# Nonlinear Global Stabilization Control for the Underactuated WAcrobot System

Authors: Shuli Gong; Ancai Zhang; Zhi Liu; Zhenxing Li; Chengdong Yang; Xinghui Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A WAcrobot is an underactuated nonlinear system that has three degrees of freedom (DOF) and two inputs. This paper discusses the global stabilization control problem for this 3-DOF underactuated system. A new control strategy is developed to solve this problem. The strategy first changes the 3-DOF WAcrobot system to be a 2-DOF reduced-order model in finite time. This 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.


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## Body

## 1. Introduction

In our daily life, many natural systems essentially have nonlinear characteristics. It is meaningful to discuss the control problem for the nonlinear systems [1–5]. 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 [6–9]. However, the reduction of inputs makes the system possess nonlinear constraints. And the constraints are usually second-order nonholonomic [10]. This means that the states of the system are in uncontrollable manifolds of the configuration space. The 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 [11–15].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. The models include Acrobot [16], Pendubot [17], TORA [18], and Pendulum-cart [19]. The 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 energy-based 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 ann-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. Their dynamic analysis and control design have been intensively discussed [25–28]. Among them, a WAcrobot model (see Figure 1) is a typical example. This mechanical system has a wheel and a double inverted pendulum. The 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. The 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.Figure 1
Physical model of the WAcrobot system.The 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. The 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. This inspires our study in this paper.The 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.


## 2. Dynamic Motion Equations of the WAcrobot System

For the physical model of a WAcrobot system shown in Figure1, θ1t, m1, J1, and L1 are the rotational angle, the mass, the moment of inertia, and the radius of the wheel, respectively. For i=2,3, θ1t, mi, Ji, and Li are the rotational angle, the mass, the moment of inertia, and the length of the ith pendulum, respectively, and Lci is the distance from the ith joint to the center of mass (COM) of the ith pendulum. Moreover, τ1t and τ2t are the input torques applied on the wheel and the 2nd pendulum, respectively, and g is the gravity acceleration constant (≈9.81m/s2).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(1)Pθ=β1cosθ2+β2cosθ2+θ3,β1=m2Lc2+m3L2g,β2=m3L3g,where θ=θ1,θ2,θ3⊤. In addition, the kinetic energy of the system is(2)Kθ,θ˙=12θ˙⊤Mθθ˙,where(3)Mθ=M11M12θM13θM21θM22θM23θM31θM32θM33,M11=α1,M22θ=α2+α4+2α6cosθ3,M33=α4,M12θ=M21θ=α3cosθ2+α5cosθ2+θ3,M13θ=M31θ=α5cosθ2+θ3,M23θ=M32θ=α4+α6cosθ2,α1=m1+m2+m3L12+J1,α2=m2Lc22+J2+m3L22,α3=m2Lc2+m3L2L1,α4=m3Lc32+J3,α5=m3L1Lc3,α6=m3L2Lc3.The Lagrangian function of the WAcrobot system is taken to beLθ,θ˙=Kθ,θ˙−Pθ. We get the Euler–Lagrange motion equations of the system as(4)ddt∂Lθ,θ˙∂θ.1−∂Lθ,θ˙∂θ1=τ1,ddt∂Lθ,θ˙∂θ˙2−∂Lθ,θ˙∂θ2=0,ddt∂Lθ,θ˙∂θ˙3−∂Lθ,θ˙∂θ3=τ2.These dynamic equations can be rewritten in the following form:(5)Mθθ¨+Hθ,θ˙+Gθ=τ,where(6)Hθ,θ˙=H1θ,θ˙H2θ,θ˙H3θ,θ˙,Gθ=0G2θG3θ,τ=τ10τ2,H1θ,θ˙=−α3θ˙22sinθ2−α5θ˙2+θ˙32sinθ2+θ3,H2θ,θ˙=−α62θ˙2+θ˙3θ˙3sinθ3,H3θ,θ˙=α6θ˙22sinθ3,G2θ=−β1sinθ2−β2sinθ2+θ3,G3θ=−β2sinθ2+θ3.DenoteΞθ,θ˙=Kθ,θ˙+Pθ to be the total energy of the WAcrobot. From (5), we easily obtain(7)Ξ˙θ,θ˙=θ˙⊤τ=θ˙1τ1+θ˙3τ2.Letx1=θ1, x2=θ.1, x3=θ2, x4=θ.2, z1=θ3, z2=θ.3, x=x1,x2,x3,x4⊤, and z=z1,z2⊤. Then, it is not difficult to obtain the state space form of (5) as(8)x˙1=x2,x˙2=Υ1x,z+Φ1x,zτ1+Ψ1x,zτ2,x˙3=x4,x˙4=Υ2x,z+Φ2x,zτ1+Ψ2x,zτ2,z˙1=z2,z˙2=Υ3x,z+Φ3x,zτ1+Ψ3x,zτ2,where(9)Υ1x,zΥ2x,zΥ3x,z=M−1θ−H1θ,θ˙−H2θ,θ˙−G2θ−H3θ,θ˙−G3θ,Φ1x,zΨ1x,zΦ2x,zΨ2x,zΦ3x,zΨ3x,z=M−1θ100001.It follows from (9) that both Ψ3x,z and Φ1x,zΨ3x,z−Φ3x,zΨ1x,z are order principal determinants of M−1θ. Since M−1θ is a positive definite matrix, we get(10)Ψ3x,z>0,Φ1x,zΨ3x,z−Φ3x,zΨ1x,z>0.


## 3. Reduced-Order Model of the WAcrobot System

The control objective discussed in this paper is to design controllersτ1 and τ2 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(11)τ2=Wz1,z2−ϒ3x,z−Φ3x,zτ1Ψ3x,z,where(12)Wz1,z2=−r2z25/3+r15/3z11/5,and r1 and r2 are positive constants. This is a relationship between control torques τ1 and τ2. Substituting (11) into (8) yields(13a)x˙1=x2,x˙2=Υ1x,z+Ψ1x,zWz1,z2−Υ3x,zΨ3x,z+Φ1x,z−Ψ1x,zΦ3x,zΨ3x,zτ1,x˙3=x4,x˙4=Υ2x,z+Ψ2x,zWz1,z2−Υ3x,zΨ3x,z+Φ2x,z−Ψ2x,zΦ3x,zΨ3x,zτ1,(13b)z˙1=z2,z˙2=Wz1,z2=−r2z25/3+r15/3z11/5.It is clear that (13a) and (13b) are a nonlinear cascade system, and the state variablez 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(14)x˙1=x2,x˙2=f1x+g1xτ1,x˙3=x4,x˙4=f2x+g2xτ1,where(15)f1x=Υ1x,0Ψ3x,0−Υ3x,0Ψ1x,0Ψ3x,0,f2x=Υ2x,0Ψ3x,0−Υ3x,0Ψ2x,0Ψ3x,0,g1x=Φ1x,0Ψ3x,0−Φ3x,0Ψ1x,0Ψ3x,0,g2x=Φ2x,0Ψ3x,0−Φ3x,0Ψ2x,0Ψ3x,0.From (10), it is easy to obtain g1x>0 for x∈ℝ4.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. The 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 τ1 such that system (14) is stabilized at x=0.Figure 2
Physical model of (14).


## 4. Design of the Stabilizing Controller for the Reduced-Order Model

This section discusses the design of a stabilizing controllerτ1 for reduced-order system (14) under the condition z=0.LetEx=Ξθ,θ˙z=0. From (1), (2), and (7), it is easy to get E0=β1+β2 and E˙x=x2τ1. A Lyapunov function for system (14) is constructed to be(16)Vx=μ12E^2x+μ22x12+μ32x22,where E^x=Ex−E0 and μi>0i=1,2,3 are constants. The derivative of Vx is(17)dVxdt=μ1E^xE˙x+μ2x1x˙1+μ3x2x˙2=μ1E^xx2τ1+μ2x1x2+μ3x2f1x+g1xτ1=μ1E^x+μ3g1xτ1+μ2x1+μ3f1xx2.We design the control lawτ1 to be(18)τ1=−γz2−μ2x1−μ3f1xμ1E^x+μ3g1x,where γ>0 is a constant. Combining (17) and (18) gives(19)dVxdt=−γx22≤0.From Figure2, it is easy to deduce that system (14) has the same dynamic properties as the Pendubot. As a result, following the same analytical procedure given in Section 4 of [22] yields that Ex⟶E0, x1⟶0, and x2⟶0 based on (16) and (19). This means that the control law (18) drives system (14) to converge the set(20)Ω=xx1=0,x2=0,Ex=E0=xx1=0,x2=0,ωx3,x4=0,where(21)ωx3,x4=M220x42−2β1+β21−cosx3.Sincex4=x˙3, we have that ωx3,x4=0 which describes a homoclinic orbit around x3=x4=0. Thus, 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:(22)x˙=Ax+Bτ1,where(23)A=010000A10000100A20,B=0B10B2,where(24)A1=∂f1x∂x3x=0=a1c3−a3c1c3,A2=∂f2x∂x3x=0=a2c3−a3c2c3,B1=g1xx=0=b1c3−b3c1c3,B2=g2xx=0=b2c3−b3c2c3,U=a1b1c1a2b2c2a3b3c3=M−10010β1+β200β201.LetΛ=B,AB,A2B,A3B. It is easy to obtain(25)detΛ=B22A2B1−A1B22=b2c3−b3c22det2Λc34.SinceM−10 is a positive definite matrix, it follows from (24) that c3>0,detU≠0, and b2c3−b3c2≠0. From (25), we have detΛ>0. As a result, linear time-invariant system (22) is controllable. So, the following Riccati matrix equation has a positive definite solution P:(26)A⊤P+PA−PBR−1B⊤P=−Q,where R>0 and Q∈ℝ4×4 is a positive definite matrix. From the quadratic optimal control theory, the control law(27)τ1=−R−1B⊤P,stabilizes linear system (22) at x=0 quickly.From the above statements, we get that the use of the control lawτ1 in (18) and (27) in sequence guarantees the global stabilization of reduced-order system (14) at x=0.Remark 1.
Note that the control lawτ1 in (18) is singular when μ1E^x+μ3g1x=0. This case occurs only when Ex<E0 (i.e., E^x<0) because μ1>0, μ3>0, and g1x>0. When Ex<E0, it follows from (1) and (2) that(28)0>E^x≥β1+β2cosx2−E0≥−2E0.
So, we have(29)μ1E^x+μ3g1x≥μ3g1x−2μ1E0.
From (29), it is easy to obtain that the singularity of τ1 in (18) does not occur if(30)μ3g1x−2μ1E0>0.
We select the control parameters such that(31)μ3>2μ1E0ρ,ρ=ming1x.
This condition guarantees (30) to hold. In other words, the control law in (18) is not singular under condition (31).Remark 2.
There are eight control parameters in this paper, i.e.,r1 and r2 in (12); μ1, μ2, and μ3 in (16); γ 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 ofr1 and r2 is selected for (12) that makes the stabilization time of (13b) to be smallest(2)
For fixedγ, a set of μ1, μ2, and μ3 is chosen to make reduced-order system (14) enter an arbitrarily small neighbourhood area of x=0(3)
A set ofR and Q is selected for the control law τ1 that stabilizes system (22) at the origin


## 5. 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](32)m1=1.22kg,m2=0.28kg,m3=0.72kg,L1=0.05m,L2=0.15m,L3=0.45m,Lc2=0.075m,Lc3=0.225m,J1=0.00153kg⋅m2,J2=0.000598kg⋅m2,J3=0.013138kg⋅m2.The initial condition was selected to be(33)θ1,θ2,θ3,θ˙1,θ˙2,θ˙3⊤=0,π,0,0,0,0.1⊤.The 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. The control parameters in (12) were selected to be r1=1andr2=5. The simulation results of subsystem (13b) are shown in Figure 3. Note that θ3 and θ.3 converge to zero in 0.25 seconds. It means that WAcrobot system (8) changes to reduced-order system (14) quickly. This demonstrates the validity of the presented theoretical results in Section 3.Figure 3
Time response ofθ3 and θ.3 with r1=1 and r2=5 in (13b).
(a)(b)To show the validity of the stabilizing controller for reduced-order model (14), the parameters in (16) and (18) were chosen as(34)μ1=350,μ2=200,μ3=15,γ=50,R=0.5,Q=0.01I4,where I4 is a 4×4 identity matrix. A simple calculation gives ρ=141.25 and 2μ1E0╱ρ=14.147. Thus, condition (31) is satisfied. Figure 4 shows the simulation results for xt by controllers (11) and (18). It is clear that θ1 and θ˙1 are stabilized at zero while θ2 and θ.2 are in a homoclinic orbit. This shows the validity of the theoretical results in Section 4.Figure 4
Time response ofθ1, θ˙1, θ2, and θ.2 by controllers (11) and (18).
(a)(b)(c)(d)Whenxt enters the small neighbourhood area of xt=0 at t=30s, the controller τ1 switches to (27) from (18). The simulation results of controllers and xt are shown in Figures 5 and 6, respectively. Note that the switch of τ1 from (18) to (27) causes small sudden changes in θ1 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. This provides guarantee for the safe and stable operation of the control system. All these demonstrate the advantages of our presented control strategy.Figure 5
Time response ofτ1 and τ2.
(a)(b)Figure 6
Time response ofθ1, θ.1, θ2, and θ.2 by controllers (11), (18), and (27).
(a)(b)(c)(d)


## 6. Conclusion

This 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. This 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.



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