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

Authors: Ancai Zhang; Jinhua She; Zhenxing Li; Guochen Pang; Zhi Liu; Jianlong Qiu
Journal: Complexity
 (2020)

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## Abstract

An underactuated horizontal spring-coupled two-link manipulator (UHSTM) is a nonlinear system that has one actuator and two degrees of freedom (DOF). This paper analyzes the nonlinear dynamics and develops a new global stabilization control method for this 2-DOF underactuated system. First, a set of suitable state variables are constructed to change the UHSTM system into a cascade nonlinear system. Second, a Lyapunov function is constructed based on the structural characteristics of the cascade system. A control law is designed to ensure the closed-loop control system to be Lyapunov stable. After that, the global asymptotical stability conditions are derived from the analysis of the closed-loop control system. Finally, numerical examples demonstrate the validity of the theoretical analysis results.


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## Body

## 1. Introduction

With the improvement of industrial automation, the manipulators have been widely used in many areas of engineering [1–4]. For a manipulator, the damage of actuators is an inevitable problem caused by the wearing of devices and the influence of the surrounding environment. The failure of actuators results in that the manipulator becomes an underactuated mechanical system [5–7], 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 [8–11]. 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 [13–15]. 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). The 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-input-disturbance (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. The 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). The 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. Therefore, the UHSTM has a good application prospect in industrial manufacturing, exploration of outer space, and other fields.Figure 1
Model of underactuated horizontal spring-coupled two-link manipulators (UHSTM).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. This 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.


## 2. Model and Nonlinear Dynamics Analysis of the UHSTM System

Figure1 shows the model of the UHSTM, where mi,Li, and Ji are the mass, the length, and the moment of inertia of the i-th link, respectively, i=1,2; Lc1 is the distance from the spring joint to the COM of the first link, and Lc2 is the distance from the active joint to the COM of the second link; qit is the rotational angle of the i-th link i=1,2; Ft 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 q1t=0. It is not difficult to get the kinetic and potential energy of the system as(1)Kq,q˙=12D11q2q˙12t+D12q2q˙1tq˙2t+12D22q˙22t,Pq=12kq12t,where qt=q1t,q2tT, q˙t=dqt/dt, and(2)D11q2=α1+α2+2α3cosq2t,D12q2=D21q2=α2+α3cosq2t,D22=α2,α1=m1Lc12+m2L12+J1,α2=m2Lc22+J2,α3=m2L1Lc2.We take the Lagrangian of the UHSTM system to beLq,q˙=Kq,q˙−Pq. The Euler–Lagrange motion equation of the system is obtained as(3a)ddt∂Lq,q˙∂q˙1−∂Lq,q˙∂q1=0,(3b)ddt∂Lq,q˙∂q˙2−∂Lq,q˙∂q2=Ft,where Ft is the control input applied to variable q2t, which is for the actuated part. It is equivalent to the following form:(4)D11q2D12q2D21q2D22q¨1tq¨2t+H1q,q˙H2q,q˙+kq1t0=0Ft,where(5)H1q,q˙=−α32q˙1t+q˙2tq˙2tsinq2t,H2q,q˙=α3q˙12tsinq2t.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 q2t and q˙2t 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 ∂Lq,q˙/∂q˙1 is equal to ∂Lq,q˙/∂q1. A simple calculation shows that ∂Lq,q˙/∂q1 is not related to q¨. So, we choose it as the third state variable of the system. Based on the expression of ∂Lq,q˙/∂q˙1 and D11q2>0, the fourth state variable of the system is chosen to be q1t+ξq2 in order to make the dynamic structure of the system in the state space be simple, where(6)ξq2=∫0q2tD12νD11νdν=∫0q2tα2+α3cosνα1+α2+2α3cosνdν=q2t2+α2−α1α1+α22−4α32arctanα1+α2−2α3α1+α2+2α3tanq2t2.In summary, the state variables of the UHSTM system are chosen to be(7)z1t=q1t+ξq2,z2t=∂Lq,q˙∂q˙1=D11q2q˙1t+D12q2q˙2t,z3t=q2t,z4t=q˙2t.Letyt=q1t,q2t,q˙1t,q˙2tT and zt=z1t,z2t,z3t,z4tT. It is easy to get(8)∂zt∂yt=1D12q2D11q2000−α3sinq2t2q˙1+q˙2D11q2D12q201000001=−D11q2<0.It means that the transformation fromyt to zt is homeomorphous. From (7), we have(9)z˙1t=q˙1t+dξq2dq2q˙2t=D11q2q˙1t+D12q2q˙2tD11q2=z2tD11z3,z˙2t=ddt∂Lq,q˙∂q˙1=∂Lq,q˙∂q1=−kq1t=−kz1t−ξq2,z˙3t=q˙2t=z4t,z˙4t=q¨2t.These give the state space equation of the UHSTM system as(10)z˙1t=z2tD11z3,z˙2t=−kz1t+kξz3,z˙3t=z4t,z˙4t=τt,where τt=q¨2t is the control input of system (10). From (4), it is not difficult to get(11)Ft=D21q2q¨1t+D22q¨2t+H2q,q˙,D11q2q¨1t+D12q¨2t+H1q,q˙+kq1t=0.Eliminating the termq¨1t from the above equations and considering τt=q¨2t yield(12)Ft=H2q,q˙−D21q2H1q,q˙+kq1tD11q2+Δq2D11q2τt,where Δq2=D11q2D22−D12q2D21q2>0. This is the relationship between the control inputs Ft and τt.The main concern of this paper is to design the control lawFt that globally stabilizes the UHSTM at yt=0. Note that yt=0 is equivalent to zt=0. So, the discussed stabilizing control objective is achieved so long as system (10) is globally stabilized at zt=0.


## 3. Design of Stabilizing Control Law

In this section, the stabilizing control of (10) at zt=0 is discussed. A control law τt 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(13)Vz=β12W2z+β22z32t+β32z42t,where βi>0i=1,2,3 are constants, and(14)Wz=kz1tz1t2−ξz3+z22t2D11z3.Note thatVz is a nonnegative function. It is easy to verify that Vz=0 when zt=0. In addition, Vz=0 gives Wz=0, z3t=0, and z4t=0 because βii=1,2,3 are positive constants. In this case, it follows from (14) that(15)Wz=kz12t2+z22t2D110=0.Equation (15) means that z1t=0 and z2t=0 since k>0 and D110>0. So, we get that zt=0 when Vz=0. In summary, Vz=0 is equivalent to zt=0. Thus, Vz is a typical positive-definite Lyapunov function for system (10). Differentiating Wz along (10) gives(16)dWzdt=dWzdz1dz1tdt+dWzdz2tdz2tdt+dWzdz3dz3tdt=kz1t−ξz3z2tD11z3+z2tD11z3−kz1t+kξz3+α3z22tsinz3tD112z3−kz1tD12z3D11z3z4t=Λzz4t,where(17)Λz=α3z22tsinz3t−kz1tD12z3D11z3D112z3.It follows from (10), (13), and (16) that(18)dVzdt=β1WzdWzdt+β2z3tz4t+β3z4tτt=β1WzΛz+β2z3t+β3τtz4t.The control law is designed to be(19)τt=−β1WzΛz−β2z3t−γz4tβ3,where γ>0 is a constant. Combining (18) and (19) gives(20)dVzdt=−γz42t≤0.SinceVz is positive definite and V˙z≤0, closed-loop control systems (10) and (19) are Lyapunov stable.


## 4. Asymptotical Stability Analysis of the Closed-Loop Control System

In order to get the final stabilizing state of closed-loop systems (10) and (19), the asymptotical stability of the system is analyzed by using LaSalle’s invariance principle. Letting V˙z≡0 gives x4t=0. It follows from the third and fourth equations of (10) that z3t=z3∗ and τt=0, where z3∗ is a constant. The further stability analysis results are summarized in the following two theorems.Theorem 1.
Ifz3t=z3∗=0, then V˙z≡0 gives zt=0.Proof.
Fromz3t=0, z4t=0, τt=0, and (19), it is easy to obtain z1tWz=0. Thus, z1t=0 or Wz=0. If z1t=0, then the first equation of (10) gives z2t=0. If Wz=0, then combining z3t=0, (14), and (15) gives z1t=z2t=0. So, V˙z≡0 gives zt=0. The proof is completed.Theorem 2.
Ifz3t=z3∗≠0 and α2≠α3, then V˙z≡0 gives z2t=0, z1t=ξz3∗, and(21)β1k2ξ3z3∗α2+α3cosz3∗2α1+α2+α3cosz3∗+β2z3∗=0.Proof.
Combiningz4t=0, τt=0, (17), and (19) yields(22)β1Wzα3z22tsinz3∗−kz1tD12z3∗D11z3∗+β2D112z3∗z3∗=0.
SinceV˙z≡0 means that Vz equals to a constant, it follows from z3t=z3∗, z4t=0, and (13) that Wz=W∗, where W∗ is a constant. From D11z3∗>0, z3∗≠0, and (22), it is easy to get that W∗≠0. As a result, (22) gives(23)α3z22tsinz3∗−kz1tD12z3∗D11z3∗=C,where C is a non-zero constant. Differentiating (23) along (10) yields(24)2α3z1t−ξz3∗sinz3∗+D12z3∗z2t=0.
We continue carrying out the analysis in the following two situations.Case 1.
z
2
t
≠
0
: it follows from (24) that(25)2α3z1t−ξz3∗sinz3∗+D12z3∗=0.
Ifsinz3∗=0, then (25) gives D12z3∗=0, i.e., α2+α3cosz3∗=0. As a result, sinz3∗=0 and α2+α3cosz3∗=0 are contradictory because α2>0, α3>0, and α2≠α3. Thus, it concludes that sinz3∗≠0. So, (25) gives(26)z1t=−D12z3∗2α3sinz3∗+ξz3∗,which is a constant. It follows from the first equation of (10) that z2t=0. This is a contradiction.Case 2.
z
2
t
=
0
: the first and second equations of (10) give that z1t=z1∗ and z1∗=ξz3∗, where z1∗ is a constant. Substituting z2t=0 and z1∗=ξz3∗ into (14) yields Wz=−kξ2z3∗/2. Combining Wz and (22) gives(27)β1k2D12z3∗ξ3z3∗2D11z3∗+β2z3∗=0.
It is equivalent to equation (21). The proof is completed.
Based on the analysis results in Theorem1 and Theorem 2, V˙z≡0 gives(28)Ω=zt=0orzt=ξz3∗,0,z3∗,0,where z3∗ 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 z3∗. To solve this problem, a condition on the control parameters β1 and β2 is presented, which ensures (21) has only one solution for z3∗. The main result is given in the following theorem.Theorem 3.
If the control parametersβ1 and β2 satisfy(29)β2>k2α3α2−α1α2+α33π32α1+α2−2α35β1,then equation (21) has only one solution z3∗=0. And thus, closed-loop control systems (10) and (19) asymptotically converge to zt=0.Proof.
Note thatz3t=q2t 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≤z3t<π (mod 2π) in this paper. Based on (21), an auxiliary function is designed to be(30)fx=β1gxα1+α2+2α3cosx+β2x,x∈−π,π,where(31)gx=k2ξ3xα2+α3cosx2,ξx=∫0xα2+α3cosνα1+α2+2α3cosνdν.
It is not difficult to obtain(32)ξx≤α2+α3α1+α2−2α3x≤α2+α3πα1+α2−2α3,(33)dgxdx=k2ξ2x23α2+α3cosx2α1+α2−2α3cosx−α3ξxsinx.
Combining (30) and (33) yields(34)dfxdx=β1dgx/dxα1+α2+2α3cosx+2α3gxsinxα1+α2+2α3cosx2+β2=β2+β1k22α3α2−α1ξ3xsinxα1+α2+2α3cosx2+3ξ2xα2+α3cosx2α1+α2+2α3cosx2≥β2+β1k2α3α2−α1ξ3xsinx2α1+α2+2α3cosx2.
It follows from (32) that(35)α2−α1ξ3xsinxα1+α2+2α3cosx2≤α2−α1ξx3α1+α2−2α32≤α2−α1α2+α33π3α1+α2−2α35.
Combining (29), (34), and (35) yields dfx/dx>0. It means that fx is a strictly monotonically increasing function during x∈−π,π. By the fact that f0=0, it is easy to conclude that fx=0 has only one solution x=0. In other words, condition (29) ensures that (21) has only one solution z3∗=0. Substituting z3∗=0 into (28) yields that Ω contains only one element zt=0. So, systems (10) and (19) asymptotically converge to zt=0. The proof is completed.
From the above statements, we get that the control lawτt in (19) globally stabilizes system (10) at zt=0 so as to both conditions, α2≠α3 and (29), are satisfied. As a result, the control law Ft obtained by substituting τt into (12) globally stabilizes the UHSTM at yt=0 under these conditions.


## 5. Numerical Examples

This section presents numerical examples to verify the effectiveness of the above theoretical results.The physical parameters of the UHSTM presented in [26] were chosen for simulation. They are listed in Table 1. The control parameters in (19) were chosen to be β1=5, β2=5, β3=0.1, and γ=1. A simple calculation gives(36)α1=1.333,α2=1.33,α3=1,k2α3α2−α1α2+α33π32α1+α2−2α35β1=0.2296.Table 1
Parameters of the underactuated horizontal spring-coupled two-link manipulator (UHSTM).
m
1
kg
m
2
kg
L
1
m
L
2
m
l
1
m
1
1
1
2
0.5
l
2
m
J
1
kg
m
2
J
2
kg
m
2
k
N
/
m
1
8.3
×
10
−
2
0.33
0.1So, both conditions,α2≠α3 and (29), are satisfied. The simulation results of the UHSTM with an initial condition,(37)q10,q20,q˙10,q˙20T=π2,−π,0,0T,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.,(38)q10,q20,q˙10,q˙20T=−1,2,0,0T.Figure 2
Time response ofqit and q˙it i=1,2 with initial condition (37).
(a)
(b)
(c)
(d)The simulation results with (38) show that our method is still effective (see Figure 3). The stabilizing time is less than 1.5 seconds. In contrast, the stabilizing time of the UHSTM in [26] is greater than 5 seconds. This shows the superiority of our presented method.Figure 3
Time response ofqit and q˙it i=1,2 with initial condition (38).
(a)
(b)
(c)
(d)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 mi, Lci, and Ji, i=1,2), white noises (peak value: ±0.15) in measurements, and the viscous friction disturbances in the joints. The results show that our method is still effective under these conditions. We give the simulation results in Figure 4 for the typical case, where m1, Lc1, and J1 are 5% smaller than their nominal values; m2, Lc2, and J2 are 5% larger than their nominal values; the peak value of white noise is ±0.1; and the friction disturbances are fv1t=−0.15q˙1t and fv2t=−0.25q˙1t in the spring joint and the active joint, respectively. Note that the UHSTM can still be stabilized at the origin. The robustness of our method is good.Figure 4
Time response ofqit and q˙it i=1,2 with initial condition (38) when parameter uncertainties, white noises, and friction exist.
(a)
(b)
(c)
(d)


## 6. Conclusions

This 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. Then, 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.



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