Review Article Workspace Classification and

Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2014, Article ID 358727, 9 pages http://dx.doi.org/10.1155/2014/358727

Review Article Workspace Classification and Quantification Calculations of Cable-Driven Parallel Robots Q. J. Duan and Xuechao Duan Xidian University, Xi’an, Shaanxi 710071, China Correspondence should be addressed to Q. J. Duan; [email protected] Received 11 November 2013; Accepted 3 May 2014; Published 27 May 2014 Academic Editor: Yuanying Qiu Copyright © 2014 Q. J. Duan and X. Duan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Large workspace is one of the promising advantages possessed by the cable-driven parallel robots (CDPR) over the conventional rigid-link robots. This paper focuses on the dynamic analysis and workspace classification based on the general motion equation of cable robot and the unilateral property of cables. The combinations of different types of two conditions lead to several different types of workspace, including static equilibrium workspace, wrench closure workspace, wrench feasible workspace, dynamic workspace, and collision-free workspace. A qualitative comparison of different types of workspaces is performed. The simulation results verify the relationship between the several types of workspaces.

1. Introduction Compared with traditional parallel manipulators, cabledriven parallel robots are endowed to increase the four characteristics—light weight, large payload handling capacity, considerably large workspace, and dynamics—by one to two orders of magnitude by virtue of the cable-band actuation. Based on these characteristics, existing and promising applications are given. Early in 1989 the NIST Robocrane system for large-scale handling [1, 2] at NIST seemed to be the first prototype. Then an ultrahigh speed Falcon was designed for fast pickand-place [3]. Along with the arrival of the 21st century, the applications for cable robots broadened based on their promising performance [4, 5]. SEGESTA, another lightweight prototype, was presented by Hiller et al. at the University of Duisburg-Essen, Germany [6, 7]. A rescue mobile cable robot system used after earthquakes [8] was developed. The IPAnema family of cable robots for medium- to largescale inspection, handling, and assembly operations [9–12] is developed in Germany. Otis developed a locomotion system [13, 14] in Canada. There were a couple of good theoretical research projects on cable robot system from the robotic lab of University Laval concurrently. The feed source of the fivehundred-meter aperture spherical radio telescope (FAST)

[15–17] is designed to be driven by cable system in China. It is under construction now. Once completed, it will be the world’s largest cable robot. A sport motion simulator device was developed at the ETH Zurich, Switzerland [18– 20]. The KNTU cable robots were studied [21] in Iran. The robot String-Man designed for gait rehabilitation focuses on force control and safety considerations [22, 23]. A cabledriven ARm EXoskeleton (CAREX) [24] for neural rehabilitation was researched by Ying Mao at the University of Delaware (U.S.A.). Another kind of under-constraint robot, automated Cable Crane [25, 26], was developed for handling at the University of Rostock, Germany. The prototype of robot MARIONET for rescue and personal assistance was developed at INRIA in France [27]. DeltaBot [28], a rigidcable hybrid ultra-high-speed cable robot, was developed by Dr. Amir Khajepour at the University of Waterloo (Canada). Spidercam [29] and Skycam [30] are two successful applications of the cable robot system. Besides these, many other prototypes and applications have been developed that we do not mention here. However, in all of these applications, one critical issue is the unidirectional nature of forces exerted by cables, which only allows them to endure tensile force when performing tasks. This means that the point in space which can be reached may not be able to meet this condition of cable in tension.

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This critical issue relates to the workspace of cable robot, which determines how to take full advantage of the robot cable in design and develop the cable robot system. In this paper, we will address the workspace classification and quantification of cable-driven robots. The organization of the paper is as follows. Section 2 presents the dynamic analysis of cable-driven robots. The different types of workspaces of cable-driven parallel robots are proposed in Section 3. Section 4 presents the quantification calculation of cable-driven robots. Numerical examples to demonstrate the workspace are given in this section. Some concluding remarks are drawn in Section 5.

2. Model and the Dynamic Analysis of Cable-Driven Parallel Robots There are two kinds of cable-driven parallel robots: one is the pure cable-driven robot, in which the end effector is supported by cables in Figure 1(a); the other is the hybrid cable-driven robot, in which the end effector is supported by cables and the rigid links in Figure 1(b). Here we give the general motion equation of cable robot. In this paper, the general motion equation of cable robot is derived based on a 6 degrees of freedom (DOFs) spatial rigid body case. Here, the following assumptions are made. (a) There is no friction in the mechanism. (b) The cables have negligible mass compared to the end effector.

the general form of a matrix 𝐽𝑇 = []𝑟×𝑛 , 𝑟 is the degree of freedom (DOF), and 𝑛 is the number of cables: 𝐽𝑇 = [

𝑊 = −𝐽𝑇 T

(T ≥ 0) .

(1)

T denotes cable force, and T = [𝑡1 𝑡2 ⋅ ⋅ ⋅ 𝑡𝑛 ]𝑇 . 𝐽𝑇 is a Jacobian matrix representing the transformation of cable forces to the wrench generated due to cable actuation. It has

(2)

where u𝑖 is unit vector along the cable and a𝑖 is the vector from center of mass of end effector to the attachment point. W can be written as 𝑚𝑥̈ [ ] 𝑚 𝑦̈ [ ] [ ] ̈ 𝑚 𝑧 [ ], W = 𝑊𝑒 + 𝐺 + [ ] 𝜔 𝜔 𝛼 𝑥 𝑥 𝑥 [ ] [𝐼 (𝛼𝑦 ) + (𝜔𝑦 ) × 𝐼 (𝜔𝑦 )] 𝜔𝑧 𝜔𝑧 ] [ 𝛼𝑧

(3)

where 𝑊𝑒 = [𝑓𝑥 𝑓𝑦 𝑓𝑧 𝑀𝑥 𝑀𝑦 𝑀𝑧 ]𝑇 denotes the external wrench vector on the end effector and 𝐺 is the gravitational vector exerted on the reference point P of the end effector. 𝑚 is the mass, 𝑔 is the acceleration due to gravity, and I is the moment of inertia of the end effector about its center of mass. It can be rewritten in a general form as ̇ 𝑋̇ + 𝐺 (𝑋) + 𝑊𝑒 = −𝐽𝑇 T 𝑀 (𝑋) 𝑋̈ + 𝐶 (𝑋, 𝑋)

(c) The cables do not stretch. According to Newton-Euler’s law, output wrenches (forces and moments) on the end effector 𝑂1 can be written as

u2 ⋅⋅⋅ u𝑛 u1 ], a1 × u1 a2 × u2 ⋅ ⋅ ⋅ a𝑛 × u𝑛

(T ≥ 0) , (4)

where 𝑋 = [𝑥 𝑦 𝑧 𝛼 𝛽 𝛾]𝑇 denotes the position and orientation of the end effector. 𝛼, 𝛽, and 𝛾 are the 𝑋-𝑌-𝑍 Euler angles. 𝑋̇ represents the twist vector which includes both the linear and angular velocities of the end effector, 𝑀(𝑋) is the mass inertia matrix, 𝐶(𝑋, 𝑋)̇ is the matrix of Coriolis and centripetal terms, and 𝐺(𝑋) represents the vector of gravity term. T is the vector of cable tensions. Henceforth, for any vector T, T > 0, T ≥ 0, and T < 0 mean that all



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the components of T are greater than zero, greater than or equal to zero, and smaller than zero, respectively, where 𝑀 (𝑋) = [

𝑚I3 03×3 ], 03×3 𝐼

03×1 [ 𝜔𝑥 𝜔𝑥 ] ] ̇ =[ 𝐶 (𝑋, 𝑋) [(𝜔𝑦 ) × 𝐼 (𝜔𝑦 )] . 𝜔𝑧 ] [ 𝜔𝑧

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𝐺(𝑋) = [0 0 − 𝑚𝑔 03×1 ]𝑇 , and I3 is the identity matrix. 0 is the zero matrix, and 𝑔 is gravity acceleration. Equations (1) and (4) are valid only if each cable force 𝑡𝑖 ≥ 0 maintains the configuration and workability of the system.

any wrench of a given set of wrenches can be applied on the platform by pulling the latter with the cables [31]. It means the end effector can physically reach the workspace when (1) all the cables are in tension and (2) all other specific motion and/or force constraints are satisfied [32]. The combinations of different types of two conditions lead to several different types of workspaces addressed in the literature, including (a) static equilibrium workspace, (b) wrench closure workspace, (c) wrench feasible workspace, (d) dynamic workspace, and (e) collision-free workspace. 3.1. Static Equilibrium Workspace. The static equilibrium workspace (SEW) is the set of postures that the end effector can attain statically (only taking gravity into account). So (1) can be written in the following form: −𝐽𝑇 T = 𝐺

3. Workspace Classification In robotic systems, workspace is always one of the most important issues, especially for the cable-driven robot research. In general, a workspace of a cable robot is defined as the set of all poses (positions and orientations) for which

(T ≥ 0) .

(6)

A number of researchers [1, 33–35] addressed the SEW. Since not all postures of the cable-driven robot are statically attainable, it is a subset of workspace for cable robots. Various names are used to term this workspace; Alp and Agrawal termed SEW as statically reachable workspace [33]. A similar



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concept termed the cable force region is also explored by Osumi et al. in [36]. 3.2. Wrench Closure Workspace. The wrench closure workspace (WCW) of a cable robot is defined as the set of poses (positions and orientations) at which the end effector can physically reach with all the cables in tension. The WCW corresponds to the set of poses for which the end effector of the mechanism is fully constrained or overconstrained by the cables. This means the number of cables must be greater than the number of DOFs of the platform. So (1) can be expressed as the following form: −𝐽𝑇 T = 𝑊𝑒 + 𝐺

(T > 0) .

(7)

A strong similarity exists between parallel cable-driven mechanisms and multifinger grasping systems with frictionless point contacts. A detailed discussion of this similarity can be found in [35, 37–40]. The first similarity is the unidirectional force whether in cables or fingers. The second similarity is the definition of the wrench matrix. In this paper, the set of force-closure poses of the end effector is called the wrench closure workspace, which Gosselin terms, rather than the force closure workspace (FCW), because the distinction

between a pure force and a wrench (combination of a force and a moment) is needed. A number of researchers [40, 41] addressed the WCW. Verhoeven and Hiller studied almost the same concept named the controllable workspace, which is defined as the set of all the poses at which the end effector can statically balance a specific set of external wrenches with all-positive cable forces [42]. Wrench closure workspace is of great interest and attractive because the parallel cable-driven robot is able to be manipulated in such workspace. Researchers also pointed out that the necessary condition for the existence of such workspace is that the cable robot system must be fully constrained or overconstrained. This means the number of cables must be greater than the number of DOFs of the platform as mentioned in the previous section [43]. 3.3. Wrench Feasible Workspace. The wrench feasible workspace (WFW) is the set of postures where the end effector can exert or balance (bounded) wrenches of a given task space with tension forces in the cables remaining within a prescribed range (usually between its minimum and maximum


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tension value) [44, 45]. In this case, (1) can be written in the following form: −𝐽𝑇 T = 𝑊𝑒 + 𝐺 with (𝑡𝑖 ≤ 𝑡𝑖 ≤ 𝑡𝑖 )

∀𝑖, 1 ≤ 𝑖 ≤ 𝑛,

(8)

where 𝑛 is the number of cables and here 𝑡𝑖 > 0. 𝑡𝑖 and 𝑡𝑖 are the minimum and maximum tension value of cable 𝑖, respectively. It is worth mentioning that T ranges from zero to the maximum tension value of cable in the definition of WFW in [40]. Wrench feasible workspace or force feasible workspace, termed by Bosscher et al. and Riechel and Ebert-Uphoff in [46, 47], is addressed as a set of postures where cable robots are needed to exert particularly required force/moment combinations to interact with environment besides maintaining their own static equilibrium. Verhoeven et al. addressed almost the same concept named acceptably controllable workspace (ACW) in [42, 48]. The ACW is defined as the set of postures where a force and torque equilibrium can be obtained with positive tensions with the constraint that all cable tensions must remain within

range of cable tension value for a tendon-based Stewart platform. What is the relationship between the WCW and the WFW? According to [34, 45], any pose X belongs to the WCW if and only if the Jacobian matrix J has full rank and its null space contains a vector z > 0 (Jz = 0). This characterization shows that the WCW depends only on the robot geometry position J only. The WFW as defined in Section 3.3 depends not only on the geometry of the robot but also on the constrains of cable force T and 𝑊. The following properties are present. Property 1. The WFW is a subset of the WCW. That is, WFW ⊂ WCW. Property 2. For any bounded set of wrenches 𝑊 and any pose X ∈ WCW, there exists a (finite) set of allowed cable tensions [T] such that X ∈ WFW. Compared with the workspace mentioned in the previous section, the WFW is able to interact with the physical real world. It is the most practical workspace for the cable-driven parallel robot systems.



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3.4. Dynamic Workspace. The dynamic workspace (DC) [49] is defined as the set of poses that the platform can reach with at least one kinematic state (position, velocity, and acceleration) at which all cables work in tension and remain within a prescribed range. In this case, (1) can be written in the following form: −𝐽𝑇 T = 𝑊 (𝑡𝑖 ≤ 𝑡𝑖 ≤ 𝑡𝑖 )

∀𝑖, 1 ≤ 𝑖 ≤ 𝑛,

(9)

where 𝑊 = [𝑀(𝑋)𝑋̈ + 𝐶(𝑋, 𝑋)̇ 𝑋̇ + 𝐺(𝑋) + 𝑊𝑒 ], and at the same time, 𝑀(𝑋) and 𝐶(𝑋, 𝑋)̇ are not zero simultaneously. It means the platform can reach points beyond the static workspace with a controlled kinematic state (e.g., a zero velocity but nonzero acceleration) [50]. It shows that the DC is a mixed space of end effector pose and kinematic state. And the cable-suspended robots are able to be controlled dynamically at the DC. Bartette and Gosselin introduced the concept of dynamic workspace in [49, 50]. Lau et al. addressed the wrench closure condition (WCC) in [51]. The WCC is defined as the set of postures, which, if a set of positive cable forces is satisfied, can be determined for any arbitrary external wrench, velocity, or acceleration of the manipulator without any upper bound to cable force. 3.5. Collision-Free Workspace. The collision-free workspace (CFW) [41] is defined as the set of poses that can be reached without collision among the end effector to cables, the cable to cable, and the cable to work piece [52]. It is clear that the CFW is the subset of the DC or the subset of the WCW. This kind of workspace can be obtained by applying fast geometrical intersection detection method [41] or by experiment method [53].

4. Quantification Calculations and Numerical Example

The base frame is located at the point A1. The coordinates of vertices are described in Figure 2. The local frame is located at the center of the moving end effector. The mass of the end effector is 1 kg. In this paper, all the following 3-dimensional workspaces were obtained at a fixed orientation where 𝛼 = 0∘ (the orientation about 𝑋), 𝛽 = 0∘ (the orientation about 𝑌), and 𝛾 = 3∘ (the orientation about 𝑍). Therefore, a set of reachable positions of this center 𝑃 at a given orientation of the end effector constitutes the different workspace under the given constraints as we defined previously. As shown in Figure 3, the SEW is generated at fixed orientations of the end effector. At the same time, the corresponding three projection views are illustrated in Figure 3. The following figures are the same style. The WCW under the moment set requirement 𝑚 = −10 Nm in 𝑌 direction is shown in Figure 4. The WFW is shown in Figure 5 with tension forces in the cables remained between 𝑡min = 3 N and 𝑡max = 70 N. The DW is shown in five layers as demonstrated in Figure 6 with the assumption that the cable tensions range from 𝑡min = 3 N to 𝑡max = 70 N and acceleration in 𝑌 direction is 9.8 m/s2 . As shown in Figures 3–5, it is realized that the wrench closure workspace and wrench feasible workspace are always smaller than static equilibrium workspace and inside the convex hull formed by the base. Wrench feasible workspace is the subset of static equilibrium workspace and wrench closure workspace. Dynamic workspace (Figure 6) shows that the end effector can reach points beyond the static equilibrium workspace with a controlled acceleration in 𝑌 direction. Generally, the collision-free workspace is often a subset of the four previously mentioned workspaces because more constraints are added.

5. Conclusion

In most studies [48, 51–54], the numerical approach is used to find out the corresponding workspace for a specific system. It means the entire structure space is discretized and exhaustively searched to find the matching workspace which is defined previously. But in this approach, the closed form expressions for the boundary of the workspaces are not available. A few researchers in [31, 47, 55, 56] proposed analytical approaches to find the SEW and WCW issue. Bosscher et al. in [46] proposed an approach to find boundaries of the WFW. However, the effectiveness is greatly degenerated when considering the constraints that the cable tension limits. The numerical approach is employed to find the different types of workspaces of the cable-driven robot in this paper. In order to show the characteristics of different types of workspaces, a 6-DOF CDPR driven by 8 cables (8-6-CDPR) is taken as an example. The illustrations taken from [57] are the symmetric designed mechanisms. It is a typical spatial parallel mechanism driven by eight cables shown in Figure 2. The base is a unit cube of 1 × 1 × 1 (m), and the end effector is a parallelepiped that has the dimension of 0.3 × 0.2 × 0.1 (m).

It is well known that the end effector’s usable workspace is essential for trajectory planning, selection, and design of robot configurations. In this paper, the workspace of cable-driven parallel robot is classified as five typical types based on the general motion equation of cable robot and working conditions without collision. They are the static equilibrium workspace, wrench closure workspace, wrench feasible workspace, dynamic workspace, and collision-free workspace. Static equilibrium workspace is determined by the structure. Wrench feasible workspace is addressed through tension conditions. It is the subset of static equilibrium workspace and wrench closure workspace. When considering at least one kinematic state (position, velocity, and acceleration) of the end effector, their dynamic workspace can be extended beyond the static workspace. The numerical example shows the effectiveness of the conclusion.

Nomenclature 𝐽𝑇 : Jacobian matrix representing the transformation of cable forces to the wrench generated due to cable actuation


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T: 𝑡𝑖 : 𝑊𝑒 : 𝐺: 𝑚: 𝑔: 𝑡𝑖 , 𝑡𝑖 :

Cable force vector Cable force in cable 𝑖 External wrench vector Gravitational vector Mass of the end effector Acceleration due to gravity The minimum and maximum tension value of cable 𝑖 𝑋, 𝑋,̇ 𝑋:̈ Position, velocities, and accelerations vector of the end effector 𝑀(𝑋): The mass inertia matrix ̇ The matrix of Coriolis and centripetal terms 𝐶(𝑋, 𝑋): I: The identity matrix 0: The zero matrix.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The authors would like to thank the National Natural Sciences Foundation of China under Grants 51105290 and 51175397 and the Fundamental Research Funds for the Central Universities (JY10000904013) for their financial support. The authors would like to thank Josephine Cameroln for her advice on the English writing in this paper.

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