论文2G精华版本 ICRA 2018 files 0461

Generating Vibration Free Rest to Rest Trajectories for Confi guration Dependent Dynamic Systems via 3 Segmented Input Shaping Dan Kielsholm Thomsen 1 2 Rune S e Knudsen1 David Brandt1 Ole Balling2and Xuping Zhang 2 Abstract This paper presents a new method to generate vibration free rest to rest RTR trajectories for confi guration dependent dynamic systems such as robots cranes or machine tools The new method named 3 Segmented Input Shaping is based on a combination of the widely known Input Shaping method and a new trajectory segmentation strategy for piece wise shaping of the trajectory The new segmentation strat egy facilitates the capability of accounting for variations in system dynamics during motion by shaping acceleration and deceleration profi les with individual frequencies In this paper the new segmentation strategy is used in combination with the bang coast bang BCB trajectory The generated trajectories are described in closed form hence requires no optimization and thereby provides strong computational performance The new method is verifi ed by numerical simulations and detailed analysis and shows great potential in vibration free RTR trajectory generation for systems with confi guration dependent dynamics I INTRODUCTION The newest development in electro mechanical systems such as robot arms or cranes shows a trend away from the traditionally heavy and rigid structures towards lightweight structures to enable fast and energy effi cent operation Lightweight structures have a higher degree of elasticity and are thereby featured with increased vibrational behavior during operation As a consequence signifi cant efforts have been devoted to vibration control technologies in recent decades 1 Very effi cient feedforward control methodolo gies have been developed for the vibration suppression of systems with time invariant dynamics The three main classes of feedforward control are trajectory generation trajectory optimization and command shaping Generation of trajectories with continuous acceleration and bounded jerk such as S curve or exponential trajectories is one approach to reducing system vibration 2 5 The downside of bounded jerk is increased traveling time On the other hand we have the BCB trajectory which is often used to obtain time optimal trajectories to minimize traveling time The BCB method is also known as Linear Segments with Parabolic Blends LSPB and has a trapezoidal velocity profi le with instantaneous acceleration changes which results in unlimited jerk and hence signifi cant vibration Another approach to vibration reduction is trajectory op timization Different trajectory optimization strategies have been implemented to generate optimal trajectories in terms of Corresponding author contact danthomsen dk Corresponding author xuzh eng au dk 1Universal Robots A S Energivej 25 DK 5260 Odense S Denmark 2Aarhus University Dept of Eng Inge Lehmanns Gade 10 DK 8000 Aarhus C Denmark 1 135 1 90 1 45 Actuated joint Fig 1 1DOF manipulator with varying mass moment of inertia residual vibration e g 6 8 The different strategies utilize a parameterized description of a trajectory and adjust the parameters using optimization algorithms to identify optimal solutions Very good vibration suppression can be achieved but optimization problems comes at a high computational cost which result in delays due to calculations which may be a problem Command shaping refers to methods which modifi es a system command such that vibrational response is mini mized Comprehensive reviews of different command shap ing methods are presented in 9 12 One command shap ing method which receives attention due to its effi ciency and simplicity is named input shaping 13 Input shaping is a method of shaping a reference command by convolving it with a set of impulses The principles of the input shaping method is briefl y explained in Section II However it is a common challenge in mechanical systems that the system dynamics are varying e g due to changes in mass distribution of the structure when a robot arm moves from one position to another The problem of varying inertia is illustrated by the 1DOF manipulator in Fig 1 Here it is seen how the mass moment of inertia seen from the actuator is dependent on actuator position Due to variations in system dynamics traditional input shaping methods for invariant dynamic behavior can not effectively reduce the level of vibration For systems with time varying dynamics it may be necessary to update the input shaping impulse sequence throughout the motion to fi t the system dynamics at different points in time There are two overall approaches to time varying com mand shaping Either the shaper is continuously updated or the trajectory is split into segments with piece wise constant shaping 14 19 A common challenge for time varying command shaping is to ensure continuous system inputs when dynamic parameters are updated Input shaping with continuously updated impulse se quences were fi rst introduced in the 1990 s by Rappole 14 and by Magee and Book 15 For complex dynamic 2018 IEEE International Conference on Robotics and Automation ICRA May 21 25 2018 Brisbane Australia 978 1 5386 3080 8 18 31 00 2018 IEEE4361 systems the continuous updating of dynamic parameters may be computationally intensive and in real time applications continuity problems arise due to time discretization The continuity issues have been addressed by different types of impulse splitting e g Murphy 16 Also different approaches to updating dynamic parameters by segmentation has been presented Kinceler and Meckl 17 split a high order polynomial position profi le into a fi nite number of intervals of equal duration and determine the shape of each interval individually by utilizing a modifi ed fi nite time Laplace Transform technique Beazel and Meckl 18 segment a bang bang trajectory such that each segment represent an equal change in vibrational frequency and shape it based on the average frequency of each segment using ver sine basis functions These segmentation strategies overcome the need for updating the dynamic parameters in every time step but requires that the trajectory is split into a signifi cant amount of segments Since continuity must be enforced at segment boundaries the large amount of segments results in a noticeable delay In this work we aim to develop a new segmentation strategy for conducting piece wise input shaping in the generation of vibration free RTR trajectories for systems with time varying dynamics We present a new strategy to trajectory segmentation which utilize only 3 segments hence limits the amount of segment boundaries This is with the goal of fast trajectories with a low level of residual vibrations and high computational effi ciency The input shaping technology are briefl y overviewed in Section II Section III presents the methodology of 3 segmented input shaping Numerical simulations are con ducted with detailed results and analysis in Section IV Summaries and discussions are given in the fi nal section II INPUT SHAPING Input shaping is a vibration suppression technology which modifi es a system input to result in vibration free system response by convolving the original input with an impulse sequence 13 The impulse sequence may be determined in several different ways but common for all methods is that a set of constraint equations is solved 11 Among the constraints are constraints on residual vibration impulse amplitudes and duration of the impulse sequence The principle of traditional input shaping is that if one system impulse results in a vibration response then it is also possible to give a second impulse to the system with the exact opposite response effectively canceling out all vibrations The simplest vibration free impulse sequence is the two impulse zero vibration ZV shaper 13 illustrated in Fig 2 Here it is seen how the fi rst impulse initiates a vibration and the second impulse eliminates this vibration An impulse sequence I t with n impulses is described by the timing and scaling A of the impulses as presented in 1 3 A A 1 A2 An 1 1 2 n 2 A1 A2 Time Response A1response A2response Total response 1 2 Fig 2 Principle of Zero Vibration Input Shaping Time Input Unshaped input x t A1 A2 Total From A1 From A2 Time Shaped input x I t 1 2 Fig 3 Input shaping by convolving input with an impulse sequence I t Aiif t i 0otherwise 3 Once a vibration free impulse sequence has been estab lished it is possible to modify any desired system input x t to be vibration free by convolving the desired input with the impulse sequence as defi ned by 4 where x I t is called the shaped input x I t n X i 1 Ai x t i 4 This process is called shaping and is illustrated by Fig 3 Here it is illustrated how the shaped input corresponds to the sum of n scaled and delayed copies of the original input The next section presents a new segmentation strategy named 3 segmented input shaping for conducting piece wise input shaping in confi guration dependent systems III 3 SEGMENTED INPUT SHAPING Our goal is to generate an RTR trajectory for a general fl exible system from position xato position xb while not introducing residual vibrations in the system output Com mon RTR trajectories have an acceleration phase a constant velocity phase and a deceleration phase This makes it pos sible to defi ne fast trajectories with bounded velocities We have observed that for this type of trajectory vibrations are induced in the acceleration deceleration phases Hence we assume that if both acceleration and deceleration is executed without residual vibrations then the complete motion will also be without residual vibrations Different acceleration strategies have been developed with different levels of smoothness to reduce vibrations 5 In this work we focus on the BCB trajectory which is illustrated in Fig 4 We choose to employ the BCB trajectory since it re quires instantaneous acceleration hence short traveling times and high level of residual vibration making it suitable to stress our new segmentation strategy The same segmentation 4362 0 0 02 0 04 d tcoast Position x t t1t2t3 xa xb 0 0 05 0 1 0 15 v Velocity x t 00 10 20 30 40 5 2 0 2 a tacc tdec Time t Acceleration x t Fig 4 Bang coast bang trajectory approach could be used in combination with any acceleration strategy Since the vibration frequency has potentially changed during the motion the acceleration and deceleration profi les must be designed separately to suppress vibrations at their individual frequencies Here it is an approximation that the frequency of vibration during acceleration is approximately constant and equal to the frequency in the initial position xa and vice versa during deceleration Therefore we segment the trajectory into 3 segments namely an acceleration segment a coast segment and a deceleration segment The acceleration and deceleration seg ments are designed by individual input shaping and they are connected by a coast segment of constant velocity An example of a resulting trajectory is illustrated in Fig 5 In the example the system is moved to a confi guration with higher frequency than the initial confi guration The example employs the 2 impulse ZV type shaper 13 for clarity The rest of this section will elaborate on the methodology of describing the position velocity and acceleration profi les for this new type of trajectory The fi rst point is to determine the duration of the shaped acceleration and deceleration segments Taccand Tdec as described by 5 6 where taccand tdecare the duration of the unshaped acceleration and deceleration and where a n and b n are the delays of latest impulses in the shaper impulse sequences based on the system dynamics of position xaand xb respectively Tacc tacc a n 5 Tdec tdec b n 6 The distance of the shaped acceleration segment Dacc can be determined by 8 where X1is the position after the acceleration segment As seen from 7 x t is shaped by the impulse sequence of position xaat the time Tacc The position after the coast segment X2 can be found from 0 0 02 0 04 Dacc Ddec Dcoast Tcoast Position X t T1T2T3 X1 X2 0 0 05 0 1 0 15 Velocity X t 00 10 20 30 40 5 2 0 2 tacc a n Tacc tdec bn Tdec Time t Acceleration X t Fig 5 Trajectory with 3 segmented input shaping 9 where t2is the starting time of the unshaped deceleration segment as illustrated in Fig 4 Then Ddeccan be determined from 10 X1 x Ia Tacc 7 Dacc X1 xa 8 X2 x Ib t2 9 Ddec xb X2 10 With Daccand Ddec in place it is straight forward to fi nd the distance and duration of the coast segment Dcoastand Tcoast as in 11 12 where d is the total traveled distance of the trajectory and v is the maximum absolute velocity of the trajectory Dcoast d Dacc Ddec 11 Tcoast Dcoast v sgn d 12 The timing of the segment boundaries can found by 13 T1 T2 T3 Tacc Tacc Tcoast Tacc Tcoast Tdec 13 And fi nally the position velocity and acceleration profi les with 3 segmented input shaping are described by 14 15 and 16 X t x Ia t for t T1 X1 sgn d v t T1 for T1 t T2 x Ib t T2 t2 for T2 t 14 X t x Ia t for t T1 sgn d vfor T1 t T2 x Ib t T2 t2 for T2 t 15 4363 X t x Ia t for t T1 0for T1 t T2 x Ib t T2 t2 for T2 t 16 In 14 16 please note the time shift of the deceleration phase by T2 t2 This time shift is very essential since it ensures continuity in the trajectory Also note that due to the strict division of segments the presented approach to trajectory generation is only applica ble when the system reaches the maximum absolute velocity i e when sgn Dcoast sgn d 0 If this is not the case e g if the d is too small some other shaping rule must take effect Hence we have now developed a closed form description of an RTR trajectory which should result in zero residual vibration in systems with confi guration dependent dynamics The description is based on our hypothesis that if accel eration and deceleration is without residual vibration then the complete motion will be without residual vibration The description also relies on the approximation of piece wise constant dynamics during acceleration and deceleration The next section presents numerical simulations which validates the performance of the new approach to vibration free trajectory generation IV NUMERICAL SIMULATION AND ANALYSIS A set of numerical simulations have been performed to verify the new method The simulations have been performed for a rotational mass spring damper system with one degree of freedom as illustrated in Fig 6 The system is intended to move from a 0rad to b 1rad with ideally no residual vibrations assuming ideal control of the system input i t First a case study is presented for a specifi c set of parameters where 3 segmented input shaping is compared to constant frequency shaping Second a parameter sweep is performed to address the robustness to variations in system dynamics for 3 segmented input shaping compared to fi nal frequency shaping J o t k cInput i t Output o t Fig 6 Rotational spring mass damper system A CASE STUDY In this case study a specifi c set of system parameters has been choosen to illustrate the similarities and differences in the behavior of the new method compared to constant frequency shaping The parameters of the dynamic system and the desired trajectory are presented in Table I The applied impulse sequences are of the 2 impulse ZV type shaper described by Singer and Seering 13 To imitate the confi guration dependent inertia of a robot the mass moment of inertia of the system J o t is made linearly dependent on the output position o t as presented TABLE I PARAMETERS OF THE SIMULATED SYSTEM AND TRAJECTORY ParameterSymbolValueUnit Mass moment of inertiaJ0 395 1 102 o t kgm2 Spring stiffnessk4000Nm Damping coeffi cientc2Nms Initial position a0rad Final position b1rad Max velocity 4s 1 Max acceleration 90s 2 TABLE II COMPARED RESULTS OF CASE STUDY MethodDelayResidual vibration Unshaped0 100 Initial frequency11 9 67 9 Average frequency17 3 41 3 Final frequency21 4 37 7 3 segmented shaping15 3 1 7 in Table I The change in position means a change in J from Ja 0 395kgm2to Jb 1 41kgm2 which results in a change in the systems damped frequency from fa d 16Hz to fb d 8 5Hz and a change in damping ratio from a 0 0063 to b 0 0033 The numerical results are presented in Fig 7 while Table II compares the different approaches on the delay of reaching fi nal position and the amplitude of oscillations in acceleration at fi nished motion o T3 By inspecting Fig 7 it is readily seen that the 3 segmented input shaping trajectory is superior to the traditional input shaping based on the dynamics of either initial average or fi nal position in terms of suppressing residual vibrations The 3 segmented input shaping trajectory reduces the amount of residual vibration to 1 7 of the amount in the unshaped trajectory while the traditional shapers still induce 68 41 and 38 of the vibrations respectively even though the fi nal ZV shaper has a higher time delay It also seems that the assumption of suppressing vibra tions in acceleration and deceleration segments individually was correct When studying the system output of fi rst two segments of the 3 segmented output one immediately notice that the output is identical to the initial frequency output and has a low level of vibration prior the deceleration segment The results also show that the vibrations are not com pletely eliminated The main source of the remaining vi brations is the rough approximation of constant vibrational frequency during acceleration and deceleration segments which it is of course not Hence this method could be improved by an enhanced frequency estimate e g by using an average frequency of the individual segments However this would require numerical equation solving which is inconsistent with the desire of high computational effi ciency B PARAMETRIC SWEEP To validate the robustness of the new method a parameter sweep has been performed Here the chosen parameters to 4364 0 0 5 1 Pos rad Original motion t 2 0