1.Introduction

In recent years,robots have been used to perform complex tasks in many places.Applications include space exploration[1],disaster rescue[2],minesweeping[3],and more.Because of the harsh and dangerous work environments,robots inevitably face the risk of damage.Therefore,many researchers have begun to investigate robot fault-tolerant control[4-8].Kawata et al.[9]proposed a fault-tolerant adaptive gait-generation method for a multilimbed robot,Mavrovouniotis et al.[10]developed a faulttolerant sliding mode control,and Zhang et al.[11]conducted research on reconstruction fault-tolerant control for underwater vehicles.Studies related to fault-tolerant control ensure the stability of robot applications.

With the development of autonomous underwater vehicles(AUVs),more and more researchers have focused on enhancing the maneuverability of underwater vehicles.After a long evolutionary period, fish have acquired sufficient flexibility to swim underwater.Inspired by this intriguing trait,Triantafyllou et al.[12] first created the biomimetic robotic fish in 1994.As a new type of AUV,robotic fish attract considerable attention from researchers due to their superior mobility[13-19].Zhong et al.[20]designed a novel robotic fish with a wire-driven active body,and Romero et al.[21]proposed a cost-effective intelligent robotic fish.Compared with traditional AUVs,robotic fish have advantages such as low noise,harmlessness to underwater organisms,and more[22].With these benefits,robotic fish have the potential to be widely used to complete complex underwater missions[23].

Research into the fault-tolerant control of AUVs is mainly divided into two categories:①sensor fault-tolerant control methods such as those discussed in Ref.[24],and②fault-tolerant control methods for the motion execution of AUVs.Ahmadzadeh et al.[25]proposed a fault-tolerant control method for AUVs in order to overcome thruster failures,and Rauber et al.[26]presented a faulttolerant control strategy for the thruster of an AUV.Existing studies on underwater fault-tolerant control seldom deal with robotic fish.However,the high maneuverability behavior exhibited by a robotic fish,such as a C-turn[27],places heavy loads on the actuators and increases the risk of damage to the robotic fish.At the same time,if the configuration of the bio-inspired central pattern generator(CPG)—which is widely applied to generate rhythmic movements for a robot with multiple degrees of freedom[28,29]—is not satisfactory,the joints of the robotic fish may become stuck during motion pattern transition.Since the fish body waves need the coordination of all the joints of the robotic fish,damage to any part of it will strongly in fluence the motion.In addition,since robotic fish operate in underwater environments,once the fault occurs,it is difficult for a robotic fish to be repaired in time.As a type of underwater vehicle,robotic fish must be autonomous and reliable.The fault-tolerant control ability of the control system is an important indicator to ensure the stable operation of a robotic fish in a faulty condition.Thus,it is necessary to investigate fault-tolerant control for robotic fish.

The main objective of this paper is to propose a fault-torrent control method to deal with the problem of a robotic fish with a stuck tail joint.In order to ensure stable motion performance for a robotic fish,a CPG-governed feedback controller was developed.We conducted dynamic modeling for a fault situation in which one of the tail joints of the robotic fish is stuck[30].Based on the modeling,we developed feedforward compensation in order to speed up convergence.Together,these two parts form the feedforwardfeedback fault-tolerant control system.

Simulations and experiments were performed in order to verify the effectiveness of the proposed control system.Existing studies on underwater fault-tolerant control are mainly aimed at underwater vehicles using a traditional propeller.Unlike previous works,this paper focuses on the problem of motion correction for a faulty robotic fish whose propulsion is generated by a multi-jointed tail.Based on an analysis of the unique motion and dynamic models of the robotic fish,a fault-tolerant control method for a multi-jointed robotic fish is proposed for the first time.Our method improves the robustness of the control system and thus lays a foundation for the practical application of robotic fish in complex environments.

The rest of this paper is organized as follows:The mechanical structure of the robotic fish is introduced in Section 2.Section 3 describes the feedback controller based on the CPG model and the feedforward compensation based on the dynamic model,and then discusses the development of the overall control system.The experiments and corresponding results are described in Section 4.Finally,Section 5 provides conclusions and looks forward to future work.

由表10可知，以上39个处理组合中单株平均合格插穗数在14-20个之间，其中处理A5B4(株行距50cm×100cm，定芽数4)最高为20个。

2.Overview of the multi-jointed robotic fish

The controlled object used in this study is a self-propelled multi-jointed robotic fish[31].Fig.1 shows the mechanical structure of the robotic fish,which follows the structure of Esox lucius.The robotic fish has a self-propulsive body with a four-joint tail and a caudal fin.The four tail joints of the robotic fish are driven by servomotors.In addition,a rigid head with two movable pectoral fins is designed to meet the requirements of threedimensional motion.In this paper,we only discuss the tolerance of a robotic fish moving in a two-dimensional plane,so the pectoral fins are always kept still.The controller of the robotic fish is embedded in its head.With the coordination of the servomotors,the robotic fish has a high peak steering speed of 670°per second and an average steering speed of 460°per second.If the tail joint of the robotic fish gets stuck,the motion direction of the robotic fish deviates severely.To determine the deviation,an inertial measurement unit(IMU)is installed in the head of the robotic fish to measure the yaw angle.

Fig.1.Mechanical structure of the self-propelled multi-jointed robotic fish.

where T=2π/w is the period and u is variable.Once the range of b is limited,the value of b c can be obtained through optimization.

3.Fault-tolerant controller design

This section proposes a fault-tolerant control method for the robotic fish.Specifically,a feedback controller based on CPG and a feedforward compensator based on a mathematical model are designed.

42例血清CRP水平升高患者中28例(66.7%)CA19-9升高，34例(81.0%)CEA升高；52例CRP正常患者中26例(50.0%)CA19-9升高，35例(67.3%)CEA升高。血清CRP水平升高组与正常组患者的CA19-9、CEA水平的差异无统计学意义(χ2=2.640，P=0.104；χ2=3.622，P=0.137)。

3.1.Feedback controller based on CPG

In general,there are two types of motion control methods for robotic fish: fish body waves curve fitting and the CPG-based method.The latter method was adopted for the robotic fish in this paper.Vertebrates can generate rhythmic signals without a central nerve through biological CPG networks.Inspired by this,the CPG model can generate control signals through the interaction of neuron oscillators in order to control the motion of a robotic fish[32].The CPG model has the advantages of strong robustness and smoothness in motion mode switching.

自二十世纪九十年代以来，计算机技术与通信技术飞速发展，智能手机、平板电脑、4G时代的普及，社会开始进入了“互联网+”时代。2013年4月，在南昌大学召开了第四届全国数字校园建设与创新发展高峰论坛，会议主题为“互联网+”时代的教学设计与教学环境，着重介绍了“互联网+”时代下翻转课堂。

The control method used in this paper is a Hopf oscillator-based CPG model with phase differences[33].The CPG model consists of multiple neuron oscillators,as shown in Fig.3.

The details of the CPG model are given below[31]:

Fig.2.Appearance of the robotic fish with color markings.

Fig.3.Topological structure of the CPG model.

where n represents the n th oscillator （n=1，...，N）and N denotes the number of the neurons in the CPG network;xn and yn represent the state variables of the n th oscillator,and˙xn and˙yn represent the derivative of xn and yn,respectively;wn and rn denote the natural frequency and amplitude of the n th oscillator,respectively;φn represents the phase difference between adjacent neurons;bn denotes the deflection factor of the n th oscillator;h1 and h2 represent the coupling coefficients;zn is the output signal of the n th oscillator;and cn is a constant coefficient.

Through appropriate adjustment of the CPG parameters,the motions of the robotic fish can be controlled.The swimming speed can be regulated by changing wn and rn,and backward swimming can be realized by settingφn.wn=w and rn=r were employed for all the oscillators in this paper,andφwas set appropriately according to Ref.[33].When bi is adjusted,the CPG model generates an asymmetric bias output signal to change the swimming direction of the robotic fish.As mentioned previously,if the tail joint gets stuck,this fault will affect the yaw of the robotic fish.Therefore,it is feasible to control bi using the yaw angle as a feedback,to ensure that the robotic fish can maintain good motion performance in the event of faults.As the head of the robotic fish always swings when swimming because of its unique motion method,it is necessary for the determined yaw angle to be properly filtered.Based on the above analysis,a proportional integral(PI)feedback controller was designed as follows:

where t denotes the moment,bi（t）denotes the bias of the i th oscillator of the robotic fish at time t,fγ（t）represents the filtering result of the yaw angle at time t,is the target yaw,e（t）denotes the PI controller input at time t,T represents the threshold and is set as T=0.17,andand are the feedback gains of the i th joint,is the proportional controller coefficient andis the integral controller coefficient.

For ,it can be calculated from the following:

In our previous work[30],a data-driven dynamic model was built for the robotic fish in that study.Using the same model as the real robotic fish,we can obtain the output of the feedback control through a simulation,as shown in Fig.5.In the simulation in the current paper,the third joint of the tail of the robotic fish is set to be stuck atπ/6,and the target yaw is set to 0.Fig.5(a)shows the straight swimming behavior of the robotic fish in the event of a stuck tail joint.When the fault occurs,the swimming direction of the robotic fish is greatly affected.The feedback control system can effectively reduce the impact of the stuck joint on the yaw,and eventually causes the output yaw angle to converge to the target yaw angle,as shown in Fig.5(b).

3.2.Feedforward compensator based on dynamic analysis

The feedback controller proposed in Section 3.1 still has many problems,including a long convergence time,which is due to the delay caused by the filter.To tackle this problem,this section proposes a feedforward compensator based on the dynamic model.

①采用焦虑自评量表（SAS）、抑郁自评量表（SDS）评价负性情绪。②采用Karnofsky功能状态评价生活质量。

作出了项目划分，工程按三级（单位、分部、单元）阶梯式地分解成框架构造，施工质量从单元工程逐步评定到分部工程，再由分部工程逐步评定到单位工程，最后得出整个项目工程的施工质量，评定工作的路子就畅通了。这样有利于从宏观上进行项目评定的总体规划，不至于在分期实施过程中从低到高评定时出现级别和归类上的混乱。

According to Ref.[32],the dynamic model of the robotic fish is specified as follows:

近日（5日），拱北海关下属闸口海关在拱北口岸查获了一名澳门女子携带并身藏4瓶拉图酒庄出产的2004年份葡萄酒入境，企图蒙混过关入境被识破。据了解，该葡萄酒在中国内地单瓶售价5000元以上。目前，该案正由海关缉私部门跟进调查处理。

（1）本声明中所涉及的文稿均指原始研究的报告或尽管2篇文稿在文字的表达和讨论的叙述上可能存在某些不同之处，但这些文稿的主要数据和图表是相同的。所指文稿不包括重要会议的纪要、学位论文、疾病的诊断标准和防治指南、有关组织达成的共识性文件、新闻报道类文稿以及在一种刊物发表过摘要或初步报道而将全文投向另一种期刊的文稿。上述各类文稿如作者要重复投稿，应向有关期刊编辑部作出说明。

Fig.4.Block diagram of the feedback control system.

Fig.5.Simulation of the system output when one of the tail joints is stuck.(a)System without control;(b)system with feedback controller.

whereis the velocity vector of the robotic fish head.M represents the whole inertial matrix of the robotic fish in the head coordinate system,Γ is the（6×1）force vector of the robotic fish head,i H0=iHi-1 i-1Hi-2...1H0 and iHi-1 denote the transformation matrix,Mi denotes the inertial matrix of the i th joint,f w，i denotes the Coriolis force and hydrodynamic force on the i th link,and f r is the remaining part of the head force.

Since the driving force of the robotic fish swimming in the water is mainly generated by the hydrodynamic force,the in fluence of f r can be ignored in calculations.In this study,the robotic fish swims in a two-dimensional plane,so it is feasible to reduce the yaw effect caused by the fault by controlling the torque of the head.The sixth component of Γ represents the torque of the head.Therefore,the sixth line of Γ is extracted as follows:

where （·）k:represents the k th line of matrix ·and Γ6 denotes the sixth component of Γ.

The swing of the head of the robotic fish is caused by the periodic signal output of the CPG.The swing period is the same as that of the CPG oscillator.Therefore,a mean filter with a threshold is designed to deal with the effect of the yaw swing.The window time of the mean filter is set to the period of the CPG oscillator.When the parameters of the CPG model change,it will take some time for its outputs to reach a steady state.Therefore,the frequency of the change control will affect the stability of the system.In order to improve the stability of the system and reduce the burden on the servo,a threshold near the target yaw angle is set.A corresponding block diagram of the feedback control system is illustrated in Fig.4.

where θi represents the angle of the i th joint and li represents the length of the i th link.

In a more explicit form,the following equations are introduced:

f w，i can be derived according to the following equation:

whereσi represents the total Coriolis force received by the i th link,and f d，i denotes the resistance of the i th link.

According to Ref.[30],the following equation can be obtained:

where V i is a（6×1）vector representing the velocity of the i th joint.Specifically,the first three dimensions of V i form the linear velocity vectors,the last three dimensions form the angular velocity vectors.ωi= （Vi）6:,Vx，i= （Vi）1:,and Vy，i= （Vi）2:;c1，i and c2，i are constants;Mi denotes the mass matrix of the i th link;and r is a variable.

By simplifying Mi to a diagonal matrix,we obtain the following:

在与急救中心联系时，要向医生告知孩子误食了什么，误食了多少，多久之前误食的。在送往医院时，要带上剩余物品及其包装，如果确实不知道孩子误食了什么，在家中有呕吐或者催吐的话，应将孩子的呕吐物一起带往医院，以便医生了解情况及时采取有效的救护措施。

where Mi，1 and Mi，2 are constants.

深圳市艾力农生态发展有限公司董事长助理、副总经理杨明波在致辞中表示，当前我国固体氮肥的使用不当以及生产工艺水平较低，造成了肥料利用率低下和资源的严重浪费，也对土壤、水资源等生态环境造成了污染和破坏，同时农民及农资经销商收益较低。针对这些问题，深圳市艾力农生态发展有限公司联合美国肥必施公司共同推出了液体缓释氮肥产品“艾力素”，在符合“减肥增效”政策要求的同时，作物应用效果良好，且提高了农民和经销商的收益，有力推动了氮肥的转型升级发展。

The complete controller is proposed as follows:

By substituting Eq.(12)into Eq.(11),Γ6 can be expressed as Γ6（t，β，b0，f）.

V i can be calculated by V0 fromThe value of Γ6 at each moment can be obtained iteratively by setting the initial value of V0 to 06×1.

A sinusoidal wave approximation can be used for the signal produced by the CPG model.For a situation in which one of the joints is stuck,the wave approximation can be expressed as follows:

where b0 is the compensation bias,f∈ ｛1 ，...，n｝marks the location of the stuck joint,and β denotes the angle of the faulty joint.

It then follows that

As previously analyzed,in order to minimize the impact of the yaw caused by the fault,a feedforward compensator was designed as follows:

那么，我们应该如何选购对人体无害的樟脑丸呢？1.闻气味。天然樟脑丸有樟脑的特殊香味，而人工合成的樟脑丸则是刺鼻的味道。2.看成分表进行初步判断。天然樟脑丸会在成分中直接标明樟脑的含量，且价格较高；人工合成的樟脑丸则能在成分表中看到对二氯苯的含量，价格相对便宜。3.将樟脑球放入水中，可漂浮起来且不易融化的是天然樟脑丸，而人工合成的樟脑丸会沉于水底，且会逐渐融化。

Fig.2 shows the appearance of the robotic fish.The whole robotic fish is covered by a waterproof skin made of an emulsion.Above the skin of the head,red and yellow color markers made of tape are placed on the outside of the rigid shell in order to help the measuring device track the head position and the posture of the robotic fish during its movements.

In order to obtain a concise form,we introduce the following denotations:

A block diagram of the fault-tolerant control system is provided in Fig.6,and Fig.7(a)shows the simulated output of the system with a feedforward compensator.The feedforward compensator can adequately correct the effect of the fault.Fig.7(b)shows the simulated output of a system with a feedforward-feedback controller,which permits eventual convergence to the target yaw with a small overshot and short convergence time when the fault occurs.The system with a feedforward-feedback controller has a shorter convergence time than the system with a feedback controller.

4.Experiments and results

To verify the effectiveness of the fault-tolerant control of the robotic fish,a motion measurement system with a global vision camera was employed[34].The global vision camera was set 190 cm above the center of the experimental setup,which was 500 cm long,400 cm wide,and 120 cm deep.Based on the image data returned by the camera,the host computer of the recording system provides the current position and velocity by identifying the colored marks on the head of the robotic fish.

Fig.6.Block diagram of the feedforward-feedback fault-tolerant control system.

Fig.7.Simulated output of the system with a feedforward compensator in a situation with a stuck tail joint.(a)System with feedforward compensator;(b)system with feedforward-feedback controller.

During this experiment,the third tail joint of the robotic fish was fixed atπ/9.Fig.8 shows the straight swimming performance of the robotic fish without fault-tolerant control.When one of the tail joints of the robotic fish was stuck,the swimming direction of the robotic fish was extremely affected;it was difficult for the robotic fish to swim normally.

Fig.9 shows the swimming performance of the robotic fish with the feedback controller.As shown in Fig.9(a),the feedback controller gave the robotic fish a satisfactory motion performance in the faulty condition.The filtered yaw error during the motion is shown in Fig.9(b).It can be seen that the system was able to stably converge to the target yaw angle.However,the system still had the disadvantage of a long convergence time.The robotic fish took nearly 7 s to adjust the yaw to the threshold.A lateral displacement of 53.4 cm occurred during the posture adjustment.The main reason for this displacement was that the filtering window for the yaw angle must be set in order to obtain the correct motion direction,by at least one period of CPG,which results in a large delay to the system and which extends the control period.

The motion performance of the robotic fish with the feedforward-feedback controller is shown in Fig.10.The parameters of the feedforward compensator were set according to Ref.[19].As shown in Fig.10(a),the feedforward compensator gave the robotic fish a smoother motion curve than the simulation with the single feedback controller.In this case,the lateral displacement of the robotic fish was limited to 24.7 cm.Fig.10(b)shows the filtered yaw error during the motion of the robotic fish.Due to the in fluence of the feedforward compensator,the robotic fish was able to quickly adapt to the fault when it occurred,and the convergence time of the system was reduced to approximately 1 s.Compared with the system with the feedback controller alone,the system with the feedforward compensator had better fault tolerance.Fig.11 provides a sequence of screenshots from a video of the robotic fish swimming in the faulty condition.

拆卸后是否重新使用塑性区螺栓，可按维修理手册中的指示来定。如果GM车规定是更换，而TOYOTA车是测量螺栓直径的收缩值来决定。例如,图7(a)是TOYOTA车2ZR-FE发动机连杆螺栓，标准直径为7.2～7.3mm，极限最小直径为7.0mm。在测量区域中使用游标卡尺测量，如果测量结果小于7.0mm，则须更换螺栓。

Fig.8.Motion of the faulty robotic fish without fault-tolerant control.(a)Initial posture;(b)swimming path.

Fig.9.Motion performance of the faulty robotic fish with the feedback controller.(a)Swimming path;(b) filtered yaw error.

Fig.10.Motion performance of the faulty robotic fish with the feedforward-feedback controller.(a)Swimming path;(b) filtered yaw error.

Fig.11.A sequence of screenshots from a video of the robotic fish swimming with a stuck tail joint.

This experiment demonstrates that the swimming performance of the robotic fish with fault-tolerant control is satisfactory when the third joint is damaged.To further validate this fault-tolerant control method for a situation in which a different joint fails,we carried out an experiment in which the second joint was set as the faulty joint.The corresponding comparative experiment is illustrated in Fig.12.As can be observed,when the fault location changed,the fault-tolerant control method was still able to ensure that the robotic fish achieved good swimming performance.To be specific,the closer the fault location is to the caudal fin,the more severe the loss of the swimming speed of the robotic fish is.This is due to the fact that the wetted surface of the caudal fin is the main source of hydrodynamic force.Also,the closer the fault is to the head,the more serious the effect of the fault is on the swimming direction of the robotic fish.

Fig.12.A comparison experiment in which the second joint was set as the faulty joint.(a)Without control;(b)fault-tolerant control.

The experimental results show that the fault-tolerant control method proposed in this paper effectively improves the performance of the robotic fish in a faulty condition.The deflection in the direction of motion that is caused by the stuck tail joint can be effectively counteracted by the feedforward-feedback controller.With the feedback controller,the control system eventually converges to the target yaw with almost no static error.Through the comparative experiment,it was found that the feedforward compensator solves the delay problem of the system.The prior knowledge obtained through the dynamic analysis allows the other joints of the robotic fish to rapidly adapt to the impact of the stuck joint.As the convergence time is shortened,the lateral displacement produced by the posture adjustment is reduced.Considering that the head of the robotic fish swings on the plane of the yaw angle,the swimming performance in this state indicates that the proposed fault-tolerant control method has some stability and anti-interference ability.

5.Conclusion and future work

In this paper,we proposed a fault-tolerant control method for a self-propelled multi-jointed robotic fish with a stuck tail joint.Based on the CPG model,the feedback controller was designed by analyzing the in fluence of the fault.In order to obtain better performance,a feedforward compensator was designed based on dynamic analysis.More specifically,controlling the specific parameters of the CPG makes the robotic fish robust to faults.The performance of each part of the fault-tolerant control system was analyzed using the simulation.Finally,the validity of the faulttolerant control algorithm in the real world was verified experimentally.The control method proposed in this paper is able to effectively control the motion of a faulty robotic fish.Hence,the fault-tolerant control of a robotic fish based on the CPG and on a mathematical model has been accomplished.

教师利用语言真实的互动情境，为学生提供一个特定的场景，引学生进入学习之中。教师也可以通过故事的讲解或以问题引发学生讨论，集中学生的注意力，使他们全身心地参与到课堂互动之中。如在《彩色世界》一课教学中，教者创设情境如下：“为庆祝十九大的胜利召开，小海龟画了一幅三角旗，你觉得怎样？如果在画图软件中画这个图形，你想如何美化它？”教者以情境激发学生的探学热情，引导学生思考如何用Logo实现画图软件中设置前景色、背景色的功能。

产出类指标是用于反映该科研项目提供的产品或服务的数量，或者科研项目的完成情况，如论文著作情况、科技成果转化情况。结果类指标用于反映农业科研项目计划达到的预期目标的程度变量，该指标主要从服务对象的角度来衡量，如资源利用率的提高等。

Our future work will focus on improving the anti-interference ability of these fault-tolerant control methods.In addition,the experiments showed that the speed of the faulty robotic fish was greatly affected;thus,it will be necessary to seek an alternative method to enhance the swimming speed tolerance under faulty conditions.We will also concentrate on adapting the faulttolerant control method to other faulty conditions,such as joint power failure.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(61725305,61633020,61633004,and 61633017),the Beijing Natural Science Foundation(4161002),and the Beijing Advanced Innovation Center for Intelligent Robots and Systems(2016IRS02).

Compliance with ethics guidelines

Yueqi Yang,Jian Wang,Zhengxing Wu,and Junzhi Yu declare that they have no conflict of interest or financial conflicts to disclose.

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