ZHU Minghong ,XIAO Shu ,and YU Fei

1.Faculty of Electrical Engineering and Computer Science,Ningbo University,Ningbo 315211,China;2.College of Intelligent Systems Science and Engineering,Harbin Engineering University,Harbin 150001,China;3.College of Mathematical Sciences,Harbin Engineering University,Harbin 150001,China

Abstract:In the applications of joint control and robot movement,the joint torque estimation has been treated as an effective technique and widely used.Researches are made to analyze the kinematic and compliance model of the robot joint with harmonic drive to acquire high precision torque output.Through analyzing the structures of the harmonic drive and experiment apparatus,a scheme of the proposed joint torque estimation method based on both the dynamic characteristics and unscented Kalman filter (UKF) is designed and built.Based on research and scheme,torque estimation methods in view of only harmonic drive compliance model and compliance model with the Kalman filter are simulated as guidance and reference to promote the research on the torque estimation technique.Finally,a promoted torque estimation method depending on both harmonic drive compliance model and UKF is designed,and simulation results compared with the measurements of a commercial torque sensor,have verified the effectiveness of the proposed method.

Keywords:harmonic drive,torque estimation,compliance model,unscented Kalman filter (UKF).

1.Introduction

Joint torque estimation technique has been used in robotic joint and manipulator controls,which are extensively applied in the fields such as various robots,measurement,analytical and test systems,and space and aircrafts [1-5].Compared with torque sensors embedded in the joints which will change the mechanical structure of joints to provide output torque directly,incremental encoders equipped with harmonic drive can only supply position information.Although a proper transmission model based on dynamic and kinematic characteristics of the joint apparatus should be built to deal with the position to obtain the output torque,encoders can avoid changing the joint mechanical structure.Besides,torque sensor measurements are susceptible to environment like temperature,humidity,and so on.These factors will not or barely influence the position measurements of encoders [6].

There exist two kinds of research ideas on the torque estimation in the manipulator joints.The first is to build or embed the torque sensors in the joints,which can help to measure the torque fast and directly.While,this method will inevitably bring changes to the structure and characteristics of the manipulator joints,and much more efforts will be put forth to eliminate or compensate the errors taken by the changes.Nevertheless,many literatures have provided research ideas on the torque sensors built or embedded in the joints with harmonic drive transmission.In [7],Jung et al.proposed a low-cost joint torque measuring method,in which characteristics of the harmonic drive were utilized to measure the external torque,specially relying on the flexspline strain measurement of harmonic drive.In [8],Pan et al.proposed a torque measurement method with a sensor built in the harmonic drive,in which a second-order ripple model was used to depict the flexspline strain,and then double compensations were adopted to cancel out ripples.In [9],Zhang et al.proposed a new 4-bar link shape torque sensor with high sensitivity to measure the joint torque of a robot manipulator,and optimized compensation techniques were employed to increase the torque sensitivity without sacrificing stiffness.These researches all took measures to compensate or promote the torque measurement precision,and their research results convinced the effectiveness of these methods.

The second research idea is based on measurements of the angle or position encoders installed on the motor side and the link side of the robotic joints with harmonic drive,which can help to calculate the joint torque indirectly without changing the structure and characteristics of the manipulator joints.While,in this method researchers need to take additional efforts to calculate the joint torques.Due to this fact,researchers proposed various joint torque measurement methods based on angle or position encoder measurements to calculate the transmission torque.In [10,11],a modular distributed control technique was presented for modular and reconfigurable robots which could immediately adjust to robot reconfigurations.Benefiting from the developed control technique with joint torque sensing,modular and reconfigurable robots could be stabilized joint by joint,and modules could not need to change the parameters setting to adapt or cooperate to other robot modules while being added or reduced.In [12],Zhang et al.proposed a joint torque estimation method depending on the structural elasticity of the robot joint with harmonic drive.In this method,the position measurements of the link side combined with an established harmonic drive model were used to help to fulfil the stiff and sensitive torque estimation.In [13],Zhang et al.proposed a joint torque estimation method which exploited the inherent structural elasticity characteristics of the robot joint with harmonic drive transmission.In the presented method,both the position measurements of the motor side and the link side and a designed compliance model of the harmonic drive were employed to accomplish the stiff and sensitive joint torque estimation without introducing a new force/torque sensor.These researches took indirect torque measurement methods mostly based on the mechanical or dynamic structures of the robot manipulators or joints,and good performances were achieved relatively.

Based on the discussion of the research ideas above and with the rapid development of the commercial highresolution absolute position encoders,it will be an important fundamental work to develop joint torque estimation techniques depending on the position measurements of the encoders embedded on the motor side and the link side of the robotic joints with harmonic drive transmission.It has higher requirements for the dynamic modeling of the system joints,and it is necessary to reasonably model and analyze all transmission models.This research takes advantage of the torque estimation model proposed by Zhang et al.,to further discuss and improve the model of the harmonic drive [10-15].Through the discussion and modeling analysis of the stiffness coefficient of flexspline and the elastic coefficient of the harmonic generator,an improved system model of harmonic drive is obtained,so that the overall torque estimation scheme with the system dynamic model can be carried out [14,15].

In our previous work [16],the basic compliance model of the robotic joint apparatus with harmonic drive has been introduced and deduced,and an immature and preliminary research idea was put forward.In this manuscript,a more mature and reasonable research framework is proposed in Subsection 3.2 and more controlled trials with different experimental methods and environments are added.Research will be taken in building the effective dynamic model of the robot joint with harmonic drive.Combing with the proposed dynamic model,the kinematic error model and the torsional deformation model of the harmonic drive are established to make compensations for the torque estimation.The two torque estimation methods based on only harmonic drive compliance model (CM) and CM with Kalman filter (KF) are separately simulated to provide a guidance to promote the research on the torque estimation technique.Then the proposed torque estimation technique depending on harmonic drive CM and unscented Kalman filter (UKF) will be provided.

To go on the research of this manuscript,structure analysis of the robot joint apparatus with harmonic drive is presented in Section 2.The torque estimation of the harmonic drive transmission-based joint apparatus is discussed in Section 3.In Section 4,experiment and verification of this research are carried out,and then the discussion and analysis of the experiment results are provided.Finally,Section 5 gives the conclusions.

2.Structure analysis of the robot joint apparatus with harmonic drive

To facilitate harmonic drive to a robot joint properly with high resolution,the first and important insertion point is to make analysis and good modeling.The basic structure of harmonic drive and integration with the robot joint will be put forward in this part,and the related model also will be discussed in detail later.In this research,the robot joint apparatus is assembled with a harmonic drive,of which the model is SHD-17-100-2SH and the specific parameters can be searched in [17].

2.1 Typical structure and kinematic model of a harmonic drive

The kinematic model of a harmonic drive system is based on its mechanical structure.To develop the kinematic model also needs to follow the transmission structure.The basic three elements—wave generator (WG),flexspline (FS),and circular spline (CS) constitute the basic structure of a harmonic drive.The integration of the three parts can be described as follows.WG is attached with the motor shaft of a brushed direct current (DC) motor by going through the WG plug,CS is connected to the joint casing,and FS is naturally clamped between CS and WG and interfaced to the joint output end.Fig.1 shows the specific structure of a harmonic drive clearly.WG is a thin ball bearing body and matched with an oval shape plug.To make the structure meshing,FS is designed as a thin cylindrical cup shape structure with external gear which is slightly smaller than the internal gear of CS [18].

Fig.1 Typical structure of a harmonic drive

In view of the typical structure of a harmonic drive above,a transmission schematic diagram can be set in Fig.2.

Fig.2 Basic kinematic model of a harmonic drive

Therefore,the harmonic drive kinematic model can be expressed as

whereNrepresents the gear ratio.θw,θf,and θcare the angular positions of WG,FS,and CS,respectively.

Based on the concept that the harmonic drive transmission is a perfectly rigid gear-reduction mechanism,an ideal kinematic model above can be developed.While in practical harmonic drive transmission applications,the viscous friction and kinematic error must be taken into consideration due to gear meshing.In the following model formulating progress,the two aspects will be discussed and balanced.

2.2 Structure of the harmonic drive transmissionbased joint apparatus

A practical integration joint apparatus is set in Fig.3.The joint marked as “Joint 1” is regarded as “test joint” in the manuscript to follow,which is interfaced with a “l(fā)inkjoint” to work as a robot arm.In the harmonic drive transmission-based joint apparatus,the transmission mechanism involves structure analysis of the three components(WG,FS,and CS) of a harmonic drive and the integration of the harmonic drive and robot joint.

Fig.3 Integration of the joint apparatus

As explained in the structure of the harmonic drive,through the plug of WG,the harmonic drive of the test joint is powered by a brushed DC motor.An optical incremental encoder with 1 024 p/rev is installed on the front end of the DC motor to measure the angular changes of the motor.On the motor side,the input torque can be directly converted by DC motor current.An absolute position encoder with 19 bits is installed on the link side to help to measure the output angle.A commercial ATI six-axis F/T (force/torque) sensor is employed to measure the load torque.To this knowledge,the schematic diagram of the test joint can be described in Fig.4.

Fig.4 Schematic diagram of the test joint

3.Torque estimation for the harmonic drive transmission-based joint apparatus

3.1 Compliance model of the harmonic drive transmission-based joint apparatus

3.1.1 Analyzing the compliance components of a harmonic drive

Based on the ideal kinematic model of a harmonic drive,a more accurate model should be developed in practical use.In practical applications of the harmonic drive system,researchers generally believe that the system nonlinearity mainly originates from three parts of a harmonic drive: kinematic error,nonlinear friction,and torsional compliance.Then the development of dynamic modeling of the harmonic drive transmission-based joint apparatus will be focused on the three aspects.A feasible method of clarifying the system dynamics is to develop an approximate model to fulfil the running characteristics of the target system.According to this thought,the system nonlinear relationship could be developed by bringing the kinematic error and compliance friction to the ideal kinematic model of the harmonic drive [19,20].

Taking the compliance friction and kinematic error into consideration,the harmonic drive compliance components can be described in Fig.5.This schematic illustration provides the transmission characteristics of the harmonic drive,and the related parameters in the scheme will be defined in the next section.

Fig.5 Scheme of harmonic drive compliance components

3.1.2 Modeling the compliance model of the harmonic drive transmission-based joint apparatus

In the harmonic drive compliance modeling,the stiffness and torsional compliance of a harmonic drive generally dominate the transmission process and output accuracy.In this research,the flexspline damping effect,hysteresis behavior,and torsional compliance of the harmonic drive are all considered.Therefore,the dynamic model of joint apparatus with hysteresis effects could be expressed by

whereJm,Bm,and θmare the rotary inertia,damping factor,and position on the motor side,respectively.τfmdenotes the friction torque,τfis the hysteresis torque,and τmrepresents the input motor torque which can be calculated by the motor model.

To analyze the system transmission characteristics,research will be focused on the scheme of the harmonic drive compliance components in Fig.5.In this scheme,θfand θwrepresent the link side and motor side angles,respectively.Δθ is assumed as the total torsional angle of harmonic drive,Δθfis the torsional angle of FS,Δθwis the torsional angle of WG,and θerrrepresents the kinematic error.Therefore,the total torsional angle of harmonic drive can be expressed by

The torsional angle of the WG Δθwis determined by the inherent elastic coefficient of WG as

whereCwandKware known values.τw=τmcan be approximately derived by the front input torque.

Then the torsional angle of FS can be derived as

In addition,the torsional angle of FS Δθfis defined by the stiffness characteristic of FS as

whereCfandKfare known values.

The total torsional angle of harmonic drive is

where θfand θwcan be obtained from the measurements of the position encoders mounted on the motor side and link side of the joint.

Kinematic error can be calculated as

wherea0,al1,bl1,ωl,aw1,aw2,bw1,bw2,and ωware known parameters.

After working out the FS torsional deformation Δθf,the torque estimation is easy to be completed by inverting (6) as

The CM based on (3)-(9) is formulated on the characteristics of the system transmission and developed with the schematic illustration in Fig.5.To explore and assess the performance of the torque estimation method with the CM,the relevant research work can be found in [11-13].

Furthermore,the performance of the torque estimation method depending on the CM also can be found in Table 1 in Section 4.In the experiment set with a relative high frequency and small amplitude input torque,the max torque difference is 1.754 N·m which is 154.4% of the max torque (1.136 N·m) of the sensor torque,and the root mean square (RMS) of torque difference is 0.602 2 N·m(about 53.0% of the max torque).This level of accuracy undoubtedly limits the method’s applications widely.To improve the accuracy,a series of attempts on the estimation algorithm have been made in the follow-up studies[21,22].

3.2 Scheme of proposed torque estimation technique

In Subsection 3.1.2,the torque estimation method only depending on the CM has been given and the estimated torque precision also has been discussed.According to this fact and to overcome the weakness of low accuracy of the output joint torque,optimization algorithms for the torque estimation are laid on the whole dynamic model of the transmission system.The designing scheme of the optimization algorithm is developed in Fig.6.

Fig.6 Scheme of the torque estimation method

It is clearly shown in Fig.6,through the harmonic drive CM and dynamic model of the system transmission,the motor-side position angle θmand torsional angleΔθfcan be calculated.And then these values will be put into the implementation at the next step.

3.3 Implementation of UKF in torque estimation scheme

According to the design thought and combining with (2),the optimization algorithm of the torque estimation method with the dynamic model depending on both harmonic drive compliance and UKF can be proposed as follows [23-28]:

wherewkandvkare Gaussian white noise with zero mean.QkandRkdenote corresponding covariance matrices.Therefore,the torque estimation optimization algorithm of this research will be designed on both harmonic drive CM and UKF [23].And the torque estimation implementation will be performed in the following.

(i) Initial values of the filter

(iv) One-step state prediction and its covariance matrix at the time stepk

(v) Calculation of one-step sample point prediction at the time stepk

(vii) Time updating

wherePXZ,PZZ,andPkare the related covariance and variance,respectively.Kkrepresents the filtering gain.

Based on the scheme and implementation of the proposed torque estimation technique in Section 3,experiment and verification will be conducted in the next section.

4.Experiment and verification

As mentioned in Subsection 3.1,the research will not only provide experiments depending on both harmonic drive CM and UKF,but also reference experiments depending on the harmonic drive CM and harmonic drive CM-KF.

In this section,verifications of the effectiveness of the proposed torque estimation optimization algorithms are carried out and the related experiment results provide the support.The experiment apparatus is designed and integrated as shown in Fig.3.To implement the process,an F/T sensor is assembled at the output side of the test joint as a reference.Therefore,the estimated joint torque is compared to the reference to assess the efficiency of the proposed torque estimation method.

In the following experiments,to show the complete experiment process,all data are kept and reflected on the graphs.However,the data in the first three seconds are regarded as inaccurate data and excluded to get a better conclusion of the effectiveness of the experimental methods.

4.1 Experiment 1: joint arranged to rotate with a relative high frequency and small amplitude torque

The experiment of test joint is designed with a sinusoidal changing torque.The joint is set to rotate in 80 with a relative high frequency and small amplitude torque.

4.1.1 Experiment based on the harmonic drive CM

The CM described in Subsection 3.1 is developed on the characteristics of the system transmission,and experiment based on the CM will be conducted to analyze the torque transmission accuracy.

The experiment results only based on CM have been provided in Fig.7 and Fig.8,and the estimated torque and related difference compared with a torque sensor are included.

Fig.7 Estimated torque and related difference based on CM compared with torque sensor

Fig.8 Estimated position difference based on CM compared with torque sensor

As shown in Fig.7 and Fig.8,the max torque difference is 1.754 N·m which is 154.4% of the max torque(1.136 N·m) of the sensor torque,and the root mean square (RMS) of torque difference is 0.602 2 N·m.The max position difference reaches 0.102 8°.

4.1.2 Experiment based on harmonic drive CM-KF

Based on the CM-KF,the experiment results are provided in Fig.9 and Fig.10.

Fig.9 Estimated torque and related difference based on CM-KF compared with torque sensor

Fig.10 Estimated position difference based on CM-KF compared with torque sensor

As shown in Fig.9 and Fig.10,the max torque difference is 2.132 N·m that is almost 187.7% of the max torque (1.136 N·m) of the sensor torque,and the RMS of torque difference is 0.194 9 N·m.The max position difference reaches 0.029 57°.

4.1.3 Experiment based on harmonic drive CM-UKF

Based on the harmonic drive CM-UKF the experiment results are provided in Fig.11 and Fig.12.

Fig.11 Estimated torque and related difference based on CMUKF compared with torque sensor

As shown in Fig.11 and Fig.12,the max torque difference is 0.579 3 N·m that is about 50.1% of the max torque (1.136 N·m) of the sensor torque,and the RMS of torque difference is 0.062 6 N·m.The max position difference reaches 0.004 222°.

Fig.12 Estimated position difference based on CM-UKF compared with torque sensor

The test results can be clearly summarized in Table 1.

Table 1 Results of experiments running with a relative high frequency and small amplitude torque

4.2 Experiment 2: joint arranged to rotate with a relative low frequency and large amplitude torque

The experiment of test joint is designed with a sinusoidal changing torque.The joint is set to rotate in 80 with a relative low frequency and large amplitude torque.

4.2.1 Experiment based on the harmonic drive CM

The experiment results only based on CM have been provided in Fig.13 and Fig.14.

Fig.13 Estimated torque and related difference based on CM compared with torque sensor

Fig.14 Estimated position difference based on CM compared with torque sensor

As shown in Fig.13 and Fig.14,the max torque difference is 2.43 N·m that is about 45.8% of the max torque(5.303 N·m) of the sensor torque,and the RMS of torque difference is 1.172 3 N·m.The max position difference reaches 0.218 8°.

4.2.2 Experiment based on harmonic drive CM-KF

Based on CM-KF,the experiment results are provided in Fig.15 and Fig.16.

As shown in Fig.15 and Fig.16,the max torque difference is 5.232 N·m that is almost 98.7% of the max torque (5.303 N·m) of the sensor torque,and the RMS of torque difference is 2.948 7 N·m.The max position difference reaches 0.063 91°.

Fig.15 Estimated torque and related difference based on CM-KF compared with torque sensor

Fig.16 Estimated position difference based on CM-KF compared with torque sensor

4.2.3 Experiment based on harmonic drive CM-UKF

Based on CM-UKF,the experiment results are provided in Fig.17 and Fig.18.

Fig.17 Estimated torque and related difference based on CMUKF compared with torque sensor

Fig.18 Estimated position difference based on CM-UKF compared with torque sensor

As shown in Fig.17 and Fig.18,the max torque difference is 1.034 N·m that is about 19.5% of the max torque(5.303 N·m) of the sensor torque,and the RMS of torque difference is 0.048 2 N·m.The max position difference reaches 0.000 260 4°.

The test results can be clearly summarized in Table 2.

Table 2 Results of experiments running with a relative low frequency and large amplitude torque

4.3 Discussion and analysis

(i) For the experiment 1 in Subsection 4.1

From Table 1,compared with the results of the experiment based on CM,the torque estimation method based on CM-KF can bring little improvement to the torque estimation of the system (the RMS of torque difference from 0.602 2 N·m to 0.194 9 N·m,while having a bigger maximum torque difference 2.132 N·m compared to 1.754 N·m).

Compared with the results of experiment based on CM,torque estimation method based on CM-UKF has effectively reduced the max torque difference from 1.754 N·m to 0.579 3 N·m,and the RMS of the difference from 0.602 2 N·m to 0.062 6 N·m which shows 89.6% improvement of the estimated torque accuracy.We also get that the max position difference is 0.004 222° (about 4.1% of max position difference in the torque estimation method based on CM),which is quite small and the method is proved to be effective to track the joint movement.

(ii) For the experiment 2 in Subsection 4.2

From Table 2,compared with the results of the experiment based on CM,the torque estimation method based on CM-KF can bring little improvement to the torque estimation of the system (max position difference from 0.218 8° to 0.063 91°,while the RMS of torque difference from 1.172 3 N·m to 2.948 7 N·m,and having a bigger max torque difference 5.232 N·m compared to 2.430 N·m).

Compared with the results of the experiment based on CM,torque estimation method based on CM-UKF has effectively reduced the max torque difference from 2.430 N·m to 1.034 N·m,and the RMS of the difference from 1.172 3 N·m to 0.048 2 N·m which shows 95.9% improvement of the estimated torque accuracy.We also get that the max position difference is 0.000 260 4° (about 0.12%of the maximum position difference in the torque estimation method based on CM),which is quite small and the method is proved to be effective to track the movement of the joint.

As has been discussed in Subsection 3.1,the system nonlinearity mainly originates from the three parts of the harmonic drive components.Based on experiments conducted with the two kinds of input (high frequency and small amplitude torque,low frequency and large amplitude torque),it is easy to see that the torque estimation method based on CM-KF can bring little or no improvement of the estimated torque accuracy,while the proposed torque estimation method based on CM-UKF really can bring great improvements of the estimated torque accuracy (89.6% and 95.9% improvements of the estimated torque accuracy based on the RMS of torque difference).

5.Conclusions

This manuscript proposes a new technique to raise the accuracy of the joint output torque based on the CM-UKF optimization algorithm.The harmonic drive compliance model is developed on the constituent components and mechanical characteristic of a harmonic drive,then the scheme of the joint torque estimation is proposed.This method is built on the dynamic characteristics of the whole system.To improve,implement,and validate the proposed method,the experiments are carried out with two other reference experiments (experiment based on CM and experiment based on CM-KF).

The experiment based on CM-KF will only give a reference to guide and help to improve the proposed torque estimation method (CM-UKF).The experiments conducted with the two kinds of input (high frequency and small amplitude torque,low frequency and large amplitude torque),have shown that the proposed torque estimation method based on CM-UKF really can bring great improvements of the estimated torque accuracy (89.6%and 95.9% improvements of the estimated torque accuracy based on the RMS of torque difference).And the max position differences are only 0.004 222° and 0.000 260 4° (about 4.1% and 0.12% of the max position difference in torque estimation method depending on CM,respectively),which are quite small.Therefore,the proposed torque estimation method (CM-UKF) is proved effective to track the joint movement,and the results have verified the effectiveness of the proposed torque estimation method.

Thereafter,more comprehensive and complex experimental environment will be considered to further improve the existing torque estimation methods.