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Trajectory Optimization of a Flexible Manipulator Using Deflection

Analysis Method

Y. MADDAHI, F. A. HASSANI

Mechanical Engineering Department

Azad University (Saveh Branch)

Robotic and Automation Research Laboratory, Islamic Azad University (Saveh Branch), Saveh, Iran

IRAN

Abstract: In practice, a robotic joint is linearly and angularly deflected when a load is applied at the end-

effector. This article presents an improved method with a combination of energy methods and the concepts of

differential relationships to more accurately calculate the static deflection at the end-effector. A systematic

approach to deflection calculation through three different Jacobians is presented. The theoretical deflection

analysis is verified by simulation results. A two-link robot is used for numerical illustration and calculation

procedure. Also the total deflection analysis of the end-effector is calculated with respect to base. Finally, the

deflection statements are minimized considering that the deflection functions are differentiable.

Key-Words: Trajectory Optimization, Flexible Manipulator, Deflection Analysis, Two Link Robot.

1 Introduction

For a lightweight robot manipulator, link deflection is

the cause of the discrepancy. Due to the distributed

weights of robotic links and a load applied at the end-

effector, each robotic link and joint are deflected.

There have been some researchers on the elastic

deflection of robotic manipulators. Whitney started

the pioneering work on the deflection and vibration

of jointed beams [1]. Derby developed a first-order

compensation analysis for link deflections [2]. The

analysis was based on the assumption of small

bending and no radical difference in the deformed

arm geometry. Zalucky and Hardt proposed a

solution to actively control the deflection using a

straightness servo [3]. The system employed two

parallel, one is to act as the manipulator link and the

other one is to carry only the bending loads.

Maghdari and Shahinpoor conducted a series of

experiments of a PUMA 560 robot manipulator to

determine the characteristics of its elastic

deformations in various geometrical configurations

and methods of operations using a dial gauge with a

resolution of 0.001 inch [4]. Fenton and Reeder also

developed an elastic deflection-compensating

algorithm, in which the method they used in solving

for the inverse kinematics of a deflected manipulator

was analogous to the method of solving for the

inverse kinematics of a rigid manipulator [5]. Tang

and Wang used a classical beam theory to compute

the linear displacements of robotic links and

considered the robot joints as torsional springs, where

the first order approximation is applied for

compliance analysis [6].

Whitney et al. pointed out five causes of robotic

positioning errors [7]. They are backlash, gear

transmission errors, joint drive compliance, cross

coupling of joint rotations, and base motion. The first

two are due to manufacturing errors, whereas the

third one is the overall compliance between the

angular encoder and the actual angular output. When

a robot is loaded at its end-effector, couple moments

in addition to the driving torques are applied to the

joints causing additional angular displacements (i.e.,

angular deflections). The classical Timoshenko's

beam theory has been employed to calculate the slope

angle (angular deflection) at the end of each robotic

link by researchers such as Derby, Tang and Wang,

and Fenton and Reeder. To overcome the problem,

this article presents a more accurate way to calculate

the angular deflections of robotic joints using one of

energy methods. Different methods used to calculate

the link deflection of a two-link robot made up of

aluminum alloy are presented. the data of simulation

study is provided to verify the calculations.

2 Transformation of forces and

displacements

Concentrated forces, distributed forces and moments

may act upon each link of a robotic manipulator. It is

necessary to transform a generalized force vector

which contains three force components and three

moment components.

Fig. 1. Coordinate frames and external

forces/moments for the two-link manipulator

A systematic way to derive the above two

equations is by means of the so-called A matrix

developed by Denavit and Hartenberg [8]. An A

matrix is a homogeneous transformation describing

the relative translation and rotation between link

coordinate systems. Thus, the position and orientation

of the second link in the base frame coordinates is

given by:

2

1

12 AAT oo = (1)

+

+−

=

1000

0100

0

0

122111212

12211212

2

0

1

SLSLCS

CLCLSC

T (2)

Where C1= Cosine of 1

θ

, C12 = Cosine of

(1

θ

+2

θ

) and S12 = Sine of ( 1

θ

+2

θ

), etc. and the

superscript c2 denotes that the weight W 2is applied

at the centroid of link 2.

Considering reactions of forces and torques in

joints, the imposed parameters are obtained

separately, as follows:

1222122 SLWmgSF c

+= (3)

1222122 CLWmgCP c

+= (4)

11111 SLWmgSF c

+= (5)

11111 CLWmgCP c

+= (6)

4 Employment of The Timoshenko’s

Beam Theory

Robotic link deflections can be calculated by at least

two methods. The Timoshenko's beam theory has

been widely used as a classical method. They are

briefly described as follows.

4.1 Deflection of a One-Link Manipulator

In using the Timoshenko’s beam theory, a link is

treated as a cantilever beam. Four possible loading

conditions, which the total deflection can be easily

superimposed for a combined loading condition. In

using the Timoshenko's beam theory, a link is treated

as a cantilever beam.

EIPLd p3/

3

= (7)

EIWLdw8/

4

= (8)

EIMLdm2/

2

= (9)

EAFLd f/

=

(10)

Since the deflection due to an axial load is much

smaller than others, it is assumed negligible

throughout this article. For the two-link robot (Fig.1),

link 2 is subjected to the concentrated load P and

distributed weight 2

W only.

Due to P and 2

Wcan be derived by substituting eqs.

(11) And (12) into (18) and (19), respectively.

22

3

212

23/ IELPCd p−= (11)

2212

4

22

28/ IECLWdw−= (12)

4.2 Deflection of two-link Manipulator

A general methodology based on the

Timoshenko's beam theory via force and

displacement transformations to calculate the end-

effector's deflection of a multilink robot is developed.

The Y or vertical component of the deflection is:

2222

2

12

3

22 24/)38( IELWPCLd y

o+−= (13)

With the addition of a moment effect, the

deflection of link I can be derived in a similar way,

whose y component is:

3

12

3

121212

2

1

2

1

3

18128( CLLWCCLPLCPLdly

o++−=

11

2

1

4

11121

2

2

2

12 24/)36 IECLWCCLLW ++ (14)

Finally, the vertical deflection of the end-effectors

of the two-link robot in base frame coordinates is

given by

y

o

y

oo ddd 21 += (15)

Using Timoshenko’s theory,the links deflections

can be calculated by:

iiiii AELFdx /= (16)

iiiii IELPdy 3/

3' = (17)

iiii IEMLdy 2/

2" = (18)

"'

iii dydydy += (19)

Where the dx i is the axial deformation of each link,

dy '

i is the vertical deflection caused by shear force

and "

i

dy is created from moment effect. Link

deflections are calculated with respect to base frame.

Fig. 3. Generalized coordinates of links

122122

'CdyCdxdx −= (20)

1111

"SdyCdxdx −= (21)

122122

'CdySdxdy += (22)

1111

"CdySdxdy += (23)

1111122122 SdyCdxSdyCdxdxt−+−= (24)

1111122122 CdySdxCdySdxdyt+++= (25)

Where t

dx and t

dy are total deflections relative to

base frame.

5 Experimental Results

To verify the theoretical deflection analysis, two-link

robot was built. It was made up of lightweight

aluminum alloy. The robot is as follows:

• Length of link 30 mm (uniform hollow square cross

section)

• Length of link 30 mm (uniform hollow square cross

section)

• Area moment of inertia of link 1: 3.56 × 10 5− m 4

• Area moment of inertia of link 2: 3.56 × 10 5− m 4

• Modulus of elasticity of link 1: 2 × 1011 pa

• Modulus of elasticity of link 2: 2 × 1011 pa

To exclude the possibilities of gear backlash, gear

transmission error, joint drive compliance, and cross

coupling of joint rotations, the robotic joints are

mechanical pin joints with no actuators. To avoid a

possible rotational slippage of a joint, two setscrews

were used to fix the joint at a specified angular

position. Various weights could be applied at the end-

effector. The procedure was repeated until all

possible arm configurations been applied. The

procedure was repeated until all possible arm

configurations and applied weights had been tested.

Two robotic links are predeflected due to their

distributed weights before a load is applied a end-

effector. The dial gauge used to measure the

deflections is zeroed after the robot has predeflected.

In order to compare the calculated deflection with the

measured ones, the distributed weight 1

W and 2

W are

not included in the theoretical calculations.

In this numerical calculation the robot data is as

follows:

m

N

WW CC 2406.19

21 ==

24

21 105.2 mAA −

×==

Moment in each stage: M=1000 N.mm

6 Minimize Link Deflections

Minimizing the deflection and displacement errors is

the one of important tasks help us to controlling the

trajectory. Derivatives of t

dx and t

dy are calculated

in experiment part.

For n=1,2,3,4,5,6 (n is the number of each stage. )

11 saF n

=

122 saF n

=

11 caP n

=

122 caP n

=

11 sbdx n

= (26)

122 sbdx n

=

1

1

'Ccdy n

=

12

2

'Ccdy n

=

9

2

"

1

"103202.6 −

×== dydy

With respect to eq. (20), 1

dy , 2

dy , can be derived.

And link deflections in base frame are as follows:

)(103202.6

)()(

112

9

11

2

12111212

SC

CSCcCSCSbdx nnt

+×−

+−+=

− (27)

)(103202.6

)()(

112

9

2

1

2

12

2

1

2

12

CC

CCcSSbdy nnt

+×+

+++=

− (28)

Derivative of t

dx , t

dy are

)(103202.6

2

))((

112

9

1212

2

1

2

1

2

12

2

12

1

CS

CSc

CSbcSbCb

d

dx

n

nnnn

t

+−×−

+−−+−=

−

θ

(29)

12

9

1212

2

12

2

12

2

103202.62)( SSCcSCb

d

dx

nn

t−

×++−=

θ

(30)

)(103202.6

))((2

112

9

111212

1

SS

SCCScb

d

dy

nn

t

+×−

+−=

−

θ

(31)

12

9

1212

2

103202.6)(2 SCScb

d

dy

nn

t−

×−−=

θ

(32)

Using equations 29 to 32 the deflection in two

directions are minimized and the actuators in given

trajectory are safety as torque.

6 CONCLUSION

in this paper the Timoshenko's beam theory was used

to calculate robotic link deflections. Simulation was

carried out on a house-made two-link robot. Although

this study shows only the most significant vertical

deflection data, the developed methodology can be

used to find all of six deflection components. From

the minimum deflection point of view, the

configuration, which produces the smallest

deflection, is considered as an optimum.

The manipulator was defined as the probability of

end-effector's pose (position and orientation) falling

within a specified range from the desired pose. Also

the deflection of end-effector is minimized and the

maximum deviation determined for given trajectory.

References

[1] D.E.Whitney, Deflection and Vibration of Jointed

Beams, design and Control of Remote Manipulators,

NASA Quarterly Report. CR-123795, July, 1972.

[2] S.Derby, The deflection and compensation of

general purpose robot arms, Mechanism and Machine

Theory,No.18,1983,pp.445-450.

[3] A. Zalucky and D. E. Hardt, Active control of

robot structure deflections, Journal of Dynamic

Systems, Measurement, and Control, Transactions of

the ASME, No.106,1984,pp.1.

[4] A. Maghdari and M. Shahinpoor,Elastic

deformation characteristics of PUMA 560 robot

manipulator, International Journal of Robotics and

Automation, No.2,1987,pp.1-5.

[5] R.G. Fenton and J. M. Reeder,Motion planning

for robots using an elastic deflection compensating

algorithm, International Journal of Robotics and

Automation,No.2,1987,pp.1-5.

[6] S.C. Tang and C.C. Wang , Computation of the

effects of link deflections and joint compliance on

robot positioning, in Proc, of the 1987 IEEE

Conference on Robotic and Automation, 1987, pp.

910-915.

[7] D.E. Whitney, C. Lozinski, and J. Rourke,

Industrial robot forward calibration method and

results,Journal of Dynamic Systems, Measurement,

and Control, Transactions of the ASME,

No.108,1986,pp.1-8.

[8] J. Denavit and R.S. Hartenberg, A kinematic

notation for lower-pair mechanisms based on

matrices. ASME Journal of Applied Mechanics,

June,1955,pp.215-221.

[9] S.P. Timoshenko and J.M.Gere ,Mechanics of

Materials, Van Nostrand Reinhold Company, 1971.