Design, Dynamics, and Workspace of a Hybrid-Driven-Based Cable Parallel Manipulator

Abstract

The design, dynamics, and workspace of a hybrid-driven-based cable parallel manipulator (HDCPM) are presented. The HDCPM is able to perform high efficiency, heavy load, and high-performance motion due to the advantages of both the cable parallel manipulator and the hybrid-driven planar five-bar mechanism. The design is performed according to theories of mechanism structure synthesis for cable parallel manipulators. The dynamic formulation of the HDCPM is established on the basis of Newton-Euler method. The workspace of the manipulator is analyzed additionally. As an example, a completely restrained HDCPM with 3 degrees of freedom is studied in simulation in order to verify the validity of the proposed design, workspace, and dynamic analysis. The simulation results, compared with the theoretical analysis, and the case study previously performed show that the manipulator design is reasonable and the mathematical models are correct, which provides the theoretical basis for future physical prototype and control system design.

1. Introduction

In manipulation systems, it is always a goal to continuously improve the range of motion, accuracy, and load capacities. Compared with the traditional mechanical structures, parallel mechanisms have the advantages of high dexterity and accuracy. Cable parallel manipulators (CPMs) are categorized as a type of parallel manipulators. They are a variant of the Gough-Stewart platform in which rigid extensible legs are replaced by cables stored spools. CPMs have several advantages over rigid-link mechanisms, such as (i) remote location of motors and controls, (ii) rapid deployability, (iii) potentially large workspaces and (iv) high load capacity; (v) reliability [1, 2]. For the preceding characteristics, the CPMs have received a lot of attention since 1980s [3]. These manipulators, referred to as the overhead crane and rotary crane, are useful for many applications in hazardous environments in order to enable the manipulation of heavy objects [4, 5].

The first written information on use of hoist mechanism in ancient Greece appeared around 530 BC, which is concerned with the construction of the first temple of Artemis in Ephesus, and temples at Selinous, temples of Apollo at Syracuse and at Corinth. After 515 BC, cranes were in common use [6, 7]. The CPMs consist of a fixed base and a centrally located end-effector, attached to moving payload, connected by cables whose tension is maintained along the tracked trajectory [8, 9]. However, due to the unilateral driving property of the cables, maintaining positive cable tension is essential to maneuver the end-effector. As a result, the number of driving cables must be more than the number of degrees of freedom (DOF) of the end-effector in order to obtain a completely restrained manipulation [10]. This type of technology is now well known, and several studies of the CPMs have been undertaken. For instance, Gosselin and Barrette [11] presented a fundamental and systematic analysis of a planar CPM. Design and performances of a 4-cable driven parallel manipulator have been presented by Ottaviano et al. [12]. Flatness-based trajectory tracking control for a three-cable suspension manipulator has been investigated by Heyden and Woernle [13]. Due to cable slippage on the drum, Baser and Konukseven developed an analytical method for predicting the transmission error [14]. Different aspects like kinematics [15], workspace characteristics [16], dimensional optimization [17], and statics [18] were extensively investigated.

As we are aware, there are relatively fewer published works on the drive system of the CPMs, which is the main contribution of this work. The existing research on the drive system of the CPMs usually uses the low power controllable motor. In the recent years, as the CPMs have been investigated widely for various applications in modern manufacturing, they are required not only for operations with high accuracy and high payload, but also for output with greater flexibility, which can change the law of output motion quickly and conveniently [19]. Nevertheless, the low power controllable motor cannot directly drive the high-loading CPM due to the restrictions of power and torque. Hence, it is necessary to carry out for a new-type driven system for the CPMs.

The hybrid-driven planar five-bar mechanism (HDPM) is a kind of machine whose drive system consists of a constant velocity (CV) motor and a servomotor [20]. The CV motor provides main power and motion required; however, it lacks high controllability. On the other hand, the servomotor acts as a motion regulation device which suffers from the limited power capacity. Therefore, the HDPM can take the advantage of the complementary characteristics of both motors to generate a programmable range of highly nonlinear output motions with high power capacities [21]. In fact, the HDPM can be seen as a type of servomechanism [22]. In this investigation the design of a hybrid-driven-based cable parallel manipulator (HDCPM) which combines four groups of HDPMs with a 4-cable parallel manipulator is presented. The HDCPM can take the advantages of the characteristics of both mechanisms to generate a high-performance output motion with high efficiency and heavy payload [23].

It is well known that dynamic performance has a great impact on the operation of the manipulator, and this is on the basis of its dynamic control. As demonstrated in [24], servomechanism dynamics constitute an important component of the complete robot dynamics. Therefore, for the HDCPM, the dynamics should not only consider the CPM, but also include the HDPM (servomechanism). However, the literature on the dynamics of the CPM system including the linkage (servomechanism) dynamics is sparse. Aiming at the HDCPM, this paper is intended to study the completely dynamic modeling including the HDPM actuator dynamics on the basis of Newton-Euler method. The latter is used to solve the inverse kinematic and dynamic problems of the HDCPM on condition that the operation path of the end-effector has been planned.

The workspace is also one of the most useful measures for the evaluation of the manipulator performances [25–27]. Pusey et al. [28] defined the workspace of a CPM as the set of points where the centre of gravity of the end-effector can be reached with admissible tensions in all cables. Apart from the tension condition, workspace of the HDCPM is also limited by geometrical conditions, which include the workspace of the HDPM actuator, length of cables, and motion range of end-effector for the HDCPM. This paper discusses the process of workspace formation of the HDCPM in detail.

In the rest of the paper, Section 2 describes the design model of the HDCPM. In the following, dynamic modeling of the HDCPM is performed based on Newton-Euler method in Section 3. Workspace analysis of the HDCPM is provided in Section 4. In Section 5, illustrative simulation studies highlight its performances. Finally, some concluding remarks and future studies are summarized in Section 6.

2. Mechanism Description

In this paper, for the purpose of analytical modeling and numerical analysis, the three-dimensional design model of the completely restrained HDCPM with three translational motions is taken as an example (see Figure 1). The HDCPM suspends an end-effector by four cables and restrains all motion degrees of freedom for the object using the cable force when the end-effector moves within the workspace. For each cable, one end is connected to the end-effector, the other one rolls through a pulley fixed on the top of the relative cable tower rack and then is fed into the HDPM. The HDCPM comprises of two modules: (1) the CPM consisting of four-cable tower racks, four cables, pulley struts, pulleys, girder, and cargo (i.e., end-effector); (2) four groups of HDPMs containing three-phase asynchronous motors, servomotors, reducers, and double crank five-bar linkage; the asynchronous motors are connected using the pulley transmission mechanisms, while the servomotors and the reducers are linked by couplings. At the same time, the pulley transmission mechanisms and reducers are joined to the double crank planar five-bar linkage. The double crank planar five-bar linkage is made up of a large crank disk, a long connecting rod, a small crank disk, and a short connecting rod. Both long and short connecting rods are articulated with each other and attached to the cable. Moreover, each tower rack of the CPM is equipped with a cable guide pulley. The mechanical configuration of the HDCPM is illustrated in Figure 2.

Figure 1 

Three-dimensional model of the HDCPM.

(a)

(b)

Figure 2 

Mechanical configuration of the HDCPM: (a) front view, (b) top view.

The working process of the HDCPM is as follows. The CV three-phase asynchronous motors and servomotors are power sources. On the one hand, the servomotor and the reducer are linked by the coupling, then the servomotors through a small crank disk are connected with a short rod connection thus adjusting the output motion of the HDPM. On the other hand, each CV three-phase asynchronous motor is connected with the pulley transmission mechanisms, then it links to a long connecting rod by a large crank disk, thus providing the main power for the HDCPM. These two types of input motions are hinged through the short and long connecting rod, so that the power distribution and other characteristics of the HDPM are improved while ensuring the output motions. Four groups of the HDPM with the same structure produce rotary motion, thus the four cables can be driven in order to realize the output motion trajectory of the end-effector.

3. Dynamic Modeling

A simple schematic sketch of the HDCPM structure model with the associated coordinate systems is depicted in Figure 3. At the bottom of one cable tower rack a global coordinate system 𝑂(𝑋𝑌𝑍) is established. The end-effector has location coordinates 𝐺(𝑥,𝑦,𝑧). The distance between each cable tower rack top 𝑀_𝑖(𝑥_𝑖,𝑦_𝑖,𝑧_𝑖) and the end-effector is 𝑙_𝑖(𝑖=1,2,3,4). The four cable tower racks have same heighth and are arrayed in a rectangle on the ground, where deformation is ignored. In order to simplify the model, the cables are treated as a massless body with no deformation, with length 𝑠. The bottom of the HDPM corresponds to the joint 𝐴 as the origin, where a local coordinate system 𝐴(𝑥^′𝑦^′𝑧^′) is established.

(a)

(b)

Figure 3 

Structure model (a) of the HDPM and schematic sketch (b) of the HDCPM.

Link 𝐴𝐵 driven by a CV motor and link ED driven by a servomotor are the two driving links (Figure 2(a)), the joint 𝐶 is the output. There are two independent constraints in the HDPM, which can be used to derive the kinematic equation of the HDPM: 𝑙^′₁cos𝜃₁+𝑙^′₂cos𝜃₂−𝑙^′₃cos𝜃₃−𝑙^′₄cos𝜃₄−𝑙^′₅𝑙=0,_′1sin𝜃₁+𝑙^′₂sin𝜃₂−𝑙^′₃sin𝜃₃−𝑙^′₄sin𝜃₄=0 (1)

In (1) 𝜃₁ and 𝜃₄ are known. Consequently 𝜃₂ and 𝜃₃ can be calculated as 𝜃₃𝜃₁,𝜃₄𝐶=arccos𝐴²+𝐵²𝜇+arctan𝜆,𝜃₂𝜃₁,𝜃₄=arctan𝜇+𝑙^′₃sin𝜃₃𝜆+𝑙^′₃cos𝜃₃, (2) where 𝐴=2𝑙^′₃𝜆, 𝐵=2𝑙^′₃𝜇, 𝐶=𝑙₂^′2−𝑙₃^′2−𝜆²−𝜇², 𝜆=𝑙^′₄cos𝜃₄−𝑙^′₁cos𝜃₁+𝑙^′₅, 𝜇=𝑙^′₄sin𝜃₄−𝑙^′₁sin𝜃₁.

Let 𝑥^′_𝐶=𝑙^′₁cos𝜃₁+𝑙^′₂cos𝜃₂=𝑙^′₅+𝑙^′₃cos𝜃₃+𝑙^′₄cos𝜃₄,𝑧^′_𝐶=𝑙^′₁sin𝜃₁+𝑙^′₂sin𝜃₂=𝑙^′₃sin𝜃₃+𝑙^′₄sin𝜃₄ (3) be the coordinates of joint 𝐶 in the local coordinate system. According to the geometric relationship of the HDCPM, 𝑙₁ can be derived as 𝑙₁=𝑆−𝑝+𝑙^′₅−𝑥^′_𝐶²+ℎ−𝑧^′_𝐶², (4) where 𝑝 represents the distance 𝐸𝑂 and 𝑠 is the total length of cable. Similarly, 𝑙₂, 𝑙₃, and 𝑙₄ can be derived, respectively.

The relationships between the cable length 𝑙_𝑖 and the end-effector location 𝐺(𝑥,𝑦,𝑧) can be easily obtained as follows: 𝑙₁=𝑥²+𝑦²+(𝑧−ℎ)²,𝑙₂=(𝑥−𝑎)²+𝑦²+(𝑧−ℎ)²,𝑙₃=(𝑥−𝑎)²+𝑦−𝑏²+(𝑧−ℎ)²,𝑙₄=𝑥²+𝑦−𝑏²+(𝑧−ℎ)² (5) resulting in 𝑙𝑥=²¹−𝑙²₂+𝑎²,𝑙2𝑎𝑦=²¹−𝑙²₄+𝑏²,2𝑏𝑧=ℎ−𝑙²₁−𝑥²−𝑦². (6)

Newton’s law for the end-effector of the HDCPM results in 𝑚𝑥̈𝑦̈𝑧̈=^4→_𝑖=1𝑇_𝑖+00−𝑚𝑔, (7) where 𝑇_𝑖(𝑖=1,2,3,4) is a tension cable force, that is, the driving force of cable 𝑖 exerted on the end-effector for the HDCPM, and 𝑇^→_𝑖 is force vector.

Thus the dynamic model can be expressed as follows: 𝑁_3×4𝑇_4×1=𝑀_3×1, (8) where 𝑁_3×4=−𝑥𝑙₁𝑎−𝑥𝑙₂𝑎−𝑥𝑙₃−𝑥𝑙₄−𝑦𝑙₁−𝑦𝑙₂𝑏−𝑦𝑙₃𝑏−𝑦𝑙₄ℎ−𝑧𝑙₁ℎ−𝑧𝑙₂ℎ−𝑧𝑙₃ℎ−𝑧𝑙₄ (9) is the matrix of coordinate transformer, 𝑇_4×1=[𝑇₁𝑇₂𝑇₃𝑇₄]^𝑇 is the vector of cable tension, 𝑀_3×1=[𝑚𝑥̈𝑚𝑦̈𝑚𝑔+𝑚𝑧]̈^𝑇 is the vector of inertia force, m is mass of the end-effector, and 𝑔 the gravitational acceleration.

The dynamic model of the HDCPM is presented in two parts. The first is directed to the structural model (CPM) above, the other one to the actuator dynamics (servomechanism). Applying the Newton-Euler equation establishes the dynamic model of the HDPM. The force analysis of each link of the HDPM is shown in Figure 4. 𝜏₁ and 𝜏₄ represent the driving torque of the CV motor on link AB and the servomotor on link ED, respectively; 𝛽= arctan ((ℎ−𝑙^′₁sin𝜃₁−𝑙^′₂sin𝜃₂)/(𝑝+𝑙^′₅−𝑙^′₁cos𝜃₁−𝑙^′₂cos𝜃₂)) is the angle between cable tension and 𝑥^′-coordinate.

(a)

(b)

(c)

(d)

Figure 4 

Free body diagrams of the HDPM: (a) link AB, (b) link BC, (c) link CD, and (d) link ED.

By the force analysis mentioned previously, the Newton-Euler formulation of the links can be listed as follows.

Link 𝐴𝐵: 𝐹_𝐴𝑥′−𝐹_𝐵𝑥′=𝑚₁𝑥̈^′₁,𝐹_𝐴𝑧′−𝑚₁𝑔−𝐹_𝐵𝑧′=𝑚₁𝑧̈^′₁,𝜏₁+𝐹_𝐴𝑥′𝑟₁sin𝜃₁−𝐹_𝐴𝑧′𝑟₁cos𝜃₁+𝐹_𝐵𝑥′𝑙^′₁−𝑟₁sin𝜃₁−𝐹_𝐵𝑧′𝑙^′₁−𝑟₁cos𝜃₁=𝐽₁𝜃̈₁. (10)

Link 𝐵𝐶: 𝑇₁cos𝛽+𝐹_𝐵𝑥′−𝐹_𝐶𝑥′=𝑚₂𝑥̈^′₂,𝑇₁sin𝛽+𝐹_𝐵𝑧′−𝐹_𝐶𝑧′−𝑚₂𝑔=𝑚₂𝑧̈^′₂,𝐹_𝐵𝑥′𝑟₂sin𝜃₂−𝐹_𝐵𝑧′𝑟₂cos𝜃₂+𝐹_𝐶𝑥′−𝑇₁cos𝛽𝑙^′₂−𝑟₂sin𝜃₂+𝑇₁sin𝛽−𝐹_𝐶𝑧′𝑙^′₂−𝑟₂cos𝜃₂=𝐽₂𝜃̈₂. (11)

Link 𝐶𝐷: 𝑇₁cos𝛽+𝐹_𝐶𝑥′−𝐹_𝐷𝑥′=𝑚₃𝑥̈^′₃,𝑇₁sin𝛽+𝐹_𝐶𝑧′−𝑚₃𝑔−𝐹_𝐷𝑧′=𝑚₃𝑧̈^′₃,𝐹_𝐶𝑧′+𝑇₁sin𝛽𝑟₃cos𝜃₃−𝐹_𝐶𝑥′+𝑇₁cos𝛽𝑟₃sin𝜃₃−𝐹_𝐷𝑥′𝑙^′₃−𝑟₃sin𝜃₃+𝐹_𝐷𝑧′𝑙^′₃−𝑟₃cos𝜃₃=𝐽₃𝜃̈₃. (12)

Link 𝐸𝐷: 𝐹_𝐷𝑥′−𝐹_𝐸𝑥′=𝑚₄𝑥̈^′₄,𝐹_𝐷𝑧′−𝑚₄𝑔−𝐹_𝐸𝑧′=𝑚₄𝑧̈^′₄,𝜏₄−𝐹_𝐷𝑥′𝑟₄sin𝜃₄+𝐹_𝐷𝑧′𝑟₄cos𝜃₄−𝐹_𝐸𝑥′𝑙^′₄−𝑟₄sin𝜃₄+𝐹_𝐸𝑧′𝑙^′₄−𝑟₄cos𝜃₄=𝐽₄𝜃̈₄, (13) where 𝑚_𝑗,  𝑙^′_𝑗, 𝐽_𝑗, 𝑟_𝑗, (𝑥^′_𝑗,𝑧^′_𝑗)(𝑗=1,2,3,4), represent the mass, length, inertia, distance between the mass centre and the joint, and coordinates of the mass centre of each link, respectively.

The equation can be summarized as 𝐴_12×14𝐻_14×1=𝐵_12×1, (14)where 𝐴_12×14=^{𝑟10−100000000000010−100000000001𝑠1−𝑟1𝑐1𝑙′1−𝑟1𝑠1𝑟1−𝑙′1𝑐100000000100010−101000000000010−10100000000𝑟2𝑠2−𝑟2𝑐2𝑙′2−𝑟2𝑠2𝑟2−𝑙′2𝑐2𝑟2−𝑙′2𝑠2𝑙′2−𝑟2𝑐2000000}_{00001010−100000000001010−100000000−𝑟3𝑠3𝑟3𝑐3−𝑟3𝑠3𝑟3𝑐3𝑟3−𝑙′3𝑠3𝑙′3−𝑟3𝑐300000000000010−100000000000010−10000000000−𝑟4𝑠4𝑟4𝑐4𝑟4−𝑙′4𝑠4𝑙′4−𝑟4𝑐401} (15)is the matrix of force transformer, 𝑠_𝑖=sin𝜃_𝑖, 𝑐_𝑖=cos𝜃_𝑖.

𝐻_14×1=[𝐹_𝐴𝑥′,𝐹_𝐴𝑧′,𝐹_𝐵𝑥′, 𝐹_𝐵𝑧′,𝐹_𝑐𝑥′,𝐹_𝑐𝑧′, 𝑇₁cos𝛽, 𝑇₁sin𝛽, 𝐹_𝐷𝑥′, 𝐹_𝐷𝑧′,𝐹_𝐸𝑥′, 𝐹_𝐸𝑧′,𝜏₁,𝜏₄]^𝑇 is the vector of resultant and active force.

𝐵_12×1=[𝑚₁𝑥̈^′₁,𝑚₁𝑔+𝑚₁𝑧̈^′₁, 𝐽₁𝜃̈₁,𝑚₂𝑥̈^′₂,𝑚₂𝑔+𝑚₂𝑧̈^′₂, 𝐽₂𝜃̈₂,𝑚₃𝑥̈^′₃,𝑚₃𝑔+𝑚₃𝑧̈^′₃, 𝐽₃𝜃̈₃,𝑚₄𝑥̈^′₄, 𝑚₄𝑔+𝑚₄𝑧̈^′₄,𝐽₄𝜃̈₄]^𝑇 is the vector of inertia force.

Taking into account (8) and (14), the dynamic model of the HDCPM is derived from the dynamic equations of the CPM and four groups of the HDPM as follows: 𝑁_3×4𝑇_4×1=𝑀_3×1,𝐴^𝑖_12×14𝐻^𝑖_14×1=𝐵^𝑖_12×1(𝑖=1,2,3,4). (16)

4. Workspace Analysis

The workspace of the HDCPM is characterized as the set of points where the end-effector can be positioned while all cables are in tension (𝑇_𝑖>0). At each point within the possible workspace, (8) describing the force in each cable is used to see if tension is obtainable. However, it is not sufficient to obtain the actual workspace of the HDCPM if it only depends on the tension condition. Accordingly, when analyzing the workspace of the HDCPM, one should also consider the following constraints such as the workspace of the HDPM and the motion range of the end-effector.(1)The workspace of the HDPM. According to (6), the set of points (𝑥,𝑦,𝑧) where the end-effector can be positioned are influenced by cable length 𝑙_𝑖. Cable length 𝑙_𝑖 is calculated by the coordinates joint 𝐶(𝑥^′_𝐶,𝑧^′_𝐶) from (4). Thus, the workspace of the joint 𝐶 in 𝑥^′-𝑧^′ plane is one of the constraints of the workspace of the HDCPM, and it can be expressed as (only the first group of the HDPM) 𝑙^′₁−𝑙^′₂<𝐿_𝐴𝐶<𝑙^′₁+𝑙^′₂,𝑙^′₃−𝑙^′₄<𝐿_𝐶𝐸<𝑙^′₃+𝑙^′₄, (17) where 𝐿_𝐴𝐶=𝑥_𝐶^′2+𝑧_𝐶^′2, 𝐿_𝐶𝐸=(𝑥^′_𝐶−𝑙^′₅)²+𝑧_𝐶^′2.(2)The motion range of the end-effector for the HDCPM can be written in the following form: 𝑙0<𝑥=²¹−𝑙²₂+𝑎²𝑙2𝑎<𝑎,0<𝑦=²¹−𝑙²₄+𝑏²2𝑏<𝑏,0<𝑧=ℎ−𝑙²₁−𝑥²−𝑦²<ℎ. (18)

A general numerical workspace generation approach is employed here. The possible motion range of joint 𝐶 in the HDPM is first discretized into a number of points. According to (17), a series of the cable length 𝑙_𝑖 can be calculated. Then, from (18), various combinations of 𝑙_𝑖 can calculate the end-effector coordinates (𝑥,𝑦,𝑧). Thus, a series of points in Cartesian coordinate system are determined. At the end, the points are checked and those satisfying the tension condition (𝑇_𝑖>0) will generate the workspace. The flowchart of generating the workspace of the HDCPM is shown in Algorithm 1.

𝑥′min=−𝑙′1+𝑙′2,𝑥′max=𝑙′3+𝑙′4+𝑙′5 𝑧′min=−max𝑙′1+𝑙′2,𝑙′3+𝑙′4,𝑧′max=max𝑙′1+𝑙′2,𝑙′3+𝑙′4 𝑥′=𝑥′min,𝑧′=𝑧′min Do While  𝑧′≤𝑧′max      Do While  𝑥′≤𝑥′max           If  𝑙′1−𝑙′2<𝐿𝐴𝐶<𝑙′1+𝑙′2and𝑙′3−𝑙′4<𝐿𝐶𝐸<𝑙′3+𝑙′4  Then            From 𝑙1=𝑠−𝑝+𝑙′5−𝑥′𝐶2+ℎ−𝑧′𝐶2,𝑙1 is determined.            Similarly, 𝑙2,𝑙3, and 𝑙4 can be calculated, respectively.            Record 𝑙𝑖           End If        𝑥′=𝑥′+Δ𝑥′       Loop        𝑧′=𝑧′+Δ𝑧′ Loop Various combinations of 𝑙𝑖 for calculating coordinates (𝑥,𝑦,𝑧) by 𝑥=𝑙21−𝑙22+𝑎2/2𝑎,𝑦=𝑙21−𝑙24+𝑏2/2𝑏,𝑧=ℎ−𝑙21−𝑥2−𝑦2 and those satisfying motion range of the end-effector (0<𝑥<𝑎,0<𝑦<𝑏,0<𝑧<ℎ) and tension condition (𝑇𝑖>0) will generate the workspace

Algorithm 1 

Flowchart of generating the workspace of the HDCPM.

5. Results and Discussions

Simulation studies were performed with the software named MATLAB 2010. A three-dimensional simulation model of the HDCPM has been established, where the four groups of HDPMs have the same structure parameters and symmetry in the three-dimensional space. The parameters of the HDCPM are listed in Table 1. All computations were carried out on a computer with a 2.99-GHz CPU and 2.00 GB memory; the circle trajectory took about 9.67 s, while the line segment trajectory requires only 2.93 s.

Table 1 

Parameters of the HDCPM.

5.1. Dynamic Simulation

This section presents two motion cases of the end-effector for dynamic simulation.

Case 1 (circle trajectory). The equation of the circle with a radius of 0.25 m parallel to the X-Y plane in global coordinate system is 𝑥=0.25cos𝑡+0.5,𝑦=0.25sin𝑡+0.5,𝑧=0.5, (19) where 0≤𝑡≤6.2(s). The end-effector moves along the trajectory with the constant velocity 𝑣=0.25 m/s as shown in Figure 5.

Figure 5 

Following trajectory of the circle motion.

Figure 6 shows cable motion in the process of simulation. The cable lengths 𝑙_𝑖 are generated by (5) and vary symmetrically. Figure 7 shows the curves of cable tension exerted on the end-effector related to the circle trajectory. Since the end-effector moves on a horizontal circle trajectory, the changes of cable tensions in Figure 7 are reasonable and the transition is smooth.

Figure 6 

Time behavior of cable length for Case 1.

Figure 7 

Curves of cable tension for Case 1.

The kinematic parameters of the HDPM can be obtained by the inverse kinematics of the HDCPM, for given end-effector position. The angle of joint 𝐴 is driven by a CV motor and assumed to be 𝜃₁=𝜔𝑡, (20) where 𝜔=1 rad/s. According to Figure 3(b) the following geometrical relation is verified: 𝑙^′₅+𝑝−𝑙^′₁cos𝜃₁−𝑙^′₂cos𝜃₂²+ℎ−𝑙^′₁sin𝜃₁−𝑙^′₂sin𝜃₂²=𝑠−𝑙₁², (21) where 𝑙₁ is known from (5). Combining (20), and (21), 𝜃₂ can be calculated as 𝜃₂𝐸=2arctan¹−𝐹²₁+𝐸²₁−𝐺²₁𝐹₁−𝐺₁, (22) where 𝐸₁=2𝑙^′₂(𝑙^′₂sin𝜃₁−ℎ), 𝐹₁=2𝑙^′₂(𝑙₁cos𝜃₁−𝑙^′₅−𝑝), 𝐺₁ = (𝑙^′₅+𝑝)²− 2𝑙^′₁(ℎsin𝜃₁+(𝑙^′₅+𝑝)cos𝜃₁)+𝑙₁^′2+𝑙₂^′2+ℎ²−(𝑠−𝑙₁)².

From (3), (20) and (22), the coordinates of joint 𝐶(𝑥^′_𝐶,𝑧^′_𝐶) can be determined as 𝑥^′_𝐶=𝑙^′₁cos𝜃₁+𝑙^′₂cos𝜃₂,𝑧^′_𝐶=𝑙^′₁sin𝜃₁+𝑙^′₂sin𝜃₂. (23)

According to (3), the coordinates of joint C (𝑥^′_𝐶,𝑧^′_𝐶) can be rewritten as 𝑥^′_𝐶=𝑙^′₅+𝑙^′₃cos𝜃₃+𝑙^′₄cos𝜃₄,𝑧^′_𝐶=𝑙^′₃sin𝜃₃+𝑙^′₄sin𝜃₄. (24)

Solving (24), one gets 𝜃₄𝐸=2arctan²−𝐹²₂+𝐸²₂−𝐺²₂𝐹₂−𝐺₂,𝜃₃𝑧=arcsin^′𝐶−𝑙^′₄sin𝜃₄𝑙^′₃, (25) where 𝐸₂=−2𝑧^′_𝐶𝑙^′₄,𝐹₂=2𝑙^′₄𝑙^′₅−2𝑥^′_𝐶𝑙^′₄ and 𝐺₂=(𝑥^′_𝐶−𝑙^′₅)²+𝑧_𝐶^′2+𝑙₄^′2−𝑙₃^′2.

According to the inverse kinematics analysis of the HDCPM above, one can calculate the angular variables for all links of the HDPM as the end-effector moves along the circle. Figure 8 shows the angle curves of the links of the HDPM. It can be noted that the angular variable law for link 𝐴𝐵 is a line, while for the other links are curves. This is in line with the motion law of the HDPM. Moreover, angular curves of all the links maintain good continuity, and the numerical values are reasonable. The angular velocity curves and angular acceleration curves of links can be generated by taking the first and second time derivatives of the angular change (𝑑𝜃_𝑖/𝑑𝑡, 𝑑²𝜃_𝑖/𝑑𝑡²). The coordinates of the joint 𝐶 in local 𝑥^′-𝑧^′ plane can be computed by (23). The running trajectory of joint 𝐶 is depicted in Figure 9.

(a)

(b)

(c)

Figure 8 

Curves of the links motion parameters of the HDPM for Case 1: (a) angle, (b) angular velocity, and (c) angular acceleration.

Figure 9 

Trajectory of joint 𝐶 in 𝑥′-𝑧′ plane for Case 1.

Figure 10 depicts the resulting force curves of the joints of the HDPM. From it one can see that the resultant force curves of the joints are smooth, which is consistent with the desired motion law.

Figure 10 

Resultant force curves of the joints of the HDPM for Case 1.

If the cables are driven only by coils and servomotors, which is a classical approach for the conventional CPM, the range and speed of obtainable output motions may be limited by certain servomotor power capacity and high cost. This problem can be solved by using hybrid-driven machines instead of coils and servomotors. For the hybrid-driven machines, the CV motor provides the majority of power supply to drive the mechanism’s motion. The servomotor is real-time controllable and offline programmable [20]. Therefore, the hybrid-driven mechanisms take the advantage of the complementary characteristics of two types of motors to generate programmable output motions with high power capacities in low cost.

The CV motor power and servomotor power of the HDPM can be calculated by 𝑃_CV1=𝜏₁×𝜃̇₁,𝑃_SV1=𝜏₄×𝜃̇₄, (26) where 𝑃_CV1 is the CV motor power of the first group of the HDPM and 𝑃_SV1 is the servomotor power of the first group of the HDPM. Similarly, the other motor power, 𝑃_CV2, 𝑃_SV2, 𝑃_CV3, 𝑃_SV3, 𝑃_CV4, 𝑃_SV4, of the HDPM can be derived, respectively.

Figure 11 shows motor absolute value power curves of four groups of the HDPMs of the HDCPM. From Figure 11, for each of the HDPM, the range of the CV motor power (𝑃_CV𝑖) is larger than that of the servomotor power (𝑃_SV𝑖). It is illustrated also that the CV motors undertake the majority of power for the HDCPM system.

Figure 11 

Motor absolute value power curves of the four groups of HDPMs of the HDCPM.

As the motion circular trajectory of the end-effector is the same, and as the structural parameters of the classical approach for the conventional CPM are the same (cables are driven only by coils and servomotors), the servomotor power of the conventional CPM can be obtained by 𝑃_𝑖=𝑇_𝑖×𝑙_𝑖̇(𝑖=1,2,3,4), (27) where 𝑃_𝑖 is the 𝑖th servomotor power, 𝑇_𝑖 represents the 𝑖th cable tension, 𝑙_𝑖̇ denotes the 𝑖th cable velocity.

Figure 12 shows the servomotor absolute value power curves of the classical approach for the conventional CPM. For comparison, absolute value power curves of the first group of HDPM of the HDCPM and the first servomotor of the conventional CPM are given in Figure 13. As it is seen, large range of servomotor power |𝑃₁| of the conventional CPM may be decreased by using the HDPM.

Figure 12 

Servomotor absolute value power curves of the conventional CPM.

Figure 13 

Comparison of absolute value power curves between the HDCPM and conventional CPM.

In order to investigate the power property of the HDCPM further, the following simulation is performed as the CV motors of the four groups of the HDPMs of the HDCPM run with 1.1 rad/s (𝜔=1.1 rad/s). Figure 14 displays curves of the link angle and velocity of the HDPM. Angular curves of all links maintain good continuity, and the numerical values are reasonable. Figure 15 shows absolute value power curves of four groups of the HDPMs of the HDCPM. Figure 16 illustrates absolute value power curves of the first group of HDPM of the HDCPM and the first servomotor of the conventional CPM. From the simulations, we can see that the CV motors provide the majority of power supply and the servomotors mainly act as a motion regulation device as depicted in Figures 11, 13, 15, and 16.

(a)

(b)

Figure 14 

Curves of the links motion parameters of the HDPM as 𝜔=1.1 rad/s: (a) angle and (b) angular velocity.

Figure 15 

Motor absolute value power curves of the four groups of HDPMs of the HDCPM as 𝜔=1.1 rad/s.

Figure 16 

Comparison of absolute value power curves between the HDCPM as 𝜔=1.1 rad/s and conventional CPM.

Case 2 (line segment trajectory). The equation of the line in global coordinate system is 𝑡𝑥=6𝑡+0.25,𝑦=+1123,𝑡𝑧=6+0.25, (28) where 0<𝑡<2(s). From (28), it can be noted that the center of gravity of the end-effector moves the geometric path that is line segment with the constant velocity 𝑣=0.25 m/s, the initial point is (0.25, 1/3, 0.25) (m). The trajectory tracking of the end-effector related to Case 2 is shown in Figure 17. From this figure the formulation tracks the planned trajectory relatively well.
Figure 18 shows the changes of cables lengths. As can be seen, the variations of the curves of cables lengths are in accordance with the reference path (28) of the end-effector.
Figure 19 depicts the curves of cable tensions for Case 2. One can see that the curves are smooth, which is the ideal data for the HDCPM system which operates steadily.
The angular variables of links of the HDPM for Case 2 can be obtained by the inverse kinematics analysis of the HDCPM. The angle curves of the HDPM links are shown in Figure 20(a). The corresponding angular velocity and angular acceleration are shown in Figures 20(b) and 20(c). Figure 21 shows the running trajectory of the joint 𝐶 in local 𝑥^′𝑜^′𝑧^′ plane.
The resulting force curves of the joints of the HDPM for Case 2 can be solved by the dynamics modeling of the HDCPM in Section 3, as shown in Figure 22. It can be noted that the resulting force curves of the joints have the same trend and change smoothly. Comparison of absolute value power curves of the HDPMs in the HDCPM and the servomotors of the conventional CPM for Case 2 are shown in Figure 23. Referring to Figure 23, it can be seen that the CV motors provide the majority of power supply and the servomotors mainly act as a motion regulation device.
From the results and discussions, illustrative simulation studies highlight its performances, which lay a foundation for the further research on optimization and real-time control.

Figure 17 

Following trajectory of the line segment motion.

Figure 18 

Curves of cable length for Case 2.

Figure 19 

Curves of cable tension for Case 2.

(a)

(b)

(c)

Figure 20 

Curves of the links motion parameters of the HDPM for Case 2: (a) angle, (b) angular velocity and (c) angular acceleration.

Figure 21 

Running trajectory of the joint 𝐶 in 𝑥′-𝑧′ plane for Case 2.

Figure 22 

Resultant force curves of the joints of the HDPM for Case 2.

Figure 23 

Comparison of absolute value power curves between the HDCPM and conventional CPM for Case 2.

5.2. Workspace Simulation

The workspace of joint 𝐶 is determined by geometric constraints (17) given in Figure 24. The simulation results of the reachable 3D workspace for the HDCPM and the projections of workspace onto the X-Y plane and X-Z plane are shown in Figure 25. The whole workspace is approximately an upside-down cone. It should be noted that most of the workspace volume is concentrated in the upper part of the tower cube, which is important for trajectory planning of the end-effector.

Figure 24 

Workspace of joint 𝐶 in 𝑥′-𝑧′ plane.

(a)

(b)

(c)

Figure 25 

Workspace of the HDCPM (a) and projections of workspace onto X-Y plane (b) and X-Z plane (c).

6. Concluding Remarks

This paper describes the theoretical investigations of a class of the completely restrained hybrid-driven based cable parallel manipulators (HDCPMs) with three translational motions. The HDCPM combines the advantages of four groups of HDPMs and a 4-cable parallel manipulator in a way to provide a solution for moving heavy objects with high efficiency and high performance. Detailed design and implementation of a virtual prototype model of the HDCPM is presented as well. Dynamic modeling of the full system including both the mechanical system and the driving system is addressed. The inverse kinematic and dynamic problems of the HDCPM system are resolved on condition that an operation path of the end-effector has been planned. Then, the main factors, which influence the workspace of the HDCPM such as workspace of the HDPM and the length of cables, are analyzed based on the structure constraints of the HDCPM. Finally, illustrative examples for trajectory tracking and workspace demonstrate that the manipulator design is reasonable and the associated models are correct.

The prototype verification of the designed HDCPM is currently under investigation. Furthermore, more work is in progress to implement control system design, design modification, and optimum synthesis for the HDCPM. In addition, the methodology presented in this work could be used as a guide for the design and analysis of other manipulators.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (50905179, 51275515), Program for Innovative Research Team in University (IRT1292), the Six Talent Peak Foundation of Jiangsu Province, and the Priority Academic Programme Development of Jiangsu Higher Education Institutions. The authors also appreciate the comments and valuable suggestions of anonymous referees and editors to improve the quality of the paper.