3.1. Simplified kinematic model

Unlike conventional cars, Epi.q is equipped with differential steering. This means that it can navigate a curve by imposing a difference in speed between the inner and outer wheels, rather than rotating them. In addition, as illustrated in Figure 4a, the presence of tripods causes lateral velocity components to be generated in curves. These lateral velocities tend to cause outward slipping and make the robot prone to understeering. The effect increases as the radius of the curve decreases.

Figure 4.

Epi.Q kinematic model. (a) Real wheels with lateral velocity, showing potential for outward slipping during curves, which causes understeering. (b) Virtual wheels without lateral velocity, simplifying the model and eliminating slip during movement.

To simplify the kinematic model, each tripod is replaced by a virtual wheel (Figure 4b), which lacks lateral velocity components and thus does not slip. This modeling approach is derived from previous work, as discussed in [Reference Botta, Cavallone, Tagliavini, Carbonari, Visconte and Quaglia9]. Given the geometry of the robot, the radius of curvature of the front module $\rho$ is a function of the speeds imposed on the front wheels. Is it possible to define the front module angular velocity $\dot {\varphi }_1$ in two ways:

\begin{gather*} \dot {\varphi }_1 = \frac {v_{\mathrm{m}}}{\rho } \\ \dot {\varphi }_1 = \frac {\Delta v}{i} \end{gather*}

where $\rho$ is the radius of curvature, $v_{\mathrm{m}}$ is the average velocity, $\Delta v$ is the difference in velocity between the inner and the outer wheelm and $i$ is the wheelbase. Consequently, the following relation can be derived:

(1) \begin{equation} \rho = \frac {v_{\mathrm{m}}}{\Delta v} \, i \end{equation}

There is a second relationship that relates the radius of curvature $\rho$ to the yaw angle $\delta$ . This relationship is valid only when conditions are stationary, meaning $\delta$ is constant. It is expressed as:

\begin{gather*} z = a + b \cos {\delta } \\ t = \frac {z}{\sin {\delta }} \\ \rho = t \cos {\delta } + b \sin {\delta } \end{gather*}

from which we obtain:

(2) \begin{equation} \rho = \frac {a \cos {\delta + b}}{\sin {\delta }} \end{equation}

where $z$ and $t$ are the geometric distances visible in the Figure 4b, $a$ and $b$ are the distance from the yaw joint and the center between the wheels of the front module and the wheels of the rear module, respectively. By equating the relations 1 and 2, we obtain a third relation that relates linear velocities to yaw angles:

(3) \begin{equation} i \, v_{\mathrm{m}} \sin {\delta } = \Delta v (a \cos {\delta } + b) \end{equation}

By rewriting the equation as a function of the motors’ angular velocities, we get:

(4) \begin{equation} i \, n_{\mathrm{m}} \sin {\delta } = \Delta n (a \cos {\delta } + b) \end{equation}

Let $n_{\mathrm{l}}$ and $n_{\mathrm{r}}$ be the angular velocities (rpm) of the left and right motors, respectively, we can write that:

\begin{gather*} n_{\mathrm{m}} = \frac {n_{\mathrm{r}} + n_{\mathrm{l}}}{2} \\ \Delta n = n_{\mathrm{r}} - n_{\mathrm{l}} \end{gather*}

Consequently, counterclockwise curves have both a positive $\Delta n$ and a positive angle $\delta$ (figure 4b). By substituting in the eq. 4 and performing a few steps, we obtain:

(5) \begin{equation} n_{\mathrm{r}} = n_{\mathrm{l}} \, \dfrac {1+\dfrac {i \sin {\delta }}{2(a \cos {\delta }+b)}}{1-\dfrac {i \sin {\delta }}{2(a \cos {\delta }+b)}} \end{equation}

Assuming:

(6) \begin{equation} K = \dfrac {1+\dfrac {i \sin {\delta }}{2(a \cos {\delta }+b)}}{1-\dfrac {i \sin {\delta }}{2(a \cos {\delta }+b)}} \end{equation}

equation 5 simplifies to:

(7) \begin{equation} \dfrac {n_{\mathrm{r}}}{n_{\mathrm{l}}} = K(\delta ) \end{equation}

which tells us that the ratio of the motor speeds is a function of the yaw angle and, through eq. 2, consequently a function of the radius of the curve we want to follow. From this point onward, $K$ is referred to as the coefficient of curvature. Equation 7 indicates that on a constant radius curve, the velocity relationship remains constant, regardless of speed, assuming no slippage. By introducing the average angular velocity, we can rewrite it as follows:

(8) \begin{equation} n_{\mathrm{r}} = \frac {2Kn_{\mathrm{m}}}{K+1} \end{equation}

The last equation can be used to calculate the speed of a single motor given the yaw angle $\delta$ and the speed of the module.

3.2. Rear control

The goal of rear module control is to enable the robot to have all-wheel drive, specifically eight wheel drive (8WD) in the case of Epi.Q. To achieve active control of the rear module, a fuzzy control system was implemented. This system takes into account both the yaw angular error and the robot’s instantaneous speed. It modulates PWM signals sent to the rear motors through the generation of two multiplicative coefficients to properly control them.

The angular error plays a crucial role since the yaw angle $\delta$ determines the robot’s steering behavior. Hence, it must be considered when executing control to maintain the correct robot configuration during movement. The angular reference is derived from the kinematic model based on the commands sent to the robot, while feedback is provided by a potentiometer mounted on the yaw joint. We can define the angular error as $e = \delta - \delta _{\mathrm{fb}}$ , where the angle is positive for counterclockwise turns.

Figure 5 illustrates several generic movement configurations that the robot can assume. The straight arrows on the wheels represent the PWM command sent to each motor, while the circular arrows on the modules qualitatively indicate the rotation of the modules themselves. In the figure, $\delta$ is the desired yaw angle, $\delta _{\mathrm{fb}}$ is the feedback yaw angle, and $e$ is the error between them.

Figure 5.

Diagram showing three steering cases for the vehicle dynamics, including adjustments for inner rear wheel (IRW) and outer rear wheel (ORW) based on the steering angle and feedback. PWM signals are adjusted to modify the curvature radius and correct steering errors in different cases.

Assuming that we have generated two PWM signals through the speed control for the front module, we can then determine how to modulate the command for the rear module. The base PWM signal is taken from the corresponding front motor. The left rear motor receives the PWM signal from the left front motor, and the same applies for the right side. We also define the rear wheels based on whether they are on the inside or outside of the curve. The inner rear wheel is referred to as IRW, and the outer rear wheel as ORW, based on the sign of the measured yaw angle $\delta _{\mathrm{fb}}$ .

There are six possible cases, but given the system’s symmetry, we only show three, with the others being mirror images along the vertical axis, where all angle signs and errors are inverted. However, the corrective actions remain the same.

In Case 1, the robot is navigating a counterclockwise curve, but the measured angle is smaller than the target, meaning $\delta \gt \delta _{\mathrm{fb}}$ . To restore the correct configuration and reduce the turning radius, we adjust the rear module’s signals by increasing the PWM on the inner rear wheel and decreasing it on the outer rear wheel, helping the rear module to tighten the robot’s turn and increase $\delta _{\mathrm{fb}}$ .

In Case 2, the situation is similar, but $\delta _{\mathrm{fb}} \gt \delta$ , so the corrective actions are inverted, as the goal here is to increase the turning radius.

In Case 3, the robot is navigating a clockwise curve but receives a command for a counterclockwise one. Speed control generates PWM for the front module, which gradually aligns with the final configuration. To accelerate this transition, the inner rear wheel is used as a brake until the robot returns to the configuration described in Case 1.

3.3. Fuzzy control

The fuzzy control logic is based on a mathematical-logical theory that introduces the concept of the gradual membership of an element to a set. Unlike traditional logic, which handles events as either true or false, fuzzy logic allows for representing a continuous range of values, providing a more flexible and realistic representation for managing complex and nonlinear systems, such as Epi.Q. This type of control is especially suitable for handling systems with multiple inputs, as it allows defining various behaviors depending on operating conditions [Reference Lee24–Reference Zimmermann28].

A typical fuzzy control system consists of three main modules: the fuzzification module, the rule application module, and the defuzzification module. Both inputs and outputs are specified by numerical (crisp) values, while the fuzzy controller operates on fuzzy variables. Thus, there are fuzzification and defuzzification interfaces, which associate inputs with their fuzzy representation and convert the fuzzy output back into crisp values. The central rule application transforms the fuzzy representations of the inputs into fuzzy output representations according to fuzzy logic rules, also known as inference rules. The defuzzification process then converts these fuzzy values back into crisp signals for output.

Fuzzification, or the conversion into fuzzy variables, is carried out through membership functions ( $\mu$ ), which define the degree of activation of a class. These functions can take various forms; common examples include triangular, trapezoidal, piecewise linear, or Gaussian functions. In our implementation, we chose to use triangular and trapezoidal membership functions for both inputs and outputs.

Figure 6.

Fuzzy control system for Epi.Q robot. Inputs include normalized angular error and speed, while outputs modulate the duty cycle applied to the inner and outer rear wheels.

Figure 6 presents the fuzzy control scheme implemented on the Epi.Q robot. The inputs are the normalized angular error ( $\overline {\mathit{e}}$ ) and the normalized average speed ( $\overline {\mathit{v}}$ ), defined as follows:

(9) \begin{align}& \overline {e} = \mathrm{sign}(\delta _{\mathrm{fb}})\,\frac {\delta - \delta _{\mathrm{fb}}}{\delta _{\mathrm{max}}} \end{align}

(10) \begin{align}&\quad\,\, \overline {v} = \left \lvert \frac {n_{\mathrm{r}} + n_{\mathrm{l}}}{2\,n_{\mathrm{max}}}\right \rvert \end{align}

where $\delta$ is the set yaw angle imposed by the control logic, $\delta _{\mathrm{fb}}$ is the feedback yaw angle measured by the potentiometer, and $\delta _{\mathrm{max}}$ is the maximum angle the robot can reach before the joint hits its mechanical stop, equal to 38 $^\circ$ . The yaw angle is therefore confined to the range [−38 $^\circ$ , 38 $^\circ$ ], while the normalized error ( $\overline {\mathit{e}}$ ) is defined in the range [−2, 2].

In the second equation, $n_{\mathrm{r}}$ and $n_{\mathrm{l}}$ are the angular velocities of the front motors, which are speed controlled. The term $n_{\mathrm{max}}$ represents the maximum angular velocity, which is 10 000 rpm. Considering the gear ratio, the tripod mechanism, and the wheel diameter, the maximum linear velocity of the robot is 2.68 m/s.

The fuzzy control output consists of two multiplicative coefficients that modulate the duty cycle applied to the motors. The coefficient $C_{\mathrm{i}}$ modulates the inner rear wheel, while the coefficient $C_{\mathrm{o}}$ modulates the outer rear wheel. $C_{\mathrm{i}}$ varies in the range [−0.8, 1.4], and $C_{\mathrm{o}}$ varies in the range [0.4, 1.4]. Negative values indicate a torque inversion by the motor to perform braking or even reverse.

Figure 7.

Membership functions for the fuzzy control system inputs ( $\overline {\mathit{e}}$ , $\overline {\mathit{v}}$ ) and outputs ( $C_{\mathrm{i}}$ , $C_{\mathrm{o}}$ ).

The membership functions of the two input variables and the two output variables in the fuzzy system are shown in Figure 7. The x-axis indicates the numerical value of each variable, while the y-axis indicates the degree of membership to each class. Specifically, the normalized error $\overline {\mathit{e}}$ (Figure 7a) is divided into five classes. Two flat regions are included to ensure full coverage of the input domain, although in most operational cases, the controller functions within a reduced range, approximately [−1,1]. The normalized velocity (Figure 7b) is divided into three classes, again with flat regions at the boundaries. Figs. 7c and 7d displays the membership classes of the outputs. The outer rear wheel coefficient ( $C_{\mathrm{o}}$ ) has five membership classes, while the inner rear wheel coefficient ( $C_{\mathrm{i}}$ ) extends these by adding three additional classes to allow for negative values.

Fuzzy control involves writing rules in the form of IF (condition) THEN (action). This means that if a certain input is activated, the corresponding output is activated according to the rules. The simplest way to visualize these rules is through a matrix, called the rule matrix, which associates an output class with every pair of input classes. Since there are two distinct outputs, two separate matrices are created. In Table I, the rules on the left side aim to increase the curvature radius of the trajectory, while the rules on the right aim to decrease it.

Table I.

Fuzzy control rule matrix used to modulate the duty cycle coefficients of the rear wheels of the Epi.Q robot. The rules are defined based on the angular error ( $\overline {\mathit{e}}$ ) and the robot’s velocity ( $\overline {\mathit{v}}$ ), allowing dynamic adjustments to enhance stability and precision during maneuvering and steering phases.

Lastly, it is worth noting that the speed-dependent component of the fuzzy control becomes particularly relevant for the inner rear wheel coefficient $C_{\mathrm{i}}$ , especially in scenarios involving negative normalized errors. Under these conditions, $C_{\mathrm{i}}$ is adjusted to contribute to increasing the curvature radius of the trajectory, allowing the vehicle to realign more effectively during high-curvature maneuvers.

Figure 8.

Fuzzy inference surfaces for the rear wheel coefficients.

Figures 8a and 8b shows the inference surfaces generated by the fuzzy control based on the rule matrices for the coefficients $C_{\mathrm{i}}$ and $C_{\mathrm{o}}$ , respectively. As can be seen, there is more aggressive control for low-speed values around zero error, and smoother control for higher speed values. This was necessary because at low speeds, the robot is much less sensitive to command variations.

The inference method used for calculating output values is Mamdani’s method, which involves the following steps. Initially, the degrees of activation for the input variable classes are calculated (as shown in Figure 9a for the membership classes of the variable $\overline {\mathit{e}}$ ). For each input value, one or two classes can be activated. The next step is to calculate the premise values ( $\mu _{\mathrm{p}}$ ), which is the intersection of fuzzy sets, according to the following formula:

(11) \begin{equation} \mu _{\mathrm{p}}(\overline {e}_i,\, \overline {v}_j) = \min (\mu _i(\overline {e}),\, \mu _j(\overline {v})) \end{equation}

Where $i$ and $j$ are the membership classes of the inputs. These premise values are used to define, based on the rule matrix, the activation degree of the output classes for each pair of input classes. Once the rule matrix is filled with the premise values, a fuzzy set union operation is applied, using the following formula:

(12) \begin{equation} \mu (u_k) = \max (\mu _{\mathrm{p}}(\overline {e_i},\,\overline {v_j})) = \max [\min (\mu _i(\overline {e}),\,\mu _j(\overline {v}))] \end{equation}

where $u$ represents a generic output, and $k$ indicates the membership class of the output.

Figure 9.

Examples of fuzzification and defuzzification used in the proposed control.

After calculating the activation values of the output classes, a clipping (or alpha-cut) operation is performed, which ”cuts” the fuzzy membership function at a specific activation level ( $\alpha$ ), representing the maximum membership degree achieved by the active fuzzy rules. The result is a simplified fuzzy function, which can be further processed to obtain a crisp value. An alternative method is scaling, which proportionally resizes the membership function to the activation degree obtained from fuzzy inference, maintaining the shape of the original membership function.

The first method (clipping) is the most common due to its simplicity, offering high computational speed and simplifying the defuzzification process. The downside is that some information is lost during the clipping operation, as the shape of the function changes. For example, a triangular function becomes trapezoidal after clipping.

The second method (scaling), though computationally more expensive, allows adjusting the influence of a fuzzy rule on the output without losing information about the original membership function distribution. Unlike clipping, which removes parts of the function, scaling preserves all characteristics of the fuzzy curve, only reducing its intensity proportionally to the degree of activation.

At this point, the aggregation of the results is carried out to obtain a single fuzzy set for the output variable. Figure 9b shows an example where a single fuzzy set is obtained from the class aggregation after applying the alpha-cut method.

Defuzzification converts the fuzzy set output into a numerical (crisp) value. This conversion can be performed using several defuzzification methods, which associate a deterministic value with the fuzzy representation of the output. The method used here is the center of gravity method, one of the most commonly used due to its accuracy. It calculates the center of gravity of the resulting fuzzy set, and the output is the x-coordinate of that point, as shown in Figure 9b.

The choice of the Mamdani inference method for the fuzzification process was primarily motivated by its interpretability and accessibility. Mamdani-type systems use linguistic labels for both inputs and outputs [Reference Mamdani and Assilian29], making them particularly suitable for implementing expert knowledge, even from domain experts without a strong background in mathematics or control theory. This approach facilitates the development of intuitive fuzzy rule bases that can be easily modified and extended based on observed behavior and experiential knowledge [Reference Lee24, Reference Lee30].

While the Sugeno method is often preferred in scenarios requiring computational efficiency and mathematical precision, particularly when the system can be effectively described using analytical functions [Reference Sugeno31–Reference Blej and Azizi33], its reliance on numerical outputs makes it less intuitive for designing interpretable rule sets involving qualitative reasoning. In contrast, the Mamdani approach provides a more descriptive framework that aligns with the main objective of this work: developing a human-readable and easily adaptable control strategy for the highly non-linear and articulated structure of the Epi.Q robot. The ability to define both inputs and outputs using linguistic variables proved essential to integrating domain knowledge and simplifying the tuning process during prototyping and experimentation.

For defuzzification, the centroid method was chosen because of its high accuracy and ability to produce smooth and stable output signals. Although this method is generally more computationally intensive than alternative defuzzification techniques, such as mean of maxima or height methods, the centroid method ensures continuous and consistent control transitions [Reference Lee30], which is especially important for an articulated system like Epi.Q operating in unstructured terrain. The controller efficiently manages the associated computational load without compromising real-time responsiveness due to the relatively low update rate required for vehicle-level control and the moderate complexity of the fuzzy rule set. As a result, the additional computational load introduced by the centroid method is well within the capabilities of the on-board processing hardware and does not compromise performance or reliability.

The objective of this study is not to demonstrate the superiority of fuzzy logic control over more commonly adopted methods such as MPC or SMC. In fact, we explicitly do not attempt a direct performance comparison with these techniques. Such methods require a reliable dynamic model of the system, and a model is currently unavailable for Epi.Q due to its complex and nonlinear behavior, which is introduced by its articulated structure, passive joint dynamics, and unpredictable terrain interactions. Without a reliable model, MPC and SMC strategies would operate under inaccurate assumptions, making any comparison unfair or misleading.

Instead, fuzzy logic is proposed here as a pragmatic and feasible alternative designed specifically to bypass the need for a precise model. The aim is to improve traction and maneuverability in real-world environments with uncertainties, not to outperform state-of-the-art methods in idealized conditions. Thus, the contribution of this work lies in demonstrating a functional and practical control implementation that enhances robot behavior in scenarios where traditional model-based approaches are difficult or impossible to deploy.

3.4. System scalability and operational robustness

The current fuzzy logic control system is designed for a modular robot composed of two articulated modules, with control rules tailored to this specific configuration. Although the system is not inherently scalable, it can be extended to accommodate additional articulated joints or modules by incorporating new input variables and expanding the rule base accordingly. However, such a modification would significantly increase both the complexity of the control logic and the computational resources required to manage the expanded system dynamics. Careful optimization and modular rule design would be necessary to maintain system performance and real-time responsiveness.

From a hardware perspective, the Epi.Q robot is equipped with four motors, each integrated with a high-reliability Hall effect encoder for accurate motor speed feedback. These encoders are known for their durability and robustness against environmental disturbances, ensuring consistent performance over extended operational periods. The robot yaw angle is measured using a high-quality conductive plastic potentiometer, selected for its environmental resistance and durability despite the use of sliding contacts.

Robustness is further enhanced by the system’s ability to tolerate partial hardware failures. In the event of a motor failure, the robot is capable of operating in a 4WD mode, where either module can assume the role of the front (driving) module. Although this mode limits the robot’s ability to perform reverse maneuvers, it preserves forward mobility, allowing the robot to maintain basic functionality and perform mission critical tasks.

This combination of scalable control architecture and resilient hardware design enables the system to operate reliably under varying conditions while providing a basis for future extensions in both complexity and capability.

4.1. Uphill tests

Figure 10 shows the setup of the test. The robot is placed on a flat wooden board whose inclination is varied. The tests were performed by imposing a speed set of 1200 rpm (0.32 m/s), with tilt angles of 12 $^\circ$ , 18 $^\circ$ , and 28 $^\circ$ . The purpose of the test is to verify the climbing ability of the robot in the two different configurations (4WD and 8WD) and to demonstrate how much the use of the rear end affects its ability to cope with steep climbs.

In this first test, with an inclination of 12 $^\circ$ , we compare the robot’s climbing ability with active fuzzy control and with only the front end.

Figure 10.

Diagram illustrating the setup of the uphill test. The robot is placed on a flat wooden board with adjustable inclination, used to test the climbing ability of the robot in different drive configurations (4WD and 8WD) at varying angles of incline.

Figure 11.

Uphill test, $\alpha = 12^\circ$ . Tripod rotations occur due to the high torque applied by the motors and the low normal load on the tripods themselves. With all-wheel drive, less torque is applied to the front tripods.

Figure 11a shows the graph of the speed of the motors. As can be seen, in 4WD mode, the robot is not able to achieve convergence in the speeds of all the motors. While the front tripods oscillate around the target speed, the rear tripods do not reach the target speed. In this mode, the rear motors are not powered, but they are connected to the wheels, so we can read the real speed of the robot.

In “Front drive only” (FDO) case, the tripods change from advancing mode to climbing mode, making them rotate without the tires transferring power to the ground. This happens because the vertical force needed to stop the tripods from rotating isn’t strong enough to balance the torque from the motors. As a result, Epi.Q cannot climb a ramp with such a slope. On the other hand, using 8WD mode and Fuzzy control, Epi.Q is able to climb this slope easily. It takes about 0.75 s to reach target speed.

Figure 11b illustrates the yaw angle during the test. In the FDO case, it is unstable, whereas with fuzzy control, it oscillates around the set value and and reaches the target after the acceleration phase.

Figure 12.

Uphill test, $\alpha = [18^\circ ,\,28^\circ ]$ . Increasing the tilt angle causes the tripods to rotate due to the low normal load on the front tripods.

Since the robot with only the front module is not able to climb a slope with an inclination angle of 12 degrees, the next tests will be performed in 8WD mode with the fuzzy control active, in order to find out what is the maximum inclination angle the robot can handle.

Figures 12a and 12b shows two tests performed with a tilt angle of $18^\circ$ and $28^\circ$ . In the first case ( $\alpha = 18^\circ$ ), the robot is able to climb the slope, but there is a lot of correction on the rear module to keep the alignment, as can be seen from the yaw angle graph and the speed graph. On such a slope, the front module’s tripods tend to rotate, but the wheels don’t switch completely.

In the second case ( $\alpha = 28^\circ$ ), the robot cannot go up the slope because the tripods of the front module switch from advancing mode to climbing mode due to the reduction of vertical load. As a result, the speeds measured by the rear module are much lower than the target speed.

Figure 13.

Uphill test with additional load, $\alpha = 28^\circ$ : motors angular velocity. The additional load prevents the tripod from rotating and allows the robot to climb.

In Figure 13, we continued the test with a tilt angle of 28 $^\circ$ by applying an additional load to the robot’s front module so that there was a greater constraint on tripod rotation. As a result, Epi.Q can climb the slope in advancing mode.

4.2. Obstacle climbing

In the obstacle climbing test (Figure 14), the robot starts on a flat concrete surface, moves forward to the step, overcomes it, and travels a short distance on gravel. After that, it reverses direction and follows the same path back to its starting position. In the diagram, the tripod is shown to scale with the step.

Figure 14.

Diagram of the obstacle climbing test. The robot starts on a flat concrete surface, approaches and climbs a step, and then moves onto gravel. It reverses the same path to test the robot’s ability to overcome obstacles in 8WD mode with fuzzy control.

The purpose of the test is to verify the robot’s obstacle-overcoming capability in 8WD mode with fuzzy control enabled. In 4WD mode, this specific test fails because once the front module passes the obstacle, it is unable to pull the rear module along.

Figures 15a and 15b show the angular velocity of the motors and the yaw angle measured at the articulated joint, respectively. The test is divided into two parts: forward climbing and backward climbing. Each part is divided into three distinct phases: approaching the obstacle, overcoming the obstacle, and withdrawing from the obstacle. The overcoming phase begins when the front module’s wheels hit the obstacle and ends when the rear module is entirely on the gravel. The same thing happens in the backward phase but the modules are reversed.Footnote ¹

Considering the forward overcoming, we can observe the transition between the approach phase and the overcoming phase. Once the front module hits the step, the tripods switch from advancing mode to climbing mode while maintaining the same angular speed of the motors (FL, FR). On the contrary, in the rear module, there is a noticeable drop in speed. During the overcoming phase, numerous corrections are made to the rear module to follow the target yaw angle. These corrections continue into the withdrawal phase due to the loose and uneven ground after the obstacle. In the backward motion, the pattern is the same but reversed, with the rear module controlled by velocity and the front by fuzzy.

Figure 15.

Obstacle climbing test. Fuzzy control actively works to maintain rear alignment when climbing obstacles and on gravel surfaces.

4.3. Steering test

Figure 16.

(a) Steering test scheme. (b) Standing start test scheme.

In the steering test (Figure 16a), the robot travels at a constant speed through a straight section (1), a curve (2), and a new straight section (3) to analyze its behavior while entering and, more importantly, exiting a curve. The test was performed in 8WD mode and 4WD mode to verify the different behaviors of the robot.

Figures 17a and 17b shows the yaw angle and the angular speed of the motors during the test, respectively. Looking first at the yaw angle graph, the main differences are in the transition between the curve and the straight section. In 8WD mode with fuzzy control enabled, Epi.Q takes about 0.5 s to reach the target angle. In contrast, in 4WD mode, there is a noticeable delay before the robot adjusts its steering, and then it takes about 2 s to reach the target angle. The time it takes Epi.Q to reach the target angle in 4WD mode results in a continuation of the curve, which makes the robot difficult to maneuver.

Looking at the angular speed graph, there is a clear difference in the robot’s behavior between 8WD and 4WD modes. In the first case, the rear module (RL, RR), with the high speed variation imposed by fuzzy control, helps to reorient the robot to quickly reach the target angle. In the second case, the rear module is just a trailer, so only the front module works to reach the final configuration.

Figure 17.

Steering test. Fuzzy control actively works to reduce system time response.

4.4. Standing start test

The standing start test (Figure 16b) consists of positioning the robot with the maximum misalignment between the modules, in this case $\delta = 38^\circ$ , and then applying a longitudinal velocity command to evaluate the acceleration phase and the realignment phase.

Figure 18.

Standing start test. Fuzzy control changes the behavior of the system during the transient.

The yaw angle graph (Figure 18a) shows that although the time to reach the target angle is identical in both scenarios, the paths differ significantly, leading to different outcomes. In the first scenario (fuzzy control), the front module of Epi.Q essentially follows a straight path, meaning that the module does not rotate from its initial position. In the second case, however, the front module rotates from its initial position because of the resistance from the rear module during the acceleration phase. When the robot reaches module alignment, the front module does not go in the same direction as in the first case.

These observations can be deduced by examining Figure 18a and Figure 18b. The yaw angle sensor records an overshoot when fuzzy control is active, which is due to the correction made by the rear module. This substantial correction helps mitigate the drift phenomenon of the front module. Moreover, an important distinction is visible in the angular speed graph: the RL wheel reaches the target in 1 s with fuzzy control activated, whereas it takes approximately 2 s without it. This discrepancy allows the front module to maintain its initial direction.

4.5. Discussion of results

In conclusion, the preliminary tests performed on the Epi.Q UGV have highlighted its strengths and areas for potential improvement under various operational conditions.Footnote ² The uphill tests showed that the fuzzy control and 8WD configuration significantly enhance Epi.Q’s climbing capabilities. The obstacle-climbing test further validated the effectiveness of the fuzzy control system, allowing Epi.Q to smoothly transition between flat and uneven surfaces, including steps, which would have been challenging in a 4WD configuration alone. The steering and standing start tests demonstrated the improved maneuverability and alignment control provided by the rear module’s fuzzy control, especially in maintaining trajectory and minimizing alignment delay after sharp turns or misalignments. Overall, these tests underscore the effectiveness of the fuzzy logic-based control in stabilizing the robot’s performance across diverse and unpredictable terrains, providing valuable insights for future optimization and refinement of the control system.