Vertical Balance of an Autonomous Two-Wheeled Single-Track Electric Vehicle

Abstract

In the dynamic landscape of autonomous transport, the integration of intelligent transport systems and embedded control technology is pivotal. While strides have been made in the development of autonomous agents and multi-agent systems, the unique challenges posed by two-wheeled vehicles remain largely unaddressed. Dedicated control strategies for these vehicles have yet to be developed. The vertical balance of an autonomous two-wheeled single-track vehicle is a challenge for engineering. This type of vehicle is unstable and its dynamic behaviour changes with the forward velocity. We designed a scheduled-gain proportional–integral controller that adapts its gains to the forward velocity, maintaining the vertical balance of the vehicle by means of the steering front-wheel angle. The control law was tested with a prototype designed by the authors under different scenarios, smooth and uneven floors, maintaining the vertical balance in all cases.

1. Introduction

In the rapidly evolving landscape of intelligent transport systems, where the main objective is to upgrade existing traffic systems, autonomous driving vehicles and the implementation of proper control strategies are increasingly gaining significance (e.g., [1,2,3]). As we witness the advent of autonomous agents and multi-agent systems reshaping the future of transport, ensuring the stability of vehicles, particularly those with only two wheels, emerges as a critical enabling technology [4]. Specially, the control strategy for the lateral stability of two-wheeled vehicles holds paramount significance, as well as facing greater difficulty [5,6].

Two-wheeled vehicles, such as motorcycles and scooters, pose unique challenges in terms of lateral stability due to their inherent design and dynamics [7]. Unlike their four-wheeled counterparts, these vehicles are more susceptible to lateral disturbances, making precise control strategies essential for safe and efficient operation [8]. The integration of intelligent control systems within the framework of these vehicles becomes a pivotal aspect in guaranteeing both rider safety and overall system performance.

In the context of autonomous agents, where vehicles navigate complex environments without direct human intervention, lateral stability control becomes even more vital. The embedded control system plays a crucial role in real-time decision-making, ensuring that the vehicle responds appropriately to varying road conditions [9], unexpected obstacles [10], and dynamic environmental factors [11]. A well-designed control strategy contributes to the vehicle’s ability to maintain stability during cornering, abrupt manoeuvres, and adverse weather conditions.

The coordination and communication between autonomous agents require a robust control strategy to harmonize the movement of vehicles, preventing potential conflicts and enhancing overall traffic flow efficiency [12]. Such is its relevance that significant public investments are being made with the aim of providing solutions to the problem. An example can be seen in the Cooperative Vehicle-Infrastructure Systems (CVIS) project, with 75 European participants, and whose main topic is eSafety Cooperative Systems for Road Transport [13]. In this sense, as the transport ecosystem transitions towards a more interconnected and collaborative model with multi-agent systems, two-wheeled vehicles become a key element in the seamless integration of different vehicles and traffic participants, thanks to their great mobility and compact size. However, the integration of this type of vehicle requires a robust lateral stability control system.

In conclusion, the control strategy for the lateral stability of two-wheeled vehicles stands at the forefront of intelligent transport systems, serving as a linchpin in the realization of autonomous agents and multi-agent systems. By embracing advanced control technologies, we pave the way for safer, more efficient, and interconnected transport, ushering in a new era of mobility that prioritizes both innovation and user well-being.

The first two-wheeled single-track gadget can be attributed to Jean Théson in 1645. Some later inventors, such as Jhon Kemp Starly [14], added new elements, such as the transmission chain, wire spokes, and a handlebar, until evolving to the current bikes and motorbikes.

Although nowadays, there is a wide variety of studies that deal with mathematical models of two-wheeled single-track vehicles [15], the first contributions were made by Whipple [16] and Carvallo [17]. Both authors defined a mechanism composed of three parts: a rear frame, a steering drive system, and wheels. All parts were hinged together. In this case, as well as in [18]’s analysis, the model assumed a front wheel with zero radius, without lateral displacement of the tires, and took into account non-holonomic constraints in the longitudinal and lateral directions.

These simplified models do not take into account certain complex dynamics such as tire contact [19], which is highly non-lineal, or the driver’s position influence [20], which would cause displacements on the centre of mass. Additionally, these dynamics vary depending on external conditions while the vehicle is moving, such as the ratio between the temperature of the road and the vehicle tires, the geometry and design of the tires, and the driver’s posture [21]. Since these parameters are difficult to model, and in many of these cases, consistent models are lacking, some studies rely on online estimations [22].

However, the use of simplified models allows authors to use ad hoc methods, i.e., obtaining a particular solution of a more complex problem. An example of this is the study of the vehicle stability through its handlebar rotation [23].

Vertical balance of autonomous two-wheeled single-track vehicles is a challenge because these types of systems are unstable and their dynamic behaviour changes with the forward velocity. Control techniques used for an inverted pendulum are also used for these types of vehicles because of its similar dynamics. Some examples of these control techniques are the following: classical PID controllers [24], adaptive fuzzy logic control [25], and neural networks [26]. But these techniques are useless when the forward velocity of the vehicle changes. Two control strategies are commonly used to solve this problem: (1) By adding mechanical systems that change the centre of mass of the vehicle [27], the most commonly used technique in stabilization involves using gyroscopes [28]. This strategy is the most used because the vertical balance of the vehicle can be achieved at low velocities, even when the vehicle is stationary [29]. Some studies were reported by the Department of Systems and Control Engineering, Tokyo Institute of Technology [30,31,32,33]. (2) By using the steering front-wheel angle [34]. This strategy is useless under certain forward velocity because it uses the acceleration of the vehicle to achieve the vertical balance [35]. Most studies about this strategy are theoretical, as reported in [36,37,38]. Some experimental results are obtained from prototypes based on commercial bikes [39,40] and roller test platforms [41,42,43,44]. Nevertheless, the dynamic behaviour of the vehicle on a roller test platform is different from moving freely and therefore, the required control system is different.

This study was proposed as a continuation of the authors’ previous works (see [7,45]), where a two-wheeled single-track vehicle was modelled following Limebeer and Sharp’s work [35]. In [45], the model of the bicycle was presented, establishing the range of variation in the main system parameters at which the model could be considered as linear and proposing a technique to find the optimal parameters of the adaptive PI controller in terms of energy consumption but guaranteeing the stability of the controlled system. The performance of the adaptive PI controller was obtained by means of simulation, checking the trajectory tracking and the energy consumption for all forward velocities inside the allowed range. In [7], the adaptive PI controller was tested to determine the energy consumption of each degree of freedom and to determine the robustness of the system in rejecting lateral disturbances. As continuation of these works, an adaptive control strategy was then developed, taking into account the worst possible conditions for the vehicle’s movement. In this novel work, the prototype presented uses the steering front-wheel angle to maintain the vertical balance of the vehicle. The designed adaptive proportional–integral controller, which adapts its gains to the forward velocity wasting the minimum energy consumption, is experimentally tested. This control strategy is tested on both smooth and uneven floors. It should be noted that the aim of this study was not to track a desired trajectory of the vehicle but maintain its vertical balance.

Two-wheeled vehicles have certain advantages over other types of vehicles such as their size, weight, and turning radius. In order to use this type of vehicle autonomously, it is necessary to address its main issue, lateral stability. Additional elements such as gyroscopes or balanced masses can be used for this purpose, which add complexity to the mechanism and increase the weight of the vehicle, thus reducing the payload capacity. The ability to control the vehicle through the roll angle allows us to maintain the lateral stability without any additional element in the vehicle.

Nevertheless, only a few works about the lateral stability control of two wheels vehicles can be found. Some of the most important ones during the last years are [46,47,48,49,50,51,52,53,54,55]. For comparison purposes,

Table 1. Comparison of lateral stability method for two-wheel vehicles.

Work	Control Approach Description	Tuning Method	Energy Optimisation
			Included?	Can Be?
[46]	Fuzzy control	Lyapunov based	No	No
[47]	Sliding-mode control	Error minimisation	No	Yes
[48]	Output-zeroing control	Algebraic identification	No	No
[49]	Coupling PD controllers	Not defined	No	No
[50]	Active disturbance rejection	Algebraic identification	No	No
[51]	LQR controller	Minimising target function	No	Yes
[52]	PD controller	Frequency domain tuning	No	No
[53]	Model predictive control	Minimising target function	No	Yes
[54]	LQR + P controller	Minimising target function	No	Yes
[55]	Model predictive control	Minimising target function	No	Yes

describes the control approach, the tuning method (if possible), and evaluate if any energy optimisation is or can be included during the tune process.

Note that while some of the works presented in Table 1 may incorporate energy optimisation into their controller tuning process, none of them directly address this issue. This aspect should be considered a fundamental prerequisite for all vehicles, particularly for electric vehicles.

The rest of the paper is structured as follows: Section 2 presents the dynamic model of the proposed two-wheeled single-track vehicle and proposes a control law accordingly to the vehicle dynamics; Section 3 presents the prototype, from the mechanics to the electric/electronics details, used for the experiments carried out in Section 4. Finally, Section 5 summarizes the conclusions of this work and proposes future ones.

2. Mathematical Modelling

This section develops the dynamic model used for the two-wheeled single-track vehicle. Once the dynamics are defined, the control law governing the control strategy is presented.

2.1. Dynamic Model

In order to model the dynamic behaviour of the two-wheeled single-track vehicle, the point-mass model proposed by Limebeer and Sharp in [35], inspired by the Boussinesq bicycle model [18], was used. The Boussinesq model assumes a front tire of zero radius, presupposes that the movement of the vehicle is limited, i.e., it does not take into account the lateral displacement of the tires, and considers non-holonomic limitations in the longitudinal and lateral directions. Moreover, the obtained model is not self-stable and can be classified according to the models that use a simplified analysis.

Figure 1 shows a scheme of the vehicle (front-wheel steering) where all the mass is concentrated at the centre of mass (CM) and the steering axis angle is 90º (zero trail).

The vehicle moves according to the following cinematic equations:

$$\dot{x}=vcos\psi$$

(1)

$$\dot{y}=vsin\psi$$

(2)

$$\dot{\psi }=\frac{vtan\alpha }{wcos\theta }$$

(3)

where $x,y$ are the Cartesian coordinates of the vehicle motion, v is the forward velocity of the vehicle, $\psi$, $\alpha$, and $\theta$ are, respectively, the yaw, steering, and roll angle, and w is the wheelbase.

On the other hand, the dynamics of the vehicle is defined as [35]

$$\begin{array}{cc}\hfill h& \ddot{\theta }=gsin\theta -\hfill \\ \hfill & \left[\left(1-h\sigma sin\theta \right)\sigma v^2+d\left(\ddot{\psi }+\dot{v}\left(\sigma -\frac{\dot{\theta }}{v}\right)\right)\right]cos\theta \hfill \end{array}$$

(4)

where h is the height of the CM, d is the distance between the CM and the rear-wheel axis, g is the gravity acceleration, and $\sigma$ is the trajectory curvature (inverse of the radius of curvature R), defined as

$$\sigma v=\dot{\psi }$$

(5)

Substituting (3) and (5) into (4) yields

$$\begin{array}{cc}\hfill h& \ddot{\theta }=gsin\theta -\hfill \\ \hfill & tan\alpha \left(\frac{v^2}{w}+\frac{d\dot{v}}{w}+tan\theta \left(\frac{vd}{\omega }\dot{\theta }-\frac{h{v}^2}{w^2}tan\alpha \right)\right)-\hfill \\ \hfill & \frac{dv\dot{\alpha }}{w{cos}^2\alpha }\hfill \end{array}$$

(6)

The control input is $\alpha$ and the output is $\theta$. In order to simplify the design of the control law, the previous nonlinear equation is linearized around the equilibrium point ($\alpha =0^{\circ }$) assuming a constant forward velocity.

$$\ddot{\theta }-\frac{g}{h}\theta =-\frac{v^2}{hw}\alpha -\frac{dv}{hw}\dot{\alpha }$$

(7)

Notice that the coefficients of $\alpha$ depend on the forward velocity, and therefore, they are different for each forward velocity. By using the Laplace transform, the following transfer function is defined

$$G\left(s\right)=\frac{\theta \left(s\right)}{\alpha \left(s\right)}=-\frac{as+b}{s^2-c}$$

(8)

where a, b y c are

$$a=\frac{dv}{wh}$$

(9)

$$b=\frac{v^2}{wh}$$

(10)

$$c=\frac{g}{h}$$

(11)

All assumptions necessary to derive the dynamic model described in Equation (8) and evaluate its effectiveness are outlined in [45].

2.2. Control Law

Because the dynamics of the vehicle changes with the forward velocity as shown in (7), we propose a real-time adaptive controller that depends on it. A gain-scheduled proportional–integral controller is designed where the forward velocity is the scheduling variable (Figure 2), where $k_p$ and $k_i$ are the proportional and integral gains of the controller, respectively. The control law in Figure 2 is $\alpha \left(t\right)=k_p\epsilon \left(t\right)+k_i{\int }_0^t\epsilon \left(\tau \right)d\tau$ with ${\alpha }_{min}\le \alpha \le {\alpha }_{max}$, $\epsilon \left(t\right)={\theta }^{*}\left(t\right)-\theta \left(t\right)$ being the error signal and ${\alpha }_{min}$ and ${\alpha }_{max}$ the physical limit of the steering angle (see Section 3.1).

A lookup table is used to choose the controller gains as a function of the velocity (Figure 3). The lookup table, which contains the optimal controller parameters, was developed in [7]. The chosen design specifications for this optimisation were: (a) the bicycle lateral stability must be achieved even in cases of disturbances such as uneven floor or wind gusts; (b) the steady-state error of the output signal must be as close to zero as possible; (c) the settling time of the output signal must be less than 3 s; (d) the system energy consumption must be minimal. More details can be found in [7]. The absolute value of the gains decreases as the velocity increases. The vertical balance of the vehicle cannot be achieved below 1.21 m/s with the proposed method. The design of the controller can be consulted in previous studies of the authors [7,45]. The resulting $k_p$ and $k_i$ values, shown in Figure 3, results at any time in a stable controlled system. In [45] the stability of the system was checked using the Routh–Hurwitz criterium.

It is important to mention that the plant to be controlled, $G\left(s\right)$, is a non-minimum-phase system with poles in the positive region of the $s-$plane; thus, the parameters of the controller $k_p$ and $k_i$ are negative. The resulting controller could also be expressed in an equivalent way in “zeros–poles–gain” form with a negative gain and minimum phase zeros/poles.

3. Description of the Experimental Prototype

Figure 4a,b show, respectively, the 3D SolidWorks design and an image of the manufactured prototype. First, we describe the mechanical structure. Second, we describe the electric/electronic system.

3.1. Mechanical Structure

The main characteristics of the prototype structure are the following: the chassis is robust to avoid structural vibrations; the height of the CM is small to facilitate the vertical balance of the vehicle; the wheels have a small radius to reduce the gyroscopic effect; and the steering axis angle is 90º (zero trail) so that the only cause that controls the roll angle $\theta$ is the steering angle $\alpha$. Table 2 shows the dimensions and mass of the vehicle as well as some constraints on $\alpha$ and $\theta$, as shown in Figure 5. The steering angle is controlled by means of a rack and pinion drive (ratio 15.7:1) as shown in the scheme of Figure 6a. On the other hand, the forward velocity of the vehicle is controlled by means of a pulley drive (ratio 4:1) as shown in the scheme of Figure 6b.

Table 2. Mechanical parameters of the prototype.

Parameter	Value [unit]
h	$0.19$ [m]
w	$0.70$ [m]
d	$0.31$ [m]
r	$0.1$ [m]
m	$7.8$ [kg]
${\alpha }_{max}$	$\pm 0.61$ [rad]
${\theta }_{max}$	$\pm 0.47$ [rad]

Table 2. Mechanical parameters of the prototype.

Parameter	Value [unit]
h	$0.19$ [m]
w	$0.70$ [m]
d	$0.31$ [m]
r	$0.1$ [m]
m	$7.8$ [kg]
${\alpha }_{max}$	$\pm 0.61$ [rad]
${\theta }_{max}$	$\pm 0.47$ [rad]

Figure 4. Mechanical details of the design and prototype: (a) 3D SolidWorks design; (b) image of the prototype; (c) vehicle supported on the wheels of the roll-over protection system.

Figure 4. Mechanical details of the design and prototype: (a) 3D SolidWorks design; (b) image of the prototype; (c) vehicle supported on the wheels of the roll-over protection system.

Figure 5. Mechanical constraints: the steering angle is in the range $\theta \in [{\theta }_{max},\pi -{\theta }_{max}]$ rad, and the rolling angle $\alpha \in [-{\alpha }_{max},{\alpha }_{max}]$ rad.

Figure 5. Mechanical constraints: the steering angle is in the range $\theta \in [{\theta }_{max},\pi -{\theta }_{max}]$ rad, and the rolling angle $\alpha \in [-{\alpha }_{max},{\alpha }_{max}]$ rad.

Figure 6. Mechanical transmission of the steering and forward systems: (a) steering drive: pinion drive transmission; (b) forward drive system: belt–pulley transmission.

Figure 6. Mechanical transmission of the steering and forward systems: (a) steering drive: pinion drive transmission; (b) forward drive system: belt–pulley transmission.

Two security devices were added to the vehicle: (1) a roll-over protection system with four small wheels, two on either side of the vehicle (Figure 4); (2) an automatic system that move up and down two small stabilizer wheels, one on either side of the rear wheel of the vehicle (Figure 7). This system keeps the stabilizer wheels down as long as a threshold velocity is not exceeded (1.21 m/s).

3.2. Electric/Electronic System

Figure 8 shows a scheme of the electric/electronic system. It is composed of four decentralized subsystems: a power system; actuation systems; a control system; and a stabilizer wheel system. The decentralized systems allowed us to reduce the computational cost. Figure 9 shows the distribution of the electronic devices in the vehicle. All subsystems are described in detail below.

3.2.1. Power Supply

We connected two lead acid batteries of 12 V in series to get 24 V (image 6 in Figure 9). In order to avoid the battery discharge and unbalance voltage, a microcontroller, model Arduino Micro (image 2 in Figure 9) monitored the voltages of the batteries.

3.2.2. Actuation Systems

The steering angle and the forward velocity of the vehicle were commanded by electric DC current motors, model maxon RE 40, without a gearbox (images 3 and 8 in Figure 9), controlled by a modular digital controller, model maxon EPOS2 70/10 (image 7 in Figure 9).

Figure 9. Distribution of the electronic devices: 1. inclinometer sensor; 2. battery monitor; 3. forward wheels’ control; 4. stabilizer wheels’ control; 5. stabilizer wheels’ actuators; 6. batteries; 7. main control board; 8. steering angle actuator.

Figure 9. Distribution of the electronic devices: 1. inclinometer sensor; 2. battery monitor; 3. forward wheels’ control; 4. stabilizer wheels’ control; 5. stabilizer wheels’ actuators; 6. batteries; 7. main control board; 8. steering angle actuator.

3.2.3. Control System

A microcontroller board, model Arduino Mega, controlled all electric/electronic systems of the vehicle. It communicated with the two EPOS through a CAN-BUS shield, with the stabilizer wheel system through a microcontroller board, model Arduino Nano, and with an inclination sensor that measured the roll angle $\theta$, located at the top of the vehicle (image 1 in Figure 9).

The implementation of the adaptive PI controller was carried out on the Arduino Mega board. A control loop was implemented and at each iteration, the forward velocity was acquired. With the instant velocity value, the $k_p$ and $k_i$ values were extracted from the lookup table and used to update every sample time the control signal, which was sent to the EPOS servocontroller. The sample time of this control loop was 1 ms, and no sample was lost during the experiments. In this sense, the computational requirement to implement the control strategy was very low, since a low-cost ATmega2560 can execute the control loop, measure the forward velocity, look for $k_p$ and $k_i$ values from the lookup table, compute these proportional and integral actions and send the control signal, $\alpha$, to the servocontrollers via the CAN-bus with a sample frequency of 1 kHz.

3.2.4. Stabilizer Wheel System

It was controlled by the Arduino Nano micro controller. An electric DC current motor, model Pololu 37D, with a gearbox, a ratio of 100:1 (image 5 in Figure 9), moved the device up/down (Figure 7). Two limit switches (${\mathrm{F}}_{\mathrm{cc}}$) detected the up/down positions.

4. Performance Analysis

Vertical balance control of the vehicle was tested in two different scenarios: a smooth floor (pallet wood) and an uneven floor (paving stones). All tests were performed by using the following sequence: (1) the vehicle started moving with the stabilizer wheel system down until the desired forward velocity was reached; (2) At that instant, the stabilizer wheel system was up, and the control system maintained the vertical balance of the vehicle during 12 s; (3) Finally, the stabilizer wheel system moved down, and the control system was deactivated.

4.1. Tests Performed on Smooth Floor

Figure 10a shows the sport hall of the university campus of Toledo, Universidad de Castilla-La Mancha, where some tests were performed. The floor is made from pallet wood as shown in Figure 10b. Figure 11 shows the results of one test with a forward velocity of 1.5 m/s. We chose that value because it was close to the threshold below which the prototype could not maintain vertical balance by means of the steering front-wheel angle, which was 1.2 m/s, and therefore, it was the worst-case scenario. Vertical balance control was easier as the forward velocity increased. A dotted line marks the instant at which the stabilizer wheel system finished going up (2 s). We see the vehicle maintaining its vertical balance (roll angle $\theta$ approximately zero) without the steering angle $\alpha$ exceeding its maximum value (0.61 rad). We also see the forward velocity control system maintaining a value close to 1.5 m/s.

A new test was performed to demonstrate that the vertical balance control also worked from an unfavourable roll angle, that is, when the vehicle was supported on the wheels of the roll-over protection system (${\theta }_{max}$) as shown in Figure 4c. This test was performed by using the following sequence: (1) the vehicle started moving supported on the wheels of the roll-over protection system with the stabilizer wheel system up until the desired forward velocity was reached (2.2 m/s); (2) at that instant, the vertical balance control was activated trying to maintain the vertical balance of the vehicle for 12 s; (3) finally, the stabilizer wheel system moved down, and the control system was deactivated. Figure 12 shows the results of this test. After 1 s, the roll angle $\theta$ was below 0.47 rad (maximum value), that is, the wheels of the roll-over protection system did not touch the floor meaning that the vertical balance control worked in a proper way. We see that the steering angle $\alpha$ reached its maximum value (0.61 rad) for most of the test to maintain the vertical balance of the vehicle. We also see that the forward velocity control system did not maintain the desired velocity during the first second after starting the vertical balance control.

4.2. Tests Performed on Uneven Floor

After testing the vertical balance control on a smooth floor, we decided to test it on an uneven floor to see how it affected the vertical balance. Figure 13a shows the place where some tests were performed (a square of the university campus of Toledo, Universidad de Castilla-La Mancha). The floor was made of rough paving stones as shown in Figure 13b. All tests were performed by following the same procedure used in the tests on the smooth floor. Figure 14 shows the results of one of the tests with a forward velocity of 1.5 m/s. We see that the control system maintained the vertical balance of the vehicle but with more effort, that is, the vehicle required larger oscillations of the steering angle than on a smooth floor. We also see that the forward velocity control system did not maintain the desired velocity during the first second after starting the vertical balance control.

In order to continue testing the robustness of the vertical balance control, we decided to hit the vehicle on one of its sides during a test with a forward velocity of 2 m/s, obtaining the results shown in Figure 15. The dotted red line indicates the moment of the impact on the vehicle. We see that large oscillations appeared in the roll angle as well as in the steering angle when we hit the vehicle, but the control maintained the vertical balance. We see again that the forward velocity control system did not maintain the desired velocity during the first second after starting the vertical balance control.

4.3. Robustness Analysis

Although the previous section showed the controller’s performance at different forward velocities, on various surfaces, and in the presence of lateral disturbances, this section presents, through simulations, a robustness/sensitivity analysis of the proposed adaptive PI controller.

For this purpose, following the procedure described in [56], a variation in the parameter of model (8) was assumed for different values of the forward velocity. Taking as reference the nominal parameters shown in Table 2, their variations were simulated inside of the allowed range shown in

Table 3. Parameters’ variation for the robustness analysis.

Uncertainty Range	$-50\%$	$-25\%$	0% (Nominal)	$25\%$	$50\%$
h (m)	0.0950	0.1425	0.1900	0.2375	0.2850
w (m)	0.3500	0.5250	0.7000	0.8750	1.0500
d (m)	0.1550	0.2325	0.3100	0.3875	0.4650

.

To illustrate the robust behaviour of the controlled system, Figure 16 represents the roll angle for a forward speed of 1.5 and 2.0 m/s², when h, w, and d parameters changed within the range presented in Table 3. Note that the adaptive PI controller had a low sensitivity to parameter changes. The parameters of the controller for the 1.5 m/s² reference forward velocity were $k_p=-3.538$ and $k_i=-11.456$, and the ones for 2.0 m/s² reference velocity were $k_p=-3.01$ and $k_i=-1.452$ (see Figure 3).

Finally, we assumed the worst scenario, where all parameters could change simultaneously. To quantify the robustness of the controller, the integral absolute error of the roll angle, ${\gamma }_{\theta }$ in Equation (12), was computed when h, w, and d simultaneously changed from −50% to 50% of their nominal values.

Table 4. Performance of the roll angle when all parameters changed within the [−50%,50%] range.

Uncertainty Range	$-50\%$	$-25\%$	0% (Nominal)	$25\%$	$50\%$
${\gamma }_{\theta }$ (v = 1.5 m/s2)	5.0930	5.2184	5.3265	5.4367	5.5369
${\gamma }_{\theta }$ (v = 2.0 m/s2)	6.8306	6.5465	6.3369	6.1680	6.0306

presents the obtained results, showing that the performance of the roll angle was not affected by the parameters’ variation.

4.4. Discussion

This section summarizes the obtained results and discusses them. The steering and roll angles, $\theta$ and $\alpha$, the forward velocity, and the instant gain values of the controller, $k_p$ and $k_i$, were recorded for the four experiments carried out ($v=1.5$ m/s and $2.0$ m/s vs. smooth and rough surfaces).

As the main target of the control proposal was to ensure the lateral stability, a tracking functional was proposed for the roll angle, $\theta$, by considering a reference of ${\theta }^{*}=0$ rad. In this way, the integral absolute error between $\theta$ and ${\theta }^{*}$ could quantify the goodness of the stability control approach:

$${\gamma }_{\theta }={\int }_0^{t_e}|{\theta }^{*}\left(t\right)-\theta \left(t\right)|dt$$

(12)

In the same way, the integral absolute error between the steering angle $\alpha$ and ${\alpha }^{*}=0$ rad indicated the efforts of the controller to ensure the lateral stability:

$${\gamma }_{\alpha }={\int }_0^{t_e}|{\alpha }^{*}\left(t\right)-\alpha \left(t\right)|dt$$

(13)

$t_e$ being the experimental time.

Additionally, each experiment was characterised by a reference forward velocity, and in an analogous way, the following integral absolute error could be defined to quantify the capability of the controlled system to follow the desired forward velocity:

$${\gamma }_v={\int }_0^{t_e}|v^{*}\left(t\right)-v\left(t\right)|dt$$

(14)

$v^{*}\left(t\right)$ being the velocity reference.

Finally, to quantify the variation in the controller parameters, $k_p$ and $k_i$, the mean value, $\mu$, and its standard deviation, $\sigma$, were determined in each experiment.

All these results are summarized in

Table 5. Comparison of lateral stability method for two-wheel vehicles.

Experiment	1	2	3	4
Terrain	smooth	smooth	rough	rough
$\theta \left(0\right)$	0	0.47	0	0
$v^{*}$ (m/s)	1.5	2.2	1.5	2.0
${\gamma }_{\theta }$	0.2507	3.1854	0.3558	0.5223
${\gamma }_{\alpha }$	1.8325	4.9026	2.2986	2.0262
${\gamma }_v$	18.0957	26.1462	17.9413	23.7291
$[{\mu }_{k_p},{\sigma }_{k_p}]$	$[-3.5385,0.1794]$	$[-2.6005,0.2192]$	$[-3.5972,0.3347]$	$[-3.0366,0.2190]$
$[{\mu }_{k_i},{\sigma }_{k_i}]$	$[-11.3138,1.9855]$	$[-1.3161,0.0877]$	$[-12.1538,4.1053]$	$[-1.5572,0.4733]$

.

Attending to the results in Table 5, some overall conclusions about the controller performance can be stated:

• The stability performance, ${\gamma }_{\theta }$, is better at low speed.
• When the forward velocity increases, the required steering angle is smaller.
• The integral action of the controller is more required to maintain the lateral stability at low velocity and over rough terrain.
• The proportional action of the controller presents a similar behaviour at any speed.
• Based on the robustness analysis presented, the adaptive PI controller presents a robust behaviour when plant parameters vary within a reasonable range of $\pm 50\%$.

5. Conclusions and Future Work

In recent years, the academic community has directed significant effort toward intelligent transport. However, this focus has predominantly centred on four-wheeled vehicles, whose autonomous driving technology is more advanced, while two-wheeled vehicles are often overlooked in this context.

Only a few studies addressing the lateral stability or autonomous navigation of two-wheeled vehicles can be found, and most of them do not consider the energy consumption in the control design process.

This paper experimentally validated an adaptive PI controller for the lateral stability of an autonomous two-wheel vehicle. The primary objective was not only to enhance the bike’s lateral stability but also to minimize energy consumption.

The proposed adaptive PI controller was tested across various forward velocities and two types of terrain: smooth and rough. The experimental results demonstrated that the proposed controller effectively maintained the vehicle’s lateral stability under all tested conditions.

Future work will expand upon this research by incorporating a trajectory tracking layer into the control system. This will enhance the system’s autonomy and demonstrate the proposed prototype’s ability to track a desired trajectory while maintaining vertical balance before integration into an intelligent transport network.