Open-loop position control of a polymer cable–driven parallel robot via a viscoelastic cable model for high payload workspaces

Abstract

A polymer cable–driven parallel robot has a wide range of potential industrial applications by virtue of its light actuator dynamics, high payload capability, and large workspace. However, due to a viscoelastic behavior of polymer cable and difficulty in actual cable length measurement, there have been inevitable position and tracking control accuracy problems such as pick and place a high payload application. In this article, to overcome control problem, we propose a model-based open-loop control with the cable elongation compensation via experimentally driven cable model and switching control logic without additional Cartesian space feedback signal. The approach suggests a five-element cable model that is made with series combination of a linear spring and two Voigt models as a function of payload and cable length that are available to be measured in real-time. Experimental results show that using the suggested method, the cable length error due to viscoelastic effect can be compensated, and thus the position control accuracy of the polymer cable–driven parallel robot improved remarkably especially in gravity direction.

Keywords

Polymer cable–driven parallel robot polymer cable viscoelastic model open-loop position control parameter estimation

Introduction

A cable-driven parallel robot (CDPR) is a special type of a parallel robot mechanism actuated by winding and unwinding flexible cables hooked up with an end-effector. Since the use of flexible light cable as an actuator can significantly reduce the actuator weight and inertia, the CDPR could be applied for wide workspace, heavy payload, and fast motion application such as low-gravity simulator, camera robot, and heavy material handling in larger workspace with low installation cost.^1–4

Recently, to magnify the advantage of the CDPR, a polymer cable–driven parallel robot (PCDPR) is developed that utilizes a polymer cable as an actuator instead of steel wire cable. However, because the PCDPR may yield a control problem caused by inaccurate cable length measurement caused by cable viscoelastic property especially for the high payload application,⁵ the PCDPR has not yet been widely applied to practical industrial application. Several researches have been tried to encounter the flexible cable’s viscoelastic effect in the CDPR control. To consider elastic behavior of cables, Huang et al.⁶ modeled the cable as series points connected via elastic massless links for the tethered space robots. Kraus et al.⁷ proposed a structured method to identify the load, estimate the resulting displacement of the platform, and improve relative accuracy by more than 50% by compensating it. Miermeister et al.⁸ developed an extended elastic cable model which included the hysteresis effect during cable force computation where the cable behavior not only depends on local parameters but also on global and time variant parameters such as the platform pose. Miyasaka et al.⁹ developed the hysteresis model for longitudinally loaded cables based on the Bouc–Wen hysteresis model. Korayem and colleagues^10–12 developed a computational method for obtaining maximum dynamic load-carrying capacity of the suspended CDPR with viscoelastic cables and suggested CDPR control algorithm derived by linear quadratic regulator (LQR), linear quadratic Gaussian (LQG), and feedback linearization control. In addition, the elasticity and uncertainties of a CDPR were neutralized by robust control, and the flexibility and uncertainties of cable suspended robot were compensated using sliding mode control.¹³ Those approaches have improved tracking performance of CDPR obviously. However, their methods need to measure some feedback signals to build up a controller, not practically applicable. Thus, using a simple but realistic viscoelastic model, which includes elastic and creep behavior of a polymer cable, is necessary to improve position control performance of PCDPR.

Advanced control algorithm of CDPR has also been proposed by some researches, such as fuzzy control and proportional–integral control¹⁴ and vision-based position control.^15,16 But most of the previous methods with feedback signal are not practically applicable to the actual PCDPR because of the sensor measurement cabling constraints and cost. Hence, improved open-loop control algorithm with precise cable deformation compensation algorithm is necessary for precise PCDPR control performance.

In this study, to improve the position control accuracy against the viscoelastic properties of a polymer cable in PCDPR, we propose a five-element viscoelastic cable model based on the experiments. And then accomplish model-based open-loop position control of the PCDPR system in experiments via the modeled polymer cable viscoelastic behavior.

This article is organized as follows. First, the description of the developed PCDPR system with its kinematic and dynamic models is shown. Second, the five-element viscoelastic modeling and parameter estimation result by experiments are provided. Then, the model-based open-loop control is explained to compensate position error for a high payload workspace. Then, the experimental results of the open-loop position control will be shown. Finally, conclusion will be addressed.

PCDPR system

Description of PCDPR system

The high payload PCDPR shown in Figure 1 is designed for the purpose of heavy object transportation, assembly process in industry. It can manipulate 6-degree-of-freedom (DOF) end-effector configurations with eight-cable connection for a wide workspace in regular hexahedron steel frame of size 4 m × 4 m × 4 m.¹⁷ Eight winches are fixed at bottom frame equipped with force sensors in each winch to measure cables’ tension. And eight pulleys, assembled with two wheels in each set, are fixed at the corners of top frame that guides all directional cables from winch to the end-effector. As a light flexible actuator of the high payload PCDPR, polyethylene (Dyneema^® cable, LIROS D-Pro 01505-0600) is used, the weight of which is 1 kg/100 m with a diameter of 6 mm. The designed maximum payload of the PCDPR is 70 kg, and the goal of this robot is to have capability of handling more than 200 kg. We use industrial real-time controller Beckhoff’s TwinCAT3 to control robot and vacuum pump for heavy payload pick-and-place application.

Figure 1.

A photograph of high payload CDPR for pick-and-place application.

Robot kinematics and dynamics

Figure 2(a) shows a schematic of conventional CDPR. Because the PCDPR is a type of parallel robot system, conventional inverse kinematics can be used to describe the joint variable from the given end-effector posture. By assuming the steady-state condition, we can derive a general form of PCDPR inverse kinematic equation, where $l_{i}$ (i = 1, 2, …, n) is the cable length vector along with the ith cable from its attaching point on the base frame $A_{i}$ to attaching point on the end-effector $B_{i}$ . $a_{i}$ and $b_{i}$ are the two constant vectors of base coordinate and end-effector coordinate, respectively. Hence, the closure vector loop for the cable length vector $l_{i} = [l_{ix}, l_{iy}, l_{iz}]^{T}$ can be described as equation (1)

l_{i} = a_{i} - p - R b_{i}

(1)

Figure 2.

A schematic of PCDPR: (a) conventional PCDPR and (b) five-element viscoelastic modeled PCDPR.

And the length of ith cable $l_{i}$ can be obtained by Euclidean norm $‖ l_{i} ‖$

l_{i} = \sqrt{{[a_{i} - p - R b_{i}]}^{T} \cdot [a_{i} - p - R b_{i}]}

(2)

where position vector $p = [x_{e}, y_{e}, z_{e}]^{T}$ describes the end-effector position and rotation matrix $R$ describes the orientation of the end-effector with its roll, pitch, and yaw angles being $α$ , $β$ , and $γ$ , respectively. Since the CDPR is over-constrained robot system with eight cables and 6-DOF end-effector, unique forward kinematic solution does not exist and various complicated computation algorithms are studied for the forward kinematics solver.^18,19 Here, we utilized the reduced Jacobian method as in equation (3)²⁰

x_{app} (t) = x (t_{0}) + {\tilde{J}}_{forw} (x (t_{0})) (l_{red} (t) - l_{red} (t_{0}))

(3)

where $x (t_{0})$ is the pose of end-effector and $l_{red} (t_{0})$ is the corresponding cable length. The position at time $t$ can be calculated using reduced Jacobian of the forward kinematics ${\tilde{J}}_{forw}$ and cable length $l_{red} (t)$ .

The standard dynamic equation of the PCDPR by assuming that the external force is applied to the end-effector can be obtained by equation (4) in global coordinate of the robot system^21,22

\begin{matrix} M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = - S T_{c} \\ and \\ S T_{c} = - [\begin{matrix} u_{1} & \dots & u_{8} \\ b_{1} \times u_{1} & \dots & b_{8} \times u_{8} \end{matrix}] T_{c} \\ = [F_{x}, F_{y}, F_{z}, T_{x}, T_{y}, T_{z}]^{T} \end{matrix}

(4)

where $q = [x, y, z, α, β, γ]^{T}$ is defined in Cartesian coordinate, $u_{i} = l_{i} / | l_{i} |$ , $T_{c} = [f_{1}, f_{2}, \dots, f_{8}]^{T}$ is the cable tension vector at the end-effector in joint coordinate, and $S$ is a structure matrix that transforms cable tension to Cartesian space force. The cable tension as a force input to the robot end-effector can be obtained from equation (5), simplified second-order winch-motor model

J_{mi} {\overset{\cdot\cdot}{θ}}_{i} + C_{mi} {\overset{\cdot}{θ}}_{i} + K_{mi} θ_{i} - r f_{i} = U_{i}

(5)

where $θ_{i}$ is a rotation angle of each winch motor, $U_{i}$ is the control input to each motor, and $J_{mi}$ , $C_{mi}$ , and $K_{mi}$ are the general motor parameters’ vector for ith motor and winch.

Numerical quantification of cable deformation effect on Cartesian space position error

Figure 2(b) depicts the suggested viscoelastic model in this article. At the actual PCDPR system, due to viscoelastic deformation of the polymer cable, cable length $l_{i}$ becomes $l_{i} + Δ l_{i}$ and the structure matrix $S (l)$ becomes $S (l + Δ l)$ , which makes end-effector position error and ultimately produces mismatched tension distribution with respect to the model of stiff cable dynamic. Table 1 shows the quantification results of kinematics computation for end-effector posture error induced by the cable length differences. But in the practical PCDPR system, although the winch-motor encoder is equipped to control the ideal cable length, because the measurement of actual cable length and the real-time end-effector posture information are not available, estimation method of the actual cable is necessary to solve the precise position control problem. Hence, we need to consider mathematical model of the cable deformation for the accurate position control.

Table 1.

Quantification of numerical computation for cable length and end-effector pose error by 65 kg payload.

Unit (m)	Cable length								End-effector (mm, °)
	l ₁	l ₂	l ₃	l ₄	l ₅	l ₆	l ₇	l ₈
Ref. 1	7.1566	7.0462	7.1457	7.0127	7.0903	7.1290	7.0722	7.0962	[0, 0, 300] [0, 0, 0]
Real 1	7.1723	7.0625	7.1615	7.0292	7.1068	7.1452	7.0887	7.1126	[0, 0, 270] [0, 0, 0]
Error	0.0157	0.0163	0.0158	0.0165	0.0165	0.0162	0.0165	0.0164	[0, 0, 30] [0, 0, 0]
Ref. 2	6.7172	7.3714	7.7195	6.9009	6.6175	7.3607	7.4692	6.8487	[−600, 210, 300] [0, 0, 18.5]
Real 2	6.7360	7.3867	7.7334	6.9186	6.6375	7.3764	7.4844	6.8671	[−600, 210, 269] [0, 0, 18.5]
Error	0.0188	0.0153	0.0139	0.0177	0.0200	0.0157	0.0152	0.0184	[0, 0, 31] [0, 0, 0]

Viscoelastic modeling for a polymer cable

The cable viscoelastic behavior based on the previous observations of cable viscoelastic properties indicates that elastic deformation is instantaneous but creep is the time-dependent characteristics under the influence of mechanical stresses. This study suggests a modified Burgers model by considering instantaneous elastic property and time-dependent creep property. Although those properties related to polymer cable elongation can be affected by various factors such as type of material property, weight of load, length of cable, and temperature.^23,24 We consider the load weight and cable length as important factors for a heavy payload PCDPR in room temperature workspace.

Mathematical cable model

Viscoelastic effect can be modeled by composing spring and damping elements in series and parallel. Among the well-known viscoelastic model, Burgers model consisted of Maxwell and Kelvin–Voigt model is commonly used for describing complicated cable behavior.²⁵ To properly model viscoelastic behavior of instantaneous response, transient creep, and long-term creep, we propose a five-element model based on modified Burgers model. The model is the serial combination of a linear spring and two Voigt models, thus having instantaneous elastic property by Hookean spring, transient creep by first Voigt model, and long-term creep by second Voigt model as shown in Figure 2(b). According to the superposition principle of the total strain at any time after applying the stress, the strain can be obtained as the sum of those three parts²⁶

ε (t) = \frac{σ}{E_{1}} + \frac{σ}{E_{2}} (1 - e^{- \frac{E_{2}}{η_{2}} t}) + \frac{σ}{E_{3}} (1 - e^{- \frac{E_{3}}{η_{3}} t})

(6)

where $σ$ is the stress for a single cable; $E_{1}$ , $E_{2}$ , and $E_{3}$ are the elastic modulus; and $η_{2}$ and $η_{3}$ are the dissipation elements in the model. By virtue of the elastic modulus definition for the given material, equation (7) is valid and we can utilize it to encounter the mass and cable length to the viscoelastic model

E = \frac{F L_{0}}{A_{0} Δ L}

(7)

where $E$ is the elastic modulus, $F$ is the force exerted on the cable, $A_{0}$ is the actual cross-sectional area through which the force is applied, and $Δ L$ is the amount of cable length changes.

Experiments for a viscoelastic cable modeling

Some methods for viscosity–elasticity experiments have been available for experimentally driven cable modeling with creep curve.^27,28 But those methods have limitations of testing equipment and experiment cable length. In this study, to derive more realistic model for the actual size PCDPR system, we used actual winch–cable–pulley system and a crane for loading and unloading weight blocks in the full designed operating range.

Figure 3 shows the experimental setup for the viscoelastic behavior test for a single cable. The cable starts from the fixed winch and passes through the pulleys to the hook of the testing mass. For loading, the crane moves down and for unloading, the crane moves up to the initial position. A precise 6-DOF displacement measurement sensor, Optical Track Sensor (OTS; model: Polaris Spectra from NDI^®; root mean square (RMS) is 0.3 mm, sampling rate is 60 Hz), is used, and its sensor tool with four ball markers is fixed at the cable (reference tool) and profile (global tool). We used two different experiments for different loads and lengths. For a fixed cable length, different payloads are applied one by one and repeated for other length cable with the same payload conditions. All the measurements are measured simultaneously. The maximum total loads can be more than 100 kg including nine weight blocks and middle bar, but by considering the safety factor, the testing weights were between 11.6 and 71.2 kg in each cable. We conducted cable length tests for three different cable lengths (5.4, 7.4, and 8.4 m) using different positions of pulleys, considering our current system configuration and operating ranges. Finally, for the given mass and cable length variation, tension and cable length changes on time domain are measured simultaneously. It is found that the parameters are considerably different for the experimented mass and cable lengths due to the nonlinear properties of polymer cable.

Figure 3.

Experimental setup for creeping test.

Model parameter estimation

The suggested viscoelastic model of strain and stress interactions in equation (6) has five parameters to be estimated. To minimize the model parameter to be estimated and enhance the model computation in real-time, sensitivity analysis was accomplished for the model parameters given by equation (8)

{| \frac{\partial ε (t)}{\partial μ} |}_{μ_{0}}

(8)

where $μ = {E_{1}, E_{2}, E_{3}, η_{2}, η_{3}}$ . The nominal parameters are obtained by considering the PCDPR at the middle of designed operating range. The results in Table 2 show that $E_{1}$ and $E_{2}$ are the dominant two parameters in model equation (6). But by one more analysis of the estimation performance comparison, for the model of $E_{1}$ parameter only, and for the model of both $E_{1}$ and $E_{2}$ parameters, shows that the estimation results were not significantly different. The root mean square error (RMSE) is 0.112 mm ± 0.049 mm for $E_{1}$ -only model and 0.119 ± 0.049 mm for $E_{1}$ and $E_{2}$ model. Then we regard $E_{1}$ as the only dominant parameter affected by mass and length variables and make it as a function of cable tension and cable length.

Table 2.

Results of the sensitivity analysis.

Sensitivity (AU)	E₁ (×10⁻¹⁴)	E₂ (×10⁻¹⁴)	E₃ (×10⁻¹⁸)	η₂ (×10⁻²¹)	η₃ (×10⁻¹⁹)
(212 N)	6.8038	8.3116	14.039	8.4056	13.575
(114 N)	8.0941	4.5210	7.6363	4.5721	7.3839

To encounter the force and length variations in the viscoelastic model, the model parameter, $E_{1}$ , is expressed as a function of cable tension $f_{i}$ and cable length $l_{i}$ with the basis of equation (7). Those values can be directly measured using force sensor and encoder equipped in the winch-motor system of the PCDPR. The surface fitting methods are used to express $E_{1}$ that can be described using polynomial function with two variables as third-order polynomial of cable tension and second-order polynomial of cable length as in equation (9)

\begin{array}{l} E_{1} (f, l) = α_{00} + α_{10} f + α_{20} f^{2} + α_{30} f^{3} + α_{01} l \\ + α_{02} l^{2} + α_{11} f l + α_{21} f^{2} l + α_{12} f l^{2} \end{array}

(9)

With the computed value of $E_{1}$ for the cable tension and length at the given position, other parameters in equation (6) are estimated from equation (10) by minimizing the error cost function. We used brute-force algorithm to find an optimal parameters

\arg \min_{E_{2}, E_{3}, η_{2}, η_{3}} ‖ ε_{model} - ε_{measured} ‖_{2}

(10)

Figure 4 shows the surface fitted elastic model where red points are the experimental points and green surface is the operating ranges of our PCDPR system. Table 3 shows the data-driven model parameters of equation (6) where the cable viscoelastic model parameters, $E_{2}$ , $E_{3}$ , $η_{2}$ , and $η_{3}$ , are constant and Table 4 shows modeled parameter of $E_{1}$ as a function of cable tension, $f_{i}$ , and cable length, $l_{i}$ in equation (9). Figure 5 shows good matches of viscoelastic model’s time response between experimental data and estimated results.

Figure 4.

Surface fitting for estimated parameter (E₁) for a single cable.

Table 3.

Parameters of polymer cable viscoelastic model.

Parameters		Values
Elastic modulus (N/m²)	E ₁	E₁(f, l)
	E ₂	9.5 × 10⁹
	E ₃	5.0 × 10¹⁰
Viscosity (N s/m²)	η ₂	1.0 × 10¹⁰
	η ₃	1.0 × 10¹³

Table 4.

Elastic modulus parameters for surface function model.

α₀₀	α₁₀	α₀₁	α₂₀	α₁₁
−7.856 × 10¹⁰	4.361 × 10⁸	2.538 × 10¹⁰	1.100 × 10⁵	−1.223 × 10⁸
α₀₂	α₃₀	α₂₁	α₁₂
−1.999 × 10⁹	2.056 × 10²	−4.343 × 10⁴	1.000 × 10⁷

Figure 5.

Experimental creep responses and the predictions of the five-element model: (a) creep responses under a series of payloads and (b) creep responses under a series of cable lengths.

Open-loop position control for high payload PCDPR

Controller design

To improve the position control performance, the experimentally driven viscoelastic model is incorporated that can estimate the cable elongation caused by the heavy payload. Advanced feedback control methodologies for CDPR have been studied,^29,30 but, in this study, we use an open-loop control. Since the open-loop compensation method is easy to implement and usually can be used in applications where the feedback measurement is not available due to end-effector sensor setting limitation. In short, we note that the given general PCDPR system has no position or force feedback sensor from end-effector. Only an encoder and a pressure sensor to measure winch rotation angle and cable tension at each winch are equipped. Actual postures of the end-effector on global coordinate and actual cable length to compute $Δ l_{i}$ are not available, and therefore the actual cable length needs to be estimated by the suggested model in equation (6).

Figure 6 shows a block diagram of the suggested open-loop control. The eight cables in the robot system are controlled simultaneously, and the combination of each cable determines end-effector pose. Each cable length $l_{id}$ is obtained from the inverse kinematics with desired posture of the end-effector $q_{d}$ . Then, the winch-motor driver and controller are used to control the desired motor output in joint space. Since cables show viscoelastic behavior while loading heavy object, the elongated cable lengths are estimated by the derived viscoelastic model at each position, where the actual tensions are measured by pressure sensors and actual cable lengths are measured by encoders. And the controller switches to compensation control mode that gives winding cable command to compensate the amount of viscoelastic elongation, $Δ l_{i}$ , are applied to the control system. Finally, eight cables are compensated individually for more accurate end-effector pose.

Figure 6.

A block diagram of an open-loop position control for cable length compensation.

To elucidate the viscoelastic effect on the controller design in terms of the force distribution of the PCDPR,^31,32 we encounter the winch-motor dynamics of equation (5) and transform the encoder space to cable length domain using $l_{i} = r θ_{i}$ , and then we can get equation (11)

J_{mi} {\overset{\cdot\cdot}{l}}_{i} + C_{mi} {\overset{\cdot}{l}}_{i} + K_{mi} l_{i} - r^{2} f_{i} = r U_{i}

(11)

However, the actual cable length with cable deformation becomes $l_{i} = r θ_{i} + Δ l_{i}$ , and the winch dynamic will have $K_{mi} Δ l_{i}$ that makes the system transient error as in equation (12), where $Δ l_{i}$ is assumed to be constant obtained at the ending time of the first position control

J_{mi} {\overset{\cdot\cdot}{l}}_{i} + C_{mi} {\overset{\cdot}{l}}_{i} + K_{mi} (l_{i} + Δ l_{i}) - r^{2} f_{i} = r U_{i}

(12)

Since we can design an open-loop force input as $U_{i} = K_{c} K_{mi} Δ l_{i}$ using viscoelastic deformation error, the system in equation (12) can compensate the position error induced by the inaccurate cable length without controller structure modification in motor driver. Let $t_{s}$ be the timing of the first position control ending, then we can compute the fixed amount viscosity deformation and relative cable length error estimated as follows

{\hat{ε}}_{i} (t_{s}) = \frac{σ_{i}}{E_{1} (f_{i}, l_{i})} + \frac{σ_{i}}{E_{2}} (1 - e^{- \frac{E_{2}}{η_{2}} t_{s}}) + \frac{σ_{i}}{E_{3}} (1 - e^{- \frac{E_{3}}{η_{3}} t_{s}})

(13)

Δ l_{i} (t) = {\hat{ε}}_{i} (t) l_{i}

(14)

The amount of force compensation can be computed as equation (15)

Δ f_{iu} = \frac{Δ l_{i} A_{0} E_{1 i} E_{2 i} E_{3 i}}{l_{i} (E_{2 i} E_{3 i} + E_{1 i} E_{3 i} (1 - e^{- \frac{E_{2 i}}{η_{2 i}} t}) + E_{1 i} E_{2 i} (1 - e^{- \frac{E_{3 i}}{η_{3 i}} t}))}

(15)

And it can be simplified to equation (16) for the given ending time that is bigger than some seconds

Δ f_{iu} ≅ \frac{Δ l_{i} A_{0}}{l_{i}} (\frac{1}{E_{1 i}} + \frac{1}{E_{2 i}})

(16)

Then, the total open-loop control input in force domain that is transformed by force distribution to end-effector frame can be expressed by $U_{i} + Δ f_{iu}$ . Here, $Δ f_{iu}$ is the additional control force necessary to compensate viscoelastic deformation converted from the estimated cable length error.

Control experimental setup

The PCDPR system described in Figure 1 is used for the position control experiments. The weight of the end-effector is 5.4 kg with the hexagonal shape payload of 65 kg (total 70.4 kg). Since one of the strong possible candidate applications of the PCDPR system is high payload transportation, we designed a reference trajectory to track as shown in Figure 7 with x-, y-, z-, and $γ$ -directional pick and place. The end-effector is picking up and placing down the object using a vacuum holder. The cable length is calculated by measured winch-drum rotation angle from winch encoder (model: MSK050B-0600-NN-M2-UG1-RNNN) and the cable tension is measured by force sensor (model: F2301) equipped at each winch system. The 6-DOF posture in Cartesian space is measured by OTS sensor that is used for the viscoelastic model experiments. All the sensor signals are measured and recorded simultaneously. In the control scheme, with the planned end-effector trajectory, $q_{d}$ , the required cable length $l_{id}$ in each cable is calculated from inverse kinematics in equation (2), where subscript $i$ denotes the number of respective cable.

Figure 7.

Experimental trajectory and results for pick-and-place experiment.

For the controller performance evaluation, three experiments were conducted by eliminating other factors’ influence of the position accuracy of the cable robot such as pulley friction and pulley kinematics: (a) we run the PCDPR without payload and only have end-effector weight in the desired path which has position error induced by other factors; (b) for the same path with high payload, open-loop control without viscoelastic deformation compensation algorithm was performed, and then (c) for the same path, open-loop control with compensation algorithm was experimented.

Experimental results

Figure 7 depicts resultant end-effector trajectory comparison between viscoelastic deformation compensated open-loop control and uncompensated position control. For the given reference trajectory (blue line), the trajectory of the viscoelastic deformation compensated control (red line) shows strongly closer behavior to the reference trajectory than the uncompensated control (green line). The measured cable length during the experiment and simulated cable length for the case of compensated control and uncompensated control are plotted together in Figure 8. The simulated cable lengths obtained by solving inverse kinematics and viscoelastic models are well matched with actual measurements. Figure 9 shows the measured cable tension during the experiments, in which it is clearly shown that cable tension is also compensated to overcome the viscoelastic deformation error induced by payload as explained in controller design.

Figure 8.

Comparison of cable lengths during pick-and-place experiment.

Figure 9.

Measured force during the pick-and-place experiments at each cable.

Figure 10 shows position control performances with respect to each coordinate system as x-, y-, z-, and γ-direction. Figure 11 shows the resultant position errors in each direction. The compensation control is activated from 14.8 to 15.8 s when the end-effector reaches around 300 mm high from home position. In this period, since the cable lengths (in Figure 8) and tensions (in Figure 9) are in the range of viscoelastic cable model boundary described in Figure 4, the proposed controller can compensate the position error correctly.

Figure 10.

Comparison of the trajectory during pick-and-place experiment: (a) in x-direction, (b) in y-direction, (c) in z-direction, and (d) in γ-direction (compensated at 14.8 s).

Figure 11.

Comparison of tracking error during pick-and-place experiment: (a) in x-direction, (b) in y-direction, (c) in z-direction, and (d) in γ-direction (compensated at 14.8 s).

The experimental results are summarized in Table 5. The results indicate that the z-direction deformation is significant than other directional motion when loading the payload; position error is 28.5 mm (9.5%) of the desired position (300 mm). By applying the viscoelastic deformation compensation control, the z-direction error is decreased to less than 1.3 mm that is 0.5% of the reference position. It illustrates that our controller can properly compensate the viscoelastic deformation of the cable when moving a heavy object. However, the position error of the proposed controller in x-, y-, and $γ$ -direction is not improved much, and it is even similar to the pick-and-place position error without payload condition experiment. It is mainly caused by the fact that the payload gravity direction is along with the z-axis of the end-effector payload and all the cables are elongated in gravity direction equally. It means that the cable elongation does not influence the other direction of heavy object except z-direction motion. Hence, the load compensation control does not improve the position control performance of cable robot in unforced direction.

Table 5.

RMSE comparison of open-loop position control results for different time span.

	x (mm)	y (mm)	z (mm)	γ (°)
Ranges	0 to 600	0 to 210	−275 to 300	0 to 18.5
Without compensation (1-s RMSE)	0.56	1.52	28.55	0.25
With compensation (1-s RMSE)	0.51	1.3	1.32	0.36
Without compensation (10-s RMSE)	0.50	1.78	29.45	0.30
With compensation (10-s RMSE)	0.54	1.52	0.75	0.36

RMSE: root mean square error.

Conclusion

In this article, an experiment based on five-element viscoelastic model is proposed to derive a mathematical model of polymer cable deformation for a PCDPR system. The dominant model parameters of the cable viscoelastic model were selected by sensitivity analysis, and the viscoelastic cable model is identified by brute-force method with single cable experiments. The comparison of the obtained model and experimental results of the polymer cable in time domain shows good matches and could be applied to controller design. Furthermore, an open-loop position control was implemented based on the identified viscoelastic cable model. Each cable deformation depending on the payload and the cable length is estimated by the derived cable model, and then the controller compensates the amount of the cable length deformation with respect to the reference position. The experimental results for the position control task under heavy payload showed that the suggested viscoelastic cable model-based open-control scheme can improve position control accuracy significantly, especially in gravity direction. Since the proposed open-loop controller can perform a satisfactory position error reduction for the PCDPR system without having Cartesian space feedback measurements, the study can contribute to implement low-cost practical industrial application of the PCDPR system.

Footnotes

Handling Editor: Yong Chen

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 research was supported by Leading Foreign Research Institute Recruitment Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT; no. 2012K1A4A3026740).

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