Kinematic and Dynamic Analysis of a Cable-Climbing Robot

Abstract

To inspect broken cables or a cracked protective layer on cable-stayed bridges, a cable-climbing robot has been proposed and designed. In this paper, the complex 3D obstacles that may be encountered on cables are theoretically described, in order to investigate the obstacle-climbing capability of the cable-climbing robot. A climbing model is then proposed and used to design the robot. In the climbing model, two driven wheels are independently supported with a spring. Kinematics and dynamics models are further derived for the obstacle-climbing capabilities of the driving and driven wheels of the robot. In addition, the robot's obstacle-climbing tracks and its obstacle-climbing performance are simulated. Payload and obstacle-climbing experiments were conducted on the climbing robot in the laboratory. Based on the results of the simulation and the experiments, we obtained the variation of the driving torque in obstacle climbing. The contribution of this paper is intended to provide a basis for the precise motion control of the robot.

Keywords

cable-climbing robot climbing model climbing ability

1. Introduction

Rod structures, such as cables, electric wires and pipelines, are being put to increasing use today. Increasing attention is being given to the development of corresponding robots, which can conduct automatic detection and maintenance operations on these rod structures, in order to prolong their service time. This suggests that the study of cable-climbing robots is of importance.

Climbing robots belong to a specialized field of mobile robots. Their main feature is mobility against the gravitational pull of the body. One important aspect to consider in the design of such robots is that they need to be lightweight and powerful enough to move upwards, supporting their own weight. Therefore, the designer should not only consider the robot's locomotion method, as in conventional mobile robots, but also its techniques for sticking to the cable.

Several different types of adhesion method have emerged. The first of these is magnetic mechanisms for climbing on ferrous surfaces via electromagnets or permanent magnets [1–3]. These adhesion mechanisms are suitable for magnetic surfaces or poles, which can generate a magnetic field. The second type is vacuum suction technologies for sticking the robot onto the walls [4–7]. This adhesion method is stable; however, it is achieved by using air compressors or some other air source. The third type of method uses armed mechanisms or micro grippers that can attach to structure surfaces such as beams, wall surface, pipes, or tubes [8–10]. This form of suction possesses a high stability and is suitable for dynamic structures.

In terms of modes of locomotion, climbing robots can be classified into three main groups. The first group comprises wheel-driven machines, which climb vertical structures by combining wheels for translation [11–14]. This movement mode is very effective, is especially suited to the inspection of long structures, and is employed in this research. The second group comprises legged climbing robots [15, 16], usually consisting of four or six legs, each of them with magnets, vacuum pumps or claws for attachment, but with limited manoeuvrability. This moving mode is suitable for the rugged or dangerous environment. The final mode of locomotion is based on the use of arms with grippers or other devices, which provides the robot with skilful mobility [12, 17–18].

In order to inspect pole-like structures, a number of similar climbing mechanisms have recently been designed. For example, Lam has proposed a tree-climbing robot called a “Treebotys [17], which is capable of climbing from a tree trunk to a branch. This robot employs several design principles, such as claw gripping and inch worm locomotion, that are adopted from arboreal animals. This robot is also equipped with artificial optimization to achieve high manoeuvrability in irregular-shaped trees. Ahmadabadi has presented a human-inspired pole-climbing robot [13], which inspired the design, static analysis, simulation, and implementation of a novel design for a naturally stable climbing robot. By using microspines that catch onto surface asperities as a basis, researchers have proposed a spiny-based bio-inspired robot called “RiSE”, for use in scansorial environments [19, 15]. By using bionics, traditional design, and module combination, manufacturers have built tree-climbing and ground-walking robots by combining six modules for legs. However, the robots manufactured using such methods require the use of numerous driving devices, resulting in complex mechanical structures. A robot called “Expliner” was proposed by HiBot Corp. for inspecting electricity lines [20]. This robot was designed to overcome cable spacers, suspension clamps and other obstacles, by actively controlling the position of its centre of mass and changing its configuration. This design features the novel functions of moving along live transmission lines and performing detailed inspections of conductors with no power interruptions, thereby reducing the risk to the operators. Aracil has developed the “Crawling Parallel Robot” [21], which can easily climb along pipeline nodes and overlapping tubular structures. The column and square rod design of this robot allows bending at any angle. Moreover, this robot can operate on surfaces with an irregular topography, such as trunk surfaces. Another useful mechanism—“UT-PCR” (University of Tehran-Pole Climbing Robot) [14]—was proposed by the Robotics and Artificial Intelligence Laboratory of the University of Tehran. UT-PCR consists of a triangular body and six limbs with ordinary wheels at their tips, and has mainly been used to clean highway lighting systems. Nevertheless, the robot is powered through an electrical wire, which can influence the stabilization when the mechanism works at high altitudes. Other studies on special pole-climbing robots have been developed and put into practice. These robots include the four-DOF climbing structure known as “PCR” [22], the ExplorerTM family of pipe robots [23], the “3DCLEVIBER” designed for 3D tubular structures [18], the robot known as “Robot V2”, which is capable of climbing poles with cylindrical or conical shapes [11], and a novel biped robot that can climb poles, trusses and trees [24]. The researchers of the present study have designed a bridge-climbing robot for smooth straight cables. This design can be seen in the references [12, 16, 26].

In extreme offshore environments, robots can carry out a wide variety of tasks, including monitoring, detecting and maintenance, processing production interventions, and cargo transport operations. One of the typical devices employed is DORIS [25]. This long-span cable-stayed bridge-climbing robot is a typical robot working at high altitude. This robot bears a large high-altitude wind load and can operate in adverse environmental conditions. In the operation of this robot, an inclined angle of the cable and the presence of obstacles are the two main conditions. In our analysis, we set the cable as vertical, which is the most difficult condition to climb. All the obstacles encountered by the robot are caused by cable surface damages. The obstacles caused by cable surface damage come in different forms, resulting from the long service time of the cable. However, these obstacles can be approximately divided into two categories: (1) damage to the cable's protective layer, such as cracks, scallops, scratches, and trunks of the protective layer; and (2) damage to the cable's steel wire, such as steel wire zinc oxidation, steel wire corrosion, and steel wire fracturing. Fig. 1 shows the two typical forms of cable damage.

Figure 1.

The typical forms of cable damage

Considering these forms of cable damage, this study first proposes a climbing model supported by independent springs to optimize the force of the robot's driving wheel. Then, the kinematics and dynamics models of the driving and driven wheels, during obstacle climbing, are analysed. Finally, the obstacle-climbing performance of the mechanism is analysed through simulations and experiments. The kinematic and dynamic analysis is the main contribution of this paper, through which the torque of the driving motor can be calculated precisely. Based on the kinematics and dynamics model, the relation between output torque and time can be obtained.

2. Independently Supported Climbing Model and the Cable-climbing Robot

In general, large trunk obstacles are generally not found on the surface of the cable. Obstacles can instead be divided into five main categories: steps, slopes, bosses, trenches, and irregular shapes (Fig. 2). Our taxonomy of these five obstacles is based on the investigations of the bridge management department and the observation of modes of cable failure over a long period. The former four types are regular obstacles, and the fifth one is irregular, which can itself be divided into several conditions. In this manuscript, we take the step obstacle as an example to analyse the robot's obstacle-climbing ability, as it is the most common obstacle. At the same total height, the step is the most difficult of the obstacles for the robot to climb.

Figure 2.

Cable obstacles

In the mechanism proposed in the literature [12], the upper and lower driven wheels are only supported by a single spring. Although this mechanism can be installed easily, its obstacle-climbing capability is not strong. To obtain a more reasonable mechanical structure, a robot is theoretically designed as a model such that the upper and lower driven wheels are individually supported by an independent spring, as shown in Fig. 3. This model consists of an equally spaced driving vehicle (A) and a passive vehicle (B) facing each other. Each vehicle possesses two wheel limbs at its two ends. Only the upper wheel of the driving vehicle is actuated by the powerful DC motor. The passive vehicle is merely applied to provide supporting force to clasp onto the cable. The driving vehicle (Fig. 4a) consists of the driving module (M) and the speed-limited descending structure (N); this serves as the power source of the entire climbing structure. The automatic descending equipment is made up of a cylinder (3), crank slide installations (4, 5), and a one-way clutch (7). When the robot descends, the driving wheel leads the crank, which drives the piston to perform the reciprocating motion in the cylinder. Then, the gas is inhaled and discharged out of the cylinder through a small hole carved at the bottom of the cylinder. Gas damping is then formed to reduce the excessive energy generated by the gravity of the robot.

Figure 3.

The robot climbing model

Figure 4.

Structures of the vehicles

The passive vehicle balances the entire structure and provides the clamping force. It possesses upper and lower swinging arms, and each swinging arm is connected to a passive wheel (23 and 24) (Fig. 4b). The upper and lower arms are compressed by the spring, to grip onto the cable and provide the clamping force once the robot is installed on the cable. When encountering the obstacle, the two arms stretch freely to allow the wheel to come into contact with the cable and adapt to the rough surface.

With four sets of connectors (AB1, AB2, AB3, and AB4), the two two-wheeled vehicles are linked in a long barrel form that is clamped around the cable. By linking the threaded holes at different distances, the linking location can be adjusted easily and installed on cables with different diameters. Wheel1 of the driving vehicle can be driven by a direct-current motor, to propel the entire structure in its upward climb. The driving and passive wheels form a “V” shape; this enlarges the contact area, reduces wear and tear, and prevents a deadlock caused by deviations to the structure. On each side of the connector, an anti-bias device is placed, which consists of an anti-deviation universal ball (C1–C8) and relevant interconnecting links (Fig. 3b). When the robot is in normal operation, the anti-bias universal balls remain a certain distance away from the cable. When the robot exhibits a deviation tendency or is critically detached from the cable, at least one group of universal balls remain in contact with the cable, in order to prevent the robot from deviating from the cableway. A 3D model of the robot can be seen in Fig. 4c. A picture of the robot is shown in Fig. 5.

Figure 5.

A picture of the robot

As shown in Fig. 6, θ₂ is the induced angle between the upper swing arm (rod CB) and the horizontal direction; β₂ is the induced angle of the spring force of wheel2 and the vertical direction; N₂ is the normal force that the cable surface or the obstacle acts on the wheel; $F_{f 2}$ is the friction force; T₂ is the spring force; F₂ is the support force of the swing arm to the driven wheel; m_w2 is the mass of wheel2; d is the diameter of the cable; and α₂ is the angle between the normal force and the horizontal line. The mass of the upper swinging arm is ignored, and the upper swinging arm is considered as a two-force bar. Using the driven wheel2 as an example, the following equations can be obtained.

Figure 6.

The force of the independent spring support

\begin{array}{l} F_{2} \sin θ_{2} = T_{2} \cos β_{2} + m_{w 2} g + N_{2} \sin α_{2} + F_{f 2} \cos α_{2} \\ F_{2} \cos θ_{2} + T_{2} \sin β_{2} + F_{f 2} \sin α_{2} = N_{2} \cos α_{2} \end{array}

(1)

The following equation can be derived from Eq. (1):

N_{2} = \frac{T_{2} \cos (θ_{2} - β_{2}) + m_{w 2} g \cos θ_{2}}{\sin (θ_{2} - α_{2}) - \frac{μ \cos (θ_{2} - α_{2})}{\sin \frac{β_{2}}{2}}}

(2)

When α₂ = 0, the smooth straight cable-climbing state can be obtained as follows:

N_{2} = \frac{T_{2} \cos (β_{2} - θ_{2}) + m_{w 2} g \cos θ_{2}}{(\tan θ_{2} - μ) \cos θ_{2}}

The action of the spring's force is to press the wheel onto the cable. Therefore, β₂ ≠ 0. If β₂ = 0, the wheel is pulled downward by the spring, not press onto the cable. According to Eq. (2), whether for smooth straight cable–climbing or obstacle-climbing, when θ₂ = β₂ (spring force is perpendicular to the upper swing arm), the horizontal component of the support force reaches the maximum value, that is, the positive pressure generated on the driving wheel achieves the maximum value, and the climbing capacity provided by the mechanism is the strongest. Fig. 7 shows the curves of positive pressure changing with θ₂ when β₂ = 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90° from top to bottom. All of the curves present the maximum values when β₂ = θ₂, that is, when the spring force is perpendicular to the swing arm.

Figure 7.

The changing law of positive pressure with the variation in swing arm angle

3. The Kinematics and Dynamics of Obstacle Climbing by the Robot Driving Wheel

Although the obstacles on the cables are very small, these obstacles also greatly influence the motion performance of the robot. Therefore, a kinematic and dynamic analysis must be performed on its obstacle climbing. Given that the positive pressure of the robot's moving wheel is provided by the spring, more factors should be considered in cable-obstacle climbing than in ground-obstacle climbing. In this study, a simplified model (linkage mechanism) is employed to analyse the kinematics issues. The upper and lower moving wheels are set in different horizontal planes to ensure that only one moving wheel is climbing an obstacle at any given time. In addition, the driving vehicle (wheel) is a fixed joint, while the passive vehicle is flexible. Therefore, the robot's obstacle-climbing status can be summarized in the following two cases:

Deflection obstacle climbing of the main robot body:

When the driving wheel1 and the driven wheel4 climb over an obstacle, the robot body will produce deflection. This is why φ₄ is formed. Obstacle climbing is here completed by the coordination motion of the upper and lower support arms of the driven trolley. In this status, the gravity centre of the robot is in plane motion (Fig. 8).

Non-deflection obstacle climbing of the main robot body:

Given that wheel1 and wheel4 are fixedly connected, when wheel2 and wheel3 climb over an obstacle, the robot body does not produce deflection. Obstacle climbing is here completed only by the rotation of the upper and lower support arms of the driven wheel around the jointed point on the trolley. In this status, the gravity centre of the main robot body is in uniform rectilinear ascending motion (Fig. 11).

To simplify the issues, the following hypotheses are drawn:

Side sliding does not exist along the wheel axis, and the sliding rotation around the wheel axis between the obstacle-climbing wheel and the cable surface—that is, robot motion—is only investigated in the x-y plane, in which “– x” refers to the motion direction of the robot.

During obstacle climbing, the robot does not rotate around the cable, and only one moving wheel climbs an obstacle at a time. In addition, a V-shaped wheel flange and the obstacle are in two-point contact.

No relative sliding exists between the robot's moving wheel and the contact point during climbing.

3.1 Kinematic Analysis of the Obstacle Climbing of the Driving Wheel

During obstacle climbing, the motion tracks of the climbing wheel's centre form an arc around the contact point. The robot's obstacle-climbing model is simplified according to these hypotheses as a 2D model in the “x-y” plane (Fig. 8), in which “– x” refers to the motion direction of the robot; φ₁ is the reduced angle of the line from the centre of the circle to the contact point, and the x direction; φ₄ is the elevation angle of the robot; φ₁₀ is the original phase angle of the original motive parts (OA); and ω_OA and ε_OA are the angular speed and angular acceleration of OA, respectively. Other robot kinematic parameters can be calculated from these parameters. Given that the driving wheel is in a uniform rotation, OA is also in a uniform rotation during obstacle climbing. Thus, the speed, acceleration, and angular acceleration of the centre of the circle of the robot's moving wheel, the centre of gravity of the driven wheels' support arms, and the centre of gravity of the robot can be determined.

Kinematics analysis for wheel1 and wheel4

The position of wheel1 and wheel4

The centroid coordinate of the driving wheel is

{\begin{cases} x_{A} = L_{O A} \cos ϕ_{1} \\ y_{A} = L_{O A} \sin ϕ_{1} \end{cases}

(3)

The vector equation of the mechanism is established as $\vec{O A} + \vec{A F} = \vec{O H} + \vec{H F}$

Transforming the vector equation into an analytical form yields

{\begin{cases} x_{F} = x_{A} + L_{A F} \cos ϕ_{4} = L_{O A} \cos ϕ_{1} + L_{A F} \cos ϕ_{4} \\ y_{F} = y_{A} + L_{A F} \sin ϕ_{4} = L_{O A} \sin ϕ_{1} + L_{A F} \sin ϕ_{4} = r - h \end{cases}

(4)

Figure 8.

Model of the obstacle climbing of the driving wheel

The following equation can then be derived:

ϕ_{4} = \arctan [\frac{y_{A} - y_{F}}{\sqrt{L_{_{A F}}^{2} - {(y_{A} - y_{F})}^{2}}}] + \frac{3}{2} π

(5)

The centroid coordinate of the AF rod is

{\begin{cases} x_{A F 2} = x_{A} + L_{A F 2} \cos ϕ_{4} \\ y_{A F 2} = y_{A} + L_{A F 2} \sin ϕ_{4} \end{cases}

(6)

The speed of wheel1 and wheel4

By calculating the first-order derivative of Eq. (3), the centroid speed of the driving wheel can be obtained as follows:

{\begin{cases} {\dot{x}}_{A} = - {\dot{ϕ}}_{1} L_{O A} \sin ϕ_{1} \\ {\dot{y}}_{A} = {\dot{ϕ}}_{1} L_{O A} \cos ϕ_{1} \end{cases}

(7)

By calculating the first-order derivative of Eq. (4), the angular speed of the AF rod and the speed of the wheel can be obtained as follows: ${\dot{ϕ}}_{4} = \frac{- {\dot{y}}_{A}}{L_{A F} \cos ϕ_{4}}$

{\begin{cases} {\dot{x}}_{F} = {\dot{x}}_{A} - L_{A F} {\dot{ϕ}}_{4} \sin ϕ_{4} \\ {\dot{y}}_{F} = 0 \end{cases}

(8)

where 3π/2<φ₄<π, it is decided by the robot structure; there isn't a singularity.

By calculating the first-order derivative of Eq. (6), the centroid speed of the AF rod can be obtained as follows:

{\begin{cases} {\dot{x}}_{A F 2} = {\dot{x}}_{A} - L_{A F 2} {\dot{ϕ}}_{4} \sin ϕ_{4} \\ {\dot{y}}_{A F 2} = {\dot{y}}_{A} + L_{A F 2} {\dot{ϕ}}_{4} \cos ϕ_{4} \end{cases}

The acceleration of wheel 1 and wheel 4

By calculating the second-order derivative of Eq. (3), the centroid acceleration of the driving wheel can be obtained as follows:

{\begin{cases} {\ddot{x}}_{A} = - {\dot{ϕ}}_{1}^{2} L_{O A} \cos ϕ_{1} \\ {\ddot{y}}_{A} = - {\dot{ϕ}}_{1}^{2} L_{O A} \sin ϕ_{1} \end{cases}

(9)

By calculating the second-order derivative of Eq. (4), the angular acceleration of the AF rod and the acceleration of wheel4 can be obtained as follows:

{\ddot{ϕ}}_{4} = \frac{- {\ddot{y}}_{A} + L_{A F} {\dot{ϕ}}_{4}^{2} \sin ϕ_{4}}{L_{A F} \cos ϕ_{4}}

{\begin{cases} {\ddot{x}}_{F} = {\ddot{x}}_{A} - L_{A F} ({\ddot{ϕ}}_{4} \sin ϕ_{4} + {\dot{ϕ}}_{4}^{2} \cos ϕ_{4}) \\ {\ddot{y}}_{F} = 0 \end{cases}

(10)

By calculating the second-order derivative of Eq. (6), the centroid acceleration of the AF rod can be obtained as follows:

{\begin{cases} {\ddot{x}}_{A F 2} = {\ddot{x}}_{A} - L_{A F 2} ({\ddot{ϕ}}_{4} \sin ϕ_{4} + {\dot{ϕ}}_{4}^{2} \cos ϕ_{4}) \\ {\ddot{y}}_{A F 2} = {\ddot{y}}_{A} + L_{A F 2} ({\ddot{ϕ}}_{4} \cos ϕ_{4} - {\dot{ϕ}}_{4}^{2} \sin ϕ_{4}) \end{cases}

(11)

Analysis of wheel2

Position of wheel2

If ACDF constitutes the robot body, then φ₅ = φ₆ = φ₇ = φ₈ = φ₄ is known. Thus, the coordinate of point C is

{\begin{cases} x_{C} = x_{A} + L_{A C} \cos ϕ_{5} \\ y_{C} = y_{A} + L_{A C} \sin ϕ_{5} \end{cases}

(12)

The coordinate of point B is

{\begin{cases} x_{B} = x_{C} + L_{C B} \cos ϕ_{2} \\ y_{B} = y_{C} + L_{C B} \sin ϕ_{2} \end{cases}

(13)

where y_B =-(h + r + d). Thus,

ϕ_{2} = π - \arctan [\frac{y_{B} - y_{C}}{\sqrt{L_{_{C B}}^{2} - {(y_{B} - y_{C})}^{2}}}]

(14)

The centroid coordinate of the CB rod is

{\begin{cases} x_{C B 2} = x_{C} + L_{C B 2} \cos ϕ_{2} \\ y_{C B 2} = y_{C} + L_{C B 2} \sin ϕ_{2} \end{cases}

(15)

Speed of wheel2

By calculating the first-order derivative of Eq. (12), the speed of point C can be obtained as follows:

{\begin{cases} {\dot{x}}_{C} = {\dot{x}}_{A} - L_{A C} {\dot{ϕ}}_{5} \sin ϕ_{5} \\ {\dot{y}}_{C} = {\dot{y}}_{A} + L_{A C} {\dot{ϕ}}_{5} \cos ϕ_{5} \end{cases}

(16)

By calculating the first-order derivative of Eq. (13), the angular speed of the CB rod and the speed of wheel2 can be obtained as follows:

{\begin{cases} {\dot{x}}_{B} = {\dot{x}}_{C} - L_{C B} {\dot{ϕ}}_{2} \sin ϕ_{2} \\ {\dot{y}}_{B} = 0 = {\dot{y}}_{C} + L_{C B} {\dot{ϕ}}_{2} \cos ϕ_{2} \end{cases}

(17)

${\dot{ϕ}}_{2} = \frac{- {\dot{y}}_{C}}{L_{C B} \cos ϕ_{2}} = \frac{- ({\dot{y}}_{A} + L_{A C} {\dot{ϕ}}_{5} \cos ϕ_{5})}{L_{C B} \cos ϕ_{2}}$ , where π/2 < φ2 < π, it is decided by the robot structure.

By calculating the first-order derivative of Eq. (15), the centroid speed of the CB rod can be obtained as follows:

{\begin{cases} {\dot{x}}_{C B 2} = {\dot{x}}_{C} - L_{C B 2} {\dot{ϕ}}_{2} \sin ϕ_{2} \\ {\dot{y}}_{C B 2} = {\dot{y}}_{C} + L_{C B 2} {\dot{ϕ}}_{2} \cos ϕ_{2} \end{cases}

Acceleration of wheel2

By calculating the second-order derivative of Eq. (12), the acceleration of point C can be obtained as follows:

{\begin{cases} {\ddot{x}}_{C} = {\ddot{x}}_{A} - L_{A C} ({\ddot{ϕ}}_{5} \sin ϕ_{5} + {\dot{ϕ}}_{5}^{2} \cos ϕ_{5}) \\ {\ddot{y}}_{C} = {\ddot{y}}_{A} + L_{A C} ({\ddot{ϕ}}_{5} \cos ϕ_{5} - {\dot{ϕ}}_{5}^{2} \sin ϕ_{5}) \end{cases}

(18)

By calculating the second-order derivative of Eq. (13), the angular acceleration of the CB rod and the acceleration of wheel2 can be obtained as follows: ${\ddot{ϕ}}_{2} = \frac{- {\ddot{y}}_{C} + L_{C B} {\dot{ϕ}}_{2}^{2} \sin ϕ_{2}}{L_{C B} \cos ϕ_{2}}$

{\begin{cases} {\ddot{x}}_{B} = {\ddot{x}}_{C} - L_{C B} ({\ddot{ϕ}}_{2} \sin ϕ_{2} + {\dot{ϕ}}_{2}^{2} \cos ϕ_{2}) \\ {\ddot{y}}_{B} = 0 = {\ddot{y}}_{C} + L_{C B} ({\ddot{ϕ}}_{2} \cos ϕ_{2} - {\dot{ϕ}}_{2}^{2} \sin ϕ_{2}) \end{cases}

(19)

By calculating the second-order derivative of Eq. (15), the centroid acceleration of the CB rod can be obtained as follows:

{\begin{cases} {\ddot{x}}_{C B 2} = {\ddot{x}}_{C} - L_{C B 2} ({\ddot{ϕ}}_{2} \sin ϕ_{2} + {\dot{ϕ}}_{2}^{2} \cos ϕ_{2}) \\ {\ddot{y}}_{C B 2} = {\ddot{y}}_{C} + L_{C B 2} ({\ddot{ϕ}}_{2} \cos ϕ_{2} - {\dot{ϕ}}_{2}^{2} \sin ϕ_{2}) \end{cases}

(20)

Analysis of wheel3

The analysis process of wheel3 is similar to that of wheel2. Through this process, the coordination, speed and acceleration of point D can be obtained as follows:

{\begin{cases} x_{D} = x_{A} + L_{A D} \cos ϕ_{6} \\ y_{D} = y_{A} + L_{A D} \sin ϕ_{6} \end{cases}

(21)

{\begin{cases} {\dot{x}}_{D} = {\dot{x}}_{A} - L_{A D} {\dot{ϕ}}_{6} \sin ϕ_{6} \\ {\dot{y}}_{D} = {\dot{y}}_{A} + L_{A D} {\dot{ϕ}}_{6} \cos ϕ_{6} \end{cases}

(22)

{\begin{cases} {\ddot{x}}_{D} = {\ddot{x}}_{A} - L_{A D} ({\ddot{ϕ}}_{6} \sin ϕ_{6} + {\dot{ϕ}}_{6}^{2} \cos ϕ_{6}) \\ {\ddot{y}}_{D} = {\ddot{y}}_{A} + L_{A D} ({\ddot{ϕ}}_{6} \cos ϕ_{6} - {\dot{ϕ}}_{6}^{2} \sin ϕ_{6}) \end{cases}

(23)

The centroid coordinate, speed, and acceleration of wheel3 can be obtained as follows:

{\begin{cases} x_{E} = x_{D} + L_{D E} \cos ϕ_{3} \\ y_{E} = y_{D} + L_{D E} \sin ϕ_{3} = - (h + r + d) \end{cases}

(24)

{\begin{cases} {\dot{x}}_{E} = {\dot{x}}_{D} - L_{D E} {\dot{ϕ}}_{3} \sin ϕ_{3} \\ {\dot{y}}_{E} = 0 = {\dot{y}}_{D} + L_{D E} {\dot{ϕ}}_{3} \cos ϕ_{3} \end{cases}

(25)

{\begin{cases} {\ddot{x}}_{E} = {\ddot{x}}_{D} - L_{D E} ({\ddot{ϕ}}_{3} \sin ϕ_{3} + {\dot{ϕ}}_{3}^{2} \cos ϕ_{3}) \\ {\ddot{y}}_{E} = 0 = {\ddot{y}}_{D} + L_{D E} ({\ddot{ϕ}}_{3} \cos ϕ_{3} - {\dot{ϕ}}_{3}^{2} \sin ϕ_{3}) \end{cases}

(26)

The centroid coordinate, speed, and acceleration of the DE rod can be obtained as follows:

{\begin{cases} x_{D E 2} = x_{D} + L_{D E 2} \cos ϕ_{3} \\ y_{D E 2} = y_{D} + L_{D E 2} \sin ϕ_{3} \end{cases}

(27)

{\begin{cases} {\dot{x}}_{D E 2} = {\dot{x}}_{D} - L_{D E 2} {\dot{ϕ}}_{3} \sin ϕ_{3} \\ {\dot{y}}_{D E 2} = {\dot{y}}_{D} + L_{D E 2} {\dot{ϕ}}_{3} \cos ϕ_{3} \end{cases}

{\begin{cases} {\ddot{x}}_{D E 2} = {\ddot{x}}_{D} - L_{D E 2} ({\ddot{ϕ}}_{3} \sin ϕ_{3} + {\dot{ϕ}}_{3}^{2} \cos ϕ_{3}) \\ {\ddot{y}}_{D E 2} = {\ddot{y}}_{D} + L_{D E 2} ({\ddot{ϕ}}_{3} \cos ϕ_{3} - {\dot{ϕ}}_{3}^{2} \sin ϕ_{3}) \end{cases}

(28)

as shown in the following:

ϕ_{3} = \arctan [\frac{y_{E} - y_{D}}{\sqrt{L_{_{D E}}^{2} - {(y_{E} - y_{D})}^{2}}}]

(29)

{\dot{ϕ}}_{3} = \frac{- ({\dot{y}}_{A} + L_{A D} {\dot{ϕ}}_{6} \cos ϕ_{6})}{L_{D E} \cos ϕ_{3}}

(30)

{\ddot{ϕ}}_{3} = \frac{- {\ddot{y}}_{D} + L_{D E} {\dot{ϕ}}_{3}^{2} \sin ϕ_{3}}{L_{D E} \cos ϕ_{3}}

(31)

Analysis of the robot body

Using the same method, the coordinate, speed, and acceleration of the centre of gravity (point P) of the robot can be obtained as follows:

{\begin{cases} x_{P} = x_{A} + L_{A P} \cos ϕ_{7} \\ y_{P} = y_{A} + L_{A P} \sin ϕ_{7} \end{cases}

(32)

{\begin{cases} {\dot{x}}_{P} = {\dot{x}}_{A} - L_{A P} {\dot{ϕ}}_{7} \sin ϕ_{7} \\ {\dot{y}}_{P} = {\dot{y}}_{A} + L_{A P} {\dot{ϕ}}_{7} \cos ϕ_{7} \end{cases}

(33)

{\begin{cases} {\ddot{x}}_{P} = {\ddot{x}}_{A} - L_{A P} ({\ddot{ϕ}}_{7} \sin ϕ_{7} + {\dot{ϕ}}_{7}^{2} \cos ϕ_{7}) \\ {\ddot{y}}_{P} = {\ddot{y}}_{A} + L_{A P} ({\ddot{ϕ}}_{7} \cos ϕ_{7} - {\dot{ϕ}}_{7}^{2} \sin ϕ_{7}) \end{cases}

(34)

Eqs. (3) to (34) constitute the kinematics equation for the robot driving wheel's obstacle-climbing capability. By using these equations, the kinematics parameters of the robot's obstacle climbing can be accurately determined.

Analysis of spring force

In obstacle climbing, the two spring forces of the driven trolley generate slight changes, which are indicated as follows:

When the driven wheel2 climbs an obstacle, T₂ = T₀ + KΔiota;, where Δι = ι₂(tgθ₂ − tgθ₂₀), that is, the positive pressure of wheel2 increases, whereas that of wheel3 remains unchanged.

The amount of spring compression when the driving wheel1 climbs an obstacle is shown in Fig. 9, where the variations in the lengths of MB and NE represent the changes in spring compression during obstacle climbing. The force of spring2 can be expressed as T₂ = T₀ + KΔι₂, where $Δ l_{2} = l_{2} t g [\frac{π}{2} - θ_{20}] - l_{2} t g [\frac{π}{2} - θ_{2} - (2 π - ϕ_{4})]$ . The positive pressure of wheel2 increases. The force of spring3 can be expressed as T₃ = T₀ + KΔι₃, where $Δ l_{3} = l_{2} t g [\frac{π}{2} - θ_{3} + (2 π - ϕ_{4})] - l_{2} t g [\frac{π}{2} - θ_{30}]$ , and the positive pressure of wheel 3 increases. θ₂₀ and θ₃₀ are the initial angles of θ₂ and θ₃, respectively, where $θ_{2} = ϕ_{2} - \frac{π}{2}$ , $θ_{3} = \frac{π}{2} - ϕ_{4}$ . In these equations, K is the spring coefficient, Δι is the spring deformation amount, T₀ is the initial value of the spring force, and ι₂ is the length of the lower swing arm.

Figure 9.

The amount of spring deformation

3.2 Dynamic Analysis of the Obstacle Climbing of the Driving Wheel

The inertia force of the component in plane motion is simplified as an inertia force, and an inertia coupled with the torque added on the centroid. The mechanism can be regarded as in equilibrium state. By employing dynamic static force analysis, the force and torque equilibrium equations for each component can be obtained as follows:

The forces of the driving wheel1

{\begin{cases} F_{v 1} + m_{1} g + F_{1} \cos ϕ_{1} = F_{f 1} \sin ϕ_{1} + m_{1} {\ddot{x}}_{A} \\ F_{f 1} \cos ϕ_{1} + F_{1} \sin ϕ_{1} = F_{h 1} + m_{1} {\ddot{y}}_{A} \\ F_{f 1} r_{1} + J_{1} {\ddot{ϕ}}_{1} = τ_{1} \end{cases}

(35)

The forces of the driven wheel4

{\begin{cases} m_{4} g + F_{f 4} = F_{F N} \cos θ_{4} + F_{F T} \sin θ_{4} + m_{4} {\ddot{x}}_{F} \\ F_{4} + F_{F N} \sin θ_{4} = F_{F T} \cos θ_{4} + m_{4} {\ddot{y}}_{F} \\ F_{f 4} r_{4} = J_{4} {\ddot{x}}_{F} / r_{4} \end{cases}

(36)

The forces of the swing arm of the driven wheel2

{\begin{cases} F_{B N} \sin θ_{2} + m_{l 2} g = F_{B T} \cos θ_{2} + F_{C x} + m_{B C} {\ddot{x}}_{B C 2} \\ F_{C y} = F_{B N} \cos θ_{2} + F_{B T} \sin θ_{2} + m_{B C} {\ddot{y}}_{B C 2} \\ F_{B T} l_{B C} = m_{B C} g \cos θ_{2} l_{B C} / 2 + J_{B C} {\ddot{ϕ}}_{2} \end{cases}

(37)

The forces of the swing arm of the driven wheel3

{\begin{cases} F_{D x} + m_{l 3} g = F_{E N} \sin θ_{3} + F_{E T} \cos θ_{3} + m_{D E} {\ddot{x}}_{D E 2} \\ F_{D y} + F_{E T} \sin θ_{3} = F_{E N} \cos θ_{3} + m_{D E} {\ddot{y}}_{D E 2} \\ m_{B C} g \sin θ_{3} l_{D E} / 2 + J_{D E} {\ddot{ϕ}}_{3} = F_{E T} l_{D E} \end{cases}

(38)

The forces of the driven wheel2

{\begin{cases} T_{2} \cos θ_{2} + m_{2} g + F_{f 2} = F_{m B T} \cos θ_{2} + F_{m B N} \sin θ_{2} + m_{2} {\ddot{x}}_{B} \\ T_{2} \sin θ_{2} + F_{m B N} \cos θ_{2} = F_{2} + F_{m B T} \sin θ_{2} + m_{2} {\ddot{y}}_{B} \\ J_{2} {\ddot{x}}_{B} / r_{2} = F_{f 2} r_{2} \end{cases}

(39)

The forces of the driven wheel3

{\begin{cases} F_{m E N} \sin θ_{3} + F_{m E T} \cos θ_{3} + m_{3} g + F_{f 3} = T_{3} \cos θ_{3} + m_{3} {\ddot{x}}_{E} \\ T_{3} \sin θ_{3} + F_{m E N} \cos θ_{3} = F_{3} + F_{m E T} \sin θ_{3} + m_{3} {\ddot{y}}_{E} \\ F_{f 3} r_{3} = J_{3} {{\ddot{x}}_{E} / r}_{3} \end{cases}

(40)

The dynamic equilibrium equation of the total mechanism

{\begin{cases} F_{m C x} + T_{3} \cos θ_{3} + F_{m F N} \cos θ_{4} + F_{m F T} \sin θ_{4} + M {\ddot{x}}_{P} = F_{m A v} + T_{2} \cos θ_{2} + F_{m D x} \\ F_{m A h} + F_{m F T} \cos θ_{4} = T_{2} \sin θ_{2} + F_{m C y} + F_{m D y} + T_{3} \sin θ_{3} + F_{m F N} \sin θ_{4} + M {\ddot{y}}_{P} \\ T_{2} \sin θ_{2} (x_{A} - x_{M}) - T_{2} \cos θ_{2} (y_{M} - y_{A}) - F_{m C y} (x_{C} - x_{A}) + F_{m C x} (y_{A} - y_{C}) - F_{m D x} (y_{A} - y_{D}) - F_{m D y} (x_{D} - x_{A}) \\ - T_{3} \cos θ_{3} (y_{A} - y_{N}) - T_{3} \sin θ_{3} (x_{N} - x_{A}) + F_{m F T} L_{A F} + (M g - M {\ddot{x}}_{P}) L_{A P} \cos (ϕ_{7} - π) - M {\ddot{y}}_{P} L_{A P} \sin (ϕ_{7} - π) \\ + J_{P} {\ddot{ϕ}}_{4} = 0 \end{cases}

(41)

Eqs. (35) to (41) constitute the dynamic equation of the robot driving wheel's obstacle-climbing capability. A force analysis of the robot components can be seen in Fig. 10.

Figure 10.

Force of the robot components

Figure 11.

Model of the obstacle-climbing capability of the robot's driven wheel

4. Kinematic and Dynamic Analysis of the Obstacle-climbing Capability of the Robot's Driven Wheel2

Given that wheel2 climbs obstacles purely by rolling, a coordinate system can be established with the contact point of wheel2 and the obstacle as the origin, where −x is the motion direction of the robot. A simplified model of wheel2 is shown in Fig. 11.

4.1 Kinematic Analysis of the Obstacle-climbing Capability of the Robot's Driven Wheel2

Displacement equation

In Fig. 11, the vector equation of the mechanism is $\vec{G B} + \vec{B C} = \vec{G H} + \vec{H C}$ . Transforming the vector equation into an analytical form yields

{\begin{cases} l_{G B} \cos α_{2} + l_{B C} \cos α_{3} = l_{G H} \\ l_{G B} \sin α_{2} + l_{B C} \sin α_{3} = l_{H C} \end{cases}

(42)

Then, ${\begin{cases} α_{2} = a r c \cos (\frac{4 A_{1} l_{G H} l_{G B} - \sqrt{16 A_{1}^{2} l_{G B}^{2} l_{G H}^{2} - 4 A_{2} A_{3}}}{2 A_{2}}) \\ α_{3} = a r c \cos (\frac{l_{G H} - l_{G B} \cos α_{2}}{l_{B C}}) \end{cases}$ is acquired.

where $A_{1} = l_{G H}^{2} + l_{G B}^{2} + l_{H C}^{2} - l_{B C}^{2}$ , $A_{2} = 4 l_{G H}^{2} l_{G B}^{2} + 4 l_{H C}^{2} l_{G B}^{2}$ , $A_{3} = A_{1}^{2} - 4 l_{H C}^{2} l_{G B}^{2}$ . In the equation, α₂ denotes the position of wheel2, that is, the angle of the generatrix with the x axle. α₃ is the angle of the supporting rod, that is, the angle of rod BC with the x axle.

The centroid coordinate of the driven wheel2 is

{\begin{cases} x_{B} = l_{G B} \cos α_{2} \\ y_{B} = l_{G B} \sin α_{2} \end{cases}

(43)

The centroid coordinate of the BC rod is

{\begin{cases} x_{B C 2} = x_{B} + l_{B C 2} \cos α_{3} \\ y_{B C 2} = y_{B} + l_{B C 2} \sin α_{3} \end{cases}

(44)

The coordinate of point C is

{\begin{cases} x_{C} = x_{B} + l_{B C} \cos α_{3} \\ y_{C} = y_{B} + l_{B C} \sin α_{3} \end{cases}

(45)

Speed equations

By calculating the first-order derivative of Eq. (42), the angular speeds of the driven wheel2 and the BC rod can be obtained as follows:

{\begin{cases} {\dot{α}}_{2} = \frac{{\dot{l}}_{G H}}{t g α_{3} l_{G B} \cos α_{2} - l_{G B} \sin α_{2}} \\ {\dot{α}}_{3} = \frac{{\dot{l}}_{G H}}{t g α_{2} l_{B C} \cos α_{3} - l_{B C} \sin α_{3}} \end{cases}

(46)

By calculating the first-order derivatives of Eqs. (43) and (44), the centroid speeds of the driven wheel2 and the BC rod can be obtained as follows:

{\begin{cases} {\dot{x}}_{B} = - l_{G B} \sin α_{2} \cdot {\dot{α}}_{2} \\ {\dot{y}}_{B} = l_{G B} \cos α_{2} \cdot {\dot{α}}_{2} \end{cases}

(47)

{\begin{cases} {\dot{x}}_{B C 2} = {\dot{x}}_{B} - l_{B C 2} \sin α_{3} \cdot {\dot{α}}_{3} \\ {\dot{y}}_{B C 2} = {\dot{y}}_{B} + l_{B C 2} \cos α_{3} \cdot {\dot{α}}_{3} \end{cases}

(48)

The speeds of centroids C, D, and P are the same, suggesting the speed of these centroids is the ascending speed of the robot.

Acceleration equations

By calculating the second-order derivatives of Eq. (42), the angular acceleration of the driven wheel2 and the BC rod can be obtained after simplification as follows:

{\begin{cases} {\ddot{α}}_{2} = \frac{t g α_{3} B_{2} - B_{1}}{l_{G B} \sin α_{2} + t g α_{3} l_{G B} \cos α_{2}} \\ {\ddot{α}}_{3} = \frac{t g θ_{2} B_{2} - B_{1}}{l_{B C} \sin α_{3} + t g α_{2} l_{B C} \cos α_{3}} \end{cases}

(49)

By calculating the second-order derivative of Eqs. (43) and (44), the centroid speeds of the driven wheel2 and the BC rod can be obtained as follows:

{\begin{cases} {\ddot{x}}_{B} = (- l_{G B} \sin α_{2} \cdot {\dot{α}}_{2})^{'} = - l_{G B} (\cos α_{2} \cdot {\dot{α}}_{2}^{2} + \sin α_{2} \cdot {\ddot{α}}_{2}) \\ {\ddot{y}}_{B} = (l_{G B} \cos α_{2} \cdot {\dot{α}}_{2})^{'} = l_{G B} (- \sin α_{2} \cdot {\dot{α}}_{2}^{2} + \cos α_{2} \cdot {\ddot{α}}_{2}) \end{cases}

(50)

{\begin{cases} {\ddot{x}}_{B C 2} = {\ddot{x}}_{B} - l_{B C 2} (\cos α_{3} \cdot {\dot{α}}_{3}^{2} + \sin α_{3} \cdot {\ddot{α}}_{3}) \\ {\ddot{y}}_{B C 2} = {\ddot{y}}_{B} + l_{B C 2} (- \sin α_{3} \cdot {\dot{α}}_{3}^{2} + \cos α_{3} \cdot {\ddot{α}}_{3}) \end{cases}

(51)

The support spring force of the driven wheel2

T_{2} = T_{20} + K l_{B C} (\tan α_{30} - \tan α_{3})

where T₂₀ is the initial pressure of the spring, and K is the stiffness coefficient.

4.2 Dynamic Analysis of the Obstacle-climbing Capability of the Driven Wheel2

As shown in Fig. 12, the equilibrium equations of force and torque are listed as follows:

Figure 12.

The force analysis of the obstacle-climbing capability of the robot's driven wheel

Forces of the driven wheel2

{\begin{cases} T_{2} \sin α_{3} + m_{2} g + F_{m B T} \sin α_{3} + F_{G x} - F_{m B N} \cos α_{3} + m_{2} {\ddot{x}}_{B} = 0 \\ F_{G y} - T_{2} \cos α_{3} - F_{m B T} \cos α_{3} - F_{m B N} \sin α_{3} - m_{2} {\ddot{y}}_{B} = 0 \\ (T_{2} + F_{m B T}) (y_{B} - y_{G}) \sin α_{3} + m_{2} g (y_{B} - y_{G}) + (T_{2} + F_{m B T}) (x_{B} - x_{G}) \cos α_{3} \\ + F_{m B N} (x_{B} - x_{G}) \sin α_{3} - F_{m B N} (y_{B} - y_{G}) \cos α_{3} - J_{2} {\ddot{α}}_{2} = 0 \end{cases}

(52)

Forces of the BC rod

{\begin{cases} F_{B N} \cos α_{3} + m_{B C} g - F_{B T} \sin α_{3} - F_{C x} + m_{B C} {\ddot{x}}_{B C 2} = 0 \\ F_{B N} \sin α_{3} + F_{B T} \cos α_{3} - F_{C y} - m_{B C} {\ddot{y}}_{B C 2} = 0 \\ F_{B T} l_{B C} - (m_{B C} g + m_{B C} {\ddot{x}}_{B C 2}) (y_{C} - y_{B C 2}) - m_{B C} {\ddot{y}}_{B C 2} (x_{C} - x_{3}) - J_{B C} {\ddot{α}}_{3} = 0 \end{cases}

(53)

By analysing the robot body, the following can be obtained.

{\begin{cases} F_{m C x} + M g + F_{3} μ + F_{4} μ = F_{f 1} + T_{2} \sin α_{3} \\ F_{m C y} + T_{2} \cos α_{3} + F_{3} + = F_{1} + F_{4} \\ T_{2} \cos α_{3} (\frac{l_{B C}}{\cos α_{3}} + L_{2}) - T_{2} \sin α_{3} (\frac{d}{2} - l_{1} + r_{2} + l_{B C} \cos α_{30}) + F_{m C y} L_{2} + F_{m C x} (\frac{d}{2} - l_{1} + r_{2} + l_{B C} \cos α_{30}) \\ - F_{3} (L_{3} + l_{D E} \cos α_{4}) + F_{3} μ (\frac{d}{2} - l_{1}) + F_{4} L_{4} - F_{4} μ (\frac{d}{2} + l_{1}) - F_{1} L_{1} + F_{f 1} (\frac{d}{2} + l_{1}) = 0 \end{cases}

(54)

Eqs. (52) to (54) constitute the dynamic equation of the obstacle-climbing capability of the robot's driven wheel2.

5. Simulation and Experiments on the Climbing Capacity

5.1 Simulation

Based on the kinematic model of the robot, the motion tracks of each wheel and swine arm, and torque during obstacle climbing, can be obtained. A previous study [12] indicates that, when the moving wheel has a diameter of 30 mm, the robot can surmount an obstacle with a vertical height of 5.2 mm. Based on the motion force model of the robot, the variation in the driving force torque during obstacle climbing can be obtained (Fig. 13), where h = 5 mm and the moving wheel radius r = 30 mm.

Figure 13.

The input torque of the obstacle climbing of the robot vs. time(s}

The driving wheel climbs obstacles by coordinating the motions of the lateral flexible supporting mechanism of the driven wheel, as shown in Fig. 14. The geological centre A of wheel1, point C, point D, and gravity centre P are all planar in motion, that is, the motional curve is an arc (Fig. 14a). The centres of the other three wheels move linearly. Figures 14b–14c illustrates the motional speeds and acceleration rates of the main reference points (A, C, D, F, P) during obstacle negotiation.

Figure 14.

Kinematic simulation of obstacle climbing by the driving wheel of the robot

Figure 14 shows the kinematics simulation results of the upper driven wheel2 of the robot. The swinging of the upper swing arm is used to climb obstacles. Figure 15a illustrates the motional curves of support point C of the upper swing arm and the centre of wheel2. Figure 15b displays the angular speed and acceleration of the upper swing arm surrounding point C.

Figure 15.

Kinematics simulation of obstacle climbing by the driven wheel of the robot

5.2 Laboratory Experiments

A climbing robot was designed and used in experiments involving inclined and vertical cables to verify the feasibility of the inspection system. The laboratory conditions were established as follows:

Two cables with lengths of 5.3 and 3.7 m, and a diameter of 100 mm, can be adjusted randomly. These are the same cables used on the Sutong Bridge, which is the longest cable-stayed bridge in the world.

Two cables with lengths of 2.3 and 3.7 m, and a diameter of 139 mm, can be adjusted vertically and individually slanted at 29°.

Numerous steel pipes with diameters ranging from 40 mm to 205 mm can be slanted at any angle.

The main technical specifications of the robot are listed in Table 1. To validate the theoretical analysis, a control system was set up. However, precise detection sensors, a server motor and feedback quantity would be required if we were to construct a closed-loop control system. High costs would also be incurred. As these main challenges would have to be faced in the development of a closed-loop control scheme, for the robot's motion along the cables in the presence of obstacles, we only set up a simple control system in the laboratory.

To validate the load capacity of the robot when climbing the smooth cable, we carried out a number of climbing experiments. To validate the obstacle-climbing ability of the robot and its variety of velocity and driving torque attributes, we also carried out obstacle-climbing experiments.

Table 1.
Technical specifications of the robot

Dimensions 392 × 205 × 220 mm Climbing speed 0–0.26 m/s

Mass 7.0 kg Maximum payload 3.5 kg

Diameter scope of the cable 60–205 mm Height of obstacle >5 mm

5.2.1 Climbing Experiments

The climbing experiments were performed to test the load capacity of the robot (Fig. 16a). The results indicate that the climbing ability of the robot almost corresponds to the various cables of different diameters. Small changes were observed in the mass of the robot and the cable diameters. Moreover, the climbing ability of the robot was nearly identical on cables with diameters ranging from 60 mm to 205 mm. Line 1 in Fig. 16b represents the nominal speed of the robot running along a vertical cable with approximately 3.5 kg of payload. The climbing speed is clearly lower than the nominal value when the payload exceeds 3.9 kg, such as in lines 2 and 3, which represent the climbing velocities with payloads of 3.9 and 4.4 kg, respectively. Lines 1, 2 and 3 represent the climbing speed of the robot with various payloads.

Figure 16.

Climbing experiments in the laboratory

As the required torque increases, the velocity decreases slightly in the process of obstacle climbing. On the other hand, the velocity fluctuation is small, since the dimensions of the obstacles are also small. The curves 4, 5 and 6 represent the climbing speed when the robot is climbing over an obstacle. In these condition, the driving wheel does not slip, indicating that the friction coefficient of the driving wheel satisfies the climbing conditions. This finding demonstrates that the payloads exceed the nominal load-bearing capacity.

5.2.2 Obstacle-climbing Experiments

When the robot is climbing upward, the electric current and voltage can be measured, and the actual power can be computed. Therefore, the output torque can be calculated according to the electrical formula. The real-time torque can be seen in Fig. 17. Points 1 and 2 denote the torque when the wheel has just come into contact with the obstacle, causing the step to appear on the curve. Fig. 17c shows a picture of the robot when climbing obstacles. The experimental outcome is in accordance with the simulation results of the input torque when the robot is climbing an obstacle.

Figure 17.

Experimental real-time torque

6. Conclusions

In this manuscript, the damage forms of the protective layer of stayed cables are described. Then, an independent spring support robot model is proposed to solve the weak obstacle-climbing capacity of the bilateral climbing robot. Based on the deflection conditions of the robot body, the kinematic and dynamic characteristics of the robot are investigated. Furthermore, the influence of the spring pressure and the support swing arm angle of the driven wheels on the obstacle-climbing capacity of the driven and driving wheels are discussed. Finally, the motion curve of the mechanism, and the torque variation curve of the driving torque during obstacle climbing, are obtained and compared through simulations and experimentation. According to the simulation and experiments, the robot can take a payload 3.9 kg while moving along the cable. The maximum driving torque is 8 Nm when climbing an obstacle, and 2 Nm when climbing on a smooth cable. The result indicates that the climbing ability of the robot satisfies the demands of cable detection.

Although the paper offers a simple solution to determining the driving torque, it should be pointed out that the paper hasn't resolved the robot's control problems completely. To ensure the stability of the robot while working at a high altitude of several hundred metres, future research should focus on a control theorem to overcome the influence of altitude, wind loading and cable vibration on the climbing ability of the robot.

Footnotes

7.

This project is supported by the National Natural Science Foundation of China (51005046), the Natural Science Research Fund of the Nanjing University of Posts and Telecommunications (NY214071), and the Jiangsu Province Natural Science Fund Project for Colleges and Universities (13KJB460013).

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Dimensions	392 × 205 × 220 mm	Climbing speed	0–0.26 m/s
Mass	7.0 kg	Maximum payload	3.5 kg
Diameter scope of the cable	60–205 mm	Height of obstacle	>5 mm

Kinematic and Dynamic Analysis of a Cable-Climbing Robot

Abstract

Keywords

1. Introduction

3.1 Kinematic Analysis of the Obstacle Climbing of the Driving Wheel

4.1 Kinematic Analysis of the Obstacle-climbing Capability of the Robot's Driven Wheel2

5.1 Simulation

Table 1. Technical specifications of the robot Dimensions 392 × 205 × 220 mm Climbing speed 0–0.26 m/s Mass 7.0 kg Maximum payload 3.5 kg Diameter scope of the cable 60–205 mm Height of obstacle >5 mm

Footnotes

7.

References

Table 1.
Technical specifications of the robot

Dimensions 392 × 205 × 220 mm Climbing speed 0–0.26 m/s

Mass 7.0 kg Maximum payload 3.5 kg

Diameter scope of the cable 60–205 mm Height of obstacle >5 mm