High-Speed Smooth Cyclical Motion of a Hydraulic Cylinder

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1 846 1 th International Conference on Fleible Autoation and Intelligent Manufacturing 00, Dresden, Gerany High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder Nebojsa I. Jaksic Departent of Industrial Engineering University of Southern Colorado Pueblo, Colorado, 81001, USA 1 ABSRAC Many traditional achining operations involve object otion, either of a part or a tool. Changes in their velocity, including starting and stopping, result in the creation of shock forces that act upon the structural and operating ebers of a achine. hese forces depend largely on the inertia of the oving ebers. At high loads, where hydraulic actuation is predoinantly used, sooth velocity changes resulting in low shock forces are not easily realizable due to the low copressibility of hydraulic liquids. High-speed cyclical otion ay be a cause of further unwanted vibrations. his research describes and ipleents a nuber of coputer-generated and coputer-controlled sooth cyclical trajectories designed for high-speed highload devices. A coputer-controlled servo-valve is used to eecute high-speed cyclical otion of a hydraulic cylinder. A coparative study of various ipleented high-speed sooth trajectories is presented. 1. INRODUCION Cyclical otion is utilized in various anufacturing operations like shaping and planing in achining, assebly, aterial handling, etc. Actuation systes with hydraulic cylinders are often used for high-load precise cyclical otions. Speed optiization of such systes is subject to achine liitations, operation characteristics and econoic requireents. A hydraulic cylinder operating at high speeds ay cause shock-induced daage to equipent. Cyclical otion can induce daaging vibrations to a achine s structural and operating eleents as well as to its control syste. his research deals with high-speed sooth cyclical otion of a hydraulic cylinder. Disregarding the effects of abrupt transitions, the iniu-tie control solution can deterine the optial trajectory that can produce the fastest possible actuator otion as a function of its velocity and acceleration liits. In practice, during the eecution of such a trajectory, vibrations are easily ecited requiring a speed reduction. In this work, coputergenerated trajectories are created to eliinate shock-induced vibrations without changing the aiu velocity liits of hydraulic cylinders. 1 Phone (719) , Fa (719) , E-ail: njaksic@uscolo.edu, WWW:

2 High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder 847 Fro a given description of a desired otion, a trajectory planning syste generates a trajectory i.e. the tie sequence of interediate configurations between an initial and a final state. his inforation is translated into a set of coand signals sent to an actuator eecuting the otion. A good trajectory is well defined for all points along a given path, does not eceed the liits of the physical syste and ay be ipleented efficiently. In addition, for sooth otion, the resulting position, velocity, acceleration [1] and second acceleration [] should be continuous functions in tie. Certain otions of a echanis cause abnoral surface wear at the bearings due to the discontinuity of their coanded velocity and/or acceleration [3]. A discontinuity in coanded velocity requires infinite acceleration that cannot be obtained physically, while a discontinuity in coanded acceleration requires infinite second acceleration (jerk), that is again, physically unachievable. Based in part on atheatical definitions [4], a sooth function having a soothness order s is defined in [5]. A given function f() at the given point has soothness of order s if s consecutive derivatives of the function at the given point, starting with the first derivative, eist and are continuous at that point. his definition is epanded for the soothness of a function in an interval where the function is defined. For any two trajectories with orders of soothness s 1 and s, the following coparative soothness criteria can be etracted fro [5]: 1. if s 1 > s, then the trajectory with s 1 is soother,. if s 1 = s, then the trajectory with the lower absolute value of the aiu discontinuity of its first discontinuous derivative is soother and 3. if s 1 = s, and the values of aiu discontinuity of their first discontinuous derivatives are siilar, then the trajectory with the fewer nuber of discontinuities is soother. Sooth trajectories are coputer generated by using two different ethods [5]. In the polynoial segents ethod that utilizes principles of ca design [3], sooth trajectories are derived by iniizing the aiu values of the discontinuities on a given tie segent, and by applying successive integration. hen, eploiting syetry, a coplete trajectory is generated segent by segent. his ethod resebles the spline function approiation. he interpolating polynoials ethod is adapted fro nuerical analysis [6]. A sooth trajectory for the first half-cycle is obtained by foring a trajectory polynoial based on prescribed boundary conditions, while the second half-cycle trajectory is derived fro the otion syetry. In sections to follow, high-speed sooth cyclical trajectories will be created, analyzed and tested. hese trajectories will be obtained by etending analyses in [5] to include the aiu velocity liits of a hydraulic cylinder. A hydraulic cylinder is chosen for its high force-to-weight ratio and fast response tie [7].. HIGH-SPEED SMOOH CYCLICAL RAJECORIES For a body at rest to be oved soothly and at high-speed fro its initial position i to soe final position f and then back to i starting at tie t = 0 and finishing at tie t =

3 848 1 th International Conference on Fleible Autoation and Intelligent Manufacturing 00, Dresden, Gerany (cyclical otion), a high-speed sooth trajectory should be generated. If the conditions were changed to require the iniu tie for otion eecution (high-speed), then the fastest way to ove a body fro a specified initial point to a specified final point and back is by applying an ipulse on position, which is physically ipossible. herefore, aiu acceleration a and aiu deceleration in as well as aiu velocity a ust be constrained. Assue for the oent that the acceleration and deceleration are constant and have different absolute values. If the cyclical otion of a body of ass is observed in the interval fro t = 0 to t = (first half of the cycle), then the aount of energy needed to accelerate the body fro rest to soe aiu velocity is equal to the aount of energy needed to decelerate the body fro that aiu velocity to rest, not including nonlinear effects. If the transition fro acceleration to deceleration occurs at tie t 1, then for (t 1 ) being a point on the trajectory at tie t 1, the total work (not including friction and gravity and assuing straight line otion) ay be epressed as t ) [ ( ) ( t )] 0; [1] a ( 1 in 1 where a and in represent a constant acceleration and constant deceleration respectively. Using the above epression to iniize the forces acting on the body, the following results ay be derived: ( t ) ( ) / ; t /. [] a in; 1 1 his result eplains, fro the dynaic point of view, how the iniization of the aiu force, and therefore iniization of the aiu acceleration, leads to the aiization of soothness for the trajectory function. he syetry between four equal tie subintervals of a cycle is also obvious. Constrained aiu acceleration or, in general, a constrained acceleration function represents one of the actuator s liitations. Although not specifically considered, the lower aiu acceleration will iply the ore desirable trajectory. he second constraint iposed by a hydraulic cylinder actuation syste is the aiu velocity that can be achieved without daaging the actuator. Let aiu velocity of an original sooth trajectory be a. Depending on the value of the actuator s aiu allowed velocity aa, there are three cases that ay affect the sooth trajectory. In the first case, when the aiu allowed velocity is greater than the aiu velocity of a sooth trajectory ( aa a ), the original sooth trajectory does not change since the velocity liit is not reached. In the second case, when the aiu allowed velocity is lower or equal to the average velocity of a given sooth trajectory ( aa avg ) where avg 0. 5 a, the desired otion is not realizable. Finally, when the aiu allowed velocity is in the range aa ( avg, a ), the original velocity profile, and therefore sooth trajectory, ust be changed to accoodate an actual velocity liit. For this case, both high-speed sooth

4 High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder 849 and sooth otion characteristics are depicted in Fig. 1. he high-speed otion is derived fro the sooth otion by assuing equal paths traversed and equal total average velocities. If t denotes the tie during which the velocity of a high-speed otion is at aiu for a half-cycle, then the average velocity for the rest of the otion in the interval [0,] avghs can be epressed in ters of the aiu allowed velocity as avghs 0. 5 aa, [3] while t can be calculated fro t f i 0.5 aa 0.5 aa. [4] Figure 1: Coparison of a Sooth (Dashed Lines) and High-speed Sooth Motion with s = Obtained by using the Polynoial Segents Method

5 850 1 th International Conference on Fleible Autoation and Intelligent Manufacturing 00, Dresden, Gerany.1. GENERAING HIGH-SPEED SMOOH MOION BY USING HE POLYNOMIAL SEGMENS MEHOD A high-speed sooth trajectory with its derivatives can be generated by adapting the ethod described in [5]. For a desired order of soothness, the piecewise constant absolute value of the first discontinuous derivative is calculated by using a set of recursive equations: a t, [5] f i aa ( ) ( n ) a t ( 1) ( n 1) a for 1 n-1, [6] a n1 t ( n1) a for n. [7] In the above epressions, n denotes the order of the trajectory polynoial epressed by n = s + 1; ( ) a a represents the distance traversed at non-aiu velocity; ( n ) denotes the aiu value of the -th derivative of the trajectory having the degree n-; and a represents the piecewise constant absolute value of the first discontinuous derivative. A part of a trajectory with its derivatives is generated for a tie s segent t [0,( ) / ] by successive integration, starting with a. ( ) n t Let (0, t) be a function in tie of the first discontinuous derivative. hen (0, t) a for 0 t s (0, t) t (0, t s ) t for s t t t s1 t t t (0, t) (0, t ) for t [8] s1 s1 s t t t (0, t) (0, t ) for t. he otion equations for a part of otion eecuted at the aiu allowed velocity are ( 0) f i aa t (1, t) t for t aa, [9]

6 High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder 851 ( 1) t (0, t) for t aa and [10] ( ) t (0, t) 0 for t and,3,..., n. [11] Using equations [5] through [11] and successive integration, otion characteristics for the tie interval [0, /] can be calculated. Motion characteristics for the tie interval [/, ] can be obtained by utilizing various otion syetries resulting in the following equations: ( n, t) f i ( n, t) position for t [1] ( ) ( n, t) ( 1) 1 ( ) ( n, t)for 1,,3,..., n for t ( n, t) ( n, t) position ( ) ( ) ( n, t) ( n, t ) for 1,,3,..., n. [13].. EXAMPLE In this section the polynoial segents ethods will be used to generate a trajectory and its derivatives for a high-speed sooth otion. Let i = 0, f = 0.15, the cycle tie = s, and the order of soothness s =. he aiu allowed velocity ust be between 0.15 /s and 0.3 /s for a feasible high-speed otion. Let aa 0.5 / s. he order of the trajectory polynoial is n = s +1 = 3, which is the order of the first discontinuous derivative. he tie t during which the velocity is at aiu is calculated using Equation [4] which yields t = s. he segent size is (-t )/ s = (-t )/4 = According to Equations [8] through [11], the segents on the interval [0,/] are [0, (-t )/4], [(-t )/4, (-t )/], and [(-t )/, /]. o find the aiu value of the first discontinuous derivative (the third derivative in this case), Equations [5], [6] and [7] are applied to obtain (3) a = 0.075, () a, = 0.5 /s, () a = 1.35 /s and (3) a = 8.1 /s 3. Noting Equation set [8], (3) 3 t (0, t) 8.1 / s for 0 t, and 4 (3) 3 t t (0, t) 8.1 / s, for t. 4 Using successive integration with all starting conditions equal to zero, the lower integration liit equal to zero and the upper integration liit of t, otion equations for the segent [0, (-t )/4] are obtained: () 3 (1, t) 8.1t / s, (, t) 4.05t / s, and (3, t) 1.35t. [15] [14]

7 85 1 th International Conference on Fleible Autoation and Intelligent Manufacturing 00, Dresden, Gerany Successive integration with starting conditions deterined by values of functions at (t )/4, the lower liit of (-t )/4 and the upper integration liit equal to t, results in otion equations for the segent [(-t )/4, (-t )/], () (1, t) ( 8.1t.7) / s, (, t) ( 4.05t.7t 0.5) / s, and 3 (3, t) ( 1.35t 1.35t 0.5t 0.015). [16] Applying equations [9], through [11] yields the otion description for the tie interval [(-t )/, /], (3) (0, t) () (0, t) 0, (0, t) 0.5 / s, and (1, t) (0.5t ). [17] Substituting equations [14] through [17] into Equation set [1], a otion description is obtained for the tie interval [/, ] (3) (3) (0, t) (0, t), () (1, t) () (1, t), (, t) (, t), and (3, t) f i (3, t), [18] where f = 0.15, and i = 0. For the tie segent [, ] equations [14] through [17] are used in Equation set [13] yielding (3) () (0, t) (1, t) (, t) (3, t) () (3) (0, t ), (1, t ), (, t ), and (3, t). [19] A graphical presentation of otion equations for the above eaple is generated using MALAB, and is shown in Figure 1., where solid lines represent high-speed sooth otion and dashed lines represent a corresponding sooth otion..3. GENERAING HIGH-SPEED SMOOH MOION BY USING HE INERPOLAING POLYNOMIALS MEHOD A set of polynoial trajectories with derivatives will be obtained based on specified boundary conditions. he polynoials will be coputed for the first quarter-cycle of the otion. Due to the eistence of a constant velocity otion segent, the polynoials will

8 High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder 853 be generated for the tie segent [0, (-t )/], where, as defined in the previous section, represents the cycle tie and t denotes the tie for a half-cycle during which the velocity is at aiu. he otion equations describing the rest of the cycle will be derived by applying equations [9] through [11], and equation sets [1] and [13] SMOOH CYCLICAL MOION Reference [5] described the first part of a sooth cyclical otion in the interval [0,]. his does not yield an easy conversion to a high-speed sooth otion description. Here, a sooth otion on segent [0, /] will be derived first. A direct consequence is the lower nuber of boundary constraints (one less), and the lower order of trajectory polynoials. For a otion with an order of soothness s, the boundary conditions are: ( n,0) ; i ( n 1,0) 0; ( n 1, ) a ; () () ( n,0) 0; ( n, ) 0; ( s) ( s) ( n s,0) 0; [0] ( n s, ) 0, where n is the order of the trajectory polynoial and a is the aiu velocity developed at tie t = /. he nuber of boundary conditions, s + 1, results in an equal nuber of polynoial coefficients thus deterining the order of the trajectory polynoial, n s. [1] he condition ( n, / ) ( f i ) / is a dependent condition and therefore it is not used. o deterine the equations of otion, the trajectory polynoial and all its continuous derivatives are epressed as ( ) i! i ( n, t) cit for 0,1,,..., s. [] ( i )! in Substituting boundary conditions for t = 0 in Equation [] leads to c i, and ci 0 for i 1,,3,..., s. [3] 0 Substituting coefficients fro Equation [3] into Equation [] and applying boundary conditions for t = / fro Equation [0] results in s1 i n i! ci ( ( i )! / ) i( s1) a / 0 s or epressed as a syste of linear equations in atri for, for 1 for,3,4,..., s, [4]

9 854 1 th International Conference on Fleible Autoation and Intelligent Manufacturing 00, Dresden, Gerany n!! ( 1)! ( ) n ( s 1) i ( ) i ( s 1) s ( n 1)! ( i 1)! s! c s n a ( ) n! 0 n ( s 1) i! i ( s 1) ( s 1)! ( ) ( ) ci ( )! ( )! ( 1 )! 0 [5] n i s n! n( s1) i! i ( s1) ( s 1)! c 0 ( ) ( ) s 1 ( n s)! ( i s)! 0! If Equation [5] is epressed as AC = B, [6] where A represents the first s s atri in Equation [5], C represents the s 1 coefficient vector and B represents the s 1 vector of boundary conditions fro Equation [5], then the solution of the above equation for the reaining polynoial coefficients is C = A -1 B. [7] After substituting all the coefficients into Equation [], the otion characteristics can be coputed for the interval [0, /]..3.. HIGH-SPEED SMOOH CYCLICAL MOION he description of a high-speed sooth cyclical otion can be easily obtained fro the description of a sooth cyclical otion. Boundary conditions at t = 0 are equal for both otions resulting in equal polynoial coefficients c i for i = 1,, 3,, s. he first tie segent in which the otion equations are derived is now [0, (-t )/], and the aiu allowed velocity aa is achieved at tie t = (-t )/, as in the polynoial segents ethod. o deterine the rest of the polynoial coefficients (c i for i = s +1,,n) in Equations [4] and [5] variable is replaced with t, and variable a is replaced with aa, and the solution obtained as in Equation [7]. As an illustration of the interpolating polynoials ethod, the eaple in Section. is revisited. Equations of a high-speed sooth otion are deterined for i = 0, f = 0.15, cycle tie = s, the order of soothness s = and aa 0.5 / s. Fro Equation [4], the tie t during which the velocity is at aiu is t = s. he order of the trajectory polynoial is n = s = 4, and fro Equation [] the equations of otion are 4 3 (4, t) c4t c3t ct c1t c0 3 (3, t) 4c4t 3c3t ct c1 () (, t) 1c4t 6c3t c. [8]

10 High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder 855 Fro boundary conditions (4,0) i 0, (3,0) 0, and (,0) 0 follows c 0 c1 c 0. Coefficients c 3 and c 4 are deterined fro Equation [5] which is adapted for high-speed sooth otion: c4.05, [9] 4 6 c3 0 resulting in c 4 = /s 4 and c 3 =.05 /s 3. he equations of otion for the tie segent [0, (-t )/] are 4 3 (4, t) t.05t (3, t) 1.15t t / s () (, t) 36.45t 1.5t / s. he equation of the third discontinuous derivative is (3) ( 1, t) 48.6t 1.5 / s 3. A graph representing high-speed sooth (solid lines) and sooth otion (dashed lines) is shown in Fig.. () [30] Figure : Coparison of a Sooth (Dashed Lines) and High-speed Sooth Motion with s = Obtained by using the Interpolating Polynoials Method

11 856 1 th International Conference on Fleible Autoation and Intelligent Manufacturing 00, Dresden, Gerany 3. EXPERIMENAL RESULS High-speed sooth cyclical otions were ipleented using a test bed consisting of a hydraulic cylinder, a MOOG servo-valve, a variable-displaceent pup run by an AC otor, and a real-tie data acquisition and control ultiprocessor syste including position, velocity and pressure transducers. o observe ore closely the behavior of the hydraulic cylinder, pressure was easured instead of acceleration. Control syste inputs were trajectories and velocity profiles generated by applying previously derived ethods, as well as sensory inforation. he output consisted of coands sent to the servo-valve. As in [5], the ipleented control algorith utilized a position feedback control schee with siplified inverse plant feedforward. Figure 3: Coanded (Dashed Lines) and Actual Velocity Profiles of a High-Speed Sooth Motion of a Hydraulic Cylinder with s = Based on the Interpolating Polynoials Method Figure 4: Pressure profile for a High-Speed Sooth Motion of a Hydraulic Cylinder with s = Based on the Interpolating Polynoials Method

12 High-Speed Sooth Cyclical Motion of a Hydraulic Cylinder 857 A nuber of high-speed otions with various orders of soothness and aiu allowed velocities were tested. Data sets recorded were actual position, velocity and pressure with corresponding coands on position and velocity. Ecept for very high coanded aiu allowed velocities, actual trajectories closely followed the coanded values (with a brief tie delay) as presented theoretically in Figures 1. and. Figure 3. presents a typical eaple of a coanded velocity profile (dashed lines) and an actual one (solid line). In the eaple, the stroke of the hydraulic cylinder was 0.154, and the aiu allowed velocity was 0.58 /s. Actual pressure inforation is presented in Figure 4. It shows siilarities with the acceleration plot in Figure. Mild overshoots at abrupt changes in pressure are observed. 4. CONCLUSIONS wo ethods in generating high-speed sooth trajectories for hydraulic cylinders were derived. Algoriths for generating otion characteristics using the polynoial segents ethod and using the interpolating polynoials ethod were presented. he trajectories generated by the interpolating segents ethod are soewhat soother than the trajectories created by the polynoial segents ethod for the sae order of soothness, since they have fewer discontinuities at the first discontinuous derivative. However, trajectory polynoials generated by the interpolating polynoials ethod are of higher order thus aking the less proising candidates for trajectories generated in real-tie for high-speed otions. he ipleentation of high-speed sooth otions was perfored using a coputer-controlled syste with a hydraulic cylinder. High-speed sooth cyclical otions of a hydraulic cylinder generated by either ethod were observed for various aiu allowed velocities and cycle ties. he ajor constraint on otion soothness at very high velocities and short strokes was contributed to the syste s hardware liits on response tie. REFERENCES [1] hopson, S. E. and Patel, R. V.: "Forulation of Joint rajectories for Industrial Robots Using B- splines", IEEE ransactions on Industrial Electronics, vol. IE-34, No., pp , [] Brady, M., Hollerbach, J. M., Johnson,. L., Lozano-Perez,. and Mason, M..: Robot Motion: Planning and Control, MI Press, Cabridge, [3] Shigley, J. E. and Uicker, J. J.: heory of Machines and Mechaniss, nd Edition, McGraw-Hill, New York, [4] Marsden, J. E. and Hoffan, M. J.: Eleentary classical Analysis, W. H. Freean, [5] Jaksic, N. I.: Generating Sooth Cyclical Motions for a Hydraulic Cylinder International Conference on Fleible Autoation and Intelligent Manufacturing 001, Dublin, Ireland, pp , 001. [6] Isaacson, E. and Keller, H. B.: Analysis of Nuerical Methods, John Wiley & Sons, New York, [7] Niksefat, N. and Sepehri, N.: Designing Robust Force Control of Hydraulic Actuators Despite Syste and Environental Uncertainties IEEE Control Systes Magazine, vol. 1, No., pp , April 001.