Abstract
The design, dynamics, and workspace of a hybriddrivenbased cable parallel manipulator (HDCPM) are presented. The HDCPM is able to perform high efficiency, heavy load, and highperformance motion due to the advantages of both the cable parallel manipulator and the hybriddriven planar fivebar 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 NewtonEuler 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 GoughStewart platform in which rigid extensible legs are replaced by cables stored spools. CPMs have several advantages over rigidlink 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 endeffector, 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 endeffector. As a result, the number of driving cables must be more than the number of degrees of freedom (DOF) of the endeffector 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 4cable driven parallel manipulator have been presented by Ottaviano et al. [12]. Flatnessbased trajectory tracking control for a threecable 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 highloading CPM due to the restrictions of power and torque. Hence, it is necessary to carry out for a newtype driven system for the CPMs.
The hybriddriven planar fivebar 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 hybriddrivenbased cable parallel manipulator (HDCPM) which combines four groups of HDPMs with a 4cable parallel manipulator is presented. The HDCPM can take the advantages of the characteristics of both mechanisms to generate a highperformance 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 NewtonEuler method. The latter is used to solve the inverse kinematic and dynamic problems of the HDCPM on condition that the operation path of the endeffector 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 endeffector 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 endeffector 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 NewtonEuler 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 threedimensional design model of the completely restrained HDCPM with three translational motions is taken as an example (see Figure 1). The HDCPM suspends an endeffector by four cables and restrains all motion degrees of freedom for the object using the cable force when the endeffector moves within the workspace. For each cable, one end is connected to the endeffector, 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 fourcable tower racks, four cables, pulley struts, pulleys, girder, and cargo (i.e., endeffector); (2) four groups of HDPMs containing threephase asynchronous motors, servomotors, reducers, and double crank fivebar 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 fivebar linkage. The double crank planar fivebar 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.
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The working process of the HDCPM is as follows. The CV threephase 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 threephase 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 endeffector.
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 is established. The endeffector has location coordinates . The distance between each cable tower rack top and the endeffector is . 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 . The bottom of the HDPM corresponds to the joint as the origin, where a local coordinate system is established.
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Link driven by a CV motor and link ED driven by a servomotor are the two driving links (Figure 2(a)), the joint is the output. There are two independent constraints in the HDPM, which can be used to derive the kinematic equation of the HDPM:
In (1) and are known. Consequently and can be calculated as where , , , , .
Let be the coordinates of joint in the local coordinate system. According to the geometric relationship of the HDCPM, can be derived as where represents the distance and is the total length of cable. Similarly, , , and can be derived, respectively.
The relationships between the cable length and the endeffector location can be easily obtained as follows: resulting in
Newton’s law for the endeffector of the HDCPM results in where is a tension cable force, that is, the driving force of cable exerted on the endeffector for the HDCPM, and is force vector.
Thus the dynamic model can be expressed as follows: where is the matrix of coordinate transformer, is the vector of cable tension, is the vector of inertia force, m is mass of the endeffector, and 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 NewtonEuler equation establishes the dynamic model of the HDPM. The force analysis of each link of the HDPM is shown in Figure 4. and represent the driving torque of the CV motor on link AB and the servomotor on link ED, respectively; arctan is the angle between cable tension and coordinate.
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By the force analysis mentioned previously, the NewtonEuler formulation of the links can be listed as follows.
Link :
Link :
Link :
Link : where , , , , , 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 where is the matrix of force transformer, , .
, , , , , , is the vector of resultant and active force.
, , , , 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:
4. Workspace Analysis
The workspace of the HDCPM is characterized as the set of points where the endeffector can be positioned while all cables are in tension (). 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 endeffector.(1)The workspace of the HDPM. According to (6), the set of points () where the endeffector can be positioned are influenced by cable length . Cable length is calculated by the coordinates joint from (4). Thus, the workspace of the joint in 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) where , .(2)The motion range of the endeffector for the HDCPM can be written in the following form:
A general numerical workspace generation approach is employed here. The possible motion range of joint in the HDPM is first discretized into a number of points. According to (17), a series of the cable length can be calculated. Then, from (18), various combinations of can calculate the endeffector coordinates . Thus, a series of points in Cartesian coordinate system are determined. At the end, the points are checked and those satisfying the tension condition () will generate the workspace. The flowchart of generating the workspace of the HDCPM is shown in Algorithm 1.

5. Results and Discussions
Simulation studies were performed with the software named MATLAB 2010. A threedimensional simulation model of the HDCPM has been established, where the four groups of HDPMs have the same structure parameters and symmetry in the threedimensional space. The parameters of the HDCPM are listed in Table 1. All computations were carried out on a computer with a 2.99GHz CPU and 2.00 GB memory; the circle trajectory took about 9.67 s, while the line segment trajectory requires only 2.93 s.
5.1. Dynamic Simulation
This section presents two motion cases of the endeffector for dynamic simulation.
Case 1 (circle trajectory). The equation of the circle with a radius of 0.25 m parallel to the XY plane in global coordinate system is where . The endeffector moves along the trajectory with the constant velocity m/s as shown in Figure 5.
Figure 6 shows cable motion in the process of simulation. The cable lengths are generated by (5) and vary symmetrically. Figure 7 shows the curves of cable tension exerted on the endeffector related to the circle trajectory. Since the endeffector moves on a horizontal circle trajectory, the changes of cable tensions in Figure 7 are reasonable and the transition is smooth.
The kinematic parameters of the HDPM can be obtained by the inverse kinematics of the HDCPM, for given endeffector position. The angle of joint is driven by a CV motor and assumed to be where rad/s. According to Figure 3(b) the following geometrical relation is verified: where is known from (5). Combining (20), and (21), can be calculated as where , , = 2.
From (3), (20) and (22), the coordinates of joint can be determined as
According to (3), the coordinates of joint C can be rewritten as
Solving (24), one gets where and .
According to the inverse kinematics analysis of the HDCPM above, one can calculate the angular variables for all links of the HDPM as the endeffector 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 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 (, ). The coordinates of the joint in local plane can be computed by (23). The running trajectory of joint is depicted in Figure 9.
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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.
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 hybriddriven machines instead of coils and servomotors. For the hybriddriven machines, the CV motor provides the majority of power supply to drive the mechanism’s motion. The servomotor is realtime controllable and offline programmable [20]. Therefore, the hybriddriven 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 where is the CV motor power of the first group of the HDPM and is the servomotor power of the first group of the HDPM. Similarly, the other motor power, , , , , , , 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 () is larger than that of the servomotor power (). It is illustrated also that the CV motors undertake the majority of power for the HDCPM system.
As the motion circular trajectory of the endeffector 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 where is the th servomotor power, represents the th cable tension, denotes the th 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 of the conventional CPM may be decreased by using the HDPM.
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 ( 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.
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Case 2 (line segment trajectory). The equation of the line in global coordinate system is
where . From (28), it can be noted that the center of gravity of the endeffector moves the geometric path that is line segment with the constant velocity m/s, the initial point is (0.25, 1/3, 0.25) (m). The trajectory tracking of the endeffector 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 endeffector.
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 in local 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 realtime control.
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5.2. Workspace Simulation
The workspace of joint 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 XY plane and XZ plane are shown in Figure 25. The whole workspace is approximately an upsidedown 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 endeffector.
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6. Concluding Remarks
This paper describes the theoretical investigations of a class of the completely restrained hybriddriven based cable parallel manipulators (HDCPMs) with three translational motions. The HDCPM combines the advantages of four groups of HDPMs and a 4cable 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 endeffector 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.