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Abstract

For a hybrid mechanism used for automobile electro-coating conveying, a chattering-free sliding mode synchronization controller is proposed to improve the synchronization performance during motion process. The Jacobi matrix of the mechanism is calculated according to kinematic analysis, and the dynamic model is established by Lagrange method. Since the mechanism consists of two sets of hybrid mechanisms bilaterally, a novel synchronization error including the synchronization error between two ends of the end-effector is designed. By combining cross-coupling control with chattering-free sliding mode control, a novel chattering-free sliding mode synchronization controller is proposed. The stability of the proposed algorithm is proved by Lyapunov stability theorem. The simulation and experimental results show that the proposed controller can effectively reduce the chattering of driving forces and further improve the synchronization performance and tracking accuracy of the system.

Keywords

Hybrid mechanism synchronization error synchronization control sliding mode control chattering free

Introduction

Hybrid mechanisms combine advantages of both serial and parallel mechanisms. Based on this mechanical superiority, a novel bilaterally symmetrical type hybrid mechanism for automobile electro-coating conveying was developed by the research group. This hybrid mechanism has simple structure, strong bearing capacity, wide applicability, and high flexibility level. Therefore, it can remedy the defects of existing conveying mechanisms with cantilever structure.¹

A hybrid mechanism usually consists of multiple active joints, and the overall synchronization performance of the system is closely related to the coordinative motions of all active joints. Most existing controllers of hybrid mechanisms did not fully consider the synchronization performance of the system. The tracking error of a certain active joint is only rectified by its loop, while other control loops will not respond.² This may cause the asynchrony of active joints and reduce overall synchronization performance, or even result in abrasion or damage of the mechanism.³ Therefore, synchronization control theory should be utilized to guarantee the synchronization performance of hybrid mechanisms. The classical synchronization control theory is mainly divided into parallel control, master–slave control, and cross-coupling control (CCC).^4–6 Among them, the CCC is first proposed by Koren⁷ for a biaxial motion platform. This synchronization control scheme adopted velocity or angle error of subsystem as additional feedback signals to reflect the load variation of any axis. The controller improved the synchronization performance via reducing the additional feedback signal. This signal is the synchronization error. After being proposed, CCC was combined with some control strategies and the synchronization error was defined in different ways. Sun et al.⁸ defined the coupling pose error by coupling the position synchronization error of the drive joint and its two neighbor joints with pose error of the end-effector. By utilizing feedforward/feedback control strategy, the coupled error converged to zero and the synchronization performance was improved. Byun and Jee⁹ applied the CCC into a 5-axis system. By utilizing a novel real-time tool orientation error model, the synchronization performance was effectively improved. Chen and Chen¹⁰ combined the trajectory tracking error and contour error of a multi-axis motion system as the synchronization error and designed a cross-coupling position command shaping controller to make it converge to zero. Thus, the synchronization performance of the system was improved. In this article, the mechanism studied consists of two sets of hybrid mechanisms bilaterally. The existing synchronization control schemes are only suitable for single mechanisms, while they cannot be directly or perfectly applied to the bilaterally symmetrical type hybrid mechanism. Therefore, a novel synchronization error needs to be designed with consideration to the synchronization error between two ends of the end-effector.

Control strategies for hybrid mechanisms are mainly divided into the kinematic control and the dynamic control. Since the dynamic control considers dynamic features of the mechanism, it can theoretically achieve better control performance compared to the kinematic control.¹¹ However, there are some limits for dynamic control, such as determination of dynamic parameters and accurate dynamic modeling. Besides, hybrid mechanisms in engineering application are usually impacted by friction, environmental noise, and other external disturbances. The impact usually results in the uncertainty problem.^12–14 Conventional control methods, such as proportional–derivative control¹⁵ and computed torque control,¹⁶ cannot effectively solve this problem, while the sliding mode control (SMC) has strong robustness and fast response speed. Therefore, it can effectively solve the uncertainty problem and is suitable for implementing hybrid mechanism.^17,18 However, since the existence of the switching function in SMC, the driving force is discontinuous, which may lead to chattering phenomena in driving forces. The existing approaches to reduce the chattering can be mainly divided into three ways: boundary layer saturation method,^19,20 estimated uncertainties method,²¹ and higher order SMC method.^22–24 However, since the introduction of the continuous function in boundary layer saturation method, the robustness of the system decreased. The latter two methods usually complicate the calculation program of controllers. This may reduce the response speed of the system. Thus, a simple and effective method is needed to reduce chattering of SMC.

This article is arranged as follows: first, the Jacobi matrix is calculated according to kinematic analysis. Second, the dynamic model of the lift turnover mechanism is established by utilizing Lagrange method. Third, the novel synchronization error is designed. Fourth, a novel chattering-free sliding mode synchronization controller is proposed and the stability of the proposed control algorithm is proved by Lyapunov stability theorem. Finally, the proposed controller is verified by simulation and actual experiment, and the results indicate that the proposed controller can effectively reduce the chattering of driving forces and further improve the synchronization performance and tracking accuracy of the system.

Kinematic analysis

The novel hybrid mechanism is shown in Figure 1, which is mainly composed of a walking mechanism and a lifting and turnover mechanism. The lifting and turnover mechanism is the main part of the mechanism, which is this article researches on. The diagram of the lifting and turnover mechanism is shown in Figure 2, where B _i (i = 1, 2, 3, 4) and P _i (i = 1, 2) are active joints.

Figure 1.

The novel hybrid mechanism.

Figure 2.

Diagram of the lifting and turnover mechanism.

According to the structure of the lifting and turnover mechanism and utilizing the rod length constraint equation, the inverse kinematic equation of the mechanism can be calculated as

${\begin{matrix} x_{1}^{2} + z_{1}^{2} = L_{1}^{2} \\ x_{2}^{2} + z_{1}^{2} = L_{1}^{2} \\ x_{3}^{2} + z_{2}^{2} = L_{1}^{2} \\ x_{4}^{2} + z_{2}^{2} = L_{1}^{2} \\ r_{2} ϕ_{1} = r_{1} β_{1} \\ r_{2} ϕ_{2} = r_{1} β_{2} \\ z_{1} = z_{2} = z \\ β_{1} = β_{2} = β \end{matrix}$ (1)

where $z_{i} (i = 1, 2)$ represents the position of the ith slider in Z-axis and $β_{i} (i = 1, 2)$ represents the anti-clockwise rotary angle around Y-axis of the ith end of the link. $x_{i} (i = 1, 2, 3, 4)$ represents the position of the ith slider in X-axis. $ϕ_{i} (i = 1, 2)$ represents the anti-clockwise rotary angle around Y-axis of the ith active wheel. $L_{1}$ is the length of the first connecting rod. $r_{1}$ is the radius length of the driven pulley. $r_{2}$ is the radius length of the driving wheel.

By differentiating equation (1) with respect to time, the equation can be derived as

$[\begin{matrix} {\overset{\cdot}{x}}_{1} \\ {\overset{\cdot}{x}}_{2} \\ {\overset{\cdot}{x}}_{3} \\ {\overset{\cdot}{x}}_{4} \\ {\overset{\cdot}{ϕ}}_{1} \\ {\overset{\cdot}{ϕ}}_{2} \end{matrix}] = [\begin{matrix} \frac{\partial x_{1}}{\partial z} & \frac{\partial x_{1}}{\partial β} \\ \frac{\partial x_{2}}{\partial z} & \frac{\partial x_{2}}{\partial β} \\ \frac{\partial x_{3}}{\partial z} & \frac{\partial x_{3}}{\partial β} \\ \frac{\partial x_{4}}{\partial z} & \frac{\partial x_{4}}{\partial β} \\ \frac{\partial ϕ_{1}}{\partial z} & \frac{\partial ϕ_{1}}{\partial β} \\ \frac{\partial ϕ_{2}}{\partial z} & \frac{\partial ϕ_{2}}{\partial β} \end{matrix}] [\begin{matrix} \overset{\cdot}{z} \\ \overset{\cdot}{β} \end{matrix}]$ (2)

Equation (2) can be written as $\overset{\cdot}{x} (t) = J \overset{\cdot}{q} (t)$ in short, where $\overset{\cdot}{x} (t)$ is the actual velocity of active joints. J is the Jacobi matrix of the lifting and turnover mechanism and $\overset{\cdot}{q} (t)$ is the actual velocity of the midpoint of the link, which can be seen as the end-effector of the mechanism.

Dynamic modeling

Lagrange method has the advantage of simple form and efficient design process of the algorithm.^25,26 The Lagrange function L is defined as L = T − P , where T is the kinetic energy and P is the potential energy of the system. The Lagrange equation can be expressed as

$\frac{d}{dt} (\frac{\partial L (q, \overset{\cdot}{q})}{\partial \overset{\cdot}{q}}) - \frac{\partial L (q, \overset{\cdot}{q})}{\partial q} = Q$

The standard dynamic equation can be expressed as

$M (q) \overset{\cdot\cdot}{q} (t) + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} (t) + G (q) = Q$ (3)

where $M (q)$ is the inertia matrix. $C (q, \overset{\cdot}{q})$ is the Coriolis and centrifugal term. $G (q)$ is the gravity term. Q is the generalized control law. $q (t)$ is the actual pose of the end-effector. $\overset{\cdot\cdot}{q} (t)$ is the actual acceleration of the end-effector.

Cartesian space dynamic modeling

The kinetic energy T of the system can be expressed as

$T = T_{P} + T_{L} + T_{T} + T_{S}$ (4)

where $T_{P}$ is the vehicle kinetic energy, $T_{L}$ is the branched chain kinetic energy, $T_{T}$ is the transmission kinetic energy, and $T_{S}$ is the kinetic energy of sliders.

The potential energy of the system P can be expressed as

$P = P_{P} + P_{L} + P_{T} + P_{S}$ (5)

where $P_{P}$ is the vehicle potential energy, $P_{L}$ is the branched chain potential energy, $P_{T}$ is the transmission potential energy, and $P_{S}$ is the potential energy of sliders.

According to equations (3)–(5), the dynamic equation of the lifting and turnover mechanism can be expressed as

$M (q) \overset{\cdot\cdot}{q} (t) + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} (t) + G (q) = Q$ (6)

where $M (q)$ is obtained from $T = (1 / 2) \overset{\cdot}{q} (t)^{T} M (q) \overset{\cdot}{q} (t)$ , $C (q, \overset{\cdot}{q}) = \overset{\cdot}{M} (q) - (1 / 2) (\partial / \partial q) (\overset{\cdot}{q} (t)^{T} M (q))$ , and $G (q) = \partial P / \partial q$ . The final result is shown as below

$M (q) = [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}]$

$C (q, \overset{\cdot}{q}) = [\begin{matrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{matrix}]$

$G (q) = [\begin{matrix} G_{1} \\ G_{2} \end{matrix}]$

$\begin{array}{l} M_{11} = m_{p} + 4 m_{l 4} + \frac{1}{2} (m_{l 1} + m_{l 2} + 4 m_{1} + 4 m_{2} + 4 m_{a}) Δ_{1} \\ + 0.5 m_{l 1} + 0.5 m_{l 2} + 2 m_{b} + m_{l 3} \end{array}$

$M_{12} = - m_{p} Δ_{2} \sin β - 2 m_{l 4} L_{4} \cos \frac{1}{2} θ \sin β$

$M_{21} = - m_{p} Δ_{2} \sin β - 2 m_{l 4} L_{4} \cos \frac{1}{2} θ \sin β$

$\begin{array}{l} M_{22} = m_{p} (\frac{1}{12} a^{2} + \frac{1}{12} c^{2} + 2 Δ_{2}^{2}) + \frac{1}{2} m_{l 3} r_{l 3}^{2} \\ + \frac{7}{3} m_{l 4} L_{4}^{2} + m_{a} R^{2} + m_{b} r^{2} \end{array}$

$C_{11} = \frac{1}{4} \overset{\cdot}{z} (m_{l 1} + m_{l 2} + 4 m_{1} + 4 m_{2} + 4 m_{a}) Δ_{m 1}$

$C_{12} = \overset{\cdot}{β} (- \frac{1}{2} m_{p} Δ_{2} \cos β - 2 m_{l 4} L_{4} \cos \frac{1}{2} θ \cos β)$

$C_{21} = \overset{\cdot}{β} (- m_{p} Δ_{2} \cos β - m_{l 4} L_{4} \cos \frac{1}{2} θ \cos β)$

$C_{22} = \overset{\cdot}{z} (\frac{1}{2} m_{p} Δ_{2} \cos β + m_{l 4} L_{4} \cos \frac{1}{2} θ \cos β)$

$G_{11} = (m_{p} + 4 m_{l 4} + m_{l 1} + m_{l 2} + m_{l 3} + 2 m_{b}) g$

$G_{21} = (- m_{p} Δ_{2} \sin β - 4 m_{l 4} \cos \frac{1}{2} θ \cos β) g$

$Δ_{1} = \frac{z^{2}}{L_{1}^{2} - z^{2}}$

$Δ_{2} = L_{4} \cos \frac{1}{2} θ + \frac{1}{2} b$

$Δ_{m 1} = \frac{2 L_{1}^{2} z}{{(L_{1}^{2} - z^{2})}^{2}}$

where $m_{p}$ is the weight of the car body. $m_{l 1}$ , $m_{l 2}$ , and $m_{l 3}$ are the weights of connecting rods 7, 11, and 16, respectively. $m_{l 4}$ is the weight of the bracket on the car body fixed frame. $m_{1}$ and $m_{2}$ are the weights of the first slider 5 and the second slider 9, respectively. $m_{a}$ and $m_{b}$ are the weights of the active wheel 13 and the driven pulley 15, respectively. a, b, and c represent the length, width, and height of the car body, respectively. $r_{l 3}$ is the radius of the link. $r_{1}$ and $r_{2}$ are the radii of the driven pulley 15 and the active wheel 13, respectively. $L_{1}$ and $L_{2}$ are the lengths of the connecting rods 7 and 11, respectively. $L_{4}$ is the oblique bracket length of the car body fixed frame 17. $θ$ represents the angle between two levers on the car body fixed frame 17. Values of parameters are shown in Table 1.

Table 1.

Parameter values.

Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
$m_{p}$	17 kg	$m_{1}$	2.0 kg	a	1.125 m	$r_{2}$	0.03 m
$m_{l 1}$	1.5 kg	$m_{2}$	2.0 kg	b	0.65 m	$L_{1}$	0.311 m
$m_{l 2}$	1.5 kg	$m_{a}$	0.5 kg	c	0.56 m	$L_{2}$	1 m
$m_{l 3}$	3.0 kg	$m_{b}$	0.25 kg	$r_{l 3}$	0.015 m	$L_{4}$	0.65 m
$m_{l 4}$	0.5 kg	$θ$	120°	$r_{1}$	0.05 m

The external disturbance and friction are considered to ensure that the dynamic model can reflect the actual system more accurately. The improved dynamic model can be expressed as

$M (q) \overset{\cdot\cdot}{q} (t) + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} (t) + G (q) + J^{T} D (t) + J^{T} F (t) = Q$ (7)

where $D (t)$ is the friction term. $D (t) = F_{c} sgn (\overset{\cdot}{x}) + B_{c} \overset{\cdot}{x}$ , where $F_{c}$ is the coulomb friction matrix and $B_{c}$ is the viscosity coefficient matrix. $x = (x_{1}, x_{2}, x_{3}, x_{4}, φ_{1}, φ_{2})^{T}$ , where $x_{i} (i = 1, 2, 3, 4)$ is the X-axis position of the ith slider. $φ_{j} (j = 1, 2)$ is the anti-clockwise rotary angle around Y-axis of the ith active wheel. $F (t)$ represents the external disturbance term.

Equation (7) has the following properties:

M is symmetric and positive definite.

$\overset{\cdot}{M} - 2 C$ is an anti-symmetric matrix.

Joint space dynamic model modeling

Since the actual poses of the end-effector are difficult to be directly measured and controlled in engineering application, the dynamic model in Cartesian space is converted to joint space.

The velocity and acceleration of the link’s midpoint have the following relationship with the active joints

$\overset{\cdot}{q} (t) = J^{+} \overset{\cdot}{x} (t)$ (8)

$\overset{\cdot\cdot}{q} (t) = J^{+} \overset{\cdot\cdot}{x} (t) - J^{+} \overset{\cdot}{J} J^{+} \overset{\cdot}{x} (t)$ (9)

where $\overset{\cdot}{x} (t)$ and $\overset{\cdot\cdot}{x} (t)$ are the velocity and acceleration of active joints.

According to equations (7)–(9), the switching relationship from the Cartesian space to the joint space can be expressed as

$M (x) = {[J^{T}]}^{+} M (q) J^{+}$ (10)

$C (x, \overset{\cdot}{x}) = {[J^{T}]}^{+} [C (q, \overset{\cdot}{q}) - M (q) J^{+} \overset{\cdot}{J}] J^{+}$ (11)

$G (x) = {[J^{T}]}^{+} G (q)$ (12)

The dynamic equation of the lifting and turnover mechanism in joint space can be expressed as

$M (x) \overset{\cdot\cdot}{x} (t) + C (x, \overset{\cdot}{x}) \overset{\cdot}{x} (t) + G (x) + D (t) + F (t) = τ$ (13)

where $M (x)$ is the inertia matrix in joint space. $C (x, \overset{\cdot}{x})$ is the corresponding Coriolis and centrifugal term. $G (x)$ is the gravity term. $τ$ is the control law vector of active joints.

Synchronization error design

The synchronization error is different from conventional tracking error. It contains not only error information of the joint itself but also the error information between joints.

Conventional synchronization error design

The position tracking error of six active joints (four translation joints and two rotation joints) can be expressed as

$\begin{matrix} e_{xi} (t) = x_{id} (t) - x_{i} (t), i = 1, 2, 3, 4 \\ e_{φ i} (t) = φ_{jd} (t) - φ_{j} (t), j = 1, 2 \end{matrix}$ (14)

where $x_{id} (t)$ and $x_{i} (t)$ , respectively, represent the desired position and the actual position of the ith active joint. $φ_{jd} (t)$ and $φ_{j} (t)$ , respectively, represent the desired position and the actual position of the jth rotation joint. Based on synchronous control theory, if the tracking error of each active joint meets the following equation, each joint is synchronized

$\begin{matrix} lim_{t \to \infty} e_{x 1} (t) = lim_{t \to \infty} e_{x 2} (t) \\ lim_{t \to \infty} e_{x 2} (t) = lim_{t \to \infty} e_{x 3} (t) \\ lim_{t \to \infty} e_{x 3} (t) = lim_{t \to \infty} e_{x 4} (t) \\ lim_{t \to \infty} e_{x 4} (t) = lim_{t \to \infty} e_{x 1} (t) \\ lim_{t \to \infty} e_{φ 1} (t) = lim_{t \to \infty} e_{φ 2} (t) \\ lim_{t \to \infty} e_{φ 2} (t) = lim_{t \to \infty} e_{φ 1} (t) \end{matrix}$ (15)

The synchronization errors of the joints can be defined as

$\begin{matrix} ε_{1} (t) = e_{x 1} (t) - e_{x 2} (t) \\ ε_{2} (t) = e_{x 2} (t) - e_{x 3} (t) \\ ε_{3} (t) = e_{x 3} (t) - e_{x 4} (t) \\ ε_{4} (t) = e_{x 4} (t) - e_{x 1} (t) \\ ε_{5} (t) = e_{φ 1} (t) - e_{φ 2} (t) \\ ε_{6} (t) = e_{φ 2} (t) - e_{φ 1} (t) \end{matrix}$ (16)

where $ε_{i} (t) (i = 1, 2, 3, 4, 5, 6)$ represents the synchronization error of the ith active joint.

Therefore, according to equation (16), the synchronization error can be expressed as

$ε (t) = H_{1} e_{x} (t)$ (17)

where $e_{x} (t) = [e_{x 1} (t), e_{x 2} (t), e_{x 3} (t), e_{x 4} (t), e_{φ 1} (t), e_{φ 2} (t)]^{T}$ and $H_{1} = [\begin{matrix} 1 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 & 0 & 0 \\ - 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 1 \end{matrix}]$ .

Novel synchronization error design

Equation (17) only takes the synchronization error of the active joints into consideration. In order to fully consider the synchronization performance of the bilaterally symmetrical type hybrid mechanism, the synchronization error between two sides of the mechanism should be considered. Therefore, a novel synchronization error is proposed.

Synchronization error between the two ends of the link

The actual pose of the two ends of the link is assumed as $q_{i} (t), (i = 1, 2)$ , and the desired pose is $q_{id} (t), (i = 1, 2)$ . Therefore, the pose error $e_{i} (t)$ can be expressed as

$e_{i} (t) = q_{id} (t) - q_{i} (t), i = 1, 2$ (18)

The control targets of the synchronization motion of the two ends of the link can be defined as

$f (q_{1} (t), q_{2} (t)) = q_{1} (t) - q_{2} (t) = 0$ (19)

$f (q_{1 d} (t), q_{2 d} (t)) = q_{1 d} (t) - q_{2 d} (t) = 0$ (20)

According to equations (19) and (20), one knows

$f (q_{1 d} (t), q_{2 d} (t)) - f (q_{1} (t), q_{2} (t)) = q_{1} (t) - q_{2} (t) - q_{1 d} (t) + q_{2 d} (t) = e_{1} (t) - e_{2} (t)$ (21)

Therefore, the novel synchronization error can be expressed as

$ε (t) = e_{1} (t) - e_{2} (t)$ (22)

If $e_{1} (t) \to 0$ and $e_{2} (t) \to 0$ , the two ends of the link are moving along the desired trajectory. If $ε (t) \to 0$ , the two ends of the link are synchronous.

Map between end-effector errors and joint errors under ideal condition

According to the kinematic analysis, one knows the following relationship

${\begin{matrix} {(\frac{x_{1} - x_{2}}{2})}^{2} + {z_{1}}^{2} = L_{1}^{2} \\ φ_{1} = \frac{r_{1}}{r_{2}} β_{1} \end{matrix}$ (23)

By differentiating equation (23), the matrix can be rewritten as

$δ Y_{1} = R_{1} δ X_{1}$ (24)

where $δ Y_{1} = [δ z_{1}, δ β_{1}]^{T}$ and $δ X_{1} = [δ x_{1}, δ x_{2}, δ φ_{1}]^{T}$ . $δ Y_{1}$ and $δ X_{1}$ , respectively, represent the errors between one end of the link and its desired position.

Therefore, tracking error of one end of the link can be expressed as

$e_{1} (t) = R_{1} {e'}_{1} (t)$ (25)

Similarly, tracking error of the other end of the link can be expressed as

$e_{2} (t) = R_{2} {e'}_{2} (t)$ (26)

where ${e'}_{i} (t) (i = 1, 2)$ represents position error of the ith end of the link. $R_{i} (i = 1, 2)$ represents the transfer matrix of position error from active joints to the ith end of the link, and the matrices can be expressed as

$\begin{array}{l} R_{1} = [\begin{matrix} \frac{x_{2} - x_{1}}{4 z_{1}} & \frac{x_{1} - x_{2}}{4 z_{1}} & 0 \\ 0 & 0 & \frac{r_{2}}{r_{1}} \end{matrix}], \\ R_{2} = [\begin{matrix} \frac{x_{4} - x_{3}}{4 z_{2}} & \frac{x_{3} - x_{4}}{4 z_{2}} & 0 \\ 0 & 0 & \frac{r_{2}}{r_{1}} \end{matrix}] \end{array}$

Therefore, position error of the end of the link can be expressed as

${e'}_{i} (t) = V_{i} e_{x} (t) (i = 1, 2)$ (27)

where $V_{i} (i = 1, 2)$ represents the relationship between desired position error in the ith side of the link and the tracking error of active joints, and the values are

$\begin{array}{l} V_{1} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}], \\ V_{2} = [\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] \end{array}$

According to equations (25)–(27) and (22), the novel synchronization error of the entire mechanism can be expressed as

$ε (t) = H_{2} e_{x} (t)$ (28)

where $H_{2} = R_{1} V_{1} - R_{2} V_{2}$ .

Controller design

The control system principle block diagram is shown in Figure 3.

Figure 3.

The control system principle block diagram.

In Figure 3, $q_{d} (t)$ is the desired pose of the link’s midpoint in the Cartesian space. In order to realize the control of active joints, it is converted to desired position $x_{d} (t)$ and desired velocity ${\overset{\cdot}{x}}_{d} (t)$ of active joints in the joint space according to inverse kinematics. $x (t)$ and $\overset{\cdot}{x} (t)$ are the actual position and velocity of active joints, respectively. $e_{x} (t)$ and ${\overset{\cdot}{e}}_{x} (t)$ are the position error and velocity error of active joints, respectively. CFSMSC is the proposed chattering-free sliding mode synchronization controller and Q is the control law calculated by the controller. The controller is designed in the joint space, while the tracking pose error $e (t)$ , tracking velocity error $\overset{\cdot}{e} (t)$ , and the novel synchronization error $ε (t)$ are calculated in the Cartesian space. Thus, $x (t)$ and $\overset{\cdot}{x} (t)$ are converted to the actual pose $q (t)$ and actual velocity $\overset{\cdot}{q} (t)$ according to forward kinematics.

The desired pose of the link’s midpoint is $q_{d} (t) = [z_{d}, β_{d}]^{T}$ . The pose error $e (t)$ and the velocity error $\overset{\cdot}{e} (t)$ can be expressed as

${\begin{matrix} e (t) = q_{d} (t) - q (t) \\ \dot{e} (t) = {\dot{q}}_{d} (t) - \dot{q} (t) \end{matrix}$ (29)

In order to realize the stable tracking of the pose of the link’s midpoint and the synchronous motion of each active joint, the synchronization error $e^{*} (t)$ is defined by combining the pose error $e (t)$ and the synchronization error $ε (t)$ . It can be expressed as

$e^{*} (t) = e (t) + λ \int_{0}^{t} ε (t) dt$ (30)

where $λ = diag (λ_{1}, λ_{2})$ , and it is a diagonal positive-definite coupling parameter matrix.

The sliding surface is designed as

$S = {\overset{\cdot}{e}}^{*} (t) + B e^{*} (t)$ (31)

where $B = diag (b_{1}, b_{2})$ , and B is invertible. b₁ and b₂ are adjustable parameters and meet Holzer Woods condition. The proposed controller is designed as follows.

By differentiating S in equation (31) and substituting equation (30) into it, one knows

$\dot{S} = {\overset{\cdot\cdot}{e}}^{*} (t) + B {\dot{e}}^{*} (t) = B \dot{e} (t) + λ B ε (t) + {\overset{\cdot\cdot}{q}}_{d} (t) - \overset{\cdot\cdot}{q} (t) + λ \dot{ε} (t)$ (32)

It can be obtained from equation (7) that

$\overset{\cdot\cdot}{q} (t) = M (q)^{- 1} (Q - C (q, \overset{\cdot}{q}) \overset{\cdot}{q} (t) - G (q) - J^{T} D (t) - J^{T} F (t))$ (33)

The trending law is set as

$\overset{\cdot}{S} = - K sgn (S)$ (34)

where $K = diag (k_{1}, k_{2})$ and $k_{1} > 0, k_{2} > 0$ .

According to equations (32)–(34), the control law based on the novel synchronization error can be derived as

$\begin{matrix} Q = M (q) (B \overset{\cdot}{e} (t) + λ B ε (t) + λ \overset{\cdot}{ε} (t) + K sgn (S) + {\overset{\cdot\cdot}{q}}_{d} (t)) \\ + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} (t) + G (q) + J^{T} (D (t) + F (t)) \end{matrix}$ (35)

where $M (q)$ is an inertia matrix. $C (q, \overset{\cdot}{q})$ is the Coriolis and centrifugal term. $G (q)$ is the gravity term. Q is the generalized control law. $D (t)$ is the friction term. $F (t)$ is the external disturbance term and sgn( S ) is the sign function.

Since the sign function sgn( S ) is discontinuous, the driving force has the problem of chattering in the actual control system. To remedy the chattering, the sign function sgn( S ) is replaced by a continuous function $S / | | S | |$ , where $S = [s_{1} s_{2}]^{T}$ , $| | S | | = (s_{1}^{2} + s_{2}^{2})^{1 / 2}$ , and $s_{i} (i = 1, 2)$ is the value of the matrix S . The control law can be finally expressed as

$\begin{matrix} Q = & M (q) (B \overset{\cdot}{e} (t) + λ B ε (t) + λ \overset{\cdot}{ε} (t) + K \frac{S}{‖ S ‖} + {\overset{\cdot\cdot}{q}}_{d} (t)) \\ + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} (t) + G (q) + J^{T} (D (t) + F (t)) \end{matrix}$ (36)

The stability theorem is considered to prove the stability of the proposed controller.

Theorem 1

Stability theorem

Consider a nonlinear uncertain dynamical system represented by equation (7). If the chattering-free sliding mode synchronization controller shown in equation (36) is applied, asymptotic robust stability of the closed-loop system in the presence of model uncertainties and disturbances is guaranteed. The stability of the designed control algorithm is demonstrated in the following.

Proof

The Lyapunov function is set as

$V = \frac{1}{2} S^{T} S$ (37)

By differentiating equation (37), $\overset{\cdot}{V}$ can be expressed as

$\begin{matrix} \dot{V} = \frac{1}{2} S^{T} \dot{S} + \frac{1}{2} {\dot{S}}^{T} S \\ = S^{T} \dot{S} \\ = S^{T} [B \dot{e} (t) + λ B ε (t) + {\overset{\cdot\cdot}{q}}_{d} (t) - \overset{\cdot\cdot}{q} (t) + λ \dot{ε} (t)] \\ = S^{T} [B \dot{e} (t) + λ B ε (t) + {\overset{\cdot\cdot}{q}}_{d} (t) - M {(q)}^{- 1} (Q - C (q, \dot{q}) \dot{q} - G (q) - J^{T} D (t) - J^{T} F (t)) + λ \dot{ε} (t)] \\ = - S^{T} K \frac{S}{‖ S ‖} \\ = - \frac{K}{‖ S ‖} (s_{1}^{2} + s_{2}^{2}) \\ = - K ‖ S ‖ \end{matrix}$ (38)

where K is a positive-definite matrix and $| | S | | \geq 0$ . Therefore, $\overset{\cdot}{V} \leq 0$ . According to Lyapunov stability theorem, the designed control algorithm is stable.

According to the SMC theorem, one knows $S \to 0$ as $t \to \infty$ . Therefore, $e^{*} (t) \to 0$ and ${\overset{\cdot}{e}}^{*} (t) \to 0$ as $t \to \infty$ . Since the relationship between $e (t)$ and $ε (t)$ is that $ε (t) = H_{2} e_{x} (t) = H_{2} J e (t)$ , as shown in equation (28), ${\overset{\cdot}{e}}^{*} (t) \to 0$ as $t \to \infty$ actually is the first-order differential equation of $e (t)$ . Therefore, $e (t) \to 0$ and $ε (t) \to 0$ as $t \to \infty$ .

Simulation results

The simulation selects ode3 mode. The sampling period is 1 ms, and the motion time is 6 s. Besides, according to the automobile electro-coating requirement, the desired motion trajectory of the link’s midpoint is set as equations (39) and (40). And the initial pose of the link’s midpoint is set as (0.25 m, 0 rad). Parameters of sliding surface and the controller are $b_{1} = 9$ , $b_{2} = 18$ , $k_{1} = 10$ , $k_{2} = 10$ , $λ_{1} = 10$ , $λ_{2} = 10$ , $D (t) = diag (3.5, 4.1)$ , $F (t) = (3 \sin (2 π t), 3 \sin (2 π t))$

$z = {\begin{matrix} 0.25 (0 s < t \leq 2 s) \\ 0.2 + 0.05 \cos [2 π (t - 2)] (2 s < t \leq 4 s) \\ 0.25 (4 s < t \leq 6 s) \end{matrix}$ (39)

$β = {\begin{matrix} π - π \cos (0.5 π t) (0 s < t \leq 2 s) \\ π (2 s < t \leq 4 s) \\ 2 π - π \cos [0.5 π (t - 4)] (4 s < t \leq 6 s) \end{matrix}$ (40)

The simulation results are shown in Figures 4 –7. In the figures, SMC represents the SMC based on the tracking error, SSMC1 represents the SMC based on the synchronization error (equation (17)), and SSMC2 represents the proposed controller.

Figure 4.

Desired tracking curve and tracking error curve of each pose component of the link’s midpoint: (a) desired tracking curve in Z direction, (b) desired tracking curve of β, (c) tracking error curve in Z direction, and (d) tracking error curve of β.

Figure 5.

Tracking error curve of each active joint in one side of the link: (a) tracking error of the first slider, (b) tracking error of the second slider, and (c) tracking error of the active wheel.

Figure 6.

Synchronization error curve of each pose component on the two ends of the link: (a) synchronization error in Z direction and (b) synchronization error of β.

Figure 7.

Driving force of each pose component on the two ends of the link: (a) driving force in Z direction and (b) driving force of β.

Figure 4 shows the desired tracking curve and tracking error curve of each pose component of the link’s midpoint. Figure 5 shows tracking error curve of each active joint in one side of the link. Figure 6 shows the synchronization error curve of each pose component on the two ends of the link. Since the proposed controller takes synchronization error between two ends of the link into consideration, the synchronization errors of the two ends of the link are much smaller than the one does not consider such errors. Besides, tracking error of each active joint is also effectively reduced with the proposed controller. Therefore, it can be concluded that the synchronization performance of the system is improved due to the novel synchronization error. Moreover, the tracking trajectories are chattering free and the tracking error is effectively reduced due to the improvement in synchronization performance.

Figure 7 shows driving force of each pose component on the two ends of the link. It is obvious that the driving forces are chattering free because the switching function is replaced by the continuous function. This is important for actual application of this controller. If the driving forces chatter violently, the motors will be burned out due to the sudden change in voltage direction. Therefore, it can be concluded that the proposed controller can effectively eliminate the chattering of driving forces and is an implementable control strategy.

Experimental results

The proposed controller was experimentally evaluated based on the bilaterally symmetrical type hybrid mechanism for automobile electro-coating conveying as shown in Figure 8.

Figure 8.

Experiment setup.

The test setup is composed of the bilaterally symmetrical type hybrid mechanism, one real-time industrial computer, one Delta Tau UMAC controller, and eight Mitsubishi Electric servo systems. During physical implementation, shift registers with proper initial values are adopted to remove the algebraic loop caused by acceleration feedback. The desired moving trajectory for the end-effector is shown as equations (39) and (40). The tracking error curves of the proposed controller are compared with the SSMC1 since its performance is better than SMC in MATLAB simulation. The experimental results are shown in Figure 9.

Figure 9.

Experimental results: (a) tracking error curve in Z direction and (b) tracking error curve of β.

It can be seen from Figure 9 that since the improvement in synchronization performance of the system, the proposed controller can effectively reduce tracking error and the chattering of the tracking trajectories.

Conclusion

For a hybrid mechanism used for automobile electro-coating conveying, the Jacobi matrix has been calculated and dynamic models in both Cartesian space and joint space have been established. Since the mechanism consists of two sets of hybrid mechanisms bilaterally, a novel synchronization error including the synchronization error between two ends of the end-effector is designed. By combining the CCC with the SMC, a novel chattering-free sliding mode synchronization controller has been proposed. In the proposed controller, the switching function has been replaced by a continuous function to eliminate the chattering and implement the application. The stability of the proposed control algorithm has been proved by Lyapunov stability theorem. The effectiveness of the proposed controller has been verified by simulations and actual experiment. It is shown from the results that the proposed controller can effectively reduce the chattering of driving forces and further improve the synchronization performance and tracking accuracy of the system.

In our opinion, the following points can be researched further. First, parameters of the controller can be further optimized. Besides, the actual pose of the end-effector is obtained by forward kinematics in this article. In the next work, the terminal pose online detection method based on machine vision can be utilized to determine the actual pose of the end-effector.

Footnotes

Academic Editor: Anand Thite

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was financially supported by the National Natural Science Foundation of China (Grant No. 51375210),the Priority Academic Program Development of Jiangsu Higher Education Institutions,Zhenjiang Municipal Industry Key Technology R&D Program (Grant No. GY2013062),and Science and Technology Program of Jingkou,Zhenjiang,China (Grant No. jkGY2013002).

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