Robust Hybrid Controller Design to Stabilise an Underactuated Robot Vehicle under Various Input

Tis paper presents the design of the control system for a robot vehicle with two wheels that mimics a double Inverted Pendulum (IP) system with an extendable payload. By expanding the degrees of freedom, the system is more fexible. Still, this model posed challenges for control of parts of the system, including the proper balancing of the intermediate body and angular displacement of both wheels and lifting the payload to the demanded height. In this paper, a hybrid control system that incorporates more than one type of controller which combined proportional integral derivative (PID), proportional derivative (PD), and fuzzy logic control (FLC) these controllers are designed for stabilising the aforementioned system. Te controller was validated by applying diferent input signals to the payload actuator to prove the control system’s stability and analyse the system’s behaviour. Te simulation results were satisfactory for the hybrid control technique. Te wheels successfully stabilised within 1.2682s. Further, the frst and second links stabilised at 9.5953s and 9.6467s, respectively. Te payload approximately produces the same input signal applied to the payload actuator. Te model was derived using the Euler–Lagrange equation. Te equations are solved using kinetic energy and potential energy which employs for motion. Simulation results of the two-wheeled robot vehicle with extendable payload designed with hybrid control systems are implemented using MATLAB/Simulink environment.


Introduction
Te proposed robot vehicle is based on a double IP which is considered a nonlinear and unstable system [1][2][3]. Te IP system and its applications play a vital role in control system engineering and research because of its highly underactuated nature and complex strong-coupling system [3,4]. In the subject of control engineering, IP problems are very common. Previous research studied the problem of stabilising IP and has refected the development of many applications based on the same concept, including the design of a classical pendulum on a cart [5][6][7]. Te IP systems have been extended and become more complicated with multiple rods which provides much fexibility [8], such as double IP [9][10][11] and triple IP [12][13][14][15]. Many pieces of research focused on applications based on IP systems, such as the Segway robot [16][17][18], the two wheelchairs [19][20][21], simple walking models [22], and the bipedal robot based on IP [23,24].
Our research considered the fve degrees of freedom model, consisting of two wheels and two links with a movable payload that can be moved to demand height. Increasing the system's degrees of freedom enables the vehicle to manoeuvre freely. Diferent input signals were applied to the movable payload to validate the system's stability. Te wheels were assumed to move within 0.8 m while both links of the intermediate body of the system were retained as upright to ensure system stabilisation. Te hybrid control system was used as a control technique to implement fve loops of model control to achieve system stability.
Tis paper is organised as follows: "system description" which describes the two-wheeled robot vehicle with movable payload and the "mathematical modelling" section, which explains the derivation of the nonlinear diferential equations utilized in the system simulation. Te section "hybrid control design" obtained the control system used for the system stabilisation. Te "Results Discussion" section discussed the simulation results. Finally, the research is concluded in the "conclusions" section.

System Description
A diagram form illustration of the assumed model [25] is shown in Figure 1. Te system consists of two wheels and an intermediate body comprised of two links connected to the payload that can move to the required height. Te robot vehicle drives these wheels using two DC motors, with a further motor used to drive the second link. A single actuator in the middle of the second link of the intermediate body lifts the payload to the required height. Te robot vehicle thus has fve degrees of freedom (DOF).
(i) Te translational motion of the two wheels (ii) Te two links of the system (iii) Te movable payload linear actuator Te angles of both links, the frst being θ 1 and the second θ 2 , were measured from the Z-axis, representing the intermediate body being in the upright position or inclined to only a small degree, depending on the controller design. Q is the linear displacement of the payload measured from the O 2 position along with the second link. Te left δ L and right δ R wheels' angular displacements cause the vehicle to move linearly in the XY plane. Tese are the main fve loops controlled to stabilise the system.

Mathematical Modelling
Te system mathematical modelling was derived using the Euler-Lagrange equation due to modelling complex, nonlinearity, and strong-coupling systems such as [19,26]. Tis method ofers a simple approach to determining a complex systems model [18]. Te Lagrange equation employs both kinetic energy and potential energy that are solved for motion [20,27] using an equation as follows: After the system was derived it yields fve nonlinear diferential equations (2) Te system was built and the dynamics were described using these fve nonlinear diferential equations. Te torques of the motors are the system's inputs including T LT T RT . Te motor derived the second link T MT , and the linear actuator force is F aT where J w , J 1 , J 2u , J 2l , and J m are the intermediate body's mass moment of inertia. Te system simulation parameters reported by [28] that based on the standard dimensions of a two-wheeled robot vehicle illustrated in Table 1.

Hybrid Control
Design. Te hybrid control system intends to improve the system performance by combing the best specifcations from the control systems used [29,30]. A hybrid control system employs the benefcial sides of the proposed controllers suggested [31]. Tis study implements hybrid controllers to improve performance. Te frst control system was the PID controller, a widely used control system that provides dependable and stable performance for most systems [32,33]. Te proposed controller is a commonly used control system because it is simple and easy to tune, further providing robust performance [34]. Ten the fuzzy logic controllers test the system stability and the improvement of this type of controller on the system. In this research, higher control eforts were exerted in PID than in FLC for the system stabilisation. However, it is essential to decrease the exerted efort of the controller to stabilise the system. According to the simulation results, the FLC was signifcantly reduced compared to PID controllers by the control efort made for system stabilisation, without impacting the stability of the system. After that, the hybrid control system intends to improve the system performance by combing the good specifcations from the PID and FLC control systems used [35].
Two types of hybrid control systems are designed for the nonlinear model for two-wheeled robot vehicles with movable payload. PID with FLC is used to control both wheels, two links of the intermediate body, while the payload actuator was controlled using PD with FLC controllers.
Te PID-PD control parameters are tuned progressively until the system stabilise [36]. Te FLC with two inputs includes error and change of error with one output used to describe a fuzzy inference system and then create the fuzzy rules [37,38]. Te linguistic variable of the two inputs and output are negative-big (NB), negative-small (NS), zero (Z), positive-big (PB), and positive-small (PS). Tese rules yield the action of the FLC parts. Te proper system tuning of the FLC to stabilise the model was developed using fve Gaussian membership functions (MF) with 25 rules base. Figure 2 illustrated the design of the model with the hybrid control system.
For PID and PD tuning, the gain parameters are shown in Table 2.
Te fuzzy logic control with 25 rules base and fve MF are shown in Table 3 depending on the desired system performance.
Te membership functions are illustrated in Figure 3.

Analysis and Discussion of Simulation Results
At this stage, the system was designed with diferent force input signals applied to the payload to validate the control system's robustness. A hybrid control system is designed to stabilise combined with PID-PD and FLC. Te system responses were tested with various input signals applied to the payload actuator.

Test 1: Simulation Results with the First Payload Input
Signal. Te frst input signal is illustrated in Figure 4. Tese simulation results of the system response using hybrid controllers are illustrated in Figure 5. Te results clearly show that the control system stabilised the two wheels robot vehicle. Te system controller efort is observed in Figure 6. It can be seen that the wheels stabilised with acceptable overshoot and peaks for the wheels, and the two links and a 4.954% overshoot were observed on the payload actuator.
Simulation results for the system response using hybrid controllers are illustrated in Figure 5.
Te exerted efort of the controllers required to stabilise the system is represented in Figure 6.  Te hybrid controller was able to help stabilise the system components with the controllers' efort as shown in Figure 6. Te wheels settle at 1.2682 s with a 4.737% overshoot. Te frst link of the intermediate body reached settling time at 0.0621 s, and the second link settled at 4.529 s, and the payload actuator produce the same input signal applied to the payload with a 4.954% overshoot.

Test 2: Simulation Results with the Second Payload
Input Signal. Te second input signal using a hybrid controller applied on the payload actuator is illustrated in Figure 7.
Simulation results for the system response using hybrid controllers are illustrated in Figure 8.
Te exerted efort of the controllers required to stabilise the system is represented in Figure 9.
Te hybrid controller was capable to stabilise the system parts with the second input signal, and the control system efort was less than the frst signal. Te payload actuator produced the same input signal applied to the payload with a 0.515% overshoot.

Test 3: Simulation Results with the Tird Payload Input
Signal. Te third input signal using a hybrid controller applied on the payload actuator is shown in Figure 10 and the system response is illustrated in Figure 11 further Figure 12 represents the control system efort used to stabilise the system.
Simulation results for the system response using the hybrid controllers are illustrated in Figure 11.
Te exerted efort of the controllers required to stabilise the system is represented in Figure 12.
Te system successfully stabilised with the third input signal, but the payload actuator required higher control efort for the payload stability, this is due to the sharp shape of the input signal. When the signal reaches the maximum point, it suddenly goes back to zero, which makes the controller require higher efort for the payload stabilisation.

Test 4: Simulation Results with the Fourth Payload Input
Signal. Te last test was with the input signal illustrated in Figure 13. In this case, the control system successfully stabilised with controller efort exerted for stabilisation, as shown in Figure 14.
Simulation results for the system response using the hybrid controllers are illustrated in Figure 15.
Te exerted efort of the controllers required to stabilise the system is represented in Figure 14.
Te hybrid controller stabilised the system parts with the fourth input signal and acceptable controller efort; Te payload actuator produced the same input signal applied to the payload with an overshoot of 59.718%.
Two-hybrid controllers are implemented, including PID with FLC and PD with FLC. Te hybrid controllers combined the excellent characteristics of the PID-PD and FLC       Comparing the simulation results to those reported by [25], the system successfully stabilised with less control efort that provided a stable response, and the wheels stabilised at 36.4 N.m. Te frst link controller efort was 4.15 N.m, which gave satisfactory responses, while in the previous study, the wheels stabilised with more than 120 N.m further more than 7 N.m controller efort exerted for the frst link.

Conclusion and Future Work
Tis paper's objective was to design a hybrid control system for a two-wheeled robot vehicle with a movable payload based on the double-inverted pendulum system. Te hybrid controller consists of PID with FLC and PD with FLC used for the system stabilisation. Te PID and PD controllers were tuned progressively while the FLC was designed with fve MFs and a 25-rules base. Te simulation results illustrated that the hybrid control system was successfully designed to control the robot vehicle. Te validation of the designed controller has been proved by applying diferent input signals to the payload actuator. Te simulation results have shown successful stable responses. Future work should thus focus on testing the control system's robustness by applying disturbances on all system parts with various amplitudes. (see Table 4).

Data Availability
Te data used in this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
Te authors declare that there are no conficts of interest in this article.  [35].

Variables and Units Description L M(t) (m)
Distance from the centre of mass to the payload L 2u(t) (m) Distance from the centre of mass to the upper part of 2 nd link L a (m) Linear actuator position L 1 (m) Te half-length of the 1 st link H(m) Te separating two wheels' distance Q(m) Te displacement of the actuator M 1 (KG) Te 1 st link mass M m (Kg) Te motor mass that drives the 2 nd link M 2l , M 2u (Kg) Te lower-upper parts mass of the 2 nd link M a (Kg) Linear actuator mass M(Kg) Mass of the payload M w (Kg) Mass of the wheels T R , T L (N · m) Torques driving left-right wheels T m (N · m) Te motor torque F a (N) Te linear actuator force F f (N) Te linear actuator frictional force F d (N) Te external force θ 1 (rad) Te angle of the 1 st link θ 2 (rad) Te angle of the 2 nd link ∅(rad) Te angle measured around the Z-axis δ R , δ L (m) Te angular displacement of both wheels J 1 (Kg · m 2 ) Mass moment of inertia of the 1 st link J 2u , J 2l (Kg · m 2 ) Mass moment of inertia of the 2 nd link upper-lower parts J a (Kg · m 2 ) Te actuator's mass moment of inertia J M (Kg · m 2 ) Te payload's mass moment of inertia J w (Kg · m 2 ) Both wheels' mass moment of inertia J IB (Kg · m 2 ) Te intermediate body's mass moment of inertia