac{120^{\circ }}{2}}-\frac{25}{2}\right)=203.4 \end{equation}

Let B ₂ represent the extreme position of the electric linear actuator I when its stroke L ₁ = 100 mm; the coordinates of B ₂ are determined as (−239.4, −247.5). Substituting the coordinates of B ₂ into Eq. (13), the following result is obtained:

(19) \begin{equation} \theta _{2}\approx 170^{\circ} \end{equation}

Therefore, under this extreme position, the coordinates of point D ₂ and the distance between the centers of the two universal wheels are determined as follows:

(20) \begin{align}& \left\{\begin{array}{l} D_{2x}=-137-104\cos \theta _{2}\approx -34.6\\[3pt] D_{2y}=-253-104\sin \theta _{2}\approx -271.2 \end{array}\right. \end{align}

(21) \begin{align}&\qquad\,\,\, l_{DD_{2}^{}}=2\times 34.6=69.2 \end{align}

Based on the geometric configuration of triangle MDDʹ, the minimum adaptable pipe diameter d of the clamping mechanism is determined as:

(22) \begin{equation} d_{\min }=2\times \left(\frac{\frac{l_{DD_{2}^{\prime}}}{2}}{\sin \frac{120^{\circ }}{2}}-\frac{25}{2}\right)=54.9 \end{equation}

Therefore, the adaptable pipe diameter range of the clamping mechanism is determined as:

(23) \begin{equation} d\in \left[54.9,203.4\right] \end{equation}

From Eqs. (12) to (23), the adaptable pipe diameter ranges calculated based on the compound drive mechanism and the clamping mechanism are combined by taking the intersection of the results from both mechanisms. After rounding the calculated diameter values, the theoretical adaptable pipe diameter range of the robot is determined as:

(24) \begin{equation} d\in \left[74,\,203\right] \end{equation}

The geometric parameters of the robot’s links, the key mechanical properties, and the primary results derived from the preceding analysis are consolidated in Table II. This table serves as a definitive reference for the specific numerical values assigned to the symbolic parameters used in the kinematic model and will be employed in the subsequent mechanical analysis.

4.1. Driving torque analysis in axial forward motion

To determine the required motor driving torque during axial forward inspection, the force analysis of the axial drive wheel system and universal wheels under clamped pipeline conditions is established, as illustrated in Figure 5.

Figure 5.

The free-body diagram of robot moving axially along the pipe.

Key parameters include: point P is Robot’s center of gravity, F _p1a , F _p1b , F _p2a , and F _p2b are Pipeline support forces acting on four universal wheels during axial motion, F _d1 and F _d2 are Pipeline support forces on two axial forward drive wheels, T_a is Minimum required axial driving torque, T _f1 and T _f2 are Rolling resistance torques of axial drive wheels and universal wheels respectively, F _f1 and F _f2 are tangential forces exerted by the pipeline on axial drive wheels and universal wheels, γ is Pipeline inclination angle, G is Robot’s gravitational force, rolling friction coefficient for axial drive wheels and universal wheels: ε = 0.01 mm, radius of circumferential drive wheels is r ₁ = 25 mm, radius of universal wheels is r ₂ = 12.5 mm, static friction coefficients between wheels and pipe wall are μ ₁ (axial drive wheels), μ ₂ (universal wheels), F_d is Pipeline reaction force on the axial drive mechanism, F _p1 and F _p2 are reaction forces on clamping arms. During axial forward inspection, the reaction forces F_d and F_p remain invariant with respect to pipe diameter variations; these reaction forces are constant parameters in the mechanical model.

During axial inspection, the robot maintains force equilibrium at any instant. Specifically, the resultant force composed of the robot’s gravity and the pipe’s normal reaction force perpendicular to the pipe axis equals zero. Simultaneously, the static friction forces generated by the axial driving wheels and caster wheels clamping onto the pipe counterbalance the component of the robot’s gravity along the pipe axis. This yields the following equilibrium equations:

(25) \begin{equation} \left\{\begin{array}{l} F_{d}=F_{d1}+F_{d2}\\ F_{p1}=F_{p1a}+F_{p1b}\\ F_{p2}=F_{p2a}+F_{p2b}\\[3pt] \left(F_{p1}+F_{p2}\right) \cos 60^{0}=F_{d}+G\cos \gamma \\[3pt] F_{d}\mu _{1}+\left(F_{p1}+F_{p2}\right)\mu _{2}=G\sin \gamma \end{array}\right. \end{equation}

From Eq. (26), the following relationship can be derived:

(26) \begin{align} \left\{\begin{array}{l} F_{P1}=F_{P2}=\dfrac{G\sin \gamma +G\mu _{1}\cos \gamma }{2\mu _{2}+\mu _{1}}\\[10pt] F_{d}=\dfrac{G\sin \gamma -2\mu _{2}G\cos \gamma }{2\mu _{2}+\mu _{1}} \end{array}\right. \end{align}

(27) \begin{align} \left\{\begin{array}{l} F_{f1}=\left(F_{d1}+F_{d2}\right)\mu _{1}=F_{d}\mu _{1}\\[3pt] F_{f2}=\left(F_{p1}+F_{p2}\right)\mu _{2}\\[3pt] T_{f1}+T_{f2}=\varepsilon \left(F_{d1}+F_{d2}+F_{p1}+F_{p2}\right) \end{array}\right. \end{align}

To ensure normal inspection operations, the robot must overcome both the rolling friction torque generated by the axial driving wheels and the universal wheels during motion, as well as the downward gravitational component along the inclined pipe. With the static friction coefficients set as μ ₁ = μ ₂ = 0.5 and the gravitational force G = 100N, combining Eqs. (26) and (27), the minimum driving torque T_a required for the dual-axis geared motor can be expressed as:

(28) \begin{align} T_{a}&=T_{f1}+T_{f2}+F_{f1}\cdot r_{1}+F_{f2}\cdot r_{2}=\varepsilon \left(F_{d1}+F_{d2}+F_{p1}+F_{p2}\right)+F_{d}\mu _{1}\cdot r_{1}+\left(F_{p1}+F_{p2}\right)\mu _{2}\cdot r_{2}\nonumber\\ &=\varepsilon \left(\dfrac{G\sin \gamma -2\mu _{2}G\cos \gamma }{2\mu _{2}+\mu _{1}}+2\times \dfrac{G\sin \gamma +G\mu _{1}\cos \gamma }{2\mu _{2}+\mu _{1}}\right)+\dfrac{G\sin \gamma -2\mu _{2}G\cos \gamma }{2\mu _{2}+\mu _{1}}\mu _{1}\cdot r_{1}+2\nonumber\\ &\quad\times \dfrac{G\sin \gamma +G\mu _{1}\cos \gamma }{2\mu _{2}+\mu _{1}}\mu _{2}\cdot r_{2}\nonumber\\ & =\dfrac{3G\left(\sin \gamma +\mu _{1}\cos \gamma \right)\varepsilon }{2\mu _{2}+\mu _{1}}+\dfrac{G\mu _{1}r_{1}\left(\sin \gamma -2\mu _{2}\cos \gamma \right)}{2\mu _{2}+\mu _{1}}+\dfrac{2G\mu _{2}r_{2}\left(\sin \gamma +\mu _{1}\cos \gamma \right)}{2\mu _{2}+\mu _{1}}-G\varepsilon \cos \gamma \end{align}

Using MATLAB, the above relationships were visualized to demonstrate the variation of the minimum required driving torque T _a for the dual-axis geared motor with respect to the pipe inclination angle γ. As shown in Figure 6, (Note: Data for γ < 30° are omitted in the figure due to their negligible magnitudes, which lack practical significance.) the required axial drive torque T_a increases with the pipe inclination angle γ. When the pipe inclination angle γ = 90°, i.e., when the pipe is in a vertical orientation, the required drive torque reaches its maximum value of approximately 0.84 N·m. This maximum value represents the minimum design torque that the dual-shaft reduction motor must deliver to ensure stable operation across all inclination angles.

Figure 6.

Relationship between γ and T _a. .

4.2. Driving torque analysis in circumferential rotation mode

A mechanical analysis of the circumferential drive wheel system and caster wheel system during clamping and rotating around the pipe is established, as illustrated in Figure 7. Point M represents the robot’s rotational center, coinciding with the pipe’s cross-sectional centroid. Key parameters include:

Figure 7.

The free-body diagram of robot rotating circumferentially around the pipe.

$F^{\prime}_{p1}$ and $F^{\prime}_{p2}$ are normal support forces exerted by the pipe on the left and right clamping arms during circumferential rotation. F _n1 and F _n2 are normal forces from the pipe acting on the left and right circumferential drive wheels, respectively. $F^{\prime}_{f1}$ and $F^{\prime}_{f2}$ are tangential reaction forces from the pipe on the circumferential drive wheel and universal wheel. $T^{\prime}_{f1}$ and $T^{\prime}_{f2}$ are rolling resistance torques generated by the circumferential drive wheel and universal wheel. T_b is the minimum required driving torque of the DC reduction motor. ω is between the lines of action of the reactive forces F _n1 and F _n2 and the robot’s symmetry axis. τ is the angular deviation between the robot’s symmetry axis and the gravitational direction.

The rolling friction coefficient of the circumferential drive wheel is set as $\varepsilon$ = 0.01 mm, and μ ₃ denotes the static friction coefficient between the circumferential drive wheel and the pipe wall. The parameter represents the distance from the robot’s center of gravity to the rotational center. To ensure that the projected angle between the contact points of the universal wheels (fixed to the two clamping arms) and the pipe center onto the Cartesian coordinate system OXY is 120°, the angular parameter ω (defined as the angle between the reaction forces and the robot’s symmetry axis) varies with the pipe diameter d. This occurs because the position of the pipe center relative to the robot shifts when clamping pipes of different diameters. The relationship between ω and d is derived from the coordinates of the pipe center before and after the robot’s motion mode transition.

The radius of the circumferential drive wheel is 25 mm. Prior to the motion mode transition, the coordinates of the pipe center M in the Cartesian coordinate system are determined from Eq. (9) as follows:

(29) \begin{equation} M_{y1}=t-25-\frac{d}{2}=\frac{\alpha -440.3}{2.021}-25-\frac{d}{2} \end{equation}

From Eq. (4), the coordinates of point I, the center of the circumferential drive wheel, can be expressed in terms of α. When the robot clamps a pipe of identical diameter and undergoes motion mode transition, the position of the pipe’s cross-sectional center remains unchanged. By analyzing the contact condition between the circumferential drive wheel and the pipe wall and combining Eq. (29), it is established that α and the pipe diameter d exhibit an implicit functional relationship, as shown in Eq. (30):

(30) \begin{equation} I_{x}+I_{y}=\left(-74-100\cos \alpha \right)^{2}+\left[-82-100\sin \alpha -M_{y1}\right]^{2}=\left(\frac{d}{2}+25\right)^{2} \end{equation}

This implicit function was input into MATLAB to generate a scatter plot of the relationship. A curve-fitting tool was then applied to approximate the functional form, with the results illustrated in Figure 8.

Figure 8.

Relationship between α and d.

Within reasonable error margins, the fitting results indicate a linear relationship between the variable α and the pipe diameter d. The fitted relationship can be expressed as:

(31) \begin{equation} \sin \omega =\frac{I_{x}}{\frac{d}{2}+25}=\frac{74+100\cos \left(\frac{247.45-d}{1.47}\right)}{\frac{d}{2}+25} \end{equation}

During rotational obstacle-avoidance maneuvers, the robot maintains force equilibrium at any instant. Specifically:

Vertical Force Balance (perpendicular to the pipe axis): the component of the robot’s gravitational force perpendicular to the pipe is balanced by the corresponding normal reaction force exerted by the pipe.

Tangential Moment Balance (at wheel-pipe contact points): the static friction torque generated by the circumferential drive wheel and universal wheel clamping onto the pipe counteracts the gravitational torque induced by the angular deviation τ between the robot’s symmetry axis and the gravitational direction.

This dual equilibrium condition ensures stable rotational motion and is mathematically expressed as:

(32) \begin{equation} \left\{\begin{array}{l} F_{p1}^{\prime}\times 66\% \cos60^{\circ}+F_{p2}^{\prime}\times 66\% \cos60^{\circ}=\left(F_{n1}+F_{n2}\right)\cos \omega +G\cos \tau \\[5pt] Gh\sin \tau =\dfrac{\left(F_{n1}+F_{n2}\right)\mu _{3}d}{2}+\dfrac{d\left(F_{p1}^{\prime}+F_{p2}^{\prime}\right)\mu _{2}}{2} \end{array}\right. \end{equation}

From Eq. (32), the following relationship can be derived:

(33) \begin{equation} \left\{\begin{array}{l} F_{n1}+F_{n2}=\dfrac{2G\left(\dfrac{h\times \sin \tau }{\varphi }-\cos \tau \mu _{2}\right)}{\mu _{3}+2\mu _{2}\cos \omega }\\[12pt] F_{p1}^{\prime}=F_{p2}^{\prime}=\dfrac{2\mathit{Gh}\cos \omega \sin \tau }{\varphi \mu _{3}+2\varphi \mu _{2}\cos \omega }+\dfrac{\mu _{3}G\cos \tau }{\mu _{3}+2\mu _{2}\cos \omega } \end{array}\right. \end{equation}

To ensure successful rotational obstacle avoidance, the circumferential drive motor must overcome both the rolling friction torques from the circumferential drive wheel system and universal wheel auxiliary system, as well as the gravitational torque generated by the elevation of the robot’s center of gravity, which causes angular deviation (τ) of the symmetry axis from the gravitational direction. The minimum driving torque T _b required for the DC reduction motor must satisfy:

(34) \begin{align} T_{b}&=\dfrac{T_{f1}^{\prime}+T_{f2}^{\prime}+Gh\sin \tau }{2}=\dfrac{\varepsilon \left(F_{n1}+F_{n2}+F_{p1}^{\prime}+F_{p2}^{\prime}\right)+Gh\sin \tau }{2}\nonumber\\& =\dfrac{\varepsilon \left[\dfrac{2G\left(\dfrac{h\times \sin \tau }{\varphi }-\cos \tau \mu _{2}\right)}{\mu _{3}+2\mu _{2}\cos \omega }+2\times \left(\dfrac{2\mathit{Gh}\cos \omega \sin \tau }{\varphi \mu _{3}+2\varphi \mu _{2}\cos \omega }+\dfrac{\mu _{3}G\cos \tau }{\mu _{3}+2\mu _{2}\cos \omega }\right)\right]+Gh\sin \tau }{2} \end{align}

During pipe clamping, the robot actively adjusts the positions of its clamping arms to ensure that the projected angle between the contact points of the universal wheels and the pipe’s centroid on the Cartesian coordinate system OXY is maintained at 120°. Consequently, when clamping pipes of varying diameters, the distance h from the robot’s center of gravity to the pipe’s cross-sectional centroid remains approximately constant, with h = 120 mm. Using the static friction coefficient μ ₃ = 0.5, Eq. (35) was visualized in MATLAB to analyze the relationship between the required driving torque T_b , pipe diameter d, and the rotational angle τ of the pipe’s centroid. As shown in Figure 9, to ensure the robot can overcome the gravitational torque throughout the entire circumferential rotation process, the drive motor must be capable of providing a torque no less than this maximum value. Analysis indicates that the maximum torque occurs at τ = 90°, where T_b reaches a peak value of approximately 1.23 N · m. Thus, the minimum drive torque required for the DC gear motor is approximately 1.23 N · m.

Figure 9.

Relationship between τ, d and T _b .

5.1. Analysis of climbing performance on inclined pipelines

Due to terrain constraints, pipelines may be installed with certain inclination angles during actual deployment, which imposes higher requirements on the climbing capability of inspection robots.

Different tilt angles of the pipeline were achieved by adjusting the components of gravitational acceleration along coordinate axes. After configuring the simulation environment, the inspection process along inclined pipelines was simulated. Numerical values of centroid deflection angles were obtained for the robot operating on 110 mm outer diameter pipelines with 60° and 90° tilt configurations, as shown in Figure 10.

Figure 10.

Simulation analysis of robot climbing performance on inclined pipelines.

As can be seen from the analysis results, during the inspection process where the robot clamps onto the pipeline and moves axially, the centroid deflection angle exhibits significant fluctuations when the robot initially grips the pipeline but rapidly stabilizes into a steady-state phase. When operating on pipelines inclined at 60°, the centroid deflection angle in the steady-state phase remains below 5°, while on pipelines inclined at 90°, this angle does not exceed 7°. Based on the overall data, the robot demonstrates stable operation on pipelines with inclination angles ranging from 0° to 90°.

5.2. Analysis of circumferential rotation obstacle avoidance performance

Under real-world working conditions, when the robot performs inspections on horizontal pipelines and encounters obstacles such as T-shaped pipeline junctions ahead, it requires a maximum rotation of 180° to complete the obstacle avoidance maneuver. In this simulation, a motion path was designed for the robot to rotate from the lowest to the highest point of the pipeline. The circumferential rotation capability of the robot was evaluated by analyzing the displacement of its centroid position and centroid velocity during this motion. The simulation of the robot rotating from the lowest to the highest point on a horizontal pipeline is illustrated in Figure 11(a).

Figure 11.

Simulation analysis of rotating obstacle avoidance for robots in horizontal pipeline.

Through post-processing, the velocity of the robot’s centroid along the non-rotational axis and its displacement along the rotational axis direction during circumferential rotation on a horizontal pipeline were plotted and exported, as shown in Figure 11(b). During the circumferential rotation motion on the horizontal pipeline, the velocity of the centroid along the non-rotational axis and its positional variation along the rotational axis exhibited significant fluctuations when the robot initially contacted the pipeline but quickly stabilized into a steady-state phase. Throughout the motion, gravity performed negative work. In the time interval of 0–6.5 s, the displacement of the robot’s centroid along the non-rotational axis changed dramatically due to the gradually increasing gravitational torque generated by the gravity acting on its centroid, reaching a maximum around 6.5 s. During 6.5–13 s, both the centroid displacement and velocity variations stabilized as the gravitational torque progressively decreased, attaining a minimum around 13 s. Over the entire process, the robot’s velocity remained stable, with a maximum centroid displacement of approximately 0.5 mm from its initial position. Given the robot’s total length of 270 mm, the relative displacement ratio was about 1.85‰.

In practical working conditions, when encountering front obstacles such as T-shaped pipes on vertical pipelines, the robot needs to perform circumferential rotational motion along the pipe. In this simulation, a motion path was designed for the robot to rotate from the right side to the left side of the pipe. The circumferential rotational capability of the inspection robot was validated by analyzing data such as the radial deflection of the centroid position and centroid velocity, as shown in Figure 12(a).

Figure 12.

Simulation analysis of rotating obstacle avoidance for robots in vertical pipeline.

Through post-processing, the velocity of the robot’s centroid perpendicular to the rotational axis and its radial deflection relative to the rotational axis during circumferential rotation on the vertical pipe were plotted and exported, as shown in Figure 12(b). During the circumferential rotation along the vertical pipe, the centroid velocity along the non-rotational axis exhibited significant fluctuations at initial pipe contact, followed by rapid stabilization. Throughout the motion, the velocity profile remained relatively smooth. Since the preset motion direction was perpendicular to the gravitational direction, a slight downward slippage of approximately 1.6 mm along the pipe occurred over the 13-second operation period. Given the robot’s total length of 270 mm, this corresponds to a deflection ratio of approximately 5.93‰. The simulation results confirm that the robot maintains stable operational performance during circumferential rotation around the pipe, demonstrating effective obstacle-avoidance capability through circumferential rotational motion.

6.1. Pipe diameter adaptability test

To rigorously evaluate the robot’s adaptability across its theoretical design range of 74–203 mm, kinematic verification and axial locomotion tests were conducted on four distinct PVC pipelines with outer diameters of 75, 110, 160, and 200 mm. The physical experimental setups, illustrating the robot’s adaptive clamping posture on each diameter, are presented in Figure 14. These specifications were selected to represent the lower bound, intermediate gradients, and upper bound of the operational scope, ensuring a comprehensive assessment of the clamping mechanism’s versatility. The adaptation to these varying curvatures was achieved by precisely regulating the extension of the Electric Linear Actuator II based on the kinematic model established in Section 3. For the compact 75 mm pipeline (Figure 14(a)), the actuator was fully retracted to fold the clamping arms inward; conversely, for the 160 and 200 mm pipelines (Figures 14(c) and 14(d)), the actuator was progressively extended to maximize the arm opening, ensuring the universal wheels maintained the optimal 120° contact geometry required for stable clamping.

Figure 14.

Experimental setup of the robot clamping onto pipelines with varying outer diameters.

Quantitative performance tests involved repeated axial traversals over a fixed 1.0-m distance for each diameter to minimize stochastic errors. The temporal data for each trial was recorded and converted into average velocity to assess locomotion consistency. The consolidated results, presented in Table II, indicate a high degree of stability across the entire diameter spectrum. The robot demonstrated robust locomotion capabilities on all four pipeline specifications, with standard deviations of speed remaining below 0.03 m/s, corroborating the reliability of the transmission system under varying geometric constraints.

Table II.

Axial movement velocity test results across varying pipe diameters.

A comparative analysis of the velocity data reveals a remarkable consistency in performance. While the average speeds for the 75 and 110 mm pipes were identical (0.195 m/s), a marginal attenuation was observed as the diameter increased further, reaching 0.193 m/s for the 160 mm pipe and 0.191 m/s for the 200 mm pipe. This slight deceleration (approximately 2.1% total drop) is physically consistent with the increased viscoelastic deformation of the rubber wheel coating induced by the altered contact patch geometry on larger curvatures. Despite this minor variation, the robot maintained stable traction without slippage across all tests. This performance validates the core advantage of the proposed dual-slider mechanism: by decoupling the clamping range from the drive unit, it ensures uniform normal force distribution and consistent traction regardless of the pipe diameter, thereby proving its effectiveness for complex industrial pipeline networks.

6.2. Evaluation of climbing performance and stability on inclined pipelines

To comprehensively evaluate the robot’s performance in inclined pipelines, a series of climbing experiments were conducted on a PVC pipeline with an outer diameter of 110 mm. The experiments focused on three key performance indicators: climbing speed stability, slippage analysis, and maximum load-bearing capacity across inclination angles of 0°, 30°, 45°, 60°, and 90°. The climbing process is illustrated in Figure 15.

Figure 15.

Robotic incline climbing capability test.

Tests demonstrated that the robot maintained stable operation in the 90°-inclined pipeline, validating its robust adaptability to complex operational conditions.

6.2.1. Climbing speed and slippage analysis

The robot was positioned at the bottom of the pipeline, and its climbing velocity was recorded after reaching a steady state. The relationship between the axial speed and the inclination angle (γ) is illustrated in Figure 16.

Figure 16.

Relationship between axial speed and pipeline inclination angle.

The results indicate a non-linear decrease in speed as the inclination increases, dropping from 0.195 m/s at 0° (horizontal) to 0.111 m/s at 90° (vertical). To quantify the stability of the locomotion, the slip ratio (λ) was introduced and calculated as follows:

(35) \begin{equation} \lambda =\frac{v_{\textit{ideal}}-v_{exp}}{v_{\textit{ideal}}}\times 100\% \end{equation}

where v _ideal is the theoretical speed determined by the wheel rotation speed and wheel radius, and v _exp is the measured experimental speed.

The analysis reveals two primary factors contributing to the speed reduction (∼43%) at vertical states:

1. 1. Motor Hard Mechanical Characteristic: As analyzed in Section 4.1 (Driving Torque Analysis), the required driving torque T_a increases with the gravitational component G sinγ. The selected DC motor (Model 5882-50ZY) exhibits a characteristic drop in rotational speed as the output torque increases.

2. 2. Micro-slippage: At γ = 90°, the slip ratio reaches approximately λ ≈ 15%. This is primarily caused by the elastic deformation of the rubber coating on the wheels and micro-slippage at the PVC interface under high traction loads. However, no continuous macroscopic slippage or "run-away" incidents were observed, confirming that the friction provided by the clamping mechanism is sufficient for vertical climbing.

6.2.3. Theoretical maximum load capacity analysis

To assess the robot’s payload potential, the maximum load capacity was estimated based on the mechanical model established in Eq. (28) and the motor parameters listed in Table I.

At a 90° inclination, the required torque for the robot’s self-weight (7.5 kg) is calculated as T _req ≈ 0.84 N·m (see Figure 6). Given the motor’s rated continuous torque of T _rated = 4.9 N·m, the available torque margin is:

(36) \begin{equation} T_{\textit{margin}}=T_{\textit{rated}}-T_{req}=4.9-0.84=4.06\,\mathrm{N}\, \mathrm{m} \end{equation}

This substantial torque margin suggests that, limited strictly by motor power, the robot could theoretically support a significantly higher payload. However, in practical applications, the load capacity is constrained by the static friction coefficient between the wheels and the pipe (μ ≈ 0.5). Conservative estimates indicate the robot can carry an additional payload of 5–8 kg (approximately 100% of its own weight) on vertical PVC pipes without experiencing gross slippage. This capacity meets the requirements for carrying standard inspection sensors and cameras.

6.3. Rotational obstacle avoidance and traversal capability test

The robot’s rotational obstacle avoidance capability distinguishes it from conventional inspection robots. To validate this functionality, a PVC pipe system with a reducing tee fitting (110 mm main pipe diameter, 75 mm branch pipe diameter) was constructed as the test platform, as shown in Figure 17.

Figure 17.

Obstacle avoidance and traversal maneuvering sequence of the robot.

When encountering the reducing T-junction obstacle (Figure 17), the robot first activated its dual-slider multi-link mechanism for locomotion mode transition (Figure 17(a)). Subsequently, the circumferential drive mechanism rotated the robot along the pipe until optimal positioning was achieved, followed by secondary mode switching (Figure 17(b–d)). Finally, the robot successfully navigated through the obstacle via circumferential propulsion (Figures 17(e) and 17(f)), thereby completing the integrated obstacle avoidance and negotiation process. This entire sequence was completed within 12.5 s. Based on prototype testing, it can be concluded that the robot is capable of completing obstacle avoidance and obstacle crossing when encountering T-shaped pipes with a diameter no greater than 3/4 of the diameter of the inspection pipe.

6.4. Comparative analysis and discussion

To rigorously validate the advanced performance of the proposed robot, a comparative analysis was conducted against representative pipeline inspection robots reported in recent literature. The comparison focuses on key performance metrics, including the drive mechanism type, adaptable pipe diameter range, adaptability ratio (defined as the ratio of maximum to minimum adaptable diameter), maximum inspection speed, and obstacle negotiation strategy. By benchmarking against these established systems, the specific engineering advantages of the novel dual-slider mechanism in balancing structural compactness with wide-range versatility are clearly elucidated. The detailed comparison is presented in Table III.

Table III.

Performance comparison between the proposed robot and existing pipeline inspection robots.