# Residual Vibration Suppression Of Two Link Flexible Engineering Essay

This paper presents development of a control algorithm for residuary quiver suppression of a two-link flexible arm and joint operator utilizing an input determining technique. In this work, a flexible operator that moves horizontally is considered. An false manner method is considered in obtaining a dynamic theoretical account of the system. The links are modelled based on Euler-Bernoulli beam theory. Two manners are selected to find the nexus bending and distortion sing to the planar manoeuvring. The simulation algorithm is developed and implemented utilizing Matlab and Simulink. Hub-angle and end-point acceleration responses at both links of the flexible operator are studied in clip and frequence spheres. The public presentation of the accountant is assessed in footings of quiver decrease as compared to the response with an unsalaried system utilizing open-loop bang-bang input. Furthermore, the hardiness of the accountant with regard to the changing in the quiver frequence of the operator is besides discussed.

Introduction

The standard premise on automaton kinematic and dynamic is the operator merely consist stiff organic structures for robot arm and articulation. However in practical state of affairs, this ideal premise merely valid for little input forces and slow motion of the automaton. A robotic operator becomes flexible if the flexural effects are important such that they can non be neglected during the accountant design phase in order for the system to run into the public presentation specifications. From the patterning point of position, flexibleness of the automaton can be assumed as concentrated at the automaton articulations and distributed along the robot links. Flexibility in articulation is common in current industrial automaton, when transmission/reduction component such as belts, overseas telegram, long shaft and harmonic thrust are used. Structural flexibleness refers to the warp of a construction under applied or inertial ( acceleration-related ) forces/torques. Arm or nexus flexibleness is dynamic behavior when the links are made of lightweight stuff and slender [ 1 ] . Lightweight automatons are developed to get the better of these drawbacks and let high velocity motions with the same or even better preciseness.

The trouble in commanding of flexible nexus and joint operator arises because the figure of input is less than the control variable. The complexness of the system theoretical account which the articulation and nexus flexibleness are present at the same clip [ 2 ] . Assorted patterning techniques have been proposed to analyze the dynamic behavior of flexible operator such as Euler-Lagrange for moral force of the system and for the flexible nexus utilizing false manner method ( AMM ) ; finite component method ( FEM ) or lump parametric quantity theoretical account. Hundreds of documents have been published on flexible-link and flexible articulation, among the most comprehensive study are, Dwivedy, S. K. and P. Eberhard [ 3 ] , Benosman, M. and G. Le Vey [ 4 ] and Ozgoli, S. and H. D. Taghirad [ 5 ] .

The demand of precise place control of flexible operators implies that residuary quiver of the system should be zero or near zero. Over the old ages, probes have been carried out to invent efficient attacks to cut down the quiver of flexible manipulatorsFeed-forward techniques for quiver suppression involve developing the control input through consideration of the physical and vibrational belongingss of the system, so that system quivers at response manners are reduced. This method does non necessitate any extra detectors or actuators and does non account for alterations in the system once the input is developed. On the other manus, feedback-control techniques use measuring and appraisals of the system states to cut down quiver. Feedback control, use measuring or appraisal of the system states as an input of the accountant beside the mention value to cut down the quiver Feedback accountants can be designed to be robust to parameter uncertainness. For flexible operators, feed frontward and feedback control techniques are used for quiver suppression and place control, severally. An acceptable system public presentation without quiver that accounts for system alterations can be achieved by developing a intercrossed accountant consisting of both control techniques [ 4 ] . Thus, a decently designed provender frontward accountant is required, with which the complexness of the needed feedback accountant can be reduced. These techniques rely on accurate system theoretical account and do the system really sensitive to patterning mistake that can impact the system public presentation [ 3 ] .

When feed frontward control methods are selected to stamp down the quiver, merely some alteration of input signal, without extra measuring component into the system. A figure of techniques have been proposed as feed-forward control schemes for control of quiver. Swigert [ 6 ] has derived a molded torsion that minimises residuary quiver and the consequence of parametric quantity fluctuations that affect the modal frequences. However, the forcing map is non time-optimal. Several research workers have studied the application of computed torsion techniques for control of flexible manipulators4. However, this technique suffers from several problems5. These are due to inaccuracy of a theoretical account, choice of hapless flight to vouch that the system can follow it, sensitiveness to fluctuations in system parametric quantities and response clip punishments for a causal input.

Bang-bang control involves the use of individual and multiple-switch bang-bang control maps ] . Bang-bang control maps require accurate choice of exchanging clip, depending on the representative dynamic theoretical account of the system. Minor modeling mistakes could do exchanging mistake, and therefore, consequence in a significant addition in the residuary quivers. Although, use of minimal energy inputs has been shown to extinguish the job of exchanging times that arise in the bang-bang input, the entire response clip becomes longer. Meckl and Seering [ 7 ] have examined the building of input maps from either ramped sinusoids or versine maps. This attack involves adding up harmonics of one of these template maps. If all harmonics were included, the input would be a clip optimum rectangular input map. The harmonics that have important spectral energy at the natural frequences of the system are eliminated. The ensuing input which is given to the system approaches the rectangular form, but does non significantly excite the resonances. The method has later been tested on a Cartesian automaton, accomplishing considerable decrease in the residuary quivers.

An attack in bid determining techniques known as input defining has been proposed by Singer and colleagues which is presently having great attending in quiver control [ 8 ] . Since its debut, the method has been investigated and extended. Input determining plants by making a bid signal that cancels the quiver it causes. That is, quiver caused by the first portion of the bid signal is cancelled by quiver caused by the 2nd portion of the bid. The method involves convoluting a coveted bid with a sequence of urges known as input maker. The molded bid that consequences from the whirl is so used to drive the system. Design aims are to find the amplitude and clip locations of the urges, so that the molded bid reduces the damaging effects of system flexibleness. These parametric quantities are obtained from the natural frequences and muffling ratios of the system. Using this method, a response without quiver can be achieved, nevertheless, with a little clip hold about equal to the length of the impulse sequence. The method has been shown to be the most effectual in cut downing quiver in flexible workss. The more urges are used, the more robust the system becomes to flexible manner parametric quantity alterations but the longer the hold introduced into the system response. Previous probes have shown that the input maker can be designed to account for patterning mistakes in natural frequences and muffling ratio. In this work bid defining is used to cut down the quiver of end point of the flexible articulations and links operator system. Zero-vibration ( ZV ) maker, zero-vibration-derivative ( ZVD ) maker and zero-vibration-derivative-derivative ( ZVDD ) maker are choice to determine the input. The simulation consequence are used to tune the input maker.

This paper is organized as ; discusses the patterning procedure of two flexible articulations and links operator, planing of bid defining, simulation consequence and the decision are drawn from the execution input determining for suppression residuary quiver.

Dynamic Modelling and Characterization

A description of flexible operator considered in this work is shown in figure 1, which consist of two flexible links and two elastic articulations. Two nexus are arrange in consecutive concatenation and are actuated at the articulations by two motors. The flexible links can be model based on Euler-Bernoulli beam theorem and the flexible articulation is dynamically considered as additive torsional spring. and stand for the stationary and traveling co-ordinates frames severally. Linkss are denoted by figure where and link-1 and link-2. stand for the torsion applied at the hub.

Figure 1 Two flexible links and articulations operator with warhead

Motor 2

Payload1

Motor 1

Flexible articulation

Flexible articulation

These parametric quantities are consider as the feature of links, is flexural rigidness, is aggregate denseness per unit length, is the nexus length. The hub parametric quantity can be represent as, , , and which are jumping stiffness for flexible articulation, hub mass, hub inactiveness and rotor inactiveness severally. An inertial warhead mass and inactiveness is connected to the distal nexus. Since the links are assumed as long and slender, the effects of rotary inactiveness and shear distortion are ignored by presuming that the cross-sectional country of the nexus is little in comparing to the nexus length. The bending warp is of i-th nexus at spacial point, . The hub angle and nexus angle are, and severally. The analysis is based on an premise ; the warp of links are little, the operator moves in horizontal plane, the gravitation consequence is non considered and the nexus length is assumed to be changeless as consequence from little distortion. The physical parametric quantity of the operator were taken from, [ 8, 9 ] as follows ;

Table 1 Physical parametric quantities of the operator

Unit of measurement

Link 1,

Link 2,

Mass denseness,

0.2

0.2

Flexural rigidness,

1

1

Length,

0.5

0.5

Hub Inertia, ,

0.1

0.1

Rotor Inertia,

0.02

0.02

Payload mass,

0.1

Payload inactiveness,

0.005

Hub mass,

0.1

0.1

Stiffness invariable, ,

100

100

## Kinematic preparation

See a planar 2 nexus flexible operator in figure 1 with revolute articulation and the gesture of the rotary motion is in horizontal plane. The co-ordinate frame with inertial frame, are set up. Rigid organic structure traveling frame are associated with frame and and the flexible organic structure traveling frame are associated with, and.

The stiff transmutation matrix and the rotary motion matrix, of the nexus at the end-point are severally,

, and

Let is the homogenous transmutation matrix between initial frame to the flexible co-ordinate system so the planetary transmutation matrix can be given by,

The absolute place vector at any point on link-i, with referred to local co-ordinate frame can be given as follow ;

And be the same point mentioning to the initial frame. The place of the beginning of with regard to is give by, and is its absolute place with regard to initial frame. Using the planetary transmutation matrix, and can be written as ;

To happen the additive absolute speed vector are merely differentiate place vector over a clip ;

## Assumed Mode Shaped

Linkss are modelled as Euler-Bernoulli beams with unvarying mass denseness ( , changeless flexural rigidness with distortion is fulfilling the following partial differential equation,

The equation ( 4 ) , is solved by enforcing boundary status at the base of each nexus. Sing a clamped-mass conA?A¬A?guration of the operator, the boundary conditions can be written as ;

Ritz enlargement is used to come close the generalised co-ordinates called the manner of system. The moral force of a flexible nexus are described by a finite-dimensional mathematical theoretical account. The finite dimensional solution can be obtained utilizing false manner method, AMM by select the manner order, . Using this method, flection can be express as superposition of mode-shapes and clip dependent modal supplantings,

Where, is the nexus figure, is the manner figure, is the manner form and is the average supplanting. are the clip changing variables associated with the false spacial manner forms of nexus. The elaborate solution of is refer in, [ 8, 9 ] .

## Derivation of equation of gesture.

The dynamic equation of gesture of a two-dimensional 2-link flexible nexus and articulations can be derived utilizing standard lagrange equation. The lagrange map, can be solved when the kinetic energy T and possible energy, U are obtained from the dynamic feature of the system. The sum of kinetic energy T, is given by the amount of the undermentioned part ;

For the hub kinetic energy, it is the combination of kinetic energy of hub nexus 1, and hub of link2, .

For hub 1 and hub 2 the kinetic energy are given by the undermentioned equation ;

The kinetic energy of nexus 1 and associate 2 are ;

The kinetic energy of the warhead can be represented by equation ;

where, .

As the articulation is flexible, the rotor kinetic energy taking into history for the sum of kinetic energy ;

The kinetic energy of the rotor-i is ;

As the gesture of the operator is assumed to be in horizontal program, gravitation consequence can be ignored, so merely possible energy, are derived from the elastic warp from both links and articulation.

The possible energy due to the nexus distortion can be present as ;

The possible energy of the nexus can be solved by mentioning the solution of utilizing AMM. Another portion of possible energy is due to the joint warp can be written as ;

where is the stiffness invariable of the joint.

The dynamic theoretical account of the two flexible links and articulations operator is obtained by replacing

and into Lagrange equation, and followed by work outing the Euler-Lagrange map ;

Where is dissipation map, is input torsion, and is refer to the generalised co-ordinate of the system with. After some algebraic use, the concluding dynamic equation can be written as the undermentioned equation ;

Where ;

Mass matrix

Modified rotor inactiveness matrix

Stiffness matrix due to the distributed flexibleness of the links

Torsional rigidness matrix of elastic articulations

Modal supplanting vector

Rotor angle vector

Link angle vector

Joint warp vector

Motor torsion vector

Coriolis and centrifugal forces

## Simulation of the flexible operator

Simulation consequence of the dynamic behavior of the two-link flexible arm operator with flexible articulation are presented in clip and frequence sphere. A bang-bang signal of amplitude 1 Nm and 0.5s breadth, shown in Figure 2, is used as an input torsion, applied at the hub of the operator. A bang-bang torsion has a positive ( acceleration ) and negative ( slowing ) period leting the operator to, ab initio, accelerate and so slow and finally halt at a mark location. System responses are monitored for continuance of 10 sec, and the consequences are recorded with a sampling clip of 0.1 msec.

System responses viz. the hub-angle, link-angle, and end-point warp for both nexus with the corresponding spectral densenesss ( SD ) are obtained and evaluated. Acceleration of hub-angle, link-angle and end-point are besides considered to formalize the frequence response and clip sphere response. These consequences were considered as the system response to the unshapen input and later will be used to measure the public presentation of the bid defining.

Figure 2 The bang-bang torsion input for both and

Due to the flexibleness of the joint, there is little faux pas between hub angle, and joint angle, show in figure 3-4. From the figure 3-4 it is observed that the joint angle are oscillate somewhat smaller than hub angle, this are consequence from revolving energy are dissipate on the flexible articulation. It s noted that the steady province link-angle degree of was achieved within rise clip 0.92 2nd and settling clip of 7.5 2nd for link-1 and steady province link-angle degree of was achieved within rise clip 0.8 2nd and settling clip of 5 2nd for link-1. The rise and settling times were calculated on the footing of response continuance of 10aa‚¬ ” 90 % and AA±2 % of the steady-state value severally. Note that quivers occur during motion of the operator, as evidenced in the hub-velocity, link-velocity and end-point residuary responses. The end-point residuary response was found to hover between 0 and 30 millimeter for link-1 and 0 and 14 millimeter for link-2. Due to the nexus constellation, figure 3-4 show link-2 give the larger rotary motion angle compared to the link-1. This can be explain as the input torsion for link-1 demand to originate the rotary motion for itself, link-2 and pay burden, in the other manus the input torsion for link-2 merely initiate the rotary motion for itself and warhead.

Figure 3: Hub and Joint Angle for link-1

Figure 4: Hub and Joint Angle for link-2

Figure 5-6 show the tip warp, tip acceleration and power spectral denseness ( PSD ) of the corresponding links. In this survey, merely two manners of frequence are considered as dominant quiver of the frequence. As demonstrated in figure 5-6, the frequences of the tip quiver, rise clip, settling clip and scope of oscillation of each nexus were obtained as table below ;

Table 2: Summary of system response to the bang-bang input refer to associate angle.

Response

Link-1

Link-2

Tip quiver frequence

Mode 1

0.5 Hz

0.6 Hz

Mode 2

5.6 Hz

2.8 Hz

Rise clip

0.82 s

0.93 s

Settling Time

6.2 s

5.6 s

Maximal oscillation scope

0mm-30mm

0mm-14mm

Figure 5 Link-1 Tip warp, tip acceleration and power spectral denseness

Figure 6 Link-2 tip warp, tip acceleration and power spectral denseness

Input determining Control Scheme

In this subdivision input determining techniques are introduced for quiver control of a flexible automaton operator. The method input defining is the procedure of convoluting the coveted input of the system with a series of impulse. The amplitude and clip location of the urges are the necessary to develop the input shaper [ 10 ] . As a simple analogy for system with quiver component, we can presume a dynamic system can be model as first order and 2nd order system. For a complex system, we can pattern the system as a combination of first order system, 2nd order and both of first order system and 2nd order system. A vibratory system of any order can be modelled as superposition of 2nd order system of transportation map. The most dominant poles of the system will find the response of the system. The dominant poles are chief focal point to design of input defining.

and

Where is the natural frequence and is the muffling ratio of the system. If the system is applied with impulse, the response of the system can be obtained as ;

Where and are the amplitude and clip of the impulse severally and undamped frequence, . Therefore for N impulse input, the response can be express as ;

where,

, and

and are the magnitudes and times at which the urges occur.

The residuary quiver amplitude of the impulse response can be obtained by measuring the response at the clip of the last impulse, as ;

where

and

In order to accomplish zero quiver after the input has ended, it is required that and in Eq. ( 6-7 ) are independently zero. Furthermore, to guarantee that the molded bid input produces the same stiff organic structure gesture as the unshapen bid, it is required that the amount of impulse amplitudes. To avoid hold, the first urge is selected at clip 0. The simplest restraint is zero quiver ( ZV ) at expected frequence and damping of quiver utilizing a two-impulse sequence. Hence by puting Eq. ( 30 ) to zero, and work outing outputs a two-impulse sequence with parametric quantities as ;

where,

The hardiness of the input maker to error in natural frequences of the system can be increased by puting, , where is the rate of alteration of with regard to. Puting the derivative to zero is tantamount of puting little alterations in quiver for alterations in the natural frequence. Therefore, extra restraints are incorporated into the equation, which after work outing outputs a three-impulse sequence with parametric quantities as ;

Where and is every bit in Eq. ( 9 ) . The ensuing maker is called a Zero Vibration and Zero Derivative ( ZVD ) maker.

The hardiness of the input maker can farther be increased by taking and work outing the 2nd derived function of the quiver in Eq. ( 5 ) . Similarly, this yields a four-impulse sequence with parametric quantities as ;

where, and is every bit in Eq. ( 3 ) . This maker called Zero Vibration and Zero Derivative- Derivative ( ZVDD ) maker. The drawback for each extra derived function is an addition the shaper continuance by one-half period of natural frequence, which is decelerating the response of the system. Multiple manners of quiver could be suppressed by convoluting multiple makers together. The restriction of these makers was shown to be their hardiness to altering design frequences. If the design frequence alterations by more than the maker ‘s built-in insensitiveness, so the input maker was no longer able to stamp down the quiver to the desired degree. To manage higher quiver manners, an impulse sequence for each quiver manner can be designed independently. Then the impulse sequences can be convoluted together to organize a sequence of urges that attenuates quiver at higher manners. For any vibratory system, the quiver decrease can be accomplished by convoluting any coveted system input with the impulse sequence. This yields a molded input that drives the system to a desired location without quiver.

Execution and Consequences

## Multi-mode input determining

Figure 7 Flexible operator with two-mode input defining.

Payload1

Motor 2

Motor 1

Bang-bang input 1

Bang-bang input 2

Two-mode maker 1

Two-mode maker 2

If the system is vibrate with multi manner frequence, the most dominant poles of the system will rule the whole response of the system. Single-mode makers can be obtained by merely stop uping

the natural frequence and muffling ratio of the system into simple equations like those derived in the

debut. The equations give the amplitudes and clip locations of the urges that comprise the input

maker. Based on the simulation consequence of two flexible nexus and joint system, the manner of frequence is choice in order to turn up the place of the impulse sequence. In this plants, two manners are choice to stand for the quiver of the nexus and articulation. A simple manner to obtain a two-mode maker is to convolute two single-mode makers together. Here the nexus tip warps are selected as the chief focal point of quiver. Two frequences will find base on simulation consequence from the bang-bang input for both nexus. The execution two-mode makers to the flexible operator are shown in figure 7.

## .

Figure 8: The unshapen bang-bang input and shaped input utilizing ZV, ZVD and ZVDD for torsion input link-1

Input determining method are applied to stamp down the quiver, three different techniques are selected ; Zero Vibration ( ZV ) , Zero Vibration Deferential ( ZVD ) , and Zero Vibration Deferential- Deferential ( ZVDD ) .

Figure 8, show the bang-bang input of 1Nm amplitude and 0.5 2nd breadth are shaped to matching input determining ZV, ZVD and ZVDD. From the figure 8, it is shown that ZVD shaper taken one-half longer than ZV maker, means the molded bid signal is one-half period longer than the ZV shaped bid. The same state of affairs besides go on when ZVD are compared with ZVD. Figure 9 support this statement, and the response of nexus angle for both nexus can be summarizes in table 3 below ;

Table 3 Comparison of settling clip of nexus angle for link-1 and link-2

Maker

Link-1,

( second )

Link-2,

( second )

unshapen

7.15

4.26

ZV

2.55

2.65

ZVD

4.02

4.15

ZVDD

4.85

4.55

When input defining is applied to a dynamic system, it will decelerate down the response of the system in term of the clip taken for the system to settle down. This the tradeoff of input determining where it can stamp down the residuary quiver and contrary addition the settling clip, . Table 3, show the input determining with matching settling clip for nexus angle of each nexus.

Figure 9: Link-1 and Link 2, hub angle and joint angle for different types of input determining

The public presentation of the input determining can be easy compared based on figure 10-11. The tip place of each nexus shows the residuary quiver. From the figure 8, the ZVDD give the lowest wave-off. PSD for tip warp of both nexus show, how the input determining suppress the residuary quiver. From the figure 10, which refer to the link1, the decrease of residuary quiver when input defining is applied can be summarized in tabular array. from the tabular array

Maker

Link-1

Link-2

ISE

Percentage of decrease

ISE

Percentage of decrease

unshapen

1.21E-03

none

2.46E-05

none

ZV

2.15E-04

82.2

3.60E-06

76.09

ZVD

8.68E-05

92.8

1.63E-06

93.38

ZVDD

4.76E-05

96.1

9.79E-07

96.03

Figure 10: Tip warp and PSD for link-1

Figure 11: Tip warp and PSD for link-2

Another method to compare the public presentation of input defining is the ability of the maker to cut down the residuary quiver which can stand for the residuary quiver in term of numerical value. Here the residuary quiver are compared when the tip place of both nexus are assumed as place mistake. ISE ( Integral of the Squared of Error ) will be used to mensurate the residuary quiver of tip warp mistake. The country under the graph can be calculated to mensurate the mistake as shown in figure 11, refer to the link-1 tip warp, .

Figure 12: ISE attack to cipher place mistake

Table 4, below show the difference of ISE value and per centum of quiver decrease of tip warp when input determining method are applied.

Table 4 Comparison of per centum decrease of tip warp and ISE value for ZV, ZVD, and ZVDD

Maker

Link-1

Link-2

ISE

Percentage of decrease

ISE

Percentage of decrease

unshapen

1.21E-03

none

2.46E-05

none

ZV

2.15E-04

82.2

3.60E-06

76.09

ZVD

8.68E-05

92.8

1.63E-06

93.38

ZVDD

4.76E-05

96.1

9.79E-07

96.03

## Robustness analysis of input determining bids for two-mode flexible systems

The hardiness of the input determining can be studied when the frequence of the residuary quiver are altering. The value of system natural frequence is really important due to the designing of input determining. In this instance the frequence quiver of tip warp for each nexus will be changed by 5 % , 10 % , and 15 % larger from the original value. Then the divergences of ISE due to the mistake frequence are compared as shown in figure 11-12 for the corresponding nexus. From the figure 13, ZVDD show the smaller divergence when the per centum of frequence mistake are addition.

Figure 13 End point residuary response of flexible operator utilizing ZVDD maker with 0, 5, 15 and 30 per cent in natural frequence.

Figure 14: Comparison o f ISE for tip warp of link-1 with frequence mistake

Figure 15: Comparison of ISE for tip warp of link-2 with frequence mistake

As a comparing table 6 show the per centum of quiver decrease compared to the frequence mistake when applied to the input determining. From the tabular array, it shows that ZVDD gives the most robust public presentation where, the divergences of ISE are smaller compared to the ZV and ZVD. To visualise the consequence of frequence mistake, figure 15 show ZVDD are most robust when the frequence value for tuning the input maker are pervert below than 30 % of its existent value.

Table 5: Percentage of quiver decrease compared to the frequence mistake

Maker

Frequency mistake

Link-1

Link-2

ISE

Percentage of quiver decrease

ISE

Percentage of quiver decrease

ZV

5 %

2.15E-04

82.2

4.70E-06

80.9

ZVD

8.68E-05

92.8

2.08E-06

91.6

ZVDD

4.76E-05

96.1

1.15E-06

95.3

ZV

15 %

2.68E-04

77.8

5.89E-06

76.1

ZVD

1.26E-04

89.5

2.98E-06

87.9

ZVDD

7.06E-05

94.1

1.71E-06

93.1

ZV

30 %

3.98E-04

67.0

8.48E-06

65.6

ZVD

2.31E-04

80.9

5.05E-06

79.5

ZVDD

1.34E-04

88.9

2.99E-06

87.8

Decisions

Input Shaping technique is utilised to stamp down quiver of the tip place of two flexible links and articulations operator. The provender frontward technique was seen successful cut down the tip quiver up to 96 % of the decrease. This technique need an accurate of the frequence manner quiver to do certain the right value of impulse map are apply when the design of the maker are take topographic point. However during the simulations, it was observed that input determining have the robust capableness despite the alterations in frequence. ZVDD show the most robust maker comparison to the others. The easiness of input determining execution and the robust capableness were demonstrated.