Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 4, pp. 911-923, Warsaw 2015
DOI: 10.15632/jtam-pl.53.4.911

PAYLOAD MAXIMIZATION FOR MOBILE FLEXIBLE MANIPULATORS
IN ENVIRONMENT WITH AN OBSTACLE

Hamidreza Heidari
College of Mechanical Engineering, Malayer University, Malayer, Iran

e-mail: hr.heidari@malayeru.ac.ir

Mohammad Haghpanahi, Moharam Habibnejad Korayem
College of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

A mobile flexible manipulator is developed in order to achieve high performance require-
ments such as high-speed operation, increased high payload to mass ratio, less weight, and
safer operation due to reduced inertia. Hence, this paper presents a method for finding the
MaximumAllowableDynamic Load (MADL) of geometrically nonlinear flexible linkmobile
manipulators. The full dynamic model of a wheeled mobile base and the mounted flexible
manipulator is consideredwith respect to dynamics of non-holonomic constraint in environ-
ment including anobstacle. In dynamical analysis, an efficientmodel is employed to describe
the treatment of a flexible structure in which both the geometric elastic nonlinearity and
the foreshortening effects are considered. Then, a path planning algorithm is developed to
find the maximum payload that the optimal strategy is based on the indirect solution to
the open-loop optimal control problem. In order to verify the effectiveness of the presented
algorithm, several simulation studies are carried out for finding the optimal path between
two points in the presence of obstacles. The results clearly show the effect of flexibility and
the proposed approach onmobile flexible manipulators.

Keywords: flexible link, nonholonomic mobile manipulator, optimal control, obstacle, path
planning

1. Introduction

Mobile manipulators that are required to have a long reach, fast motion and reduced weight
typically also possess significant structural flexibility. For example, mobile flexiblemanipulators
have important application in space stations, manufacturing automation, nuclear contaminated
environments, andmany other areas. A common task for mobile robots is handling heavy loads
from one place to another, particularly for wheeledmobile flexiblemanipulators when operating
in high speeds with long arms. For such systems, to make an effective use of robotic systems, it
is important to consider the path planning of the system for finding full-load motion in point-
-to-point maneuvers since it increases the productivity and economic usage of robotic systems.
However, kinematic anddynamic analysis of such anonholonomicwheeledmobile robot (WMR)
is challenging due to complexwheel/manipulator interactions, flexibility and kinematic constra-
ints. An efficient model should be employed to describe the treatment of a flexible structure in
which both the geometric elastic nonlinearity and the foreshortening effects are considered.
In this investigation, the optimal strategy is based on the indirect solution to the open-loop

optimal control problem. In the open loop optimal control, in spite of the closed-loop ones,many
difficulties like process nonlinearities and all types of constraints can be explicitly considered
because of the off-line computation of optimal trajectories. On the other hand, the indirect
solution method appears to be a well suited approach for this kind of problems, which is based



912 H. Heidari et al.

on Pontryagin’s minimum principle. Combining this approach with an iterative algorithm, the
optimal paths andmaximum load forWMM in the presence of obstacles can be achieved.
Many researchers have studied the problems of mobile manipulators for the last few years.

The dynamic model for links in most of these researches is often based on rigid or small deflec-
tion theory, but for applications like light-weight links, high-precision elements or high speed
maneuver, it is necessary to capture the deflection caused by nonlinear terms. Seraji (1998) re-
ported a simple on-line approach formotion control ofmobilemanipulators using the augmented
Jacobian matrix. Yamamoto and Yun (1994) focused their research on the modeling and com-
pensation of the dynamic interaction between the manipulator and the mobile platform of a
mobile manipulator, and developed a coordination algorithm based on the concept of a prefer-
red operating region. Korayem et al. (2012) andXi andFenton (1991) designed an algorithm for
motion planning of flexiblemanipulators in quasi-static operations. A concisemotion expression
for flexible manipulators was developed to reflect the contributions of joint motions and link
deflections to themotion of the end effector by three respective Jacobians. It was found that the
algorithmwas efficient and accurate formotion planning of flexible linkmanipulators. Damaren
and Sharf (1995) presented and classified different types of inertial and geometric nonlinearities
in thedynamical equation for flexiblemultibody systems.Theyobserved that for sufficiently fast
maneuvers of the flexible-linkmanipulators, the ruthlessly linearized approximation completely
inadequate.
Several papers tried to give an answer to the pathplanningproblemand calculate theMADL

for rigid andflexiblemanipulators. For instance,Wang et al. (2001) developed an algorithm that
maximized the robot payload while taking into account realistic constraints such as joint torque
limits and velocity bounds. The governing optimal control problemwas converted into a direct,
SQPparameter optimization inwhich the joint trajectorieswere definedbyB-spline polynomials
along with a time-scale factor. Park (2003) presented a method for generating the path of a
redundant flexible manipulator which significantly reduced residual vibration in the presence of
obstacles. The proposed method was based on an optimized path that was constructed from
a combined Fourier series and polynomial with coefficients of each harmonic term selected to
minimize the residual vibration.
Oneof themostpopularmethods for obstacle avoidance is theartificial potential fieldmethod

(Castro et al. 2002). In contrast to manymethods, the robotmotion planning through artificial
potential fields (APF) is able to take into account simultaneously the problems of obstacle
avoidance and trajectory planning. The first use of the APF concept for obstacle avoidance
was presented by Khatib (1986). He proposed the force involving an artificial repulsion from
the surface which should be non-negative, continuous and differentiable. More recently, a new
version such as the repulsive artificial potential field has been proposed (Agirrebeitia et al.,
2005). In most of the previous works, flexibility and nonholonomic constraints of the wheeled
mobile manipulator in the path planning problem have not been considered. Hence, this paper
proposes amethod for planning the trajectory of the nonholonomicmobile flexible manipulator
for determining the maximum allowable load and considering the effect of flexibility to achieve
the specified point to point maneuver in the presence of an obstacle.

2. Kinematic and dynamic model of the mobile flexible manipulator

In this Section, for the sake of modeling and analysis, a mobile flexible manipulator comprising
a manipulator flexible arm mounted on a nonholonomic mobile base is considered, as shown
in Fig. 1. The motion of the system has to be decomposed into the motion of the flexible
manipulator and themotion of the base. The unidirectional platform shown inFig. 2 is a typical
example of a nonholonomic WMR which has two rear driving wheels and two castor wheels.



Payload maximization for mobile flexible manipulators... 913

The two driving wheels are powered by DC motors and have the same wheel radius r. The
point Pb is the origin of WMR axis, which is located at the intersection of the longitudinal
x-axis and the lateral y-axis. L0 and b are length and width or WMR body, respectively. The
origin of the inertial frame {X,Y} is shown as O and as such allows the position of the WMR
to be completely specified through the following vector of generalized coordinates with respect
to {X,Y}, qb = [Xb,Yb,ϕ,θr,θl], where Xb and Yb are the coordinates of the center of mass.
The orientation of the WMR frame from the inertial frame is denoted by ϕ. θr and θl are the
angular displacements of the right and left drivingwheel, respectively. Due to the nonholonomic
nature of the system, the constraint equation obeys the ideal no-slip condition. The rolling and
the knife edge constraint equations for this system can be found in Yamamoto and Yun (1994).
In the next Section, these constraints will be explained.

Fig. 1. Schematic view of a mobile flexible manipulator

Fig. 2. Nonholonomic wheeled mobile robot platform

The global position vector of the end-effector r can be defined by appropriately considering
the position vector of the corresponding local coordinate in the global reference system as

ri = rb +rm/b = rb +
0
i−1r+R





x+u
v
w





(2.1)

where rb is the position of the mobile platform, R is the transformation matrix and u, v, w
denote the longitudinal and transverse displacements.
The analysis of the flexible link can be modeled by slender elastic beams. In this investiga-

tion, a more efficient computationally model is employed, in which both the geometric elastic



914 H. Heidari et al.

nonlinearity and the foreshortening effects are considered. The model takes into account the
distinction between the longitudinal displacement due to axial deformation, denoted as s, and
the longitudinal displacement that can occur due to the foreshortening effect, denoted by ufs.
The longitudinal displacement caused by transverse deflection of the neutral axis of the beam
can be expressed as (Korayem et al., 2012)

ufs =−
1
2

x∫

0

[(∂v
∂x

)2
+
(∂w
∂x

)2]
dx (2.2)

The assumed field of displacements for u can be written as

u=



u
v
w


=



s+ufs
v
w


 (2.3)

The general expression of the strain energy is written in terms of s,v and w, as below (Korayem
et al., 2012)

U =
EA

2

l∫

0

(∂s
∂x

)2
dx+

EI

2

l∫

0

(∂2v
∂x2

)2
dx+

EI

2

l∫

0

(∂2w
∂x2

)2
dx (2.4)

where E, A, I, and l denote Young’s modulus, cross-sectional area, moment of inertia of the
cross section, and length, respectively. This formulation brings nonlinear inertia terms and a
constant stiffness matrix in the equations of motion.
TheLagrangianmethod is utilized to formulate the dynamic equations governing themotion

ofmobile flexiblemanipulator systems. In order to derive dynamic equations, the kinetic energy
and the potential energy are computed for the entire system. The kinetic energy for the overall
system is obtained by computing the kinetic energy for each element ij and then by summing
over all the elements. The potential energy of the manipulator is obtained by computing the
strain energy for each element ij due to elasticity and gravity of any link. After calculation
of these energies, by applying the Lagrangian multipliers procedure and performing some alge-
braic manipulations, the compact form of the governing equations of a two-link flexible mobile
manipulator can be obtained from




Mbb Mbm Mbf
Mmb Mmm Mmf
Mfb Mfm Mff








q̈b
q̈m
q̈f



+




Cb(qb,qm,qf, q̇b, q̇m, q̇f)
Cm(qb,qm,qf, q̇b, q̇m, q̇f)
Cf(qb,qm,qf, q̇b, q̇m, q̇f)



+




Gb(qb,qm,qf)
Gm(qb,qm,qf)
Gf(qb,qm,qf)



=




Fb −AT(q)λ

Fm
0





(2.5)

whereM is the nonlinear mass matrix,C is the vector of Coriolis and centrifugal forces,G de-
scribes the gravity effects, and A denotes the nonholonomic constraints. The generalized coor-
dinates q and the generalized forceF are the following vectors

q= [qb,qm,qf] = [Xf,Yf,ϕ,θr,θl,θ1,θ2,qf1, . . . ,qfn]

F= [Fb,Fm,0] = [0,0,0,τwr,τwl,τ1,τ2,0, . . . ,0]
(2.6)

The WMRs are called nonholonomic mobile robots because of their no-slip kinematic constra-
ints. The vehicle is prevented from sliding sideways relative to its instantaneous heading, and
each drive wheel is assumed to roll without slipping. These three independent nonholonomic
constraints are represented as

Ẏf cosϕ− Ẋf sinϕ−dϕ̇ =0
Ẏf sinϕ+ Ẋf cosϕ+ bϕ̇ = rθ̇r

Ẏf sinϕ+ Ẋf cosϕ− bϕ̇ = rθ̇l
(2.7)



Payload maximization for mobile flexible manipulators... 915

The compact form of the nonholonomic constraints can be written as

A(q)q̇=0 (2.8)

and

A=



−sinϕ cosϕ −d 0 0 0 0 0 . . . 0
−cosϕ −sinϕ −b r 0 0 0 0 . . . 0
−cosϕ −sinϕ b 0 r 0 0 0 . . . 0


 (2.9)

where r is the radius of the driving wheel, b is the distance of two wheels and d is the distance
between the front and rear wheels.
By defining the matrix B(q) , which is the null space of the matrix A(q), the Lagrange

multipliers can be eliminated

A(q)B(q) = 0 (2.10)

One choice ofB(q) is as follows

B=




(r/2b)(bcosϕ−dsinϕ) (r/2b)(bcosϕ+dsinϕ) 0 0 0 . . . 0
(r/2b)(bsinϕ+dcosϕ) (r/2b)(bsinϕ−dcosϕ

)
n 0 0 0 . . . 0

(r/2b) −(r/2b) 0 0 0 . . . 0
1 0 0 0 0 . . . 0
0 1 0 0 0 . . . 0
0 0 1 0 0 . . . 0
0 0 0 1 0 . . . 0
0 0 0 0 1 . . . 0
...

...
...
...
...
... 0

0 0 0 0 0 0 1




(2.11)

as well as q̇ can be expressed as follows

q̇=B(q)v (2.12)

where

v= [θ̇r, θ̇l, θ̇1, θ̇2, q̇f1, . . . , q̇fn]
T (2.13)

By differentiating equation (2.12)

q̈=B(q)v̇+ Ḃ(q)v (2.14)

after performing some algebraic manipulations, the dynamic equation of the mobile flexible
manipulator is

BT(q)M[B(q)v̇+ Ḃ(q)v]+BT(q)(C+G)=BT(q)F (2.15)

Finally, the dynamic equations in the state space are as follow

ẋ=

[
Bv

(BTMB)−1(−BTMḂv−BTC)

]
+

[
0

(BTMB)−1

]
F (2.16)

By using these equations, the optimal trajectory planning problem can be formulated.



916 H. Heidari et al.

3. Optimization strategy

Path planning in the case of a mobile flexible manipulator is complex due to flexibility of the
manipulator arm. In this Section, an indirect solution to the optimal control problem is applied
for the off-line global trajectory planning of the mobile flexible manipulator. The purpose of
the optimal control problem is to determine the control u(t) that minimizes the performance
indexJ(u). In this investigation, the specificobjective functionalJ is to obtain the optimal paths
withminimum effort and vibration. The general expressionwhichminimizes the cost functional
means that (Wang et al., 2001)

minJ =

tf∫

t0

L
(
X(t),U(t), t

)
dt =

1
2
‖X1‖2WP +

1
2
‖X2‖2WV +

1
2
‖U‖2R (3.1)

Here, the integrand L(·) is a smooth differentiable function in the arguments, X(t) and U(t)
denote the state space form of the generalized coordinate and the joint torque, respectively.
‖X‖2

K
= XTKX is the generalized squared norm, WP , WV are symmetric, positive semi-

-definite (k×k) weightingmatrix, andR is the symmetric, positive definite (k×k)matrix. The
designer can decide on the relative importance among the angular position, angular velocity,
vibration amplitude and control effort by the numerical choice of penalty matrices WP , WV
andR. In order tominimize theobjective function subjected to thenonlineardynamic equations,
the well-known Pontryagin minimum principle is used. By introducing the cost vector ψ, the
Hamiltonian function of the system can be defined as

H(X,U,ψ, t)= L(X,U)+ψTẊ (3.2)

ThePMPthen implies that the necessary condition for a localminimum is that H beminimized
with respect to u(t) at all times. If it is assumed that the set of admissible inputs is bounded
U−i ¬ u∗i ¬ U

+
i , this condition is equivalent to

Ẋ=
∂H

∂ψ
ψ̇ =
−∂H
∂X

0=
∂H

∂U
(3.3)

The considered boundary conditions are

X(ti)=Xi X(tf)=Xf (3.4)

where X(ti) and X(tf) represent positions and velocities of the links at the beginning and at
the end of the maneuver. The optimal trajectory is then obtained by solving the 2n differential
equations

ẋ∗(t)=
∂H

∂p

(
x∗(t),u∗(t),p∗(t), t

)

ṗ∗(t)=−∂H
∂x

(
x∗(t),u∗(t),p∗(t), t

)

H
(
x∗(t),u∗(t),p∗(t), t

)
¬ H

(
x∗(t),u(t),p∗(t), t

)

(3.5)

The control values are limited with the upper and lower bounds. One of the most commonly
used motors for actuating the joints of small and medium size mobile robots are permanent
magnetDCmotors. Typical speed-torque characteristics of DCmotors inwhich the relationship
between speed and torque is linear are defined as below (Wang et al., 2001)

U
(+)
allow
=K1−K2q̇ U

(−)
allow
=−K1−K2q̇ (3.6)



Payload maximization for mobile flexible manipulators... 917

where

K1 =
[
τs1 τs2 . . . τsm

]T
K2 = diag

[
τs1
ω1

τs2
ω2

. . . τsm
ωm

]T

the stall torque (torque generated by the motor when fully “ON” but unable to move) and no-
-load speed (output speed of the motor when runningwithout load) are denoted by τs and ωm,
respectively.

3.1. Maximum payload algorithm

The set of dynamic equations, the governing optimal control problem and the boundary
conditions lead to the standard form of a two-point boundary value problem (TPBVP). The
collocation method is one of the basic ways of solving TPBVP. The method iterates on the
initial values of the co-state until the final boundary conditions are satisfied by the following
desired accuracy

h(X(tf), tf)=
1
2
‖X1(tf)−X1f‖2Wp +

1
2
‖X2(tf)−X2f‖2WV ¬ ε (3.7)

In this Section, an algorithm is proposed to find the maximum payload shown in Fig. 3. The
proposedmethod considers torque bounds and is based on increasing the payload until one point

Fig. 3. The algorithm for calculation of the maximum payload

of one of the actuators torque reach the upper or lower torque bounds. As shown in Fig. 3, the
proposedalgorithm includes two stages; thefirst stage index i increases the tipmassmpuntil the
actuators torque reach the upper or lower torque constraints. The desired accuracy ε inTPBVP
solutionmust be satisfied for the payload in each step.A further increase in the payload exceeds
the torque limits. Consequently, the desired accuracy ε inTPBVPsolution could not be satisfied
and the boundary conditions at yhe final timemay be obtained incorrectly. At this status, while
onepoint of one of the actuators torque reach theupperor lower torque bounds; the second stage



918 H. Heidari et al.

index k decreases the payload until themaximum payload for the supposed penalty matrices is
obtainedwith the accuracy ε. Theaccuracy of themaximumpayload mpmax calculation depends
on the value e. The iteration number is denoted by s symbol in this algorithm.

3.2. Path planning in the presence of an obstacle

In order formobile flexiblemanipulators to successfully carry out tasks, especially in carrying
heavy loads on different trajectories, the Artificial Potential Fields (APFs) is used throughout
the robot workspace with each point in the workspace having an associated potential. The idea
used in APF-based obstacle avoidance is to position a mobile manipulator in the workspace
such that the overall potential encountered by the mobile manipulator is minimized while still
accomplishing the desired task. A repulsive potential formulation based on the distance between
parts of the WMM and obstacles is used in the cost function for obstacle avoidance. The most
commonly used repulsive potential takes the form (Khatib, 1986)

Urep =






1
2
η
( 1
ρ(q,qobs)

− 1
ρ0

)
if ρ(q,qobs)¬ ρ0

0 if ρ(q,qobs) > ρ0
(3.8)

where η is a positive weightingmatrix, ρ(q,qobs) denotes theminimal distance from the robot q
to the obstacle, qobs denotes the point on the obstacle such that the distance between this point
and the robot is minimal between the obstacle and the robot, and ρ0 is a positive constant
denoting the distance of influence of the obstacle.
The obstacle avoidance problem is formulated in terms of collision avoidance of the base,

links and joints with the obstacles. In order to add the penalty function to the performance
index in order to guarantee free-collision motion of the mobile body, the distance between the
center of the mobile base and the center of the obstacle will be

ρb = ‖Pobs −Pb‖=
√
(Xobs −Xb)2+(Yobs −Yb)2 (3.9)

By assuming the links as lines, the minimal distance between ij link and the center of the
obstacle can be calculated as

ρij =
1

√
(Xj −Xi)2+(Yj −Yi)2

∣∣∣∣∣det
[
Xj −Xi Xobs −Xi
Xj −Xi Yobs −Yi

]∣∣∣∣∣ (3.10)

Theposition of parts in theworkspace is denotedbyXi andYi. Therefore, theobjective function
to guarantee the free-collision motion can be defined as

J(u)=

tf∫

t0

L(X,U, t) dt =
1
2
‖X1‖2W1 +

1
2
‖X2‖2W2 +

1
2
‖U‖2

R
+
1
2

∥∥∥
1
ρ
−
1
ρ0

∥∥∥
2

η
(3.11)

The new objective function is used to obtain the trajectory optimization problem to avoid
collision of theWMMparts with the obstacles. Figure 4 shows the wheeledmobile robot in the
presence of an obstacle.

4. Simulation results

A simulation study has been carried out to investigate further the validity and effectiveness of
the mobile flexible manipulators in finding the optimal path between two points with different
objective functions.A two-link planarmanipulator is considered. It ismountedon adifferentially
drivenmobile base at pointF on themain axis of thebase (Fig. 4).Theparameters of themobile
flexible manipulator are given in Table 1.



Payload maximization for mobile flexible manipulators... 919

Fig. 4.WMM in the presence of an obstacle

Table 1. Simulation of parameters

Parameter Value (base) Value (manipulator)

Length [m] L0 =0.4 L1 = L2 =1
Mass [kg] mb =94 m1 = m2 =3, mp =0.5
Cross section area [m2] – A1 = A2 =4 ·10−8
Moment of inertia [m4] Ib =6.609 I1 =0.416, I2 =0.0625
Young’s modulus of the material [N/m2] – E1 = E2 =2 ·1010

4.1. First case study: optimal path for minimum effort

The motion planning problem is to find the optimal trajectory with minimum effort. The
main motivation behind the minimum effort is to find a path to reduce the amount of torques
and hence to lower energy consumption. According to the algorithm presented in Fig. 3, the
general solution method is based on increasing the payload from its minimum value mpmin up
to themaximumpayload can be found. Therefore, in this case, the initial payload, initial values
of the co-state vector, accuracy values, and penalty matrices are considered as follows

mpmin =0.5kg ψ(0)= 0 e =0.1 ε =0.0001

Wp =WV =0 R= diag(1)
(4.1)

This cost function is typical for systems that need to conserve energy during a particular ope-
ration. The actuator constants are given as follows

K1 =
[
20 20 50 50

]T
N ·m

K2 = diag
[
1.5 1.5 2.5 2.5

] N ·m · s
rad

(4.2)

The system is initially at rest, thus the mobile base is initially at the point (xF = 0.75m,
yF =−0.5m, ϕ =0◦) andmoves to its final position (xF =1.6m, yF =−0.2m, ϕ =15◦). The
initial conditions of the manipulator are θ1(0) = 1.5rad, θ2(0) = 2rad, θ̇1(0) = 0, θ̇2(0) = 0
(pointA inFig. 5a) and thefinal conditions are θ1(tf)=−0.86rad, θ2(tf)= 1.09rad, θ̇1(tf)= 0,
θ̇2(tf) = 0 (point B in Fig. 5a) during the overall time tf = 1.9s (see Fig. 5a) and also the
remaining boundary conditions are equal to zero.
A comparative study is carried out between the rigid model and flexible models (linear and

nonlinear models) as shown in Fig. 5b. The optimal angular positions of the link and wheels,
corresponding to the minimum effort are shown in Figs. 6a and 6b.



920 H. Heidari et al.

Fig. 5. The optimal paths between point A and B via minimum effort (a) and for different models (b)

Fig. 6. The optimal angular positions of the first and second joints (a) and the right and left wheels (b)

The simulation results presented in Fig. 7 illustrate the optimal controls to carry the maxi-
mumpayload, which also show the upper and lower bounds of the actuator torque capacity. By
increasing the payload from mpmin to mpmax, the required torque grows until one point of one
of the actuators torque reach the upper or lower torque bounds. It can be seen that, the second
motor reaches its maximum capacity. In this case study, the maximum payload is obtained to
be mpmax =2.45kg, while by considering the rigid link andNikoobin’smethod (Korayem et al.,
2012), themaximumpayload is found to bempmax =8.25kg. This difference is due to flexibility
of the link which increases the oscillation of torque curves.

Fig. 7. Minimum effort of the first and secondmotors within upper and lower acceptable boundaries

4.2. Second case study: minimum vibration trajectory

In the motion planning of flexible robots, obtaining the minimum vibration trajectory is
one of themost frequently encountered problems. The optimization objective is tominimize the
vibration excitation during motion. By increasing the weighting factors corresponding to the
derivative of flexural displacements (q̇f1, . . . , q̇fn), the vibrational motions will be suppressed.
The bounds of the motor capacity are not considered. Hence, the proper penalty matrices are
selected to be R = diag(0.01) and Wp = 0, WV = diag(0,0,1, . . . ,1). The load must be
carried from the initial point with coordinate (xe =0.5m, ye =−0.08m) to the final point with



Payload maximization for mobile flexible manipulators... 921

coordinate (xe =3.31m, ye =−0.25m).Theoptimal trajectory between these twopoints during
the overall time tf =1.9s is desired for the rest-to-rest maneuver. The other conditions remain
the same with the previous case study. The characteristics of the parts are shown in Table 1.

Fig. 8. The optimal paths via minimum vibration

Fig. 9. The optimal flexural deflections of the first link (a) and the optimal flexural displacements of the
second link (b)

Thesimulation results (small and largemodels) are illustrated inFig. 8.Theobtainedoptimal
flexural deflections, corresponding to theminimum vibration are shown in Figs. 9a and 9b. The
simulation results show that a significant reduction inmanipulator vibration can be achieved by
employing the proposed optimization procedure. As expected, it can be seen, by decreasing the
amplitude vibration, that the linear andnonlinearmodels come closer together. Also, it is shown
that application of the proper input torquemay decrease the end effector vibration significantly.

4.3. Third case study: trajectory optimization in the presence of an obstacle

Recently, there has been a great deal of interest in path planning for autonomous mobile
manipulators in the presence of an obstacle because of their ability to replace objects in a wide
workspace. Hence, the problem is how to find a feasible trajectory for all components in order
to carry themaximumpayload in environment with an obstacle. The characteristics of the non-
holonomic mobile manipulator, penalty matrices, accuracy value, and the actuator constants
are the same as in the first case study. Moreover, the augmented functional obstacle avoidance
is considered in the cost function. The task is considered to move the end effector from the
initial point p0 = (0.5,−0.08) to the final configuration pf = (3.31,−0.25) for the rest-to-rest
maneuver during the overall time tf = 1.9s. Also, there is an obstacle with robs = 0.05m at a
point with coordinates pobs = (xobs = 1.1m, yobs = −0.55m). The simulation parameters are
shown in Table 1.
In this condition, the maximum payload is found to be 2.05kg. To avoid the obstacle, the

manipulatormoves far from it and causes a decrease in the allowable payload. The schematic of
the obstacle and the obtained optimal paths of the end-effector and themobile base are shown in
Fig. 10a. A comparative study is performed between the rigidmodel and flexible models (linear
and nonlinear models) as shown in Fig. 10b.



922 H. Heidari et al.

Fig. 10. The schematic view of obtained optimal paths in the presence of an obstacle (a) and the
obtained optimal paths of the end-effector andmobile base for different models (b)

5. Conclusions

The main objective of this investigation is to determine the trajectory optimization for cal-
culating the MADL of flexible-link mobile manipulators in the point-to-point maneuver in the
presence of an obstacle, based on the indirect solution to the optimal control problem.The effect
of dynamic interaction between the flexible manipulator and the mobile platform is considered
to characterize themotion of a nonholonomicmobilemanipulator with the compliant link capa-
ble of large deflection, in which both the geometric elastic nonlinearity and the foreshortening
effects are considered. This model leads to a constant stiffness matrix and makes the formula-
tion particularly efficient in computational terms and numerically more stable than alternative
geometrically nonlinear formulations based on lower-order terms. Pontryagin’s minimum prin-
ciple is used to obtain the optimality conditions, which leads to a standard form of a two-point
boundary value problem. An augmented objective function based on an artificial potential field
is considered to avoid the obstacle during point-to-point maneuvers. Several simulation studies
on a nonholonomic wheeledmobilemanipulator are carried out for finding theMADL and opti-
mal paths with different objective functions like minimum effort and minimum vibration. The
numerical results indicate the effect of employing trajectory optimization in the performance
improvement of the mobile flexible manipulator. It is shown that the presence of the obstac-
le causes the manipulator moves far from it; therefore, it reduces the maximum load-carrying
capacity. Moreover, in this method, the designer can compromise between different objectives
by considering proper penalty matrices, and may choose the proper trajectory between various
paths. The obtained results illustrate the power and efficiency of the model in overcomming
the highly nonlinear nature of the large deflection and optimization problem, which with other
methods may be very difficult or impossible to achieve.

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Manuscript received March 12, 2015; accepted for print May 8, 2015