Robust Approach for Global Stabilization of a Class of Underactuated Mechanical Systems in Presence of Uncertainties

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Introduction
During the last three decades, the control of underactuated systems specifcally Underactuated Mechanical Systems (UMSs) have attracted researchers to work in this domain [1]. Te main reasons for UMSs research and interest are as follows: (i) a large number of practical applications; (ii) importance and usefulness of these systems in areas such as marine, aerospace, mechatronics, and robotics; (iii) inventions and development of new types of UMSs like miniaturized robots and fexible link manipulators used for diferent kinds of applications; and (iv) a pursuit for the ever demanding benefts such as reduction of power consumption, weight, cost, and complexity of UMSs. Te above-mentioned systems are highly nonlinear in nature and their control is the subject of major interest for control domain scientists. Many of these systems, e.g., the Ball-and-Beam [2], TORA [3], and the Cart-Pole [4] are considered to be a benchmark for nonlinear systems for comparing and assessing diferent types of control techniques. Due to their vast application area and diverse use, UMSs are considered to be the subjects of excellent research works.
Te underactuated nature of these systems, i.e., a lesser number of independent control variables than the state variables, causes great difculties for the recognition of the efective control design scheme for such systems. Because of this, previously well-developed and elegant nonlinear design control strategies like direct backstepping and exact feedback linearization [5] cannot be directly applicable to such systems in many cases. Due to the underactuated nature of these systems, a feedback technique based on partial linearization [6,7] may be one possible solution, but make the controller difcult to construct due to coupling in the system's unactuated and actuated parts. Alternatively, normal forms [8], based on the Lagrangian of the systems, were devised to decouple the unactuated and actuated parts. Henceforth, normal form and partial feedback linearzation have considerably been utilized as frst-choice control framework for UMSs in most works. Apart from the conventional energy-based techniques [9], more general strategies based on the Hamiltonian or Lagrangian characteristic is adopted to solve the controller problem of UMSs [10,11].
As the full linearization of the UMSs is not possible, another widely used technique is the approximate inputoutput linearization [12]. In this technique, a linear sliding surface approach is used to compensate the complex nonlinear terms by designing the higher-order compensating sliding mode control (HOCSMC). Tese strategies are also suitable for the class of UMSs that are approximate inputoutput linearizable, for example, the Cart-Pole and the Balland-Beam system, but they cannot be applied to the whole class of UMSs. A systematically unique design strategy is usually used for such systems such as IWP [13], TORA [14], the Furuta Pendulum [15], the Overhead Crane [16], and the Beam-and-Ball [17] due to complex dynamics. In literature, a design framework is also proposed which enclose systems in a specifc class, e.g., controlled Lagrangian [18], equivalent-input disturbance [19], sliding mode [20], energy-based [21], IDA-PBC [22], output feedback stabilization [18], dynamic surface control [23], and adaptive approaches [14].
Te swingup control is well studied by many researchers and the literature is vast. Diferent control design techniques used to achieve swingup include: without state feedback using oscillatory inputs [24], energy-based [25], energybased control combined with neural and fuzzy neural networks [26], Lyapunov-based [27], intelligent control [28], fuzzy control [29], partial feedback linearization [30], adaptive sliding mode [31], virtual holonomic constraintsbased design [32], and an impulse-momentum approach [33]. On the other hand, methods for balancing control are typically limited to the linearization of the dynamics around the equilibrium point and then using linear control techniques such as linear quadratic regulator (LQR) and pole placement to design control. Tis approach is adopted in most works, for example, [33,34].
Te reason for using linear methods is that underactuation makes difcult the application of standard nonlinear control methods such as exact feedback linearization and backstepping for the design of balancing control. However, a balancing control obtained using the abovementioned linear control techniques has limited practical usefulness due to the small region of attraction. Further, the control is not robust to uncertainties and external disturbances; and hence, results in degraded system performance. A balancing control obtained from robust nonlinear control techniques like sliding mode control (SMC) [35][36][37][38][39] will not sufer from these drawbacks. Tis is the motivation behind this work.
Te techniques based on sliding mode control (SMC) is a frst line of choices and are widely applied to UMSs due to their robustness properties against the uncertainties and their capability to control highly nonlinear uncertain systems. Te design simplicity, order reduction and invariance properties make the SMC techniques more desirable to be applied on UMSs. In this work, a broader and wide range of two classes of UMSs are considered. First, by using Partial feedback linearization and nonlinear state transformation, the frst class dynamics are transformed to cascade strict feedback normal form and of the second class to cascade nontriangular quadratic normal form [7]. Second, it is quite difcult using the traditional nonlinear control design techniques like backstepping to use due to the nontriangular quadratic normal form. Hence, a robust design framework based on sliding mode control is proposed. Te proposed SMC framework not only simplifes the controller design but has also improved performance as well as robustness. Tird, a sliding manifold for reduced-order nonlinear subsystem known as Lagrangian zero dynamics is proposed, such that the reduced-order system stability is guaranteed. Lastly, Lyapunov theory is utilized to prove the overall stability of the closed loop system. Moreover, to overcome the limitations in the SMC design framework, a more general swingup laws are designed for a Wheel and Furuta Pendulums examples. Tese control laws are used to swingup the unactuated pendulum from the downward stable equilibrium position and then stabilize it in the presence of external disturbance to the upward unstable equilibrium position using the designed sliding mode control laws. Illustrative application examples with numerical simulations of the Inertia-Wheel Pendulum and the TORA of the frst class, and for the second class, the Furuta and the Overhead Crane is presented to prove the efciency of the proposed design methodology.
Te rest of the article is illustrated as follows. Problem formulation and UMSs model verifcation are depicted in Section 2. In Section 3, a sliding surface and a design of sliding mode controller are presented. Section 4 illustrates the proposed method importance with the help of application examples. Simulation results and a detailed discussion on the efectiveness of the proposed strategy are presented in Section 5. Finally, Section 6 concludes the work.

Model Description
A generalized equation, describing the dynamics of a mechanical system, is presented in the following equation: (1) In equation (1), ξ ∈ R n represents the confguration vector, whereas M(ξ) ∈ R n×n denote the inertial matrix. C(ξ, ξ . ) ∈ R n×n represents the Coriolis and centrifugal terms, G(ξ) ∈ R n×1 is the gravitation efect, F(ξ) ∈ R n×m represents the input vector, and u ∈ R m is the input control. Moreover, uncertainties in the systems model are shown by ∇(ξ, ξ . , t) ∈ R n .

Complexity
Tese generalized dynamics has two possibilities of actuation.
(1) Te dynamics are termed as fully-actuated if rank(F) � m � n. (2) (2) Te dynamics will be underactuated if In this work, two classes of systems are described by equation (1) with M(ξ 2 ) are considered, frst with F(ξ) � [0, I m ] T and second with F(ξ) � [I m , 0] T . For the frst case, split the vector ξ ∈ R n into unactuated ξ 1 ∈ R n− m and actuated ξ 2 ∈ R m vectors, which gives the following representation of the nominal dynamics.
For the second case, by dividing the vector ξ ∈ R n result in nominal dynamics as follows: ), g 1 (ξ 1 , ξ 2 ), and g 2 (ξ 1 , ξ 2 ) on states. Table 1 shows this functional dependence for application examples.
In general, the dynamics of the form in equations (4) and (5), though second-order nonlinear subsystems are not considered suitable for synthesizing controllers even in the absence of uncertainties. Input and state transformation greatly simplify this challenging control design problem. Te dynamics in equations (4) and (5) can only be partially linearized using the collocated and noncollocated PFL. Furthermore, nonlinear state transformations are used to transform the dynamics in equations (4) and (5) into normal forms. Although higher-order UMSs in the form of equations (4) and (5) can be transformed into normal forms [7], there exist explicit transformations for two degrees of freedom systems outlined below.
For the frst class, equation (4), the manipulated/control input is generally presented, with a new/heroic control v, as follows: Te following set of expressions and variables [7] transform the dynamics in equation (4) into the strict feedback normal form: where � denotes d/dξ 2 . Tese transformed normal forms can be represented as general lumped parameter system, as follows: where z ∈ R (n− m) and η ∈ R m for the frst class, equation (4), and z ∈ R (m) and η ∈ R n− m for the second class, equation (5), and D(z, _ z, η, _ η, t) illustrate the lumped efect of all the uncertainties.
To proceed with the design of sliding manifold/surface and discontinuous SMC, some assumptions are made about dynamic behavior of the system. Assumption 2. Te zero dynamics (equation (14a)) are at rest initially i.e., f(. Assumption 3. Te relative degree and controllability can be guaranteed if (zf/zη) ≠ 0 and (zf/zη) ≠ 0.
In the next section, a siding surface and sliding mode control for a general system (14) is presented.

Design Methodology
Stability analysis of the Lagrangian zero dynamics in (14a) leads to diferent design scenarios, as detailed below, for the stabilization of overall dynamics in (14). In each scenario, the stability of the overall system is dependent upon the stability of corresponding zero dynamics. Tus, each scenario requires a stable switching manifold and a discontinuous controller to force and confne the dynamics to that manifold.
If the zero dynamics in equation (15), with η as output, are not stable then the control design procedure is carried out as in design scenario 2 below.

Design Scenario 2.
Te frst-order zero dynamics, when z is an output and provided that z and all its total time derivatives are zero, are given by the following equation: Te overall system's (equation (14)) stability is then depending upon stability of equations (18) and (14a). Moreover, a control algorithm can be synthesized for the z-dynamics provided the zero dynamics (equation (18)) are stable.
Te dynamics in equation (14a) can be forced to follow a predefned trajectory, as shown in the following equation: where a, b ∈ R are strictly positive numbers. Tis enforcement is achieved with the introduction of a sliding manifold as follows: Te enforcement of sliding modes (i.e., σ 2 � 0), in the given systems' state space, reduces the overall z-subsystem to linear second-order dynamics, as coined in the following equation: Te discontinuous control law that will force σ 2 � 0 is formally presented in Teorem 5. where will guarantee sliding modes in the state space of system represented by equation (14) Proof. A positive defnite and radially unbounded Lyapunov candidate function (V(t)) and its frst total time derivative along the trajectories of equation (14) is described in the following equation: Substituting v from the control law (22), the above expression for _ V(t) becomes: which can be written, using σ 2 sign(σ 2 ) � |σ 2 |, as follows: and alternatively as follows: which is strictly negative defnite. Tis ensures that the trajectories will reach the switching manifold (σ 2 ) from any initial condition and will be confned there after. Moreover, Hurwitz nature of σ 2 ensure that the trajectories will slide towards the equilibrium. Hence, stability of the overall system is confrmed. □ Remark 6. Te above design procedure is valid if _ η explicitly appears in the function f i.e., the system under consideration has relative degree one. If this is not the case then design is carried out according to design scenario 3 below. Complexity 5
Te only solution to the above statement is η � 0 which confrms stability of the overall system.
To compensate for the relative degree, a novel dynamic sliding manifold is defned.
Now, the control problem is to design a law which will force sliding modes in σ.
Theorem . Let the assumptions, made above, are intact then for the dynamics of under-actuated mechanical system, expressed in the normal form (equation (14)), the control law with strictly positive design constants Γ 1 , Γ 2 , and smoothening parameter 0 < c < 1, will guarantee stable sliding modes in the manifold σ (see equation (27)).
Proof. Te proof of this theorem is intuitive (similar to previous theorem) and is thus omitted here.
□ Remark 8. Te parameters Γ 1 and Γ 2 are responsible for the robustness properties of the algorithm while c provides the necessary smoothness of the control signal. An increase in Γ i s enhance robustness (a disturbance of relatively higher magnitudes can be coped) but at the cost of excessive control actions which, in practical cases, can be dangerous for actuators. Te parameter c, on the other hand, is a trade-of between robustness and performance i.e., taking it closer to 1 will increase smoothness but at the cost of robustness.
Remark 9. Te inherent advantages include a generic methodical structure of the proposed controller for a class of underactuated mechanical systems. In practical applications, the proposed algorithm will be efective due to the smooth, yet robust, and control action. Moreover, the overall implementation will not be requiring any extra circuitry (in case of analog implementation) or machine cycles (in case of digital implementation) as the algorithm does not require extra diferentiators and/or integrators. In addition, the algorithm deals with the reduced dynamics which in turn facilitate practical implementation.

Remark 10.
A smooth control action, reduced dynamics in the design process and existence and smooth switching between swingup and main stabilizing controller are the factors causing the necessary performance enhancement Te overall design methodology has been summarized in Figure 1 while application examples in the next section demonstrate the efectiveness and elaborate the design procedure of the proposed technique.

Application Examples
Two application examples are presented for each of the two classes. Te schematics and Euler-Lagrange Equations of Motions (EOMs), for the above mentioned systems, are outlined in Figure 2 and Table 1, respectively.

IWP.
Te IWP, consisting of a rotating uniform inertia wheel and a pendulum (see Figure 2(a)) demand stabilization of the pendulum at the upright unstable equilibrium (ξ 1 � 0) with a control torque u which is applied to the rotating wheel [18,23,40,41]. Parameters of the IWP are shown in Table 2.
Te dynamic equations, governing IWP, followed from Table 1 and equation (4), are listed in the following: Using the input and state transformations in (6) and (7), the normal form in equation (9) for the IWP becomes: where
where us(.) represent a shifted unit step signal.

Swingup.
Te IWP has two equilibriums: a stable one (downward, ξ 1 � π) and an unstable one (upward, ξ 1 � 0). Between the two at midway (when pendulum is horizontal),   (34)). Tus, an additional controller, v swingup , is introduced to take the pendulum from ξ � π to ξ � 0. Te v, coined above, is then efective for stabilization. A partial linearization of IWP dynamics and solution for equation (4a), for € ξ 2 , gives the following expression: Applying a noncollocated Partial Feedback Linearizing (PFL) control  to equation (4b) transforms the IWP as follows: It may be noticed that the ξ 1 dynamics are now represented by a double integrator system, and selecting v swingup as a PD control will stabilize it.
Te control v swingup combined with u PFL (38) can stabilize ξ 1 from ξ 1 � π to ξ 1 � 0, however the ξ 2 − dynamics, obviously, become unstable. Once ξ 1 is stabilized, v swingup becomes zero which is obvious, however, the PFL control u PFL also becomes zero which is in accordance with the expression in equation (38). Tis implies the ξ 2 − dynamics, in accordance with equation (37) (and equivalently, equations (39b), (31a), and (31b)), become Te closed loop response of IWP, having v swingup in the loop (K d � 11, K p � 30) with initial conditions ξ(0) � [π, 0, 0, 0] T , is shown in Figure 4. A double integrator efect may be noticed in Figure 4 Te strategy is to use v swingup to bring the pendulum from ξ 1 � π to the vicinity of ξ 1 � 0 and then switch to v (equation (34)) for stabilization. A robust stabilization and an efective switching between the controllers are demonstrated in Figure 5. Te disturbance in Figure 5(c) may be treated as was explained in Figure 3(c). Figure 2(b). Being a challenging nonlinear benchmark it has been used widely by researchers for validating their algorithms [14,18,23,43,44].

TORA. TORA, frst introduced in [42], is shown in
A TORA is composed of two main components: a rotating actuator and a translational oscillating platform. Te control problem is the stabilization of oscillating platform (regulation of ξ 1 , translational displacement). For this purpose, the control input is a torque u which is applied to the rotational actuator. Te parameters used for simulations are outlined in Table 3.
Using Table 1 in equation (4), the dynamics of TORA are as follows: Using the input and state transformation in (6) and (7), the normal form in equation (9) for the TORA becomes: where k 1 � (k/m 1 + m 2 ), k 2 � (km 2 e/(m 1 + m 2 ) 2 ). Te dynamics in (43a) show that f(z, η) � − k 1 z+ k 2 sin(η) and the design scenario 3 is applicable. We note that in equation (43a), k 1 > 0, and hence, the frst term − k 1 z on the right hand side is naturally helpful in the stabilization of the system. Terefore, we leave this term and choose the sliding manifold, in accordance with equations (20) and (29), as the following: with σ 2 � f(η) + a _ z and f(η) � k 2 sin(η). Applying the design procedure in Teorem 7, the control v in equation (30), in terms of the actual states (ξ 1 , _ ξ 1 , ξ 2 , _ ξ 2 ) of system (42) is calculated as follows: 10 Complexity

Furuta Pendulum (FP).
An FP is composed of two main components: a rotating arm and an inverted pendulum (see Figure 2(c)). Te FP has also been widely exercised by the research community due to its challenging dynamic nature [7,15,[48][49][50][51].
Te control problem is the stabilization of inverted pendulum at an unstable equilibrium (ξ 2 � 0). For this purpose the control input is a torque u which is applied to the rotating arm. Te parameters used for simulations are outlined in Table 4.

Swingup.
Similar to the case of TORA, the controller v will become singular if we take the pendulum from ξ 2 � π to ξ 2 � 0. Tus, a swingup control is designed here as well to take the pendulum to a vicinity of ξ 2 � 0. Te controller v in equation (50) with u in equation (10) can then be used as a balancing controller.
Since we cannot partially linearize ξ 2 − dynamics due to the singularity problem in the PFL control in equation (10), thus a partial linearization of the FP dynamics with respect to ξ 1 and evaluating € ξ 2 from equation (5b) gives the following expression: incorporation of this in equation (5a), the following PFL control is designed with a new control v swingup .
Te u PFL reduces the FP dynamics to the following form.
Once again choosing v swingup as a PD-like controller, with strictly positive gains K d and K p , will stabilize the above dynamics.
14 Complexity Te strategy is to use v swingup to swingup the pendulum and once ξ 2 some vicinity of the unstable equilibrium then the SMC based v (see equation (51)), is activated to stabilize the system. Te swingup from unstable to a stable equilibrium and robust stabilization is demonstrated in Figure 9.

Overhead Crane (OC).
An OC consists of a payload having mass m which is suspended with a rope of length L (the rope is assumed to be massless). An end of the rope is fxed with trolley of mass M (see Figure 2(d)). Some research highlights on OC can be found in [16,[52][53][54][55][56].
Te controller objective is to move the trolley under an action of control input u such that the payload is transported with precision and speed. Moreover, the swing should be minimal at the payload. Te parameters of the OC are given in Table 5.
Dynamics of the OC are extracted from equation (5) with the use of Table 1 as follows:  16 Complexity Using the input state transformations in (10) and (11), the normal form in equation (13) for the above dynamics is coined in the following: It may be noted that the above normal form representation is a clear case of design scenario 3.
Some realistic assumptions can be made, keeping in view practical use of OC.

Overview of the Simulation Results
A brief overview of the results, obtained above, is presented in this section. Figure 3, it is revealed that the proposed algorithm robustly stabilizes the pendulum at an unstable equilibrium in 2 seconds while the wheel comes to rest in approximately 5 sec. Te results are in agreement with [7]; however, a negligible efect of the sinusoidal disturbance is evident in the response.

IWP. In
In addition, the closed loop responses, with a swingup controller in the loop, are presented in Figures 4 and 5. It may be noticed in Figure 4 that the ξ 1 − dynamics have been stabilized but the ξ 2 − dynamics are unstable. Tis problem is shown to be solved in Figure 5 which shows swingup, from ξ 1 � π to ξ 1 � 0, and then robust stabilization with the proposed SMC algorithm in the presence of matched external disturbance.
In summary, the proposed control schemes robustly stabilize the inherent nonlinear dynamics by neglecting the sinusoidal disturbance. While doing this a noticeable fact is that the control action is still smooth and bounded. Figure 6. A robust stabilization and fnite-time convergence (approximately in 6 sec) are achieved. Te nonlinear benchmark specifcations [45] are met, i.e., boundedness of control efort (u < 0.04 N.m in our case) and stability in the closed loop. Te results are in agreement with [46] as well. Te robustness of the proposed algorithm is also demonstrated by introduction of a sinusoidal disturbance. It may be noted that the oscillations, evident in Figure 6, are of the order of 10 − 4 and hence can be considered negligible.  Figure 8. Tese results seem to stabilize the ξ 1 − dynamics but ξ 2 − dynamics are still unstable. Te issue is shown to be coped with in Figure 9, showing a successful swingup from ξ 2 � π to ξ 2 � 0. Moreover, the trajectories are then robustly stabilized by the proposed SMC controller ((51)) in the presence of an introduced sinusoidal matched disturbance. Figure 10 shows closed loop response of the Overhead Crane with SMC Controller (56) with parameters a � 0.2041, b � 0.1020, c � 0.5, c 0 � 0.5, Γ 1 � 0.1, and Γ 2 � 0.15. Te Crane transports the payload to the desired position ξ 1de s � 25 (m) in less than 15 seconds while the payload swing angle ξ 2 remains within the desired range of |ξ 2 | < (π/18) � 0.1745 radians, i.e., within 10 degrees. Small and even matched external disturbance d(t) � 0.25 sin (0.4πt) slightly disturbs the payload swing angle.

OC.
In brief, simulation results verify the enhanced control performance and robustness of the designed controllers. Moreover, the control action is considerably smooth. Both smoothness and robustness of control action are highly demanded for mechanical control systems operating under uncertainty conditions. Te results are in agreement with and improved to previous works while the design procedure is comprehensive and simple.

Conclusion
We presented a robust sliding mode control design framework for underactuated mechanical systems. Two classes of underactuated mechanical systems were considered. Using control input and state transformations, the dynamics were transformed either into a cascade strict feedback normal form or into a cascade nontriangular quadratic normal form with both forms consisting of a set of reduced-order nonlinear sub-systems and a reduced-order linear subsystems. Sliding manifold and sliding mode control were designed for the reduced-order nonlinear subsystems such that stability of the overall system is also guaranteed. Illustrative application examples of the IWP and TORA of the frst class and the Furuta Pendulum and the Overhead Crane of the second class were presented with numerical simulations. Simulation results verifed the high performance and robustness of the proposed control design strategy. Chattering in the control action, inherently associated with sliding mode but undesired for and practically not applicable in mechanical control systems, was Complexity 19 signifcantly minimized by the designed control laws. Simulation results also verifed the smoothness of the control action needed for mechanical control systems. Additionally, to overcome limitations of the designed sliding mode control laws, swingup control laws were designed in the case of the IWP and the Furuta Pendulum to swingup the unactuated pendulum from the downward stable equilibrium position and stabilize it at the upward unstable equilibrium position. Successful swingup and stabilization were demonstrated in the presence of external disturbance using the designed sliding mode control laws. Te possible future avenues include an implementation of the proposed algorithm on the practical test benches. In addition, an extension for systems with higher Degrees-of-Freedom (DoF) can also be considered.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.