Smooth jerk-bounded optimal path planning of tricycle wheeled mobile manipulators in the presence of...

International Journal of Advanced Robotic Systems, www.intechweb.org Vol. 8, No. 1 (2011) ISSN 1729-8806, pp 10-20 www.intechopen.com

Path Planning of Mobile Elastic Robotic Arms by Indirect Approach of Optimal Control

Moharam Habibnejad Korayem1, Hamed Rahimi Nohooji1, 2 and Amin Nikoobin3

1 Robotic Research Lab, College of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran 2 Islamic Azad University, Damavand Branch, Damavand, Iran 3 Department of Mechanical Engineering, Semnan University, Semnan, Iran

Abstract Finding optimal trajectory is critical in several

applications of robot manipulators. This paper is applied

the open‐loop optimal control approach for generating

the optimal trajectory of the flexible mobile manipulators

in point‐to‐point motion. This method is based on the

Pontryaginʹs minimum principle that by providing a two‐

point boundary value problem is solved the problem.

This problem is known to be complex in particular when

combined motion of the base and manipulator, non‐

holonomic constraint of the base and highly non‐linear

and complicated dynamic equations as a result of flexible

nature of links are taken into account. The study

emphasizes on modeling of the complete optimal control

problem by remaining all nonlinear state and costate

variables as well as control constraints. In this method,

designer can compromise between different objectives by

considering the proper penalty matrices and it yields to

choose the proper trajectory among the various paths.

The effectiveness and capability of the proposed

approach are demonstrated through simulation studies.

Finally, to verify the proposed method, the simulation

results obtained from the model are compared with the

results of those available in the literature.

Keywords Elastic mobile robotic arms, Optimal motion

planning, Pontryaginʹs minimum principle.

1. Introduction

Mobile manipulators due to their extended workspace

offer an efficient application for wide areas. However,

these systems are usually ʺpower on boardʺ with limited

capacity. Hence, using light links can minimize the inertia

and gravity effects on actuators. This helps the mobile

manipulator to work in energy ‐ efficient manner and

pilots us to use of flexible manipulators. The platform path planning was investigated for a given

end‐effector path with assuming that the robot is

redundant (Chen M.W. & Zalzala A.M.S., 1997). Then, the

robot configuration was determined by minimizing the

overall actuation torque and maximizing the

manipulability criterion. By employing an appropriate

mapping, Papadopoulos et al. transformed the three

differentials of the nonholonomic constraint into two

differentials and designed the path via polynomial

functions (Papadopoulos E. G. & Poulakakis J., 2001).

Tanner by implementing a potential field technique using

point‐world topology proposed a methodology to motion

planning for multiple mobile manipulators cooperation

that is applicable to articulated, non point nonholonomic

robots with guaranteed convergence properties (Tanner

H.G., 2003). Sun et al. discussed the load distribution and

joint trajectory planning of two robots with one flexible

link (Sun Q., 1999). The control of two‐link flexible

manipulators was studied on the basis of quasi‐static

11 International Journal of Advanced Robotic Systems, Vol. 8, No. 1 (2011)

equation (Matsuno F. & Hatayama M., 1999). Subudhi

and Morris presented a dynamic modeling technique for

a manipulator with multiple flexible links and flexible

joints based on a combined Euler–Lagrange formulation

and assumed modes method. Then, they controlled such

system by formulating a singularly perturbed model and

used it to design a reduced‐order controller (Subudhi B.

& Morris A.S., 2002). Path planning of a flexible

manipulator was formulated as a discrete‐time open‐loop

optimal control problem which its solution was done via

discrete dynamic programming (Wilson D.G. et al., 2004).

The assumed mode expansion method was used to derive

the dynamic equation of fixed base flexible manipulator

(Green A. & Sasiadek J.Z., 2004).

In above‐mentioned works, only base mobility or link

flexibility has been considered, and the synthesis of the

mobile base with flexible links has not been studied.

However, as explained before, the flexible links

manipulator mounted on the mobile platform can exhibit

many advantages. Such systems require less material,

consume less power, require smaller motors and they

reduce the torque capacity; hence, they can improve the

efficiency of the system noticeably. The maximum

payload of flexible mobile manipulator was determined

along the given trajectory using finite element approach.

So, the study did not consider finding the optimal path

(Korayem M.H. et al., 2008). A computational algorithm

was presented to determine the maximum payload. In

this study, linearizing the dynamic equations and

constraints are done on the basis of Iterative Linear

Programming (ILP) approach for flexible mobile

manipulators (Korayem M.H. & Ghariblu H., 2004). But,

in ILP method, final boundary conditions are not satisfied

exactly and obtained results have a significant error in

final time, so the end effector can not place in the desired

target point. Furthermore, in this method, the linearizing

procedure and its convergence to the proper answer is a

challenging issue, especially when nonlinear terms are

large and fluctuating, e.g. in problems with consideration

of flexibility in links or having high‐speed motion. As a

result, in the previous ILP based works, the link flexibility

has not been considered either in the dynamic equations

or in the simulation procedure.

Optimal control can be used in both open loop and close‐

loop strategies. However, because of the off‐line nature of

the open loop optimal control in spite of the close‐loop

ones, many difficulties such as system nonlinearities and

all types of constraints may be catered for and

implemented easily, so it generally used in analyzing

nonlinear systems such as trajectory optimization of

different types of robots (Mohri et al., 2001; Furuno S. et

al., 2003; Sentinella M. R. & Casalino L., 2006). This

method is solved by direct and indirect approaches.

However, since direct method leads to the approximate

solution also this approach is time consuming and quite

ineffective due to the large number of parameters involved

(Wilson D.G. et al., 2004; Hull D.G., 1997; Park K. J., 2004),

indirect methods is a good candidate for the cases where

the system has a large number of degrees of freedom or

optimization of the various objectives is targeted (Kirk DE.,

1970; Bessonnet G. & Chessé S., 2005; Bertolazzi E. et al.,

2005; Callies R. & Rentrop P., 2008). This method is often

applied to the generation of highly accurate trajectories and

in combination with the structural analysis of optimal

control problems in robotics.

Korayem et al. used indirect solution of open‐loop

optimal control approach to find the maximum load

carrying capacity of a rigid link mobile manipulators for a

given two‐end‐point task (Korayem M.H. et al., 2009). In

spite of the previous studies based on ILP approach, they

satisfied boundary conditions exactly and solved the

optimization problem explicitly. Also, the complete form

of the obtained nonlinear equation was used in their

study and unlike the previous studies linearizing the

equations was not required. However, they have studied

the rigid‐link manipulator and they have not considered

the flexibility of links.

In the presenting paper, with applying an indirect

solution of optimal control problem, path planning of

flexible mobile manipulator is studied based on

Pontryaginʹs minimum principle. The paper firstly deals

with the modeling of the general flexible mobile

manipulators. Although path planning of flexible

manipulators is much more complicated task than the

rigid ones, the method handles highly non‐linear and

complicated dynamics of the system. After that,

nonholonomic constraints are defined and dynamic

equations are obtained in the state space form. Then, the

optimal control problem that with employing of

Pontryaginʹs minimum principle supports the execution

of the optimization solution of model is expressed. In

comparison with other method the open‐loop optimal

control method does not require linearizing the

equations, differentiating with respect to joint parameters

and using of a fixed‐order polynomial as the solution

form. Moreover, via changing the penalty matrices

values, various optimal trajectories with different

specifications can be obtained which able the designer to

select a suitable path through a set of obtained paths.

Then, an application example with two‐link mobile

manipulator with flexible links is presented and

discussed to demonstrate the effective performance of the

proposed approach. Also, in order to validate the

method, simulations are performed for the high‐stiffness

flexible link manipulator and compared with the

equivalent rigid links done in 0(Korayem M.H. et al.,

2009). The comparison shows a good agreement between

the results of this study and those reported in the

literature. Lastly, the paper is concluded with

highlighting the feature properties of the proposed

model.

Moharam Habibnejad Korayem, Hamed Rahimi Nohooji and Amin Nikoobin: Path Planning of Mobile Elastic Robotic Arms by Indirect Approach of Optimal Control 12

2. Modeling of a general mobile manipulator

with multiple flexible links

In order to path planning of a flexible mobile

manipulator, a suitable dynamic modeling of the system

is needed. Thus, concepts such as base mobility and link

flexibility must be explained exactly. The generalized

coordinates of the flexible mobile manipulator consist of

three parts, the generalized coordinates defining the

mobile base motion Tbnbbb )q,,q,q(qb

21= , the

generalized coordinates of the rigid body motion of links T

kr )q,,q,q(q

21= and the generalized coordinates that

related to link flexibility T

knknnf )q,,q,,q,,q,q,,q,q(qfff

122111211= , where

bn and fn are base degree of freedom and number of

manipulator mode shapes, respectively. In the mobile

manipulators, the concept of redundancy expresses when

the number of the overall rigid system degrees of

freedom ( bm nnn += ) is strictly greater than the end

effector degree of freedom in Cartesian space (m). Where,

mn is stood for the rigid manipulator degree of freedom.

Thus, the mechanical system is redundant if mn > and

the order of redundancy is mnr . There are different

techniques, which can be applied to a robotic system to

solve the redundancy resolution. Some of these

techniques are based on an optimization criterion such as

overall torque minimization, minimum joint motion and

so on. Hence, Seraji has used r additional user‐defined

kinematic constraint equations as a function of the motion

variables (Seraji H., 1998). This method results in simple

and online coordination of the control of a mobile

manipulator during motion. The presenting study follows

this method. Hence, some additional suitable kinematic

constraint equations to the system dynamics are applied.

Results are in simple and on‐line coordination of the

mobile manipulator during the motion. These constraints

undertake the robot movement only in the direction

normal to the axis of the driving wheels along with

previously specification of the base trajectory during the

motion.

The robotic systems with flexible links are continuous

dynamical systems characterized by an infinite number of

degrees of freedom and are governed by nonlinear

coupled, ordinary and partial differential equations. The

exact solution of such systems is not feasible practically

and the infinite dimensional model imposes severe

constraints on the design of controllers as well. However,

they are discretized using assumed modes, finite

elements or lumped parameter methods.

In the lumped parameter model, which is the simplest

one for analysis purpose, the manipulator is modeled as

spring and mass system, which does often not yield

sufficiently accurate results (Zhu G. et al. 1999; Kim J.S. &

Uchiyama M., 2003). Many authors used the finite

element method where the elastic deformations are

analyzed by assuming a known rigid body motion and

later superposing the elastic deformation with the rigid

body motion 0(Korayem M.H. et al., 2008). However, in

order to solve a large set of differential equations derived

by the finite element method, a lot of boundary

conditions have to be considered, which are, in most

situations, uncertain for flexible manipulators (Yue S. et

al., 2001). In assumed mode model formulation, the link

flexibility is usually represented by a truncated finite

modal series in terms of spatial mode eigen functions and

time‐varying mode amplitudes. Hence, using this modes

of vibration to model robot dynamics establish an

appropriate interaction between flexural vibrations and

nonlinear dynamics of system. The assumed mode

method and the finite element method use either the

Lagrangian formulation or the Newton–Euler recursive

formulation to model the problem. In the presenting

research, all the flexible links are assumed Euler–

Bernoulli beams, so the shearing and rotary inertia effects

are neglected. In addition, assumed modes method with

the generalized Euler–Lagrange formulation is used to

derive the dynamic equations of flexible manipulators.

For a general k‐link flexible robot, the vibration )t,x(v ii of

each link can be obtained through truncated modal

expansion, under the planar small deflection assumption

of the link as

k.,..,i)t(e)x(φ)t,x(vik

jijiijii 1==

1=

∑ , (1)

where ik is the number of modes used to describe the

deflection of link i; )x(φ iij and )t(eij are the thj mode

shape function and thj modal displacement for the thi

link, respectively.

Hence, applying the Lagrangian assumed modes method

the dynamic equation of flexible mobile manipulator

could be obtained as:

.

),,(

),,(

),,(

),,,,,(

),,,,,(

),,,,,(

f

r

b

frbf

frbr

frbb

frbfrbf

frbfrbr

frbfrbb

f

r

b

fffrfb

rfrrrb

bfbrbb

U

U

U

qqqG

qqqG

qqqG

qqqqqqC

qqqqqqC

qqqqqqC

q

q

q

MMM

MMM

MMM

(2)

By defining TTf

Tr

Tb )q,q,q(q =

as the vector of generalized

coordinate of the system, Eq. (2) can be written in

compact form as

( ) ,U)q(G)q,q(CqqM =++ (3)

where M is the Inertia matrix, C is the vector of Coriolis

and centrifugal forces, G describes the gravity effects and

U is the generalized force inserted into the actuator.


3. State‐space form of dynamic equation

Consider an n DOFs rigid mobile manipulator with

generalized coordinates n,...,,i,qq i 21 and a

task described by m task coordinates m,...,,j,rj 21

with m < n. By applying h holonomic constraints and c

non‐holonomic constraints to the system, chr +=

redundant DOFs of the system can be directly

determined. Therefore, m DOFs of the system is remained

to accomplish the desired task. As a result, we can

decomposed the generalized coordinate vector as

Tnrr qqq , where rq is the redundant generalized

coordinate vector determined by applying constraints

and nrq is the non‐redundant generalized coordinate

vector. Also, considering the flexible link manipulators

instead of the rigid ones, their related generalized

coordinates, fq , are added to the system; therefore, the

overall decomposed generalized coordinate vector of

system obtain as Tnrfr qqq , where nrfq is the

combination vector of nrq and fq .

In this case, the system dynamics can also be

decomposed into two parts: one is corresponding to

redundant set of variables, rq and the remained

nonholonomic set, nrfq . That is,

nrf

r

nrfnrf

rr

nrf

r

nrf,nrfnrf,r

nrf,rr,r

U

U

GC

GC

q

q

MM

MM

, (4)

where by considering the second row in order to path

optimization procedure leads to

BqAU nrfnrf += . (5)

Using redundancy resolution rq will be obtained as a

known vector in terms of the time (t). Therefore A is

obtained as a function of time and nrfq and B as a

function of time, rq and nrfq .

By defining the state vector as

TnrfnrfT qqXXX 21 , (6)

Eq. (5) can be rewritten in state space form as

2

12

2

1

F

F

UXDXN

X

X

XX

)()( , (7)

where 1= MD and ))X(G)X,X(C(MN 1211 += . Then,

optimal control problem is determined the position and

velocity variable X1(t) and X2(t), and the joint torque U (t)

which optimize a well‐defined performance measure

when the model is given in Eq. (7).

4. Formulation of the optimal control problem

Pontryaginʹs minimum principle provides an excellent

tool to calculate optimal trajectories by deriving a two‐

point boundary value problem. Let the trajectory

generation problem be defined as determining a feasible

specification of motion, which will cause the robot to

move from a given initial state to a given final state. The

method presented in this article adapts in a

straightforward manner to the generation of such

dynamic profiles.

There are known that nonlinear system dynamics stated

as Eq. (7) be expressed in the term of states (X), controls

(U) and time (t) as

)t,U,X(fX = (8)

Generating optimal movements can be achieved by

minimizing a variety of quantities involving directly or

not some dynamic capacities of the mechanical system. A

functional is considered as the integral

∫ft

t

dttUXLuJ0

),,()( (9)

where the function L may be specified in quite varied

manners. There are initial and terminal constraints on the

states:

ff XtXXtX )()( 00 (10)

There may also be certain pragmatic constraints

(reflecting such concerns as limited actuator power) on

the inputs. For example:

)()( max tUtU (11)

According to the minimum principle of Pontryagin (Kirk

D.E., 1970), minimization of performance criterion at Eq.

(9), is achieved by minimizing the Hamiltonian (H) which

is defined as follow:

),,(),,,(),,,,( tUXfΨtmUXLtmΨUXH Tpp (12)

where TTT tψtψtΨ )()()( 21 is the non‐zero costate

time vector‐function.

Finally, according to the aforementioned principle,

stating the costate vector‐equation

XHΨ T , (13)

in addition to the minimality condition for the

Hamiltonian as

0 UH (14)

ΨHX , (15)

leads to transform the problem of optimal control into a

non‐linear multi‐point boundary value problem.

Consequently, for a specified payload value, substituting

obtained computed control equations from Eqs. (14) and

Eq. (11) into Eqs. (13) and (15), leads to sixteen nonlinear

ordinary differential equations which with sixteen


boundary conditions given in Eq. (10), constructs a Two

Point Boundary Value Problem (TPBVP). Such a problem

is solvable with available commands in different software

such as MATLAB and MATEMATHICA.

5. Implementation

In this section, implementation is performed for the

flexible mobile manipulator consists of a two flexible link

manipulator attached at point ),( ff yxF on the main

axis of a differentially driven vehicle as shown in Figure

1. A concentrated payload of mass pm is connected to the

second link.

1

θ0

Base Path

X0

Y0

1r

2r )y,x(G bb

)y,x(F ff

)y,x(E ee

pm

0L

2v

1v

End Effector Trajectory

2

Figure 1. Two‐link mobile manipulator with flexible links

In order to define the mobile manipulator with two

flexible links the mechanical system generalized

coordinates can be chosen as

][][ 2111210 eeθθθyxqqqq fffrb ,

where ][ 0θyxq ffb is the base generalized

coordinates vector, ][ 21 θθqr is the joint angles

vector and ][ 2111 eeq f is the vector of link modal

displacements. Here, one mode shape per each link is

considered to show the simplified model of the system

with sensible accuracy. By defining br , 1r , 2r and mpr as

the position vectors of the base, first link, second link and

payload respectively, according to Fig. 1, the expressions

of these vectors can be expressed as:

where iLv is the vibration of the end of each link,

1010 θθθ and 210210 θθθθ . ),( txv ii and )( tviL

can be expressed as follows:

teφvteφv

texφvtexφv

LLLL 2121211111

212212111111

,

,. (17)

where with considering the simply support mode shape,

ijφ and ijLφ can be computed as:

i

iiij

L

xπjxφ sin)(

121 jandiπjφijL ,sin

(18)

where i and j refer to the number of link and mode shape

respectively.

Now, using combined Euler–Lagrange formulation and

assumed modes method, the set of system dynamic

equations is derived as follows:

0

02

1

0

7

6

5

4

3

2

1

21

11

2

1

0

77

6766

575655

47464544

3736353433

272625242322

17161514131211

τ

τ

T

F

F

c

c

c

c

c

c

c

e

e

θ

θ

θ

y

x

m

mmSym

mmm

mmmm

mmmmm

mmmmmm

mmmmmmm

y

x

f

f

.

(19)

where because of the motion is in the horizontal plane, the

gravity effects, ),,( frb GGGG will not taking into account.

The position of the end effector in the world reference

frame RFw can be specified with ex and ey as shown in

Figure 1. Therefore operational coordinated of the end

effector can be chosen as ),( ee yxE and the end effector

degrees of freedom in the Cartesian coordinate system

will be m = 2. The rigid system degree of freedom is equal

to n = 5, hence the system has redundancy of order

3 mnr . Thus, this system needs three additional

kinematical constraints for resolving the redundancy

problem and guarantees soft and well‐organized

movement. There is one nonholonomic constraint for the

mobile base that undertakes the robot movement only in

the direction normal to the axis of the driving wheels:

00000 θLθyθx ff )cos()sin( . (20)

Hence, the number of kinematical constraints which must

be applied to system for redundancy resolution is

decreases to two constraints. In this case, with the

previously specification of the base trajectory during the

motion, the user‐specified additional constraints can be

considered. The base position coordinates at point

),,( ff yxF gives zf Xx 1 and zf Xy 2 . Where, zX1 and

zX2 are functions in terms of time. Now, by considering

fifth order polynomial function for the base trajectory

along a straight‐line path from ( )( 0fx , )( 0fy ) to

)θcos(v)θsin(L)θcos(v)θsin(Ly

)θsin(v‐)θcos(L)θsin(v‐)θcos(Lx

r

)θcos(v)θsin(x)θcos(v)θsin(Ly

)θsin(v‐)θcos(x)θsin(v‐)θcos(Lx

r

)θcos(v)θsin(xy

)θsin(v‐)θcos(xxr

)θsin(Ly

)θcos(Lxr

2102L210101L101f

2102L210101L101f

mp

21022102101L101f

21022102101L101f

2

101101f

101101f1

00f

0fb

2

2

0

(16)


( )( ff tx , )( ff ty ) during the overall time ft the

redundancy solution of the system can be complete.

Velocity at start and stop time is considered to be zero.

According to the base motion, fx and fy are known,

therefore if the base angle at initial time )( 00θ be

specified, angular position and angular velocity of the

base ))(),(( tθtθ 00 , can be determined by solving non‐

holonomic constraint equations. Note that, the initial base

angle is considered to be zero, )( 00θ =0.

Now, after resolving the redundancy of the system and

remaining the non‐redundant set of dynamic equations

(Eq. (5)), by defining the state vectors as

.

,TT

TT

xxxxQX

xxxxQX

86422

75311

(21)

The state space form of the dynamic equations of the

system can be written as

4122212 ...;)(, iifxxx iii , (22)

where )( if2 can be obtained from Eq. (7). These

equations must be completed with boundary conditions

that consist in the specified initial and final states as

)( 01x =1.554 rad , )( 03x =1.998 rad ;

)( ftx1 = ‐ 0.858 , )( ftx3 = 1.09. (23)

Other boundary conditions are assumed to be zero

0(Korayem M.H. et al., 2009). A performance index is

expressed as some functional evaluated over the

trajectory. In the presenting paper, the desired path is

optimally designed so that the required movement can be

achieved while minimum effort is obtained and velocities

are bounded at proper magnitude. Hence, the objective

function is considered as:

4

1i

22ii

222

211 xwur+ur

2

1=L . (24)

The main advantage of this cost functional is that the

designer can decide on the relative importance among the

objective function. For example, in objective function

explained in Eq.(24), between the angular velocity and

control effort. In this case, by the numerical choice of iw

and ir the desire movement can be achieved. This, results

to choose the most appropriate path among various

optimal paths that satisfy the designing requirements.

Now, by considering the costate vector as

821 ψψψΨ ... , the Hamiltonian function is formed as

8

1iii

4

1i

22ii

222

211 xψxwτr+τr

2

1H , (25)

where 81 ,..,., ixi can be substituted from Eq. (22). Using

Eq. (13), differentiating the Hamiltonian function with

respect to the states, result in costate equations as follows:

81 ,,,

ix

Hψ

ii (26)

The control function in the admissible interval can be

computed according to Eq. (14), by differentiating the

Hamiltonian function with respect to the torques ),( 21 ττ

and setting the derivative equal to zero. The actuators

that are used in this study are the permanent magnet D.C.

motor with the final bound of control as

1222212222

1111111111

xSτuxSτu

xSτuxSτu

;

; (27)

where miii ωτS / . iτ and miω are the stall torque and

maximum no‐load speed of thi motor respectively.

Typically, applying the Pontryaginʹs minimum principle

results a two‐point boundary value problem. Indeed,

substituting the computed control equations in the state

and costate equations (22) and (26) yields a 4n ordinary

differential equation, where the functions X (t) and Ψ (t)

must satisfy the 4n boundary conditions.

Numerical libraries offer efficient computing codes for

solving non‐linear problems of this type. We used the

routine BVP4C of the MATLAB® library, which is based

on the collocation method. The details of the numerical

technique used in MATLAB® to solve the TPBVP are

given in (Shampine L. F. et al.).

6. Illustrative examples

In this section, simulations are carried out at two cases. In

the first case, path planning is performed for the different

values of penalty matrices. In the second case, simulation

is done for the high stiffness mobile robot and results are

compared with the rigid mobile robot (Korayem M.H. et

al., 2009). The necessary parameters used in the

simulations are summarized in the Table 1.

Parameter Value Unit

Length of Links L1 = L2 = 0.5 m

Mass of Links m1 =5 ; m2=3 kg

Mass of Base mb=1 kg

Moment of Area of

Links I1 = I2 = 5e‐11 4m

Module of Elasticity

of Links E1 = E2=2e11 2kg.m

Max. no Load Speed

of Actuators 1sw =6.45 ; 2sw =2.4 rad/s

Actuator Stall Torque 1sτ =34.67 ; 2sτ =12.21 N.m

Table 1. Simulation Parameters

The load must be carried from an initial point with

coordinate (xe = 0.3m, ye = 0.5m) to the final point with

coordinate (xe = 2.45m, ye = 0.5m). The optimal trajectory

between these two points during the overall time

91.ft second is desired. It should be noted that the

final load position is not feasible without the base motion.

Simultaneously, the mobile base is initially at point


( fx = 0.75m, fy = 0.2m, fθ = 0) and moves to final

position ( fx = 1.6m, fy = 0.5m). L0 is supposed to be 40

cm. (Korayem M.H. et al., 2009).

In the first case, the payload is considered to be 2 kg. The

purpose is to find the optimal path of payload in such a

way that the more suitable amount of control value be

applied and the angular velocity values of motors be

bounded in srad /.51 .

By considering penalty matrices as: 1021 . ww ,

043 ww and 21 rr = diag (0.01) the first path with

appropriate amount of control value are determined, but

the angular velocities are greater than 51. rad/s. Hence,

reducing the maximum velocity magnitude must be

considered. Therefore, for decreasing the joint velocities,

at cost functional defined in the Eq. (24) the penalty

matrix corresponding to them must be increased. A range

of values of W = (w, w, 0, 0) used in simulation are given

in Table 2. ir remains without changes.

End effector trajectories in the XY plane are shown in

Figure 2. Figure 3 shows the angular positions of joints

with respect to time. This graph shows that by increasing

the w, the angular position change to approach

approximately to a straight line. Figure 4 shows the

angular velocities of the first and second joints. It can be

found that by increasing the w, the final values of angular

velocity reduce from ‐2.3 rad/s to ‐1.5 rad/s. By growing

the w, the angular velocities reduce greatly for the first to

second cases whereas at the third case a little reduction

has been occurred in spite of great increase in w.

4 3 2 1 case

200 10 1 0.1 w

Table 2. The values of w used in the simulation

The computed torqueses are plotted in Figure 5. As it can be

seen, increasing the w causes to raise the torques, so that for

the last case the torque curves reach to their bounds at the

beginning and end of the path. This result is predictable,

because, increasing the w decreases the proportion of ir and

the result of this is increasing the control values.

Mode shapes are plotted in Figure 6, shows the flexible

nature of the system in case 3. Finally, arm motions with

related end effector and base trajectories in this case are

plotted in Figure 7.

0.5 1 1.5 2 2.5

0.5

0.6

0.7

0.8

0.9

1

1.1

x(m)

y(m

)

End Effector Trajectory

case1case2case3case4

Figure 2. End effector trajectory in XY plane

0 0.5 1 1.5-100

-50

0

50

100

Time (s)

Ang

ular

Pos

ition

- J

oint

1(d

egre

e)


0 0.5 1 1.560

80

100

120

140

Time (s)

Ang

ular

Pos

ition

- J

oint

2 (

degr

ee)


Figure 3. Angular positions of joints


0 0.5 1 1.5-2.5

-2

-1.5

-1

-0.5

0

0.5

Time (s)

Ang

ular

Vel

ocity

- J

oint

1(r

ad/s

)


0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

1

Time (s)

Ang

ular

Vel

ocity

- J

oint

2(r

ad/s

)


Figure 4. Angular velocities of joints

0 0.5 1 1.5-40

-30

-20

-10

0

10

20

30

40

50

60

Time (s)

Tor

que

- M

otor

1 (

N-m

)

case1case2case3case4upper boundlower bound

0 0.5 1 1.5-20

-15

-10

-5

0

5

10

15

20

Time (s)

Tor

que

- M

otor

2 (

N-m

)

case1case2case3case4upper boundlower bound

Figure 5. Torques of motors

0 0.5 1 1.5-4

-2

0

2

4

6x 10

-3

Time (s)

Mod

e S

hape

- L

ink

1 (m

)

0 0.5 1 1.5-6

-4

-2

0

2x 10

-3

Time (s)M

ode

Sha

pe -

Lin

k 2

(m)

Figure 6. Mode shapes of links

0.5 1 1.5 2 2.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x(m)

y(m

)

Robot Configuration

Figure 7. robot configuration in the Cartesian plane

It can be concluded that, the first path is the optimal path

with the least control values, whereas its angular velocity

is the largest magnitude and the last path is the optimal

path with the smallest amount of velocities, while its

control values is the largest amount. Finally, for the

considered problem, the optimal path is the third path,

which its velocity magnitude is bounded in srad /.51

interval and the torque values is the lowest.

In contrast, the optimal path with lower effort with greater

velocity and the optimal path with greater effort with

lower velocity are obtained using this method; accordingly,


based on the objective contrast principle, there is not a

unique solution that satisfies all the desired objectives

simultaneously. Consequently, in this method, designer

compromises between different objectives by considering

the proper penalty matrices and obtain the finest path that

appropriately cope with the designing requirements.

In order to verify the validity of the study, the case proposed

in literature is developed with elastic behavior in links

(Korayem M.H. et al., 2009). The rational comparison can be

made between rigid manipulator and flexible ones, by

increasing the stiffness at the flexible link. It leads to reduce

the flexible manners of the system and approaches it into

rigid links. Hence, the simulation is done by increasing the

moments of area of each link in previous case study from 1e‐

11 to 6e‐9. The payload is considered to be 20.2 kg (Korayem

M.H. et al., 2009). Figures 8‐10 show the comparison

between angular positions, angular velocities and torque

curves in the rigid and high stiffness case, respectively; as

seen from these figures, both simulation outputs are in a good

agreement, which validates the method studied in this paper.

Finally, mode shapes in this case is shown in Figure 11.

0 0.5 1 1.5-100

-50

0

50

100

150

Time (s)

An

gula

r P

osit

ion

- Jo

ints

1 a

nd

2 (

deg

ree)

Joint 1

Joint 2

rigid high stiffness case

Figure 8. Angular positions of joints

0 0.5 1 1.5-4

-3

-2

-1

0

1

2

Time (s)

An

gula

r V

eloc

ity-

Joi

nts

1 a

nd

2 (

rad

/s)

Joint 1

Joint 2


Figure 9. Angular velocities of joints

0 0.5 1 1.5-60

-40

-20

0

20

40

60

Time (s)

Tor

qu

e -

Mot

or 1

(N

-m)


0 0.5 1 1.5-20

-15

-10

-5

0

5

10

15

20

Time (s)

Tor

qu

e -

Mot

or 2

(N

-m)


Figure 10. Torque curve of actuators

0 0.5 1 1.5-20

-15

-10

-5

0

5x 10

-6

Time (s)

Mo

de

Sh

ape

- L

ink

2(m

)

0 0.5 1 1.5-15

-10

-5

0

5x 10

-6

Time (s)

Mo

de

Sh

ape

- L

ink

1(m

)

Figure 11. Mode shapes of links 1 and 2


7. Conclusion

In this paper, path planning problem of a flexible mobile

manipulator in point‐to‐point motion is formulated based

on the open‐loop optimal control approach. The first

objective of paper is to state the dynamic optimization

problem under a quite generalized form for the

optimality solution that commonly encountered in the

field of path planning of mechanical manipulators. The

second objective consists in developing the method for

optimizing the applicable case studies, which results.

First, concepts such as redundancy and link flexibility are

explained and the general flexible links mobile

manipulator is modeled. Then, Pontryagin’s minimum

principle is called for a Hamiltonian state equations and

optimal path with general objective function is designed

based on TPBVP. Finally, the method is implemented on

two case studies and obtained results are compared with

those reported in literature. It illustrates the power and

capability of the method to overcome the high

nonlinearity nature of the optimization problem in spite

of using complete form of the obtained nonlinear

equations. Highlighting the main contribution of the

paper can be presented as:

The proposed approach can be adapted to any general

serial manipulator including both non‐redundant and

redundant systems with link flexibility and base

mobility.

In this approach the nonholonomic constraints do not

appear in TPBVP directly, unlike the method given in

literature (Mohri A. et al. 2001; Furuno S. et al. 2003).

This approach allows completely nonlinear states and

control constraints treated without any

simplifications.

This paper is the first path generation study that

applied at the flexible mobile manipulators with un‐

predefined trajectory of end‐effector.

The obtained results illustrate the power and

efficiency of the method to overcome the high

nonlinearity nature of the optimization problem,

which with other methods, it may be very difficult or

impossible.

In this method, boundary conditions are satisfied

exactly, while the results obtained by ILP method

have a considerable error in final time.

In this method, designer is able to choose the most

appropriate path among various optimal paths by

considering the proper penalty matrices.

The optimal trajectory and corresponding input control

obtained using this method can be used as a reference

signal and feed forward command in the closed‐loop

control of such manipulators.

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