Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 2 (September 2016), 116-123

Effects of System Parameters and Controlled Torque on
the Dynamics of Rigid-Flexible Robotic Manipulator
Sachindra Mahto

Mechanical Engineering Department, NERIST, Arunachal Pradesh-791109, India
E-mail: smh@nerist.ac.in
www.nerist.ac.in

Abstract
This work illustrates the effects of various system parameters on the dynamics of flexible link of revolute-jointed
rigid-flexible manipulator. Flexible link is considered as a Euler-Bernoulli beam and finite element based on
Langrange approach is employed for dynamic analysis. A comparative study is carried out for comparative
dynamic response for the variation of system parameters and controlled torque excitation.
Keywords: Rigid-flexible revolute manipulator, Euler-Bernoulli beam, Shape optimization, Finite element method,
Parametric study

1. Introduction
In the last few decades, dynamics and control of theflexible manipulators have received a considerable
research attention. Flexible robotics systems haveseveral advantages over the convention systems.
However, due to lightweight of flexible systems,
stiffness is low and there is a serious problem ofvibration. Sometimes, to retain the advantages of theflexible system, some of the links are made flexible andsome are rigid. Still the dynamic behavior improvement,
optimal design and proper control strategy is theresearch interest for flexible manipulator.
Most of the researchers contributed their works indifferent ways of dynamic modeling and control aspectsof flexible robotic manipulators. Sunada and Dubowsky[1] developed a lumped parameter FEM model for
analyzing the complete behaviour of industrial robotic
manipulator with complex-shape flexible links. Fakuda
and Arakawa [2] studied the modeling and dynamic
characteristics of two-link flexible robotic arms andcontrolled the vibration by taking into account thegravity, payload, and the coupled vibration between thefirst and second arm. Usoro et al. [3] developed finite
element models to describe the deflection of a planar
multi-link model. Ower and Vegte [4] used a
Lagrangian approach to model the planar motion of amanipulator consisting of two flexible links and tworotary joints. Benati and Morro [5] developed a
Lagrangian approach for the dynamics of chain withflexible links..
Bayo [6] used FEM to deal with multi-link flexible
manipulator considering Timoshenko beam theory andincluding nonlinear coriolis and centrifugal effects forthe elastic behavior. Jonker [7] presented a nonlinear
finite element based formulation for analyzing the
dynamic behavior of flexible manipulators. De Luca andSiciliano [8] presented closed-form equations of motion
for planar flexible multi-link robot arm. Morris andMadani [9] studied the accurate modeling based on
Lagrange-Euler formulation of a two link flexible
manipulators.

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Sachindra Mahto

Dogan and Iftar [10] carried out the modeling andcontrol of two-link robot manipulator whose first link is
rigid and second link is flexible. Everett et al. [11]showed that it is possible to design a two-link flexible
manipulator that has nearly position invariant firstnatural frequency with wide separation between the first
two natural frequencies, to have its behaviour like rigid
manipulator. Chen [12] developed a linearized dynamic
model for multi-link planar flexible manipulator. Yanget al. [13] studied the tip trajectory tracking control for
flexible multi-link manipulator using Lagrangian
assumed mode method. Zhang et al. [14] derived apartial differential equation model for a flexible
two-link manipulator using Hamilton's principle.
Some researchers presented their works on shape
optimization of static/rotating beams. Karihaloo andNiordson [15] determined the optimum tapering of a
cantilever beam carrying an end mass to maximize
fundamental frequency. Wang [16] addressed optimum
design of a single link manipulator to maximize its
fundamental frequency. Yoo et al. [17] used theassumed mode method for dynamic modelling ofrotating flexible manipulator for modal analysis andshape optimization to increase the fundamental
frequency of the beam. Dixit et al. [18] presented afinite element model of single link robotic manipulator
for revolute as well as prismatic joint. They used SQP
for optimizing beam shapes under different optimization
conditions.
From the above survey, it is observed that there is nomuch research contribution for the effects of systemparameters in the dynamics of rigid-flexible robotic
manipulator. In this paper, the dynamic behavior ofrobotic manipulator is presented for the variation ofsystem parameters and improvement through control
strategy.
2. Modeling and Solution Technique
The finite element formulation has been adopted here as
described by Usoro et al. [3]. Fig. 1(a) shows
rigid-flexible manipulator comprised of two links, twojoints and tip load. The links are clamped on the joints.
Shoulder joint (joint 1) is located at the origin of XOY
represents the stationary co-ordinate frame and elbow
joint (joint 2) is located at the origin of X1O1Y1 andX2O1Y2 which represent the moving co-ordinate frames.

manipulator is considered slender. Flexible link istreated as a Euler-Bernoulli beam and gravity force isneglected.
Consider a point P in the ith element on the
manipulator at a distance ' x ' from the elbow hub. The
point P attains the position P ′ with respect toinertial frame of reference (XOY) after having rigid
body motion θ1(t) and θ2(t) of shoulder and elbow joint
respectively and flexural deflection w ( x, t ). Flexural
deflection w ( x, t ) of point P is approximated in
finite element technique using Hermitian shapefunctions as
w ( x, t ) = N1w2i −1 + N 2 w2i + N 3 w2i +1 + N 4 w2i + 2 =  N  {W } , (1)where N  =  N1 N 2 N 3 N 4 and{W } =  w2i −1 w2i
T

w2i +1w2i − 2  .

Fig. 1. (a) Configuration of rigid-flexible manipulator,(b) Typical ithelement with six dof

In FEM formulation, the manipulator is divided into
finite number of elements with each element having six
thdegrees of freedom. Detail of i element of the link isshown in Fig. 1(b), where θ1 and θ2 are the hub rotationof shoulder and elbow joints respectively, w2i-1, w2i and
w2i+1, w2i+2 are the transverse deflection and slope at thefirst and second node of the element respectively. The
position vector of P′ with respect to inertial system
XOY is given by Eq. 2. For the prediction of approximate
dynamic behaviour of the optimized beams, smaller
excitation torque is considered for linearization of systemmodelling. Global position vector of the point P ′ undersmaller angular and flexural displacement is given by

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Effects of System Parameters22
a + L1 + b + (i − 1) h + xX  
1 h ∂ w
T 1h'' T ''
e

V
=
EI
d
x
W
{
}
∫
∫ EI  N   N  dx {W } .
,W
r = op′=  Y  = ( a + L1 )θ1 + ((i − 1) h + x + b)(θ1 + θ 2 ) +  N =
i
{
}
2 



2 0  ∂x 
20

1 
1
  

(9)

(2)
In finite element method, variables are converted into
nodal variables.
′ f (θ1 ,θ 2 , w2 i −1 , w2 i , w2 i +1 , w2 i + 2 ) .
(3)
=
r op
=
then
Z = θ1 θ 2 w2 i −1 w2 i w2 i +1 w2 i + 2  ,absolute velocity of the point P of the flexible link is
obtained as

Thus, elemental stiffness matrix is given byh
K ie = EI ∫  N ''  T  N ''  dx
0Let ∂r   T=
Z .
∂t  ∂Z ∂r

K ie =(4)

2.1 Kinetic Energy Computation of the Link Elementth

The Kinetic energy of the i element of the link isgiven by ∂r T ∂r 
e 1h
(5)
=
Ki
⋅ dx .
∫ m2 
20
∂t ∂t

0 0
0 0

EI 0 0


0
0
12

0
0
6h
2

0
0
−12

0 
0 

6h
2

.

2



3
h 0 0 6 h 4 h −6 h 2 h 0 0 −12 −6h 12 −6h 0 0 6 h

2h

2

(10)

−6 h 4 h 

2.3 Lagrange’s Equation of Motion in Discretized
FormFirst link is rigid and posses kinetic energy only.
However, being second link flexible it posses bothkinetic and potential energy. The kinetic energy and theWe havepotential energy of the system are obtained by
T
T
r
r
∂r
∂r
∂
∂




T
computing energy of rigid link and the energy of each
(6)
Z
Z .
⋅
=




element of the flexible link and then summing over all∂t ∂t ∂Z   ∂Z 
the elements. The global mass matrix and global
Substituting Eq. 6 in Eq. 5, we have
stiffness matrix can be obtained asT ∂r   ∂r   
e 1 T h
1 T
1 T
(7)
=
Ki
Z  ∫ m2, ∂Z  ⋅  ∂Z dx  Z=
T =qM q and V
q
K q
(11)
2
0


2
2
and the elemental mass matrix is given byrespectively.
ThHere
w1 w2 . . w2 n +1 w2 n + 2 
[ q ] = qq ∂r   ∂r 
1
2
e
=
Mi
∫ m2   ⋅  dx
isthe
globalnodal
vector.
The Lagrangian of the system0 ∂Z   ∂Z 
is given by L = T - V and then Lagrange’s equations of M 11 M 12 M 13 M 14 M 15 M 16 
motion of this dynamic system may be written as M 21 M 22 M 23 M 24 M 25 M 26 ∂  ∂L  ∂Le M 31 M 32
(12)
−
=
Fq ,
 . (8)Mi = 
∂t  ∂q  ∂q
Pi ( 4X4)
 M 41 M 42
where Fq is the generalized force vector. Due to M 51 M 52

modelling linearization, global mass and stiffness M 61 M 62matrices are constant and equation of motion of
All the constants of the above matrix in Eq. 8 areundamped system is expressed as
obtained by proper integration.
(13){F } .
[ M ]{q} + [ K ]{q} =[ ] [ ][ ]2.2 Elastic Potential Energy of the Link Elementth

The potential energy of the i element of link due toelastic deformation is given by[ ] [ ][ ]Global load vector {F} and global nodal displacement

vector {q} for ' n ' number of finite elements are givenTby {F } = τ 1 τ 2 0 0.....0 0  .

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3. Optimization Procedure
To retain and optimize the advantage, upper limit of the
optimized mass (M*) is constraint to the prescribed mass
(M). In the rigid-flexible robotic system, flexible link is
considered for the shape optimization. X = [d1 d2 ….....
dn]T is the design vector with di indicating diameter ofthe ith finite element of link 2.

link lengths in its natural frequencies, static tip
deflection, hub angles, and dynamic response areplotted in Fig. 2 to Fig. 5. Natural frequencies of theflexible links decrease with increase of its length andvice versa. Static link deflection increases with increaseof its length. Hub angles increase with decrease of its
length and vice-versa. Residual vibration increases withincrease of length of the flexible link.

Table 1. Different Optimization Problems
Optimization Problem/Objective
Constraint
Maximization of fundamental beam
M* -M≤0frequency
UB
Permissible Bound : X < X<XLB

Minimum and maximum diameter of the beam elements(X) are denoted by XLB , XUB respectively. The MATLAB
function “fmincon” employing sequential quadratic
programming (SQP) technique is used for constrained
optimization of nonlinear function.
Fig. 2. Beam frequencies of flexible link for different link
lengths

4. Results and Discussion
Modeling of this system is highly complex andnonlinear in nature. However, linearized model ispreferred and considered in this work to predict its
approximate dynamic behavior. Structural dimensionsof this revolute-jointed rigid-flexible robotic
manipulator for numerical experiments are taken as
lengths 0.75m and 0.75m, diameters 0.03m and 0.01m,
mass densities 2710 kg/m3 and 2710 kg/m3 , hub radii
0.02m and 0.01m, hub inertias 0.03 kgm2 and 0.03 kgm2
for first and second link respectively. Young's modulus
7.11x1010 N/m2 and mass of the motor 0.1 kg is
considered.
Excitation torque τ=τmsin2πt Nm is considered forboth the joints for 2 seconds action. Torque amplitude
3.0 Nm and 1.0 Nm are considered for shoulder andelbow joints respectively. Static load of 1N is
considered at the tip of the manipulator for comparative
static beam deflections. The ratio of payload (Mp) to themass of second link (M2) is denoted by µ2; ratio of motor
mass (Mm) to mass of the first link (L1) is denoted by µ1
and ratio of the second link length (L2) to first link
length (L1) is denoted by (L*).

Fig. 3. Static deflection of flexible link due to 1N force at tip
for different link lengths

4.1 Effects of Different Link Length
As the second link is flexible, its length is varied withrespect to the first link length. Effects of variation ofFig. 4. Hub angles due to excitation torques for different link
lengths

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119Effects of System Parameters

Fig. 7. Dynamic flexible tip deflection for different payloads
Fig. 5. Vibration residual of the tip of flexible link for
different link lengths

4.2 Effects of Different Payload, Motor Mass and HubInertia
Robotic system means to take load and due to flexible
links, there is always flexural vibration. Changes in the
dynamics of the manipulator due to the change of
payload, motor mass and hub inertia are plotted in Fig. 6
to Fig. 11. Hub angle decrease and residual vibration offlexible link tip increases with the increase of payload
and vice-versa. By increasing the motor mass which isacting as a payload for link 1 has significant effect onshoulder joint angle for a given set of torques but lesseffect on elbow joint, however residual vibration
increases. By increasing the hub-inertia of the shoulder
joint, hub angles of shoulder joint decreases and elbow
joint increase for a given set of applied torque and very
less effect on the residual vibration of the flexible linktip.

Fig. 8. Hub angles for different motor mass for set of applied
TorquesFig. 9. Dynamic flexible tip deflection for different motor
Mass

Fig. 6. Hub angles for different payloads for set of applied
torquesFig. 10. Hub angles for different hub inertia for set of applied
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Sachindra Mahto

Fig. 11. Dynamic flexible tip deflection for different
hub inertia
Fig. 14. Comparison of natural frequencies

4.3 Effects of Mass Distribution of Link
Optimal design has a great importance in engineering
applications. Shape optimization is done as per the
optimization problem (Table 1) to increase thefundamental frequency and the optimized shape of theflexible link is shown in Fig. 12, where the mass of thelink is re-distributed. It is observed that mass isconcentrated more towards the root side in shape
optimization. Static link deflection due to 1N force attip, natural frequencies of the optimized flexible link areplotted in Fig. 13 and Fig. 14 and its dynamics (hubangles and residual vibration) are plotted in Fig. 15 andFig. 16 respectively for set of excitation torque. It isobserved that parameters viz., static tip deflection,
natural frequencies, hub angle and residual vibration are
improved.

Fig. 15. Comparison of joint angles for set of applied torques

Fig. 16. Comparison of tip residuals for set of applied torques

4.4 Effects of Controlled Torque
Yigit [19] presented the position and derivative (PD)control torque for single link revolute-jointed flexible
manipulator as given below
τ PD =− K p (θ − θ f ) − K v (θ).
(14)

Fig. 12. Optimized shape of flexible link

The feedback gains K p and K v depend upon theequivalent rigid system parameters and are expressed as
1

3

2

2

(15)

K v =2( J h + mL + M p L ) f n .

3

2

(16)

Kp =
( J h + mL + M p L ) f n ,Fig. 13. Static beam deflection due to 1 N force at the tip of
optimized link

3

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1

3

Effects of System Parameters

where 'm' is the mass per unit length, 'L' is the link
length, 'Jh' , 'θf' is the final angular position, and 'fn ' isthe fundamental frequency. PD controller is able tostabilize the system but vibration of the flexible beam
can not be controlled. Ge et al. [20] extended the workof Yagit [19] and introduced energy-based robust (EBR)
control law and added the nonlinear deflection feedback
to improve the performance of the PD controller byadding the nonlinear control term as given byttt= PD − K f y ( L, t ) ∫ θ (σ ) y ( L, σ )dσ ,
EBR
0

(17)

where K f is the gain constant of robust control, σ isthe dummy variable, θ is the hub angular accelerationand y is the deflection at the tips. Above control law
is also applicable for multi-link robotic revolute jointed
systems.
Elbow angle due to the controlled torqueis plotted in Fig. 17 and residual vibration is shown in
Fig. 18. It is observed that hub angle is achieved fasterby controlled torque and tip vibration is lesser than thesinusoidal excitation.

5. Conclusion
In this work, finite element analysis of revolute-jointed
rigid-flexible manipulator has been performed throughlinear modeling. Classical nonlinear optimization isused to solve the constrained shape optimization. From
the numerical experiments, it is observed that there is agreat role of system parameters (link lengths, payloads,
hub-inertia, link shape) in its system dynamics andshould be analyzed properly during the design. It is also
observed that controlled torque improves the system
dynamics further.
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5.
Fig. 17. Comparison of elbow angles for different excitation
6.

7.

8.
Fig. 18. Comparison of dynamic tip response for different
excitation

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