Stabilization Control for the Giant Swing Motion of the Horizontal Bar Gymnastic Robot Using Delayed Feedback Control

Abstract

Open-loop dynamic characteristics of an underactuated system with nonholonomic constraints, such as a horizontal bar gymnastic robot, show the chaotic nature due to its nonlinearity. This chapter deals with the stabilization problems of periodic motions for the giant swing motion of gymnastic robot using chaos control methods. In order to make an extension of the chaos control method and apply it to a new practical use, some stabilization control strategies were proposed, which were, based on the idea of delayed feedback control (DFC), devised to stabilize the periodic motions embedded in the movements of the gymnastic robot. Moreover, its validity has been investigated by numerical simulations. First, a method named as prediction-based DFC was proposed for a two-link gymnastic robot using a Poincar section. Meanwhile, a way to calculate analytically the error transfer matrix and the input matrix that are necessary for discretization was investigated. Second, an improved DFC method, multiprediction delayed feedback control, using a periodic gain, was extended to a four-link gymnastic robot. A set of plural Poincare maps were defined with regard to the original continuous-time system as a T-periodic discrete-time system. Finally, some simulation results showed the effectiveness of the proposed methods.

1. Introduction

In the past decades, scientists have found that unusual and unexpected evolution patterns arise frequently in numerous natural and model nonlinear systems in physics, chemistry, biology, engineering, medicine, economics, and so forth, since Lorenz [1] in 1963, May [2] in 1976, and others reported chaotic behavior in very simple dynamical models. The most peculiar aspect of these patterns is their random-like behavior, although the systems are deterministic, in other words, the deterministic nature of these systems does not make them predictable [3]. This behavior is known as deterministic chaos, or simply chaos. It is considered that chaos is due to sensitive dependence on the initial conditions. This property implies that two trajectories emerging from two different close-by initial conditions separate exponentially in the course of time. Due to this property, and due to the fact that, in general, experimental initial conditions are never known perfectly, chaotic systems cannot be predicted by their long-term behaviors.

This feature of their critical sensitivity to initial conditions, often regarded as a troublesome property, made chaos undesirable in engineering control practice, and most experimentalists considered such characteristics as something to be strongly avoided since it restricts the operating range of many electronic and mechanic devices. Besides that, chaotic systems exhibit two other important properties. One is that there is an infinite number of unstable periodic orbits embedded in the underlying chaotic set. The other is that the dynamics in the chaotic attractor is ergodic; that is to say, the system visits ergodically small neighborhood of every unstable fixed point during its temporal evolution [4].

Although the existence of steady states and an infinity of different unstable periodic orbits embedded in chaotic motion is not usually obvious in free-running chaotic evolution, these orbits offer a great potential advantage if one wants to control a chaotic system. The presence of chaos may be a great advantage for control in a variety of situations. In a nonchaotic system, small controls typically can only change the system dynamics slightly. However, in a chaotic system, one can choose between a rich variety of dynamical behaviors. It is then not surprising that the matter of controlling chaotic systems has come under detailed investigation by several different scientific communities. Chaotic dynamics consists of a motion where the system moves in the neighborhood of one of the unstable periodic orbits (UPO) for a while, then falls close to a different UPO and remains for a limited time. Thus, it allows to exploit a single dynamical system for the production of a large number of different periodic behaviors so that the single system can carry out different performances with different yields [5]. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny, to avoid significant modification of the system’s natural dynamics.

Since Ott, Grebogi, and Yorke [6], in 1990, pointed out the existence of many unstable periodic orbits (UPOs) embedded in chaotic attractors that raise the possibility of using very small external forces to obtain various types of regular behavior, Pyragas [7] in 1992, proposed a so-called “Delayed Feedback Control (DFC)” idea that an appropriate continuous controlling signal formed from the difference between the current state and the delayed state is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit, several techniques were devised for controlling chaos [8–21] during the past years and applied to various systems [22–31]. It is worth noting that in spite of the enormous number of applications among the chaos control, very few rigorous results are so far available. Most results are justified by computer simulations rather than by analytical tools. Therefore, many problems remain unsolved.

On the other hand, recently in the field of mechanical and control engineering, the open-loop link mechanism, in terms of diversity and ease of movement, has been used in industrial robots, although a large control input is needed due to speeding up, and the opposing force to the joint and the base increases. Therefore, the actuator with a large driving performance is needed, and the mass of the entire mechanism is increased. It is thought that the control method using the opposing force and gravity is important to achieve the desired movement with high speed and high efficiency by the limited driving performance. The joint type movement robot is actively researched in recent years, and the achievement of dynamic walking and running with high speed and efficiency becomes a problem. Although living creatures, such as humans or animals, have achieved the high-speed movement by using the open-loop link mechanism, it is thought that this is because that a good movement possessing the energy efficiency and small joint drive power and small impact power, and so forth in the gravitational filed had been acquired adaptively during the process of evolution. For example, the human biped locomotion is performed by dynamic link mechanism coupled to the pendulum motion [32, 33], and brachiation (movement across branches) used by the monkeys is a specialized form of arboreal locomotion that can be seen as a continuous pendulum motion [34, 35]. In addition, humans have competed for the manipulation of their own multi-degree of freedom link mechanism through sports or physical exercise, and so forth, and the achievement of a complex movement meeting a specific initial terminal requirement as a technique.

The systems, such as humans and robots, which control the internal force to achieve motions by using reaction force from the outside world, can be modeled as an underactuated link mechanism. These systems usually have nonholonomic second-order constraints due to the presence of passive joints, which makes development of control methods a difficult task. In recent years, studies on underactuated mechanical systems that possess fewer actuators than degrees of freedom have received increased interest [36–40].

Open-loop dynamic characteristics of an underactuated system with nonholonomic constraints show the chaotic nature that even though there are small differences in the initial conditions, the amplitude grows to be a completely different movement for its nonlinearity due to centrifugal force, Coriolis force, and gravity. Motions control of underactuated systems known also as nonholonomic control, such as wheeled robot positioning control [41], aerial posture control of space robots [42], and positioning control of underactuated link mechanism robot [43, 44], have been studied. However, a generalized control method has not yet been established to this kind of system due to the difficulties in analysis, and the fact that for an underactuated system, it cannot be directly controlled because of the generalized coordinates of its passive parts. Control methods, which utilize empirical skills catching the motion characteristics and proper motion of the system, are desirable. From the above-mentioned viewpoint, in order to achieve a robot with a high-speed, highly effective optimal motion, and the skill in movement seen in living creatures, fundamental researches on the control method and the trajectory planning method using periodic free movement for the linkage with a passive joint have been done [45–51].

The remaining parts of this chapter are organized as follows: Section 2 discusses a kind of DFC method, PDFC, for a two-link gymnastic robot by using of a Poincaré section. In the Section 3, an improved DFC method, Multiprediction Delayed Feedback Control, is extended to a four-link gymnastic robot. Section 4 gives some numerical simulations to show the effectiveness of the proposed method. Section 5 summarizes the chapter.

2. Delayed feedback control for a two-link horizontal bar gymnastic robot

The two-link horizontal bar gymnastic robot is a highly simplified model of a human gymnast on a high bar, where the underactuated first joint models the gymnast’s hands on the bar and the actuated second joint models the gymnast’s waist [52]. Studies on such underactuated mechanical systems that possess fewer actuators than degrees of freedom, as gymnastic robot with the underactuated first joint, have received considerable interest in recent years. Open-loop dynamic characteristics of such a linkage as this kind of system, which is classified as an underactuated systems with nonholonomic constraints, show the chaotic nature that even though there are small differences in initial conditions, the amplitude grows to be a completely different movement for its nonlinearity due to centrifugal force, Coriolis force, and gravity. Since a generalized control method has not yet been established to this kind of system for the difficulties in analysis, their control problems are challenging.

While applying the original DFC to the continuous system, such as gymnastic robot, stability analysis of the closed-loop system becomes a very difficult task because of the time-delay dynamics described by a difference-differential equation. A modified delayed feedback control method, called Prediction-based DFC (PDFC), is proposed to stabilize the giant swing motion of two-link horizontal bar gymnastic robot.

2.1. Two-link gymnastic robot model

The two-link horizontal bar gymnastic robot is a simplified model of a human gymnast on a high bar, where the first joint being passive models the gymnast’s hands on the bar and the second joint comprising of an actuator models the gymnast’s hips. The formula m_i, l_i, a_i, I_i(i = 1, 2) in Figure 1 denotes the ith link’s mass, length, distance from joint to its center of mass, inertia moment around its center of mass, respectively.

Figure 1.

Two-link gymnastic robot model.

The equation of motion of the gymnastic robot is as follows:

$\left[\begin{array}{c}\hfill u_1\hfill \\ \hfill u_2\hfill \end{array}\right]=\underset{M}{\underbrace{\left[\begin{array}{cc}\hfill M_{11}\hfill & \hfill M_{12}\hfill \\ \hfill M_{21}\hfill & \hfill M_{22}\hfill \end{array}\right]}}\left[\begin{array}{c}\hfill {\ddot{x}}_1\hfill \\ \hfill {\ddot{x}}_2\hfill \end{array}\right]+\underset{C}{\underbrace{\left[\begin{array}{c}\hfill C_1\hfill \\ \hfill C_2\hfill \end{array}\right]}}+\underset{G}{\underbrace{\left[\begin{array}{c}\hfill G_1\hfill \\ \hfill G_2\hfill \end{array}\right]}}\text{,}$E1

where x = (x₁, x₂)^T ∈ R² is the generalized coordinate vector, u = (u₁, u₂)^T ∈ R² is the joint torque vector, and M ∈ R^2×2, C ∈ R^2×1, G ∈ R^2×1 denote the inertia matrix, Coriolis matrix, and gravitational matrix, respectively.

$M_{11}=I_1+I_2+m_1{a}_1^2+m_2\left(l_1^2+a_2^2+2l_1{a}_{2}cos{x}_2\right)$E2

$M_{12}=I_2+m_2\left(a_2^2+l_1{a}_{2}cos{x}_2\right)$E3

$M_{21}=I_2+m_2\left(a_2^2+l_1{a}_{2}cos{x}_2\right)$E4

$M_{22}=I_2+m_2{a}_2^2$E5

$C_1=-\left({\stackrel{.}{x}}_2^2+2{\stackrel{.}{x}}_1{\stackrel{.}{x}}_2\right)m_2{l}_1{a}_{2}sin{x}_2$E6

$C_2={\stackrel{.}{x}}_1^2{m}_2{l}_1{a}_{2}sin{x}_2$E7

$G_1=g\left\{\left(m_1{a}_1+m_2{l}_1\right)sin{x}_1+m_2{a}_{2}sin\left(x_1+x_2\right)\right\}$E8

$G_2=g{m}_2{a}_{2}sin\left(x_1+x_2\right)$E9

In addition, since the first joint cannot generate active torque, the following constraint must be satisfied.

$u_1\text{≡}h\left(\ddot{\mathbf{x}},\stackrel{.}{\mathbf{x}},\mathbf{x}\right)=0$E2

In order to facilitate the analysis, we rewrite the motion Eq. (1) into the following equation of state:

$\dot{x}=\left[\begin{array}{c}\hfill \stackrel{.}{\mathbf{x}}\hfill \\ \hfill M^{-1}\left(\mathbf{x}\right)\left(-C\left(\mathbf{x},\stackrel{.}{\mathbf{x}}\right)-G\left(\mathbf{x}\right)\right)\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill M^{-1}\left(\mathbf{x}\right)E\hfill \end{array}\right]u\text{≡}f\left(x,u\right)\text{,}$E3

where

$\begin{array}{c}M\left(\mathbf{x}\right)=\left[\begin{array}{cc}\hfill c_1+c_2+2dcos{x}_2\hfill & \hfill c_2+dcos{x}_2\hfill \\ \hfill c_2+dcos{x}_2\hfill & \hfill c_2\hfill \end{array}\right],\\ C\left(\mathbf{x},\stackrel{.}{\mathbf{x}}\right)=\left[\begin{array}{c}\hfill -\left({\stackrel{.}{x}}_2^2+2{\stackrel{.}{x}}_1{\stackrel{.}{x}}_2\right)dsin{x}_2\hfill \\ \hfill {\stackrel{.}{x}}_1^{2}dsin{x}_2\hfill \end{array}\right],\\ G\left(\mathbf{x}\right)=\left[\begin{array}{c}\hfill g_{1}sin{x}_1+g_{2}sin\left(x_1+x_2\right)\hfill \\ \hfill g_{2}sin\left(x_1+x_2\right)\hfill \end{array}\right],\\ E=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]\text{.}\end{array}$E4

Here, $x={\left[\mathbf{x},\stackrel{.}{\mathbf{x}}\right]}^T\in {\mathrm{R}}^4$ denotes the state vector, and u ∈ R¹ represents the control input. Meanwhile, the variables among the above equations are defined as follows:

$c_1=I_1+m_1{a}_1^2+m_2{l}_1^{2}C$E13

$c_2=I_2+m_2{a}_2^{2}C$E14

$d=m_2{l}_1{a}_{2}C$E15

$g_1=g\left(m_1{a}_1+m_2{l}_1\right)C$E16

$g_2=g{m}_2{a}_2\text{.}$E17

2.2. Prediction-based delayed feedback control

As is known, the stability of a periodic orbit of the original continuous-time system is closely related to the stability of the fixed point of the corresponding Poincaré map. Since the stability of the periodic orbit means that the sequence of points converges to a fixed point in the phase plane, the objective system can be expressed as the following difference equation for the discrete-time systems.

$\tilde{x}\left(k+1\right)=A\tilde{x}\left(k\right)+Bu\left(k\right)\text{≡}f\left(\tilde{x}\left(k\right),u\left(k\right)\right)$E5

where k, $\tilde{x}\in {\mathrm{R}}^4$, u ∈ R¹ denote the discrete time, the state error, and the control input, respectively. And A ∈ R^4×4 is named as the error transfer matrix (ETM), B ∈ R^4×1 is called the Input Matrix.

2.2.1. Solution for obtaining the error transfer matrix A

Here, let $y_1={\stackrel{.}{x}}_1$, $y_2={\stackrel{.}{x}}_2$. Then the state vector of two-link gymnastic robot can be written in the following form:

$x=\left(x_1,x_2,y_1,y_2\right)$E6

As shown in Figure 2, the (k + 1)th state vector x_p,k + 1 can be calculated by one-cycle integration in accordance with the control law by defining the kth state vector passing through the section P as x_p,k.

Figure 2.

Poincaré map of a closed periodic trajectory.

This relationship representing this state transfer map is defined as ϕ as follows:

$x_{p,k+1}=\varphi \left(x_{p,k}\right)$E7

If given an initial condition from which a periodic orbit can be formed, it will return to the same point after one period. Hence, the following equation holds:

$x_{p,ref}=\varphi \left(x_{p,ref}\right)$E8

While a difference occurs in the initial state, the motion will stray away from the periodic orbit. Suppose that there was an error corresponding to e_p,k in kth state vector, the (k + 1)th state x_{p,k + 1} can be described as

$x_{p,k+1}=\varphi \left(x_{p,ref}+e_{p,k}\right)\text{.}$E9

Using the above equation’s Taylor series at e_p,k and neglecting the higher-order terms can yield

$e_{p,k+1}=\frac{\partial \varphi \left(x_{p,ref}\right)}{\partial x_p}{e}_{p,k}\text{≡}A{e}_{p,k}\text{.}$E10

It is obvious that ∂ϕ(x_p,ref)/∂x_p is equivalent to the ETM A defined in Eq. (5). It can be judged from this equation that if the absolute value of the ETM’s eigenvalue >1, then the error increases with each cycle, and if <1, the motion will approach asymptotically the periodic orbit. However, because the state transfer mapping cannot be stated explicitly as a function, in previous studies, this ETM had been computed based on numerical differentiation. In this work, we obtained ETM with an analytical method by using variational equation.

Now consider the objective system given by the equation of motion in Eq. (3). First, given the case of u = 0, the objective system becomes

$\stackrel{.}{x}=f\left(x\left(t\right)\right)\text{≡}{\left[f_1,f_2,f_3,f_4\right]}^T$E11

Assume the state x to be the following vector function

$x\left(t\right)=\phi \left(t,x\right)\text{≡}{\left[{\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4\right]}^T\text{,}$E12

where f_i, φ_i (i = 1 ∼ 4) are the functions of x. The solution of equation passing through x(0) at t = 0 is defined as follows:

$x\left(0\right)=x_0\text{.}$E13

Moreover, the periodic solution satisfies the following equation.

$x\left(T\right)=x\left(0\right)=\phi \left(T,x_0\right)$E14

Here, a Poincaré map is defined as follows:

$P:\begin{array}{l}{\mathbf{R}}^n\to {\mathbf{R}}^n\left(n=4\right)\hfill \\ x_0\mapsto x_1=P\left(x_0\right)=\phi \left(T,x_0\right)\hfill \end{array}$E28

Then, the T-period sequence of points obtained by sampling the trajectory from the initial value x₀ is

$\left\{x_0,x_1,.,x_k,\dots \right\}=\left\{x_0,P\left(x_0\right),P^2\left(x_0\right),.,P^k\left(x_0\right),\dots \right\}\text{.}$E15

Thus, it is obvious that the periodic solution is shown as a point passing through the same position. Substituting x = φ(t, x₀) into Eq. (11), we obtain an equation for the variable φ(t, x₀), which can be written as

$\frac{d\phi \left(t,x_0\right)}{dt}=f\left(\phi \left(t,x_0\right)\right)\text{.}$E16

Next, taking differentiation for Eq. (16) at x₀, we can obtain

$\frac{\partial }{\partial x_0}\left(\frac{d\phi }{dt}\left(t,x_0\right)\right)=\frac{\partial }{\partial x_0}\left(f\left(\phi \left(t,x_0\right)\right)\right)\text{.}$E17

The differential order of the left-hand side in the above equation can be modified as

$\frac{d}{dt}\left(\frac{\partial \phi }{\partial x_0}\left(t,x_0\right)\right)=\frac{\partial f}{\partial x}\left(\phi \left(t,x_0\right)\right)\frac{\partial \phi }{\partial x_0}\left(t,x_0\right)\text{≡}\tilde{A}\left(t\right)\frac{\partial \phi }{\partial x_0}\left(t,x_0\right)\text{.}$E18

Thus, it becomes a time-varying linear matrix differential equation with the form of

$\frac{dX\left(t,x_0\right)}{dt}=\tilde{A}\left(t\right)X\left(t,x_0\right)\text{,}$E19

which is called the Variational Equation if defining $X\left(t,x_0\right)=\frac{\partial \phi }{\partial x_0}\left(t,x_0\right)$. From Eqs. (11) and (12), the above equation is equivalent to

$\frac{d}{dt}\left[\begin{array}{l}\partial {\phi }_1/\partial x_{i,0}\hfill \\ \partial {\phi }_2/\partial x_{i,0}\hfill \\ \partial {\phi }_3/\partial x_{i,0}\hfill \\ \partial {\phi }_4/\partial x_{i,0}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill \partial f_1/\partial x_1\hfill & \hfill \partial f_1/\partial x_2\hfill & \hfill \partial f_1/\partial y_1\hfill & \hfill \partial f_1/\partial y_2\hfill \\ \hfill \partial f_2/\partial x_1\hfill & \hfill \partial f_2/\partial x_2\hfill & \hfill \partial f_2/\partial y_1\hfill & \hfill \partial f_2/\partial y_2\hfill \\ \hfill \partial f_3/\partial x_1\hfill & \hfill \partial f_3/\partial x_2\hfill & \hfill \partial f_3/\partial y_1\hfill & \hfill \partial f_3/\partial y_2\hfill \\ \hfill \partial f_4/\partial x_1\hfill & \hfill \partial f_4/\partial x_2\hfill & \hfill \partial f_4/\partial y_1\hfill & \hfill \partial f_4/\partial y_2\hfill \end{array}\right]\left[\begin{array}{l}\partial {\phi }_1/\partial x_{i,0}\hfill \\ \partial {\phi }_2/\partial x_{i,0}\hfill \\ \partial {\phi }_3/\partial x_{i,0}\hfill \\ \partial {\phi }_4/\partial x_{i,0}\hfill \end{array}\right]\text{,}$E20

where x_i,0 refers to the solution at t = 0 of the ith state corresponding to the state vector x defined in Eq. (12).

Note that X(0, x₀) = I. Carrying out integrals in numerical integration to the above variational equations over the interval t ∈ [0, T], one column of the matrix X(T, x₀) can be obtained. Hence, repeating it four times, all of its values can be calculated. It should be emphasized here that to solve these equations since the value of Ã(t) is required, however, Ã(t) contains x(t) in terms of the relevant time, they have to be solved together with Eq. (11) simultaneously. From Eqs. (10) and (15), it is not difficult to be verified that X(T, x₀) can be regarded as ETM.

2.2.2. Solution for obtaining the input matrix B

Note that Eq. (3) can be rewritten into

$\stackrel{.}{x}=\tilde{f}\left(x,u\right)=f\left(x\right)+g\left(x\right)u\text{,}$E21

where

$g\left(x\right)=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill M^{-1}\left(x\right)E\hfill \end{array}\right]\text{≡}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill g_3\hfill \\ \hfill g_4\hfill \end{array}\right]\text{.}$E22

The input torque u can be described as

$u=\mathrm{rect}\left(t\right)u_0\text{,}$E23

where

$\mathrm{rect}\left(t\right)=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathrm{if}kT<t\le kT+\tau \left(k=0,1,2,\dots \right)\hfill \\ \hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\text{,}\right.$E24

Here, u₀ is the control input during (t₀, τ), while x = x₀ and τ are control parameters.

Since the solution of equation at x = x(0) can be stated as

$x\left(t\right)=\phi \left(t,x_0,u_0\right)\text{.}$E25

Define a Poincaré map as follows:

$\tilde{P}:\begin{array}{l}{\mathrm{R}}^n\to {\mathrm{R}}^n\left(n=4\right)\hfill \\ x_0\mapsto x_1=P\left(x_0,u_0\right)=\phi \left(T,x_0,u_0\right)\hfill \end{array}$E40

The continuous system described by Eq. (21) will change to be a discrete system by mapping of $\tilde{P}$. Substituting x = φ(t, x₀, u₀) into Eq. (21), the following equation for the variable φ(t, x₀) can be obtained.

$\frac{d\phi \left(t,x_0,u_0\right)}{dt}=f\left(\phi \left(t,x_0,u_0\right)\right)+g\left(x\right)\mathrm{rect}\left(t\right)u_0$E26

It is obvious here that by taking differentiation at x₀, its result will be equivalent to Eq. (18). Consider here the case of differentiation at u₀, which can yield

$\frac{\partial }{\partial u_0}\left(\frac{d\phi }{dt}\left(t,x_0,u_0\right)\right)=\frac{\partial }{\partial u_0}\left(f\left(\phi \left(t,x_0,u_0\right)\right)\right)+g\left(x\right)\mathrm{rect}\left(t\right)\text{.}$E27

The differential order of left-hand side in the above equation can also be changed.

$\frac{d}{dt}\left(\frac{\partial \phi }{\partial u_0}\left(t,x_0,u_0\right)\right)=\frac{\partial f}{\partial x}\left(\phi \left(t,x_0,u_0\right)\right)\frac{\partial \phi }{\partial u_0}\left(t,x_0,u_0\right)+g\left(x\right)\mathrm{rect}\left(t\right)$E28

Let X(t, x₀, u₀) = ∂φ/∂x₀(t, x₀, u₀). Similar to previous subsection, the above equation is equivalent to the following equation.

$\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}\hfill \partial {\phi }_1/\partial u_0\hfill \\ \hfill \partial {\phi }_2/\partial u_0\hfill \\ \hfill \partial {\phi }_3/\partial u_0\hfill \\ \hfill \partial {\phi }_4/\partial u_0\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill \partial f_1/\partial x_1\hfill & \hfill \partial f_1/\partial x_2\hfill & \hfill \partial f_1/\partial y_1\hfill & \hfill \partial f_1/\partial y_2\hfill \\ \hfill \partial f_2/\partial x_1\hfill & \hfill \partial f_2/\partial x_2\hfill & \hfill \partial f_2/\partial y_1\hfill & \hfill \partial f_2/\partial y_2\hfill \\ \hfill \partial f_3/\partial x_1\hfill & \hfill \partial f_3/\partial x_2\hfill & \hfill \partial f_3/\partial y_1\hfill & \hfill \partial f_3/\partial y_2\hfill \\ \hfill \partial f_4/\partial x_1\hfill & \hfill \partial f_4/\partial x_2\hfill & \hfill \partial f_4/\partial y_1\hfill & \hfill \partial f_4/\partial y_2\hfill \end{array}\right]\left[\begin{array}{l}\partial {\phi }_1/\partial u_0\hfill \\ \partial {\phi }_2/\partial u_0\hfill \\ \partial {\phi }_3/\partial u_0\hfill \\ \partial {\phi }_4/\partial u_0\hfill \end{array}\right]\\ +\mathrm{rect}\left(t\right){\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill g_3\hfill & \hfill g_4\hfill \end{array}\right]}^T\end{array}$E29

Note that X(0, x₀, u₀) = 0. Carrying out integrals in numerical integration to Eq. (29) over the interval t ∈ [0, T], it is not difficult to obtain the value of the matrix X(T, x₀, u₀) which can be regarded as the input matrix B defined in Eq. (5).

2.2.3. Stability of periodic orbits

Since the value of both the matrices A and B can be calculated analytically, i.e., the continuous system of two-link gymnastic robot can be expressed in a discrete model described in Eq. (5). In order to study the stabilization of gymnastic robot’s continuous system by delayed feedback control, it is sufficient to consider the discrete model.

Consider the following Prediction-based feedback control:

$u\left(k\right)=K\left\{x\left(k\right)-x_P\left(k\right)\right\}\text{,}$E30

where K ∈ R^1×4 is a feedback gain, x(k) is the state vector at k-step, x_P(k) denotes one period future states of uncontrolled system which can be obtained from Eq. (3). Moreover, notice that the following almost equality holds.

$x\left(k\right)-x_P\left(k\right)\approx \tilde{x}\left(k\right)-f\left(\tilde{x}\left(k\right),0\right)$E31

Here $f\left(\tilde{x}\left(k\right),0\right)$ stands for the state error at (k + 1)th step. From Eq. (5), it is obvious that $f\left(\tilde{x}\left(k\right),0\right)=A\tilde{x}\left(k\right)$ holds.

Therefore, the closed-loop system can be described by

$\begin{array}{c}\tilde{x}\left(k+1\right)=A\tilde{x}\left(k\right)+BK\left(I-A\right)\tilde{x}\left(k\right)\\ =\left\{A+BK\left(I-A\right)\right\}\tilde{x}\left(k\right)\text{.}\end{array}$E32

Let

$\widehat{K}=K\left(I-A\right)\text{,}$E33

then the stabilization problem is reduced as follows:

Given a system Eq. (5), find a feedback gain $\widehat{K}$ that places the closed-loop poles of the system in the set

$\Lambda =\left\{z\in \mathrm{C}:\left|z\right|<1\right\}\text{.}$E34

By using the pole placement technique, it is not difficult to obtain the value of $\widehat{K}$, and if det(I − A) ≠ 0, then the feedback gain K is given by

$K=\widehat{K}{\left(I-A\right)}^{-1}\text{.}$E35

Thus, the above design procedure can be applied to analyze the stability of two-link gymnastic robot system.

3. Delayed feedback control for a four-link horizontal bar gymnastic robot

As is known, the simplified two-link robot shown in the previous section is not an ideal physical model of a human gymnast on a high bar. In order to mimic gymnastic routine more realistically, the complicated robot model with higher degrees of freedom (DOF) needs to be considered. An improved method based on PDFC which control a three-link gymnastic robot via a periodic gain has been proposed [51]. In this section, the Multiprediction Delayed Feedback Control (MDFC), is extended to a more complicated gymnastic robot with four DOF.

3.1. Four-link gymnastic robot model

Figure 3 shows a four-link horizontal bar gymnastic robot model, which consists of four links and four joints. Shoulder, hips, and knees are active, while the pivot connecting the hand and bar is a passive joint. Assume m_i, l_i, a_i, I_i (i = 1, 2, 3, 4) to be the link mass, length, distance from joint to its center of mass, and inertia moment around its center of mass, respectively.

Figure 3.

Four-link gymnastic robot model.

The motion equation of the robot is

$\left[\begin{array}{c}\hfill u_1\hfill \\ \hfill u_2\hfill \\ \hfill u_3\hfill \\ \hfill u_4\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill M_{11}\hfill & \hfill M_{12}\hfill & \hfill M_{13}\hfill & \hfill M_{14}\hfill \\ \hfill M_{21}\hfill & \hfill M_{22}\hfill & \hfill M_{23}\hfill & \hfill M_{24}\hfill \\ \hfill M_{31}\hfill & \hfill M_{32}\hfill & \hfill M_{33}\hfill & \hfill M_{34}\hfill \\ \hfill M_{41}\hfill & \hfill M_{42}\hfill & \hfill M_{43}\hfill & \hfill M_{44}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\ddot{x}}_1\hfill \\ \hfill {\ddot{x}}_2\hfill \\ \hfill {\ddot{x}}_3\hfill \\ \hfill {\ddot{x}}_4\hfill \end{array}\right]+\left[\begin{array}{c}\hfill C_1\hfill \\ \hfill C_2\hfill \\ \hfill C_3\hfill \\ \hfill C_4\hfill \end{array}\right]+\left[\begin{array}{c}\hfill G_1\hfill \\ \hfill G_2\hfill \\ \hfill G_3\hfill \\ \hfill G_4\hfill \end{array}\right]\text{,}$E36

where x = (x₁, x₂, x₃, x₄)^T ∈ R⁴ is the generalized coordinate vector, (u₁, u₂, u₃, u₄)^T ∈ R⁴ is the joint torque vector, and M_ij, C_i, G_i (i, j = 1, 2, 3, 4) are, respectively, the terms of inertia matrix, Coriolis matrix, and gravitational matrix. Each item of M_ij, C_i, G_i is as follows. Note that M_ij = M_ji, (i ≠ j).

$\begin{array}{l}\text{M}11=m_2\left(2a_2{l}_1\cos \left(x_2\right)+a_2^2+l_1^2\right)+m_3\{2a_3{l}_1\cos \left(x_2+x_3\right)+2a_3{l}_2\cos \left(x_3\right)\\ +a_3^2+2l_1{l}_2\cos \left(x_2\right)+l_1^2+l_2^2\}+m_4\{2a_4{l}_1\cos \left(x_2+x_3+x_4\right)+2a_4{l}_2\cos \left(x_3+x_4\right)+2a_4{l}_3\cos \left(x_4\right)\\ +a_4^2+2l_1{l}_2\cos \left(x_2\right)+2l_1{l}_3\cos \left(x_2+x_3\right)+2l_2{l}_3\cos \left(x_3\right)+l_1^2+l_2^2+l_3^2\}\\ +a_1^2{m}_1+I_1+I_2+I_3+I_4\text{;}\end{array}$E52

$\begin{array}{l}\text{M12}\mathrm{<mo>=</mo>}{m}_2\left(a_2{l}_1\cos \left(x_2\right)+a_2^2\right)+m_3\{a_3{l}_1\cos \left(x_2+x_3\right)+2a_3{l}_2\cos \left(x_3\right)+a_3^2+l_1{l}_2\cos \left(x_2\right)\\ +l_2^2\}+m_4\{a_4{l}_1\cos \left(x_2+x_3+x_4\right)+2a_4{l}_2\cos \left(x_3+x_4\right)+2a_4{l}_3\cos \left(x_4\right)+a_4^2\\ +l_1{l}_2\cos \left(x_2\right)+l_1{l}_3\cos \left(x_2+x_3\right)+2l_2{l}_3\cos \left(x_3\right)+l_2^2+l_3^2\}+I_2+I_3+I_4\mathrm{<mo>;</mo>}\end{array}$E53

$\begin{array}{l}\text{M13}\mathrm{<mo>=</mo>}{m}_3\left(a_3{l}_1\cos \left(x_2+x_3\right)+a_3{l}_2\cos \left(x_3\right)+a_3^2\right)+m_4\{a_4{l}_1\cos \left(x_2+x_3+x_4\right)\\ +a_4{l}_2\cos \left(x_3+x_4\right)+2a_4{l}_3\cos \left(x_4\right)+a_4^2+l_1{l}_3\cos \left(x_2+x_3\right)\\ +l_3\left(l_2\cos \left(x_3\right)+l_3\right)\}+I_3+I_4\mathrm{<mo>;</mo>}\end{array}$E54

$\mathrm{M}14=m_4\left(a_4{l}_{1}cos\left(x_2+x_3+x_4\right)+a_4{l}_{2}cos\left(x_3+x_4\right)+a_4{l}_{3}cos\left(x_4\right)+a_4^2\right)+I_4\text{;}$E55

$\begin{array}{l}\text{M22}\mathrm{<mo>=</mo>}{m}_3\left(2a_3{l}_2\cos \left(x_3\right)+a_3^2+l_2^2\right)+m_4\{2a_4{l}_2\cos \left(x_3+x_4\right)+2a_4{l}_3\cos \left(x_4\right)\\ +a_4^2+2l_2{l}_3\cos \left(x_3\right)+l_2^2+l_3^2\}+a_2^2{m}_2+I_2+I_3+I_4\mathrm{<mo>;</mo>}\end{array}$E56

$\mathrm{M}23=m_3\left(a_3{l}_{2}cos\left(x_3\right)+a_3^2\right)+m_4\left\{a_4{l}_{2}cos\left(x_3+x_4\right)+2a_4{l}_{3}cos\left(x_4\right)+a_4^2+l_2{l}_{3}cos\left(x_3\right)+l_3^2\right\}+I_3+I_4\text{;}$E57

$\mathrm{M}24=m_4\left(a_4{l}_{2}cos\left(x_3+x_4\right)+a_4{l}_{3}cos\left(x_4\right)+a_4^2\right)+I_4\text{;}$E58

$\mathrm{M}33=m_4\left(2a_4{l}_{3}cos\left(x_4\right)+a_4^2+l_3^2\right)+a_3^2{m}_3+I_3+I_4\text{;}$E59

$\mathrm{M}34=m_4\left(a_4{l}_{3}cos\left(x_4\right)+a_4^2\right)+I_4\text{;}$E60

$\mathrm{M}44=a_4^2{m}_4+I_4\text{;}$E61

$\begin{array}{l}\text{C1}\mathrm{<mo>=</mo>}{m}_3\{-a_3{l}_2{\dot{q}}_3\left(2\left({\dot{q}}_1+{\dot{q}}_2\right)+{\dot{q}}_3\right)\sin \left(x_3\right)-a_3{l}_1\left({\dot{q}}_2+{\dot{q}}_3\right)\left(2{\dot{q}}_1+{\dot{q}}_2+{\dot{q}}_3\right)\sin \left(x_2+x_3\right)\\ -l_1{l}_2{\dot{q}}_2\left(2{\dot{q}}_1+{\dot{q}}_2\right)\sin \left(x_2\right)\}-a_2{l}_1{m}_2{\dot{q}}_2\left(2{\dot{q}}_1+{\dot{q}}_2\right)\sin \left(x_2\right)\\ +m_4\{-a_4{l}_3{\dot{q}}_4\left(2\left({\dot{q}}_1+{\dot{q}}_2+{\dot{q}}_3\right)+{\dot{q}}_4\right)\sin \left(x_4\right)\\ -a_4{l}_2\left({\dot{q}}_3+{\dot{q}}_4\right)\left(2{\dot{q}}_1+2{\dot{q}}_2+{\dot{q}}_3+{\dot{q}}_4\right)\sin \left(x_3+x_4\right)\\ -a_4{l}_1\left({\dot{q}}_2+{\dot{q}}_3+{\dot{q}}_4\right)\left(2{\dot{q}}_1+{\dot{q}}_2+{\dot{q}}_3+{\dot{q}}_4\right)\sin \left(x_2+x_3+x_4\right)\\ +l_2{l}_3{\dot{q}}_3^2\left(-\sin \left(x_3\right)\right)-2l_2{l}_3\left({\dot{q}}_1+{\dot{q}}_2\right){\dot{q}}_3\sin \left(x_3\right)\\ +l_2{l}_3{\dot{q}}_3^2\left(-\sin \left(x_3\right)\right)-2l_2{l}_3\left({\dot{q}}_1+{\dot{q}}_2\right){\dot{q}}_3\sin \left(x_3\right)\end{array}$E62

$\begin{array}{l}\text{C2}\mathrm{<mo>=</mo>}{a}_2{l}_1{m}_2{\dot{q}}_1^2\sin \left(x_2\right)+m_3\{a_3{l}_1{\dot{q}}_1^2\sin \left(x_2+x_3\right)-2a_3{l}_2{\dot{q}}_3{\dot{q}}_1\sin \left(x_3\right)-a_3{l}_2{\dot{q}}_3^2\sin \left(x_3\right)\\ -2a_3{l}_2{\dot{q}}_2{\dot{q}}_3\sin \left(x_3\right)+l_1{l}_2{\dot{q}}_1^2\sin \left(x_2\right)\}+m_4\{a_4{l}_1{\dot{q}}_1^2\sin \left(x_2+x_3+x_4\right)-2a_4{l}_3{\dot{q}}_4{\dot{q}}_1\sin \left(x_4\right)\\ -2a_4{l}_2{\dot{q}}_3{\dot{q}}_1\sin \left(x_3+x_4\right)-2a_4{l}_2{\dot{q}}_4{\dot{q}}_1\sin \left(x_3+x_4\right)-a_4{l}_3{\dot{q}}_4^2\sin \left(x_4\right)-2a_4{l}_3{\dot{q}}_2{\dot{q}}_4\sin \left(x_4\right)\\ -2a_4{l}_3{\dot{q}}_3{\dot{q}}_4\sin \left(x_4\right)-a_4{l}_2{\dot{q}}_3^2\sin \left(x_3+x_4\right)-a_4{l}_2{\dot{q}}_4^2\sin \left(x_3+x_4\right)-2a_4{l}_2{\dot{q}}_2{\dot{q}}_3\sin \left(x_3+x_4\right)\\ -2a_4{l}_2{\dot{q}}_2{\dot{q}}_4\sin \left(x_3+x_4\right)-2a_4{l}_2{\dot{q}}_3{\dot{q}}_4\sin \left(x_3+x_4\right)+l_1{l}_2{\dot{q}}_1^2\sin \left(x_2\right)+l_1{l}_3{\dot{q}}_1^2\sin \left(x_2+x_3\right)\\ -2l_2{l}_3{\dot{q}}_3{\dot{q}}_1\sin \left(x_3\right)-l_2{l}_3{\dot{q}}_3^2\sin \left(x_3\right)-2l_2{l}_3{\dot{q}}_2{\dot{q}}_3\sin \left(x_3\right)\}\mathrm{<mo>;</mo>}\end{array}$E63

$\begin{array}{l}\text{C3}\mathrm{<mo>=</mo>}{m}_3\left\{a_3{l}_1{\dot{q}}_1^2\sin \left(x_2+x_3\right)+a_3{l}_2{\left({\dot{q}}_1+{\dot{q}}_2\right)}^2\sin \left(x_3\right)\right\}\\ +m_4\{a_4{l}_1{\dot{q}}_1^2\sin \left(x_2+x_3+x_4\right)-2a_4{l}_3{\dot{q}}_4{\dot{q}}_1\sin \left(x_4\right)+a_4{l}_2{\left({\dot{q}}_1+{\dot{q}}_2\right)}^2\sin \left(x_3+x_4\right)\\ -a_4{l}_3{\dot{q}}_4^2\sin \left(x_4\right)-2a_4{l}_3{\dot{q}}_2{\dot{q}}_4\sin \left(x_4\right)-2a_4{l}_3{\dot{q}}_3{\dot{q}}_4\sin \left(x_4\right)\\ +l_1{l}_3{\dot{q}}_1^2\sin \left(x_2+x_3\right)+l_2{l}_3{\dot{q}}_1^2\sin \left(x_3\right)+2l_2{l}_3{\dot{q}}_2{\dot{q}}_1\sin \left(x_3\right)+l_2{l}_3{\dot{q}}_2^2\sin \left(x_3\right)\}\mathrm{<mo>;</mo>}\end{array}$E64

$\begin{array}{l}\text{C4}\mathrm{<mo>=</mo>}{m}_4\{a_4{l}_1{\dot{q}}_1^2\sin \left(x_2+x_3+x_4\right)+a_4{l}_3{\dot{q}}_1^2\sin \left(x_4\right)+2a_4{l}_3{\dot{q}}_2{\dot{q}}_1\sin \left(x_4\right)+2a_4{l}_3{\dot{q}}_3{\dot{q}}_1\sin \left(x_4\right)\\ +a_4{l}_3{\dot{q}}_2^2\sin \left(x_4\right)+a_4{l}_2{\left({\dot{q}}_1+{\dot{q}}_2\right)}^2\sin \left(x_3+x_4\right)+a_4{l}_3{\dot{q}}_3^2\sin \left(x_4\right)+2a_4{l}_3{\dot{q}}_2{\dot{q}}_3\sin \left(x_4\right)\}\mathrm{<mo>;</mo>}\end{array}$E65

$\begin{array}{l}\text{G1}\mathrm{<mo>=</mo>}g\{m_2\left(a_2\sin \left(x_1+x_2\right)+l_1\sin \left(x_1\right)\right)\\ +m_3\left(a_3\sin \left(x_1+x_2+x_3\right)+l_1\sin \left(x_1\right)+l_2\sin \left(x_1+x_2\right)\right)\\ +m_3\left(a_3\sin \left(x_1+x_2+x_3\right)+l_1\sin \left(x_1\right)+l_2\sin \left(x_1+x_2\right)\right)\\ +l_2\sin \left(x_1+x_2\right)+l_3\sin \left(x_1+x_2+x_3\right))+a_1{m}_1\sin \left(x_1\right)\}\mathrm{<mo>;</mo>}\end{array}$E66

$\begin{array}{l}\text{G2}\mathrm{<mo>=</mo>}g\{m_3\left(a_3\sin \left(x_1+x_2+x_3\right)+l_2\sin \left(x_1+x_2\right)\right)+m_4(a_4\sin \left(x_1+x_2+x_3+x_4\right)\\ +l_2\sin \left(x_1+x_2\right)+l_3\sin \left(x_1+x_2+x_3\right))+a_2{m}_2\sin \left(x_1+x_2\right)\}\mathrm{<mo>;</mo>}\end{array}$E67

$\mathrm{G}3=g\left\{m_4\left(a_{4}sin\left(x_1+x_2+x_3+x_4\right)+l_{3}sin\left(x_1+x_2+x_3\right)\right)\right.\left.+a_3{m}_{3}sin\left(x_1+x_2+x_3\right)\right\}\text{;}$E68

$\mathrm{G}4=a_{4}g{m}_{4}sin\left(x_1+x_2+x_3+x_4\right)\text{;}$E69

Similar to the motion equation of the two-link robot, Eq. (36) can be rewritten as follows.

$\stackrel{.}{x}=\left[\begin{array}{c}\hfill y\hfill \\ \hfill -M^{-1}\left(x\right)\left(C\left(x\right)+G\left(x\right)\right)\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill M^{-1}\left(x\right)E\hfill \end{array}\right]\left[\begin{array}{c}\hfill u_2\hfill \\ \hfill u_3\hfill \\ \hfill u_4\hfill \end{array}\right]\text{≡}f\left(x,u\right)\text{.}$E37

3.2. Multiprediction delayed feedback control

As is discussed in the reference [51], multiprediction delayed feedback control is based on a new discretization method to increase the control performance by introducing a notion of plural Poincaré maps, (P_i, P₂,..., P_N) which divide the first link angle x₁ ∈ (−π, π) into N sections as shown in Figure 4.

Figure 4.

Image of Poincaré maps [51].

By the ith (i = 1, 2,..., N) Poincaré map, the four-link horizontal bar gymnastic robot system can be discretized into the following

$x\left(k+1,i\right)=A_{i}x\left(k,i\right)+B_{i}u\left(k,i\right)\text{,}$E38

where k is the discrete time and x(k, i)(i = 1, 2,..., N) ∈ Rⁿ, A_i ∈ R^n×n(n = 8), B_i ∈ R^n×3 denote, respectively, the state error, the error transfer matrix, and the input matrix in terms with the ith Poincaré map. Furthermore, u(k, i) is defined as

$u\left(k,i\right)=K_i\left\{x\left(k,i\right)-x\left(k+1,i\right)\right\}={\widehat{K}}_{i}x\left(k,i\right)\text{.}$E39

Here, the equivalence of ${\widehat{K}}_i\text{≡}{K}_i\left(I-A_i\right)$ and K_i = K_i + N ∈ R^3×n are satisfied for each i. By introducing the variable of control period, τ, and defining u₀ as the control input to the system during (t₀, τ) while x = x₀, the input torque u is described as

$u=\mathrm{rect}\left(t\right)u_0\text{,}$E40

where rect(t) is a rectangular function as Eq. (41).

$\mathrm{rect}\left(t\right)=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathrm{if}\left(kT+t_i\right)<t\le \left(kT+t_i+\tau \right)\left(k=0,1,2,\dots \right)\hfill \\ \hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right.$E41

Here, t_i is the time at ith Poincaré map P_i(i = 1, 2,..., N).

Summarize the state variables at each one of the ith Poincaré section into one vector as follows:

$X\left(k,i\right)={\left[x\left(k,i\right),\dots ,x\left(k,N\right),x\left(k+1,1\right),\dots ,x\left(k+1,i-1\right)\right]}^T\in {\mathrm{R}}^{nN}$E42

Therefore, the close-loop system relating to X(k, i) can be stated as the following discrete-time system with periodic N.

$X\left(k,i+1\right)=F\left(i\right)X\left(k,i\right)\text{,}$E43

where,

$F\left(i\right)=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill I\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill I\hfill \\ \hfill A_i+B_i{K}_i\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{.}$E44

As it is known, the local stability of such a system can be determined by the eigenvalue of the monodromy matrix as follows:

${\Phi }_F\left(i+N,i\right)\text{≡}F\left(i+N-1\right)\cdots F\left(i+1\right)F\left(i\right)\text{.}$E45

The solution of the error transfer matrix A_i and the input matrix B_i is similar to the determination of the two-link gymnastic robot, so only the difference is shown below.

3.3. Solution for obtaining the error transfer matrix A_i

Here, let $y_1={\stackrel{.}{x}}_1$, $y_2={\stackrel{.}{x}}_2$, $y_3={\stackrel{.}{x}}_3$, $y_4={\stackrel{.}{x}}_4$. Then the state vector of four-link gymnastic robot becomes

$x=\left(x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4\right)\text{.}$E46

Consider the objective system given by the equation of motion in Eq. (37). First, given the case of u = 0, the objective system becomes

$\stackrel{.}{x}=f\left(x\left(t\right)\right)\text{≡}{\left[f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8\right]}^T$E47

Assume the state x to be the following vector function

$x\left(t\right)=\phi \left(t,x\right)\text{≡}{\left[{\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4,{\phi }_5,{\phi }_6,{\phi }_7,{\phi }_8\right]}^T\text{.}$E48

where f_ĩ, φ_ĩ(ĩ = 1 ∼ 8) are the functions of x. Introducing the following definition,

$\begin{array}{r}{\phi }_{x_{\tilde{i},0}}^m=\frac{\partial {\phi }_m}{\partial x_{\tilde{i},0}}\left(\tilde{i}=1\sim 8\right),\\ f_{x_j}^m=\frac{\partial f_m}{\partial x_j},f_{y_j}^m=\frac{\partial f_m}{\partial y_j}\left(m=1\sim 8,j=1\sim 4\right)\end{array}$E49

the variational equation becomes

$\frac{d}{dt}\left[\begin{array}{c}\hfill {\phi }_{x_{\tilde{i},0}}^1\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^2\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^3\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^4\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^5\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^6\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^7\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^8\hfill \end{array}\right]=\left[\begin{array}{cccccccc}\hfill f_{x_1}^1\hfill & \hfill f_{x_2}^1\hfill & \hfill f_{x_3}^1\hfill & \hfill f_{x_4}^1\hfill & \hfill f_{y_1}^1\hfill & \hfill f_{y_2}^1\hfill & \hfill f_{y_3}^1\hfill & \hfill f_{y_4}^1\hfill \\ \hfill f_{x_1}^2\hfill & \hfill f_{x_2}^2\hfill & \hfill f_{x_3}^2\hfill & \hfill f_{x_4}^2\hfill & \hfill f_{y_1}^2\hfill & \hfill f_{y_2}^2\hfill & \hfill f_{y_3}^2\hfill & \hfill f_{y_4}^2\hfill \\ \hfill f_{x_1}^3\hfill & \hfill f_{x_2}^3\hfill & \hfill f_{x_3}^3\hfill & \hfill f_{x_4}^3\hfill & \hfill f_{y_1}^3\hfill & \hfill f_{y_2}^3\hfill & \hfill f_{y_3}^3\hfill & \hfill f_{y_4}^3\hfill \\ \hfill f_{x_1}^4\hfill & \hfill f_{x_2}^4\hfill & \hfill f_{x_3}^4\hfill & \hfill f_{x_4}^4\hfill & \hfill f_{y_1}^4\hfill & \hfill f_{y_2}^4\hfill & \hfill f_{y_3}^4\hfill & \hfill f_{y_4}^4\hfill \\ \hfill f_{x_1}^5\hfill & \hfill f_{x_2}^5\hfill & \hfill f_{x_3}^5\hfill & \hfill f_{x_4}^5\hfill & \hfill f_{y_1}^5\hfill & \hfill f_{y_2}^5\hfill & \hfill f_{y_3}^5\hfill & \hfill f_{y_4}^5\hfill \\ \hfill f_{x_1}^6\hfill & \hfill f_{x_2}^6\hfill & \hfill f_{x_3}^6\hfill & \hfill f_{x_4}^6\hfill & \hfill f_{y_1}^6\hfill & \hfill f_{y_2}^6\hfill & \hfill f_{y_3}^6\hfill & \hfill f_{y_4}^6\hfill \\ \hfill f_{x_1}^7\hfill & \hfill f_{x_2}^7\hfill & \hfill f_{x_3}^7\hfill & \hfill f_{x_4}^7\hfill & \hfill f_{y_1}^7\hfill & \hfill f_{y_2}^7\hfill & \hfill f_{y_3}^7\hfill & \hfill f_{y_4}^7\hfill \\ \hfill f_{x_1}^8\hfill & \hfill f_{x_2}^8\hfill & \hfill f_{x_3}^8\hfill & \hfill f_{x_4}^8\hfill & \hfill f_{y_1}^8\hfill & \hfill f_{y_2}^8\hfill & \hfill f_{y_3}^8\hfill & \hfill f_{y_4}^8\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\phi }_{x_{\tilde{i},0}}^1\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^2\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^3\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^4\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^5\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^6\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^7\hfill \\ \hfill {\phi }_{x_{\tilde{i},0}}^8\hfill \end{array}\right]\text{,}$E50

where x_ĩ,0 refers to the solution at t = 0 of the ĩth state corresponding to the state vector x defined in Eq. (48).

Note that the following equation holds.

$\begin{array}{l}A=\left[\begin{array}{cccccccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill & \hfill a_4\hfill & \hfill a_5\hfill & \hfill a_6\hfill & \hfill a_7\hfill & \hfill a_8\hfill \end{array}\right]\\ a_{\tilde{i}}={\left[\begin{array}{cccccccc}\hfill {\phi }_{x_{\tilde{i},0}}^1,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^2,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^3,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^4,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^5,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^6,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^7,\hfill & \hfill {\phi }_{x_{\tilde{i},0}}^8\hfill \end{array}\right]}^T\left(\tilde{i}=1\sim 8\right).,\end{array}$E51

Carrying out integrals in numerical integration to the variational equations of Eq. (50) over the interval t ∈ [0, T] by eight times, each item of the error transfer matrix, A_i(i = 1, 2,..., N), can be calculated.

3.4. Solution for obtaining the input matrix B_i

Note that Eq. (37) can be rewritten as

$\stackrel{.}{x}=\tilde{f}\left(x,u\right)=f\left(x\right)+g\left(x\right)u\text{.}$E52

Here,

$g\left(x\right)=\left[\begin{array}{c}\hfill 0_{4\times 3}\hfill \\ \hfill M^{-1}\left(x\right)E\hfill \end{array}\right]\text{≡}\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill g_{12}\hfill & \hfill g_{13}\hfill & \hfill g_{14}\hfill \\ \hfill g_{22}\hfill & \hfill g_{23}\hfill & \hfill g_{24}\hfill \\ \hfill g_{32}\hfill & \hfill g_{33}\hfill & \hfill g_{34}\hfill \\ \hfill g_{42}\hfill & \hfill g_{43}\hfill & \hfill g_{44}\hfill \end{array}\right]\text{.}$E53

Since the solution of equation at x = x(0) can be written as

$x\left(t\right)=\phi \left(t,x_0,u_0\right)\text{.}$E54

The continuous system described by Eq. (53) will change to be a discrete system by mapping of ${\tilde{P}}_i$. Substituting Eq. (54) into Eq. (53), the following equation for the variable φ(t, x₀) can be obtained.

$\frac{d\phi \left(t,x_0,u_0\right)}{dt}=f\left(\phi \left(t,x_0,u_0\right)\right)+g\left(x\right)\mathrm{rect}\left(t\right)u_0$E55

Here, consider the case of differentiation at u₀, which can yield

$\frac{\partial }{\partial u_0}\left(\frac{d\phi }{dt}\left(t,x_0,u_0\right)\right)=\frac{\partial }{\partial u_0}\left(f\left(\phi \left(t,x_0,u_0\right)\right)\right)+g\left(x\right)\mathrm{rect}\left(t\right)\text{.}$E56

Let

${\phi }_{u_{j,0}}^m=\frac{\partial {\phi }_m}{\partial u_{j,0}}\left(m=1\sim 8,j=2,3,4\right)\text{.}$E57

Similar to previous subsection, Eq. (53) is equivalent to the following equation.

$\begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}\hfill {\phi }_{u_{j,0}}^1\hfill \\ \hfill {\phi }_{u_{j,0}}^2\hfill \\ \hfill {\phi }_{u_{j,0}}^3\hfill \\ \hfill {\phi }_{u_{j,0}}^4\hfill \\ \hfill {\phi }_{u_{j,0}}^5\hfill \\ \hfill {\phi }_{u_{j,0}}^6\hfill \\ \hfill {\phi }_{u_{j,0}}^7\hfill \\ \hfill {\phi }_{u_{j,0}}^8\hfill \end{array}\right]=\left[\begin{array}{cccccccc}\hfill f_{x_1}^1\hfill & \hfill f_{x_2}^1\hfill & \hfill f_{x_3}^1\hfill & \hfill f_{x_4}^1\hfill & \hfill f_{y_1}^1\hfill & \hfill f_{y_2}^1\hfill & \hfill f_{y_3}^1\hfill & \hfill f_{y_4}^1\hfill \\ \hfill f_{x_1}^2\hfill & \hfill f_{x_2}^2\hfill & \hfill f_{x_3}^2\hfill & \hfill f_{x_4}^2\hfill & \hfill f_{y_1}^2\hfill & \hfill f_{y_2}^2\hfill & \hfill f_{y_3}^2\hfill & \hfill f_{y_4}^2\hfill \\ \hfill f_{x_1}^3\hfill & \hfill f_{x_2}^3\hfill & \hfill f_{x_3}^3\hfill & \hfill f_{x_4}^3\hfill & \hfill f_{y_1}^3\hfill & \hfill f_{y_2}^3\hfill & \hfill f_{y_3}^3\hfill & \hfill f_{y_4}^3\hfill \\ \hfill f_{x_1}^4\hfill & \hfill f_{x_2}^4\hfill & \hfill f_{x_3}^4\hfill & \hfill f_{x_4}^4\hfill & \hfill f_{y_1}^4\hfill & \hfill f_{y_2}^4\hfill & \hfill f_{y_3}^4\hfill & \hfill f_{y_4}^4\hfill \\ \hfill f_{x_1}^5\hfill & \hfill f_{x_2}^5\hfill & \hfill f_{x_3}^5\hfill & \hfill f_{x_4}^5\hfill & \hfill f_{y_1}^5\hfill & \hfill f_{y_2}^5\hfill & \hfill f_{y_3}^5\hfill & \hfill f_{y_4}^5\hfill \\ \hfill f_{x_1}^6\hfill & \hfill f_{x_2}^6\hfill & \hfill f_{x_3}^6\hfill & \hfill f_{x_4}^6\hfill & \hfill f_{y_1}^6\hfill & \hfill f_{y_2}^6\hfill & \hfill f_{y_3}^6\hfill & \hfill f_{y_4}^6\hfill \\ \hfill f_{x_1}^7\hfill & \hfill f_{x_2}^7\hfill & \hfill f_{x_3}^7\hfill & \hfill f_{x_4}^7\hfill & \hfill f_{y_1}^7\hfill & \hfill f_{y_2}^7\hfill & \hfill f_{y_3}^7\hfill & \hfill f_{y_4}^7\hfill \\ \hfill f_{x_1}^8\hfill & \hfill f_{x_2}^8\hfill & \hfill f_{x_3}^8\hfill & \hfill f_{x_4}^8\hfill & \hfill f_{y_1}^8\hfill & \hfill f_{y_2}^8\hfill & \hfill f_{y_3}^8\hfill & \hfill f_{y_4}^8\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\phi }_{u_{j,0}}^1\hfill \\ \hfill {\phi }_{u_{j,0}}^2\hfill \\ \hfill {\phi }_{u_{j,0}}^3\hfill \\ \hfill {\phi }_{u_{j,0}}^4\hfill \\ \hfill {\phi }_{u_{j,0}}^5\hfill \\ \hfill {\phi }_{u_{j,0}}^6\hfill \\ \hfill {\phi }_{u_{j,0}}^7\hfill \\ \hfill {\phi }_{u_{j,0}}^8\hfill \end{array}\right]\\ +\mathrm{rect}\left(t\right)\times {\left[\begin{array}{cccccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill g_{1j}\hfill & \hfill g_{2j}\hfill & \hfill g_{3j}\hfill & \hfill g_{4j}\hfill \end{array}\right]}^T\end{array}$E58

Note that the following equation holds.

$\begin{array}{l}B=\left[\begin{array}{ccc}\hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \end{array}\right],\\ b_{\tilde{i}}={\left[\begin{array}{cccccccc}\hfill {\phi }_{u_{j,0}}^1,\hfill & \hfill {\phi }_{u_{j,0}}^2,\hfill & \hfill {\phi }_{u_{j,0}}^3,\hfill & \hfill {\phi }_{u_{j,0}}^4,\hfill & \hfill {\phi }_{u_{j,0}}^5,\hfill & \hfill {\phi }_{u_{j,0}}^6,\hfill & \hfill {\phi }_{u_{j,0}}^7,\hfill & \hfill {\phi }_{u_{j,0}}^8\hfill \end{array}\right]}^T\left(\tilde{i}=1,2,3\right)\text{.}\end{array}$E59

Carrying out integrals in numerical integration to Eq. (58) over the interval t ∈ [0, T] by three times, it is not difficult to obtain the value of the input matrix B_i(i = 1, 2,..., N).

4. Numerical simulations

This section shows some simulation results to verify the validity of the proposed prediction-based delayed feedback control for a two-link horizontal bar gymnastic robot. The parameters of the robot model are shown in Table 1. The stability of giant swing motion with period T = 0.7(s) was examined.

		1st link	2nd link
Mass	mi (kg)	39.5	20.7
Moment of inertia	Ii (kgm^2)	3.25	1.56
Link length	li (m)	1.2	0.879
Offset of mass center	ai (m)	0.756	0.377

Table 1.

Link parameter values.

By means of the shooting method, an initial condition was obtained such as x(0) = (π, 0.0, 8.352, − 1.19) that can achieve a free giant swing motion with a period of 0.7(s). During the first 30 s starting from this initial condition, the phase portrait of the uncontrolled system is depicted in Figure 5 in the phase plane (X, $\stackrel{.}{x}$) which refers to the angle and angular velocity of the link, respectively. It can be seen from Figure 5 that the ideal phase plane orbit of the giant swing motions traces out a closed curve.

Figure 5.

Phase portrait of $X-\stackrel{.}{x}$ (no error).

To see the stability of periodic orbits via delayed feedback control, numerical simulation was exercised from the initial condition x(0) = (π, 0.1, 9.252, − 2.39) that contained errors. First, consider the uncontrolled case, the 30 s trajectory of the orbits in the phase plane filled up a section of the phase space as in Figure 6. It can be confirmed that there was no periodicity without control.

Figure 6.

Phase portrait of $X-\stackrel{.}{x}$ (with error).

Next, consider the system controlled by the proposed method, with which the error transfer matrix A was obtained as follows:

$A=\left[\begin{array}{rrrr}\hfill 1.0054& \hfill 0.0521& \hfill 0.5986& \hfill 0.0954\\ \hfill -0.0377& \hfill 0.7685& \hfill -0.1017& \hfill 0.0210\\ \hfill 0.2982& \hfill 1.8534& \hfill 1.0041& \hfill 0.0399\\ \hfill -1.7310& \hfill -10.6479& \hfill -0.0170& \hfill 0.7728\end{array}\right]\text{.}$E93

The input matrix B in the case of τ = 10(ms) is obtained as follows:

$B=\left[\begin{array}{r}\hfill -0.000167\\ \hfill 0.000552\\ \hfill -0.004464\\ \hfill 0.024859\end{array}\right]\text{.}$E94

Based on the value of the matrix A and B shown in the above, the feedback gain K was obtained by assigning all of its poles of the system matrix $A+B\widehat{K}$ in the unit disk. Generally, in a linear system, the poles of the system matrix $A+B\widehat{K}$ are assigned as close as possible to the origin in order to decrease the convergence time; however, in such a nonlinear system as gymnastic robot, it is found that it becomes to show the divergence while the poles of $A+B\widehat{K}$ is in the disk with ratio equals 0.6 approximately. Here, two cases were studied by assigning the poles larger than 0.6 as follows.

$\left\{\begin{array}{lll}\mathit{Case}I:\hfill & \mathit{pole}=\left(0.75,0.73,0.71,0.69\right),\hfill & K=\left(-2242.7,-128.82,48526,8447.7\right),\hfill \\ \mathit{Case}II:\hfill & \mathit{pole}=\left(0.90,0.88,0.835,0.815\right),\hfill & K=\left(-206.94,354.01,1577.1,296.42\right)\text{.}\hfill \end{array}\right.$E95

In Case I, the numerical simulation results via calculated feedback gain K are shown in Figure 7. Figure 7(a) and (b) show the trajectory of the orbits in the phase plane (x, $\stackrel{.}{x}$), of which Figure 7(a) depicts the trajectory of the first 50 s, while Figure 7(b) plots that of 50 ∼ 60(s). Meanwhile, the time history of control input is shown in Figure 7(c) from which it can be seen that the control input diminished with time. It is obvious that from Figure 7(c), the control input u began to vanish from about t = 30(s), and from Figure 7(b) the periodicity began to appear from approximately t = 50(s).

Figure 7.

Phase portrait and joint torque (Case I): (a) Phase portrait of $X-\stackrel{.}{x}$(t=0~50s); (b) Phase portrait of $X-\stackrel{.}{x}$ (t=50~60s); (c) Joint torque.

However, on further analysis conducted, it was found that one can change the type of giant swing type by changing the pole placement of the matrix $A+B\widehat{K}$. This phenomenon is shown in Figure 8 which depicts the change histories of link angle, angular velocity, and input control with 160 s in Case II. The first 150 s trajectory of the orbits in the phase plane (x, $\stackrel{.}{x}$) is depicted in Figure 8(a) and the remaining is shown in Figure 8(b). Moreover, the time history of input torque is shown in Figure 8(c).

Figure 8.

Phase portrait and joint torque (Case II): (a) Phase portrait of $X-\stackrel{.}{x}$ (t=0~150s); (b) Phase portrait of $X-\stackrel{.}{x}$ (t=150~160s); (c) Joint torque.

It could be said that in Case II, the proposed method showed the ability of controlling the giant-swing motion to an unknown periodic orbit, which is known as one of the advantages of delayed feedback control.

Moreover, Figures 9 and 10 depict the stick diagrams based on one period data after the giant swing motion converging to a stable orbit in Case I and Case II, in which the figures of (a), (b), and (c) plot, respectively, the angle, angular velocity, and stick figure. From the above two figures, it can be seen that the difference between the assigned poles caused periodic giant swing motion to change to a different type.

Figure 9.

Angle, angular velocity, and stick (Case I).

Figure 10.

Angle, angular velocity, and stick (Case II).

5. Conclusion

This chapter studied the behavior of the giant swing motions of a two-link and a four-link horizontal bar gymnastic robot systems via delayed feedback control. First, it has been discussed as a solution to calculate analytically the error transfer matrix and the input matrix by which the stability of DFC for the original two-link gymnastic robot system is equivalent to that of the corresponding discrete system. Moreover, it was introduced as a method by which a feedback gain to ensure the stability of original system can be determined. Second, a modified DFC method, MDFC, has been extended to a four-link gymnastic robot. Plural Poincaré maps were defined so that the stability of the close-loop system can be evaluated based on the theory of monodromy matrix. Finally, the simulation results showed its effectiveness.

Acknowledgments

This work is supported in part by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and in part by the Scientific Research Foundation for Young Teachers, Shanghai JiaoTong University.