high-speed smooth cyclical motion of a hydraulic...

846 12th International Conference on Flexible Automation and Intelligent Manufacturing 2002, Dresden, Germany

High-Speed Smooth Cyclical Motion of a Hydraulic Cylinder

Nebojsa I. Jaksic Department of Industrial Engineering

University of Southern Colorado Pueblo, Colorado, 81001, USA1

ABSTRACT

Many traditional machining operations involve object motion, 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 members of a machine. These forces depend largely on the inertia of the moving members. At high loads, where hydraulic actuation is predominantly used, smooth velocity changes resulting in low shock forces are not easily realizable due to the low compressibility of hydraulic liquids. High-speed cyclical motion may be a cause of further unwanted vibrations. This research describes and implements a number of computer-generated and computer-controlled smooth cyclical trajectories designed for high-speed high-load devices. A computer-controlled servo-valve is used to execute high-speed cyclical motion of a hydraulic cylinder. A comparative study of various implemented high-speed smooth trajectories is presented.

1. INTRODUCTION

Cyclical motion is utilized in various manufacturing operations like shaping and planing in machining, assembly, material handling, etc. Actuation systems with hydraulic cylinders are often used for high-load precise cyclical motions. Speed optimization of such systems is subject to machine limitations, operation characteristics and economic requirements. A hydraulic cylinder operating at high speeds may cause shock-induced damage to equipment. Cyclical motion can induce damaging vibrations to a machine’s structural and operating elements as well as to its control system.

This research deals with high-speed smooth cyclical motion of a hydraulic cylinder. Disregarding the effects of abrupt transitions, the minimum-time control solution can determine the optimal trajectory that can produce the fastest possible actuator motion as a function of its velocity and acceleration limits. In practice, during the execution of such a trajectory, vibrations are easily excited requiring a speed reduction. In this work, computer-generated trajectories are created to eliminate shock-induced vibrations without changing the maximum velocity limits of hydraulic cylinders. 1 Phone (719) 549-2112, Fax (719) 549-2519, E-mail: [email protected], WWW: http://faculty.uscolo.edu/njaksic

High-Speed Smooth Cyclical Motion of a Hydraulic Cylinder 847

From a given description of a desired motion, a trajectory planning system generates a trajectory i.e. the time sequence of intermediate configurations between an initial and a final state. This information is translated into a set of command signals sent to an actuator executing the motion. A good trajectory is well defined for all points along a given path, does not exceed the limits of the physical system and may be implemented efficiently. In addition, for smooth motion, the resulting position, velocity, acceleration [1] and second acceleration [2] should be continuous functions in time.

Certain motions of a mechanism cause abnormal surface wear at the bearings due to the discontinuity of their commanded velocity and/or acceleration [3]. A discontinuity in commanded velocity requires infinite acceleration that cannot be obtained physically, while a discontinuity in commanded acceleration requires infinite second acceleration (jerk), that is again, physically unachievable.

Based in part on mathematical definitions [4], a smooth function having a smoothness order sm is defined in [5]. “A given function f(x) at the given point has smoothness of order sm if sm consecutive derivatives of the function at the given point, starting with the first derivative, exist and are continuous at that point”. This definition is expanded for the smoothness of a function in an interval x where the function is defined. For any two trajectories with orders of smoothness sm1 and sm2, the following comparative smoothness criteria can be extracted from [5]:

1. if sm1 > sm2, then the trajectory with sm1 is smoother,

2. if sm1 = sm2, then the trajectory with the lower absolute value of the maximum discontinuity of its first discontinuous derivative is smoother and

3. if sm1 = sm2, and the values of maximum discontinuity of their first discontinuous derivatives are similar, then the trajectory with the fewer number of discontinuities is smoother.

Smooth trajectories are computer generated by using two different methods [5]. In the polynomial segments method that utilizes principles of cam design [3], smooth trajectories are derived by minimizing the maximum values of the discontinuities on a given time segment, and by applying successive integration. Then, exploiting symmetry, a complete trajectory is generated segment by segment. This method resembles the spline function approximation. The interpolating polynomials method is adapted from numerical analysis [6]. A smooth trajectory for the first half-cycle is obtained by forming a trajectory polynomial based on prescribed boundary conditions, while the second half-cycle trajectory is derived from the motion symmetry.

In sections to follow, high-speed smooth cyclical trajectories will be created, analyzed and tested. These trajectories will be obtained by extending analyses in [5] to include the maximum velocity limits of a hydraulic cylinder. A hydraulic cylinder is chosen for its high force-to-weight ratio and fast response time [7].

2. HIGH-SPEED SMOOTH CYCLICAL TRAJECTORIES

For a body at rest to be moved smoothly and at high-speed from its initial position xi to some final position xf and then back to xi starting at time t = 0 and finishing at time t = 2T


(cyclical motion), a high-speed smooth trajectory should be generated. If the conditions were changed to require the minimum time for motion execution (high-speed), then the fastest way to move a body from a specified initial point to a specified final point and back is by applying an impulse on position, which is physically impossible. Therefore, maximum acceleration maxx�� and maximum deceleration minx�� as well as maximum velocity maxx� must be constrained.

Assume for the moment that the acceleration and deceleration are constant and have different absolute values. If the cyclical motion of a body of mass m is observed in the interval from t = 0 to t = T (first half of the cycle), then the amount of energy needed to accelerate the body from rest to some maximum velocity is equal to the amount of energy needed to decelerate the body from that maximum velocity to rest, not including nonlinear effects. If the transition from acceleration to deceleration occurs at time t1, then for x(t1) being a point on the trajectory at time t1, the total work (not including friction and gravity and assuming straight line motion) may be expressed as

;0)]()([)( 11 �� txTxxmtxxm minmax �� [1]

where maxx�� and minx�� represent a constant acceleration and constant deceleration respectively. Using the above expression to minimize the forces acting on the body, the following results may be derived:

.2/;2/)()(; 11 TtTxtxxx minmax �� [2]

This result explains, from the dynamic point of view, how the minimization of the maximum force, and therefore minimization of the maximum acceleration, leads to the maximization of smoothness for the trajectory function. The symmetry between four equal time subintervals of a cycle is also obvious.

Constrained maximum acceleration or, in general, a constrained acceleration function represents one of the actuator’s limitations. Although not specifically considered, the lower maximum acceleration will imply the more desirable trajectory. The second constraint imposed by a hydraulic cylinder actuation system is the maximum velocity that can be achieved without damaging the actuator.

Let maximum velocity of an original smooth trajectory be maxx� . Depending on the value of the actuator’s maximum allowed velocity maxax� , there are three cases that may affect the smooth trajectory. In the first case, when the maximum allowed velocity is greater than the maximum velocity of a smooth trajectory ( maxmaxa xx �� ), the original smooth trajectory does not change since the velocity limit is not reached. In the second case, when the maximum allowed velocity is lower or equal to the average velocity of a given smooth trajectory ( avgmaxa xx �� ) where maxavg xx �� 5.0 , the desired motion is not realizable. Finally, when the maximum allowed velocity is in the range

),( maxavgmaxa xxx �� , the original velocity profile, and therefore smooth trajectory, must be changed to accommodate an actual velocity limit. For this case, both high-speed smooth


and smooth motion characteristics are depicted in Fig. 1. The high-speed motion is derived from the smooth motion by assuming equal paths traversed and equal total average velocities.

If tm denotes the time during which the velocity of a high-speed motion is at maximum for a half-cycle, then the average velocity for the rest of the motion in the interval [0,T]

avghsx� can be expressed in terms of the maximum allowed velocity as

maxaavghs xx �� 5.0� , [3]

while tm can be calculated from

maxa

maxaifm x

Txxxt

�

�

5.0

5.0��

� . [4]

Figure 1: Comparison of a Smooth (Dashed Lines) and High-speed Smooth Motion with sm = 2 Obtained by using the Polynomial Segments Method


2.1. GENERATING HIGH-SPEED SMOOTH MOTION BY USING THE POLYNOMIAL SEGMENTS METHOD

A high-speed smooth trajectory with its derivatives can be generated by adapting the method described in [5]. For a desired order of smoothness, the piecewise constant absolute value of the first discontinuous derivative is calculated by using a set of recursive equations:

,)()0(mmaxaifmax txxxnx �� [5]

,for)1(2)( )1()( n-1m1mnxtT

mnx maxm

m

m

maxm

��

�

�� [6]

.for)1(2)0( )1(1

)( nmxtT

x maxn

m

n

maxn

�

�

��

�

[7]

In the above expressions, n denotes the order of the trajectory polynomial expressed by n = sm + 1; maxnx )()0( represents the distance traversed at non-maximum velocity;

maxm mnx )()(

� denotes the maximum value of the m-th derivative of the trajectory having

the degree n-m; and maxnx )0()( represents the piecewise constant absolute value of the first

discontinuous derivative. A part of a trajectory with its derivatives is generated for a time segment ]2/)(,0[ sm

mtTt �� by successive integration, starting with maxnx )0()( .

Let ),0()( tx n be a function in time of the first discontinuous derivative. Then

smm

maxnn tT

txtx2

0for)0(),0( )()( ��

1)()(

22for)

2,0(),0(

�

��

�� sm

msm

msm

mnn tTt

tTtTtxtx

211)()(

22for)

2,0(),0(

��

��

�� sm

msm

msm

mnn tTt

tTtTtxtx [8]

.22

for)2

,0(),0( 22)()( mmmnn tT

ttTtT

txtx�

��

��

�

The motion equations for a part of motion executed at the maximum allowed velocity are

,22

for2

),1()0( TttT

txTxxx

tx mmaxa

maxaif��

��

��

� �

�

[9]


22for),0()1( Tt

tTxtx m

maxa ��

� � and [10]

.,...,3,2and22

for0),0()( nmTttT

tx mm��

�� [11]

Using equations [5] through [11] and successive integration, motion characteristics for the time interval [0, T/2] can be calculated. Motion characteristics for the time interval [T/2, 2T] can be obtained by utilizing various motion symmetries resulting in the following equations:

��

��

�

��

��

nmtTmnmxmtmnmx

tTnxixfxtnxTtT

,...,3,2,1for),()(1)1(),()(position),()0(),()0(

2for [12]

��

��

��

��

.,...,3,2,1for),(),(position)2,(),(2for )()(

)0()0(

nmTtmnxtmnxtTnxtnxTtT mm [13]

2.2. EXAMPLE

In this section the polynomial segments methods will be used to generate a trajectory and its derivatives for a high-speed smooth motion. Let xi = 0, xf = 0.15 m, the cycle time 2T = 2 s, and the order of smoothness sm = 2. The maximum allowed velocity must be between 0.15 m/s and 0.3 m/s for a feasible high-speed motion. Let smxmaxa /225.0�� .

The order of the trajectory polynomial is n = sm +1 = 3, which is the order of the first discontinuous derivative. The time tm during which the velocity is at maximum is calculated using Equation [4] which yields tm = 0.333 s. The segment size is (T-tm)/2sm = (T-tm)/4 = 0.16667. According to Equations [8] through [11], the segments on the interval [0,T/2] are [0, (T-tm)/4], [(T-tm)/4, (T-tm)/2], and [(T-tm)/2, T/2]. To find the maximum value of the first discontinuous derivative (the third derivative in this case), Equations [5], [6] and [7] are applied to obtain x(0)(3)max= 0.075 m, x(1)(2)max,= 0.225 m/s, x(2)(1)max= 1.35 m/s2 and x(3)(0)max= 8.1 m/s3. Noting Equation set [8],

.24

for,/1.8),0(

and,4

0for/1.8),0(

3)3(

3)3(

mm

m

tTt

tTsmtx

tTtsmtx

��

��

��

[14]

Using successive integration with all starting conditions equal to zero, the lower integration limit equal to zero and the upper integration limit of t, motion equations for the segment [0, (T-tm)/4] are obtained:

.35.1),3(and,/05.4),2(,/1.8),1( 3)0(2)1(2)2( mttxsmttxsmttx �� [15]


Successive integration with starting conditions determined by values of functions at (T-tm)/4, the lower limit of (T-tm)/4 and the upper integration limit equal to t, results in motion equations for the segment [(T-tm)/4, (T-tm)/2],

.)0125.0225.035.135.1(),3(

and,/)225.07.205.4(),2(

,/)7.21.8(),1(

23)0(

2)1(

2)2(

mttttx

smtttx

smttx

��

��

��

[16]

Applying equations [9], through [11] yields the motion description for the time interval [(T-tm)/2, T/2],

.)0375.0225.0(),1(

and,/225.0),0(

,0),0(),0(

)0(

)1(

)2()3(

mttx

smtx

txtx

��

�

��

[17]

Substituting equations [14] through [17] into Equation set [12], a motion description is obtained for the time interval [T/2, T]

),,3(),3(

and),,2(),2(

),,1(),1(

),,0(),0(

)0()0(

)1()1(

)2()2(

)3()3(

tTxxxtx

tTxtx

tTxtx

tTxtx

if ��

��

��

��

[18]

where xf = 0.15, and xi = 0. For the time segment [T, 2T] equations [14] through [17] are used in Equation set [13] yielding

).2,3(),3(

and),,2(),2(

),,1(),1(

),,0(),0(

)0()0(

)1()1(

)2()2(

)3()3(

tTxtx

Ttxtx

Ttxtx

Ttxtx

��

��

��

��

[19]

A graphical presentation of motion equations for the above example is generated using MATLAB, and is shown in Figure 1., where solid lines represent high-speed smooth motion and dashed lines represent a corresponding smooth motion.

2.3. GENERATING HIGH-SPEED SMOOTH MOTION BY USING THE INTERPOLATING POLYNOMIALS METHOD

A set of polynomial trajectories with derivatives will be obtained based on specified boundary conditions. The polynomials will be computed for the first quarter-cycle of the motion. Due to the existence of a constant velocity motion segment, the polynomials will


be generated for the time segment [0, (T-tm)/2], where, as defined in the previous section, T represents the cycle time and tm denotes the time for a half-cycle during which the velocity is at maximum. The motion equations describing the rest of the cycle will be derived by applying equations [9] through [11], and equation sets [12] and [13].

2.3.1. SMOOTH CYCLICAL MOTION

Reference [5] described the first part of a smooth cyclical motion in the interval [0,T]. This does not yield an easy conversion to a high-speed smooth motion description. Here, a smooth motion on segment [0, T/2] will be derived first. A direct consequence is the lower number of boundary constraints (one less), and the lower order of trajectory polynomials. For a motion with an order of smoothness sm, the boundary conditions are:

,0)2

,(;0)2

,2(;)2

,1(

;0)0,(;0)0,2(;0)0,1(;)0,(

)()2()1(

)()2()1()0(

��

��

TsmnxTnxxTnx

smnxnxnxxnx

smmax

smi

��

�

[20]

where n is the order of the trajectory polynomial and maxx� is the maximum velocity developed at time t = T/2. The number of boundary conditions, 2sm + 1, results in an equal number of polynomial coefficients thus determining the order of the trajectory polynomial,

.2smn � [21]

The condition 2/)()2/,()0(if xxTnx �� is a dependent condition and therefore it is

not used. To determine the equations of motion, the trajectory polynomial and all its continuous derivatives are expressed as

� ��

��

�

m

ni

mii

m smmtcmi

itmnx ,...,2,1,0for)!(

!),()( . [22]

Substituting boundary conditions for t = 0 in Equation [22] leads to

smicxc ii ,...,3,2,1for0and,0 �� . [23]

Substituting coefficients from Equation [23] into Equation [22] and applying boundary conditions for t = T/2 from Equation [20] results in

� ��

�

�

��

��

��

�

�

�

��1 )1(

,...,4,3,2for1for

02/)2/(

)!(!sm

ni

smmaxsmi

i smmmTxTc

mii � , [24]

or expressed as a system of linear equations in matrix form,


��

�

�

��

�

��

�

��

�

�

��

�

�

�

��

�

�

��

�

�

�

��

��

��

0

00

)2

(

!0)!1()

2(

)!(!)

2(

)!(!

)!1()!1()

2(

)!(!)

2(

)!(!

!)!1()

2(

)!1(!)

2(

)!1(!

1)1()1(

)1()1(

)1()1(

�

�

�

�

��

��

��

��

�� smmax

sm

i

n

smismn

smismn

smismnTx

c

c

c

smTsmiiT

smnn

msmsmT

miiT

mnn

smsmT

iiT

nn

[25]

If Equation [25] is expressed as

AC = B, [26]

where A represents the first sm x sm matrix in Equation [25], C represents the sm x 1 coefficient vector and B represents the sm x 1 vector of boundary conditions from Equation [25], then the solution of the above equation for the remaining polynomial coefficients is

C = A-1B. [27]

After substituting all the coefficients into Equation [22], the motion characteristics can be computed for the interval [0, T/2].

2.3.2. HIGH-SPEED SMOOTH CYCLICAL MOTION

The description of a high-speed smooth cyclical motion can be easily obtained from the description of a smooth cyclical motion. Boundary conditions at t = 0 are equal for both motions resulting in equal polynomial coefficients ci for i = 1, 2, 3,…, sm. The first time segment in which the motion equations are derived is now [0, (T-tm)/2], and the maximum allowed velocity maxax� is achieved at time t = (T-tm)/2, as in the polynomial segments method. To determine the rest of the polynomial coefficients (ci for i = sm +1,…,n) in Equations [24] and [25] variable T is replaced with T – tm, and variable maxx� is replaced with maxax� , and the solution obtained as in Equation [27].

As an illustration of the interpolating polynomials method, the example in Section 2.2 is revisited. Equations of a high-speed smooth motion are determined for xi = 0, xf = 0.15 m, cycle time 2T = 2 s, the order of smoothness sm = 2 and smxmaxa /225.0�� . From Equation [4], the time tm during which the velocity is at maximum is tm = 0.333 s.

The order of the trajectory polynomial is n = 2sm = 4, and from Equation [22] the equations of motion are

.2612),2(

234),3(

),4(

232

4)2(

122

33

4)1(

012

23

34

4)0(

ctctctx

ctctctctx

ctctctctctx

��

��

��

[28]


From boundary conditions 0)0,2(and,0)0,3(,0)0,4( )2()1()0(�� xxxx i follows

.0210 �� ccc Coefficients c3 and c4 are determined from Equation [25] which is adapted for high-speed smooth motion:

��

��

��

�

��

��

��

�

��

�

0025.2

6433333.1

3

4cc

, [29]

resulting in c4 =-3.0375 m/s4 and c3 = 2.025 m/s3. The equations of motion for the time segment [0, (T-tm)/2] are

./5.1245.36),2(

/075.615.12),3(

025.20375.3),4(

22)2(

23)1(

34)0(

smtttx

smtttx

mtttx

��

��

��

[30]

The equation of the third discontinuous derivative is 3)3( /5.126.48),1( smttx �� . A graph representing high-speed smooth (solid lines) and smooth motion (dashed lines) is shown in Fig. 2.

Figure 2: Comparison of a Smooth (Dashed Lines) and High-speed Smooth Motion with sm = 2 Obtained by using the Interpolating Polynomials Method


3. EXPERIMENTAL RESULTS

High-speed smooth cyclical motions were implemented using a test bed consisting of a hydraulic cylinder, a MOOG servo-valve, a variable-displacement pump run by an AC motor, and a real-time data acquisition and control multiprocessor system including position, velocity and pressure transducers. To observe more closely the behavior of the hydraulic cylinder, pressure was measured instead of acceleration. Control system inputs were trajectories and velocity profiles generated by applying previously derived methods, as well as sensory information. The output consisted of commands sent to the servo-valve. As in [5], the implemented control algorithm utilized a position feedback control scheme with simplified inverse plant feedforward.

Figure 3: Commanded (Dashed Lines) and Actual Velocity Profiles of a High-Speed Smooth Motion of a Hydraulic Cylinder with sm = 2 Based on the Interpolating Polynomials Method

Figure 4: Pressure profile for a High-Speed Smooth Motion of a Hydraulic Cylinder with sm = 2 Based on the Interpolating Polynomials Method


A number of high-speed motions with various orders of smoothness and maximum allowed velocities were tested. Data sets recorded were actual position, velocity and pressure with corresponding commands on position and velocity. Except for very high commanded maximum allowed velocities, actual trajectories closely followed the commanded values (with a brief time delay) as presented theoretically in Figures 1. and 2. Figure 3. presents a typical example of a commanded velocity profile (dashed lines) and an actual one (solid line). In the example, the stroke of the hydraulic cylinder was 0.1524 m, and the maximum allowed velocity was 0.58 m/s. Actual pressure information is presented in Figure 4. It shows similarities with the acceleration plot in Figure 2. Mild overshoots at abrupt changes in pressure are observed.

4. CONCLUSIONS

Two methods in generating high-speed smooth trajectories for hydraulic cylinders were derived. Algorithms for generating motion characteristics using the polynomial segments method and using the interpolating polynomials method were presented. The trajectories generated by the interpolating segments method are somewhat smoother than the trajectories created by the polynomial segments method for the same order of smoothness, since they have fewer discontinuities at the first discontinuous derivative. However, trajectory polynomials generated by the interpolating polynomials method are of higher order thus making them less promising candidates for trajectories generated in real-time for high-speed motions. The implementation of high-speed smooth motions was performed using a computer-controlled system with a hydraulic cylinder. High-speed smooth cyclical motions of a hydraulic cylinder generated by either method were observed for various maximum allowed velocities and cycle times. The major constraint on motion smoothness at very high velocities and short strokes was contributed to the system’s hardware limits on response time.

REFERENCES

[1] Thompson, S. E. and Patel, R. V.: "Formulation of Joint Trajectories for Industrial Robots Using B-splines", IEEE Transactions on Industrial Electronics, vol. IE-34, No. 2, pp. 192 -199, 1987.

[2] Brady, M., Hollerbach, J. M., Johnson, T. L., Lozano-Perez, T. and Mason, M. T.: Robot Motion: Planning and Control, MIT Press, Cambridge, 1983.

[3] Shigley, J. E. and Uicker, J. J.: Theory of Machines and Mechanisms, 2nd Edition, McGraw-Hill, New York, 1994.

[4] Marsden, J. E. and Hoffman, M. J.: Elementary classical Analysis, W. H. Freeman, 1993.

[5] Jaksic, N. I.: ”Generating Smooth Cyclical Motions for a Hydraulic Cylinder” International Conference on Flexible Automation and Intelligent Manufacturing 2001, Dublin, Ireland, pp. 167 – 176, 2001.

[6] Isaacson, E. and Keller, H. B.: Analysis of Numerical Methods, John Wiley & Sons, New York, 1994.

[7] Niksefat, N. and Sepehri, N.: “Designing Robust Force Control of Hydraulic Actuators Despite System and Environmental Uncertainties” IEEE Control Systems Magazine, vol. 21, No. 2, pp. 66 – 77, April 2001.

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