# Deployment/Retrieval Modeling of Cable-Driven Parallel Robot

Authors: Q. J. Duan; J. L. Du; B. Y. Duan; A. F. Tang
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

A steady-state dynamic model of a cable in air is put forward by using some tensor relations. For the dynamic motion of a long-span Cable-Driven Parallel Robot (CDPR) system, a driven cable deployment and retrieval mathematical model of CDPR is developed by employing lumped mass method. The effects of cable mass are taken into account. The boundary condition of cable and initial values of equations is founded. The partial differential governing equation of each cable is thus transformed into a set of ordinary differential equations, which can be solved by adaptive Runge-Kutta algorithm. Simulation examples verify the effectiveness of the driven cable deployment and retrieval mathematical model of CDPR.


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## Body

## 1. Introduction

Cable systems arise in many practical applications, such as bridges, underwater systems, aircraft decoy systems, and tethered satellite systems. When cables are utilized to replace links to the feed cabin to track radio source in 500 m aperture spherical radio telescope, we get Cable-Driven parallel robot (CDPR), which is developed from parallel and serial cable-driven robot. Conventional robots with serial or parallel structures are impractical for some applications since the workspace requirements are higher than what the conventional robots can provide. CDPR which uses cables instead of links to manipulate objects reduces the structural weight considerably in [1, 2]. The cables are so light that actuators of CDPR have only to drive their loads. Furthermore, CDPR gives a wide range of motion, because drums of the mechanism can wind long cables. For the above reasons, cable-driven mechanisms have received attention and have been recently studied since the 1980s [3]. Some reported researches on CDPR are NIST Robocrane [4], ultrahigh speed robot, tendon-driven Stewart platform, parallel wire mechanism for measuring a robot pose, controller designs for CDPR [5], and wrench-feasible workspace analysis for CDPR [6].Particularly, dynamical characteristics of this type of mechanism have been well studied, such as the manipulation problem of load by multiple wires [7, 8], and the cables are considered to be massless. However, the cable will extend when stretched, and an error in length of 0.1% can give rise to an error in force of about 50%, thus accurate modeling of the cables is crucial [9].Methods for approximating the cable motion and its equilibria have been studied for many years. Cable-body systems have been modeled using continuum models based on partial differential equations for strings, as well as lumped mass method in [10]. In complex applications, the lumped mass method representation is usually the preferred choice for detailed simulation work [11] and is employed in our work.In this paper, we present a deployment and retrieval cable mathematical model using a lumped parameter representation. First, the general representation of the cable model is given. Then, according to the 50 m scaled model (Figure1), based on the inverse kinematics analysis, the inverse dynamic formulation of deployment and retrieval CDPR is established. Finally, numerical simulations illustrate the performance of the proposed method.Figure 1
The 50 m scaled model.


## 2. Mathematical Model for Cable

A simple schematic of the CDPR for 50 m scaled model representing the coordinate systems is shown in Figure1, and cable coordinate system is showed in Figure 2. The CDPR is considered that is kinematically and statically determined. In the design, there is a cable tower/winch pair at each vertex of a regular hexagon of radius α, which actuates six cables that are linked to a cabin. Therefore, the cabin can translate and rotate in the inertial frame. Let the origin of the inertial frame OXYZ be the center of the six towers [12].Figure 2
Fixed and cable coordinate system.In this model, the cable is assumed to be idea extensible and uniform in mass and the torsional deformation is ignored. The cable is approximated through the linear stress-strain relationship. The section is circle, and the cable is deployed from one end mass only. A representation of the model, as well as the generalized coordinates used to describe the motion, is shown in Figure2. Let (τ,n,andb) represent the unit vectors of the cable frame. The rotation matrix oR transforms S from the cable frame to the inertial frame OXYZ. s represents the length coordinate of the unstretched cable measured form its downstream end. When the cable is stressed, the length is S(s,t). Cable satisfies the following relation: dS=(∂S/∂s)ds. According to the mechanics of elasticity relationship between Strain ε and Stress T, we get: ε=T/EA0 and ε=(dS-ds)/ds⇒∂S/∂s=1+ε.In the above equation,r(s,t) is the position vector of a point on the continuous cable, E is Young’s modulus, and A0 is the unstressed cross-sectional area. Considering a cable element located at point P at time t, Its inertial position is given by the vector r(s,t):[t0,∞)×[0,L]→R3, which satisfies the relation r(s,t)=[x(s,t),y(s,t),z(s,t)]T.(2.1)τ=∂r∂S,(2.2)τ(s,t)=∂s∂S∂r∂s=11+ε∂r∂s=11+ε(∂x∂si+y∂y∂sj+∂z∂sk).The axial strain along the curve is approximated by(2.3)ε=12(∂r∂s·∂r∂s-1)=12[(∂x∂s)2+(∂y∂s)2+(∂z∂s)2-1].Considering a cable elementds moving in a steady state and applying Newton’s law to this body [13–17], we get:(2.4)Md2r(s,t)dt2=dTds+∑F.In the above equation,T is the tension at the right end of cable’s element, M is the diagonal inertia matrix of the concentrated mass, d2r(s,t)/dt2 is accelerate, and F is the total external force, both corresponding at the unit of cable length, including gravity, damping force, and aerodynamic drag forces.Starting from the significant operatord/dt=(∂/∂t)+(L̇∂/∂t), the velocity and the acceleration of the point P of curvilinear coordinate s are (2.5)drdt=∂r∂t+∂r∂sL̇,(2.6)d2rdt2=∂2r∂t2+2∂2r∂s∂tL̇+∂2r∂s2L̇2+∂r∂sL̈,
where L̇=ds/dt is the cable velocity, and L̈=d2s/dt2 is the cable accelerate when the cable moves slowly. For the purposes of simplicity, we assume  L̇2≈0 and L̈≈0 when the cable moves slowly. Then from (2.6) we get(2.7)d2rdt2=∂2r∂t2+2∂2r∂s∂tL̇.
The dynamical equations for the cable are derived from (2.4) and (2.7).By neglecting inertial force, the bending moments, and torsional moments of the cable in steady movement, a governing (2.4) for the cable can be expressed as(2.8)dTdS+B+D=0.
Note that, T=Tτ,dT/dS=(Tdτ/ds+τdT/ds)(1+ε)-1 where τ is the intensity of the vector tension T. Within a slow motion of the cable, in what follows the radial force Tdτ/ds can be considered as negligible. B is the net gravitational force per unit length, D is aerodynamic drag forces. Substituting the force [16, 17] in (2.8) and rearranging the terms, we get(2.9)-m0gk+0.5ρaπd0CF1+ε|μτ|μτ,-0.5ρad0Cd1+ε|μ-μτ|(μ-μτ)+(Tdτds+τdTds)=0,
where m0 is the mass per unit unstressed cable length. d0 is diameter of the unstressed cross-section. ρa is air density. μ=dr/dt-J,J is wind velocity. μτ is the tangential component of μ in local cable coordinates. Cd is the cable drag coefficient to the wind. CF is drag coefficient surface friction factor. CF=CmCd/π, when the cable cross section is circle, Cm=0.02~0.05,andCd=1.2 [18].Given conditions, time-domain simulation of the cable dynamics is obtained by propagating (2.9) in time through the use of a suitable temporal integration algorithm. In this work, adaptive Runge-Kutta algorithm is used.


## 3. A Three-Dimensional Dynamic Model of Deployment/RetrievalCable

### 3.1. A Lumped Mass Method for Cable Dynamic Model

The cable is discretized into a series ofN elastic segments joined at nodes, which are numbered by i from 0 at the downstream end to N at the upper end. The cable length coordinates satisfy 0=s0<s1<⋯<si<⋯<sN=S. The mass of each segment is lumped in equal amounts at each of its two boundary nodes, by analogy the external forces are also lumped in this way. These segments are required to be small enough, so that the forces acting on them are approximately uniform over their length.Applying Newton’s second law to the nodei, we get the governing equation(3.1)Miẍi=F=ΔTi+Wi+Di,
where the mass matrix Mi=miI,I represents 3×3 unit square matrix, mi=(ρlui-1,j+ρlui,j+1)/2,ρ is the mass per unit cable length, and lu is the unstressed cable length between units. The subscript i and i+1 denote the physical terms between node i and i+1. As a whole cable, it is x=[x1y1z1⋯xiyizi⋯xn-1yn-1zn-1]T, where  xi=[xi-1yi-1zi-1⋯xi+1yi+1zi+1]T,ẍ is the acceleration of the node.The tensionΔTi of the node i is:
(3.2)ΔTi=Ti,j+1-Ti-1,j,Ti,j+1=A0Eεi,j+1τi,j+1,εi,j+1=(li,i+1-lui,i+1)/lui,i+1, where A0 is the area of unstressed cross section, and li,i+1=|ri+1-ri| is the length between node i and i+1. It can be expressed in recursive form as [19]. Different methods lead to the same results.Wi is the gravity of node [20] i:(3.3)Wi=-migk,Di [16] is the wind force of node i:(3.4)Di=(Di-1,i+Di,i+1)2.


### 3.2. Boundary and Initial Conditions

vd is the velocity of deployment/retrieval cable, and it can be controlled. When the cable is deployed, vd<0. When the cable is retrieved, vd>0. Refer to Figure 3. When the cable is deployed or retrieved, its length at any time t can be determined by(3.5)sid(t)=s0-∫0tvd1+εdτ,Figure 3
Discrete model of the cable deployment/retrieval.wheres0 is the initial length of the cable at startup.


### 3.3. Boundary Conditions of Cable Deployment/Retrieval

The cable force at the node 0, which is connected to the cabin, can be calculated in (3.1) (3.6)M0ẍ0=F0=ΔT0+W0+D0+f0,
where M0=(wlu0,1/2)I,ΔT0=A0Eε0,1τ0,1,W0=m0gk,D0=D0,1/2,andf0=F0.Refer to Figure3. when the velocity of the node connected to pulley is vrp, the cable length at any time t can be determined by：(3.7)S={s-∫0tvrp1+εdt,retrieval,s+∫0tvrp1+εdt,deployment,where theε is the function of tension of node N and changes with time. Here we ignore the variations.The position and velocity of the end node can be expressed as(3.8)xN=xpulley,ẋN=vrpτrp.The subscriptrp is the mark of deployment/retrieval. τrp is the tangent vector of the cable end node. τN,N-1 is used to the numerical calculation.The initial condition of the simulation is steady states of the cable movement. According to the governing equation, we get(3.9)dẋidt=Mi-1(ΔTi+Wi+Di)dxidt=ẋi(i=1~N-1).With the boundary condition and the initial condition, the nonlinear equations can be solved.


## 3.1. A Lumped Mass Method for Cable Dynamic Model

The cable is discretized into a series ofN elastic segments joined at nodes, which are numbered by i from 0 at the downstream end to N at the upper end. The cable length coordinates satisfy 0=s0<s1<⋯<si<⋯<sN=S. The mass of each segment is lumped in equal amounts at each of its two boundary nodes, by analogy the external forces are also lumped in this way. These segments are required to be small enough, so that the forces acting on them are approximately uniform over their length.Applying Newton’s second law to the nodei, we get the governing equation(3.1)Miẍi=F=ΔTi+Wi+Di,
where the mass matrix Mi=miI,I represents 3×3 unit square matrix, mi=(ρlui-1,j+ρlui,j+1)/2,ρ is the mass per unit cable length, and lu is the unstressed cable length between units. The subscript i and i+1 denote the physical terms between node i and i+1. As a whole cable, it is x=[x1y1z1⋯xiyizi⋯xn-1yn-1zn-1]T, where  xi=[xi-1yi-1zi-1⋯xi+1yi+1zi+1]T,ẍ is the acceleration of the node.The tensionΔTi of the node i is:
(3.2)ΔTi=Ti,j+1-Ti-1,j,Ti,j+1=A0Eεi,j+1τi,j+1,εi,j+1=(li,i+1-lui,i+1)/lui,i+1, where A0 is the area of unstressed cross section, and li,i+1=|ri+1-ri| is the length between node i and i+1. It can be expressed in recursive form as [19]. Different methods lead to the same results.Wi is the gravity of node [20] i:(3.3)Wi=-migk,Di [16] is the wind force of node i:(3.4)Di=(Di-1,i+Di,i+1)2.


## 3.2. Boundary and Initial Conditions

vd is the velocity of deployment/retrieval cable, and it can be controlled. When the cable is deployed, vd<0. When the cable is retrieved, vd>0. Refer to Figure 3. When the cable is deployed or retrieved, its length at any time t can be determined by(3.5)sid(t)=s0-∫0tvd1+εdτ,Figure 3
Discrete model of the cable deployment/retrieval.wheres0 is the initial length of the cable at startup.


## 3.3. Boundary Conditions of Cable Deployment/Retrieval

The cable force at the node 0, which is connected to the cabin, can be calculated in (3.1) (3.6)M0ẍ0=F0=ΔT0+W0+D0+f0,
where M0=(wlu0,1/2)I,ΔT0=A0Eε0,1τ0,1,W0=m0gk,D0=D0,1/2,andf0=F0.Refer to Figure3. when the velocity of the node connected to pulley is vrp, the cable length at any time t can be determined by：(3.7)S={s-∫0tvrp1+εdt,retrieval,s+∫0tvrp1+εdt,deployment,where theε is the function of tension of node N and changes with time. Here we ignore the variations.The position and velocity of the end node can be expressed as(3.8)xN=xpulley,ẋN=vrpτrp.The subscriptrp is the mark of deployment/retrieval. τrp is the tangent vector of the cable end node. τN,N-1 is used to the numerical calculation.The initial condition of the simulation is steady states of the cable movement. According to the governing equation, we get(3.9)dẋidt=Mi-1(ΔTi+Wi+Di)dxidt=ẋi(i=1~N-1).With the boundary condition and the initial condition, the nonlinear equations can be solved.


## 4. Numerical Results of CDPR

(1)
In the first simulation, the cabin tracks a moving target with constant velocityv=2 m/s and constant pose (α=0,β=0,γ is not limited). The initial point is (0.0, 0.0, 9135.0, 0.0, 0.0, 0.0)T (mm, rad). The geometric path is a circle with the radius of 2 m, parallel to the x-y plane. (0, 1, -1470, 0, 0, 0)T (mm, rad) is the error between current position and start position. Figure 4 shows the simulation result.(2)
In the second simulation, the cabin moves from the equilibrium position (2.00, 1.00, 16.0, 0.0889,-0.1756, 3.108)T (m, rad) to the end position  (2.50, 1.50, 16.2, 0.135, -0.263, 3.121)T (mm, rad) in 10 s. The trace of each coordinate is line. There is no error between the equilibrium position and start position. Figure 5 shows the simulation result.Trace and error of circle path.
(a)(b)(c)(d)Error in line path.
(a)(b)(c)(d)(e)(f)


## 5. Conclusions

In this study, a computational dynamics model is developed to simulate the dynamics of variable length cable-driven parallel robot (CDPR) system. Numerical results show the effectiveness of the model to capture the complex cable dynamics where the cable length is varying for it is gradually deployment or retrieval. When there is no error between current position and start position, the error values attenuate slowly, owing to little damp of cable system. When there is error between current position and start position, the dynamic characteristic of cable is significantly increased. We can find the changes of cabin position under impact loading, and the relevant researches and developing tendency [21] on the error between current position and start position versus trace errors will be researched.



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*Source: 909527-2010-07-21.xml*