谐波传动机器人关节的自适应转矩估计【中文5640字】
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谐波传动机器人关节的自适应转矩估计【中文5640字】,谐波,传动,机器人,关节,自适应,转矩,估计,估量,中文
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1译自：Mechanical Systems and Signal Processing,Volume 96,November 2017,Pages 1-15.Adaptive torque estimation of robot joint with harmonic drive transmissionZhiguo Shi:School of Computer and Communication Engineering, University of Science and Technology, Beijing 100083, ChinaYuankai Li:School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, ChinaGuangjun Liu:Department of Aerospace Engineering, Ryerson University, CanadaAbstract:In this paper, a torque estimation method with adaptive robustness and optimality adjustment according to load variation is proposed for robot joint with harmonic drive transmission. Based on a harmonic drive model and a redundant adaptive robust Kalman filter (RARKF), the proposed approach can adapt torque estimation filtering optimality and robustness to the load variation by self-tuning the filtering gain and self-switching the filtering mode between optimal and robust. The redundant factor of RARKF is designed as a function of the motor current for tolerating the modeling error and load-dependent filtering mode switching. The proposed joint torque estimation method has been experimentally studied in comparison with a commercial torque sensor and two representative filtering methods. The results have demonstrated the effectiveness of the proposed torque estimation technique.Key words:harmonic drive；load variation；torque estimation；joint torque estimation technique1、 IntroductionHarmonic drives, invented in the 1950s 1, are widely used in robotic systems, due to their desirable features of near-zero backlash, compactness, light weight, high torque capacity, high gear ratio and coaxial assembly. These distinctive characteristics of harmonic drives vindicate their widespread applications, especially in electrically-driven robot manipulators.Joint torque feedback (JTF) has been widely recognized to improve the performance of robot control in the robotics community 2,3. JTF is typically used in motion control of robot manipulators to suppress the effect of load torques 4 and can be used in the dynamic control 2of robots without the need of computing the robot inverse dynamics 3. Joint torque sensing or estimation is also essential for controlling robot force and compliance, as well as for collision detection.Conventionally, joint torque sensors or a multi-axis force/torque (F/T) sensor is utilized in robot force control. When estimating joint torques using F/T sensor at the robot wrist, extensive calculations are needed, and the results are affected by computation delays and model errors 5. There are several techniques of direct joint torque sensing, e.g., joint torque sensors based on elastic elements that are placed in the output transmission line of each joint of the robot 6,7.In 8, a linear encoder is used to measure the torsional deformation of the additional elastic body of a joint torque sensor. As measurement accuracy is in inverse proportion to sensor stiffness, low sensor stiffness is desirable in order to achieve high measurement resolution, which leads to complicated joint dynamics. Another joint torque sensing technique is based on the method proposed by Hashimoto et al. 9. Joint torque sensing is achieved by mounting strain gages directly on the harmonic drive, which is usually referred to as built-in torque sensing for harmonic drives 911. The torsional compliance of harmonic drives lends itself to torque sensing, but it is difficult to achieve durable accurate torque measurement with this technique due to signal ripples and amplifier drifts 12.Torque estimation using harmonic drive compliance model provides an effective way of torque estimation for robots with harmonic drives, which is shown in 13, a previous work of our laboratory. The joint torque estimation technique uses link-side position measurement along with a proposed harmonic derive model to realize stiff and sensitive torque estimation, obtaining joint torque with minimal mechanical modifications. However, in 13, a low-pass filter is used to filter the output of the harmonic drive model. While the low pass filter provides a simple and convenient way to resist high frequency noises, the torque estimation response speed is limited by the filter bandwidth.Traditionally, optimality is mainly considered in controlling robot force and compliance. In 16, a neural network learning using approximate dynamic programming is developed to achieve optimal control. In 10, to cancel the torque measurement ripples due to the gear 3teeth meshing, standard Kalman filter estimation is used. By on-line implementation of the Kalman filter, it has been demonstrated with experiments that this method is a fast and accurate way to filter torque ripples and torque due to misalignment. However, standard Kalman filter only guarantees optimality for the torque estimation, while the robustness is also indispensable.As robustness conflicts with optimality, robust performance improvement is often associated with loss of optimality. In the meantime, strong robustness is not essential all the time, and it may cause excessive loss of optimality. Torque estimation represents an efficient, economical and convenient to obtain joint torque with minimal mechanical modifications. In 18, joint torque estimation is based on EMG signals, in which matrix modifier is applied to make the controller adaptable to every upper-limb posture of any users. In this paper, a torque estimation method is developed for robot joint with harmonic drive based on a harmonic drive model and a redundant adaptive robust Kalman filter, in that Kalman filter based filtering method is commonly used and has been proved to be conveniently realized in practice.The proposed method provides desirable tolerant capability to the modeling error and enables dynamic balance of optimality and robustness according to the load variation. The proposed method consists of a modeling part and a filtering part. The modeling part models the harmonic drive compliance on the basis of link-side absolute encoder readings, motor-side encoder readings and the motor current. In the torque estimation, the motor current and the output of the modeling part provide the inputs to the filtering part that consists of a redundant adaptive robust Kalman filter. The proposed method has been studied experimentally in comparison with a commercial torque sensor and two representative filtering methods, and the experimental results have demonstrated the superior effectiveness of the proposed robot joint torque estimation method.2、Overall structure of torque estimationThe overall structure of the proposed torque estimation method is as shown in Fig. 1. The redundant adaptive robust Kalman filter (RARKF) approach is introduced in the filtering part. Generally, the joint loading is directly reflected by the motor current, which contains low 4noise and is fast in response. Therefore, the redundant factor of RARKF is designed as a function of the motor current to deal with modeling error and disturbances, and more importantly, to balance optimality and robustness according to the load variation.Fig. 1. Overall structure of the torque estimation.As a result of joint load variation, motion control, friction, or external forces, the torque of a robot joint may vary dramatically. A robust filter can allow timely joint torque estimation. However, there exists a trade-off between robustness and optimality. Sometimes it is desirable to smoothen the torque estimation at the sacrifice of robustness. The proposed RARKF based method adaptively switches the filtering mode between the optimal and robust according to the joint loading, which can effectively balance the estimation accuracy and response speed. The RARKF makes the switching function work stably within the range of a certain redundancy and switch the filtering mode between the optimal and robust when necessary.3、Modeling of harmonic drive torqueA harmonic drive consists of three main components: wave generator (WG), flex spline (FS) and circular spline (CS). Typically the WG is connected to the motor shaft, the CS is connected to the joint housing, and the FS is sandwiched in between (CS and WG) and connected to the joint output. The WG consists of an elliptical disk (rigid elliptical inner-race), called wave generator plug, and an outer ball bearing.The wave generator plug is inserted into the bearing, thereby giving the bearing an elliptical shape as well. The FS fits tightly over WG; when the WG plug is rotated, the FS deforms and 5molds into the shape of the rotating ellipse but does not rotate with WG.3.1. Harmonic drive kinematic modelThe ideal input/output kinematic relationship, as explained in 20, equates the angular positions at the components of the harmonic drive:w=Nf (1)where w is the wave generator position, f is the flexspline output position, and N is the gear ratio. The static force balance can be described as:w=1Nf (2)where w is the torque at the wave generator and f is the flexspline output torque. Eqs. (1) and (2) represent the harmonic drives ideal linear input/output relationship in which the harmonic drive transmission is treated as a perfectly rigid gear reduction mechanism. However, the empirical measurements of the input/output relationship provided in the cited literature show that the output is not linearly related to the input 21.The causes of this nonlinearity are torsional compliance in the harmonic drive components, nonlinear friction forces, and the kinematic error that is due to gear meshing and machining errors. Given this ideal kinematic relationship which describes the motion and force constraints present in harmonic drives, the remaining effects can be incorporated by modeling compliance, friction and kinematic error.The harmonic drive torsional compliance is due to flexibility in both flexspline and wave generator 13,22. By taking the torsional compliance of the wave generator into account, the hysteresis behavior of the harmonic drive is captured without the need for a separate hysteresis model. The compliance behavior of the harmonic drive 13 is illustrated in Fig. 2, with consideration of the flexspline and wave generator compliance.6Fig. 2. Kinematic representation of a harmonic drive showing the three ports.The torsional angle of the flexspline can be defined as:f=fo-fi (3)where fi denotes the angular position at the flexspline geartoothed circumference, fo denotes the flexspline angular position at the load side which is measured using the linkside encoder.The wave generator torsional angle is defined as:w=wo-wi (2)where wo, wi denote the positions of the wave generator outside part (ball-bearing outer rim) and the center part (wave generator plug), respectively. Note that fi and wo are not available as only fo and wi are measured by the link-side encoder and the motor-side encoder, respectively.The kinematic error is defined as the measured flexspline output minus the expected flexspline output (wave generator displacement multiplied by the gear ratio) 21. Therefore, the kinematic error, can be expressed as:=fi-woN (5)As in 13, the total torsional angle of the harmonic drive is written as:=fo-wiN (6)Substituting Eqs. (3)(5) into Eq. (6), we obtain:=f+wN+ (7)Considering the harmonic drive friction, the expression in Eq. (2) becomes:7w=1N(f-ft) (8)where ft is the harmonic drive friction torque as seen with respect to the output side of the transmission. For practical purposes and simplicity, the Stribeck friction model 23 has been proven useful and validated on a robot arm for joint friction compensation 24, which is utilized in harmonic drive torque estimation and written asft=Fc+(Fs-Fc)e-(v/s)+Fvvsign(v) (9)where v is the relative velocity, Fs denotes static friction, Fc denotes the minimum value of the Coulomb friction, s and Fv are lubricant and load parameters, and is an additional empirical parameter.3.2 Kinematic error modelThe effective torque estimation can be achieved only if one compensates for the effects of the kinematic error. The kinematic error can be measured for one complete output revolution using the high resolution link side absolute encoder. To determine the kinematic error, , the test joint is rotated clockwise and counterclockwise one complete output revolution with no payload 13. The total torsional deformation cw and ccw are measured during this process.cw=f+wcwN+ (10)ccw=f+wccwN+ (11)where wcw and wccw are the wave generator torsional deformation in the clockwise and counterclockwise direction respectively, which are calculated using both link-side and motor-side encoders measurements by the way of rotating in both full directions with no payload.Since the output torque is equal to zero, the flexspline torsional deformation is also equal to zero, f=0. With the assumption that the torsional deformation of the wave generator is symmetric, i.e.,wcw=-wccw (12)the kinematic error can be determined from the following equation8=(cw+ccw)/2 (13)3.3 Harmonic drive compliance modelA typical harmonic drive stiffness curve 20 is shown in Fig. 3, which features increasing stiffness with displacement and hysteresis behavior.Fig. 3. Typical stiffness and hysteresis curve of a harmonic drive.The total harmonic drive torsional deformation comprises deformation of both the flexspline and the wave generator. Based on experimental observation, the harmonic drive torsional deformation is largely contributed by the flexspline torsional compliance. The flexspline torsional compliance is described in 20, where f is approximated by a piecewise linear function of the output torque.f=fK1,f11K1+(f-1)K2,1f21K1+(2-1)K2+(f-2)K3,f2 (14)where K1, K2, K3, 1, and 2 are given by the manufacture. The slope of the curve shown in Fig. 3 indicates the harmonic drive stiffness.The curve is approximated by three straight-line segments with stiffness of K1, K2, and K3. Stiffness K1 applies for flexspline torque of 0 to 1; stiffness K2 applies for flexspline torque ranging from 1 to 2; and stiffness K3 applies for flexspline torque greater than 2.The harmonic drive torsional deformation can be from /2 to /2 at zero torque output. In 13, to replicate the hysteresis shape of this stiffness curve, the wave generator local elastic coefficient is approximated as:9Kw=Kw0eCw|w| (15)where Kw0 and Cw are constants to be determined. The wave generator torsional angle can be calculated using the following equation:w=0wdwKw (16)Substituting Eq. (15) into Eq. (16), we obtain:w=sign(w)CwKw0(1-e-Cw|w|) (17)The total torsional deformation of the harmonic drive is measured using both link-side and motor-side encoders. The wave generator torque w is approximated by the motor torque command.The relationship between f and f is presented in Eq. (14), in which f is calculated by combining Eq. (17) with Eq. (7), shown as:f=-sign(w)CwNKw0(1-e-Cw|w|)- (18)Using the inverse of Eq. (14), the flexspline output torque f, the joint torque to be estimated, can be presented as:f=K1f,ff11+K2(f-f1),f1ff22+K3(f-f2),ff2 (19)where f1=1/K1 and f2=1/K1+(2-1)/K2.4、Time delayThe torque sensor and link-side encoder are sampled every 2 ms in the current setup. In addition, software execution will cost additional clock time, causing time delay in the torque estimation.The time delay in torque estimation is evaluated by applying two kinds of external force: one is a sudden external force, and the other is a sustained constant force. A sudden external force 10is added to the previous experiment, where the moving links and joints attached to the base joint introduce only gradual load change. To introduce fast torque changes, external torque is applied in both directions.The experimental procedure is the same as the previous experiment except for the introduction of sudden external forces. The results are compared with those of KF and LPF, as shown in Fig. 11.Fig. 11. Results comparison of link-joint with external force.Zoomed view of the portion, from 44.5 s to 47 s as indicated in Fig. 11, is shown in Fig. 12. In this stage, the responses of KF are very slow, almost impossible to reflect the external force. The response of LPF is still relatively slow. RARKF works in its robust mode, giving faster response than LPF.11Fig. 12. Zoomed view of the portion indicated in Fig. 11.The response to external force in torque estimation has a great effect on collision detection and motion control of robot joints control system. There is an obvious time delay in torque estimation in Refs. 5,13, while LPF is used in 13. Load variation adaptive filter bandwidth can be introduced to improve the performance of LPF, while it is desired to increase the filter bandwidth with increased load and decrease the filter bandwidth with decreased load.In 10, Kalman filter is used to provide torque ripple elimination and misalignment torque compensation, which is proved to be an effective method. However, it will greatly increase the time delay to the external disturbance. The load variation adaptive mechanism can be used to improve the KF method.Further zooming on the indicated portion from 44.8 s to 45.2 s generates the results as shown in Fig. 13. In this stage, the time delay of RARKF is approximately 9 ms, while the error is about 0.5 N m. The time delay of LPF is approximately 70 ms and KF almost cannot reveal the changes of the external force.Fig. 13. Time delay in the estimated torque with external force.In another case, experiment with a sustained constant external force is conducted to observe the time delay, as shown in Fig. 14. Zoomed view of the portion, from 53.5 s to 56.5 s, is indicated in Fig. 14, shown in Fig. 15. In this stage, KF responds very s
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