An Experimental Study on Harmonic Drives

Despite widespread industrial application of harmonic drives modelling and control of such systems has not been fully addressed In this theoretical experimental study of harmonic drive systems a systematic way to capture and rationalize the dynamics of the system is proposed Simple and accurate models for their compliance hysteresis and friction are given and model parameters are estimated from experimental data using least square approximation A statistical measure of consistency is de ned by which the reliability of the estimated parameter for di erent operating condition as well as the accuracy and integrity of the proposed model is quanti ed The validity of the modelling scheme is evaluated by comparing the experimental and simulation results Two separate simulations are developed for the harmonic drive system operating in restrained and unrestrained motion The problem of torque control of harmonic drive is addressed in the H controller design framework It has been shown that an empirical nominal model estimated from system fre quency responses provides necessary requirements for controller design It is illustrated that H framework is a very rich and powerful theory to ensure the stability and performance in the presence of modelling uncertainties An H torque controller is implemented on the system and the time and frequency domain performances of the closed loop system are shown to be exceptionally good Final Report to I S E Introduction Since its inception in the harmonic drive has found widespread acceptance among practition ers This mechanical transmission occasionally called strain wave gearing employs a continuous de ection wave along non rigid gear to allow for gradual engagement of gear teeth Figure Because of this unconventional gear tooth meshing action harmonic drives can deliver high reduc tion ratios in a very small package In fact the radical mechanical operation of this gear train de es conventional understanding of gear behaviour and creates a new arena for exploration and understanding Figure Harmonic drive components The harmonic drive exhibits performance features both superior and inferior to conventional gear transmissions Its performance advantages include high torque capacity concentric geometry lightweight and compact design zero backlash high e ciency and back drivability Harmonic drive systems su er however from high exibility resonance vibration friction and structural damping nonlinearities The lack of literature relative to that for conventional gear transmission also inhibit their use The unique performance features of the harmonic drive have captured the attention of designers in many elds It has been used in industrial and space robots assembly equipment and mea suring instruments as well as heavy duty applications such as machine tools and printing presses Additionally space and aircraft systems often employ harmonic drives for their light weight and compact geometry However the performance disadvantages of harmonic drive motivates the on going research on better understanding of the harmonic drive systems and nding control schemes which reduce the unwanted dynamics In many application speci cally in robotics torque is often taken to be the control input The physical variable being manipulated however is not torque but armature current in a DC motor for instance For harmonic drive systems the relation between output torque and input current possess nonlinear dynamics due to the exibility Coulomb friction and structural damping of harmonic drive The main objective of this research is to improve this input output relation through compensation for nonlinearities and to convert the system to a torque source with a at frequency response over a wide bandwidth Moreover how accurate we should model the system to obtain a desired performance is another question to be answered Finally we would like to experiment the controller on a single robot arm whose joint is actuated by harmonic drive and to Final Report to I S E observe its performance in practice Throughout its short existence the harmonic drive has enjoyed increasing international at tention from designers as well as researchers The Russians were perhaps the rst who initiated substantial research on the dynamic behavior of harmonic drives More recently Tuttle and Seering performed an extensive e ort to model the sti ness posi tioning accuracy gear tooth meshing mechanism and friction of the harmonic drive Their experimental observations show that the velocity response to step commands in motor current are not only contaminated by serious vibration but also by unpredictable jumps The velocity response observations were used to guide the development of a series of models with increasing complexity to describe the harmonic drive behavior Their most complex model included kinematic error nonlinear sti ness and gear tooth interface with frictional losses Kircanski and Goldenberg have also attempted to model the harmonic drive in detail They used the drive system in contact with a sti environment in contrast to unrestrained motion experiments used in and illustrated that in this case nonlinear sti ness hysteresis and friction are more tractable Simple models for soft windup hysteresis and friction were proposed and the parameters were identi ed by restrained motion experiments Hsia Legnani Marilier and Sey erth are among others who attempted to model the sti ness friction and position accuracy of harmonic drive systems All these researchers noted the inherent di culties in nding an accurate model for the system Brigdes et al Kaneko et al Kazerooni Hogan and Chapel and Su are representative of researchers who worked on the control of harmonic drive system Bridges used a very simple linear model for the system with PD torque control His results show some improvement in tracking error but insu cient performance near resonant frequency Kaneko also based his analysis on a simple model of the system but included nonlinear sti ness in the system He then applied a feedforward loop to adjust for nonlinear sti ness and then a pure gain torque feedback to shape the performance Kazerooni considers a simple linear system for the harmonic drive and used a sensitivity loopshaping technique to design a linear controller for the system Hogan proposed impedance control for robots with harmonic drive systems to deal with the dynamic interaction induced in contact tasks Chapel applied H control design methods to the analysis and design of impedance control laws In this report rst the experimental setup of the harmonic drive is described Then the results obtained on the system modelling and identi cation is illustrated and the delity of the model is veri ed by simulations The results on robust torque control of system in locked load and free load cases is outlined next and nally the report is concluded by the summary of the results and possible future research horizons Experimental Setup A picture of the setup and its schematic are illustrated in Figure The harmonic drive is driven by a DC motor and a load inertia is used to simulate the robot arm for unrestrained motion Also a positive locking system is designed such that the output load can be locked to the ground The detail design drawings of the setup are collected in Appendix C In this experimental setup the exspline is xed to the ground and the output is carried by the circular spline The DC motor is a brushless Kollmorgen Inland motor model RBE A Its weight is grammes its maximum rated torque is Nm and its torque constant is Nm amp Final Report to I S E DC Harmonic Drive Motor Load Servo Amplifier Figure A picture of the experimental setup and its schematics The servo ampli er is a FAST Drive Kollmorgen model FD E The motor was equipped with a pulse rev encoder and the servo amp includes a DSP to estimate the motor velocity from position readings of the encoder However the encoder resolution for our low speed experiments was not enough to obtain an accurate estimate of the velocity Therefore a tachometer has been purchased form Kollmorgen Inland and has been replaced the encoder The detail drawings of the tachometer assembly are given in Appendix C The harmonic drive is from CFS series of HD Systems Inc with gear ratio and rated peak torque of Nm To measure the torque a pair of strain gauges has been mounted on the exspline by I S E and the output signal is ampli ed by using Entran ampli er We replaced the ampli er to a variable gain ampli er which has the capability of resetting the o set The output encoder is Renco model RM with pulse rev resolution The technical speci cation of the system components are collected in Appendix D Motor velocity and current and the output torque and position are measured by the measurement units To obtain an estimate of the output velocity a Kalman lter estimator on encoder readings is employed These signals were processed by several data acquisition boards and monitored by a Challenger C processor executing compiled computer C codes Tachometer Calibration To calibrate the tachometer we mounted an encoder on the free side of the motor shaft and experiment the motor for di erent inputs The output signal of the tachometer is then integrated o line in Matlab and compared to the encoder readings Using least square approximation for each experiment we estimate the optimum tachometer gain which best ts the integrated velocity signal and the encoder position signal Matlab le tach ls m given in Appendix A provides the details of the least square approximation The nal gain is obtained from the average value of the estimated gain for di erent experiments and the consistency measure of the gain is calculated by the ratio of the standard deviation to the nal gain The nal gain is obtained by this method for eight experiments and the results are given in Table The tachometer gain is rad sec volts and the consistency measure is less than which shows an accurate calibration pro