2.1. Overall configuration

A systematic design of the robot’s core mechanical structure and transmission system is imperative, both for stable climbing on the dedicated T-shaped guide rail (Figure 1) and for adaptive lifting force adjustment with reliable fall prevention. This chapter presents the overall configuration and the key component designs developed to fulfill these objectives.

Figure 1.

A cross-sectional view of a T-shaped guide rail.

The robot consists of four main subsystems as shown in Figure 2: a multi-stage compound transmission, a rubber-wheel clamping climbing module, a self-balancing anti-fall mechanism, and an electrical control unit. All subsystems are integrated within the robot frame, forming a compact overall structure. The power transmission and self-balancing anti-fall mechanism are positioned in the mid-rear section, with the rubber-wheel clamping climbing module symmetrically mounted along the midline of the chassis. Each subsystem is elaborated in the following sections.

Figure 2.

Schematic of the robot system composition.

2.2. Analysis of rubber-wheel clamping climbing module

This design employs a three-point clamping mechanism, as shown in Figure 3, whose core principle lies in utilizing its geometric configuration to achieve passive mechanical self-locking. The three wheels are arranged at specific angles to naturally envelop the rail profile, while the robot’s self-weight is automatically resolved into three mutually distinct normal forces. These forces constitute a stable mechanical self-locking system in space, with their resultant consistently oriented toward the interior of the guide rail. This ensures continuous and reliable contact pressure and friction between the wheels and the guide rail under any orientation, without requiring active control, thereby significantly simplifying the control system and enhancing inherent safety and reliability. Furthermore, the left and right driving wheels symmetrically clamp onto a T-shaped guide rail, providing lateral guiding constraints that prevent the robot from side-slipping or derailing during vertical operations.

Figure 3.

Principle of three-point clamping mechanism.

Figure 3. Long description

Panel A: A diagram showing the central rubber wheel positioned between the left and right drive wheels. The contact area between the wheels is highlighted. Panel B: A diagram depicting the three-point clamping mechanism in action. The central rubber wheel is in contact with a track, and the left and right drive wheels are connected to a transmission shaft. The forces acting on the wheels are indicated by arrows labeled F_n.

The geometric configuration of the wheel arrangement predetermines the force distribution, resolving the robot’s self-weight into positive normal forces that generate sufficient friction to counteract gravity and achieve self-locking. Through rational design of the geometric profiles and wheel angles, the self-locking condition can be satisfied, enabling stable self-locking inherently through the geometric configuration itself.

In practical applications, local rail irregularities such as welding seams, corrosion products, or manufacturing tolerances may alter the wheel–rail contact conditions. Owing to the three-point clamping configuration, the robot remains statically constrained even when one contact point experiences a temporary disturbance. The remaining contact points redistribute the contact forces and contribute to maintaining rail engagement and traction capability. Consequently, local rail irregularities primarily influence the instantaneous force distribution rather than the overall stability of the system. The stability of the robot is evaluated according to the support-region criterion defined by the three wheel–rail contact points rather than requiring the projection of the resultant force to pass through the geometric centroid of the support triangle. Although the resultant force line may shift away from the centroid of the support triangle, static stability is maintained as long as its projection remains within the support region formed by the three wheel–rail contact points. Accordingly, the influence of local rail irregularities is primarily reflected by the redistribution of the wheel–rail contact forces rather than by the loss of stability. When the resultant force approaches the boundary of the support region, the resulting overturning moment is constrained by the geometric confinement of the T-shaped guide rail. Figure 4 schematically illustrates the stability of the robot under normal and disturbed contact conditions.

Figure 4.

Schematic illustration of the robot stability under different contact conditions.

The three wheels can be coordinately driven to provide multi-point synergistic traction, demonstrating excellent dynamic fault tolerance. In the event of a single wheel encountering slippage, the remaining two wheels continue to supply effective traction due to the non-collinear wheel arrangement and the sustained clamping forces. Crucially, the plane defined by the three wheels facilitates adaptive motion adjustment through its inherent geometric constraints. This allows a slipping wheel to naturally regain effective frictional contact after traversing a local obstacle, enabling an automatic recovery of traction force. This mechanism significantly improves the system’s operational robustness in the guide rail environments.

2.3. Configuration of self-balancing anti-fall mechanism

The self-balancing anti-fall mechanism serves as a critical safety guarantee for the system. Its core is a passively rotatable eccentric cylindrical shaft, installed at the bottom of the robot and normally kept in close contact with the guide rail. The schematic diagram of its structure is shown in Figure 5. When the robot climbs vertically along the transmission line rail while carrying tools or operating in coordination with inspectors, and experiences accidental sliding or loss of traction, the eccentric shaft will automatically rotate by a predetermined angle under the combined action of gravity and the pre-load of the mechanism. This rapid transition from a conforming state to mechanical jamming with the guide rail structure achieves instantaneous self-locking without external control commands, effectively arresting motion and preventing a fall.

Figure 5.

Schematic diagram of self-balancing anti-fall mechanism.

By adjusting the angle between the eccentric cylindrical shaft and the guide rail, the robot’s lifting force and climbing velocity can be modulated. Acting as a mechanical advantage controller, this mechanism converts rotational input into precise vertical motion: increasing the eccentric angle raises the velocity but reduces the lifting force, and vice versa. This power-focused configuration allows the robot to handle heavier payloads or greater resistance at lower climbing velocity, enhancing mission flexibility.

2.4. Multi-stage compound transmission system

As the core drive unit, the multi-stage compound transmission efficiently delivers motor power and torque to the climbing wheels through spur gears and belt pulleys. This design ensures precise rotational velocity synchronization across the entire set of wheels, significantly reduces transmission losses, and coordinates the motion of all drive points, thereby avoiding mechanical jamming and uneven wear caused by velocity mismatch. Figure 6 illustrates the structural layout of the robot’s transmission system, where the orange arrow indicates the power transmission path.

Figure 6.

Power transmission of multi-stage compound transmission system.

2.5. Reliability design and performance analysis of the mechanism

2.5.1. Kinematics analysis

The robot’s velocity relationship can be expressed by Eq. (1). This indicates that the robot’s overall translational velocity can be modulated by adjusting either the effective rolling radius R_i of the climbing wheels or the rotational velocity n_i of the drive motors. This dual-parameter control strategy provides operational flexibility, allowing the system to maintain optimal climbing performance under varying load conditions and guide rail surface properties:

(1) \begin{equation} v=2\pi R_{i}n_{i} \end{equation}

where v, R_i , and n_i denote the robot’s climbing velocity, the effective rolling radius of the climbing wheels, and the rotational velocity of the drive motors, respectively.

2.5.2. Quasi-static equilibrium analysis

This section analyzes the force distribution during climbing (Figure 7) to assess mobility performance, which is critical for stability and safety. Under the constant-velocity assumption, a quasi-static force-equilibrium model is established to analyze the force distribution during climbing. The proposed force-equilibrium model is used to determine the required wheel torque and to evaluate the robot’s capability to maintain climbing equilibrium. Based on the proposed force-equilibrium model together with the robot mass, wheel geometry, wheel–rail friction, and rolling resistance, the required motor torque for climbing equilibrium on T-shaped guide rails is determined. The parameters used in this analysis are summarized in Table I.

Figure 7.

Force distribution in the climbing process.

Figure 7. Long description

A diagram of the force distribution during climbing. The diagram features two circular objects labeled M1 and M2, connected by a vertical line. Various forces and moments are indicated with arrows and labels. Key components include Fax, Fay, Fn1, Ms1, Fs1, Fs2, Fby, Fbx, Fn2, Ms2, and Ms1. The distances H1, H2, and H3 are marked horizontally, and the height h is marked vertically. The diagram includes the X?Y coordinate system.

Since the left and right front drive wheels are structurally symmetric and subjected to identical loading conditions, only one side drive wheel is considered in the following force-equilibrium analysis.

Table I.

Mechanical parameters used in the proposed force-equilibrium model.

Table I. Long description

A table listing mechanical parameters for a force-equilibrium model. The table has two columns: Symbol and Description. It contains 14 rows with the following data: Row 1: Symbol, m; Description, Mass of each wheel. Row 2: Symbol, m_total; Description, Total mass of the robot. Row 3: Symbol, g; Description, Gravitational acceleration. Row 4: Symbol, M_i (i = 1,2); Description, Driving torque of each wheel. Row 5: Symbol, M_si (i = 1,2); Description, Rolling-resistance moment of each wheel. Row 6: Symbol, F_ni (i = 1,2); Description, Normal force acting on the front and rear wheels. Row 7: Symbol, μ; Description, Coefficient of friction. Row 8: Symbol, r; Description, Wheel radius. Row 9: Symbol, G; Description, Gravitational force of the robot. Row 10: Symbol, δ; Description, Rolling resistance arm. Row 11: Symbol, F_fh (j = a,b); Description, Horizontal reaction force at the front (a) and rear (b) wheel shafts. Row 12: Symbol, F_fv (j = a,b); Description, Vertical reaction force at the front (a) and rear (b) wheel shafts.

The force-equilibrium equations for the front wheel are as follows:

(2) \begin{equation} Fs_{1}+F_{ax}=0 \end{equation}

(3) \begin{equation} F_{n1}+F_{ay}=0 \end{equation}

(4) \begin{equation} M_{1}-Fs_{1}r-Ms_{1}=0 \end{equation}

The wheel–rail friction force acts opposite to the relative slipping tendency at the contact interface and is bounded by:

(5) \begin{equation} F_{s1}\leq \mu F_{n1} \end{equation}

The force-equilibrium equations for the rear wheel are as follows:

(6) \begin{equation} F_{s2}+F_{bx}=0 \end{equation}

(7) \begin{equation} F_{n2}+F_{by}=0 \end{equation}

(8) \begin{equation} M_{2}-F_{s2}r-M_{s2}=0 \end{equation}

The available wheel–rail traction force is bounded by:

(9) \begin{equation} F_{s2}\leq \mu F_{n2} \end{equation}

Force equilibrium equation for the entire system:

(10) \begin{equation} 2F_{s1}+F_{s2}-G=0 \end{equation}

(11) \begin{equation} F_{n1}-F_{n2}=0 \end{equation}

(12) \begin{equation} G=m_{\text{total}}g \end{equation}

The rolling-resistance moment of the drive wheel is expressed as:

(13) \begin{equation} M_{s1}=\delta F_{n1} \end{equation}

(14) \begin{equation} M_{s2}=\delta F_{n2} \end{equation}

3.1. Control principle and architecture

The proposed control framework aims to achieve human–robot cooperative climbing assistance rather than direct human-motion tracking. During climbing, the robot continuously regulates its climbing velocity and assistive support force according to the relative human–robot position, thereby maintaining a safe interaction distance while providing continuous load-bearing support and anti-fall protection. Through the coordinated regulation of climbing velocity and assistive support force, the robot assists the user in a safe and adaptive manner throughout the climbing process. The overall principle of the proposed human–robot cooperative climbing-assistance strategy is illustrated in Figure 8.

Figure 8.

Principle of human–robot cooperative climbing assistance based on eccentric-angle regulation.

As illustrated in Figure 8, the proposed climbing-assistance strategy is based on the force transmission principle of the self-balancing anti-fall mechanism. During operation, the measured relative human–robot position is used to determine the corresponding eccentric-angle adjustment. The variation of the eccentric angle simultaneously influences the climbing velocity and the assistive support force, thereby establishing the physical relationships exploited by the proposed controller. Specifically, an increase in relative position results in a larger eccentric angle, leading to lower climbing velocity and higher assistive force, whereas a decrease in relative position produces the opposite effect. Accordingly, the interaction among the relative human–robot position, eccentric angle, climbing velocity, and assistive support force provides the physical basis for the proposed force–position cooperative control strategy. Based on these relationships, the controller regulates the climbing velocity through eccentric-angle adjustment to maintain a safe relative human–robot position while providing adaptive assistive support.

The proposed system is equipped with a wireless remote-control unit consisting of a handheld transmitter and a receiver module. Through the remote controller, the operator manually issues the commands for upward movement, downward movement, stopping, and system setting. The corresponding command definitions are summarized in Table II. Accordingly, the operator is responsible for selecting the climbing direction and issuing the commands for motion initiation, stopping, and reversal. Once a motion command is received, the proposed controller continuously regulates the climbing velocity according to the relative human–robot position throughout the climbing process.

Table II.

Remote-control command definitions.

Table II. Long description

A table with four rows and three columns. The columns are labeled Remote-control command, Key value, and IO. The rows are as follows: Row 1: Upward movement, RC_UP_MASK, P0.0. Row 2: Downward movement, RC_DOWN_MASK, P0.1. Row 3: Stop, RC_STOP_MASK, P0.2. Row 4: Setting, RC_SET_MASK, P0.3. Row 5: All buttons, RC_MASK, P0.0/P0.1/P0.2/P0.3.

The proposed force–position adaptive controller adopts a dual closed-loop control architecture consisting of an inner current loop and an outer position loop. The inner current loop measures the motor current through the controller’s ADC module and regulates the PWM duty cycle using a Proportional-Integral-Derivative (PID) controller, achieving a response time of less than 0.5 ms. The outer position loop estimates the relative human–robot position from the tilt sensor and generates the target climbing velocity using the proposed velocity-adaptive strategy.

3.2. Real-time load analysis model

Load force estimation is realized through the interaction between the motor current sampling and the proposed model. The system acquires the motor current at a frequency of 1 kHz via the current sense amplifier and the ADC module of the controller, with a sampling resolution of 10 bits and a common-mode rejection ratio (CMRR) of over 80 dB. The current–force mapping model is established through experimental calibration using the least-squares method:

(15) \begin{equation} F\left(t\right)=k_{I}\cdot I\left(t\right)+b_{I}+\Delta F_{dyn} \end{equation}

In Eq. (15), k_I represents the current-force conversion coefficient, b_I denotes the static offset, and ΔF_dyn is the dynamic compensation term introduced to compensate for variations in the force-transmission characteristics caused by changes in climbing conditions and external disturbances, whose detailed calculation formula is given below:

(16) \begin{equation} \Delta F_{dyn}=m\cdot \ddot{\!x}+c\cdot \dot{\!x}+k\cdot \left(x-x_{0}\right) \end{equation}

In Eq. (16), m is the mass of the robot, c is the damping coefficient, k is the elastic coefficient, x is the real-time distance, and x ₀ is the initial position. The mechanical equation is converted into a frequency-domain transfer function through the Laplace transform to achieve real-time calculation of dynamic load force.

3.3. Force–position cooperative control framework

The force–position cooperative control framework adopted in this section is established according to the force-transmission characteristics of the climbing system. The interaction among the assistive support force, climbing velocity, and eccentric angle provides the physical basis for the proposed force–position cooperative control strategy.

For clarity, the principal control variables are defined as follows. The assistive support force F, climbing velocity v, and eccentric angle θ are selected to describe the operating state of the climbing-assistance robot. The driving torque T generated by the motor and the eccentric-angle adjustment Δθ are regarded as the control inputs. The relative human–robot position, obtained from the relative-position detection device, serves as the feedback variable for the proposed controller, while the assistive support force and climbing velocity are treated as the control outputs.

During climbing, the relative-position detection device continuously monitors the relative human–robot position. According to the detected relative-position state, the controller updates the target climbing velocity. The motor velocity is subsequently regulated by a PID controller to track the target velocity. Consequently, the robot continuously maintains the relative human–robot position within a predefined safe range while providing adaptive climbing assistance.

3.4. Relative-position-based velocity-adaptive control

3.4.2. Relative position detection mechanism

The safety belt of the fall prevention device is equipped with a relative position detection device, which mainly consists of three tilt sensors. The detection device is connected to the main device through an electrical interface. The device estimates the relative position level between itself and the worker by detecting the status of the three tilt sensors in combination with its own movement direction and adjusts the climbing velocity accordingly.

The tilt sensors equipped in this device are switch-type sensors, as shown in Figure 9. Let θ ₁ denote the sensor tilt angle and θ ₂ represent the switching threshold; the sensor is in the OFF state when θ ₂ > θ ₁ and in the ON state when θ ₂ < θ ₁.

Figure 9.

Schematic diagram of the tilting sensor status.

3.4.3. Relationship between tilt sensor states and robot motion

The tilt sensors are switch-type devices that output 0 in the conducting (ON) state and 1 in the disconnected (OFF) state. The three sensors are numbered according to their positions: sensor 3 is closest to the robot, sensor 1 is closest to the worker, and sensor 2 is in between. Their combined states form a 3-bit status variable, where bit 2 corresponds to sensor 3, bit 1 to sensor 2, and bit 0 to sensor 1. Based on this status and the movement direction (upward or downward), the relationship between the robot’s velocity and the worker’s velocity can be inferred, as summarized in Table III.

Table III.

The relationship between the movement of the device and the tilt sensor.

Table III. Long description

A table with 8 rows and 5 columns. The columns are labeled Number, Tilt sensor 1, Tilt sensor 2, Tilt sensor 3, Rising, and Descending. The table shows the state of three tilt sensors and the corresponding movement of the device. Row 1: Number 1, Tilt sensor 1 0, Tilt sensor 2 0, Tilt sensor 3 0, Rising Too fast, Descending Too slow. Row 2: Number 2, Tilt sensor 1 0, Tilt sensor 2 0, Tilt sensor 3 1, Rising Fast, Descending Slow. Row 3: Number 3, Tilt sensor 1 0, Tilt sensor 2 1, Tilt sensor 3 0, Rising Not applicable, Descending Not applicable. Row 4: Number 4, Tilt sensor 1 0, Tilt sensor 2 1, Tilt sensor 3 1, Rising Slow, Descending Fast. Row 5: Number 5, Tilt sensor 1 1, Tilt sensor 2 0, Tilt sensor 3 0, Rising Not applicable, Descending Not applicable. Row 6: Number 6, Tilt sensor 1 1, Tilt sensor 2 0, Tilt sensor 3 1, Rising Not applicable, Descending Not applicable. Row 7: Number 7, Tilt sensor 1 1, Tilt sensor 2 1, Tilt sensor 3 0, Rising Not applicable, Descending Not applicable. Row 8: Number 8, Tilt sensor 1 1, Tilt sensor 2 1, Tilt sensor 3 1, Rising Too slow, Descending Too fast.

3.4.4. Velocity-adaptive control strategy

The states of the tilt sensors are as shown in Figure 10, totaling six valid states.

Figure 10.

Schematic diagram of the tilt sensors.

The goal of the proposed velocity-adaptive strategy is to dynamically adjust the robot’s climbing velocity according to the status of the relative-position detection device, thereby maintaining an appropriate and safe relative position between the robot and the worker. As described in Section 3.1, the robot operates under the motion commands issued by the operator, while the proposed controller is responsible only for adaptive climbing-velocity regulation during the climbing process.

After startup, the outer position loop estimates the relative human–robot position from the tilt sensor and updates the target climbing velocity using the proposed velocity-adaptive strategy, while the inner current loop regulates the motor current through the PID controller to ensure stable motor operation under varying load conditions.

Within each control period, the status of the relative-position detection device is read. Based on the robot’s movement direction, the detected relative-position state, and prior knowledge, the target climbing velocity is updated accordingly. The target climbing velocity is modified only when the detected relative-position state changes and is always constrained within the allowable operating range of the robot. According to the detected relative-position state, the target climbing velocity is increased or decreased until the relative position returns to the desired range. This adaptive adjustment is repeated during each control cycle until the relative position falls within a reasonable range.

The proposed controller employs the dual closed-loop control architecture. The outer position loop generates the target climbing velocity using the proposed rule-based velocity-adaptive strategy, while the inner-loop PID controller continuously regulates the motor velocity to track the corresponding reference velocity. During climbing, the reference climbing velocity is constrained within the allowable operating range:

(17) \begin{equation} v_{\min }\leq v_{r}\leq v_{\max } \end{equation}

where v_r denotes the target climbing velocity, and v _min and v _max denote the allowable minimum and maximum climbing velocities, respectively. Since the target climbing velocity is adjusted only when the relative-position state changes, abrupt velocity variations during state transitions are avoided. Consequently, smooth climbing operation is achieved under different operating conditions. The robustness of the proposed controller is further demonstrated by the experimental results under different payload conditions and external disturbances presented in Section 5.

3.5. Force–position cooperative regulation strategy

A hierarchical control strategy is employed to achieve force–position coordination.

Force control layer: In the force control layer, we calculate the load force in real time based on current sampling. When $F \gt F_{max}$ , the overload protection logic is triggered, and soft stop is realized through linear attenuation of the PWM duty cycle at an attenuation rate of 5%/ms; when $F \lt F_{min}$ , the system enters the velocity-up mode.

Position control layer: The rate of change of relative displacement is detected by an array of tilt sensors (three-axis switch sensors). When the displacement deviation exceeds ±5 cm, velocity-adaptive adjustment is initiated.

Force–position collaboration layer: Based on the interaction among eccentric angle, force, and velocity, adjustment rules are established: when F increases and v decreases, the eccentric angle is reduced to enhance force output; when F decreases and v exceeds the target value by 15%, the eccentric angle θ is increased to reduce driving power. The relationship between sensor states and velocity adjustment is shown in Table IV.

Table IV.

The relationship between the status of the tilt sensor and velocity regulation.

Table IV. Long description

A table with three columns and three rows. The columns are labeled Sensor status, Relative velocity direction, and Adjustment strategy. Row 1: Sensor status, 001; Relative velocity direction, Robot is faster than the worker; Adjustment strategy, Slow down by 10 percent. Row 2: Sensor status, 010; Relative velocity direction, Synchronization; Adjustment strategy, Maintain the current velocity. Row 3: Sensor status, 100; Relative velocity direction, The worker is faster than the robot; Adjustment strategy, Growth rate: 15 percent.

3.6. Adaptive PID parameter tuning

In response to changes in control characteristics under different load conditions, a PID parameter adaptive adjustment mechanism is designed.

Load force zoning: The load force is divided into three zones: light load (F< 40 N), medium load (40 N ≤ F ≤ 60 N), and heavy load (F> 60 N).

Parameter table mapping: The optimal PID parameters are obtained for each interval through offline experiments.

Light load zone: KP = 15, KI = 1.2, KD = 2;

Medium load zone: KP = 12, KI = 0.8, KD = 3;

Heavy load zone: KP = 8, KI = 0.5, KD = 4;

Dynamic correction factor: A correction factor α is introduced to perform real-time fine-tuning of the PID parameters, which is defined as:

(18) \begin{equation} \alpha =1+\beta \left(F-F_{ref}\right) \end{equation}

The real-time fine-tuning rules for the PID parameters are expressed as:

(19) \begin{equation} KP^{*}=KP\cdot \alpha \end{equation}

(20) \begin{equation} KI^{*}=KI\cdot \alpha \end{equation}

(21) \begin{equation} KD^{*}=KD\cdot \alpha \end{equation}

3.7. Experimental verification and performance analysis

In the 10 m T-shaped guide rail experiment, when the load force jumps from 30 N to 70 N, the response of the force–position cooperative algorithm is as follows:

When the current rises from 1.8 A to 4.2 A, the force estimation error is less than 3%.

The eccentricity angle automatically decreases from 30° to 25°, and the velocity fluctuation range is less than 4%.

After dynamic adjustment of PID parameters, the system regulation time is less than 0.5 s and the overshoot is less than 8%. The force–position cooperative control accuracy under different loads is shown in Table V.

Table V.

The accuracy of force–position cooperative control under different loads.

Table V. Long description

A table titled ‘The accuracy of forceposition cooperative control under different loads’ with four columns and three rows. The columns are labeled ‘Load (N)’, ‘Force control error (%)’, ‘Velocity control error (%)’, and ‘Adjustment time (ms)’. The rows present data for loads of 30 N, 50 N, and 70 N. Row 1: Load (N), 30; Force control error (%), 2.1; Velocity control error (%), 1.3; Adjustment time (ms), 350. Row 2: Load (N), 50; Force control error (%), 1.8; Velocity control error (%), 2.8; Adjustment time (ms), 420. Row 3: Load (N), 70; Force control error (%), 2.5; Velocity control error (%), 4.4; Adjustment time (ms), 480.

3.8. Velocity coordination analysis for human–robot cooperative climbing

The performance of the proposed human–robot cooperative climbing assistance strategy is illustrated through the velocity coordination between the user and the robot, as shown in Figure 11. The control objective is not exact trajectory tracking; instead, the robot adjusts its climbing velocity to maintain a safe relative position and provide stable assistive support during climbing.

Figure 11.

The robot follows the functional velocity curve.

3.9. Control flowchart

This algorithm realizes the coordinated adaptive adjustment of load force and position during the climbing process through a closed-loop process of hardware sampling, mechanical modeling, logical judgment, and control output, ensuring the stability and safety of the system under different working conditions. The algorithm execution process is illustrated in Figure 12.

Figure 12.

Algorithm execution process.

4.1. Contact modeling and simulation framework

Dynamic simulation of the robot was performed using Automatic Dynamic Analysis of Mechanical Systems (ADAMS) dynamics software to investigate the variation trends of motor torque and angular velocity during its movement along vertical T-shaped guide rail surfaces. Contact modeling between the robot assembly, the T-shaped guide rail, and the climbing surface was implemented in ADAMS, as illustrated in Figure 13. The contact model enables the prediction of wheel–rail interaction forces, friction effects, and impact phenomena during climbing, thereby providing the basis for evaluating the dynamic behavior of the proposed robot.

Figure 13.

Schematic diagram of robot motion simulation.

4.2. Motor torque and angular velocity during climbing

To investigate the motor dynamic characteristics during the robot’s ascent, directional reversal, and descent phases, the simulated torque and angular velocity profiles were extracted, with the corresponding results presented in Figure 14. This figure depicts the motor torque variation (green curve) and angular velocity profile (blue curve), where positive/negative angular velocity values denote clockwise/counter-clockwise motor rotation, respectively.

Figure 14.

Simulated angular velocity and torque responses during robot climbing.

Ascending motion (0–3 s):

The motor starts at t = 0 s, with the rotor initially at rest. To overcome the initial inertial load, both motor torque and angular velocity surge rapidly to their peak values during startup. After this transient peak phase, the system enters a steady operating state: the motor torque stabilizes at approximately 0.22 N · m, providing the driving torque required for steady climbing. The angular velocity remains steady at +445 deg/s, ensuring a constant-velocity ascent of the robot.

Directional reversal (3–3.4 s):

The control command reverses the motor rotation to initiate braking. This induces an abrupt change in the direction of the wheel-rail contact force, while the driving torque and wheel angular velocity drop sharply, enabling the system to decelerate and reverse its motion direction.

Descending motion (3.4–7 s):

Post-reversal, torque and angular velocity rapidly stabilize. During controlled descent, angular velocity is sustained at −450 deg/s (± 2% fluctuation). The motor mainly provides a braking torque, which stabilizes at approximately 0.05 N · m.

4.3. Characteristics of wheel contact forces

To further investigate the dynamic interaction between the robot and the vertical T-shaped guide rail during locomotion, the contact force profiles of the robot’s wheels were simulated and analyzed, with the results presented in Figure 15. Three curves denote contact forces: blue (mid-mounted drive wheel), green (right drive wheel), and purple (left drive wheel).

Figure 15.

Simulated wheel–rail contact force responses during robot climbing.

Initial motion (0–0.5 s)

During inertial torque startup, all wheels exhibit significant contact force oscillations due to static equilibrium overcoming and dynamic stability establishment. As the primary load-bearing component, the mid-mounted drive wheel – being proximal to the center of mass (COM) and subjected to the maximum dynamic load – exhibits the largest contact force fluctuation amplitude, with a peak value of F _max = 48 N. The left driving wheel has a peak contact force of F _max = 33 N, while the right driving wheel has a peak contact force of F _max = 35 N.

Ascending motion (0.5–3 s)

During ascent, left and right drive wheels display congruent contact force trends with minimal fluctuation (mean F = 26.8 N). In contrast, the mid-mounted drive wheel shows amplified oscillations (mean F = 37.4 N) due to its single-point support geometry and overturning moment effects, reflecting dynamic adjustments for vertical alignment against gravity and guide rail irregularities.

Directional transition (3–3.5 s)

During braking and descent initiation, the counter-torque generated by motor reversal couples with gravity, triggering an instantaneous surge in the contact forces of all wheels and thus significantly elevating the peak stresses at the wheel-rail interfaces and component joints.

Descending motion (3.5–7 s)

After the robot completes the turnaround operation, the front–rear positional layout of the mid-mounted drive wheel and the left/right drive wheels are swapped. The mid-mounted drive wheel sustains significantly higher forces (mean F = 68.4 N) with pronounced oscillations, while left/right drive wheels show attenuated variations (mean F = 41 N) with minimal fluctuations.