Research on Climbing Robot for Transmission Tower Based on Foot-End Force Balancing Algorithm

Abstract

This paper aims to introduce robot technology to carry out the safety inspection of transmission towers in long-distance power transmission, so as to improve the safety and efficiency of inspection. However, aiming at the problem that the existing climbing robots are mainly used for large load applications, which leads to the large size and lack of flexibility of the robot, we propose an innovative solution. Firstly, a lightweight quadruped climbing robot is designed to improve portability and operational flexibility. Then, a one-dimensional force sensor is added at the end of each leg of the robot, and a special swing phase trajectory is designed. The robot can judge whether the electromagnetic adsorption is effective and avoid potential safety hazards. Finally, based on the principle of virtual model control (VMC), a foot-end force balancing algorithm is proposed to achieve uniform distribution and continuous change in force, and improve safety and load capacity. The experiments show that the scheme has a stable climbing ability in the environments of angle steel, vertical ferromagnetic plane and transmission tower.

1. Introduction

Transmission towers are indispensable infrastructures in power delivery systems, whose stability and safety are crucial to the operation of power grids [1]. The conventional manual inspection of transmission towers is inefficient and hazardous to workers [2]. In recent years, using robots as substitutes for human work to conduct autonomous inspection and maintenance of transmission equipment has become a research hotspot. The introduction of robotic technologies can significantly mitigate risks for workers, improve inspection efficiency and reduce costs under the growing power demands. Therefore, investigating applications of robotic technologies in transmission tower inspection bears great practical significance.

Currently, the mainstream climbing robots include tracked climbing robots [3], wheeled climbing robots [4] and legged climbing robots. Tracked and wheeled climbing robots [5] can provide steady climbing speed and travel on walls or pipes. Bio-inspired climbing robots with adsorption capabilities, such as biped [6] and multi-legged wall-climbing robots [7,8], can perform facade painting and cleaning tasks on buildings. To accelerate mobility, soft climbing robots [9] can be employed. Some researchers also developed linkage-based wall-moving robots [10], which can maneuver and turn swiftly on flat walls. However, the landing spots on transmission pylons are limited, and there are many obstacles like bolts and angle steels. Due to the more complex environment compared to building facades, the above robot types are inapplicable for power transmission scenarios.

Quadruped robots are more suitable for the complicated climbing tasks on transmission pylons compared to other robot morphologies. Climbing robots deployed on transmission towers need a certain load capacity, so research on adsorption mechanisms is necessary. The adsorption techniques include mechanical fixing [11], electromagnetic adsorption [12,13] and material adhesive properties [14]. Considering the demands on both mobility and loading capacity for climbing robots on transmission towers, as well as cost-effectiveness, electromagnetic adsorption is more favorable.

This study proposes an innovative solution to enhance the safety, load-bearing capacity and operation efficiency of climbing robots for transmission tower inspection. First, we designed a lightweight and compact quadruped robot, which improves the mobility compared to conventional bulky climbing robots, as shown in Figure 1. Each leg is equipped with an electromagnetic chuck to provide strong ground-holding forces on the tower surface. Combined with a specially designed swing gait trajectory, the robot can actively judge the adsorption state and ensure firm attachment during climbing. Third, based on the principle of VMC, we developed an algorithm for balancing endpoint forces to achieve uniform force distribution and smooth transitions, hence improving the safety and load capacity.

2. Problem Description and System Design

2.1. Problem Description

Whether the electromagnets can effectively adsorb onto the ferromagnetic surface is a critical issue during the climbing of quadruped robots, which directly affects the safety, stability and reliability. In practical operation, there may exist slight deviations between the actual and expected attitude of each leg end, which can easily lead to the ineffective adsorption of electromagnets onto the ferromagnetic plane. This situation is difficult to avoid since deviations can accumulate from mechanical errors or joint motor errors. Generally, the leg end of quadruped robots is spherical or arc-shaped, which can adapt to such minor attitude deviations. Therefore, the contact area between each leg end and the ground can be regarded as a point or small patch [15]. However, this study employs electromagnets as the adsorption structure, so the contact area between each leg end and the ferromagnetic plane is larger. If there is a certain angle deviation between the adsorption plane of the electromagnet and the ferromagnetic plane, the electromagnet will fail to adsorb effectively. Without an autonomous detection of effective electromagnet adsorption, potential safety hazards may arise. Therefore, solving the electromagnet adsorption issue is crucial for ensuring the stable operation and safety of the robot.

Another problem is that, during the gait transition of the quadruped robot climbing, the forces exerted on each leg can be discontinuous, which affects the stability and mobility. As illustrated in Figure 2, it is a schematic of the force analysis when the quadruped robot climbs an inclined ferromagnetic plane.

When the leg is adsorbed on the ferromagnetic plane, as shown by the RH leg, each leg end will be subjected to static friction f, support force F and torque M. When the RF leg transitions from the swing phase to the support phase and just adsorbs onto the ferromagnetic plane effectively, the forces exerted on the robot are

$$\begin{array}{c}\hfill mgcos\alpha =f_{LF}^{\prime }+f_{LH}^{\prime }+f_{RH}^{\prime }.\end{array}$$

(1)

At this point, we assume that the force exerted on each leg is uniform (the forces on each leg are definitely unequal in practice), that is,

$$\begin{array}{cc}\hfill f_{LF}^{\prime }=f_{LH}^{\prime }=f_{RH}^{\prime }& =\frac{1}{3}mgcos\alpha ,\hfill \\ \hfill f_{RF}^{\prime }& =0.\hfill \end{array}$$

(2)

Without taking any measures, when lifting the next leg, i.e., the moment the LH leg is lifted, the forces exerted on the robot are

$$\begin{array}{c}\hfill mgcos\alpha >f_{LF}^{\prime }+f_{RH}^{\prime }=\frac{2}{3}mgcos\alpha .\end{array}$$

(3)

Therefore, the self-weight of the robot will cause some disturbance to itself until the next force equilibrium.

$$\begin{array}{cc}\hfill f_{LF}^{″}& =(1+{\mathsf{\mu }}_1)\frac{1}{3}mgcos\alpha ,\hfill \\ \hfill f_{LH}^{″}& =0,\hfill \\ \hfill f_{RH}^{″}& =(1+{\mathsf{\mu }}_2)\frac{1}{3}mgcos\alpha ,\hfill \\ \hfill f_{RF}^{″}& ={\mathsf{\mu }}_3\frac{1}{3}mgcos\alpha ,\hfill \end{array}$$

(4)

where ${\mathsf{\mu }}_1$, ${\mathsf{\mu }}_2$, ${\mathsf{\mu }}_3$ are friction distribution coefficients, which are determined by the robot’s current posture and mechanical structural errors, and ${\mathsf{\mu }}_1+{\mathsf{\mu }}_2+{\mathsf{\mu }}_3=1$. Similarly, when the gait switches twice more, the friction force exerted on the RF leg will increase significantly. Experiments show that the friction force on this leg can be greater than the sum of the friction forces on the other three legs.

If using only the traditional quadruped gait, each transition of the gait will cause a sudden change in the frictional force acting on each leg, resulting in discontinuity of the forces acting on the foot. To enhance the stability and movement efficiency of the robot, a unique control algorithm is necessary to make the force acting on the foot of each leg more uniform and continuous.

2.2. Mechanical Structure Design

The schematic diagram of the quadruped climbing robot designed in this paper is shown in Figure 3. The structure emulates a spider-like form, aiming to effectively lower the robot’s center of gravity. As the operating environment involves a ferromagnetic area, the climbing mechanism utilizes electromagnet adhesion. In order to reduce the robot’s mass, servo motors with higher torque density were selected as joint motors, and the body structure is constructed using nylon, a material known for its high structural strength. Each leg of the robot possesses four degrees of freedom, which ensures maximum accessibility in spatial movement while ensuring that the electromagnetic adhesion surface remains parallel to the climbing surface. Springs and one-dimensional force sensors [16] are installed between the electromagnet and joint 4. The direction of the one-dimensional force sensor is aligned with the X-axis direction in the foot coordinate system (referred to as the X-axis direction in the foot coordinate system in subsequent text), as illustrated in Figure 3 by the $rf$ coordinate system.

Due to factors such as servo accuracy, assembly errors and mechanical structural inaccuracies, the actual posture of each leg of the quadruped robot may not perfectly match the desired posture. To ensure the effective adhesion of the electromagnet to the ferromagnetic surface, we introduced springs as passive joints between joint 4 and the electromagnet. Under certain pressure, these passive joints are capable of adaptively compensating for positional errors at the foot.

When the quadruped robot climbs on a vertical or inclined plane, the support legs may experience uneven frictional forces, leading to instability. Thus, it is necessary to implement force control for each leg. To achieve force feedback control at the foot, force sensors were integrated between joint 4 and the spring. External forces acting on the foot propagate along the X-axis direction onto the electromagnet, transmitted through the spring to the force sensor. The spring also serves as a low-pass filter, reducing vibrations transmitted to the foot.

Typically, the foot of a quadruped robot experiences forces in three directions (excluding moments or torques). However, multi-axis force sensors (such as three-axis or six-axis force sensors) are costly, especially for applications with small volumes and masses. During climbing, the quadruped robot primarily experiences frictional forces along the X-axis direction. Consequently, this paper employs a simple one-axis force sensor, detecting forces in the direction of the projection of joint 2 and joint 3 torques on the adhesion surface of the electromagnet.

2.3. Kinematic Model

In order to facilitate the control of the multi-jointed robot, it is essential to calculate the inverse kinematic solution for each leg based on the known target position and posture, thereby determining the rotational angles for each joint. The application of inverse kinematics transforms the robot’s control from the four joints in the leg to the control of the foot-end coordinates ${(P_x,P_y,P_z)}^T$, significantly simplifying subsequent algorithm design. Figure 4 displays a three-view diagram of the left-front leg of the robot designed in this study.

In the diagram, $L_1$, $L_2$, $L_3$ and $L_4$ represent the mechanical lengths of link 1, link 2, link 3 and link 4, respectively. ${\theta }_1$, ${\theta }_2$, ${\theta }_3$ and ${\theta }_4$ represent the rotation angles of joints 1, 2, 3 and 4, respectively. ${\theta }_{2f}$ denotes the deviation angle considered in the mechanical design. We define counterclockwise rotation as the positive direction for joint rotation. The leg coordinate system L is utilized here, with the origin of this coordinate system coinciding with the position of joint 2. The origin shown in Figure 4 is chosen for illustration purposes and does not precisely represent the joint 2 position. According to geometric relationships, the following direct kinematic equations can be derived:

$$\begin{array}{cc}\hfill P_x=& L_{2}c{\theta }_{1}s({\theta }_2+{\theta }_{2f})-L_{3}c{\theta }_{1}s({\theta }_2+{\theta }_3)-\hfill \\ \hfill & L_{4}c{\theta }_{1}s({\theta }_2+{\theta }_3+{\theta }_4),\hfill \\ \hfill P_y=& L_{2}s{\theta }_{1}s({\theta }_2+{\theta }_{2f})-L_{3}s{\theta }_{1}s({\theta }_2+{\theta }_3)-\hfill \\ \hfill & L_{4}s{\theta }_{1}s({\theta }_2+{\theta }_3+{\theta }_4),\hfill \\ \hfill P_z=& L_{2}c({\theta }_2+{\theta }_{2f})-L_{3}c({\theta }_2+{\theta }_3)-\hfill \\ \hfill & L_{4}c({\theta }_2+{\theta }_3+{\theta }_4),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\theta }_4=& -{\theta }_2-{\theta }_3.\hfill \end{array}$$

(6)

In the above equations, $s{\theta }_1$ denotes $sin{\theta }_1$, $c{\theta }_1$ represents $cos{\theta }_1$ and so forth. Equation (6) represents the kinematic constraint to ensure that link 4 remains parallel to the Z-axis, thereby guaranteeing that the electromagnet’s adhesion surface remains parallel to the ferromagnetic surface. Through the inverse operations applied to Equations (5) and (6), the inverse kinematic equations can be derived as follows:

$$\begin{array}{cc}\hfill {\theta }_1=& atan2(P_y,P_x),\hfill \\ \hfill {\theta }_2=& \frac{\pi }{2}-atan2(L_4+P_z,L)-\hfill \\ \hfill & acos\left(\frac{L_2^2+L_1^2-L_3^2}{2L_1{L}_2}\right)-{\theta }_{2f},\hfill \\ \hfill {\theta }_3=& -acos\left(\frac{L_2^2+L_3^2-L_1^2}{2L_2{L}_3}\right)+{\theta }_{2f},\hfill \\ \hfill {\theta }_4=& -{\theta }_3-{\theta }_2.\hfill \end{array}$$

(7)

Considering ease of computation, Equation (8) has been simplified:

$$\begin{array}{cc}\hfill L& =\sqrt[]{z^2+x^2},\hfill \\ \hfill L_t& =\sqrt[]{{(L_4+P_z)}^2+L^2}.\hfill \end{array}$$

(8)

2.4. System Control Framework

The overall control framework for the quadruped climbing robot in this paper is depicted in Figure 5. The controller primarily consists of three sub-modules. Different sub-modules are employed for legs in different phases, each with distinct parameters and statuses. The foot force balancing control module manages the forces acting on the robot’s foot to prevent discontinuities in foot forces. The swing trajectory module provides the desired posture information for the swinging leg to control its motion.When in the electromagnet adhesion phase, the swing trajectory module operates in conjunction with the electromagnet adhesion signal detection module to assess the effective adhesion of the electromagnet. The controller computes the control posture for each leg, performs inverse kinematic calculations and then controls each joint motor to rotate to the corresponding angle.

3. Main Research Content

3.1. Detection of Effective Electromagnet Adhesion

When the quadruped robot climbs on a vertical plane, the lack of effective adhesion of any leg to the angle steel or iron plate may lead to safety accidents. Hence, during the climbing process, it is essential to determine through feedback detection whether the swinging leg has achieved effective adhesion on the angle steel or ferromagnetic surface.

3.1.1. Swing Phase Motion Trajectory

The design objective of the swing phase motion trajectory is to allow the quadruped robot’s foot to lift, avoid obstacles and possess good resilience during the swinging process to minimize energy loss [17]. In the robot designed in this paper, the swing phase trajectory’s design aligns with the effective adhesion detection of the electromagnet, aiming for a smoother transition of the trajectory into a force-balanced state.

To maintain the continuity of foot position, velocity and acceleration [18], the swing phase motion trajectory is divided into two stages: from the initial time 0 to the highest point time $t_m$, and from $t_m$ to the end of the swing phase at time $t_{sw}$. In the trajectory design, we use a fourth-order polynomial and a fifth-order polynomial to represent the foot’s motion in the X-axis and Z-axis directions, respectively.

Let us consider the phase from 0 to $t_m$. To ensure a smooth transition between the swing and support phase trajectories, the following boundary conditions are set:

$$\begin{array}{cc}\hfill S_{sw1,x}\left|{}_{t=0}\right.=& x_s,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill S_{sw1,x}\left|{}_{t=t_{sw}}\right.=& 0,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{S}_{sw1,x}}{\mathrm{d}t}\left|{}_{t=0}\right.=& 0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \frac{{\mathrm{d}}^2{S}_{sw1,x}}{{\mathrm{d}}^{2}t}\left|{}_{t=0}\right.=& 0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{S}_{sw1,x}}{\mathrm{d}t}\left|{}_{t=t_{sw}}\right.=& v_x,\hfill \end{array}$$

(13)

where $x_s$ represents the position of the starting point of the support phase, and $v_x$ is the velocity along the X-axis at time $t_m$, as depicted in Figure 6. Based on these constraints, we derive a fourth-order polynomial trajectory for the quadruped robot’s foot in the X-axis direction.

$$\begin{array}{c}\hfill x_{sw1}=\frac{v_x{t}_m+3x_s}{t_m^4}{t}^4-\frac{v_x{t}_m+4x_s}{t_m^3}{t}^3+x_s.\end{array}$$

(14)

The same method can be applied to obtain the trajectories for other swing phases, as follows:

$$\begin{array}{cc}\hfill z_{sw1}=& \frac{6(z_f-z_s)}{t_m^5}{t}^5-\frac{15(z_f-z_s)}{t_m^4}{t}^4\hfill \\ \hfill & +\frac{15(z_f-z_s)}{t_m^3}{t}^3,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill x_{sw2}=& \frac{v_x}{t_f^3}{t}^4-\frac{v_x}{t_f^2}{t}^3+v_{x}t,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill z_{sw2}=& -\frac{6(z_f-z_m)}{t_m^5}{t}^5+\frac{15(z_f-z_m)}{t_m^4}{t}^5\hfill \\ \hfill & -\frac{10(z_f-z_m)}{t_m^3}{t}^3+z_f.\hfill \end{array}$$

(17)

Taking the LF leg as an example, the foot motion trajectory of the quadruped climbing robot in this paper is shown in Figure 6.

The leg motion during the swing and electromagnetic adhesion phases for the quadruped climbing robot is represented by the designed swing trajectory, as depicted by the red and yellow paths in Figure 6. Upon completion of the swing phase, the robot’s foot will continue moving along the negative direction of the X and Z axes in the trajectory coordinate system, preparing to adhere the electromagnet to the ferromagnetic surface. The inclusion of the yellow trajectory enables continual variations in leg posture to ensure successful adhesion and align with force sensors to detect the effective adhesion at the foot.

3.1.2. Electromagnet Adhesion Signal Detection

Upon transforming the conventional compound hypocycloid motion trajectory to the polynomial trajectory obtained, the force curve measured by the X-axis force sensor at the foot is depicted in Figure 7.

From the graph, it can be observed that the force gradually decreases when the leg is in the swing phase. This decrease is primarily due to the force sensed by the sensor, which is determined by the gravity of the electromagnet and the foot’s acceleration during this phase. During the swing phase, the gravity of the electromagnet acts in the negative direction of the X-axis. However, when the leg transitions to the electromagnetic adhesion phase, if the electromagnet successfully adheres, the force sensed by the force sensor gradually increases, as depicted in the green time segment in Figure 7. Therefore, the sliding window average sequence algorithm [19] can be employed to effectively identify the state of successful adhesion of the electromagnet.

The computation method for the sliding window average sequence algorithm is as follows:

$$\begin{array}{c}\hfill {\widehat{y}}_{min,t}=min\left({\widehat{y}}_t\right),{\widehat{y}}_t=\frac{1}{w}\sum _{j=i-w+1}^i{y}_j,\end{array}$$

(18)

where $y_j$ represents the raw data, ${\widehat{y}}_t$ signifies the average value obtained at the current time t through the sliding window algorithm, w represents the size of the sliding window and ${\widehat{y}}_{\min ,t}$ denotes the historical minimum value within the sliding window up to the current time. By evaluating the ${\widehat{y}}_t$ calculated for each window, the determination of effective electromagnet adhesion can be made by assessing $|{\widehat{y}}_t-{\widehat{y}}_{\min ,t}|>T_F$.

3.2. Foot Force Equilibrium Control

For traditional gait control, when a leg is in the support phase, the force it experiences in the X-axis direction is excessive, placing high demands on the adhesion force of the electromagnetic iron and the ferromagnetic surface. In the context of a transmission tower environment, the adhesion force of the electromagnetic iron drastically reduces, leading to a significant decrease in friction. While increasing the size of the electromagnetic iron can boost adhesion force, it can also impose excessive load on the joint motors due to its increased size and weight. Therefore, the introduction of the foot-end force balance control algorithm aims to balance the forces experienced by the supporting legs during the climbing process, augmenting the robot’s load-bearing capacity and enhancing its stability.

Considering the robot as a unified entity, an aggregate of external forces equivalent to or surpassing the robot’s weight causes an upward shift in its center of gravity and a tendency to move upward. The traction force can be measured by the force sensor. This research-designed robot features a relatively uniform mass distribution, making it unsuitable to be considered as a single mass model [20]. This complexity in modeling prevents the straightforward estimation of the center of mass deviation [21,22]. However, in contrast, the leg mass is relatively minor, allowing the entire system to be visualized as a body with four electromagnetic irons. Consequently, the upward movement of the center of mass primarily reflects the elevation of the body and changes in the position of the swinging legs.

Assuming that all legs are in the support phase and adhered to the transmission tower, when the external forces acting on the robot exceed its gravitational force, the center of the mass will shift upward. A damper was designed based on the principles of VMC [23]. It calculates the desired force based on the robot’s motion speed, as shown in Equation (19).

$$\begin{array}{cc}\hfill F_d& =\frac{1}{3}[B_u(\dot{u_d}-\dot{u})+Gsin\alpha sin\beta ],\hfill \end{array}$$

(19)

where $B_u$ represents the damping coefficient, $\dot{u_d}$ signifies the desired speed and $\beta$ is the inclination angle of the angle steel. Therefore, the foot-end force of each leg is related to the expected traction force calculated by Equation (19). In the gait of this robot, at any given moment, only three legs provide upward support. To ensure that each leg bears uniform force, an incremental PID control scheme is employed. Due to the fact that it is not feasible to exert torque control on the servo motors, this study indirectly controls the posture of each leg, thereby indirectly regulating the force experienced by each leg.

$$\begin{array}{cc}\hfill x_{su}& =x_{su,last}+K_p(\dot{F_d}-\dot{F})+K_i(F_d-F),\hfill \end{array}$$

(20)

where $x_{su}$ represents the posture of the supporting leg along the X-axis, $K_p$ and $K_i$ denote PID parameters, $F_d$ is the desired force and F represents the actual force exerted. When a leg is in the support phase, external forces are required to support the movement of the entire body. However, when the leg is in the swing phase, the force at the foot is almost determined by the acceleration. Thus, during gait transition, a transitional state, called the force balance phase (as depicted in Figure 6), is introduced.

The force balance phase is a transition phase for the leg forces, indicating that the leg intended to support will gradually bear the force, while the leg planned to lift will gradually release the force. The postures of each leg along the X-axis are as follows:

$$\begin{array}{c}\hfill x=\left\{\begin{array}{ccc}{x}_{sw},\hfill & & 0<t<t_f,\hfill \\ x_{last}+K_p(\dot{F_d}-\dot{F})+K_i(F_d-F),\hfill & & t_f<t<t_{su},\hfill \\ x_{last}-K_p\dot{F}-K_{i}F,\hfill & & t_{su}<t<T,\hfill \end{array}\right.\end{array}$$

(21)

where $0<t<t_f$ represents the swing phase and the electromagnetic adhesion phase, $t_f<t<t_{su}$ denotes the force balance phase and the support phase and $t_{su}<t<T$ represents the force release phase. Throughout the process of quadrupedal climbing, the swing leg significantly affects the overall center of mass of the robot, leading to substantial interference with the force control of the supporting legs. Hence, minimizing abrupt changes in posture for the swing leg is essential.

4. Experiments

The physical connection diagram between the one-dimensional force sensor and the spring of the robot designed in this study is shown in Figure 8.

Table 1. Performance parameters of the quadruped climbing robot.

The Degrees of Freedom for Each Leg	The Weight of the Robot	The Minimum Retracted Size	Climbing Step Distance	Crawling Speed	The Maximum Safe Load Capacity
4	849 g	56 × 138 × 105 mm	80 mm	1.2 m/min	700 g

shows the performance parameters of the robot.

4.1. Climbing Experiment

The graph in Figure 9 depicts the force curve along the X-axis when using the traditional composite cycloidal motion as the foot swinging trajectory. When a leg ends the swing phase and just adheres to the ferromagnetic plane, it provides support in the Z-axis direction for the entire robot, with minimal force in other directions. In Figure 9, at the final stage of the LF leg’s swing phase, it shows that the electromagnetic attraction is effective, resulting in a stabilized force in the X-axis direction. However, it is observed that the leg did not experience significant force, indicating a lag in providing sufficient friction to maintain the robot’s movement promptly.

Based on the characteristics of lag and the “walk” gait, at the instant when a leg finishes the electromagnetic adhesion, only the friction of two legs is supporting the stability of the robot. Clearly, the friction from these two legs is insufficient to fully counteract gravity, and the excess gravity is shared among the other three legs in the support phase. As seen in Figure 9, there is a discontinuity in the force exerted on the legs during the transition from the swing phase of the LF leg to the start of the swing phase of the RB leg. This discontinuity signifies an abrupt change in the force experienced by the legs, in line with the force analysis presented in Equation (4). Such an abrupt change may cause the robot to oscillate, reducing overall stability.

It should be noted that the sum of the forces acting on the four force sensors is less than 6N, which is inconsistent with the total weight of the robot for 850 g. First of all, in general, the angle steel above the transmission tower is inclined, so when the quadruped robot designed by us looks up and climbs on the angle steel, the gravity along the climbing direction is only a part of the gravity component, which is expressed as $mgcos\alpha cos\beta$ in this paper. From here, it can be seen that the force measured by the output sensor is not equal to the total weight of 850 g. Secondly, the most important point is that the force sensor is installed between joint 4 and the electromagnet. When each leg of the quadruped robot is adsorbed on the angle steel, it can be regarded as the electromagnet and the angle steel as a whole, and the remaining part of the robot is a whole. Therefore, the weight measured by the force sensor at this time will only include the part that removes the electromagnet, and it so happens that the electromagnet is relatively heavy. This is also the reason why the influence of leg movement on the center of gravity of the robot cannot be ignored. In summary, the total force measured by the force sensor has a large deviation from the total weight of the robot.

The following Figure 10 illustrates the curves detected by the X-axis force sensors at the foot of each leg during quadruped climbing with force balance control.

Due to reasons such as the joint motors being servo motors and assembly errors, the force curves for different legs are different at the same phase, but the overall trends are similar. It is noticeable that, within one gait cycle, the maximum force exerted by each leg is roughly equal. Compared to Figure 9, when no force balance control is implemented, the maximum friction between the electromagnet and the angle steel can reach up to 4N, but after applying force balance control, this friction is reduced to 2N. We observe a significant reduction in the amplitude of forces acting along the X-axis for each leg. This advantage allows the robot to carry larger loads or opt for smaller electromagnet sizes.

During the swing phase, the swinging legs still have a certain impact on the force control of the entire body. The small mass of the legs in the quadruped robot can be ignored, but the electromagnet still has a certain effect on the robot’s center of gravity. In the electromagnet adhesion phase, we can see that effective adhesion to the electromagnet is still rather sensitive.

Through Figure 10, we can see that the entire robot’s climbing is smoother and safer after implementing force balance control. Without force balance control, when one leg of the quadruped climbing robot transitions from the support to the swing phase, the legs in the support phase will experience a certain gravitational impact, causing vibrations in the entire body, which is detrimental to the robot’s stability.

Figure 11 shows the timestamps for robot climbing. Here, we have selected 4 s as a gait cycle.

4.2. Safe Load Experiment

The load capacity of the quadruped climbing robot is primarily influenced by two aspects: the maximum joint drive force at the leg end and the maximum friction between the electromagnet and the adhesive surface. The maximum joint drive force at the leg end depends on the mechanical structure and the maximum torque of the joint motor. The maximum friction between the electromagnet and the adhesive surface is affected by multiple factors, including the magnetic strength of the electromagnet, the coefficient of friction, and the distance between the electromagnetic adhesive surface and the ferromagnetic plane.To ensure the safe use of the robot, this study defines the safe load as the load that the robot can carry while only two legs are adhering. Through the force balance control algorithm developed in this paper, the friction on each leg’s electromagnet can be balanced, significantly enhancing the robot’s safe load capacity.

Figure 12 displays the results of the load capacity tests for the quadruped climbing robot. When the robot climbs on a 6 mm thick vertical iron plate, its load capacity is mainly affected by the motor torque, with a maximum safe load capacity of 700 g. Climbing on a 3 mm thick angle steel surface primarily affects the load capacity due to the maximum friction between the electromagnet and the adhesive surface, resulting in a maximum load capacity of 500 g. When climbing on an angle steel tower, due to the thicker anti-rust coating on the tower, there is a larger distance between the adhesive surface and the ferromagnetic plane, resulting in lower magnetic force of the electromagnet and thus, reduced friction. Consequently, a stronger electromagnet (15 kg) was utilized. As per the experimental results, the robot can carry a load of 200 g, with the primary influencing factor for maximum carrying load being the maximum torque of the joint motor [24].

5. Conclusions

This paper presents a lightweight and compact quadruped climbing robot capable of crawling at a speed of 1.2 m/min and supporting a maximum load of 700 g. To address the challenge of effective adhesion of the electromagnetic system during climbing, a one-dimensional force sensor was incorporated at the robot’s foot, and a specialized trajectory for swing phase motion was designed. Leveraging feedback from the one-dimensional force sensor and the VMC algorithm, a foot force balance control algorithm was formulated to address the discontinuous force issues experienced by the robot’s feet during climbing, thus enhancing the robot’s climbing stability. Considering that the surfaces of power transmission towers are typically coated with anti-corrosion layers, significantly affecting the adhesion capabilities of the electromagnets, the foot force balance control algorithm described in this paper can considerably enhance the robot’s safety and load-bearing capacity.