Continuum manipulators are widely used in minimally invasive surgical robot systems (MISRS) because of their flexibility and compliance, while their modelling and control are relatively difficult and complex. This paper proposes an improved hysteresis model of a notched continuum manipulator based on the classical Bouc–Wen model, which can reduce errors and increase the accuracy of the kinematic-mechanics coupled model. Then parameters are identified by the mean of genetic algorithm (GA). Hysteresis phenomenon of the mentioned manipulator is actually caused by many factors such as the hysteresis property of Hyperelastic Nitinol Alloy (HNA), the elastic deformation of tendon and the friction between the tendon and the tube. The results of both static and dynamic experiments show that the introduced hysteresis model can eliminate the positional difference between forward and reverse bending processes, and thus improve the forecast precision of deformation during motion. This model can also be used to compensate modelling errors caused by hysteresis of other similar systems.

Continuum robots, different from conventional discrete joints robots, do not contain rigid links and identifiably rotational joints, whose deformation bases on intrinsic elastic and compliant character (Robinson and Davies, 1999). Because of their small volume, light weight, flexible movement and compliant interaction, continuum manipulators are widely used in MISRS (Burgner-Kahrs et al., 2015). In the past twenty years, lots of novel and useful continuum manipulators were reported. Simaan and Taylor (2004) designed a tendon-driven continuum manipulator which contains several elastic continuum backbones and spacer disks. Webster et al. (2007) and Dupont et al. (2010) presented a concentric tube manipulator comprising several nested pre-curved tubes. Li and Du (2013) proposed a continuum manipulator which consists of multiple section set backbones composed of short links. In addition, notching different type incisions on a hyper elastic tube was also utilized to generate continuum manipulators (Du et al., 2015a; Gao et al., 2017). The modelling for continuum manipulators is a challenge in academia because of their special deformation characterises. A variety of modelling methods were developed which can be divided into two groups (Walker, 2013). Some researchers adopted overall modelling to map driving force and deformation using diverse theorems: energy method such as the principle of virtue power and the transformation of potential energy (Tatlicioglu et al., 2007; Rone and Ben-Tzvi, 2014), Cosserat rod theory (Jones et al. 2009; Rucker and Iii, 2011; Giorelli et al., 2012), constant curvature assumption (Camarillo et al., 2008; York et al., 2015; Wu et al., 2017), etc. Moreover, some researchers established a unit model first and then built an integrate deformation model considering unit kinematics and mechanics character (Du et al., 2015b; Kato et al., 2015).

However, the hysteresis of the continuum manipulator attracts less attention compared with the mechanics and kinematics. Actually, the hysteresis phenomenon widely exists in the nature which affects control accuracy or even stability of most mechanical systems seriously, so many researches are conducted on this phenomenon (Liu et al., 2014). Seyfferth et al. (1995) built a mechanical model for the harmonic drive robotic transmissions considering compliance, friction and hysteresis. Kwok et al. (2007) established a hysteresis model based on hyperbolic tangent non-linear function to improve the model precision of magnetorheological fluid dampers. Van Damme et al. (2008) proposed a Preisach based hysteresis model to solve the specific shape of the force-contraction characteristic of pneumatic muscles. Ruderman et al. (2009) designed a novel approach which combines distributed Preisach model and generalized Maxwell–Slip friction model to explore elastic robot joints with hysteresis and backlash. Xiao and Guo (2010) conducted a research on modelling and parameter identification of the hysteresis existed in the micromanipulator for human scale teleoperation system. Xu (2013) presented an approach to realizing hysteresis identification and compensation of piezoelectric actuators using least squares support vector machine based hysteresis model. Miyasaka et al. (2016) proposed a hysteresis model for longitudinally loaded cables which can be applied for control of cable driven systems such as the RAVEN II surgical robotic research platform.

Most of the kinematics modelling of continuum manipulators ignored hysteresis problems, Kang et al. (2013) demonstrated the sigmoid shape in length versus pressure curve caused by hysteresis when model pneumatically actuated continuum manipulator, but they ignored this phenomenon in order to simply the model. Kato et al. (2016) took hysteresis of a tendon-driven continuum robot for neuroendoscopy into account by adding friction coefficient into their model instead of using fitting parameters, and this research only worked out the hysteresis caused by the friction. However, the hysteresis effect of our proposed continuum manipulator has many factors, including the hysteresis property of HNA and the elastic deformation of tendon (Cruz-Hernandez and Hayward, 1998), as well as the friction and backlash happened in the tendon transmission (Swevers et al., 2000). Therefore, consideration of hysteresis problem of the notched continuum manipulator driven by the tendon during modelling is important to improve the modelling accuracy, and also the designed hysteresis model and parameter identification method in this paper is effective.

Our previous work designed and modelled a notched continuum manipulator driven by tendons (Du et al., 2015b). This paper gives a further research, which mainly focuses on solving the nonlinear position difference caused by hysteresis character of the continuum manipulator between forward and reverse bending processes under the same tendon displacement. A brief review of the overall structure, mechanics model and kinematics model of the notched continuum manipulator is introduced in Sect. 2. In Sect. 3, a theoretical analysis based on experimental results is presented, and then a hysteresis model of the continuum manipulator based on the classical Bouc–Wen model is established. Also, the unknown parameters are identified by GA in this section. A prototype is machined and the experiment platform is constructed, and then static and dynamic experiments are conducted in Sect. 4 to verify the effectiveness of the improved hysteresis model. Finally, this paper is summarized in Sect. 5.

This section reviews the previous contributions which plays a fundamental role in the subsequent research, including the overall structure, mechanics model and kinematics model of a single notched ring (SNR) unit and the integrated manipulator (Du et al., 2015b).

The schematic projection of the notched continuum manipulator is shown in Fig. 1. The overall structure is made of HNA tube on which several V-shape cuts are processed by low-speed wire cut electrical discharge machining (LS-WEDM), and two tendons are fixed at the tip of the manipulator to transmit driving force. The compliant character is guaranteed by intrinsic super-elasticity of the material. The sufficient inner cavity can also provide enough volume for tendons of other joints as well as signal wires. The continuum manipulator can be considered as an integrated structure with several SNRs. This design ensures relatively small volume, easy processing, long life and good biocompatibility. In addition, the symmetrical notches can supply bending capability in a specific direction and guarantee enough stiffness in other directions so that it improves the stability of the continuum manipulator.

Schematic projection of the notched continuum manipulator.

The proposed continuum manipulator is driven by tendons, whose total deformation is integrated by all SNRs' deformation. Since kinematics and mechanics are coupled in the proposed continuum manipulator, our previous work developed an innovative model describing the manipulator deformation from “micro” to “macro”. A SNR is presented in Fig. 2 which is divided into deformed and un-deformed regions. In the 3-D space, for each ring in the SNR, there are two symmetrical deformed arc structures with small central angle, i.e., the main deformation area concentrates on these arc structures with small central angle. Then each deformation arc is simplified into two 3-D Timoshenko beam elements with three nodes as shown in Fig. 2. Finally, the mechanics model is concentrated on the simplified Timoshenko beam elements, which is built based on Timoshenko beam theory (Du et al., 2015b).

Simplified Timoshenko beams model of a SNR.

Based on the mechanics, the relationship between the deformation of the SNR,
the deformation of continuum manipulator and the driving force can be
expressed as follows,

Besides, an equivalent kinematics model of the continuum manipulator based on mechanics model is proposed. The continuum manipulator is decomposed into several equivalent rigid links and discrete joints. The equivalent kinematics coordinate system is established as shown in Fig. 3.

When the continuum manipulator is equivalent to the series connection of
discrete joints, the forward kinematics model can be built via
Denavit–Hartenberg (D–H) method,

In addition, due to the redundancy of the continuum manipulator, it is
complicated to obtain explicit expression of angle deformation when the
distal position of the manipulator is given. The curve fitting method is
used to solve the inverse kinematics. Finally, the relationship between
driving space (displacement of tendon), joint space (deformation angle of
the manipulator) and Cartesian space (distal position) is finally
established,

Equivalent kinematics coordination system of the notched continuum manipulator.

The hysteresis characteristic of HNA and the elastic deformation of tendon generate the hysteresis characteristics of the continuum manipulator. This section concentrates on the nonlinear deformation characteristic caused by the hysteresis effect. The hysteresis effect of the system is analysed firstly, and then a hysteresis model of the manipulator is established based on the classical Bouc–Wen model. Finally, the parameters of the hysteresis model are identified.

To explore the systematic hysteresis character of the continuum manipulator, static and dynamic experiments are implemented and the experiment setup will be discussed in Sect. 4. The main factors that influence the hysteresis effect of the manipulator includes: the elasticity and backlash of tendon, the friction force between the tendon and the manipulator, and the hysteresis behaviour of HNA.

Static experimental results under discrete signal.

Figure 4 is the results of the static experiment when the continuum
manipulator is actuated at regular 0.1

A further analysis of the experimental results under 20

Hysteresis loop of the notched continuum manipulator.

Figure 6 shows the results of the dynamic experiment, in which the bending angle curve is described in the time domain. An obvious difference exists between the theoretical deformation and the actual deformation. The HNA hysteresis, friction force, tendon elasticity and other effects that generate the difference, in which the manipulator cannot reach the peak theoretical value due to the hysteresis accumulation under sinusoidal driving signal. Besides, when the manipulator is actuated continuously, the time delay happens between actual deformation and theoretical deformation.

Dynamic experimental results under sinusoidal signal.

The static and dynamic experimental results show that hysteresis phenomenon does exist in the movement of continuum manipulator. To improve the modelling precision and reduce the hysteresis disturbance, the hysteresis compensation is necessary. In this section, a hysteresis model is developed to compensate the errors in the forward and reverse bending.

The classical Bouc–Wen algorithm model uses a nonlinear differential equation with uncertain parameters to describe the systematic hysteresis character. By selecting different model parameters, the hysteresis curve of various shapes can be simulated, including viscous stage and slip stage (Smyth et al., 2002). This method has the advantages of few parameters, simple format and convenient application, which is widely used in the modelling and control of hysteresis structure in engineering field (Ismail et al., 2009).

The Bouc–Wen model is composed of a hysteretic system, a spring and a damper
in parallel as shown in Fig. 7, whose mechanical property can be expressed as
(Wen, 1976),

The Eq. (4) can be considered as the sum of linear stiffness component and
nonlinear hysteresis components. Based on the Eq. (5), once the parameters

Diagram of the Bouc–Wen model.

The classical Bouc–Wen model given above is used to describe the hysteresis
relationship between the restoring force and the displacement of the
nonlinear system, which divides the mechanical properties into the systemic
theoretical movement component

According to the Bouc–Wen model given by Eq. (5), the expression

According to the kinematics and the hysteresis models in Eqs. (6) and (7),
four unknown parameters (

This paper uses the genetic algorithm (GA) to identify the parameters of the nonlinear hysteresis model which is good at solving multivariable optimization and global optimization problem (Tavakolpour et al., 2010). The method can easily and effectively identify the unknown parameters in the hysteresis model based on the data obtained in the static experiment. To identify the parameters of the hysteresis model, two steps are necessary. Firstly, the fitness function should be defined. Secondly, the GA toolbox parameters including the fit function number of variables, initial population, generation, stall generation, and function tolerance should be defined.

Fit function: @fitnessfunc;

Number of variables: 4;

Initial population: [1000, 1000, 1000, 1000];

Generation: 1000;

Stall generation: inf;

Function tolerance:

This section explores the hysteresis model of the continuum manipulator by comparing the improved model with the experiment results, and both the static and dynamic experiments are employed to verify the hysteresis model.

The experiment platform is shown in Fig. 8. For each tendon, one side is fixed at the tip of the continuum manipulator, and the other side is rotated around and fixed on an output wheel. Two wheels are driven by one motor and are installed on the output shaft in the opposite direction. The displacement of tendons is feedback by the digital encoder integrated on the motor. The image of the continuum manipulator is taken by the GigE Industrial CCD camera produced by Point Gery Company in Canada, and MATLAB is involved in processing images. The side wall profiles (including the endpoint positions) of the continuum manipulator are extracted firstly, and then the bending angles in different positions can be calculated.

Experiment platform for the hysteresis model of the notched continuum manipulator.

In static experiment, data are collected point by point, the displacement of
tendon changes from

The hysteresis model of the continuum manipulator is obtained by putting the
identified parameter of Eq. (10) back into Eq. (7), so that an improved
hysteresis model is obtained. By using the fourth order Runge–Kutta method,
the compensated theoretical bending angle of the continuum manipulator under
the displacement

Comparison between actual and theoretical bending angle under discrete signals.

In the dynamic experiment, the driving signal is set as a 1

The results of various error indicators under different driving signals are
shown in Table 1. Based on the actual bending angels and the predicted
angles, the maximum absolute error

The improved hysteresis model has better prediction with large deformation of the continuous manipulator, while under small tension the error is relatively obvious. Generally speaking, the nonlinear hysteresis model established in this paper can simulate both the desirable tendon displacement-bending angle and the undesirable hysteresis loop. The maximum relative error is about 3 %, which is mainly due to the asymmetrical hysteresis value generated by the assembly error of two driving tendons.

Comparison between actual and theoretical bending angle under sinusoidal signals.

Errors analysis of the improved hysteresis model.

This paper focuses on the hysteresis phenomena of the continuum manipulator which is caused by driving backlash, system friction, tendon elasticity and HNA hysteresis. A hysteresis model is designed based on the Bouc–Wen model, and GA method is utilized to identify the unknown parameters. The static and dynamic experiments show that the improved hysteresis model can predict the actual deformation effectively. The contribution of this paper is building an improved hysteresis model which can describe the positional difference between forward and reverse bending process under the same tendon displacement. This method can also be applied to other similar systems to compensate predicted errors caused by hysteresis. In the future, to increase the hysteresis accuracy, an asymmetric hysteresis model will be used to compensate the errors caused by hysteresis.

Data can be made available upon reasonable request. Please contact Wenlong Yang (yangwl@hit.edu.cn).

The authors declare that they have no conflict of interest.

This research is supported in part by the National Nature Science Foundation of China under Grant No. 61673139, the State Key Lab Self-planned Project under Grant No. SKLRS201701A, and the Programme of Introducing Talents of Discipline to Universities under Grant No. B07018. Edited by: Chin-Hsing Kuo Reviewed by: two anonymous referees