Nonlinear Dynamics Analysis and Global Stabilization of Underactuated Horizontal Spring-Coupled Two-Link Manipulator

School of Automation and Electrical Engineering, Linyi University—Key Laboratory of Complex Systems and Intelligent Computing in Universities of Shandong, Linyi, Shandong 276000, China School of Engineering, Tokyo University of Technology, Hachioji, Tokyo 192-0982, Japan School of Automation, China University of Geosciences, Wuhan, Hubei 430074, China Department of Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia


Introduction
With the improvement of industrial automation, the manipulators have been widely used in many areas of engineering [1][2][3][4]. For a manipulator, the damage of actuators is an inevitable problem caused by the wearing of devices and the influence of the surrounding environment. e failure of actuators results in that the manipulator becomes an underactuated mechanical system [5][6][7], in which the system has less number of actuators than degrees of freedom (DOF). Since an underactuated manipulator consumes less energy, it has a simple structure and is lightweight, and it also has good potential applications in many fields. It is meaningful to study the control of this kind of mechanical system [8][9][10][11]. However, this problem is challenging due to the complex inherent nonlinear dynamics and nonholonomic constraints possessed by this kind of system [12].
Over the past few years, a great deal of efforts have been made on the control of underactuated manipulators [13][14][15]. Among them, a two-link manipulator that has only one actuator is the simplest example. According to whether the manipulator is affected by gravity or not, the underactuated two-link manipulators are divided into two categories: underactuated vertical two-link manipulators (UVTM) and underactuated horizontal two-link manipulators (UHTM). e UVTM moves in a vertical plane [16], which includes Acrobot [17] and Pendubot [18]. Many control strategies have been presented to solve the stabilizing control of the UVTM, e.g., a partial feedback linearization method [19,20], a fuzzy control method [21], an energy-based method [22,23], and an equivalent-inputdisturbance (EID) method [24]. In contrast, the UHTM moves in a horizontal plane and is not affected by the gravity. It is clear that the UVTM becomes the UHTM in a weightless environment. Although the UHTM has the same mechanical configuration as the UVTM, they have quite different dynamic properties. e absence of gravity makes the UHTM system possess some unique properties. For example, any point in the motion space is an equilibrium point of the UHTM, and the system is not local controllable and even not small-time local controllable around the equilibrium point [25]. All these make the control of the UHTM be a very difficult problem.
In order to solve this control problem, some attempts have been made to find a gravity substitute for the UHTM. In [26], a spring was added around the passive joint of the UHTM, and an underactuated horizontal spring-coupled two-link manipulator (UHSTM) was constructed (see Figure 1). e UHSTM uses the spring to create a source of potential energy. It makes the control of the underactuated horizontal manipulator be easy to solve. In addition, the cost of a spring is much lower than an actuator, and the elasticity of the spring does not disappear in a weightless environment. erefore, the UHSTM has a good application prospect in industrial manufacturing, exploration of outer space, and other fields.
Currently, there are few results about the control of the UHSTM system. In order to promote its application in the engineering areas, it is necessary to analyze the dynamics and to develop more control methods for this mechanical system. is is the main motivation of this study. In this paper, a set of suitable state variables are selected for the UHSTM system based on its nonlinear dynamics. It changes the UHSTM into a cascade nonlinear system. And then, the global stabilization control problem for the cascade system is concerned. Based on the structural characteristics of the cascade system, a positive-definite Lyapunov function is constructed. A control law is designed to guarantee the closed-loop control system to be Lyapunov stable. After that, the asymptotical stability of the control system is discussed by LaSalle's invariance principle. Moreover, the conditions that ensure the system to globally converge to the origin are presented. Finally, numerical examples verified the effectiveness of the presented theoretical results. Figure 1 shows the model of the UHSTM, where m i , L i , and J i are the mass, the length, and the moment of inertia of the i-th link, respectively, (i � 1, 2); L c1 is the distance from the spring joint to the COM of the first link, and L c2 is the distance from the active joint to the COM of the second link; q i (t) is the rotational angle of the i-th link (i � 1, 2); F(t) is the input torque applied on the active joint; and k is the elastic coefficient of the spring. Assume that the spring is fully relaxed when q 1 (t) � 0. It is not difficult to get the kinetic and potential energy of the system as

Model and Nonlinear Dynamics Analysis of the UHSTM System
where q(t) � [q 1 (t), q 2 (t)] T , _ q(t) � dq(t)/dt, and D 11 q 2 � α 1 + α 2 + 2α 3 cos q 2 (t), We take the Lagrangian of the UHSTM system to be L(q, _ q) � K(q, _ q) − P(q). e Euler-Lagrange motion equation of the system is obtained as d dt where F(t) is the control input applied to variable q 2 (t), which is for the actuated part. It is equivalent to the following form: where Active joint Center of mass (COM) k It is clear that (3a) and (3b) is a complicated nonlinear system. In order to make the control design of the system be easy to solve, it is necessary to choose a set of state variables to change the system into the state space form. Note that q 2 (t) and _ q 2 (t) are actuated variables of the UHSTM system. As the partial feedback linearization method developed in [20], we choose them to be two state variables of the system. In addition, it follows from (3a) that the first derivative of So, we choose it as the third state variable of the system. Based on the expression of zL(q, _ q)/z _ q 1 and D 11 (q 2 ) > 0, the fourth state variable of the system is chosen to be q 1 (t) + ξ(q 2 ) in order to make the dynamic structure of the system in the state space be simple, where In summary, the state variables of the UHSTM system are chosen to be It means that the transformation from y(t) to z(t) is homeomorphous. From (7), we have ese give the state space equation of the UHSTM system as where τ(t) � € q 2 (t) is the control input of system (10). From (4), it is not difficult to get Eliminating the term € q 1 (t) from the above equations and considering where Δ(q 2 ) � D 11 (q 2 )D 22 − D 12 (q 2 )D 21 (q 2 ) > 0. is is the relationship between the control inputs F(t) and τ(t). e main concern of this paper is to design the control law F(t) that globally stabilizes the UHSTM at y(t) � 0. Note that y(t) � 0 is equivalent to z(t) � 0. So, the discussed Complexity stabilizing control objective is achieved so long as system (10) is globally stabilized at z(t) � 0.

Design of Stabilizing Control Law
In this section, the stabilizing control of (10) at z(t) � 0 is discussed. A control law τ(t) is designed by using Lyapunov stability theory. Note that (10) is a cascade nonlinear system. Based on the structural characteristics of this system, a function is constructed to be where β i > 0(i � 1, 2, 3) are constants, and Note that V(z) is a nonnegative function. It is easy to verify that V(z) � 0 when z(t) � 0. In addition, V(z) � 0 gives W(z) � 0, z 3 (t) � 0, and z 4 (t) � 0 because β i (i � 1, 2, 3) are positive constants. In this case, it follows from (14) that Equation (15) means that z 1 (t) � 0 and z 2 (t) � 0 since k > 0 and D 11 (0) > 0. So, we get that z(t) � 0 when V(z) � 0. In summary, V(z) � 0 is equivalent to z(t) � 0. us, V(z) is a typical positive-definite Lyapunov function for system (10). Differentiating W(z) along (10) gives where It follows from (10), (13), and (16) that e control law is designed to be where c > 0 is a constant. Combining (18) and (19) gives Since V(z) is positive definite and _ V(z) ≤ 0, closed-loop control systems (10) and (19) are Lyapunov stable.
Case 2. z 2 (t) � 0: the first and second equations of (10) give that z 1 (t) � z * 1 and z * (22) gives It is equivalent to equation (21). e proof is completed. Based on the analysis results in eorem 1 and eorem 2, where z * 3 is the solution of (21). According to LaSalle's invariance theorem [27], closed-loop control systems (10) and (19) asymptotically converge to the largest invariant set contained in Ω.
In order to clearly describe the stabilizing state of the control system, it is necessary to solve (21). Note that (21) is a complicated transcendental equation. It is difficult to obtain an analytical expression for z * 3 . To solve this problem, a condition on the control parameters β 1 and β 2 is presented, which ensures (21) has only one solution for z * 3 . e main result is given in the following theorem.
Proof. Note that z 3 (t) � q 2 (t) stands for the angle position of the second link of the UHSTM (see Figure 1). It is a cyclic variable with a period 2π. So, we can assume that −π ≤ z 3 (t) < π (mod 2π) in this paper. Based on (21), an auxiliary function is designed to be It is not difficult to obtain Combining (30) and (33) yields It follows from (32) that Combining (29), (34), and (35) yields df(x)/dx > 0. It means that f(x) is a strictly monotonically increasing function during x ∈ [−π, π). By the fact that f(0) � 0, it is easy to conclude that f(x) � 0 has only one solution x � 0.
From the above statements, we get that the control law τ(t) in (19) globally stabilizes system (10) at z(t) � 0 so as to both conditions, α 2 ≠ α 3 and (29), are satisfied. As a result, the control law F(t) obtained by substituting τ(t) into (12) globally stabilizes the UHSTM at y(t) � 0 under these conditions.

Numerical Examples
is section presents numerical examples to verify the effectiveness of the above theoretical results. e physical parameters of the UHSTM presented in [26] were chosen for simulation. ey are listed in Table 1. e control parameters in (19) were chosen to be β 1 � 5, β 2 � 5, β 3 � 0.1, and c � 1. A simple calculation gives So, both conditions, α 2 ≠ α 3 and (29), are satisfied. e simulation results of the UHSTM with an initial condition, are shown in Figure 2. Note that the UHSTM is quickly stabilized at the origin. For the sake of comparison, we also carried out the simulation for the UHSTM with the same initial condition as in [26], i.e., e simulation results with (38) show that our method is still effective (see Figure 3). e stabilizing time is less than 1.5 seconds. In contrast, the stabilizing time of the UHSTM in [26] is greater than 5 seconds. is shows the superiority of our presented method.
In order to verify the practicality of our method in a real environment, the simulations were carried out by considering parameter uncertainties ( ± 5% of their nominal values for m i , L ci , and J i , i � 1, 2), white noises (peak value: ± 0.15) in measurements, and the viscous friction disturbances in the joints. e results show that our method is still effective under these conditions. We give the simulation results in Figure 4 for the typical case, where m 1 , L c1 , and J 1 are 5% smaller than their nominal values; m 2 , L c2 , and J 2 are 5% larger than their nominal values; the peak value of white noise is ± 0.1; and the friction disturbances are f v1 (t) � −0.15 _ q 1 (t) and f v2 (t) � −0.25 _ q 1 (t) in the spring joint and the active joint, respectively. Note that the UHSTM can still be stabilized at the origin. e robustness of our method is good.

Conclusions
is paper concerned the nonlinear dynamics analysis and the global stabilization control of a 2-DOF underactuated system called UHSTM. A set of suitable state variables were constructed that change the system into a cascade nonlinear system. en, a stabilizing control law was designed for the cascade system based on the Lyapunov theory. After that, the conditions were presented in order to guarantee the global asymptotical stability of the closed-loop control system at the origin. Numerical examples verified the effectiveness of the method. In the future, we will further study how to   extend the method to stabilize the underactuated horizontal multilink manipulator and other underactuated nonlinear systems.

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.