Analysis and control for a new reconfigurable parallel mechanism

Abstract

This article proposes a new reconfigurable parallel mechanism using a spatial overconstrained platform. This proposed mechanism can be used as a machine tool. The mobility is analyzed by Screw Theory. The inverse kinematic model is established by applying the closed-loop equation. Next, the dynamic model of the presented mechanism is established by Lagrange formulation. To control the presented mechanism, some controllers have been used. Based on this dynamic model, the fuzzy-proportion integration differentiation (PID) controller is designed to track the trajectory of the end effector. For each limb, a sliding mode controller is applied to track the position and velocity of the slider. Finally, some simulations using ADAMS and MATLAB are proposed to verify the effectiveness and stability of these controllers.

Keywords

Reconfigurable parallel mechanism dynamic model adaptive sliding mode control fuzzy-PID controller trajectory tracking

Introduction

Over the past few decades, developing reconfigurable parallel mechanism has become a noticeable trend for researchers due to the necessity of flexible industrial requirements.^1
–3 The reconfigurable parallel mechanism can change its configurations to obtain different performances, and it also has the advantages of traditional parallel mechanism, like highly dynamic movement and low inertia and positioning errors.^4,5 Based on these attractive characteristics, the reconfigurable parallel mechanism (RPM) has been widely used in many industrial areas, such as machine tools,^6

–9 3-D printers,^10,11 gripper,¹² rehabilitation manipulator,¹³ and sensors.¹⁴

Recently, parallel mechanism with reconfigurable structure or function has been applied in modern industrial fields. Therefore, many researchers have done a lot of work to design and manufacture variable reconfigurable parallel mechanism. The initial presented reconfigurable parallel mechanisms change their configurations by attaching or detaching some modular components.^15,16 While, it’s not convenient to change their performances. Thus, using metamorphic mechanism, some reconfigurable parallel mechanisms are designed to develop variable performances.^17,18 These mechanisms can obtain different configurations by altering some linkages into different working modes. Similar to this way, some metamorphic joints are presented to change their configurations to develop different mobility variations.^19,20 However, the first way needs to attach or detach some components and the rest ways need to pass the singularity points to obtain different configurations, which means all the above methods need manual intervention. In this case, some reconfigurable parallel mechanisms with continuous changing characteristic are proposed.^21

–24

As the significant characteristic, kinematic performance has a great impact on the application of the mechanism.²⁵ Thus, some scholars focus on this topic and have done some work on it. For the planar parallel mechanism, Liu et al.²⁶ addressed a performance evaluation and kinematic optimization to make this planar manipulator obtain the larger workspace and more stable transmissibility. A novel performance called dexterous stiffness has been proposed to optimize a parallel mechanism by Zhang and Gao,²⁷ which can show the mean value and the standard deviation of the stiffness distribution. And, the mean value represents the average stiffness of the parallel manipulator over the workspace, while the standard deviation indicates the stiffness variation relative to the mean value.

It’s well-known that controller is a great important to operation of the manipulator. To control the parallel mechanism, the dynamic model should be established, firstly. Thus, several theoretical methods are proposed to solve dynamic problem. Based on Newton–Euler method, several works have been done. For a three-degree of freedom (DOF) micro/nano parallel mechanism, the flexure hinges are regarded as the revolute joints with constant torsional stiffness, then, the dynamic model is analyzed.²⁸ Considering the friction between joint components caused by joints clearances, the dynamic model of a 3-RRR parallel mechanism with six imperfect joints is established.²⁹ Based on this research, it is reasonable that the proposed mechanism should be operated in smaller loads, moderate velocity, and simple moving trajectory to ensure the accuracy and stability of the system. The principle of virtual work is also an efficient method for the dynamic analysis of parallel mechanism.³⁰ A 3-PUU parallel mechanism with 2R1T DOFs is presented by Wang et al., and the simplified dynamic model is established. Thus, the inverse dynamic of the proposed mechanism is carried out with the principle of virtual work.³¹ Based on the virtual work principle,³² the dynamic model of a novel spherical robot called a 3(RPSP)-S mechanism is obtained. Lagrangian equation is another way to establish the dynamic model of the parallel mechanisms. A nonlinear elastic dynamics model of 4-UPS-UPU flexible parallel mechanism is establish to realize the dynamic response analysis and chaos identification of the proposed spatial parallel mechanism, which is acquired by Lagrange equation.³³ For a redundantly actuated parallel mechanism, the Lagrange formulation is employed to establish the dynamic model of the proposed mechanism with multiple operation modes.³⁴ Besides, the principle of Kane is also applied to analyze the dynamic model of the parallel mechanism.^35,36

The trajectory control for a parallel manipulator is important section in the industrial application. And, the control system of the parallel mechanism is typical nonlinear control system. Compared with linear systems, the nonlinear control system is imperfect and complex. Many methods for nonlinear systems are proposed to reduce the effect of nonlinear uncertainties, like nonlinear adaptive control,³⁷ adaptive sliding mode control,³⁸ fuzzy-PID control,³⁹ and so on. For general parallel mechanism, the prismatic joint is usually employed by the ball-screw system. For a piston air motor ball-screw system, the hybrid sliding mode position method is applied to accomplish this system accurate position performance.⁴⁰ To improve the machining accuracy of ball-screw feed-drive systems, a novel sliding mode controller with a nonlinear sliding surface is presented. And the energy in machining of feed-drive systems is considered to be reduced by this controller.⁴¹ For controlling the whole parallel mechanism, uncertain and nonlinear system, fuzzy logic is considered to be an efficient and effective tool to manage it.⁴² For a cable-driven parallel mechanism, Li et al.⁴³ presented a novel type-2 fuzzy neural network to approximate the inverse dynamics of the process and to handle uncertainties. A novel intelligent control scheme is presented to control the highly nonlinear 3-RRR planer parallel mechanism.⁴⁴ Using this fuzzy-PD controller, the trajectory of this planer parallel mechanism is tracked precisely compared to the conventional linear resolved acceleration control. Aiming to a high nonlinear and unknown disturbances system, a novel reconfigurable parallel mechanism, a sliding mode controller is designed for the ball-screw system of the prismatic joint. And a fuzzy-PID controller is designed to track the trajectory of the end effector.

In this article, a novel reconfigurable parallel mechanism is proposed by using a spatial overconstrained parallel mechanism. The mobility and structure of the proposed mechanism is introduced. Thus, the inverse kinematic of the proposed mechanism is calculated. To verify the correctness of the kinematics model, the simulation is performed in the Online Appendix using ADAMS. The dynamic model both for the reconfigurable section and parallel section is established by Lagrange method. Next, the position and velocity of the ball-screw system are both tracked by using an adaptive sliding mode controller. And the trajectory of the end-effector is tracked by using a fuzzy-PID controller. To build the control system, MATLAB Simscape Multibody Toolbox and ADAMS-MATLAB co-simulation^45,46 are becoming the useful tool. In the fourth section, the control system is built using Simscape Multibody Toolbox, whereas the ADAMS-MATLAB co-simulation approach is used to build the closed-loop system in the Online Appendix. The simulations are presented at the end of each section. Finally, the conclusions are obtained.

Manipulator design and mobility analysis

The reconfigurable parallel mechanism consists of two sections: (1) the reconfigurable section, which is designed by the spatial multi-loop overconstrained parallel mechanism; (2) the parallel section, which is designed by a traditional 3-DOF parallel mechanism. As shown in Figure 1, the two section both have three identical limbs, and they are connected by three prismatic joints. For the parallel section, these prismatic joints are actuators to drive the end effector. Considered the stiffness of the reconfigurable parallel mechanism, the reconfigurable parallel mechanism has three platforms. The bottom platform is the fixed platform located at the ground. The middle platform is the end effector of the reconfigurable section and the top platform is the end effector of the parallel section. Besides middle and fixed platforms, the reconfigurable section includes three limbs and three sliders. The joints between fixed platform and sliders are prismatic and each limb uses revolution joints to connect slider and middle platform. The parallel section also has three identical limbs and each limb includes prismatic joint and two universal joints. By driving the reconfigurable section, the structural parameters of the parallel section can be changed, thus, the different configurations of the reconfigurable parallel mechanism can be obtained.

Figure 1.

The CAD model of the reconfigurable parallel mechanism.

The schematic of reconfigurable section is shown in Figure 2. The reconfigurable section is also a parallel mechanism which has three identical limbs. Based on its structure, the middle platform can only move along the z-axis. And its mobility can be analyzed by Screw Theory.

Figure 2.

The schematic of the reconfigurable mechanism.

To analyze its mobility, the coordinates can be given as ${[\begin{matrix} x_{a i} & y_{a i} & z_{a i} \end{matrix}]}^{T}$ , ${[\begin{matrix} x_{b i} & y_{b i} & z_{b i} \end{matrix}]}^{T}$ , and ${[\begin{matrix} x_{c i} & y_{c i} & z_{c i} \end{matrix}]}^{T}$ $(i = 1, 2, 3)$ . Thus, the twist system of the limb $A_{i} B_{i} C_{i}$ can be written as

$_{i} = {\begin{matrix} {[\begin{matrix} 0 & 0 & \begin{matrix} \begin{matrix} 0; & - a_{i} \end{matrix} & b_{i} & 0 \end{matrix} \end{matrix}]}^{T} \\ {[\begin{matrix} b_{i} & a_{i} & \begin{matrix} \begin{matrix} 0; & - a_{i} z_{b i} \end{matrix} & - b_{i} z_{b i} & - a_{i} z_{b i} + b_{i} y_{b i} \end{matrix} \end{matrix}]}^{T} \\ {[\begin{matrix} b_{i} & a_{i} & \begin{matrix} \begin{matrix} 0; & - a_{i} z_{c i} \end{matrix} & - b_{i} z_{c i} & - a_{i} z_{c i} + b_{i} y_{c i} \end{matrix} \end{matrix}]}^{T} \end{matrix} (i = 1, 2, 3)

where a_i and b_i are the directions of prismatic joint.

The wrench system of this limb can be written as

$_{i}^{r} = {\begin{matrix} {[\begin{matrix} \frac{b_{i}}{a_{i}} & 1 & \begin{matrix} \begin{matrix} 0; & 0 \end{matrix} & 0 & 0 \end{matrix} \end{matrix}]}^{T} \\ {[\begin{matrix} 0 & 0 & \begin{matrix} \begin{matrix} 0; & 0 \end{matrix} & 0 & 1 \end{matrix} \end{matrix}]}^{T} \\ {[\begin{matrix} 0 & 0 & \begin{matrix} \begin{matrix} 0; & - \frac{b_{i}}{a_{i}} \end{matrix} & 1 & 0 \end{matrix} \end{matrix}]}^{T} \end{matrix} (i = 1, 2, 3)

According to equation (2), the twist system of reconfigurable section can be calculated as

$ = {[\begin{matrix} \begin{matrix} 0 & 0 & 0; & 0 \end{matrix} & 0 & 1 \end{matrix}]}^{T}

Finally, the mobility of the reconfigurable section is proved.

The schematic of the parallel section is shown in Figure 3. The mobility of the parallel section can be analyzed by Screw Theory as well. The coordinates of the points D_i , E_i , and F_i are given as ${[\begin{matrix} x_{D i} & y_{D i} & z_{D i} \end{matrix}]}^{T}$ , ${[\begin{matrix} x_{E i} & y_{E i} & z_{E i} \end{matrix}]}^{T}$ , and ${[\begin{matrix} x_{F i} & y_{F i} & z_{F i} \end{matrix}]}^{T}$ . Thus, the twist system of limb $D_{i} E_{i} F_{i}$ can be written as

$_{i} = {\begin{matrix} {[{\begin{matrix} 0 & 0 & 0; & l_{i} & m_{i} & n \end{matrix}}_{i}]}^{T} \\ {[\begin{matrix} - m_{i} & l_{i} & 0; & l_{i} z_{E i} & m_{i} z_{E i} & - l_{i} x_{E i} - m_{i} y_{E i} \end{matrix}]}^{T} \\ {[\begin{matrix} a_{i} & b_{i} & c_{i}; & - c_{i} y_{E i} + b_{i} z_{E i} & c_{i} x_{E i} - a_{i} z_{E i} & - b_{i} x_{E i} + a_{i} y_{E i} \end{matrix}]}^{T} \\ {[\begin{matrix} a_{i} & b_{i} & c_{i}; & - c_{i} y_{F i} + b_{i} z_{F i} & c_{i} x_{F i} - a_{i} z_{F i} & - b_{i} x_{F i} + a_{i} y_{F i} \end{matrix}]}^{T} \\ {[\begin{matrix} - m_{i} & l_{i} & 0; & l_{i} z_{F i} & m_{i} z_{F i} & - l_{i} x_{F i} - m_{i} y_{F i} \end{matrix}]}^{T} \end{matrix}

where l_i , m_i , and n_i are the directions of prismatic joint; a_i , b_i , and c_i are the direction of one of the universal joint. The direction of the other joint is ${[\begin{matrix} - m_{i} & l_{i} & 0 \end{matrix}]}^{T}$ . In universal joint, the two axes are orthogonality, thus,

a_{i} = \frac{l_{i} b_{i}}{m_{i}}

Figure 3.

The schematic of the parallel section.

So, the wrench system of the limb $D_{i} E_{i} F_{i}$ can be given as

$_{i}^{r} = {[\begin{array}{l} 0 & 0 & 0; & - \frac{c_{i} l_{i}}{a_{i} l_{i} + b_{i} m_{i}} & - \frac{c_{i} m_{i}}{a_{i} l_{i} + b_{i} m_{i}} & l_{i} \end{array}]}^{T} (i = 1, 2, 3)

According to equation (6), the twist system of the parallel section can be written as

$ = {\begin{array}{l} {[\begin{array}{l} 0 & 0 & 0; & 0 & 0 & 1 \end{array}]}^{T} \\ {[\begin{array}{l} 0 & 0 & 0; & 0 & 1 & 0 \end{array}]}^{T} \\ {[\begin{array}{l} 0 & 0 & 0; & 1 & 0 & 0 \end{array}]}^{T} \end{array}

Equation (7) shows that the parallel section has three translations along x, y, and z axes.

Kinematic and dynamic analysis

Inverse kinematics

When the parallel section is working, the reconfigurable section is locked at a certain position. Namely, the kinematics model of the parallel section is different, while the reconfigurable parallel is located at different positions. To solve the inverse kinematics of the parallel section, the first step is to obtain the mathematical model of the reconfigurable section. For the reconfigurable section, the input parameter is m and the output parameter is z_r . When the prismatic joint is driving, the middle platform will be located at a certain altitude and the structural parameters of the parallel section will be changed as well.

Based on Figure 2, the closed-loop vector equation of the reconfigurable section can be written as

\vec{O B_{1}} + \vec{B_{1} C_{1}} = \vec{O O_{1}} + \vec{O_{1} C_{1}}

Thus, the kinematics model of the reconfigurable can be calculated as

z_{r} = \sqrt{d^{2} - {(m - r_{2})}^{2}}

where d is the length of the link $C_{i} B_{i}$ and r ₂ is the radius of the middle platform’s radius.

For the parallel section, the inverse problem is that the position of the end effector is known to solve the lengths of three driving joints. As shown in Figure 3, the point O ₂ is in the center of the end effector and the point O is the original point of the fixed coordinate system. In this article, the coordinate of O ₂ is ${[\begin{matrix} x_{1} & y_{1} & z_{1} \end{matrix}]}^{T}$ which is known to solve the lengths $l_{i} (i = 1, 2, 3)$ ; b is the length of the identical link $E_{i} F_{i}$ ; r is the radius of the end effector’s radius; a is the length of link $D_{i} E_{i}$ which is the displacement between slider and axis of the revolute joint.

Based on the structural constraints, the closed-loop vector equation can be given as

\vec{O B_{1}} + \vec{B_{1} D_{1}} + \vec{D_{1} E_{1}} + \vec{E_{1} F_{1}} = \vec{O O_{2}} + \vec{O_{2} F_{1}}

And the vector of $\vec{O E_{1}}$ can be calculated as

\vec{O E_{1}} = {[\begin{matrix} m - l_{1} cos (θ) + a sin (θ) & 0 & l_{1} sin (θ) + a cos (θ) \end{matrix}]}^{T}

where θ is the angle between $B_{1} C_{1}$ and $A_{1} O$ , thus, it can be given as

θ = arctan (\frac{z_{r}}{m - r_{2}})

In the moving coordinate system, the coordinate of point O ₂ is ${[\begin{matrix} x^{'} & y^{'} & z^{'} \end{matrix}]}^{T}$ , and the moving platform can be seen as a regular triangle, thus, the vector of $\vec{O_{2}^{'} F_{1}^{'}}$ can be given as

\vec{O_{2}^{'} F_{1}^{'}} = {[\begin{matrix} x^{'} + r & y^{'} & z^{'} \end{matrix}]}^{T}

To transfer the vector from moving coordinate system to the global coordinate system, the vector should be multiplied a rotation matrix. Fortunately, the type of the DOF of the moving platform is only the translation, namely, the rotation matrix can be given as

R = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

Thus, the coordinate of F ₁ in the global coordinate system can be calculated as

F_{1} = R \cdot F_{1}^{'} + O_{2}^{'}

Based on the geometrical constraint, the length of the identical link $E_{1} F_{1}$ is constant, thus, following equation can be derived

‖ \vec{E_{1} F_{1}} ‖ = b

Substituting equations (13) and (15) into equation (16), the inverse kinematics of the parallel section can be obtained.

For the further analysis, the structural parameter is listed in Table 1.

Table 1.

The structural parameters of the mechanism.

Parameter	Value	Parameter	Value
c	0.046 m	Θ	30°
d	0.223 m	A	0.025 m
r	0.018 m	b	0.185 m
r ₂	0.042 m

Dynamics model for the overconstrained mechanism

Based on the Virtual Work Theory, the inverse dynamics equation of the overconstrained mechanism can be given as

{(- m_{w} a - m_{w} g e)}^{T} δ z_{r} + (F_{r} - m_{h} a_{m}) δ m = 0

where $e = {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T}$ , m_w is the mass including the middle platform and the equivalent section of the link $B_{i} C_{i} (i = 1, 2, 3)$ , and m_h is the mass including the sliders and the equivalent section of the link $B_{i} C_{i} (i = 1, 2, 3)$ . $F_{r}$ is the driving force.

Based on equation (9), the Jacobian matrix of the overconstrained mechanism can be written as

{\dot{z}}_{r} = \frac{\sqrt{d^{2} - {(m - r_{2})}^{2}}}{m - r_{2}} \dot{m}

Thus,

δ z_{r} = \frac{\sqrt{d^{2} - {(m - r_{2})}^{2}}}{m - r_{2}} δ m

Substituting equation (19) in equation (17) yields

F_{r} = \frac{{(m_{w} a + m_{w} g e)}^{T} (m - r_{2})}{\sqrt{d^{2} - {(m - r_{2})}^{2}}} + m_{h} a_{m}

Dynamics model for the parallel section

Compared with the overconstrained mechanism, the inverse dynamics model of the parallel mechanism is more complex. According to its structural characteristic, the Lagrangian Theory is suitable to be applied. For the parallel section, there are three redundant coordinates, x, y, and z, employed for the first type of the Lagrangian equations. Then, x, y, z and $l_{i} (i = 1, 2, 3)$ are the generalized coordinates. Thus, the Lagrangian equations of the first type can be given as

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) - \frac{\partial L}{\partial q_{j}} = Q_{j} + \sum_{i = 1}^{k} λ_{i} \frac{\partial Γ_{i}}{\partial q_{j}} (j = 1 \dots 6)

where $Γ_{i} (i = 1, 2, 3)$ is the constraint function. For the proposed mechanism, it has three limbs, thus, $k = 3$ . And $λ_{i} (i = 1, 2, 3)$ is the Lagrangian multiplier. Note that the generalized forces, $Q_{i} (i = 1, 2, 3)$ , represent the x, y and z components of an external force exerted at the moving platform. $Q_{i} (i = 4, 5, 6)$ is the force of the driving joint.

To solve three Lagrange multipliers, equation (21) can be written as

\sum_{i = 1}^{k} λ_{i} \frac{\partial Γ_{i}}{\partial q_{j}} = \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) - \frac{\partial L}{\partial q_{j}} - \overset{Λ}{Q_{j}}

where $\overset{Λ}{Q_{j}} (j = 1, 2, 3)$ represents the generalized force contributed by an externally applied force exerted on the moving platform, $Γ_{i} (i = 1, 2, 3)$ is the constraint equation of the ith limb, and L is the energy of the whole parallel mechanism including kinetic and potential energies.

For simplifying the dynamics analysis, the mass of each rod can be assumed to divide into two endpoints, F_i and E_i , denoted as m_b . Thus, the total energy of the parallel section can be given as

L = \frac{1}{2} m_{p} ({\dot{x}}^{2} + {\dot{y}}^{2} + {\dot{z}}^{2}) + \frac{3}{2} m_{b} ({\dot{x}}^{2} + {\dot{y}}^{2} + {\dot{z}}^{2}) + \frac{1}{2} (m_{a} + m_{b}) ({\dot{l}}_{1}^{2} + {\dot{l}}_{2}^{2} + {\dot{l}}_{3}^{2}) - (3 m_{b} + m_{p}) z g - \sum_{i = 1}^{k} (m_{b} + m_{a}^{i}) l_{i} g sin (θ)

where m_p is the mass of the parallel platform and $m_{a}^{i} (i = 1, 2, 3)$ is the mass of the slider.

Based on equation (16), the three constraint equations can be given as

Γ_{i} = {‖ \vec{E_{i} F_{i}} ‖}^{2} - b^{2} = 0 (i = 1, 2, 3)

Then, the Lagrangian multiplier, $λ_{i}$ , can be calculated by substituting equations (23) and (24) into equation (22). Thus, the actuator force can be determined directly from the second type of Lagrangian equations, which can be written as

F_{j} = \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) - \frac{\partial L}{\partial {\dot{q}}_{j}} - \sum_{i = 1}^{k} λ_{i} \frac{\partial Γ_{j}}{\partial q_{j}} (j = 1, 2, 3)

Based on equation (25), the equation can be rewritten as matrix formulation, which is

F = D (l) \ddot{l} + C (l, \dot{l}) \dot{l} + G (l)

where $D (l)$ is the mass matrix of the system, $C (l, \dot{l})$ is the Lagrangian multipliers matrix, $G (l)$ is the gravitational matrix, and F is the driving force.

In this article, the Lagrangian multipliers matrix can be written as

C (l, \dot{l}) = A^{- 1} B

where $A = {[\frac{\partial Γ_{i}}{\partial q_{j}}]}^{T} (j = 1, 2, 3)$ and $B = {[\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) - \frac{\partial L}{\partial q_{j}} - \overset{Λ}{Q_{j}}]}^{T} (j = 1, 2, 3)$ .

Control system design

Adaptive sliding mode control

Control model and design of adaptive controller

For the parallel mechanism, each limb is nonlinear uncertainties control system. In this article, the driving joint is the prismatic joint, but the motor is a DC servo disc motor. Thus, a ball-screw system is used through a helical coupling. And the driving system setup is illustrated in Figure 4. In this case, the rotational dynamics of ball screw can be described as follows⁴⁷

(J_{rot} + m_{1} l_{p}^{2}) \frac{\ddot{l}}{l_{p}^{2}} + F (·) = u (t)

where $J_{r o t}$ is the total rational inertia of ball screw and motor, l_p is the lead pitch of the ball screw, $F (·)$ is an equivalent frictional torque, and $u (t)$ is output force to the system. For the ball-screw system, its frictional torque dominates the system and the transmission mechanism has high stiffness, thus, the reduced-order model for the ball-screw system can be given as

(Δ M) \ddot{l} (t) + F (·) = u (t)

where $Δ M$ is an effective mass of ball screw, motor, and equivalent rational inertia of the slider mass; $F (·)$ is the equivalent friction force caused by the motor and transmission connection; and $u (t)$ is the motor force.

Figure 4.

Conceptual model of the ball-screw system.

For a certain ball-screw system, $Δ M$ cannot be determined clearly and the mathematical model of $F (g)$ cannot be established, precisely. To avoid these problems, a novel adaptive sliding mode control is applied.

Assuming 1

The uncertain parameter $Δ M$ is limited and it has a certain range, which is

Δ M \in Ω = {Δ M; 0 < Δ M_{min} \leq Δ M \leq Δ M_{max}}

Assuming 2

The uncertain nominal $F (g)$ has boundaries, which is

| F (g) | \leq D

Thus, the sliding surfaces in this article is chosen as

s = \dot{e} + c e = \dot{l} - {\dot{l}}_{d} + c e

where l_d is the desired position and $e = l - l_{d}$ is the error between actual and desired position. And c is a positive number. The next step of controller design is to choose a control law with variable parameters such that it makes the Lyapunov function a decreasing function of time. In this article, the control law is considered as

u (t) = \hat{l} ({\ddot{x}}_{d} - c \dot{e}) - k_{s} s - η sign (s)

where $\hat{l}$ is the estimated value of l, $k_{s} > 0$ , $η > F (\cdot)$ .

Stability analysis

Stability is a significant index for the control system. Considered as a nonlinear uncertain dynamical system, the ball-screw system is described as equation (29) and the adaptive control law, equation (33), is applied. At the same time, asymptotic stability of the closed-loop system in the presence of model uncertainties and disturbance must be guaranteed.

Proof

To prove the stability of the proposed controller and to derive an estimation law for the unknown equivalent friction force $F (g)$ , thus, Lyapunov function is given as

V = \frac{1}{2} l s^{2} + \frac{1}{2 γ} {\tilde{l}}^{2}

where $\tilde{l} = \hat{l} - l$ and $γ > 0$ .

From equation (34), the derivative of Lyapunov function is as

\dot{V} = l s \dot{s} + \frac{1}{γ} \tilde{l} \dot{\tilde{l}}

Substituting equation (32) into equation (35) yields

\dot{V} = s (l \ddot{l} - l {\ddot{l}}_{d} + l c \dot{e}) + \frac{1}{γ} \tilde{l} \dot{\hat{l}}

Then, equation (36) can be rewritten as

\dot{V} = s (u + F (g) - l ({\ddot{l}}_{d} - c \dot{e})) + \frac{1}{γ} \tilde{l} \dot{\hat{l}}

where u is the control law shown in equation (33) and $F (g)$ is the equivalent friction force.

Substituting equation (33) into equation (37) yields

\dot{V} = - k_{s} s^{2} - η | s | + F (\cdot) s + \tilde{l} (s ({\ddot{l}}_{d} - c \dot{e}) + \frac{1}{γ} \dot{\hat{l}})

In this article, the adaptive control law is designed as

\dot{\hat{l}} = - γ s ({\ddot{l}}_{d} - c \dot{e})

Thus, equation (38) can be written as

\dot{V} = - k_{s} s^{2} - η | s | + F (g) s

Thus,

\dot{V} \leq - k_{s} s^{2} \leq 0

Based on equation (41), if $\dot{V} = 0$ , thus s must be equal to zero. According to LaSalle Theory, this closed-loop control model is asymptotical stability system, which means that when $t \to \infty$ , $s \to 0$ . And the convergence speed of this system is determined by k_s . Besides, equation (41) also ensure that the states will be confined to the surface $s = 0$ for all future time since leaving the surface requires $\dot{V}$ to be positive. Based on the definition of s in equation (32), the results can be concluded. To avoid the exaggerated $u (t)$ or $\hat{l} \leq 0$ , a projector adaptive arithmetic is applied to limit the range of $\hat{l}$ , thus, equation (39) can be rewritten as

\dot{\hat{l}} = P_{l} (- γ s ({\ddot{l}}_{d} - c \dot{e}))

And

P_{l} (x) = {\begin{matrix} 0 ​ ​ ​ ​ \hat{l} \geq l_{max} and x > 0 \\ 0 ​ ​ ​ ​ \hat{l} \leq l_{max} and x < 0 \\ x otherwise \end{matrix}

where $\hat{l} \in [l_{min}, l_{max}]$ .

Based on the proposed adaptive sliding mode controller, the block diagram of the controller is given as Figure 5.

Figure 5.

Block diagram of the adapt sliding mode controller with perturbation force.

PD control

For this ball-screw system, a PD controller is presented based on equation (29). As a comparison, a PD control law can be expressed via the following equation

u (t) = k_{p} e (t) + k_{d} \frac{d e (t)}{d t}

In which

e (t) = l (t) - l_{d} (t)

where $l (t)$ is the actual displacement of the ball-screw system and $l_{d} (t)$ is desired displacement, $e (t)$ denotes the error in the linear displacement signal. k_p and k_d are proportional and derivative gains of the PD controller, respectively.

Simulation result

In this simulation, it’s assumed that $Δ M = 1.5 kg$ , $c = 15$ , $k_{s} = 15$ , $γ = 500$ , $η = 2.01$ and the equivalent friction force, $F (\cdot)$ , is given as

F (\cdot) = 0.5 \dot{l} + 1.5 sign (\dot{l})

And the input signal is the length of the prismatic joint calculated by equation (16) and the trajectory of the end effector is given in equation (47). While, the input is too complex to simulate slowly. To simplify the signal, curve fitting is applied in this article.

{\begin{matrix} x = r^{'} cos (t) + c_{1} \\ y = r^{'} sin (t) + c_{2} \\ z = c_{3} \end{matrix}

Based on Fourier Fitting Method, the fitting result of three driving joints is shown in Figure 6. The Fourier formulation is given as

f (x) = a + b cos (x \cdot w) + c sin (x \cdot w)

Thus, the coefficients of three limbs are listed in Table 2. And the result is shown in Figure 6. The points in Figure 6(a), (b), and (c) are the actual positions of the driving joints and the blue, green, and red lines represent the fitting curves based on Fourier Fitting Method. Their trends are similar and the fitting curves can replace the discrete points very well. Thus, the fitting equations can be regarded as the input function.

Table 2.

The coefficients of three limbs.

	Limb 1	Limb 2	Limb 3
a	0.1151	0.1151	0.115
b	1.955 × 10⁻⁵	−0.0173	0.01714
c	−0.2001	0.01005	0.01019
w	1	1	1

Figure 6.

The comparison between driving joints and fitting curves, (a) first joint, (b) second joint, and (c) third joint.

The results of the sliding mode control are shown in Figures 7 to 9. In this PD control, $k_{p} = 100$ and $k_{d} = 50$ . Thus, the result of the PD control is shown in Figure 10. Figure 7(a) and (b) show the actual signal and actual tracking, respectively. Figure 7(c) and (d) show the local enlarging graphs marked in the (a) and (b). The orange dashed box expresses the displacement and light blue dashed box is the velocity. From Figure 7, it can be seen that the actual signal is tracking desired signal during the simulation time and the fluctuation of actual signal is in a small range. Similar to Figure 7, Figure 8 shows the error between actual and desired signal of both displacement and velocity. (a) shows the error of displacement and (b) is the error of the velocity. The section of orange dashed box is shown in (c). The error of displacement is in a small range, which is $[- 0.01 m, 0.01 m]$ . Thus, the error can be ignored. The red dashed box is shown in (d). The error of velocity is between $- 0.02 m / s$ and $0.02 m / s$ , thus, the error of velocity is very small, as well. Figure 10 is the results of PD control. The tracking time is more than 2 s and the errors are bigger than sliding mode no matter displacement or velocity. The error of displacement is in the range $[- 0.1 m, 0.1 m]$ and the range of error of velocity is $[- 0.04 m / s, 0.04 m / s]$ . Thus, it can be realized that the effectiveness of the proposed controller by comparing the robot tracking results in the case of using the adaptive sliding mode controller and PD controller as compared to Figures 7 and 10. Figure 9 shows the other results of the sliding mode controller: (a) is the control input of this controller; (b) is the phase trajectory of the sliding mode; (c) shows the changes of the equivalent mass; while (d) is the trend of the equivalent frictional force.

Figure 7.

The desired and actual displacement and velocity: (a) displacement, (b) velocity, (c) local enlarging graphs of displacement, and (d) local enlarging graphs of velocity.

Figure 8.

The errors of both displacement and velocity: (a) displacement, (b) velocity, (c) local enlarging graphs of displacement, and (d) local enlarging graphs of velocity.

Figure 9.

Other results: (a) control input, (b) phase trajectory, (c) equivalent mass, and (d) equivalent force.

Figure 10.

The results of PD control: (a) displacement, (b) velocity, (c) error of displacement, and (d) error of velocity.

Fuzzy-PID control

For the trajectory tracking, the mathematical dynamic model of the parallel mechanism is established in equation (26). Based on equation (26), $D (l)$ and $G (l)$ can be determined by the structural and mass parameters, while $C (l, \dot{l})$ cannot be expressed clearly by an analytical solution. And, based on equation (27), $C (l, \dot{l})$ has a range, which can be given as

C (l, \dot{l}) \in Ω = {Δ C (l, \dot{l}); 0 < Δ C_{min} \leq Δ C \leq Δ C_{max}}

In this case, traditional PID controller with fixed control parameters is not suitable for this reconfigurable parallel mechanism. Thus, fuzzy PID controller with changing control parameter can be applied to control this mechanism.

Traditional PID controller

Similar to equation (44), the transfer function of a traditional PID controller for the proposed mechanism can be given as

u (t) = K_{p} e (t) + K_{i} \int e (t) d t + K_{d} \frac{d e (t)}{d t}

Here, $u (t)$ is the control signal and $e (k)$ is the error between the reference and the process output.

Design of fuzzy PID controller

For this reconfigurable parallel mechanism, its control scheme is a kind of tracking control. Here calculate the forces of the prismatic joints that the end-effector is moving with the desired trajectory in the Cartesian coordinate system. Specially, we hope that the end effector can follow the desired trajectory and the prismatic joints are given in terms of forces. Almost always, these driving forces are determined using the feedback from the mathematical dynamic model to calculate the forces required for a desired trajectory. The PID parameters of the whole parallel mechanism are tuned by comparing both the error and differential of the error between actual and desired trajectory of the end effector.⁴⁸ For each limb, the proposed sliding mode control is applied, thus, the control scheme of the whole parallel mechanism is shown in Figure 11.

Figure 11.

The whole control scheme.

It is assumed that K_p and K_d are in estimated ranges $[K_{p, min}, K_{p, max}]$ and $[K_{d, min}, K_{d, max}]$ , respectively. In this article, K_p , K_d , and K_i can be calculated by following equations

K_{p} = (K_{p, max} - K_{p, min}) K_{p}^{'} + K_{p, min}

K_{d} = (K_{d, max} - K_{d, min}) K_{d}^{'} + K_{d, min}

K_{i} = \frac{K_{p}^{2}}{α K_{d}}

where $K_{p}^{'}$ , $K_{d}^{'}$ , and α can be obtained by a set of fuzzy rules.

For the fuzzy controller, the input parameters are $e (t)$ and $\dot{e} (t)$ . The membership functions setting for these parameters are shown in Figure 12. Both $e (t)$ and $\dot{e} (t)$ have seven linguistic sets defined as negative big (NB), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive medium (PM), and positive big (PB). The Mamdani type is applied to generate the best possible conclusion. For the defuzzification process, the center of gravity method is used to calculate the crisp control commands. For this fuzzy controller, there are two input variables represented by Triangular membership function, thus, the total number of rules is 49, which are listed in Tables 3 to 5. And the surfaces of these fuzzy rules are shown in Figures 13 to 15.

Figure 12.

Fuzzy membership functions.

Table 3.

Fuzzy rules for K_p .

		$\dot{e} (t)$
		NB	NM	NS	ZO	PS	PM	PB
$\dot{e} (t)$	NB	B	B	B	B	B	B	B
	NM	S	B	B	B	B	B	S
	NS	S	S	B	B	B	S	S
	ZO	S	S	S	B	S	S	S
	PS	S	S	B	B	B	S	S
	PM	S	B	B	B	B	B	S
	PB	B	B	B	B	B	B	B

Table 4.

Fuzzy rules for K_d .

		$\dot{e} (t)$
		NB	NM	NS	ZO	PS	PM	PB
$\dot{e} (t)$	NB	S	S	S	S	S	S	S
	NM	B	B	S	S	S	B	B
	NS	B	B	B	S	B	B	B
	ZO	B	B	B	B	B	B	B
	PS	B	B	B	S	B	B	B
	PM	B	B	S	S	S	B	B
	PP	S	S	S	S	S	S	S

Table 5.

Fuzzy rules for α.

		$\dot{e} (t)$
		NB	NM	NS	ZO	PS	PM	PB
$\dot{e} (t)$	NB	NB	NB	NB	NB	NB	NB	NB
	NM	NS	NS	NB	NB	NB	NS	NS
	NS	S	NS	NS	NB	NS	NS	S
	ZO	B	S	NS	NS	NS	S	B
	PS	S	NS	NS	NB	NS	NS	S
	PM	NS	NS	NB	NB	NB	NS	NS
	PP	NB	NB	NB	NB	NB	NB	NB

Figure 13.

The rules’ surface for α.

Figure 14.

The rules’ surface for K_d .

Figure 15.

The rules’ surface for K_p .

Simulation result

In this simulation, the simulation time is 10 s. The traditional PID control gains are tuned by the expert experience, thus, the parameters are chosen as follows: proportional gain $K_{p} = 5000$ , differential gain $K_{d} = 100$ , and integral gain $K_{i} = 2$ . For the fuzzy PID control, the initial parameters are chosen as the same as traditional one. The range of K_d is $[50, 150]$ and the range of K_p is $[4700, 5300]$ . And the desired trajectory of the end effector is listed in equation (47). The dynamic parameters of the reconfigurable parallel mechanism are listed in Table 6.

Table 6.

Dynamic parameters of the proposed mechanism.

Parameter	Value	Parameter	Value
c	0.046 m	θ	31°
d	0.223 m	m_b	0.028 kg
r	0.018 m	m_p	0.053 kg
r₂	0.042 m	m_a	0.055 kg
a	0.025 m	g	9.8 kg/s2
b	0.1851 m

Using Simscape Multibody toolbox in Simulink, the results of the simulation are shown in Figures 16 to 25 and Online Supplementary Figures 26 to 34. Figure 16 shows the trajectories of the end effector under fuzzy-PID control and traditional PID. It can be seen from it, the fuzzy PID control can track the desired trajectory better than traditional PID control and its error along $z - a x i s$ is smaller than traditional PID control. Figure 17 shows the errors of both traditional PID and fuzzy-PID controller along $x - a x i s$ . The error of fuzzy-PID along $x - a x i s$ is smaller than traditional PID, obviously. And this fuzzy-PID control corrects the fluctuation appearing in the traditional PID. Similar to Figure 17, Figure 18 shows the errors of both fuzzy-PID and traditional PID controller along $y - a x i s$ . It can be seen that fuzzy-PID controller can correct the fluctuation. Figures 19 and Figure 21 show the three driving joints’ trends for length, velocity, and force under traditional PID control, separately. In Figure 19, the lengths of driving joints have flops near fourth second. In Figure 20, the velocities of driving joints have big flops near fourth second and slight fluctuation between sixth second and eighth second. Similar to Figure 20, it can be seen from Figure 21 that some fluctuations are appeared between sixth second and eighth second, obviously. Figure 22, Figure 24 show the three driving joints’ trends for length, velocity, and force under fuzzy-PID control, separately. Compared with Figure 19, the lengths of driving joints in Figure 22 is smoother, and the flop is eliminated by this fuzzy-PID control. Figure 23 shows the velocities of driving joints under fuzzy-PID control. Compared to Figure 20, the velocities of driving joints using fuzzy-PID control are smoother than using traditional PID control. Figure 24 are the forces of driving joints under fuzzy-PID control. It can be seen that the fluctuations of driving force by using Fuzzy-PID control appeared are more slightly than by using traditional PID control, obviously. To eliminate the scales of the PID parameters, a method is used, which can be written as

P (t) = \frac{K (t)}{K (t = 1)}

where $P (t)$ is the percentage of PID parameters and $K (t)$ is the PID parameters at tth second.

Figure 16.

The trajectory of the end effector: (a) fuzzy-PID and (b) traditional PID.

Figure 17.

The error along $x - a x i s$ .

Figure 18.

The error along $y - a x i s$ .

Figure 19.

The lengths of driving joints under traditional PID control.

Figure 20.

The velocities of driving joints under traditional PID control.

Figure 21.

The forces of driving joints under traditional PID control.

Figure 22.

The lengths of driving joints under fuzzy-PID control.

Figure 23.

The velocities of driving joints under fuzzy-PID control.

Figure 24.

The forces of driving joints under fuzzy-PID control.

Thus, the trends of three PID parameters are shown in Figure 25. The blue, red, and yellow lines represent K_d , K_i , and K_p , separately. Between 0.1th second and 1.8th second, the fuzzy rules are applied to tune these PID parameters and after 1.8th second, the parameters become stable.

Figure 25.

The changing trends of fuzzy-PID control parameters.

Conclusions

In this article, a reconfigurable parallel mechanism based on spatial overconstrained mechanism is proposed based on Screw Theory. This reconfigurable parallel mechanism can change its structural parameters continuously, which makes it has variable performances. The CAD model and its kinematic mathematical model have been established. Next, the dynamic model of this reconfigurable parallel mechanism has been established by utilizing the Lagrange formulation. To control the limb of this mechanism, an adaptive sliding mode has been designed to track the position and velocity of the desired driving curves. The control law and adaptive law have both been presented, as well. Compared with traditional PD control, a simulation has been presented to verify the efficiency of this adaptive sliding controller in Simulink. For the whole reconfigurable parallel mechanism, a fuzzy-PID control has been applied to track the trajectory of the end effector by using Simscape Multibody toolbox in Simulink. Finally, a simulation has been proposed to verify the effectiveness and correctness of this fuzzy-PID control. Compared with traditional PID control, this fuzzy-PID controller can work more smoothly and flexibly.

Supplemental material

Supplemental Material, Appendix - Analysis and control for a new reconfigurable parallel mechanism

Supplemental Material, Appendix for Analysis and control for a new reconfigurable parallel mechanism by Guanyu Huang, Dan Zhang, Hongyan Tang, Lingyu Kong and Sumian Song in International Journal of Advanced Robotic Systems

Footnotes

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: The work was financially supported by the Youth Foundation of Zhejiang Lab (Grant No. 2020NB0AA02), the Natural Science Foundation of Zhejiang Province of China (Grant No. LQ20E050008), the Leading Innovation and Entrepreneurship Team of Zhejiang Province of China (Grant No. 2018R01006), the Natural Sciences and Engineering Research Council of Canada (NSERC), and the York Research Chairs (YRC) program.

ORCID iD

Guanyu Huang

Supplemental material

Supplemental material for this article is available online.

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