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Abstract

With the advantages of simple structure, low power consumption, high ratio of load to self-weight, and good flexibility, the cable-driven robots are especially suitable for medical instruments and home service. Due to the alignment of cables, there is a problem of motion coupling between the manipulator joints, leading to the degradation of control accuracy. Moreover, the hysteresis and the dead zone of the cable driving unit make it difficult to control the joint accurately. To effectively solve above-mentioned motion-coupling problem, a novel motion-decoupling mechanism is proposed and investigated. An experimental setup is established to verify the kinematic decoupling characteristics. Based on the “static friction + Coulomb friction” model, the friction model is established and verified. Thus, studies on the friction model have provided theoretical basis for the accurate force coordinated control of the cable-driven system.

Keywords

Cable driven cable alignment manipulator passive decoupling friction model

Introduction

Nowadays, cable-driven method is widely utilized in manipulators involved in human–machine interaction.^1–4 Cable-driven multi-joints manipulators usually place their driving units at the base. Motion and force are transmitted by light cables. Compared with traditional manipulators driven by motors and reducers, cable-driven manipulators consume less energy to complete a certain task benefiting from lower movement inertia.⁵ Moreover, the hazard of breakage is reduced thanks to the light weight. On the other hand, cable-driven manipulators perform well in shock-absorbing and impulse-reducing thanks to cable’s flexibility.⁶ Thus, cable-driven manipulators can effectively improve safety in human–machine interaction.⁷

Since driving units are placed at the base of the manipulator, cables have to go through a carved path to drive the end-effector leading to motion decoupling and degeneration of the control performance.⁸ Therefore, novel cable-wound mechanism is a research hotspot recently. Cable-wound mechanism is essential to realize passive decoupling between joints, which can improve driving accuracy. Morgan Quigley realized active decoupling in a four-joint manipulator based on algorithm.⁹ JK Lee and colleagues^10,11 proposed a novel decoupling mechanism by combining parallel rods and movable pulley, which realized the passive decoupling between wrist and end-effector, as well as elbow and back joints. Zhao and Nelson¹² proposed a cable-decoupling mechanism for revolutionary joint based on planetary gear and further simplified the mechanism. A medical forcep was proposed and investigated by Burbank.¹³ The proposed forceps were driven by four cables based on a novel driving mode, which can realize many motions of forceps, such as opening and closing, pitch motion, and wrist’s rotation.

In this article, the coupling of multi-joint manipulator is analyzed. A novel cable-wound mechanism is proposed to avoid coupling among different joints of manipulator. A 2-degree-of-freedom test platform is designed to verify the decoupling mechanism and analyze the factors that influence decoupling effects. Based on the model of static friction and Coulomb, the friction model of this decoupling mechanism is studied using infinitesimal method. Finally, the effectiveness of our proposed mechanism and the friction model are verified through some experiments.

Cable-driven motion-decoupling design

Cable motion-coupling analysis of multi-joints

In cable-driven manipulators, the joint motion is coupled with the motion of the joint which driving cables traverse. In order to facilitate the discussion of the coupling relationship between two joints, the joint which driving cables traverse is known as proximal joint, the joint which cables drive is known as distal joint. Figure 1 shows the alignment diagram of cables.

Figure 1.

Routings of cables in two adjoining joints.

Cable 1 drives proximal joint and link 2 to rotate around the central shaft. Cable 2 starts from the base and traverses proximal joint. Meanwhile, it drives distal joint to rotate around its own central shaft. Due to the effect of the guide pulley on link 2, when cable 1 drives the proximal joint to rotate by $θ_{1}$ degrees, cable 2 will wind or unwind around the guide pulley on the central axis of proximal joint. As shown in Figures 2 and 3, due to the rotation of proximal joint, cable 2-1 will unwind around the guide pulley by the length of $θ_{1} r$ ; meanwhile, cable 2-2 will wind around the guide pulley by the length of $θ_{1} r$ , where $r$ is the radius of pulley groove on proximal joint. It is reflected as that the end of cable 2-1 and the end of cable 2-2 extend and retract by the length of $θ_{1} r$ . If the radius of the pulley groove on distal joint is defined as $r_{1}$ , the rotation angle of distal joint is $θ_{1} r / r_{1}$ . By the geometric analysis of the expression, it can be seen that the two joints are linearly coupled. Figure 3 shows the routing of cable 2. It can be seen that distal joint will rotate by the angle of $θ_{1} r / r_{1}$ because of the rotation of proximal joint.

Figure 2.

Analysis of motion coupling in cable-driven system.

Figure 3.

Routing of cable 2.

The routing of cable is shown in Figure 4 when proximal joint is rotary joint. Cable 3 drives proximal joint and link 2 rotates around the central axis, while cable 4 drives distal joint.

Figure 4.

Routing of cable in rotary joint.

As shown in Figure 5, cable 3 drives proximal rotary joint to rotate by the angle of $θ_{1}$ , which means link 2 rotates around the central axis of proximal rotary joint by the angle of $θ_{1}$ . Cable 4 will twist and wind due to the effect of pulley. The right ends of cable 4-1 and 4-2 will move a distance to the left, and the displacement changes nonlinearly as it rotates. Meanwhile, the cable tension is getting bigger, which makes it difficult to drive distal joint. Thus, this kind of cable routing cannot meet the requirement of reliable and stable joint drive.

Figure 5.

Analysis of motion coupling in rotary joint.

It can be seen from the above analysis that if the proximal joint is a pitch rotation joint, the motion of proximal joint and distal joint is linearly coupled. Otherwise, if the proximal joint is a rotary joint, the motions are nonlinearly coupled, which means the distal joint cannot drive the robot reliably and stably.

Therefore, it is necessary to find a new way of cable winding to realize the passive decoupling of the movement between proximal joint and the distal joint to realize stable and reliable drive. Meanwhile, universal algorithm of common industrial robot control can be directly applied in cable-driven robot if the motion is decoupled.

Motion-decoupling modular joints design

In this article, a novel motion-decoupling modular joint is proposed to achieve the independent movement of proximal joint and distal joint. The proposed decoupling modular mechanism is shown in Figure 6.

Figure 6.

Routings of cable in motion-decoupling modular.

The mechanism includes driving cable of distal joint, fixed wheel, following wheel, decoupling cable, driving wheel, and driving cable of proximal joint from left to right. Both the following wheel and the driving wheels are mounted on the central shaft by bearings and the axial displacements are fixed. There are two decoupling cables and four ends of these two cables. Two left ends are fixed to the fixed wheel after traversing two through holes of fixed wheel. Similarly, two right ends are fixed to the driving wheel after traversing two through holes of driving wheel. These two decoupling cables are both along the same guiding groove. The wiring of driving cable is shown in Figure 6.

Figure 7 shows the working diagram of the motion-decoupling module. The fixed wheel is fixed to the joint base. The driving wheel is fixedly connected with joint link 1. The proximal joint is driven by a motor placed at the base. The distal joint is driven by a distal articulated driving cable, which travels from the motor at the base, passes through the motion-decoupling module, and is fixed with the connecting rod at the distal joint.

Figure 7.

The working diagram of motion-decoupling module.

The proximal joint is rotated at an angular velocity $ω$ , which means that the driving wheel of the decoupling module rotates at an angular velocity $ω$ with respect to the fixed wheel. Since the ends of the decoupling cable are respectively fixed on the fixed wheel and the driving wheel, the decoupling cables will restrict the follower wheel to rotate, and the follower wheel’s rotation angular velocity is 1/2 the joint rotation angular velocity with the same direction. The driving cables of distal joint go in the same routings of decoupling cables and the radius of the groove is $r$ . When the driving wheel rotates in the direction shown in Figure 7 at the angular velocity $ω$ , left driving cable 1 of distal joint will unwind from the follower wheel along the guide groove and the displacement of right end is $+ ω Δ t • r$ . Meanwhile, the right driving cable 2 will wind along the guide groove and the displacement of right end is $- ω Δ t • r$ . As shown in Figure 8, $ω$ refers to angular velocity and $r$ refers to the radius of the guide groove of driving cables on follower wheel. The upward direction of cable is taken as positive.

Figure 8.

The working diagram of motion decoupling of driving cable.

Due to the decoupling cables, the speed of the end of the driving cable of distal joint is $2 r$ times the speed of the follower wheel, and the displacement of the end of these two cables is $ω Δ t / 2 \cdot 2 r = ω Δ tr$ . The displacement of the end of left driving cable is $- ω Δ t • r$ , while the displacement of the end of right driving cable is $+ ω Δ t • r$ .

Combining the above two factors that cause the relative displacement of the end of the driving cable, it can be seen that the left and right driving cables of distal joint have no relative displacement on the right side of the end with the decoupling module. As a result, the driving cables of distal joint do not have the relative movement due to the rotation of the front end joint, which means motion decoupling is realized.

Verification of motion decoupling

Decoupling verification platform system

The overall prototype of the 2-degree-of-freedom decoupling verification platform is shown in Figure 9. Two articulated drive units are mounted at the base, and the drive wheels of the respective joints are driven by a cable. In the figure, the dashed line indicates the motion transmission chain of the driving cable of proximal joint, and the white solid line indicates the motion transmission chain of the driving cable of distal joint. The upper left end is the cable motion-decoupling module, and the rear end of the drive rope is wrapped around the decoupling module. The two ends of the prototype are equipped with two joint encoders, respectively, to detect the movement angle of the two driving wheels. Encoder 1 measures the movement angle of the proximal joint and encoder 2 measures the movement angle of the distal joint.

Figure 9.

The overall prototype of decoupling verification platform.

An experiment to verify the effectiveness of the cable-driven decoupling module was designed and performed. First, the two driving cables of the cable-drive system are pre-tensioned. In the experiment, the driving motor of the proximal joint is in the position mode, and the distal joint motor is always in the holding state. The position control command is applied to the motor of the proximal joint to drive the proximal joint for sinusoidal reciprocating rotation. The pulse signals of the joint encoders 1 and 2 are acquired in real time by reading the external encoder registers of the two servo drives.

Experimental results and analysis

Corresponding experimental results are given in Figure 10. As shown in Figure 10, the distal joint presents obvious movement as the proximal joint moves without the decoupling module, and the movement change ranges from −10° to +10°. Meanwhile, the movement of the distal joint was effectively eased after the adoption of the proposed decoupling mechanism, and the movement change ranges from −2° to +2°.

Figure 10.

Decoupling performance verification results.

Since the driving motor of the distal joint is always in the brake state, the distal joint after decoupling should always remain relatively stationary with the proximal joint. But at the time (t = 0.06–0.16 s, 0.84–0.94 s), the distal loading wheel appears mutation, and in t = 0.15–0.8 s and t = 1–1.65 s process, there appears angle offset. Figure 11 shows a curve showing the movement of the corner of the distal joint with time in a cycle T. At position 1 (t = 0.06–0.16 s) and position 2 (t = 0.84–0.94 s) as shown in Figure 11, the driving direction of the proximal joint is changed, resulting in the switching of the tension state of the two driving cables of the distal joint. The distal loading wheel will be relative to the central axis of the hysteresis rotation, that is, the distal joint relative to the proximal joint angle hysteresis, corresponding to the distal joint position 1 of the mutation, position 2 of the mutation with the rationale.

Figure 11.

The actual movement angle of the distal joint.

From the analysis in Figure 11, it was found that the cable produced a slack amount of about 1.26 mm. The relaxation of the driving cable occurs in the process of driving; there may be two main reasons: first, the cable’s initial pre-tightening force is not enough, leading to the loose side getting too loose; second, error of decoupling unit manufacturing and assembly, resulting in the trace path of the two driving cables in the coupling unit, is not symmetrical.

In the experiment, by changing the initial pre-tightening force of the driving cable and analyzing the decoupling performance of the system, we found that the decoupling unit has a tendency to decrease the amount of mutation of the distal joint under a certain pre-tightening force. But as the initial pre-tightening force increases, the amount of mutation in the commutation increases. The reason for this phenomenon is that as the initial pre-tightening force of the system increases, the asymmetry error of the two driving cables in the decoupling unit will be more likely to reflected on the side of the loading wheel of the distal joint, resulting in an increase in the amount of corner change of the distal joint’s driving wheel.

In order to improve the decoupling performance of the decoupling module and reduce the amount of mutation (positions 1 and 2) of the distal joint, there can mainly be two ways: first, minimizing the error of manufacturing and assembly; second, dynamic controlling the tension of the cable-driven system, which means reduce or eliminate the amount of motion slack of the cable dynamically during the driving process to avoid a large amount of joint mutations in the joint movement commutation.

In position 3 (t = 0.15–0.8 s) and position 4 (t = 1–1.65 s) in Figure 11, there is an angle offset at the distal joint, and the end of the cable produces an offset of about 0.21 mm. In the decoupling module, there is a friction between the driving cable and the chute contact surface on the guide disk, resulting in a tensile loss in the direction of force, that is, the tension of the driving cable at each end of the decoupling module is not equal. In position 3 and position 4, the cable tension of the tight side cable on both sides of the decoupling module is different, in the tight side of the cable, near the distal loading wheel is a tight edge, the end of the driving motor is the edge under tight. In the experiment, it was found that the internal tension of the under tight edge was increasing when t = 0.15–0.8 s and t = 1–1.65 s. It is deduced that the continuous extension of the tight side cable results in a corner lag shift of positions 3 and 4 of the distal joint.

In order to improve the decoupling performance of the decoupling module in the cable-driven system, we change the tensile stiffness conditions of different cables according to the angular deviation at 3 and 4 position 3 and 4 and then carry out experimental analysis.

As shown in Figure 12, the curve of the decoupling-distal joint of the 304 stainless steel wire cable under the condition that the initial pre-tensioning force is 20 N, the diameter is 0.8 mm, and 1.2 mm is changed with time.

Figure 12.

Distal joint corner curve under different stretch stiffness conditions.

As can be seen from Figure 12, the distal joint angle offset of the driving cable with a diameter of 1.2 mm is smaller than the driving cable with a diameter of 0.8 mm. So increasing the tensile rigidity of the driving cable can reduce the angular offset of the distal joint to some extent.

The experimental analysis shows that first, increasing the initial pre-tensioning force of the cable-driven system and minimizing the error of manufacturing and assembly, besides, controlling the dynamic tension of the cable-driven system, during the driving process, reducing or eliminating the amount of motion slack of the cable dynamically; second, improving the tensile stiffness of the driving cable can improve the decoupling performance of the motion-decoupling module to some extent, in the choice of driving cable, we should consider the toughness and tensile stiffness of the two performance indicators in the compromise, select the appropriate cable.

Friction model of designed motion-decoupling mechanism

Due to using sliding chute of the decoupling mechanism to restrict the path of cables, the nonlinear friction between cables and the sliding surface is created in the process of transmission, which will cause the loss of power and energy. The problems of dead zone and hysteresis are also introduced¹⁴ that make accurate control performance worse. In order to improve the control effect, the transmission model of cable-sliding needs to be established. In this article, the static transmission model based on coulomb friction is adopted.

Theoretical modeling

To establish the transmission between single cable and sliding chute, the following hypotheses are put forward:

The masses and the moment of inertia of the cables are negligible.

The deformation of cable infinitesimal under a tensile force obeys Hooke’s law. And the deformation is within the elastic range.

The friction between cable and sliding chute is mainly due to the tension exerted on both ends of cable. The force analysis of cable infinitesimal in the decoupled unit is about to be carried out. As shown in the left of Figure 13, take the cable on right side for an example, the direction of force transferring is left-to-right along the sliding chute of the radius R . Take the intersection between sliding chute and the far left of cable as a starting point, $x = 0$ . And the cable is routed to the point at $x = L / 2$ , alone the left surface of the sliding chute. Thus, the contact point between the right surface of the sliding chute and the upper of the cable is located at $x = L / 2$ . Then, the cable is routed to the point at $x = L$ alone the right surface of sliding chute.

Figure 13.

The force analysis of the cable infinitesimal.

The force analysis of the cable infinitesimal from the starting point $[x, x + dx]$ transferring on the sliding surface is shown on the left of Figure 13. The relative sliding happens between cable and sliding surface, and the stress analysis of the cable infinitesimal is carried out

${\begin{matrix} F_{n} (x, t) = F (x, t) \sin \frac{\overset{\cdot}{θ}}{2} + F (x + \overset{\cdot}{x}, t) \sin \frac{\overset{\cdot}{θ}}{2} \\ sign (v) • F_{f} (x, t) = F (x, t) \cos \frac{\overset{\cdot}{θ}}{2} - F (x + \overset{\cdot}{x}, t) \cos \frac{\overset{\cdot}{θ}}{2} \\ \overset{\cdot}{F} (x, t) = F (x + \overset{\cdot}{x}, t) - F (x, t) \\ F_{f} (x, t) = μ F_{n} (x, t) \end{matrix}$ (1)

where $d θ$ is the wrap angle between the cable infinitesimal element and the sliding chute, $dx$ is the arc length of the cable infinitesimal element, $F (x, t), F (x + dx, t)$ are the tension of the cable infinitesimal element initiator and terminal at time t. $d F (x, t)$ is the tension difference between the initiator and terminal of the cable infinitesimal element at time t, v is the rope speed along the sliding chute, $F_{n} (x, t)$ is the normal support force of the cable from the sliding chute at the time t, $F_{f} (x, t)$ is Coulomb friction resulting from the relative motion between the cable and the sliding chute at the time t

$F (x, t) = F (0, t) \exp [- sign (v) μ • θ]$ (2)

where $θ$ is the wrap angle between the cable and the sliding chute, $μ$ is the friction coefficient, $F (0, t)$ is the tension of cable initiator of the decoupled unit at time t, and $F (x, t)$ is the cable tension at the x away from the initiator at time t. From the above equation, the friction between the cable and the sliding chute just depends on the friction coefficient $μ$ and the wrap angle $θ$ between cable and sliding chute and is irrelevant to the routing radius of the cable along the sliding chute.

Model validation

In order to validate the above friction model of the transmission between cable and sliding chute, the model verification platform is constructed as shown in Figure 9. Tension sensors 1, 2, 3, 4 are installed which are used to obtain the tension values of the decoupled unit on both sides.

Figure 14 shows the structure diagrammatic of verification experimental platform of the friction model. Four measuring units of the cable tension are installed on the both sides of the motion-decoupling module. Tension sensors 1, 3 are used to measure the tension of the tight side cable; tension sensors 2, 4 are used to measure the tension of the loose side cable.

Figure 14.

The structure diagrammatic of verification experimental platform of the friction model.

Experimental verification and analysis

After pretension loading in the cable-driven system, the rear joint is driven to do rotary motion. In the meantime, the tension data $F_{1}$ , $F_{3}$ of the tension sensors 1, 3 respectively are collected in real time. In this experiment, the system parameters are given in Table 1. Figure 15 shows the transmission characteristics of the cable input/output forces. x-axis and y-axis represent the tension values of tension sensors 1 and 3, respectively.

Table 1.

The system parameters of the decoupled cable module.

System parameters	Value
Static friction coefficient $μ$	0.13
Wrap angle $θ$ (rad)	$5 π / 3$
Sliding chute radius $r$ (m)	0.15

Figure 15.

The transmission characteristics of the cable input/output forces.

As shown in Figure 15, in the transmission process of the cable-driven system, the transmission condition of the input/output force can be divided into the following four parts and variation of $F_{1}$ and $F_{3}$ in the four parts is shown in Table 2.

Table 2.

The variation of $F_{1}$ and $F_{3}$ in the four parts.

Part	$F_{1}$	$F_{3}$
A	↗	—
B	↗	↗
C	↘	—
D	↘	↘

In the cable transmission process of Part A, the internal tension $F_{1}$ measured by tension sensor 1 is increased. The cable undergoes elongation when stressed, but the tension $F_{3}$ on the right side of the decoupled module is still remain the same. The right segment of cable is not response to the input force, which result in the dead zone of force/displacement output. In the cable transmission process of Part B, the tension $F_{3}$ varies according to the change of the system driving force. Besides, $F_{3}$ increases linearly with the increase of tension $F_{1}$ , and the linear relationship can be described approximately by the formula: $F_{3} = e^{0.68} • F_{1}$ . In the cable transmission process of Part C, the internal tension $F_{1}$ measured by tension sensor 1 is decreased, and the cable undergoes contraction. The cable segment on tension sensor 3 is not response to $F_{1}$ , which result in the dead zone of force/displacement output as well. In the cable transmission process of Part D, the tension $F_{3}$ varies according to the change of the system driving force. $F_{3}$ decreases linearly with the decrease of tension $F_{1}$ , and the linear relationship can be described approximately by the formula: $F_{3} = e^{- 0.68} • F_{1}$ .

From the above analysis, the input/output response relationship of the cable-sliding chute approximately accords with the static cable-sliding chute friction model. And the result proofs the correctness of the friction transmission model.

As shown in Figure 15, when the backend joint of the decoupled module system is repeatedly driven, there are two transitional intervals, the force/displacement dead zone: Part A and Part C. In order to realize the accurate control of the cable-driven decoupled system, it is necessary to compensate for the dead zones.

Conclusion

A novel scheme of cable routing was proposed in this article, which can achieve the motion decoupling of multi manipulators joints driven by cables. Experiments were designed to verify the performance of the decoupling mechanism, and improvements were given accordingly. The cable friction model of the decoupling mechanism was built and experiments were designed to verify its effectiveness, which provided the theoretical basis for the accurate motion control of the cable driving system.

Footnotes

Handling Editor: Xianzhi Zhang

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 study was supported by “the Fundamental Research Funds for the Central Universities” (no. NS2017053).

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