Typical configuration analysis of a modular reconfigurable cable-driven parallel robot

Abstract

To obtain better flexibility and multifunction in varying practical applications, several typical configurations of a modular reconfigurable cable-driven parallel robot are analyzed in this article. The spatial topology of the modular reconfigurable cable-driven parallel robot can be reconfigured by manually detaching or attaching the different number of modular branches as well as changing the connection points on the end-effector to satisfy diverse task requirements. The structure design of the modular reconfigurable cable-driven parallel robot is depicted in detail, including the design methodology, mechanical description, and control architecture. The inverse kinematics and dynamics of the modular reconfigurable cable-driven parallel robot considering diverse configurations are derived according to the vector closed rule and Lagrange method, respectively. The numerical simulation and related experiments of a typical configuration are achieved and analyzed. The results verify the effectiveness and feasibility of the inverse kinematics and dynamics models for the modular reconfigurable cable-driven parallel robot.

Keywords

Reconfigurable cable-driven parallel robot typical configuration analysis structure design inverse kinematics and dynamics

Introduction

Cable-driven parallel robots (CPRs) are a robotic manipulator designed to control the position and orientation of its end-effector within the system’s workspace driven by flexible cables.¹ The CPRs are a particular class of parallel robots in which flexible cables are utilized to take the place of rigid links in the robot structure. Compared with rigid-link parallel robots, the CPRs have many potential advantages in terms of large workspace, high payload to weight ratio, low inertia, high speed and acceleration capability, simple and lightweight mechanical structure, ease of assembly and disassembly, and easy reconfigurability and scalability.^2
–4 Based on the abovementioned attractive characteristics, the CPRs have been extensively investigated by numerous researchers and widely applied in a great many of engineering fields in recent decades, such as large spherical radio telescope,⁵ rehabilitation robot,^6
–8 3-D printer,^9,10 material and cargo hoisting,¹¹ wind tunnel test,¹² aerial camera,^13,14 high-speed assembly and pick-and-place operation,^15,16 haptic interface,¹⁷ ultrahigh speed robot,¹⁸ building construction,¹⁹ rescue operations,²⁰ hazardous environment,²¹ airplane maintenance,²² and such specific purpose as back-heating or painting in ship building and aircraft construction.³ Each of the aforementioned diverse applications has specific requirements, most of them are focused on a single operational task of the CPRs, which lead to different CPRs configurations. Therefore, the need for enlarging the role of traditional CPRs within the last few decades has caused the development of the ones that are modular and reconfigurable.

Modularity retains cost low by reusing common components, while reconfiguration extends the capabilities of what is realizable from a single-goal mechanism. The integration of these two design concepts has resulted in the development of modular reconfigurable robots (MRRs), especially in rigid serial robots, but modular reconfigurable parallel robots (MRPRs) have been developed recently.²³ The MRRs are made up of many basic modular components that can be rapidly assembled into different figurations to perform all kinds of tasks to satisfy different requirements.²⁴ The merits of the MRRs are given as follows: (1) shorten development cycle; (2) rapid design, manufacture, and low cost; and (3) adapt the changing operational needs and reduce maintenance.²⁵ In general, the MRRs can be classified into three categories according to the types of the module reconfiguration which include manual reconfiguring, self-reconfiguring, and self-assembly. Manual reconfiguring robots are in fact modular robots, which can only be reconfigured by manual assistance. Self-reconfiguring robots can perform reconfiguration by themselves after a robot system is assembled with some form of manual assistance. However, self-reconfiguring robots cannot achieve self-assembly. Self-assembly robots are the highest level of reconfigurability robot that can assemble and disassemble from their own modules automatically.²⁴

Some researchers have focused on the design and analysis of the modular reconfigurable cable-driven parallel robot (MRCPR), which can not only increase the number of features of the CPRs but also expand the workspace of the CPRs. As the same time, the MRCPR can also overcome the drawback of the CPRs related to the potential collisions, including cables and cables, cables and obstacles, and cables and surrounding environment. Li et al.²⁶ proposed a modular and reconfigurable cable-driven robotic grasper for grasping varying unknown objects in cluttered environments, which combines the features of full actuation and under actuation. Izard et al.²⁷ developed a reconfigurable cable robot to explore novel concepts for CPRs and illustrated its capabilities in industrial tasks. The specific view on the different components and the capabilities of reconfiguration were presented, as well as examples of layouts meant for various research and industrial projects. Gagliardini et al.²⁸ introduced a reconfigurable CPR to be employed in industrial operations on large structure. The reconfigurable CPRs can modify their geometric parameters to adapt their own features and be intended to sandblast and paint a large tubular structure. Gagliardini et al.²⁹ dealt with reconfigurable CPRs whose cable connection points on the base frame can be positioned at a possibly large but discrete set of possible locations. Anson et al.³⁰ investigated two planar mobile base configurations of the CPRs including the rectangular configuration and circular configuration, and their results were compared with as traditional fixed-base system. Compared with the MRPRs, very limited research has focused on the MRCPR. Thus, it is extremely necessary to analyze the proposed MRCPR. It is shown that the proposed MRCPR not only provides innovation in modular and reconfigurable CPRs design but also motivates further investigation of modular and reconfiguration CPRs.

The main motivation of this article is to investigate the structure design, typical configuration, inverse kinematics, and dynamics models of the MRCPR considering diverse configurations to satisfy the performance requirements in terms of modularity and reconfiguration, flexibility, and multifunction for the CDPRs in varying practical applications. Other motivation with respect to the MRCPR including the trajectory tracking, safety monitoring, and obstacle avoidance have been investigated in author’s former works.³¹ The contributions of this article are as follows: (1) Several typical configurations of the MRCPR are analyzed, which can be obtained by manually detaching or attaching the different number of modular branches and changing the connection points on the end-effector to meet diverse task requirements. (2) The inverse kinematics of the MRCPR considering diverse configurations is formulated according to the vector closed rule. (3) The dynamics of the MRCPR considering diverse configurations is derived based on Lagrange method. (4) The effectiveness and feasibility of the inverse kinematics and dynamics models of the MRCPR are verified through the numerical simulation and experiments.

The rest of this article is organized as follows. The structure design of the MRCPR including the design methodology, mechanical description, and control architecture is described in the second section. Several typical configurations of the MRCPR are presented in the third section. According to the vector closed rule, the inverse kinematics of the MRCPR is derived in the fourth section. Based on the Lagrange method, the dynamics of the MRCPR is derived in the fifth section. The numerical simulation and related experiments of a typical configuration are conducted in the sixth section. Conclusions are summarized in the last section.

Structure design

Design methodology

The idea of designing the MRCPR is based on the use of modular branches as basic modular components. This design methodology applied in the article is based on the methodology proposed for the design for reconfigurability (DFR). The design of the MRCPR based on the DFR can be depicted according to the axiomatic design theory.³² The design problem is to seek a set of design parameters ${D P}$ that can meet a set of function requirements ${F R}$ via the structure of a design matrix $[D M]$ as

{F R} = [D M] {D P}

where ${F R}$ represents the required degree of freedoms (DOFs) of the end-effector, ${D P}$ denotes a set of modular branch, and $[D M]$ denotes the contribution of each design parameter (modular branch) to each function requirement (DOF).

Thus, the objective of designing the MRCPR is defined that maximizes the number of the function requirements $(F R)$ while minimizes the number of the modular branches $(D P)$ .This problem can be depicted as a bi-objective design optimization problem which is expressed as³³

\underset{k}{M a x} \underset{n}{M i n} ({F R (k), D P (n)})

where n denotes the number of modular branches and k denotes DOFs of the end-effector.

Mechanical description

The three-dimensional model and physical prototype of the MRCPR are shown in Figures 1 and 2, respectively. The main components of the MRCPR include a circular orbit, 6 sets of telescopic and rotatable modular branches, 6 cables, 6 tension sensors, 12 rope displacement sensors, an end-effector, a hydraulic station, a control cabinet, an industrial personal computer (IPC), and so on. Each modular branch mainly consists of a mobile platform, 2 servomotors, 2 motor seats, a hydraulic cylinder, 2 rope displacement sensors, a fixed pulley, a fixed pulley stent, 4 bevel gears, a cylindrical gear, 4 shafts, a gear shaft, a drum, 4 guide wheels, 17 bearings, 2 couplings, and so on. The assembly and disassembly diagram of the modular branch is shown in Figure 3.

Figure 1.

Three-dimensional model of the MRCPR. MRCPR: modular reconfigurable cable-driven parallel robot.

Figure 2.

Physical prototype of the MRCPR. MRCPR: modular reconfigurable cable-driven parallel robot.

Figure 3.

Assembly and disassembly diagram of modular branch.

Each modular branch can be rotated along the circular orbit by gear transmission, and the height of each modular branch can be changed by a hydraulic cylinder. The mobile platform and drum of each modular branch are driven by a servomotor, respectively. Four guide wheels of each modular branch are mounted inside the mobile platform, which can increase the stability and smoothness of the modular branch that is rotated along the circular orbit. For each cable, one end is attached to the corresponding connection point on the end-effector, while the other end passes through the fixed pulley on the top of the modular branch and twines on a drum fixed on the mobile platform. Notice that the huge disarray of cables will appear with respect to the fixed pulley stents during the movement of the end-effector approaching the borders of the workspace. So each fixed pulley on the top of the modular branch is installed on an omnidirectional fixed pulley stent to adapt to the changes of the end-effector positions. The layout guides the cable vertical to the drum axis for reliable coiling and uncoiling of the cable. A tension sensor is attached to each cable in series near the end-effector to test the cable tension. Two rope displacement sensors are installed on the top of each modular branch, which can accurately measure cable length and hydraulic cylinder height, respectively.

According to the diverse task requirements, the spatial topology of the MRCPR can be reconfigured by manually detaching or attaching the number of modular branches and changing the connection points on the end-effector. Compared with the traditional CPRs, the proposed MRCPR can be reconfigurable into a number of different configurations that are constructed by six identical modular branches. By detaching 1, 2, 3, and 4 modular branches separately, the MRCPR can be reconfigured from 6-DOF to 4, 3, or 2, respectively.

Control architecture

The system control architecture diagram of the MRCPR is shown in Figure 4. It can be decomposed into three layers, that is, the topmost host computer, intermediate programmable logic controller (PLC), and the bottommost controller including $n (n = 2, 3, 4, 5, or 6)$ hydraulic cylinders and $2 n (n = 2, 3, 4, 5, or 6)$ electric drives. The host computer consists of a set of LabVIEW and an IPC which is adopted from the Advantech IPC-610H (China). The host computer mainly devotes to the kernel algorithms, for example, uploading the configuration parameters, trajectory planning, computation of $n (n = 2, 3, 4, 5, or 6)$ coordinate drum positions for cable coiling or uncoiling, optimization cable tensions, setting the motion mode of shafts, switching on the enabler for servomotors, eliminating the alarm information, and so on. In addition, the host computer consists of a few modules including user interface, database management, communication, and so on. Meantime, the LabVIEW is implemented into an IPC-based real-time operating system.

Figure 4.

Control architecture diagram of the MRCPR. MRCPR: modular reconfigurable cable-driven parallel robot.

It is can be connected through RS232 bus between the host computer and the intermediate PLC. The PLC consists of a Mitsubishi FX2N-48MR controller, power module, I/O and O/I modules, and so on. Its task is that gathers process parameters, performs real-time calculations, and carries out operation commands. The bottommost controller consists of $n (n = 2, 3, 4, 5, or 6)$ hydraulic cylinders and $2 n (n = 2, 3, 4, 5, or 6)$ electric drives. Each hydraulic cylinder is controlled by an electromagnetic proportional valve, and hydraulic oil is supplied by a hydraulic station. Each mobile platform is driven by a servomotor that is controlled by a servomotor driver. Each cable is coiled or uncoiled by a drum driven by a servomotor that is controlled by a servomotor driver.

The PLC and motor drivers can command velocity, position, or torque set values for the motors. The current control loop and inner torque are embedded in all motor drivers. The data of each cable length and each hydraulic cylinder height are detected by a rope displacement sensor. The data of each cable tension are detected by a tension sensor. All of the above data are sampled by a data acquisition card and processed by the PLC.

Typical configuration

Several typical configurations of the MRCPR can be obtained by reconfiguring the different number of modular branches and changing the connection points on the end-effector, which are performed by manual assistance, as shown in Figure 5.

Figure 5.

Several typical configurations of the MRCPR. (a) Configuration 1, (b) configuration 2, (c) configuration 3, (d) configuration 4, (e) configuration 5, and (d) configuration 6. MRCPR: modular reconfigurable cable-driven parallel robot.

Typically, CPRs with n-DOFs driven by m cables can be generally classified into three main categories: incompletely restrained CPRs (IRCPRs) when $n + 1 > m$ , completely restrained CPRs (CRCPRs) when $n + 1 = m$ , and redundantly restrained CPRs (RRCPRs) when $n + 1 < m$ .⁵ In Figure 5, configuration 1 consists of two modular branches and two cables, and it can accomplish two translational motions in the two-dimensional plane. For the 2-DOF CPRs, ${F R}$ can be specifically defined as ${x, z}^{T}$ . Configuration 2 consists of three modular branches and three cables, and it can carry out three translational motions in the three-dimensional space. Configuration 3 consists of four modular branches and four cables, and it can also realize three translational motions in the three-dimensional space. For the two 3-DOF CPRs, ${F R}$ can be specifically defined as ${x, y, z}^{T}$ . Configuration 4 consists of five modular branches and five cables, and it can fulfill three translational motions and one rotational motion in the three-dimensional space. For the 4-DOF CDPR, ${F R}$ can be specifically defined as ${x, y, z, β}^{T}$ . Configurations 5 and 6 consist of six modular branches and six cables, and they can achieve three translational motions and three rotational motions in the three-dimensional space. For the 6-DOF CDPR, ${F R}$ can be specifically defined as ${x, y, z, α, β, γ}^{T}$ . In all configurations, $x, y, z$ denote the three translational motions of the end-effector which move along X-, Y-, and Z-axes in global coordinate frame, respectively, and $α, β, γ$ represent the three rotational motions of the end-effector which rotate about Z-, Y-, and X-axes in global coordinate frame, respectively. In addition, the heights and/or angles of modular branches of each configuration can be changed when the hydraulic cylinders and/or mobile platforms of the modular branches are not locked. Thus, there are more motions increased which compare with corresponding original configurations. Simultaneously, the MRCPR has the characteristics of the IRCPRs, CRCPRs, and RRCPRs. In other words, the parameters of varying configurations of the MRCPR can be changed according to specific task requirements, which can improve flexibility and save costs for optimization.³⁴

Now that the proposed MRCPR in this article can alter its configurations by manual assistance, the pivotal is to find out a unified approach to deal with its diverse inverse kinematics and dynamics.

Inverse kinematics analysis

The schematic of the MRCPR is shown in Figure 6. Inverse kinematics solution is considered according to the position and orientation of the end-effector to solve $i (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)$ cable lengths. Two coordinate frames are set up in Figure 6 to depict the kinematic model of the MRCPR. A global coordinate frame $P X Y Z$ is fixed to the center point P of the circular orbit surface, which the X-axis is from point P to point C ₀, the Z-axis is perpendicular to the plane of the circular orbit upwards, and the Y-axis is perpendicular to the plane formed by the X- and Z-axes. A local coordinate frame $Q X Y Z$ is fixed to the center point Q of the bottom surface of the end-effector, which the X-axis is parallel to $P C_{0}$ , the Z-axis is perpendicular to the bottom surface of the end-effector upwards, and Y-axis is perpendicular to the plane formed by the X- and Z-axes. All coordinate frames are right-hand coordinate frame in this article.

Figure 6.

Schematic of the MRCPR. MRCPR: modular reconfigurable cable-driven parallel robot.

The end-effector is driven by $i (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)$ cables that connection points are B_i , and the other end of cables pass through the fixed pulley connecting to the drum driven by a servomotor, respectively, that anchor points are A_i . C_i represents the projection of points A_i on the circular orbit surface. R denotes the radius of the circular orbit. H_i represents the height of the ith modular branch. $φ_{i}$ denotes the angle between $P C_{0}$ and $P C_{i}$ . r ₁ represents the radius of the top surface of the end-effector. r ₂ represents the radius of the bottom surface of the end-effector. h denotes the thickness of the top surface and bottom surface of the end-effector.

The angles of modular branches are expressed as

φ_{i} = \frac{2 π (i - 1)}{n} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

The coordinates of anchor points $A_{i} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)$ in the global coordinate frame $P X Y Z$ are described as

{}^{P}A ​_{i} = {[R cos φ_{i} R sin φ_{i} H_{i}]}^{T} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

The coordinates of connection points $B_{i} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)$ in the local coordinate frame $Q X Y Z$ are described as

{}^{Q}B ​_{i} = {\begin{matrix} {[\begin{matrix} r_{1} cos ϕ_{i} & r_{1} sin ϕ_{i} & h \end{matrix}]}^{T} (i = 1, 2, \dots n, n = 2, 3, o r 4) \\ {[\begin{matrix} r_{1} cos ϕ_{i} & r_{1} sin ϕ_{i} & h \end{matrix}]}^{T} (i = 1, 3, 5, n = 5) \\ {[\begin{matrix} r_{2} cos ϕ_{i} & r_{2} sin ϕ_{i} & 0 \end{matrix}]}^{T} (i = 2, 4, n = 5) \\ {[\begin{matrix} r_{1} cos ϕ_{i} & r_{1} sin ϕ_{i} & h \end{matrix}]}^{T} (i = 1, 3, 5, n = 6) \\ {[\begin{matrix} r_{2} cos (ϕ_{i} + η) & r_{2} sin (ϕ_{i} + η) & 0 \end{matrix}]}^{T} (i = 2, 4, 6, n = 6) \end{matrix}

where η denotes the starting angle of connection points $B_{i} (i = 2, 4, 6)$ .

Thus the coordinates of connection points $B_{i} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)$ in the global coordinate frame $P X Y Z$ are expressed as

{}^{P}B ​_{i} = u + {}_{Q}^{P}R \cdot {}^{Q}B ​_{i}

u = {[x, y, z]}^{T}

where $u$ denotes the location vector which the origin Q of the local coordinate frame $Q X Y Z$ with respect to the global coordinate frame $P X Y Z$ .

{}_{Q}^{P}R = {}_{Q}^{P}R ​_{X Y Z} (γ, β, α) = [\begin{matrix} c α c β & c α s β s γ - s α c γ & c α s β c γ + s α s γ \\ s α c β & s α s β s γ + c α c γ & s α s β c γ - c α s γ \\ - s β & c β s γ & c β c γ \end{matrix}]

where $s α$ is shorthand for $sin α$ , $c α$ for $cos α$ , and so on. γ denotes the pitch angle, β denotes the roll angle, and α denotes the yaw angle. The pitch, roll, and yaw angles are defined for the end-effector in space as the rotations about the X-, Y-, and Z-axes attached to the end-effector, respectively. ${}_{Q}^{P}R _{X Y Z} (γ, β, α)$ is the rotate matrix which the local coordinate frame $Q X Y Z$ with respect to the global coordinate frame $P X Y Z$ .

The length vector of the ith cable is expressed as

l_{i} = B_{i} A_{i} = {}^{P}A ​_{i} - {}^{P}B ​_{i} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

Substituting equations (4) and (6) into equation (9) yields

l_{i} = B_{i} A_{i} = {}^{P}A ​_{i} - u - {}_{Q}^{P}R \cdot {}^{Q}B ​_{i} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

that is, the length of the ith cable is written as

l_{i} = A_{i} B_{i} = \sqrt{l_{i}^{T} l_{i}} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

Therefore, the velocity of end-effector of the MRCPR can be expressed as

\dot{q} = {\begin{matrix} {[\dot{x} \dot{y}]}^{T} (n = 2) \\ {[\dot{x} \dot{y} \dot{z}]}^{T} (n = 3) \\ {[\dot{x} \dot{y} \dot{z}]}^{T} (n = 4) \\ {[\dot{x} \dot{y} \dot{z} \dot{β}]}^{T} (n = 5) \\ {[\dot{x} \dot{y} \dot{z} \dot{α} \dot{β} \dot{γ}]}^{T} (n = 6) \end{matrix}

The Jacobian matrix of the MRCPR is defined as the relationship between the velocity of end effector and the velocity of cables. Then the relationship between the velocity of cables ${\dot{l}}_{i}$ and the velocity of end-effector $\dot{q}$ can be written as

{\dot{l}}_{i} = J \cdot \dot{q} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

Jacobian matrix of the MRCPR can be obtained according to equation (10). Thus Jacobian matrix $J$ can be expressed as

J = {\begin{matrix} [\begin{matrix} \frac{\partial l_{i}}{\partial x} & \frac{\partial l_{i}}{\partial z} \end{matrix}] (i = 1, \dots, n, n = 2) \\ [\begin{matrix} \frac{\partial l_{i}}{\partial z} & \frac{\partial l_{i}}{\partial y} & \frac{\partial l_{i}}{\partial z} \end{matrix}] (i = 1, \dots, n, n = 3) \\ [\begin{matrix} \frac{\partial l_{i}}{\partial x} & \frac{\partial l_{i}}{\partial y} & \frac{\partial l_{i}}{\partial z} \end{matrix}] (i = 1, \dots, n, n = 4) \\ [\begin{matrix} \frac{\partial l_{i}}{\partial x} & \frac{\partial l_{i}}{\partial y} & \frac{\partial l_{i}}{\partial z} & \frac{\partial l_{i}}{\partial β} \end{matrix}] (i = 1, \dots, n, n = 5) \\ [\begin{matrix} \frac{\partial l_{i}}{\partial x} & \frac{\partial l_{i}}{\partial y} & \frac{\partial l_{i}}{\partial z} & \frac{\partial l_{i}}{\partial α} & \frac{\partial l_{i}}{\partial β} & \frac{\partial l_{i}}{\partial γ} \end{matrix}] (i = 1, \dots, n, n = 6) \end{matrix}

Dynamics analysis

This section deals with the dynamics equation of the MRCPR, which definitely depicts the relationship between force and motion. It is extremely important for us to consider the dynamics equation in the process of the MRCPR design, control algorithm, and MATLAB simulation. Lagrange method is adopted to derive the dynamics equation of the MRCPR, which can be obtained by subtracting the total potential energy from the total kinetic energy of the system. To begin with, assuming that end-effector is a rigid object. The total kinetic energy K of the system is the sum of the translational kinetic energy K ₁ and the rotational kinetic energy K ₂ of the end-effector.

Inertial matrix of the end-effector with respect to the point Q can be written as

I_{Q} = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]

The diagonal elements of the inertial matrix, $I_{x x}, I_{y y}, I_{z z}$ , are called the principal moments of inertia about the X-,Y-, and Z-axes, respectively. The off-diagonal elements of the inertial matrix, $I_{x y,} I_{x z}$ , and so on, are called the cross products of inertia. Assuming that the mass distribution of the end-effector is symmetric with respect to the local coordinate frame $Q X Y Z$ , then the cross products of the inertia are equal to zero.

Therefore, the inertial matrix can be rewritten as

I_{Q} = [\begin{matrix} I_{x x} & 0 & 0 \\ 0 & I_{y y} & 0 \\ 0 & 0 & I_{z z} \end{matrix}]

The total kinetic energy of the end-effector is given in Cartesian coordinates as

K = K_{1} + K_{2} = \frac{1}{2} m \cdot v_{​}^{T} \cdot v_{​} + \frac{1}{2} ω_{​}^{T} \cdot {}_{Q}^{P}R \cdot I_{Q} \cdot {}_{Q}^{P}R ​^{T} \cdot ω

Likewise, the only source potential energy of the end-effector is gravity, it can be expressed as

U_{​} = m g z

where m denotes the mass of the end-effector. g represents the acceleration due to gravity. Lagrange function of the system can be obtained as

L = K - U = \frac{1}{2} m \cdot v^{T} \cdot v + \frac{1}{2} ω_{​}^{T} \cdot {}_{Q}^{P}R \cdot I_{Q} \cdot {}_{Q}^{P}R ​^{T} \cdot ω - m g z

The generalized coordinates of the MRCPR are defined as

{x, y, z, α, β, γ} = {q_{1}, q_{2}, q_{3}, q_{4}, q_{5}, q_{6}}

Thus v, $ω$ , and $q$ can be separately expressed as

v = {\begin{matrix} {[\begin{matrix} \dot{x} & \dot{z} \end{matrix}]}^{T} = {[\begin{matrix} {\dot{q}}_{1} & {\dot{q}}_{3} \end{matrix}]}^{T} (n = 2) \\ {[\begin{matrix} \dot{x} & \dot{y} & \dot{z} \end{matrix}]}^{T} = {[\begin{matrix} {\dot{q}}_{1} & {\dot{q}}_{2} & {\dot{q}}_{3} \end{matrix}]}^{T} (n = 3, 4, 5, 6) \end{matrix}

ω = {\begin{matrix} {[0]}^{T} (n = 2, 3, 4) \\ {[\dot{β}]}^{T} = [{\dot{q}}_{5}] (n = 5) \\ {[\begin{matrix} \dot{α} & \dot{β} & \dot{γ} \end{matrix}]}^{T} = [\begin{matrix} {\dot{q}}_{4} & {\dot{q}}_{5} & {\dot{q}}_{6} \end{matrix}] (n = 6) \end{matrix}

q = {\begin{matrix} {[\begin{matrix} x & z \end{matrix}]}^{T} = {[\begin{matrix} q_{1} & q_{3} \end{matrix}]}^{T} (n = 2) \\ {[\begin{matrix} x & y & z \end{matrix}]}^{T} = {[\begin{matrix} q_{1} & q_{2} & q_{3} \end{matrix}]}^{T} (n = 3) \\ {[\begin{matrix} x & y & z \end{matrix}]}^{T} = {[\begin{matrix} q_{1} & q_{2} & q_{3} \end{matrix}]}^{T} (n = 4) \\ {[\begin{matrix} x & y & z & β \end{matrix}]}^{T} = {[\begin{matrix} q_{1} & q_{2} & q_{3} & q_{5} \end{matrix}]}^{T} (n = 5) \\ {[\begin{matrix} x & y & z & α & β & γ \end{matrix}]}^{T} = {[\begin{matrix} q_{1} & q_{2} & q_{3} & q_{4} & q_{5} & q_{6} \end{matrix}]}^{T} (n = 6) \end{matrix}

Lagrange function of the MRCPR can be given as

L = {\begin{matrix} \frac{1}{2} m ({\dot{q}}_{1}^{2} + {\dot{q}}_{3}^{2}) - m g q_{3} (n = 2) \\ \frac{1}{2} m ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) - m g q_{3} (n = 3) \\ \frac{1}{2} m ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) - m g q_{3} (n = 4) \\ \frac{1}{2} m_{​} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) + \frac{1}{2} I_{y y} {\dot{q}}_{5}^{2} - m g q_{3} (n = 5) \\ \begin{array}{l} \frac{1}{2} m_{​} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) + \frac{1}{2} {I_{x x} {({\dot{q}}_{4} c q_{4} c q_{5} + {\dot{q}}_{5} s q_{4} c q_{5} - {\dot{q}}_{6} s q_{5})}^{2} + I_{y y} [(c q_{4} s q_{5} s q_{6} - s q_{4} c q_{6}) {\dot{q}}_{4} + (c q_{4} c q_{6} + s q_{4} s q_{5} s q_{6}) {\dot{q}}_{5} \\ + {\dot{q}}_{6} c q_{5} s q_{6}]^{2} + I_{z z} {[(c q_{4} s q_{5} c q_{6} + s q_{4} s q_{6}) {\dot{q}}_{4} + (s q_{4} s q_{5} c q_{6} - c q_{4} s q_{6}) {\dot{q}}_{5} + {\dot{q}}_{6} c q_{5} c q_{6}]}^{2}} - m g q_{3} (n = 6) \end{array} \end{matrix}

Next, if Lagrange equation of the MRCPRs will be derived then two conditions hold:

first, the total kinetic energy is a quadratic function of the vector $\dot{q}$ . Thus, the total kinetic energy of the system is expressed as

K = \frac{1}{2} \sum_{i, j}^{n} d_{i j} (q) {\dot{q}}_{i} {\dot{q}}_{j} : = \frac{1}{2} {\dot{q}}^{T} D (q) \dot{q} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)

where the $n \times n$ “inertia matrix” $D (q)$ is symmetric and positive definite for each $q \in R^{n}$ .

Second, the total potential energy is independent of $\dot{q}$ . Thus, the total potential energy of the system is written as

U = U (q)

The Euler–Lagrange equation of the system can be derived as follows. Since

L = K - U = \frac{1}{2} \sum_{i, j} d_{i j} (q) {\dot{q}}_{i} {\dot{q}}_{j} - U (q)

in which

\frac{\partial L}{\partial {\dot{q}}_{s}} = \sum_{j} d_{s j} {\dot{q}}_{j}

\frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{s}} = \sum_{i} d_{s j} {\ddot{q}}_{j} + \sum_{j} \frac{d}{d t} d_{s j} {\dot{q}}_{j} = \sum_{j} d_{s j} {\ddot{q}}_{j} + \sum_{i, j} \frac{\partial d_{s j}}{\partial q_{i}} {\dot{q}}_{i} {\dot{q}}_{j}

\frac{\partial L}{\partial q_{s}} = \frac{1}{2} \sum_{i, j} \frac{\partial d_{i j}}{\partial q_{s}} {\dot{q}}_{i} {\dot{q}}_{j} - \frac{\partial U}{\partial q_{s}}

Therefore, the Euler–Lagrange equation can be expressed as

\sum_{j} d_{s j} {\ddot{q}}_{j} + \sum_{i, j} {\frac{\partial d_{s j}}{\partial q_{i}} - \frac{1}{2} \frac{\partial d_{i j}}{\partial q_{s}}} {\dot{q}}_{i} {\dot{q}}_{j} - \frac{\partial U}{\partial q_{s}} = τ_{s}

By utilizing the Lagrange equation and exchanging the order of summation, it can be shown that

\sum_{i, j} {\frac{\partial d_{s j}}{\partial q_{i}}} {\dot{q}}_{i} {\dot{q}}_{j} = \frac{1}{2} \sum_{i, j} {\frac{\partial d_{s i}}{\partial q_{j}} + \frac{\partial d_{s j}}{\partial q_{i}}} {\dot{q}}_{i} {\dot{q}}_{j}

Thus

s u m_{i, j} {\frac{\partial d_{s j}}{\partial q_{i}} - \frac{1}{2} \frac{\partial d_{i j}}{\partial q_{s}}} {\dot{q}}_{i} {\dot{q}}_{j} = \sum_{i, j} \frac{1}{2} {\frac{\partial d_{s i}}{\partial q_{j}} + \frac{\partial d_{s j}}{\partial q_{i}} - \frac{\partial d_{i j}}{\partial q_{s}}} {\dot{q}}_{i} {\dot{q}}_{j}

The terms

c_{i j s} : = \frac{1}{2} {\frac{\partial d_{s i}}{\partial q_{j}} + \frac{\partial d_{s j}}{\partial q_{i}} - \frac{\partial d_{i j}}{\partial q_{s}}}

are known as Christoffel symbols. Finally, if $ψ_{s}$ can be defined as

ψ_{s} = \frac{\partial U}{\partial q_{s}}

Then the Euler–Lagrange equation can be written as

\sum_{i} d_{s j} (q) {\ddot{q}}_{j} + \sum_{i, j} c_{i j s} (q) {\dot{q}}_{i} {\dot{q}}_{j} + ψ_{s} (q) = τ_{s} (s = 1, 2, \dots, n)

Equation (36) can be rewritten in matrix form as

D (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) = τ

τ = - J^{T} \cdot T

Here, the k,j-th element of the matrix $C (q, \dot{q})$ is defined as

\begin{array}{l} c_{s j} = \sum_{i = 1}^{n} c_{i j s} (q) {\dot{q}}_{i} \\ = \sum_{i = 1}^{n} \frac{1}{2} {\frac{\partial d_{s j}}{\partial q_{j}} + \frac{\partial d_{s i}}{\partial q_{j}} - \frac{\partial d_{i j}}{\partial q_{s}}} {\dot{q}}_{i} \end{array}

where $τ$ represents the external force and external torque acting on the end-effector. $T = {[t_{1}, t_{2}, \dots, t_{i}]}^{T} (i = 1, 2, \dots, n, n = 2, 3, 4, 5, or 6)$ denotes cable tensions, $D (q)$ is the inertia matrix for the system, $C (q, \dot{q})$ denotes the vector of Centrifugal and Coriolis terms, and $G (q)$ represents the vector of potential energy with respect to gravity terms.³⁵

Numerical simulation and experiments

Configuration 2 is selected from all typical configurations of the MRCPR, in which the performance of kinematics and dynamics can be verified by numerical simulation and a series of experiments. The angles and heights of three modular branches are locked during the course of the experiments. The change curves of cable lengths and cable tensions are obtained from the different angles and heights of three modular branches, which the results are compared and analyzed. The errors of cable lengths and cable tensions can be separately obtained by comparing simulation results and experiments results.

{\begin{matrix} x = 150 * cos (p i * t / 20) \\ y = 150 * sin (p i * t / 20) \\ z = 3 * t + 220 \end{matrix} (0 \leq t \leq 60 s)

The trajectory equation of end-effector is a space cylinder spiral line, as shown in equation (40) and Figure 7.

Figure 7.

Space cylinder spiral line.

Meanwhile, the specific geometric parameters of the MRCPR are listed in Table 1.

Table 1.

Geometric parameters of the MRCPR.

Symbol	Quantity	Value
R	Radius of the circular orbit	520 (mm)
H_i	Height of the i-th modular branch	660–760 (mm)
$ϕ_{i}$	Angle of between $P C_{0}$ and $P C_{i}$	0–360 (°)
d	Diameter of cable	0.6 (mm)
ρ	Density of cable	7.9 (g/cm³)
E	Young’s modulus	$1.6 \times 10^{11} (Pa)$
r ₁	Radius of the top surface of the end-effector	80 (mm)
r ₂	Radius of the bottom surface of the end-effector	90 (mm)
h	Thickness of the end-effector	54 (mm)
m	Mass of the end-effector	5 (kg)

MRCPR: modular reconfigurable cable-driven parallel robot.

Cable lengths

The first group of cable lengths experiments is illustrated in Figure 8.

Figure 8.

The change curves and error change curves of cable lengths ( $ϕ_{1} = 0^{\circ}, ϕ_{2} = 120^{\circ}, ϕ_{3} = 240^{\circ}$ ).

The second group of cable lengths experiments is illustrated in Figure 9.

Figure 9.

The change curves and error change curves of cable lengths ( $ϕ_{1} = 0^{\circ}, ϕ_{2} = 90^{\circ}, ϕ_{3} = 240^{\circ}$ ).

The third group of cable lengths experiments is illustrated in Figure 10.

Figure 10.

The change curves and error change curves of cable lengths ( $ϕ_{1} = 0^{\circ}, ϕ_{2} = 150^{\circ}, ϕ_{3} = 240^{\circ}$ ).

Cable tensions

The first group of cable tensions experiments is illustrated in Figure 11.

Figure 11.

The change curves and error change curves of cable tensions ( $ϕ_{1} = 0^{\circ}, ϕ_{2} = 120^{\circ}, ϕ_{3} = 240^{\circ}$ ).

The second group of cable tensions experiments is illustrated in Figure 12.

Figure 12.

The change curves and error change curves of cable tensions ( $ϕ_{1} = 0^{\circ}, ϕ_{2} = 90^{\circ}, ϕ_{3} = 240^{\circ}$ ).

The third group of cable tensions experiments is illustrated in Figure 13.

Figure 13.

The change curves and error change curves of cable tensions ( $ϕ_{1} = 0^{\circ}, ϕ_{2} = 150^{\circ}, ϕ_{3} = 240^{\circ}$ ).

Three groups of experiments have been carried out to verify the kinematics and dynamics capability of configuration 2, which are shown in Figures 8 to 13. All experiments are based on the trajectory, which is a space cylinder spiral line. In Figures 8 to 13, cables 1, 2, and 3 are separately denoted by red, green, and blue lines, and the actual curve and desired curve of cable lengths and cable tensions are, respectively, represented by solid line and dotted line.

Three groups of experiments are compared and analyzed when the angles of three modular branches are (1) $ϕ_{1} = 0^{\circ}$ , $ϕ_{2} = 120^{\circ}$ , $ϕ_{3} = 240^{\circ}$ ; (2) $ϕ_{1} = 0^{\circ}$ , $ϕ_{2} = 90^{\circ}$ , $ϕ_{3} = 240^{\circ}$ ; (3) $ϕ_{1} = 0^{\circ}$ , $ϕ_{2} = 150^{\circ}$ , $ϕ_{3} = 240^{\circ}$ . Each group of experiment consists of four different schemes when the heights of three modular branches are (1) $H_{1} = 660 mm$ , $H_{2} = 660 mm$ , $H_{3} = 660 mm$ ; (2) $H_{1} = 760 mm$ , $H_{2} = 660 mm$ , $H_{3} = 660 mm$ ; (3) $H_{1} = 760 mm$ , $H_{2} = 760 mm$ , $H_{3} = 660 mm$ ; (4) $H_{1} = 760 mm$ , $H_{2} = 760 mm$ , $H_{3} = 760 mm$ . Each scheme consists of the change of cable lengths and cable tensions and the error change of cable lengths and cable tensions.

In each group of experiment, the change curves of cable lengths and cable tensions are similar to sinusoid. As shown in Figures 8(a) to 10(a), three cable lengths gradually become shorter with the elevating of the end-effector from the initial point of the trajectory to the final point. It is obvious that cable lengths gradually become longer at the initial point of the trajectory when the heights of three modular branches are increased from 660 mm to 760 mm. With the height of each modular branch is changed from 660 mm to 760 mm, the cable length corresponding to the modular branch becomes longer. However, the second cable length becomes shorter when the angle $ϕ_{2}$ is changed from $120^{\circ}$ to $90^{\circ}$ , while the second cable length becomes longer when the angle $ϕ_{2}$ is changed from $120^{\circ}$ to $150^{\circ}$ . As shown in Figures 11(a) to 13(a), three cable tensions gradually become larger with the elevating of the end-effector from the initial point of the trajectory to the final point. With the height of each modular branch is changed from 660 mm to 760 mm, the cable tension corresponding to the other two modular branches is affected greatly. The change curve of the third cable tension is affected greatly when the angle $ϕ_{2}$ is changed from $120^{\circ}$ to $90^{\circ}$ , while the change curve of the first cable tension is affected greatly when the angle $ϕ_{2}$ is changed from $120^{\circ}$ to $150^{\circ}$ . Based on the above experimental results, the structure design of the MRCPR is feasible, the mechanical design and control architecture are reasonable, and its precision can meet the requirement. Modular branches are collaborated well with each other.

The error change curves of cable lengths and cable tensions separately display a similar shape which are shown in Figures 8(b) to 10(b) and Figures 11(b) to 13(b). The end-effector trajectory error is caused by diverse error sources, which include kinematic error, servo error, and deformation. Machining, assembly, and operation can lead to kinematic error, which is the primary error sources.³¹ Notice that these deviations can be eliminated by the proposed robust iterative learning controller in author’s former works.¹¹ Therefore, the end-effector of the MRCPR can be elevated within the accepted error scope smoothly.

Conclusions

In this article, a MRCPR with several typical configurations has been analyzed. The proposed MRCPR can be reconfigurable into a number of different configurations that are constructed by six identical modular branches. The spatial topology of the MRCPR can be reconfigured by manually detaching or attaching the different number of modular branches and changing the connection points on the end-effector to meet diverse task requirements. By detaching 1, 2, 3, and 4 modular branches separately, the MRCPR can be reconfigured from 6-DOF to 4, 3, or 2, respectively. The modular branches can be not only rotated along the circular orbit but also raised or lowered by hydraulic cylinders. The structure design of the MRCPR including the design methodology, mechanical description, and control architecture is depicted in detail. The MRCPR with several typical configurations has the features of the IRCPRs, CRCPRs, and RRCPRs. The inverse kinematics and dynamics of the MRCPR considering varying configurations including 2, 3, 4, and 6-DOF are derived by the vector closed rule and Lagrange method, respectively. The numerical simulation and related experiments of the typical configuration are achieved and analyzed. The results not only verify the collaboration of modular branches each other but also confirm the effectiveness and feasibility of the inverse kinematics and dynamics models for the MRCPR.

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: This work was supported by the National Natural Science Foundation of China (91748109 and 51575150).

ORCID iD

Tao Zhao

Bin Zi

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