Nonlinear Global Stabilization Control for the Underactuated WAcrobot System

A WAcrobot is an underactuated nonlinear system that has three degrees of freedom (DOF) and two inputs. (is paper discusses the global stabilization control problem for this 3-DOF underactuated system. A new control strategy is developed to solve this problem. (e strategy first changes the 3-DOF WAcrobot system to be a 2-DOF reduced-order model in finite time. (is transforms the stabilizing control of the WAcrobot system into that of the reduced-order model. After that, nonsingular control laws that globally stabilize the reduced-order model at the origin are designed. It guarantees the stabilizing control objective of the WAcrobot to be achieved. Finally, a simulation experimental example demonstrates the validity of the presented theoretical results. Simulation results show the advantage of our strategy over others.


Introduction
In our daily life, many natural systems essentially have nonlinear characteristics. It is meaningful to discuss the control problem for the nonlinear systems [1][2][3][4][5]. An underactuated mechanical system is a typical example of nonlinear systems, which has fewer control inputs than the number of systems' degrees of freedom (DOF). Compared with the fully actuated system, the underactuated system has the characteristics of light weight, low energy consumption, and flexible movement because of the reduction of actuators. Such systems can be widely used in health care, space exploration, transportation, military, and other fields [6][7][8][9]. However, the reduction of inputs makes the system possess nonlinear constraints. And the constraints are usually second-order nonholonomic [10]. is means that the states of the system are in uncontrollable manifolds of the configuration space. e control design of an underactuated mechanical system is challenging in the nonlinear control area. Many researchers have devoted their efforts to the study on underactuated systems in the past few years [11][12][13][14][15].
Since the 1990s, the control problems for the simplest underactuated system (that is, the 2-DOF underactuated system) have been attracting much attention. In order to explore the nonlinear control strategy, many experimental models of 2-DOF underactuated systems have been presented in the laboratory. e models include Acrobot [16], Pendubot [17], TORA [18], and Pendulum-cart [19]. e stabilization control issue is a commonly addressed problem for these 2-DOF underactuated mechanical systems. And many stabilizing control methods have been developed, e.g., a partial feedback linearization method [20], an energybased method [21], a nonsmooth Lyapunov function method [22], an equivalent-input disturbance method [23], and a trajectory tracking strategy [24].
With the deepening of the research work, the control problems presented by an n-DOF (n ≥ 3) underactuated system need to be further explored since multi-DOF systems are more consistent with the reality of natural systems.
However, this problem is not easy to solve because the nonlinear dynamics and constraint equations of underactuated systems become more complicated with the increase of the DOF. To solve the control problems for multi-DOF underactuated systems, it is necessary to study the simplest case, i.e., n � 3. Recently, some 3-DOF underactuated system models have been presented. eir dynamic analysis and control design have been intensively discussed [25][26][27][28]. Among them, a WAcrobot model (see Figure 1) is a typical example. is mechanical system has a wheel and a double inverted pendulum. e wheel rolls in a horizontal plane driven by an actuator. And the double pendulum freely rotates in a vertical plane driven by an actuator at the second joint. e first joint of the pendulum is passive. Note that the WAcrobot is a 3-DOF underactuated system, and it is the combination of a wheel and an Acrobot. e WAcrobot has a good application prospect in the mobile robotic area. It is a perfect test bed for illustrating a nonlinear control algorithm for multi-DOF underactuated systems. So, it is meaningful to study the problems presented by this mechanical system. In [29], the swing-up and stabilization control of the WAcrobot were discussed. e authors used noncollocated feedback linearization and linear quadratic regulator technique to achieve the system's control objective. In addition to this paper, there are no related research results about this system. Since the WAcrobot possesses common inherent nonlinear features of multi-DOF underactuated systems, it is necessary to explore more control methods for it. is inspires our study in this paper. e global stabilization control problem for the WAcrobot is addressed in this paper. By using the inherent dynamic coupling relationship, we first change the WAcrobot into a 2-DOF reduced-order system in finite time. And then, the global stabilization of the reduced-order model is discussed. Stabilizing control laws are developed, and the condition that avoids the singularity in the control law is also presented. Finally, numerical experiments demonstrate the validity of the theoretical analysis results. Simulation results show the advantage of our strategy over the results in [29]. Moreover, the presented strategy gives a good guidance for the solution of motion control of other multi-DOF underactuated mechanical systems.

Dynamic Motion Equations of the WAcrobot System
For the physical model of a WAcrobot system shown in Figure 1, θ 1 (t), m 1 , J 1 , and L 1 are the rotational angle, the mass, the moment of inertia, and the radius of the wheel, respectively. For i � 2, 3, θ 1 (t), m i , J i , and L i are the rotational angle, the mass, the moment of inertia, and the length of the ith pendulum, respectively, and L ci is the distance from the ith joint to the center of mass (COM) of the ith pendulum. Moreover, τ 1 (t) and τ 2 (t) are the input torques applied on the wheel and the 2nd pendulum, respectively, and g is the gravity acceleration constant ( ≈ 9.81 m/s 2 ). We take the plane in which the center of the circle is located on as the zero potential energy reference plane. A simple calculation gives the potential energy of the WAcrobot as In addition, the kinetic energy of the system is where e Lagrangian function of the WAcrobot system is taken to be L(θ, _ θ) � K(θ, _ θ) − P(θ). We get the Euler-Lagrange motion equations of the system as ese dynamic equations can be rewritten in the following form: where to be the total energy of the WAcrobot. From (5), we easily obtain Let It follows from (9) that both Ψ 3 (x, z) and

Reduced-Order Model of the WAcrobot System
e control objective discussed in this paper is to design controllers τ 1 and τ 2 to globally stabilize WAcrobot system (8) at [x, z] ⊤ � 0. Note that the nonlinear dynamics of (8) is very complicated. In order to simplify the structure of system (8), we design where W z 1 , z 2 � −r 2 z 5/3 2 + r 5/3 1 z 1 and r 1 and r 2 are positive constants. is is a relationship between control torques τ 1 and τ 2 . Substituting (11) into (8) yields It is clear that (13a) and (13b) are a nonlinear cascade system, and the state variable z is separated from others. According to the results in [30], the finite-time stabilization of subsystem (13b) at z � 0 is achieved. As a result, system (13a) and (13b) becomes the following form in finite time _ where From (10), it is easy to obtain g 1 (x) > 0 for x ∈ R 4 . In fact, nonlinear system (14) is the zero dynamics of cascade system (13a) and (13b). In other words, (14) is a reduced-order system of the WAcrobot. e physical model of (12) is shown in Figure 2. In order to achieve the control objective of the WAcrobot, it is necessary to design a controller τ 1 such that system (14) is stabilized at x � 0.

Design of the Stabilizing Controller for the Reduced-Order Model
is section discusses the design of a stabilizing controller τ 1 for reduced-order system (14) under the condition z � 0.
us, the state variable x in Ω can enter any arbitrarily small neighbourhood area of x � 0. In this area, system (14) approximates the following linear system: where where Figure 2: Physical model of (14).
where R > 0 and Q ∈ R 4×4 is a positive definite matrix. From the quadratic optimal control theory, the control law stabilizes linear system (22) at x � 0 quickly. From the above statements, we get that the use of the control law τ 1 in (18) and (27) in sequence guarantees the global stabilization of reduced-order system (14) at x � 0.
Remark 1. Note that the control law τ 1 in (18) is singular when is case occurs only when E(x) < E(0) (i.e., E(x) < 0) because μ 1 > 0, μ 3 > 0, and (1) and (2) that So, we have From (29), it is easy to obtain that the singularity of τ 1 in (18) does not occur if We select the control parameters such that is condition guarantees (30) to hold. In other words, the control law in (18) is not singular under condition (31).

Remark 2.
ere are eight control parameters in this paper, i.e., r 1 and r 2 in (12); μ 1 , μ 2 , and μ 3 in (16); c in (19); and R and Q in (26). In order to make the control design process clear, a selected guideline for these parameters is given as follows: (1) An optimal set of r 1 and r 2 is selected for (12) that makes the stabilization time of (13b) to be smallest (2) For fixed c, a set of μ 1 , μ 2 , and μ 3 is chosen to make reduced-order system (14) enter an arbitrarily small neighbourhood area of x � 0 (3) A set of R and Q is selected for the control law τ 1 that stabilizes system (22) at the origin

Numerical Example
A numerical example is presented in this section in order to verify the validity of our presented theoretical results. In the simulation experiment, the physical parameters of the WAcrobot were chosen to be [29] e initial condition was selected to be e physical meaning of (33) is that the double inverted pendulum of the WAcrobot begins to move from the straight-down position with a small velocity, while the wheel is stationary at the origin. e control parameters in (12) were selected to be r 1 � 1 and r 2 � 5. e simulation results of subsystem (13b) are shown in Figure 3. Note that θ 3 and θ . 3 converge to zero in 0.25 seconds. It means that WAcrobot system (8) changes to reduced-order system (14) quickly.
is demonstrates the validity of the presented theoretical results in Section 3.
To show the validity of the stabilizing controller for reduced-order model (14), the parameters in (16) and (18) were chosen as Mathematical Problems in Engineering where I 4 is a 4 × 4 identity matrix. A simple calculation gives ρ � 141.25 and 2μ 1 E(0)╱ρ � 14.147. us, condition (31) is satisfied. Figure 4 shows the simulation results for x(t) by controllers (11) and (18). It is clear that θ 1 and _ θ 1 are stabilized at zero while θ 2 and θ . 2 are in a homoclinic orbit. is shows the validity of the theoretical results in Section 4. When x(t) enters the small neighbourhood area of x(t) � 0 at t � 30 s, the controller τ 1 switches to (27) from (18). e simulation results of controllers and x(t) are shown in Figures 5 and 6, respectively. Note that the switch of τ 1 from (18) to (27) causes small sudden changes in θ 1 and θ . 1 at t � 30 s. Moreover, it is clear that the WAcrobot can be quickly stabilized at the origin in 40 seconds, and the maximal value of inputs is less than 1.1 Nm. Compared with the results in [29], the value of inputs in this paper is more smaller, and the motion process of the WAcrobot system is more smooth and natural. is provides guarantee for the safe and stable operation of the control system. All these    (11) and (18). 6 Mathematical Problems in Engineering demonstrate the advantages of our presented control strategy.

Conclusion
is paper addressed the global stabilization of a 3-DOF underactuated WAcrobot system. A novel stabilizing control strategy was developed. First, the 3-DOF WAcrobot was changed to be a 2-DOF reduced-order model in finite time.
is makes the stabilizing control of the WAcrobot be easy to handle. And then, the stabilizing control laws were designed that globally stabilize the reduced-order system at the origin. And the condition that avoids the singularity in the stabilizing control law was also presented. Finally, the numerical simulation example showed the validity of the proposed theoretical results. In the future, we will extend the new strategy to the stabilization of other multi-DOF underactuated systems [31] and will also further explore other stabilizing control methods for the WAcrobot system.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Acknowledgments
is work was supported in part by the Natural Science Foundation Program of Shandong Province under Grant no.