Redundancy Resolution for Singularity Avoidance of Wheeled

Redundancy Resolution for Singularity Avoidance of Wheeled Mobile Manipulators

Adel Abbaspour

Research Assistant, Faculty of Mechanical Engineering, K. N. Toosi University of Technology

Tehran, Iran

[email protected]

Hadi Zare Jafari

Research Assistant, Faculty of Mechanical Engineering, K. N. Toosi University of Technology

Tehran, Iran

[email protected]

Mohammad Ali Askari Hemmat Research Assistant, Faculty of Mechanical

Engineering, Concordia Univeristy Montreal, Canada

[email protected]

Khalil Alipour Assistant Professor, University of Tehran

Faculty of New Sciences and Technologies

Tehran, Iran

[email protected]

ABSTRACT Mobile robots consist of a mobile platform with

manipulator can provide interesting functionalities in a

number of applications, since, combination of platform and

manipulator causes robot operates in extended work space.

The analysis of these systems includes kinematics redundancy

that makes more complicated problem. However, it gives more

feasibility to robotic systems because of the existence of

multiple solutions in a specified workspace. This paper

presents a novel combination of evolutionary algorithms and

artificial potential field theory for motion planning of mobile

manipulator which guaranteed collision and singularity

avoidance. In the proposed algorithm, the developed concepts

of potential field method are applied to obstacle avoidance

and interaction of mobile base with manipulator is used as a

new idea for singularity avoidance ability of intermediate

links for mobile operations. For this purpose, kinematic and

dynamic modeling is derived to define redundant solutions.

Afterward, methods from potential field theory combine with

evolutionary algorithms to provide an optimum solution

among possibly of redundancy resolution scheme. A controller

based on dynamic feedback linearization is augmented to

track the selective motion trajectory. Simulation results verify

obstacle avoidance, singularity avoidance for the

manipulators and asymptotic convergence of the robots

errors.

NOMENCLATURE 𝑥𝑐 The coordinate (𝑥) of point 𝑃𝑐

𝑦𝑐 The coordinate (𝑦) of point 𝑃𝑐

𝜃1 Joint angle of first link

𝜃2 Joint angle of second link

𝜃3 Joint angle of third link

𝜃𝑟 Rotation angle of right wheel of platform

𝜃𝑙 Rotation angle of left wheel of platform

𝑃 Center point of differential axle

𝑃𝑐 Center of gravity of mobile platform

d

Distance of 𝑃 to 𝑃𝑐

R

Wheel radius of platform

b

Half of distance between wheels

𝑙1 Length of the first link

𝑙2 Length of the second link

𝑙3 Length of the third link

𝜏𝑟 Right wheel torque of platform

𝜏𝑙 Left wheel torque of platform

𝜏1 Torque of first joint

𝜏2 Torque of second joint

𝜏3 Torque of third joint

𝜑 Orientation of the platform

𝜂 Coefficient of the repulsive function

𝜉 Coefficient of the attractive function

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-38639

1 Copyright © 2014 by ASME

mailto:[email protected]



1. INTRODUCTION

Robots, which include the combinations of the mobile

robots and manipulators, are widely used for different

applications such as material handling, spray painting,

mechanical and electronic devices assembly, material removal

and payload transport [1,2, 12]. To realize all the above

functionalities, the fundamental problem is getting a robot to

move from an initial configuration to a desired final

configuration without collides with any obstacles. However,

for a given motion, there might be singularity postures which

can affect the performance of a robot, since; the inverse

kinematics of a mobile manipulator does not have a

meaningful solution. A redundant mobile manipulator can

avoid singular postures by exploiting the set of all possible

motions and its extra DOFs than that required for a given main

task [3]. Mobile manipulators thus have adaptability and

flexibility for executing complex or even dexterous tasks.

Motion planning for nonholonomic wheeled mobile

manipulators has always been a challenging problem which

attracted significant attention in recent years [4]. No general

solutions have been proposed for online navigation of mobile

manipulators, partly due to the complexity of the problem and

the fact that robots should have ability to avoid singularity and

obstacle simultaneously [5, 6]. Motion planning strategies for

mobile robots are based on differential geometry [6], flatness

properties [5] of nonhonolomic systems have been exploited

[7] and other forms of input parameterization can lead to multi

rate [8] and time varying control laws [6].

However, researchers have developed various algorithms

to avoid singularities in the workspace of fixed base

manipulators [14] or platform type parallel manipulators [3, 8]

but these methods cannot be applied in mobile base

manipulators. On the other hand, other proposals for avoiding

obstacles and singular points have included the use of

dumping element perturbations [5], and regulation of the

dumping element [10]. More recent research on redundant

manipulators has involved minimum time path-tracking

control [9], online kinematic control algorithms [11], and a

real-time approach to singularity avoidance in resolved motion

rate control [6] to solve problems of finding an arm form that

does not collide with an obstacle and how to use redundant

flexibility.

Obstacle avoidance refers to the methodologies of

shaping the robot’s path to overcome unexpected obstacles

and some of the techniques developed for path planning are

classified as geometric path planners [6] with several

variations [10] and solution to the path planning problem is

generally sought by graph searching techniques. An

alternative methodology is artificial potential field approach

which has been the most popular technique used to avoid

obstacles [13]. This technique is used to guide the robot to a

given goal at the same time that the robot is avoiding

obstacles.

Moreover, robot control can be defined as the task of

getting the robot to follow a planned desired motion. Robot

control, involves taking the planned motion as the desired

motion and the planned torque inputs to produce the

appropriate input torques to asymptotically follow the desired

motion. In [13] was found a method to compute input

acceleration based on a nonlinear feedback control law

derived from a previous work [15], where the accelerations of

the robot are computed based on a reference mobile robot of

the same class. These accelerations are applied to the robot

dynamic extension model to compute torque commands. This

way is possible to control the kinematic model of the robot

and extend the effects to the dynamic model by computing

motor torque commands for the robot. The method in [13]

shows asymptotic stability guaranteed by Lyapunov theory.

Due to the lack of detailed information of the singular

posture in a Wheeled Mobile Manipulator (WMM), the robot

may specify a trajectory that partly lies close to the singular

points. In this paper, the objective of the singularity avoidance

algorithm would be to find an alternate path, close to the

specified one. In view of the above, by considering the base

mobility, number of arms formations to achieve the desired

position approach to infinity. For this aim, multi objective

genetic algorithm is employed for finding position of robot

mobile base with respect to avoiding the singularity in joints

and collision avoidance.

The rest of this paper is organized as follows: The

kinematics and dynamics model of a mobile manipulator are

derived as the case of study system in Section 2. In Section 3,

the problem of motion planning which contains obstacle and

singularity avoidance and achieving to target is proposed and

the proposed algorithm for desired path planning for the

mobile manipulator is obtained. Then, a control law based on

feed-back linearization has been designed for tracking the

desired path in Section 4. In Section 5, the relative merits and

drawbacks of the proposed method are detailed in simulation

results. Finally, Section 6 concludes the paper with a

discussion of the results and future works.

2. SYSTEM MODELING

Mobile manipulators are built based on diverse mobile

platform designs which are different by the driving

mechanisms. The most commonly available mobile platforms

apply either a differential drive which employs two

independently actuated driven wheels with a common axis,

and casters or spherical wheels that add stability to the mobile

platform. In this section, a robotic system made of a

nonholonomic mobile platform with two independently

actuated driven wheels carrying a manipulator is considered

and differential kinematics and dynamics equations for these

types of mobile manipulator systems is derived (see Fig. 1).

The manipulator is mounted on the center of mass of the

platform and all of the manipulator joints are revolute and

separately actuated. First link can rotate around Z axis, and the

other links can rotate about their joints that are perpendicular

to the plane of motion in the figure shown. This design allows

the end-effecter to work in a 3D task space.

Kinematics Modeling

The configuration of a nonholonomic mobile manipulator

can be 𝐪 = [𝐪𝑝 𝐪𝑚 ]T , where 𝐪𝑝 ∈ 𝑛𝑝 and 𝐪𝑚 ∈ 𝑛𝑚 are

the generalized coordinates of a mobile based and manipulator

respectively. These variables can describe the task space of a

nonholonomic mobile manipulator which are related to 𝐪 with

the kinematics map:


𝐗 = 𝓕 (𝐪p , 𝐪m) (1)

Figure 1: The mobile manipulator and the platform

where 𝐗 ∈ s describe the task space. By considering the

pure rolling of mobile base robot wheels on the ground, the

nonholonomic mobile base platform kinematic model is then

given by:

�̇�𝑝 = 𝐒(𝐪𝑝)𝐯𝑝 (2)

where 𝐯p ∈ p

are the velocities input of a

nonholonomic mobile base with p < np and the columns of

np × p matrix 𝐒(𝐪p) span the admissible velocity space at

each platform configuration. From a kinematic point of view,

the manipulator is instead a completely unconstrained system,

i.e.:

�̇�𝑚 = 𝐯𝑚 (3)

where 𝐯𝑚 ∈ 𝑚 are the velocity inputs for the

manipulator, therefore, the velocity input vector of the

nonholonomic mobile manipulator is then, 𝐯 = [𝐯pT 𝐯m

T ]T∈

p+m. When the total number of velocity inputs

(equivalently, of DOF’s) exceeds the dimension of the task,

i.e., p + nm > 𝑠 , a kinematic (motion) redundancy occurs.

This is the relevant basic definition which is used in the rest of

this paper for the aforementioned algorithm. So differentiating

the Eq. (1), we have:

�̇� =𝜕𝓕

𝜕𝐪𝑝 �̇�𝑝 +

𝜕𝓕

𝜕𝐪𝑚 �̇�𝑚 = 𝐉𝑝(𝐪)𝐒(𝐪𝑝)𝐯𝑝 + 𝐉𝑚(𝐪)𝐯𝑚

= [𝐉𝑝(𝐪)𝐒(𝐪𝑝) 𝐉𝑚(𝐪)] [𝐯𝑝𝐯𝑚]

= 𝐉(𝐪)𝐯

(4)

The s × (p + nm ) matrix 𝐉 will be simply called the

Jacobian of the nonholonomic mobile manipulator. A

configuration 𝐪 is singular if rank( 𝐉(𝐪)) < 𝑠. The addition

of a mobile base platform to a given manipulator can often

delete many of the singularities that would affect the

manipulator taken alone and this fact is illustrated in Motion

Planning Section.

For the system which is depicted in Fig. (1), position of

the robot in an inertial Cartesian frame {X, Y, Z} is completely

specified by the vector 𝐪 = [𝑥𝑐 𝑦𝑐 𝜑 𝜃1 𝜃2 𝜃3 ]T ∈

6; 𝐪p = [𝑥𝑐 𝑦𝑐 𝜑]T ∈ 3 & 𝐪m = [𝜃1 𝜃2 𝜃3 ]T ∈

3

, where (𝑥𝑐,𝑦𝑐) and 𝜑 are the coordinates of the point Pc ,

and the orientation of the mobile platform with respect to the

inertial frame respectively and 𝜃1 , 𝜃2 ,𝜃3 denotes the joint

angles of manipulator links. The platform is driven by two

driving fixed wheels; consequently mobile manipulator is

subjected to a nonholonomic constraint due to non-slipping

condition imposed on wheels from the surface of motion [16].

This constraint yields to the following relation:

𝑥̇𝑐 𝑠𝑖𝑛(𝜑) − �̇�𝑐 𝑐𝑜𝑠(𝜑) + 𝑑�̇� = 0 (5)

Therefore, it can be written in the following matrix form

of 𝐀(𝐪𝑝)�̇�𝑝 = 𝟎. Let 𝐒(𝐪𝑝) be a full rank (3 × 2) matrix

formed by a set of smooth and linearly independent vector

fields spanning the null space of matrix 𝐀(𝐪𝑝), i.e.

𝐒T(𝐪𝑝)𝐀T(𝐪𝑝) = 𝟎. It is easy to choose matrix 𝐒(𝐪𝑝) as:

𝐒(𝐪𝑝)

=

[ 𝑅

2𝑏(𝑏𝑐𝑜𝑠(𝜑) − 𝑑𝑠𝑖𝑛(𝜑))

𝑅

2𝑏(𝑏𝑐𝑜𝑠(𝜑) +𝑑𝑠𝑖𝑛(𝜑))

𝑅

2𝑏(𝑏𝑠𝑖𝑛(𝜑) + 𝑑𝑐𝑜𝑠(𝜑))

𝑅

2𝑏(𝑏𝑠𝑖𝑛(𝜑) − 𝑑𝑐𝑜𝑠(𝜑))

𝑅

2𝑏−𝑅

2𝑏 ]

(6)

In the above equation, R is the radius of the driving wheel

and 2b is the distance of two differential wheels. The vectors

of S(qp) are closed related to the nature of the constraints as is

pointed out in [16].

Since, 𝑣 = [𝑣𝑝𝑇 𝑣𝑚

𝑇 ]𝑇= [�̇�𝑟 �̇�𝐿 �̇�1 �̇�2 �̇�3]

𝑇 ∈ 5 , the

degree of kinematic redundancy is 2 in Cartesian space.

Moreover, θ̇r, θ̇L are the velocities of the right and left wheels

respectively. Considering that l1, l2, l3 as the length of

manipulator links, the coordinates of the end-effecter can be

written as:


{

𝑥𝑒 = 𝑥𝑐 + 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3 ) 𝑐𝑜𝑠(𝜑 + 𝜃1 ) + 𝑙2 𝑐𝑜𝑠(𝜑 + 𝜃1 ) 𝑐𝑜𝑠(𝜃2 )

𝑦𝑒 = 𝑦𝑐 + 𝑙3 𝑐𝑜𝑠(𝜃2 +𝜃3) 𝑠𝑖𝑛(𝜑 + 𝜃1 ) + 𝑙2 𝑐𝑜𝑠(𝜃2 ) 𝑠𝑖𝑛(𝜑 + 𝜃1)

𝑧𝑒 = 𝑙1+ 𝑙2 𝑠𝑖𝑛(𝜃2 ) + 𝑙3 𝑠𝑖𝑛(𝜃2 + 𝜃3 )

(7)

By derivation of Eq. (7), the end-effecter velocities are

derived as the following equation:

{

𝑥̇𝑒 = 𝑥̇𝑐 − 𝐴1(�̇� + �̇�1) − 𝐴2�̇�2 − [𝑙3 𝑠𝑖𝑛 (𝜃2 + 𝜃3) 𝑐𝑜𝑠(𝜑 + 𝜃1)]�̇�3

�̇�𝑒 = �̇�𝑐 + 𝐵1(�̇� + �̇�1) − 𝐵2 �̇�2 − [𝑙3 𝑠𝑖𝑛 (𝜃2 + 𝜃3) 𝑠𝑖𝑛(𝜑 + 𝜃1)]�̇�3

𝑧 ̇𝑒 = [𝑙2 𝑐𝑜𝑠(𝜃2 ) + 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3)]�̇�2 + 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3) �̇�3

(8)

where,

𝐴1 = [𝑙2 𝑐𝑜𝑠(𝜃2 )𝑠𝑖𝑛(𝜑 +𝜃1 )

+ 𝑙3𝑐𝑜𝑠(𝜃2 + 𝜃3 ) 𝑠𝑖𝑛(𝜑 + 𝜃1 )] 𝐴2 = [𝑙2𝑠𝑖𝑛(𝜃2 )𝑐𝑜𝑠(𝜑 + 𝜃1)

+ 𝑙3𝑠𝑖𝑛(𝜃2 + 𝜃3 ) 𝑐𝑜𝑠(𝜑 + 𝜃1 )] 𝐵1 = [𝑙2 𝑐𝑜𝑠(𝜃2 )𝑐𝑜𝑠(𝜑 + 𝜃1 )

+ 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3 )𝑐𝑜𝑠(𝜑 + 𝜃1 )] 𝐵2 = [𝑙2𝑠𝑖𝑛(𝜃2 ) 𝑠𝑖𝑛(𝜑 + 𝜃1)

+ 𝑙3 𝑠𝑖𝑛(𝜃2 + 𝜃3 )𝑠𝑖𝑛(𝜑 + 𝜃1 )]

By choosing �̇� = [𝑥̇𝑒 �̇�𝑒 �̇�𝑒 𝑥̇𝑐 �̇�𝑐 ]T, the above equations

can be expressed as following matrix form:

�̇� = 𝐉(𝐪)𝐯 (9)

For more details about the matrix 𝐉(𝐪), refer to appendix I.

Dynamics Modeling

To attain the dynamics modeling of the wheeled mobile

manipulator system, two methods can be considered. The first

method is Newton-Euler approach, which this method deals

with the coupling correlation of forces acted on the joints and

the displacements of links, but it is not very easy to deal with

when there are many joints. Lagrange method is the second

approach, which is the equilibrium equation of energy. The

second method is used to obtain the equations of motion. By

using the total kinetic and potential energies in Lagrange

equations and simplifying, the equations of motion will be

obtained in the following matrix form:

𝐌(𝐪)�̈� + 𝐂(𝐪, �̇�) + 𝐆(𝐪) = 𝐁(𝐪)𝛕 + 𝐀T(𝐪)𝛌 (10)

where 𝐌(𝐪) is the symmetric, positive definite inertia

matrix, 𝐂(𝐪, �̇�) is the centripetal and coriolis vector, 𝐆(𝐪) is

the gravitational vector, 𝛌 is Lagrange multiplied coefficient,

𝐁(𝐪) is the input transformation matrix and 𝛕 =[𝜏𝑟 𝜏𝑙 𝜏1 𝜏2 𝜏3]

T is torques vector of wheels and manipulator

actuators. More details about the kinetic and potential energy

are presented in appendix II.

Substituting Eq. (6) into 𝓗(𝐪) = [𝐒 (𝐪𝑝) 𝟎𝟑×𝟑𝟎𝟑×𝟐 𝐈𝟑×𝟑

]6×5

, and

then multiplying by 𝓗T(𝐪), the constraint forces 𝐀T(𝐪)𝛌 can

be eliminated. Therefore complete equations of motion of the

nonholonomic mobile platform are given by:

𝐌(𝐪) �̇� + 𝐂(𝐪, �̇�) + 𝐆(𝐪) = 𝐁(𝐪)𝛕

(11)

where,

𝐌(𝐪) = 𝓗T𝐌𝓗

C(q, q̇) = ℋTMℋ̇v + ℋTC

G(q) = ℋTG

B(q) = ℋTB

3. MOTION PLANNING

The results of this paper are motivated by the problem of

singularity postures of robotic manipulator in an environment

with obstacles under additional task constraints. Though, the

main objective of this section is to introduce a new

architecture in order to make the mobile base navigate amidst

of the obstacles while the manipulator avoids singularity

points and end-effector reach to a specified target

simultaneously. The general architecture is shown in Fig. (2) which contains

several levels. At the first level, the system path is planned, for

doing this, the location of the target and obstacles are given to the

artificial potential field algorithm. As a result, the desired

trajectory of the robot will be determined which is an input for the

control system. At the next level, the generated commands as a

control law assign to the end-effector to track the desired

trajectory in the task space. For conformity of mobile base and

end-effector, motion feedback can produce with �̃� by the control

system where the result is the nonlinear gain called α(�̃�). Motion

feedback is the input to the path generator which can adjust the

velocity of the end-effector and mobile base in tracking the

desired trajectory.

Discussions of singularity avoidance ability include three

topics, i.e., singular configuration of the manipulator which

should be avoided, optimization of performance criteria, e.g.

energy consumption and minimum traveling angle of the

joints. In the case of a redundant mobile manipulator, the

number of inverse kinematic solutions may become infinite

and closed form solutions are impossible to find in general.

One of the methods available for solving these problems are

intelligent optimization approaches such as Genetic

Algorithm.

Figure 2: Basic architecture of the proposed approach

In the mentioned architecture, a multi objective Genetic

Algorithm is applied to find the best displacement for the

MULTI OBJECTIVE

GENETIC ALGORITHM

SINGULARITY AVOIDANCE

MINIMUM ENERGY CONSUMPTION

MINIMUM ANGLE TRAVELING

ARTIFICIAL

POTENTIAL FIELD ALGORITHM

ENVIRONMENT

Location of

Target

Location of

Obstacles

PATH

GENERATO

R

CONTROLLER

WHEELED

MOBILE

MANIPULATO

R

𝛼(�̃�)

𝐪 𝐗𝑑𝑒𝑠

𝐗


mobile platform which can delete many of the singularities

that would affect the manipulator taken alone. The

contribution in multi objective Genetic Algorithm is motivated

by the following challengeable problems:

(I) The characteristics of inverse kinematic solution

gained by the Genetic Algorithm inversion scheme depends on

the fitness value of final solution gotten from genetic

algorithm. Therefore, stopping criteria of genetic algorithm

must be set in a way that genetic algorithm generation process

to be continued until end-effector position error from desired

point satisfied pre-determined accurate value.

(II) In order to satisfy the joint angle constraints, it is

necessary to provide the appropriate constraints to limit

genetic algorithm solutions to valid interval for joint angles

and having minimum angle travel from current angle vector to

singularity avoided angle vector.

(III) Since a redundant manipulator can reach a given

target through multiple joint angle vectors, it is desirable to

find the best solution based on minimum energy consumption

for moving to proposed position. Singularity avoided posture

with minimum energy consumption is found with genetic

algorithm. Therefore, as robot close to singularity postures,

longer displacement is required. Meanwhile, in singularity

avoided situations, the proposed displacement is zero because

of minimum energy consumption objective.

(IV) The proposed displacement must consider the

environment properties such as obstacle and target position, so

obstacle avoidance algorithm should also take to account

avoiding from singularity postures. In this work, to avoid the

obstacle and singularity avoidance algorithm conflict , defined

points for displacement by GA create an attraction forces in

artificial potential algorithm for mobile platform which can

provide alternate and simpler solutions to overcome

singularity problems.

The objectives of genetic algorithm are minimizing

energy consumption and minimum traveling angle with

constraint of singularity avoidance. The consequences of local

learning will also be analyzed in detail in this section aims to

learn a local solution around the current robot configuration.

In the simulation section, it will be shown that this.

Singularity Avoidance Algorithm

Existing potential field methods are based on the

assumption that the system can be represented by a point in

the workspace. In its simplest form, such potential field

method can be implemented quickly and provide acceptable

results without requiring many refinements. In fact, this

method is suitable for real-time implementation requiring only

local gradient information and without requiring global

information. While the target is ideally at the minimum, all

obstacle, or walls, are assumed to create high potential hills.

The robot is under the influence of forces exerted from

potential fields and will move in the configuration space based

on the direction of total forces exert on it. Target location

exerts an attractive force and obstacles exert repulsive forces

on the robot. Therefore sum of both potentials, determines the

direction and speed of travel for the mobile manipulator.

Typically, the attractive potential field and the repulsive

potential fields are formulated separately, and the total

potential field of the workspace is obtained by linear

superposition of the two fields. The overall potential is the

sum of these two types of potentials and can be written as:

Utotal (𝐪) = Uatt (𝐪) + Urep (𝐪) (12)

where Utotal(𝐪) denotes the total artificial potential

field; Uatt (𝐪) denotes the attractive potential field and

Urep (𝐪) is the repulsive potential field. While many different

attractive potential functions have been proposed in the

literature, the applicable attractive potential field takes the

form of:

Uatt (𝐪)

=

{

1

2𝜉1‖𝒒𝑡𝑎𝑟 − 𝒒𝑟𝑜𝑏‖

𝑚1 , 𝑖𝑓‖𝒒𝐺𝐴 − 𝒒𝑟𝑜𝑏‖ > 𝐷

1

2𝜉1‖𝒒𝑡𝑎𝑟 − 𝒒𝑟𝑜𝑏‖

𝑚1 +1

2𝜉2‖𝒒𝐺𝐴 − 𝒒𝑟𝑜𝑏‖

𝑚2 ,

𝑖𝑓‖𝒒𝐺𝐴 − 𝒒𝑟𝑜𝑏‖ < 𝐷

(13)

where 𝜉𝑖 is a positive scaling factor, 𝐪tar and 𝐪rob denote

the position for the target and the robot, respectively.

Moreover, 𝐪GA is the result of multi objective Genetic

Algorithm which means, GA produce a local and temporary

target for mobile platform in order to avoid singular postures

while define optimized (above-mentioned) displacement. D is

defined tolerance for adding attractive aspect to singularity

avoided posture to the attractive potential field. ‖𝐪tar− 𝐪rob ‖

and ‖𝐪GA − 𝐪rob‖ are the Euclidean distances between the

robot and the targets, which are only a function of position,

and m𝑖 is any positive number greater than zero.

Various types of repulsive potential functions were

proposed in the literatures. A common problem found in most

potential field methods, the goals non-reachable with obstacle

nearby, or the GNRON problem, identified by S.S. Ge and Y.J

Cui [17], can be handled using their proposed potential fields.

This function takes the following form:

Urep(𝐪) = {

1

2η (

1

ρRo−1

ρ0)2

ρRTn , 𝑖𝑓 ρRo ≤ 0

0, 𝑖𝑓 ρRo > 0

(14)

where 𝜂 is a positive scaling factor, 𝜌Ro = ‖𝐪obs − 𝐪rob‖

is the shortest Euclidean distance between the robot from the

obstacle surface, and 𝜌0 is the limit distance of the repulsive

potential field influence. In the mentioned equation, 𝜌RT

denotes the minimal Euclidean distance from robot to the

target. The introduction of the term 𝜌RTn ensures that the total

potential will reach its global minimum, if and only if the

robot reaches the target where 𝜌RT = 0.

End-Effector – Platform Path Planning

The gradient information is used for path planning of an

artificial potential field created using potential functions as an

input force in the system’s dynamics equations that drives the

robot to its desired destination while avoid collisions with

obstacles. For all the points beside the target point in the


configuration space, gradient information always exists and

serves as the input to the controller. This is simply letting:

𝐮 = −Kf∇𝐪Utotal (15)

In the kinematics or dynamics equations of the robot,

where 𝐮 is the input to the rrllcrnnoc; 𝐪 is the position vector

of the robot, Kf is a positive diagonal scaling matrix, and

Utotal is the total potential field of the workspace defined in

previous section. For the end-effecter we consider 𝐪𝑒 =[𝑥𝑒 𝑦𝑒 𝑧𝑒]

T and for the platform 𝐪𝑐 = [𝑥𝑐 𝑦𝑐 ]T, so the

velocities of the end-effecter and the platform can be obtained

from Eq. (16) and then the orientation of the platform will be

obtained from the nonholonomic constraint in Eq. (5).

�̇� = −Kf∇𝐪Utotal (16)

Note that, the value of 𝑚𝑖, 𝜉𝑖, η and Kf must be properly

select such that it will not create a sudden drop or exceed the

speed limit of the robot, however it is inevitable in online

motion planning. For avoiding this, velocity of the mobile

platform and end-effector should be coordinated in motion.

For this aim, the motion planning part should get feedback

consist of errors of end-effector and platform. By defining

error formation vector as form as [13]:

𝐳𝑒 = [

𝑥e𝑦e𝑧e

], 𝐳𝑐 = [𝑥𝑐𝑦𝑐] , 𝐙 = [𝐳𝑒 𝐳𝑐 ]T ,

𝐳𝑑𝑒 = [

𝑥𝑑𝑒𝑦𝑑𝑒𝑧𝑑𝑒

] , 𝐳𝑑𝑐 = [𝑥𝑑𝑐𝑦𝑑𝑐

] ,𝐙d = [𝐳𝑑𝑒 𝐳𝑑𝑐 ]T

�̃� = 𝐙d − 𝐙

(17)

while 𝐳𝑑𝑒 and 𝐳𝑑𝑐 denote vector of desired position of the

end-effector and platform respectively, 𝐳𝑒 and 𝐳𝑐 are the

vector of real position for end-effector and platform. Now,

motion feedback function can obtain as form as:

β(Z̃) = Z̃T Z̃ (18)

In order to include motion feedback we augment gradient

information by a nonlinear gain depend on the quantity �̃�. For

designing this nonlinear gain, we would like the robot to slow

down and stop as the robot get out of desired path or exceed

the speed limit. If the robot moves with an appropriate

velocity or track the desired trajectory, it is desired to move

towards its final goal, therefore, this function is given by:

�̇� = −α(�̃�) ∗ ∇𝐪UTotal (19)

where �̇� is the velocity of the robot and α(�̃�) is the

nonlinear gain as define by Eq. (18) as form as:

α(�̃�) =1

Kf ∗ β(�̃�) +1K

(20)

By selecting proper positive values for Kf, K the velocity

of the robot can control such that it will not exceed the speed

limit and have a smooth path.

4. MODEL-BASED CONTROL DESIGN

To complete the ideas which is presented in Fig. (2), the

focus of this section is the development of a dynamic-level

controller for the wheeled mobile manipulator that allows the

end-effector and platform to track a desired trajectory at the

task space. For achieving this, the dynamic model should be

rewritten in task space (Cartesian Space), so the nonlinear

control method separates the equations of motion into the end-

effector and platform independently.

In Eq. (4), the task space velocities are defined by the

vector �̇� and the independent joint-space velocities by 𝐯. For a

kinematically redundant system, a resolved acceleration for

the independent joint rates can be obtained by differentiating

of Eq. (9):

�̇� = 𝐉−1(�̈� − 𝐉̇𝐯) (21)

The final dynamic model equation of the system can be

achieved by substituting the Equation (21) in (11) and then

expressed in task space in the following form:

𝛕 = 𝐌x(�̈�) + 𝐂x + 𝐆x (22)

where,

𝐌x = (𝐁(𝐪))−1(𝐌(𝐪)𝐉−1)

Cx = (B(q))−1(C(q, q̇) − M(q)J−1J̇v)

Gx = (B(q))−1(G(q))

Since we use dynamically consistent equations for the

controller, equivalent-controlled closed loop control law in the

independent task space may be written, �̈� = �̈�d + 𝐤v�̇�+ 𝐤p𝐞,

where, �̈�d is the desired acceleration and 𝐞 = 𝐗d −𝐗 , so the

error dynamics is:

�̈� + 𝐤v�̇� + 𝐤p𝐞 = 𝟎 (23)

Therefore, by selecting gain matrices 𝐤v , 𝐤p and 𝐤 i

diagonal and positive definite guarantee the error dynamics

asymptotically converges to the origin. Finally, Computed

Torque Control law which affects the end-effector and

platform of the robot can be derived from Eq. (22):

𝛕eq = 𝐌x(�̈�d + 𝐤v�̇� + 𝐤p𝐞) + 𝐂x + 𝐆x (24)

5. SIMULATION RESULTS

In this section a nonholonomic mobile manipulator is

simulated in MATLAB to show how the combination of

mobile platform and redundant manipulator can eliminate the

set of singular points by performing extensive test. To discuss

about the singularity and obstacle avoidance algorithms which

proposed in section 3, four obstacles are chosen in such a way

that the robots have to pass through them.

The initial position of the end-effector of the robot is 𝑥𝑒 =−2.2, 𝑦𝑒 = −4.5 with 𝑧𝑒 = 0.4 and the mobile base 𝑥𝐶 =−2.2, 𝑦𝐶 = −4.5 with 𝜑 = 0 and the desired trajectory of the

end-effector and mobile base of the robot which artificial

potential field produce it, start from 𝑥𝑑𝑒 = −2, 𝑦𝑑𝑒 = −4 with

𝑧𝑑𝑒 = 0.5 𝑥𝐶𝑑 = −2, 𝑦𝐶𝑑 = −4 with 𝜑𝑑 =π

2. Center


coordinates (X,Y) and radii (R) of the obstacles are defined in

table III. The parameters of the artificial potential field are 𝜉 =0.5,𝜂 = 0.8, 𝑚1 = 𝑚2 = 𝑛 = 2.

TABLE 1: T HE PROPERTIES OF THE LINKS

Link Length Mass Center of

gravity 𝐈𝐱𝐱 𝐈𝐲𝐲 𝐈𝐳𝐳

1 0.5 0.6 0.25 0.04 0.04 0.0013

2 0.5 0.6 0.25 0.04 0.04 0.0013

3 0.5 0.6 0.25 0.04 0.04 0.0013

TAB . 2: T HE PROPERTIES OF THE PLATFORM

Distance from P to Pc d = 0.2 [m]

Distance from wheel to P b = 0.3 [m]

Radius of wheel r = 0.1 [m]

Mass of platform M = 70 [kg]

Moment of inertia of platform IZ = 6.12 [kg.m2]

TABLE III: OBSTACLES PROPERTIES

O bstacles X Y R

Obstacle 1 5.5 -1 2

Obstacle 2 -3 7 3

Obstacle 3 5.5 10 2

Obstacle 4 16 10 3

Units of length, moment of inertia and mass are (m),

(kg. m2) and (kg) respectively. The controller gains were

chosen so that the closed-loop system exhibits a critical

damping behavior and have been conducted by employing the

dynamical equations of the composite mobile robot system to

validate the designed control law given by eq. (24). Figures

(5) and (7) illustrate the path traversed by the mobile base and

End-effector in 3D and 2D respectively. As expected, the

mobile base starts from initial location (−2.2, −4.5) and

passes through target location at (6.5,14). End-effector starts

from initial location (0.4, −2, −4) and passes through target

location at (6.5,14,0.7). The Figure (4), (5) illustrate the

errors of the End-effector and platform respectively.

By using the mentioned algorithm which is proposed in

Section 3, it is possible to find inverse kinematic solutions for

all points within a reasonable amount of time. Figures (3) and

(4) are the joint angles and displacement of the mobile base

respectively. Therefore, by these variations, the robot avoids

from singular points and move towards to the goal while

avoids from obstacles.

Figure 3: Variation of joint angles versus time

Figure 4: Mobile base displacement

Figure 5: Desired path and path of End-effector and position of the obstacles

0 20 40 60 80 100-80

-60

-40

-20

0

20

40

60

Time[s]

Arm

s A

ng

les [D

eg

ree

]

1

2

3

0 20 40 60 80 100-5

0

5

10

15

Time[s]

Po

stio

n o

f R

ob

ot B

ase

[m

]

x-axis Positon of Robot Base

Y-axis Positon of Robot Base

-5 0 5 10 15 20

-5

0

5

10

15

0

1

2

X [m]

Y [m]

Z [m

]


Figure 6: Errors of the End-effector position

Figure 7: Desired path and path of platform and position of the obstacles in

X, Y plane

Figure 8: Errors of the platform position

6. CONCLUSIONS

In this paper, an efficient inverse-forward scheme based

on evolutionary algorithm is used to solve the inverse

kinematic problem of a redundant manipulator. In this method,

an evolutionary algorithm is used to find optimum solution to

escape from singularity and approach to desired target of end-

effector position. First of all, optimum singularity avoided

postures are found by evolutionary algorithm and then,

artificial potential field considered singularity avoided

postures as attractive potential points . The artificial potential

field is used to navigate the wheeled mobile manipulator in an

environment containing both repulsive and attractive

potentials that steer the mobile manipulator to avoid the

obstacles and singularity postures and attract to the target

simultaneously. Kinematics and dynamics modeling of robot

manipulator is derived to define redundant solutions. A

dynamic controller is designed for motion control of a mobile

manipulator as a nonholonomic system in an environment

with obstacles. The powerful aspects of evolutionary

algorithms and artificial potential field are combined for the

purpose of singularity and obstacle avoidance problems in

robot manipulator control. The simulation analysis has

demonstrated that the proposed intelligent redundant

manipulator can effectively solved singularity and obstacle

avoidance problems.

REFERENCES

[1] Farbod Fahimi, “Autonomous Robots, Modeling, Path

Planning, and Control”, Springer Science Business Media,

LLC 2009. [2] Viswanathan, A., Jouaneh, M., Datseris , P. and Palm, W.,

“A manufacturing system for automated production of

polystyrene molds”, IEEE Robotics & Automation

Magazine, Vol. 3, No. 3, 1996.

[3] S. Bhattacharya, h. Hatwal and a. Ghosh, “Comparison of

an Exact and an Approximate Method of Singularity

Avoidance in Platform Type Parallel Manipulators”, Mech.

Mach. Theory Vol. 33, No. 7, pp. 965±974, 1998.

[4] Dietrich, A., Wimbock, Albu-Schaffer; Hirzinger, G.

“Reactive Whole-Body Control: Dynamic Mobile

Manipulation Using a Large Number of Actuated Degrees

of Freedom” Robotics & Automation Magazine, IEEE

Volume: 19, Issue: 2, 2012.

[5] Herbert G. Tanner , Savvas G. Loizou and Kostas J.

Kyriakopoulos “Nonholonomic Navigation and Control of

Cooperating Mobile Manipulators” Journal of Transactions

on Robotics and Automation, IEEE, Volume: 19, Issue: 1,

2003.

[6] Swagat Kumar, Laxmidhar Behera, T.M. McGinnity

“Kinematic control of a redundant manipulator using an

inverse-forward adaptive scheme with a KSOM based hint

generator”, Journal of Robotics and Autonomous Systems

pp. 622-633, 2010.

[7] Glenn D. White, Rajankumar M. Bhatt, Venkat Krovi,

“Dynamic Redundancy Resolution in a Nonholonomic

Wheeled Mobile Manipulator”, Journal of Robotica,

Volume: 25, Issue: 2, 2007.

[8] E. Macho, O. Altuzarra, E. Amezua, A. Hernandez,

“Obtaining configuration space and singularity maps for

parallel manipulators”, Journal of Mechanism and Machine

Theory Volume: 44, 2009.

[9] M.H. Korayem, M. Nazemizadeh, V. Azimirad, “Optimal

trajectory planning of wheeled mobile manipulators in

cluttered environments using potential functions”, Journal

of Scientia Iranica, 2011.

[10] Mamoru Minami, Hiroshi Tanaka and Yasushi Mae

“Avoidance Ability of Redundant Mobile Manipulators

0 20 40 60 80 100-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time [s]

Err

or

Error of X [m] of End-Effector

Error of Y [m] of End-Effector

Error of Z [m] of End-Effector

-5 0 5 10 15 20-5

0

5

10

15

X [m]

Y [m

]

0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time [s]

Err

or

Error of X [m] of mobile base

Error of Y [m] of mobile base

Error of [rad] of mobile base


During Hand Trajectory Tracking”, Journal of Advanced

Computational Intelligence and Intelligent Informatics,

Vol.11 No.2, 2007. [11] Alessandro De Luca, Giuseppe Oriolo Paolo,

Robuffo Giordano “Kinematic Modeling and

Redundancy Resolution for Nonholonomic Mobile

Manipulators”, Proceedings of the 2006 IEEE International

Conference on Robotics and Automation Orlando, Florida,

2006.

[12] Henzeh Leeghim, Hyoung Lee, Dong-Hun Lee, Hyochoong Bang and Jong-Oh Park, “Singularity Avoidance of Control Moment Gyros by Predicted Singularity Robustness: Ground Experiment”, IEEE Transactions on Control Systems Technology, Vol. 17, No. 4, 2009.

[13] Adel Abbaspour, S. Ali A. Moosavian, Hadi Zare and Khalil Alipour, “Formation Control and Obstacle Avoidance of Cooperative Wheeled Mobile Robots,” First RSI/ISM International Conference on Robotics and Mechatronics, IEEE, ICRoM 2013, Sharif University, Tehran, Iran.

[14] Oriol Bohigas, Michael E. Henderson, Llu ı́s Ros, Montserrat Manubens, and Josep M. Porta, “Planning Singularity-Free Paths on Closed-Chain Manipulators”, IEEE Transactions on Robotics, Vol. 29, No. 4, 2013.

[15] Alexander Dietrich, Thomas Wimb¨ock, Alin Albu-Sch¨affer, and Gerd Hirzinger, “Singularity Avoidance for Nonholonomic, Omnidirectional Wheeled Mobile Platforms with Variable Footprint”, IEEE International Conference on Robotics and Automation, 2011.

[16] Adel Abbaspour, Khalil Alipour and S. Ali. A Moosavian, “Finding Near-Optimal Formation of Cooperative Wheeled Mobile Robots in Payload Transportation,” 20th Annual International Conference on Mechanical Engineering, ISME 2012, Shiraz, Iran.

[17] S. S. Ge and Y. J. Cui “New Potential Functions for Mobile Robot Path Planning”, IEEE Transactions on Robotics and Automation, vol. 16, no. 5, 2000.

Appendix I: The details of matrix J:

[ J11 J12 J13 J14 J15J21 J22 J23 J24 J250J41J51

0J42J52

000

J3400

J3500 ]

And:

J11 =𝑅

2𝑏[𝑏𝑐𝑜𝑠(𝜑) − 𝑑𝑠𝑖𝑛(𝜑)] −

𝑅

2𝑏[𝑙2 𝑐𝑜𝑠(𝜃2) 𝑠𝑖𝑛(𝜑 + 𝜃1) +

𝑙3 𝑐𝑜𝑠(𝜃2 +𝜃3) 𝑠𝑖𝑛(𝜑 +𝜃1)]

J12 =𝑅

2𝑏[𝑏𝑐𝑜𝑠(𝜑) + 𝑑𝑠𝑖𝑛(𝜑)] +

𝑅

2𝑏[𝑙2 𝑐𝑜𝑠(𝜃2) 𝑠𝑖𝑛(𝜑 + 𝜃1) +

𝑙3 𝑐𝑜𝑠(𝜃2 +𝜃3) 𝑠𝑖𝑛(𝜑 +𝜃1)]

J13 = −[𝑙2 𝑐𝑜𝑠(𝜃2) 𝑠𝑖𝑛(𝜑 + 𝜃1) + 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3) 𝑠𝑖𝑛(𝜑 + 𝜃1)]

J14 = −[𝑙2 𝑠𝑖𝑛(𝜃2) 𝑐𝑜𝑠(𝜑 + 𝜃1) + 𝑙3 𝑠𝑖𝑛(𝜃2 +𝜃3) 𝑐𝑜𝑠(𝜑 + 𝜃1)]

J15 = −[𝑙3 𝑠𝑖𝑛(𝜃2 + 𝜃3) 𝑐𝑜𝑠(𝜑 + 𝜃1)]

J21 =𝑅

2𝑏[𝑏𝑠𝑖𝑛(𝜑) + 𝑑𝑠𝑖𝑛(𝜑)] +

𝑅

2𝑏[𝑙2 𝑐𝑜𝑠(𝜃2) 𝑐𝑜𝑠(𝜑 + 𝜃1)+

𝑙3 𝑐𝑜𝑠(𝜃2 +𝜃3) 𝑐𝑜𝑠(𝜑 + 𝜃1)]

J22 =𝑅

2𝑏[𝑏𝑠𝑖𝑛(𝜑) − 𝑑𝑠𝑖𝑛(𝜑)] −

𝑅

2𝑏[𝑙2 𝑐𝑜𝑠(𝜃2) 𝑐𝑜𝑠(𝜑 + 𝜃1)+

𝑙3 𝑐𝑜𝑠(𝜃2 +𝜃3) 𝑐𝑜𝑠(𝜑 + 𝜃1)]

J23 = 𝑙2 𝑐𝑜𝑠(𝜃2) 𝑐𝑜𝑠(𝜑 + 𝜃1)+ 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3) 𝑐𝑜𝑠(𝜑 +𝜃1)

J24 = −[𝑙2 𝑠𝑖𝑛(𝜃2) 𝑠𝑖𝑛(𝜑 + 𝜃1) + 𝑙3 𝑠𝑖𝑛(𝜃2 +𝜃3) 𝑠𝑖𝑛(𝜑 +𝜃1)]

J25 = −[𝑙3 𝑠𝑖𝑛(𝜃2 + 𝜃3) 𝑠𝑖𝑛(𝜑 + 𝜃1)]

J34 = 𝑙2 𝑐𝑜𝑠(𝜃2)+ 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3)

J35 = 𝑙3 𝑐𝑜𝑠(𝜃2 + 𝜃3)

J41 =𝑅

2𝑏[𝑏𝑐𝑜𝑠(𝜑) −𝑑𝑠𝑖𝑛(𝜑)]

J42 =𝑅

2𝑏[𝑏𝑐𝑜𝑠(𝜑) +𝑑𝑠𝑖𝑛(𝜑)]

J51 =𝑅

2𝑏[𝑏𝑠𝑖𝑛(𝜑) + 𝑑𝑐𝑜𝑠(𝜑)]

J52 =𝑅

2𝑏[𝑏𝑠𝑖𝑛(𝜑) − 𝑑𝑐𝑜𝑠(𝜑)]

We can obtain the determinant of the J as form as:

𝑑𝑒𝑡(J) = 0.003(cos(𝜃3) + 1)(sin(𝜃2 + 𝜃3) − sin (𝜃2))

Appendix II: The model of dynamic

Let 𝑟1 , 𝑟2 , 𝑟3 denote the distance between joints and the center

of mass of links, as shown in fig. 1, the coordinates of the

center of mass of first link can be obtain:

{

𝑥1 = 𝑥𝑐𝑦1 = 𝑦𝑐𝑧1 = 𝑟1

The coordinates of the center of mass of second link can be

express as follows:

{𝑥2 = 𝑥1+ 𝑟2 𝑐𝑜𝑠 (𝜃2 )𝑐𝑜𝑠(𝜑 + 𝜃1)𝑦2 = 𝑦1 + 𝑟2𝑐𝑜𝑠 (𝜃2 )𝑠𝑖𝑛(𝜑 + 𝜃1 )

𝑧2 = 𝑙1+ 𝑟2 𝑠𝑖𝑛(𝜃2 )

The coordinates of the center of mass of third link is:

{𝑥2 = 𝑥1+ 𝑙2 𝑐𝑜𝑠(𝜃2 ) 𝑐𝑜𝑠(𝜑 +𝜃1 ) + 𝑟3 𝑐𝑜𝑠(𝜃2 + 𝜃3 )𝑐𝑜𝑠(𝜑 + 𝜃1 )

𝑦2 = 𝑦1 + 𝑟2 𝑐𝑜𝑠(𝜃2 ) 𝑠𝑖𝑛(𝜑 + 𝜃1 ) + 𝑟3 𝑐𝑜𝑠(𝜃2 + 𝜃3 ) 𝑠𝑖𝑛(𝜑 + 𝜃1 )

𝑧2 = 𝑙1+ 𝑙2 𝑠𝑖𝑛(𝜃2 ) + 𝑟3𝑠𝑖𝑛 (𝜃2 + 𝜃3 )


The totally kinematic energy can be written as form as:

𝑇 =1

2(𝑚𝑝 + 2𝑚𝑤)(𝑥̇𝑐

2+ �̇�𝑐2)+

1

2(𝐼𝑧𝑝 + 2𝐼𝑧𝑤 +

2𝑚𝑤(𝑑2+ 𝑏2)) �̇� 2+

1

2(𝐼𝑤𝑦 �̇�𝑟

2) +

1

2(𝐼𝑤𝑦 �̇�𝐿

2) +

2𝑚𝑤𝑑�̇�(𝑥̇𝑐 𝑠𝑖𝑛(𝜑) − �̇�𝑐 𝑐𝑜𝑠(𝜑)) +1

2𝑚1(𝑥̇1

2+ �̇�12 + �̇�1

2)+1

2𝐼𝑧𝑧(�̇�1+ �̇�1)

2+

1

2𝑚2(𝑥̇2

2+ �̇�22 + �̇�2

2)+1

2𝐼𝑥𝑥 (�̇�2

2) +

1

2𝐼𝑦𝑦 ((�̇� + �̇�1)𝑐𝑜𝑠 (𝜃2 ))

2

+1

2𝐼𝑧𝑧 ((�̇� + �̇�1)𝑠𝑖𝑛 (𝜃2 ))

2

+1

2𝑚3(𝑥̇3

2+ �̇�32+ �̇�3

2)+1

2𝐼𝑥𝑥(�̇�2 + �̇�3)

2+

1

2𝐼𝑦𝑦 ((�̇� +

�̇�1)𝑐𝑜𝑠 (𝜃2 ))2

+1

2𝐼𝑧𝑧 ((�̇� + �̇�1)𝑠𝑖𝑛 (𝜃2 ))

2

And the totally potential energy is:

𝑈 = 𝑚1𝑔𝑟1 +𝑚2𝑔(𝑙1+ 𝑟2 𝑠𝑖𝑛(𝜃2 )) + 𝑚3𝑔(𝑙1+𝑙2𝑠𝑖𝑛(𝜃2 ) + 𝑟3𝑠𝑖𝑛 (𝜃2 + 𝜃3 ))

The dynamical equations of the composite mobile robot arm

system are obtained from Lagrange's equations:

𝑑

𝑑𝑡(𝜕ℒ

𝜕�̇�𝑘) − (

𝜕ℒ

𝜕𝑞𝑘) = 𝑓𝑘 − ∑ 𝜆𝑗

𝑚𝑗=1 𝑎𝑗𝑘

Where ℒ(𝑞, �̇�) = 𝑇(𝑞, �̇�) − 𝑈(𝑞), 𝑓𝑘 are the forces or torques

acted on the platform and the links, 𝜆𝑗 is the Lagrange

multiplied coefficient. Now we can determine the dynamic

equation of the robot.

𝐌(𝐪) �̈� + 𝐂(𝐪, �̇�) +𝐆(𝐪) = 𝐁(𝐪)𝛕 +𝐀T (𝐪)𝛌


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Redundancy Resolution for Singularity Avoidance of Wheeled