Keywords

1 Introduction

Traditional rigid robots consist of discrete joints as well as rigid links and are widely employed in industrial production. However, when it comes to performing tasks in unstructured environment, these rigid robots are incompetent due to their limited flexibility and compliance. To extend robots’ working environment, researchers, inspired by continuum structures in nature, such as elephant trunks, lizard tongues, and octopus tentacles [1], are paying more and more attention to continuum robots. This kind of robots have structural compliance, making them have promising potential to operate in unstructured, irregular and highly constrained environment, including minimally invasive surgery (MIS) [2] and rescue operations in collapsed buildings [3]. In conclusion, continuum robots with have attracted more and more attention as they are more suitable for operations requiring for high flexibility and compliance.

Since the concept of continuum robot was put forward by Robinson and Davies [4], various actuators have been proposed to realize the movement of continuum robots. The thermal shape memory alloys (SMAs) actuators [5, 6] are easy to control and can generate large contractile stress, but their temperature control characteristic slows down their motion frequency and cycle time. The electro-active polymers (EAPs) [7] have the advantages of relatively high efficiency and light weight. However, extremely high voltage is often required and their output force and displacement are usually limited. Fluid-driven actuators are also widely used in continuum robots. These fluid-driven actuators, either pneumatically [8] or hydraulically [9], can achieve large deformation, displacement and strength. Nevertheless, because of the leakage phenomenon and the compressibility of fluid, it is difficult for these kinds of actuators to realize high positioning accuracy. Cable-driven continuum robots [10], have high resolution when transmitting contracting movement, but low stiffness when taking compressive and lateral forces. The elastic Ni-Ti rods [11] improve the overall stiffness of the robot, however, at the cost of lower compliance in comparison with cable-driven robot. Through above analysis, it is found that every actuating method has its advantages and disadvantages, which means that it is difficult to balance the performance on workspace, precision, strengths, stiffness as well as compliance.

To deal with this problem, some researchers tried to change the stiffness of robots, making them maintain compliant and flexible when navigating through complicated environment and become stiff and accurate when manipulating targets. Brown et al. [12] delineated a universal stiffness-variable robotic gripper based on the jamming of granular material. Li et al. [13] proposed the principle of passive particle jamming, making the vacuum power not necessary. And the variable stiffness of continuum robot was also achieved by Shiva et al. [14] utilizing inherently antagonistic actuation scheme. Apart from that, smart materials have been applied to realize variable stiffness as well. Taniguchi et al. [15] produced two types of soft magnetic materials and developed new soft actuators with magnetic intelligent fluids, of which phase change can lead to stiffness change. Finally, another effective method to achieve comprehensive performance of continuum robots is hybrid actuation. Kang et al. [16] arranged additional steel cables inside pneumatic muscles. The pneumatic muscles are designed to realize preliminary positioning, while the embedded cables are responsible for fine adjustment of position. Although this hybrid continuum robot is able to improve positioning accuracy and, at the same time, maintain large displacement capability, considering the fact that cables cannot bear compressive and lateral forces, the stiffness and the precision it improves is actually limited. Similar hybrid actuation method is used in this paper. The pneumatic muscles are kept to achieve large-scale flexible movement while elastic rods, whose stiffness is higher in the axial and lateral directions compared with cables, are used to realize high positioning accuracy. Such type of hybrid actuation method is promising to achieve comprehensive performance, such as high flexibility, large workspace, high positioning precision and large strengths, into one continuum robot.

The working principles as well as actuation and control methods of continuum robots are quite different from conventional rigid robots, which means that the existing kinematics, statics and dynamics models built based on rigid bodies are no longer applicable for continuum robots. To investigate the motion and control of such robots, quantities of researchers have been trying to build the mathematical models of continuum robots. Jones and Walker [17] presented the kinematic models of continuum robots based on the assumption of piece constant curvature (PCC), which are widely recognized by related researchers. Xu and Simaan [18] used elliptic integrals to derive an analytic formulation for kinematics and statics of continuum robots. Renda et al. [19] established a dynamic model according to discrete Cosserat beam theory. Kang et al. [20] developed a dynamic model for a multi-segmented pneumatic continuum robot based on the Newton-Euler equation. They also presented a static model for a hybrid continuum robot using an improved Kirchhoff rod theory [21]. All in all, these models considered the continuum actuator as one body structure. However, the actuators of hybrid continuum robot mentioned in [21] include both pneumatic artificial muscles and elastic rods, which are actually two structures arranged in parallel. To handle this problem, this paper builds astatic finite element model of such hybrid continuum robot to describe the individual structure of pneumatic muscles and rods in one actuator and identify the performances of each actuation method.

This paper is organized as follows. Firstly, the prototype is briefly introduced in Sect. 2. Subsequently, the static finite element model is represented in Sect. 3. And the model is validated in Sect. 4. Finally, conclusions are drawn in Sect. 5.

2 Prototype

The prototype of hybrid continuum robot is presented in this section. As shown in Fig. 1, the prototype consists of a continuum arm and a driving unit.

Fig. 1.
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The overall structure of the prototype

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The continuum arm is composed of three parallel hybrid actuators and three disks, whose function is to keep the hybrid actuators together and parallel to each other. The hybrid actuators are elastic rods embedded artificial pneumatic muscles (eREMs), whose structure is shown in Fig. 2. The eREM consists of an elastic rod, a nylon braided mesh, a rubber tube, a top connector, a bottom connector and five blocks. The function of the top connector is to fix the elastic rod with the top of the pneumatic muscle using aluminum collet and hot melt adhesive. The bottom connector, connecting the airway, pneumatic muscle and elastic rod, is hollow to allow the elastic rod to pass through. And a rubber seal is placed to prevent air leaking. Equally distributed along the elastic rod, the five blocks are used to make sure the elastic rod will always be coaxial with the central axis of the pneumatic muscle.

Fig. 2.
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The overview and exploded view of the eREM

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Moreover, the driving unit below the continuum arm includes linear motors, sensing system and mode switching mechanisms. The detailed design of the continuum robot can be found in our previous work [21].

3 The Static Finite Element Model

3.1 Structure Simplification

The simplified structure of the hybrid continuum robot is demonstrated in Fig. 3. Three parallel hybrid actuators and one top disk is kept. To make the structure of the robot easy to simulate, the kept hybrid actuators only consist of elastic rods, nylon braided meshes and rubber tubes in the finite element model. Besides, the elastic rods made of Ni-Ti alloy are fixed to the top of the pneumatic muscles while going through and sliding along the bottom of the pneumatic muscles. And the bottom of three pneumatic muscles is fixed.

Fig. 3.
figure 3

The basic structure of the hybrid continuum robot

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3.2 Working Principle

Large-Scale Flexible Movement for Preliminary Positioning.

The large-scale flexible movement is achieved pneumatically and the elastic rods act in a passive mode. The excellent performance, including large output force, high speed, big stroke and flexibility, of pneumatic actuators are fully utilized for large-scale movement. As shown in Fig. 3, the elasticity of the inner rubber tube is large while the elasticity of the outer braided meshes, which are made of nylon, is extremely small. With the initial angle θ between the nylon fiber and the eREM axis greater than 54°44′, the outer nylon braided meshes can limit the radial expansion of the eREMs, making the eREMs elongated in the axial direction when inflated. Since there are three parallel eREMs, when eREMs are elongated with different pressures, Pi (i means the ith eREMs), the robot will bend. Moreover, the robot will elongate along its axis if all three eREMs are pressurized in the same degree at the same time (P1 = P2 = P3). Besides, twisting will not happen and is not considered for the ideal robot model. Assuming P is a vector whose elements are pressures of different eREMs, as in (1), different P will result in different curved configurations of the continuum robot.

$$ \varvec{P}\, = \,\left[ {P_{1} \;P_{2} \;P_{3} } \right] $$
(1)

Fine Adjustment of Position.

After the preliminary positioning is realized and the end of the continuum arm has arrived near the target working point, the elastic rods work in an active mode and drive the robot to achieve fine adjustment of position with high positioning accuracy. With the bottom of three eREMs fixed, the robot will also bend when the elastic rods are pushed or pulled with different displacement. The length of the elastic rods between the top and bottom of the eREMs is called driving length, li (i means the ith eREMs). Assuming L is a vector whose elements are driving length of different rods, as in (2), different L will also result in different curved configurations of the continuum robot. But considering the fact that the rods are only used for fine adjustment of position, the variation range of li is limited to only 5 mm for the continuum robot mentioned in this paper.

$$ \varvec{L}\, = \,\left[ {l_{1} \;l_{2} \;l_{3} } \right] $$
(2)

3.3 Parameters Setting

Material Parameters.

The inner rubber tubes of eREMs are made of rubber with large elasticity. In this model, the rubber is in a state of small deformation under normal working conditions. In order to simplify the analysis, the non-linear elastic material is approximated as liner material. While the outer braided meshes are made of nylon with extremely small elasticity. Besides, the elastic rods are made of Ni-Ti alloy. The material of the top disk, which can be fabricated by 3D printer, is UV Curable Resin. Therefore, the common parameters of these materials, shown in Table 1, are set in the finite element model.

Table 1. Material parameters
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Geometric Parameters.

The geometric parameters of the rubber tubes, braided meshes, elastic rods and disk are demonstrated in Table 2.

Table 2. Geometric parameters
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3.4 Modeling and Analysis Method

The static finite element model is built in the commercial software Abaqus 6.14. Three processes are simulated in the finite element model. Firstly, the large-scale flexible movement are simulated with different pressure combinations set in the eREMs. Secondly, the same pressure combinations of large-scale flexible movement are kept and at the same time elastic rods are driven, to simulate the process of the fine adjustment after preliminary positioning. Thirdly, an external force is applied to the end of the robot arm to learn the different influence on the configurations of the robot actuated pneumatically or by the elastic rods. Besides, the gravity and frictions are all considered in the model. The braided meshes limits the radical expansion of rubber tubes, so that rubber tubes will not expand like the inflated balloon but extend along the radical.

To record the configurations of the continuum arm, an absolute coordinate system, defined as reference coordinate system, is built as shown in Fig. 4. Six sampling points along the continuum arm are set in the model so that their coordinates can be obtained. Besides the mechanical properties of the model, such as stress and strain, can also be acquired in Abaqus.

Fig. 4.
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The reference coordinate system and the sampling points set in the model

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Figure 5(a) is the analysis result of large-scale flexible movement with the vector P set as [100 kPa, 50 kPa, 50 kPa] while Fig. 5(b) is the analysis result of fine adjustment with the vector L set as [262 mm, 260 mm, 260 mm] and vector P same as that in Fig. 5(a). The simulated results of Fig. 5(a) and (b) are similar and difficult to distinguish by the naked eyes, since the fine adjustment actuated by the rods is with 5 mm. Moreover, Figs. 6 and 7 are analysis results of the continuum robots when an external force F = 1.5 N is applied to the end of the robot under the circumstances when the elastic rods are in passive and active modes. A qualitative conclusion can be made that the hybrid actuation is beneficial to the repeatability of the robot. This because the structure stiffness of the robot improves in the locked mode.

Fig. 5.
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The analysis result of large-scale flexible movement and fine adjustment

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Fig. 6.
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The analysis result of large-scale flexible movement and fine adjustment when an external force pointing the negative direction of x axis is applied

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Fig. 7.
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The analysis result of large-scale flexible movement and fine adjustment when an external force pointing the positive direction of x axis is applied

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4 Model Validation

In order to verify the accuracy and reliability of the finite element model, multiple sets of simulated configurations and experimental configurations of above prototype under the same driving conditions were compared.

4.1 Experimental Setup

To get the configurations of the prototype, it is necessary to measure the coordinates of sampling points in Fig. 4, so an electromagnetic sensing system was built. The electromagnetic sensing system (3D Guidance tranSTAR, manufactured by Ascension Technology Corporation), whose measurement error is within 0.5 mm, consists of a magnetic field emitter, an electronic converter and an electromagnetic sensor. The prototype was placed in the magnetic field emitted by the magnetic field emitter. With the aid of the electromagnetic position sensor, the coordinates of the sampling points in the magnetic field coordinate system whose origin is built on the magnetic field emitter could be accurately measured. Then a coordinate transformation from the magnetic field coordinate system to the reference coordinate system shown in Fig. 4 was implemented, for the convenience of comparing the simulated configurations with the experimental configurations of the prototype. The experimental setup and corresponding simulated results are shown in Fig. 8.

Fig. 8.
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Experimental setup and corresponding simulated results

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4.2 Result Comparison of Simulation and Experiment

The model was validated in totally three cases.

Case (a).

The continuum robot in case (a) is in the large-scale movement mode and the P is set as [200 kPa, 50 kPa, 50 kPa]. The comparing result is shown in Fig. 9(a).

Fig. 9.
figure 9

The comparing results of case (a), (b) and (c)

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Case (b).

The continuum robot in case (b) is also in the large-scale movement mode and the P is also set as [200 kPa, 50 kPa, 50 kPa]. Besides, a 1 N external force pointing to the positive direction of x-axis is then applied to the top disk of the robot. The comparing result is shown in Fig. 9(b).

Case (c).

The continuum robot in case (c) firstly move in the large-scale movement mode (the P is also set as [200 kPa, 50 kPa, 50 kPa]) and then the system is switched to the fine adjustment mode. Besides, a 1 N external force in the same direction as in case (b) is also applied to the top disk of the robot. The comparing result is shown in Fig. 9(c).

Due to the ideal finite element model is based on simplified structure and ignores the effect of twisting and torsion, there must be deviations of the sampling points coordinates between the simulated configurations and the experimental configurations. But the errors are within 10 mm (4.76% of the initial length of the actuator), which means that the finite element model built in this paper can predict the configurations of the hybrid continuum robots accurately.

Moreover, according to the comparing results in Fig. 8, it can be seen that the stiffness of the continuum robot in case (c) is higher than that in case (b) because the elastic rods in case (c) are in active mode. These results prove that the performance and function of each actuation method should be identified respectively. Many uncertainties remain such as friction, leakage, hysteresis and measurement errors.

5 Conclusion

Considering the fact that various actuation methods all have their advantages and disadvantages, it is important, but difficult, to achieve the comprehensive performance of continuum robots. The robot prototype discussed in this paper, which could either be driven pneumatically or by elastic rods, took advantage of a hybrid actuation method to deal with the performance problem. As the actuators of the robot contain two different structures, a dynamic finite element model was built to predict the configurations of the continuum robot. The proposed finite element model could identify the two structures of the actuators independently, rather than regarding them as one single structure, used in most existing models. The comparison between the simulated and experimental results verified the reliability of the finite element model and the necessity to model the two actuating structures separately.

Finally, the division of the mesh, can also have an impact on the stiffness and performance of the robot. Future work will investigate the exact influence of meshing of parts. The hybrid actuation indeed improved the positioning precision, but at the expense of light weight of the whole system. It can be optimized continuously in the future.