International Journal of Engineering Insights: (2023) Vol. 1, Nro.1, Regular Paper

https://doi.org/10.61961/injei.v1i1.6

Simultaneous motion and shape control of redundant mobile

manipulators

Mar

ıa del Carmen Claudio · Viviana Moya · Emanuel Slawi

nski · Vicente Mut

Received: 28 August 2023 / Accepted: 29 October 2023

Abstract: This work proposes the design of a simultaneous

motion and shape control of a redundant mobile manipula-

tor. The method is based on the construction of a variable

weighting matrix that allows to control of speciﬁc sections

of this type of robot composed of a mobile platform and

manipulator arm. The weight matrix allows for prioritizing

desired parts of the robot through cost functions that depend

on an additional variable deﬁned in this work, which can

be changed automatically according to some pre-established

criteria or by a human operator if the robot is teleoperated.

In addition, simulations are included to show the usefulness

of the proposal.

Keywords Weighting Matrix · Redundant Robot · Robot

Shape Control · Jacobian Matrix

1 Introduction

The advancement of robotics has facilitated the execution

of tasks that previously could only be performed by hu-

man operators, either by replacing them in repetitive and

tedious tasks or by showing themselves as complementary

in dangerous scenarios [1]. In this context, a robot’s control

actions may be generated internally (from automatic refer-

ences from sensors) or may come from a distant environ-

ment, where redundant robots are generally used, e.g. a mo-

bile manipulator.

Considering that the robot can execute a task in an au-

tonomous way, the difﬁculty of the mission may require more

degrees of freedom (DoF) than apparent, either for safety or

design requirements. In this case, the amount of additional

Mar

ıa del Carmen Claudio

Inmersoft Technologies

Quito, Ecuador

Tel.: +593 98 787 3246

mcclaudio@inmersoft.com

Emanuel Slawi

nski · Vicente Mut

Instituto de Autom

atica, Universidad Nacional de San Juan

San Juan, Argentina

Viviana Moya

Universidad Internacional del Ecuador

Quito, Ecuador

degrees of freedom would allow to execution of secondary

actions, where appropriate control techniques would allow

to use of certain joints (e.g. the mobile platform) more fre-

quently compared to other DoFs (e.g. the robotic manipu-

lator) [2]. Such techniques are developed in order to fulﬁll

additional goals to the primary one, such as obstacle avoid-

ance, singularity avoidance, energy saving, maximum ma-

nipulability, and so on. Therefore, having full control over

the redundancy characteristics of the system could facilitate

the proper execution of the [3] mission.

On the other hand, if in isolated cases of the task, it is

required to generate higher magnitude control commands

to the mobile platform and lower magnitude commands to

the robotic manipulator, these strategies must be embedded

in the remote control algorithm. However, handling such

a large number of degrees of freedom requires additional

physical and mathematical analysis. Based on the literature,

two common ways are proposed to solve this problem: sec-

tion the robot into two main parts and control them indepen-

dently or relate the whole robotic system (mobile manipu-

lator) through a single Jacobian matrix and control it with

some technique.

The ﬁrst way is to control each section of the robot in-

dependently (i.e. in the mobile manipulator, to control either

the platform or the robotic manipulator in isolation), treat-

ing it as a set of mechanically linked mechanisms where

one dynamically perturbs the other [4]. This solution, al-

though obvious and easily implementable, poses discrete ac-

tivations and deactivations of each part of the robot, addi-

tionally needing to generate references to each section of

the system. The most notorious disadvantages are the need

for non-uniﬁed mathematical modeling, controllers for each

of the sections, discrete switching that could generate con-

trol instabilities, and possibly the involvement of more than

one human operator with different robotic controls (the lat-

ter two in cases of the remote control).

For the second form, completely activating/deactivating

different parts of the robot depending on the situation would

not be feasible by directly using the Jacobian matrix that

relates Cartesian reference velocities to joint velocities. In

this aspect, using strategies such as shape control (as a pri-

21 International Journal of Engineering Insights, (2023) 1:1

mary objective) to command the entire robotic system in a

uniﬁed manner is a straightforward solution, however, there

are technique-speciﬁc constraints to be able to position the

end effector in a desired state, given that the reach of the

operating end to a desired position would be located in the

null space of the homogeneous solution [3]. An appropri-

ate solution to this problem is to modify the behavior of

the Jacobian by multiplying it by a weighting matrix, which

weighs sections of the matrix to give/remove priority to cer-

tain joints. The modiﬁcation of the behavior of a pondered

Jacobian (or weighted) is a concept widely used in the liter-

ature. In this aspect, works such as [5] use this technique to

limit the positions of the joints with the objective of limit-

ing their achievable range, forcing them to work in safe lim-

its (preventing the actuators from exceeding positions that

would damage them). In this type of work, the weight ma-

trix is based on derivable functions that cause zero veloci-

ties when the joints reach the safe limits, with a continuous

reduction until reaching zero (causing complete immobility

in the effector). Similarly, [6] proposes the use of this tech-

nique to avoid collisions between parts of the robot, where

each link is represented by a protective sphere. The danger-

ous approach between links (and therefore spheres) causes

the deceleration of all the actuators involved, all this based

on a function that depends on the diameters and distances

between the virtual spheres. In addition to the weight ma-

trix, this body of work has in common the continuous way in

which joint positions are constrained, which depends on an

internal factor (range of maximum and minimum positions

for [5] and collision between sphere surfaces for [6]). In the

literature, no proposal is determined that uses this technique

to weight robot sections through a weight matrix, speciﬁ-

cally using an additional variable that depends on task vari-

ables or can be modiﬁed by an operator.

The main contribution of this work is the design of a

weighting matrix that includes cost functions for the con-

tinuous selection of sections of a highly redundant robot,

speciﬁcally a redundant mobile manipulator robot (MM). In

this way, by means of the weight matrix, it is proposed to

control either isolated sections of the MM (when the switch-

ing variable is at its extremes), or to give execution priority

to desired parts of the mechanism (when the variable is set

to intermediate values), considering that the switching value

is task-dependent and can be modiﬁed in real time.

2 Mathematical Foundations

2.1 MM velocity relationships

The differential of the geometric representation with respect

to time makes it possible to obtain the relationship between

the joint velocity and the Cartesian velocity of the robotic

Table 1: DH parameters of the manipulator under consider-

ation

Link q d a α

1 φ + q

h 0 π/2

2 q

+ π/2 0 l

3 q

+ π/2 lb 0 π/2

4 q

+ π/2 + atan(30/264) 0 l

5 q

+ 3π/4 − atan(30/258) 0 −l

π/2

6 q

l2 0 −π/2

7 q

0 0 π/2

8 q

l3 0 0

9 0 0 0 π/2

10 π/2 0 0 0

system,

˙x = J(q) ˙q (1)

where J(q) is the matrix relating these velocities and is gen-

erally referred to as the Jacobian of the whole robot, while

˙x is the Cartesian velocity with respect to a global reference

frame. Fig. 1 shows a redundant mobile manipulator. Then,

to take into account the characteristics of the robotic mech-

anism, one can deﬁne a generalized variable of the form

q =





, (2)

where q

∈ R

represents the integral of the linear and

angular velocities of the moving platform and q

∈ R

represents the angular positions of the robotic arm.

To determine the kinematic behavior of the robotic ma-

nipulator, Denavit Hartenberg representation (DH) is used

[7]. Table 1 shows the DH parameters to deﬁne the geo-

metric relationship of the robotic manipulator used in this

work. As for the mobile platform [8], the possibility of inde-

pendently controlling all four wheels of a skid-steering ve-

hicle greatly extends the applications that these robots can

have, especially when the applications must be developed

outdoors and with non-regular surfaces. The position of a

skid-steering robot [9] can be represented by



x y φ



, (3)

where



x y



are the coordinates of O

(arm reference sys-

tems) with respect to the global reference frame (< W >),

while φ is the angle of rotation with respect to < W >. Ki-

netically, the mathematical representation of a skid-steering

robot can be expressed as follows.

˙q

= S(q

)v + A(q

, (4)

where v =





represents the linear and angular veloc-

ity of the robot, v

is the lateral velocity of the robot (gen-

erally produced by sliding), while S(q

) and A(q

) are

22 International Journal of Engineering Insights, (2023) 1:1

Fig. 1: Mobile Manipulator robot

deﬁned as:

S(q

) =





cos(φ) −a sin(φ)

sin(φ) a cos(φ)

0 1





, (5)

A(q

) =





−sin(φ)

cos(φ)





. (6)

In this way, the velocity of the robotic structure of the

mobile manipulator can be deﬁned as:

η = ˙q =



˙q



∈ R

, (7)

where n = n

+ n

, while η

(con i = 1, 2, ..., n) is the

i-th velocity of the MM.

Thus, the displacement velocity of the Operating End

(O.E) in Cartesian space is represented as follows:

˙x

= J(q)η, (8)

where J(q) ∈ R

m×(n

)

is a Jacobian matrix that re-

lates the joint velocities to the Cartesian velocities of the

complete robotic mechanism.

2.2 Matrix Design W

Assuming that we have a single Jacobian of the whole robot,

we need to modify the solutions given by the inverse kine-

matics based controller. Cost functions governed by a  vari-

able solve the given problem. Thus, the selection of robot

sections of interest is achieved through the construction of a

diagonal weight matrix W. Depending on the application,

the construction of the matrix W() uses both functions

placed on the elements of its diagonal. Consider the case of

a mobile manipulator, divided into two parts of interest: 1)

the mobile platform, and 2) the robotic manipulator. Thus,

a number of α references is required to control the moving

platform, while β references are required to control the ma-

nipulator.

Thus, the weight matrix for selecting degrees of freedom

depending on  for the MM can be constructed with w

ele-

ments. The scalars of w

are deﬁned by:



() f or i = 1 : α

() f or i = α + 1 : α + β,

(9)

in this way,

W() =







0 0 0 0 0

0 0 0 0

0 0 w

0 0 0

0 0 0 w

α+1

0 0

0 0 0 0

0 0 0 0 0 w

α+β







The solution considers a pair of candidate cost functions

denoted by f

() and f

(), both with different properties

given the application requirements.

Let us consider f

() as a decreasing function that has a

large value when  tends to zero, and tends to unity when the

adjustable variable  also tends to 1; while we deﬁne f

()

as an increasing function that tends to a large value when

 tends to unity, and tends to unity when  tends to zero.

Among many possibilities, we adopt the following functions:

() =

 + b

, (10)

() = a

+a

+ 1, (11)

where a

and b

are constants to adjust the shape of the func-

tions.

The behavior of the inverse of W is directly related to

the ﬁnal solution of the controller (as will be seen below).

Thus, the change from 0 to 1 in  continuously deactivates

the moving platform, while activating the manipulator. At

the extremes of , each of the robots is either activated or

completely deactivated.

2.3 Resolution of the redundancy of inverse kinematics

The displacement velocity of an operating end can be calcu-

lated using differential kinematics ( ˙x = J(q) ˙q), however,

23 International Journal of Engineering Insights, (2023) 1:1

for practical purposes the parameters set are given by the

inverse kinematics solution:

η = J

†

(q) ˙x, (12)

where J

†

(q) is the pseudo-inverse of J(q). Given the redun-

dancy of robotic systems, an exact solution for η does not

normally exist, but an approximation can be found through

the use of least squares by calculating k ˙x − J(q)ηk, obtain-

ing the following solution:

η = J

†

(q) ˙x + (I − J(q)

†

J(q))z

, (13)

where I − J

†

J is the projection on the null space generated

by J(q) and z

is an arbitrary vector in the homogeneous

part of the solution which may include secondary objectives,

e.g. operating end orientation, energy saving or maximum

manipulability.

Considering that there are two control variables: x

and

, in our case, we propose to ﬁnd a solution that allows

us to meet both objectives through the redundancy of the

robotic system. Similar to how (13) is obtained, a least-squares

solution allows to use the null space of the redundant sys-

tem, having:

η =J

†

˙x

+ (I − J

†

)

†

( ˙x

− J

†

˙x

)

+ (I − J

†

)(I −

†

∗

(14)

where

†

∆

= J

(I − J

†

), with

†

being the pseudoinverse

and z

∗

is an arbitrary joint velocity vector that could

contain tertiary targets.

Now, based on [10], the constructed weight matrix is

used to compute the Moore-Penrose right-handed pseudoin-

verse. In this way, the inverses of the Jacobian are calculated

as follows:

†

= W

−1

]

−1

, (15)

†

= W

−1

[

−1

]

−1

. (16)

In order to correct the numerical error, information ob-

tained from the robot in real time allows us to calculate both

position and orientation errors. From (14), results the ﬁnal

closed-loop inverse kinematics solution

η = J

†

˙x

ref

+ (I − J

†

)

†

( ˙x

ref

− J

†

˙x

ref

), (17)

where ˙x

ref

and ˙x

ref

are references in Cartesian coordi-

nates generated by the human operator through a local robot.

2.4 Control references

The  variable can be calibrated considering several factors,

automatically or teleoperated. For this work, we assume that

the master robot has an additional degree of freedom that

allows a continuous online modiﬁcation of this parameter

and that the operator conﬁgures it depending on the require-

ments of the task. On the other hand, the reference to follow

is generated by a position controller, based on the position

error. Thus, we have:

˙x

ref

= K

) = K

− x

), (18)

with x

being the desired positions; K

is a positive def-

inite matrix that regulates the position errors. On the other

hand, we have established that the redundancy of the MM al-

lows the inclusion of secondary objectives. Keeping in mind

that the secondary target is set in the null space of the overall

solution, this requirement is fulﬁlled as long as it does not

interfere with the main solution. For our case, the orienta-

tion of the end-effector can be modiﬁed in-line according to

task requirements and deﬁned as follows:

˙x

ref

= K

, (19)

where K

is a gain matrix and e

are the orientation errors

deﬁned through the unitary quaternion [11] and are calcu-

lated as follows:







+ υ

rυ

− r

υ − S

(υ)υ



, (20)

where (r, υ) and (r

, υ

) are the real and desired orientation

expressed in quaternions [11], respectively, with the fulﬁll-

ment that e

= e

+ e

= 1, while S

(υ) is a skew-

symmetric matrix deﬁned as:

(υ) ,





0 −υ

−υ





. (21)

3 Results

The execution of a task is shown in this section to determine

the performance of the proposal. For this, three points to

be reached are deﬁned, where  is modiﬁed in real time at

different periods of execution.

To reach the ﬁrst point, the robot starts from a known po-

sition, with a default conﬁguration (Figs. 2, 3). For the ﬁrst

5 seconds,  = 0 is set, i.e., the platform is fully activated

and the manipulator deactivated. In this time frame, posi-

tion errors tending to zero are present for the X-Y axes (Fig.

4), while the error in Z remains constant since the participa-

tion of the MM is required to achieve zero errors. The range

of e

= 0 is performed in the next 5 seconds of execution

24 International Journal of Engineering Insights, (2023) 1:1

Fig. 2: Reconstruction of execution, lateral view

Fig. 3: Reconstruction of execution, front view

since  = 1 is set and the manipulator is activated, while the

platform remains stationary. All this behavior of the robotic

structure is shown in Figs. 5 and 6. These initial ten seconds

show an approach of the platform to the desired point, after

which it switches ( practically unobtrusively) to the MM.

The second point range runs from seconds 10 to 20.

Differently from the ﬁrst point range, this period consid-

ers an involvement of both the platform and the manipulator

in an indicated period. In the second ten, the new desired

point generates position errors that the controller tries to re-

solve. From 10 to 12.3 seconds,  enables only the moving

platform, i.e.  = 0. Subsequently, the weighting value is

 = 0.6, which enables velocities to be induced at all joints

of the robot. In this period (at seconds 12.3 to 15), it can

be observed how the platform moves backward (Fig. 5) to

facilitate the MM reaching the target (Fig. 6), nulling the

position errors in all axes (Fig. 4).

25 International Journal of Engineering Insights, (2023) 1:1

Fig. 4: Position errors

Fig. 5: Platform velocities

Finally, the range of the third point indicates a continu-

ous involvement of both the moving platform and the ma-

nipulator. In this case, continuous motion of all joints (Figs.

5 and 6) and a trend of errors (Fig. 4) to zero similar in all

axes is observed, given a conﬁguration of  = 0.4.

For their part, quaternion-based orientation errors are

shown in Fig. 7. Since the use of redundancy of the robotic

system is posited, a desired orientation of the operating end

is conﬁgured throughout the task. In this case and since this

secondary objective is projected onto the null space of the

solution, the position range may or may not be achieved.

However, given the characteristics of the robotic system and

as shown in Fig. 7, the errors tend to zero as long as  allows

the manipulator to be activated.

Fig. 6: Robotic manipulator positions

Fig. 7: Orientation errors

4 Conclusions

The main contribution of this work is the design of a weight

matrix that includes cost functions for the continuous selec-

tion of sections of a highly redundant robot. In this way, it

is proposed to control either isolated sections of a mobile

manipulator robot, or by means of the weight matrix, to give

higher execution priority to desired parts of the mechanism,

based on a task-dependent variable that can be modiﬁed in

real time.

The activation of parts of a redundant robotic system is

possible through the use of a weighting matrix. As shown

in the development of this proposal, the design of a ma-

trix that allows to continuously switch sections of interest of

the robot while deactivating others is possible. Additionally,

since the switching factor can vary in real time, the applica-

26 International Journal of Engineering Insights, (2023) 1:1

tion of such techniques can be useful for both autonomous

and teleoperated control.

Acknowledgements The authors would like to thank the German Stu-

dent Exchange Service (DAAD), and the Instituto de Autom

atica, a

dual-dependent institute: Universidad Nacional de San Juan and CON-

ICET - Argentina.

Conﬂict of interest

The authors declare that they have no conﬂict of interest.

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ıa del Carmen Claudio, Viviana

Moya, Emanuel Slawi

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