1 Introduction

It is a kind of generalized rigid 3D pendulum that consists of a rigid body supported by a fixed and frictionless pivot with three rotational degrees,acted by a constant gravitational force. Symmetry assumptions are shown to lead to the spherical 2D pendulum if the angular velocity around the axis of symmetry is zero,and the 3D rigid pendulum is referred as a Lagrange top if the angular velocity around the axis of symmetry is a non-zero constant. Many literatures about axially symmetry 3D rigid pendulum have been published.

In 2004,Shen et al.^[1] proposed the reduced attitude model of 3D rigid with Euler’s angles formulation. Chaturvedi et al.^[2] studied the Poincarémapping with the reduced attitude model. Chang and Ge^[3] studied the chaotic motion of the 3D rigid pendulum in the inertial coordinate frame with the Poincaré surface of section method. The chaoticmotion of the 3D rigid pendulum makes the controller’s design more difficult and the research more significant. The stabilization of the 3D rigid pendulum has been studied in recent years. Chaturvedi et al.^[4] designed the attitude controllers for the 3D rigid pendulum at inverted equilibrium position with angular velocity feedback and attitude feedback. Santillo et al.^[5] realized the attitude stabilization for the 3D rigid pendulum at hanging equilibrium position with angular velocity feedback.

In 1995,Jackson and Grosu^[6] and Jackson^[7] proposed the open-plus-closed-loop (OPCL) control method for the complex dynamic system,and then Chen and Liu’s group^{[8, 9, 10, 11]} researched the OPCL approach and its applications for chaotic motion of nonlinear oscillators. To avoid the singular problem and complex trigonometric function of Lagrange’s equation,the quaternion matrix equation Euler’s dynamic equation are used to describe the 3D rigid pendulum,and then we analyze the dynamic characteristic with the Poincaré section.

The present paper continues our research on the control of an asymmetric 3D rigid pendulum. For complex motion of the 3D rigid pendulum,an OPCL controller,which contains two parts: the open-loop part to construct an ideal trajectory and the closed-loop part to calm the 3D rigid pendulum down,is designed. 2 Mathematical model of 3D rigid pendulum

The 3D rigid pendulum rotates around a fixed and frictionless pivot O with three degrees. An inertial coordinate frame O-XYZ has its origin at the pivot O,and the first two coordinate axes X and Y lie in the horizontal plane along the inertia principal axis direction while the third coordinate axis Z is vertical in the direction of gravity. The origin of the body fixed coordinate frame O-X′Y ′Z′ is also located at the pivot. In this frame,the inertia matrix J is constant. A schematic of a 3D rigid pendulum is shown in Fig. 1. Consider Euler’s dynamic equation of the 3D rigid pendulum^[1]

where J = diag(J₁ J₂ J₃) is the inertia matrix of the 3D rigid pendulum,and m is the total mass. Symbol g denotes the constant acceleration due to gravity,and ρ is the body-fixed vector from the pivot O to the centre of mass C_m. The unit vector e₃ = (0 0 λ)^T,where λ = ±1. When λ = 1 means that Z-axis is along the direction of gravity in the inertial frame,R^Te₃ is the direction of gravity in the pendulum-fixed frame. When λ = -1 means that Z-axis is along the reverse direction of gravity in the inertial frame,R^Te₃ is the reverse direction of gravity in the pendulum-fixed frame. ω ∈ R³ is the angular velocity of the 3D rigid pendulum,and u ∈ R³ is the input control moments. For the vectors a and b ∈R³,accordingly,the cross product a × b is a × b = b,where is the skew-symmetric matrix as follows: The parameter R in (1) stands for the direction cosine matrix which is described by quaternion as follows: The attitude kinematics equations described by quaternion are where the quaternion has to satisfy the following holonomic equation: The angular velocities described by the quaternion from (2) can be obtained as The nonholonomic constraint (speed) equation is Form (4),we can obtain Taking the second derivative of (3),we have Substituting (4) and (6) into (1) and combining with (7),a second-order ordinary differential equation which consists of quaternion and whose derivative function will be produced,can be expressed as follows: where = [u^T 0]^T,and The determinant of M(q) is 16J₁J₂J₃ > 0. The components of F(q,) are as follows: where = [u^T 0]^T,and 3 Chaotic motion of 3D rigid pendulum

In this section,we present the chaotic motion of an uncontrolled 3D rigid pendulum. It is well known that chaotic motion is very sensitive to the initial conditions. The chaotic motion will be observed if we choose proper initial values with q(0) and (0). Consider an uncontrolled 3D rigid pendulum,the parameters in (8) are J = diag(40 45 50) kg · m²,g = 9.8 m/s²,l = 0.5 m,m = 140 kg,and λ = -1. We choose q(0) = (0 0 0 1)^T and (0) = (0.1 0.1 0 0)^T. We observe the vector ω in the inertial coordinate frame. Figure 2 shows the angular velocity phase diagram of the 3D rigid pendulum. Figures 3-5 show us the time history of the angular velocities.

Figure 6 is the Poincaré section of the angular velocities,and we can conclude that the 3D pendulum appears the chaotic motion. At first,the pendulum is located in the inverted equilibrium position,then it takes on chaotic motion if the initial value of is a nonzero vector,and the quaternion changes irregularly with time.

4 Design of OPCL controller

Consider the chaotic motion of the 3D rigid pendulum,an OPCL controller is designed in this section. From (8),we can get

By assuming = M^-1 and = (υ₁ υ₂ υ₃ υ₄)^T,we have Equations (10)-(12) can be written in the vector form as where q = [q₁ q₂ q₃]^T,f = [f₁ f₂ f₃]^T,and υ = [υ₁ υ₂ υ₃]^T.

Define the error as e =q-p,where p stands for the control objective function. Introduce the OPCL law^[12]

where -h(p,,t) is the open loop part,the rest of them are the closed loop parts,A and B are the coefficient matrices of the controller. Substituting (15) into (14),we can get the error function of the system Select the Lyapunov function as follows: Its derivative function is where I is an unit matrix. If we make B = -I and A < 0,then we can get (e,) = ^TA < 0. According to the Lyapunov stable theory,q₁,q₂,and q₃ are globally asymptotically stable. From = M,we can get 5 Simulation results and analysis 5.1 Inverted equilibrium

Considering the chaotic characteristics of the 3D rigid pendulum,we simulate the inverted equilibrium of the 3D rigid pendulum in this section. When the 3D rigid pendulum reaches the inverted equilibrium,and λ = -1,the reduced attitude Γ = [0 0 -1]^T. Then,we have 2(q₄² + q₃²) -1 = 1,i.e.,q₄² + q₃²= 1,q₁ = 0,and q₂ = 0. From (19),it can be seen that q₄ ≠ 0. Therefore,the valuesq₁ = 0,q₂ = 0,q₃ = 0,and q₄² = 1 are introduced as the final equilibrium state. For the inverted equilibrium of the 3D rigid pendulum,the parameters in (8) are J = diag(40 45 50) kg · m²,g = 9.8 m/s²,l = 0.5 m,and m = 140 kg. We choose the same initial conditions q(0) = (0 0 0 1)^T and (0) = (0.1 0.1 0 0)^T.

The initial values of quaternion and its derivative have to satisfy the constraint from (5). For the OPCL control law in (15),the objective function as p = (0 0 0)^T is selected,and A = B = -I is used as the coefficient matrix. When the controller is opened at the 11st second,the evolution of quaternion for the 3D rigid pendulum at inverted equilibrium is shown in Fig. 7. From Figs. 8 and 9,it can be concluded that the 3D rigid pendulum finally reaches the inverted equilibrium. Figure 10 shows us the input control moment of 3D rigid pendulum for the inverted equilibrium.

5.2 Hanging equilibrium

Considering the same pendulum,we simulate the hanging equilibrium of the 3D rigid pendulum in this section. When the 3D rigid pendulum reaches the hanging equilibrium,and λ = 1,the reduced attitude Γ = [0 0 1]^T. Then,we have 2(q₄²+q₃²)-1 = 1,i.e.,q₄²+q₃² = 1,q₁ = 0,and q₂ = 0. For the hanging equilibrium of the 3D rigid pendulum,we choose the initial conditions q(0) = (0 0.6 0.8 0)^T. and (0) = (0.1 0 0 0.1)^T. We select the objective function as p = (0 0 0)^T and A = B = -I as the coefficient matrix. When the controller is opened at the 8th second,Fig. 11 shows the evolution of quaternion of the 3D rigid pendulum at the hanging equilibrium. From Figs. 12 and 13,we can conclude that the 3D rigid pendulum finally reaches the hanging equilibrium. Figure 14 shows us the input control moment of the 3D rigid pendulum for the hanging equilibrium.

6 Conclusions

In order to avoid the singular phenomenon of Euler’s angular velocity equation,the quaternion kinematic equation is used to describe the 3D rigid pendulum in this paper. We design an OPCL controller for chaotic motion of the 3D rigid pendulum at the inverted equilibrium position. The OPCL controller contains two parts: the open-loop part to construct an ideal trajectory and the closed-loop part to calm the 3D rigid pendulum down. Simulation results show that this control law can fulfill the attitude control for the 3D rigid pendulum at the inverted and hanging equilibrium positions.