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Trajectory Optimization of a Flexible Manipulator Using Deflection Analysis Method

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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.
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.
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Deflection and Vibration of Jointed Beams, design and Control of Remote Manipulators
  • D E Whitney
D.E.Whitney, Deflection and Vibration of Jointed Beams, design and Control of Remote Manipulators, NASA Quarterly Report. CR-123795, July, 1972.
  • S P Timoshenko
  • J M Gere
S.P. Timoshenko and J.M.Gere,Mechanics of Materials, Van Nostrand Reinhold Company, 1971.