A FUZZY LOGIC APPROACH USED IN THE INVERSE

KINEMATIC ALGORITHM OF A SPACE ZERO-G FREE

FLYING ROBOT

Andrea Bulgarelli

, Alessio Aboudan

, Carlo Menon

, Massimo Trifoglio

and Fulvio Gianotti

Institute for Space Astrophysics and Cosmic Physics (IASF) Bologna, INAF, Via Gobetti 101, 40129 Bologna, Italy

CISAS “G. Colombo”, University of Padova, Via Venezia 15, 35131 Padova, Italy

Menrva Group, School of Engineering Science, Simon Fraser University

8888 University Dr., Burnaby, BC, V5A 1S6, Canada

Keywords: Redundant space robot manipulator, Inverse kinematic control, Hierarchical fuzzy system controller, Fixed

attitude restricted motion problem.

Abstract: A fuzzy algorithm for the fixed attitude restricted motion problem of free-flying robots is proposed in this

paper. One of the main applications is to guide the robotic arm of a space servicing satellite: in such a

mission, one of the priorities is to reduce disturbances on the satellite attitude induced by robotic arm

movements so as not to perturb the pointing position of the satellite. A robot whose base has both mass and

inertia of the same order of magnitude of its robotic arm is considered - in this configuration the disturbance

of the satellite attitude is not negligible. Objective is to plan the robot’s arm motion in such a way as the

end-effector tracks a desired trajectory while disturbances on the base’s attitude are minimized. This

objective is achieved by taking into account the coupling between the arm and the floating base of the robot

in the kinematic inversion of the guiding control, controlling the gain matrix of the subtask introduced in the

kinematic inversion equation by means of a fuzzy algorithm. The proposed strategy combines the

advantages of the inverse kinematic algorithm and a fuzzy logic approach.

1 INTRODUCTION

The advancement of space robotics is currently

recognized by space agencies as a key strategy to

reduce costs of space exploration. Robotic systems

can advantageously be used during robotic

exploration missions, on-orbit servicing operations

Internal Vehicular Activities (IVA) and Extra-

Vehicular Activities (EVA) substituting astronauts

during operations. The use of a floating robot with a

dexterous arm is considered to be a viable option for

on-orbit servicing (Hirzinger, 2000; Tatsch, 2006;

Thronson, 2008).

Hence, controlling autonomous robotic systems

is of outmost importance and research is currently

focusing on the development of algorithms, which

allow performing safe and complex robotic

manoeuvres in space. Many space agencies have

funded important programs for developing space

robotic systems (Hirzinger, 1994; Yoshida, 2003;

Roderick, 2004; Marzwell, 2001; Culbertson, 2003;

Oda, 2008).

In this paper, we tackle the problem of controlling a

platform to be used in space servicing missions. In

the framework of this research, a spacecraft

equipped with a robotic arm is considered to be a

free-flying robot where the spacecraft is the floating

base of the robot - no active thrusters are considered

in this study. The goal is to plan the robot’s arm

motion in such a way that the end-effector tracks a

desired trajectory while disturbances on the base’s

attitude are minimized. This problem is known as

Fixed Attitude Restricted (FAR) motion problem

(Gu, 1993; Sagara, 2008; Boning, 2010,

Khaloozadeh, 2010; Rastegaria, 2010). We

investigated the problem in which both mass and

inertia of the base are comparable to those of the

robotic arm - the disturbance produced by the arm

on the base can therefore be significant if a non-

suitable controller is used. The goal is achieved by

taking into account the coupling between the arm

and the base of the robot in the kinematic inversion

of the guiding control and by combining the

advantages of inverse kinematic algorithms and a

429

Bulgarelli A., Aboudan A., Menon C., Trifoglio M. and Gianotti F..

A FUZZY LOGIC APPROACH USED IN THE INVERSE KINEMATIC ALGORITHM OF A SPACE ZERO-G FREE FLYING ROBOT.

DOI: 10.5220/0003653304290434

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 429-434

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Fuzzy Logic approach.

This work follows an experimental phase in

which our working group has built a free-floating

robot that was tested during the Sixth Student

Parabolic Flight Campaign sponsored by the ESA

(Menon, 2003; Menon, 2004; Cocuzza, 2004;

Menon, 2005).

In Section 2 we introduce the fixed attitude

restricted motion problem. In Section 3 the fuzzy

algorithm is presented. Section 4 reports the

simulation performed to test the algorithm and the

main results. Finally, Section 5 draws the conclusion

of this work.

2 KINEMATIC CONTROL

The free-floating problem (FFP), which we

considered, consists in the description and control of

a system in which (i) the position and orientation of

the spacecraft respect to an inertial coordinate

system are well-known in the initial state, (ii) there

are no external forces or torques about the center of

mass, hence the conservation of momentum and the

equilibrium of forces and moments hold strictly true,

(iii) there are no attitude control devices such as

reaction wheels or thrusters - internal forces are

generated only by joint motors, (iv) the robot acts in

a zero-gravity environment. In order to solve the

FFP, the linear and angular momentum conservation

laws are used. The considered system is

nonholonomic, namely the satellite orientation is not

only a function of the joint configuration, but also a

function of the path taken to reach such an

orientation. As a consequence, nonholonomy offers

the possibility to perform a reorientation of the

satellite using the motion of the only robotic arm

(Nenchev, 1988; Nenchev, 1992). This could be

useful in docking operations to save fuel and reduce

pollution near the target satellite. This kind of

manoeuvres can be employed to augment the

workspace of the robot too; it allows turning the

satellite into a different orientation, bringing back

the manipulator into its reference configuration.

Another possibility resulting from nonholonomy is

that manoeuvres can be sought to reduce attitude

variation of the spacecraft.

A closed loop differential kinematic inversion

algorithm can be adopted (Sciavicco, 1988;

Nenchev, 1992; Siciliano, 1993) and additional

subtasks introduced. In out context, n is the number

of degrees of freedom of our problem, m

is the

dimension of operating space of the arm and m

the dimension of operating space of the base or

spacecraft; for a 3D environment, m

=6. If



is the

velocity vector of the arm joints, it is represented by

the following equation:

()( ) ( )

aAG p d AGAGccc

qJqKex IJJJKe

=++−



(1)

where q is the vector of joint positions (arm and

base), J

is the Analytic Generalized Jacobian

(Umetani, 1989; Menon, 2003),

is its pseudo-

inverse, e is error of the end-effector (the difference

between desired and real position), x

is the desired

position and orientation of working space, K

and K

are gain matrixes (K

is positive definite), I is the

identity matrix, J

is the Jacobian associated to the

constraint task error, e

is the error of the constraint

task defined as e

–q

where q

is a generalized

position variable of the free floating base of the

robot that is constrained to the desired trajectory q

The matrix

−

)

projects the joint velocity

contribution into the null space of the generalized

Jacobian in order to separate the constraint and end-

effector tasks.

Since the aim is to keep fix a rotational angle of

the base, the constraint error can be expressed as

=-q

. In order to obtain high performance of this

algorithm, K

should be selected properly (Menon,

2005): the variables of interest should be handled in

a flexible manner for planning optimal trajectories

while taking into account the constraints of the

problem; a fuzzy algorithm capable to online adjust

the elements of the K

matrix, and therefore manage

the motion of the robot is implemented.

In an experiment performed with a free-floating

robot that was tested during the Sixth Student

Parabolic Flight Campaign sponsored by the ESA

(Menon, 2003; Menon, 2004; Cocuzza, 2004;

Menon, 2005) we have proved the real possibility of

keeping the robot base stationary during arm

operations with a K

matrix fixed. This work is an

improvement of the experimental results: the main

purpose of the proposed solution is to handle the K

matrix during the trajectory evolution and controls

any instability that can be induced to the system.

Specifically, a Mamdami Fuzzy Inference System

(FIS) is developed to control the values of K

during

the evolution of the trajectory of the space robot.

The use of fuzzy systems has been extensively used

in other works e.g. (Gu, 1993; Antonelli, 2003;

Radaideh, 2003; De Santis, 2008; Zou, 2009; Fu,

2009) - in this work we have used this approach to

control the motion of a free-floating robot operating

in a zero-g environment.

The representation of the robot is obtained by

using the reference frame proposed in Menon

FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications

430

(2005): the entire system is modelled as a single

robot with a fixed base. To describe the position and

orientation of the base, a fictitious arm, which is a

sequence of frames and joints linking the inertial

reference frame to the base itself, is introduced.

Because the free-floating base has six degree of

freedoms (DOF), we considered a fictitious arm with

three prismatic and three revolute joints, called

virtual joints.

3 FUZZY LOGIC ALGORITHM

The correct determination of the K

matrix of Eq.1 is

of fundamental importance to effectively solve the

constraint minimization problem for the following

reasons: (1) an unstable dynamic response of the

robotic system and an increased position error of the

end-effector could be caused by selecting too large

values of the elements of the K

matrix; (2) imposed

constrains could not be satisfied if elements of the K

matrix assume too small values.

The proposed fuzzy algorithm selects the

elements of the K

matrix in order to minimize the

error e

(see Eq.1). This algorithm combines the

advantages of the inverse kinematic algorithm (see

Section 2) with a Fuzzy Logic approach with a

hierarchical structure having a conventional

differential inverse kinematic algorithm at the

bottom layer (Eq.1) and a tuner at an upper-level: at

each step the kinematic is inverted using the K

matrix calculated by fuzzy algorithm in the previous

step. This work is related to the top-level tuner using

a fuzzy approach: this tuner uses a set of linguistic

rules for adjusting the constraint error gain matrix

during the kinematic inversion. The performance of

this fuzzy tuner is evaluated hereafter on the basis of

simulation results. The proposed algorithm takes as

input the constraint error e

=-q

defined in Section 2.

3.1 Detection of Oscillations

Possible dynamic oscillations of the robotic system

are detected by monitoring the oscillation of the

joint velocity



; at this purpose a moving average

filter is used.

During the calculation of the differential inverse

kinematics, for each time step the moving average of

the joint velocity vector is calculated, and compared

with its actual value. If the moving average is

crossed twice by the joint velocity the joint velocity



is determined to be oscillating. The number of

joints (for the arm) and virtual joints (for the base),

which are oscillating, becomes an input of the fuzzy

algorithm.

The crisp inputs of the FIS are the constraint

error e

and the number of oscillating joints. The

crisp output is the increment or the decrement of the

current K

elements.

Two linguistic variable are considered:

constraint error = {zero (Z), positive small (PS),

positive (P), positive medium (PM), positive large

(PL)} and oscillation = {OFF, ON}, where OFF

means no oscillations. If the constraint error is

greater than 0.1 rad/s, the error is considered PL.

The output linguistic variable is the gain increment =

{Negative Large (NL), Negative (N), Zero (Z),

Positive Small (PS), Positive (P), Positive Large

(PL)} that is the increments of K

elements; the

fuzzy sets used for this output linguistic variable are

not symmetric. In fact, for a better management of

the instability for a more rapid decrement of the K

the robotic system can react in a better way to the

system instability that can be reached during the

increment of the Kc.

The membership functions used for the input and

output linguistic variable are reported in Fig. 1.

The fuzzy rules used are the follows: (1) if

(constraint error is Zero) or (constraint error is

Positive Small) and (oscillation is OFF then (gain

increment is Positive Small); (2) if (constraint error

is Positive) and (oscillation is OFF) then (gain

increment is Positive); (3) if (constraint error is

Positive Large) and (oscillation is OFF) then (gain

increment is Positive Large); (4) if (oscillation is

ON) then (gain increment is Negative Large).

The main idea of the algorithm is the following:

if the system has no oscillations, the increment of

gain matrix is in the same direction of the constraint

error: this enable the controller to increment the

weight of the constraint that we have added to

resolve the FAR motion problem, but when the

instability is reached the controller reacts and

decrements rapidly the gain matrix.

4 SIMULATIONS

The test bed for the evaluation and comparison of

the algorithm presented in this paper is based on a

Matlab simulator developed by the authors. This

simulator enables the simulation of (i) rigid body

motion, (ii) direct and inverse kinematic (iii)

differential kinematic, and (iv) control problem. It is

possible to build various types of redundant or non-

redundant robots with fixed or free-floating base and

with revolute or prismatic joints.

A FUZZY LOGIC APPROACH USED IN THE INVERSE KINEMATIC ALGORITHM OF A SPACE ZERO-G FREE

FLYING ROBOT

431

Figure 1: Fuzzy membership input and output functions.

From left to right: (a) constraint error; (b) oscillation of

joints; (c) output K

The geometrical parameters of the robot, which were

selected for the simulation, were those of a robotic

prototype tested by the authors during the Sixth

Student ESA Parabolic Flight Campaign. This

prototype was a free-floating robot with 4 DOFs

operating in a 3D zero-g space environment. Table 1

reports the links of the parameters of the simulated

robot.

For all links the mass is 4 kg. The mass of the

base is 16 kg and its inertia is:

0.5760 0 0

0 0.5760 0

0 0 0.5760

⎛

⎝

⎜

⎞

⎠

⎟

kg m

(3)

With a 4-DOF robotic arm, it is possible to control

the position or orientation of the end-effector and

keep fixed the rotation of the base about one axis:

this means that the K

becomes a scalar. The end-

effector (EE) position and the yaw angle of the base

were chosen as desired parameters to be controlled

in this study.

Table 1: Parameters of the simulated robot.

Link

Arm

length

Arm center

of mass

Inertia tensor

7, 8, 9 1 m 0.5 m

0.3358 0 0

0 0.3358 0

0 0 0.0450

⎛

⎝

⎜

⎞

⎠

⎟

kgm

10 0.5 m 0.25 m

0.1892 0 0

0 0.0450 0

0 0 0.1892

⎛

⎝

⎜

⎞

⎠

⎟

kgm

About 1000 random trajectories of the EE were

used as input for the kinematic inversion algorithms.

To test if the generated trajectories were physically

consistent, the calculated trajectories were passed to

the robot simulator with a robust control algorithm

that control the end-effector position and the yaw

angle of the base (Menon, 2005) and compared with

the desired input. The fuzzy algorithm uses the same

robust controller but it adapts the K

matrix. A max

rotation of 0.4 rad in 10-15 s (the last of simulated

trajectories) is considered for the base yaw angle.

The integration step time is 0.01 s.

The following parameters were considered to

evaluate the performance of the algorithm and

reported in Table 2: (i) EE pos.: norm of the end-

effector position error (difference between the

desired and generated trajectory), (ii) maximum

(Max d.) and mean deviation (Mean d.) of the base

yaw angle with respect to a zero rotation, (iii)

Oscill.: mean value of the oscillation occurring

during the kinematic inversion (this was considered

to be a suitable parameter to evaluate the stability),

that is the number of joints that oscillate for each

step time divided for the number of steps.

Table 2 reports the main results of the performed

simulations with the proposed fuzzy algorithms with

respect to the approach of Eq. 1 (called classical

approach in this context) with a fixed K

, where the

first column reports the K

value used as a constant

(for the closed loop inverse algorithm) or the starting

value for the fuzzy algorithm.

From Table 2 it is possible to note that the mean

and maximum deviation from the zero trajectory of

the base yaw angle is smaller than the classical

approach; in addition, the number of oscillations of

the fuzzy algorithm is comparable to the one

obtained with a classical approach, with the

exception of cases with initial K

≥1000, in which the

algorithm acts as a system stabilizer. This is a very

interesting behaviour and an important result of this

algorithm; in fact the algorithm is able to reduce or

eliminate large joint oscillations by increasing K

enabling the optimization of the fixed attitude

restricted motion subtask; this behaviour provides

justification of the smaller end-effector position

error obtained by using the fuzzy algorithm.

Some examples of trajectories are shown on the

left–hand side of Fig. 3, 4 and 5. Each Figure

contains a trajectory with (i) orientation of the yaw

angle of the base not constrained, (ii) orientation

constrained with K

=1000

fixed (Eq.1), (iii)

orientation of the proposed fuzzy algorithm; the

right panel shows the K

value selected by the fuzzy

algorithm. Fig. 3 reports an example of trajectory in

FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications

432

Table 2: Comparison between closed loop algorithm with fixed and fuzzy-adaptive K

which the subtask target is reached but the fuzzy

algorithm is more effective. Fig. 4 shows an

example of trajectory in which the subtask is

maintained only with this proposed fuzzy algorithm.

The trajectory of Fig. 5 is stable without task

optimization but the subtask objective is not

reached; if we use a fixed K

the trajectory is not

stable; it is stabilized with the proposed fuzzy

algorithm and the subtask objective is reached. A

shown in these figures the performance of the fuzzy

algorithm is higher than the closed loop inverse

algorithm of Eq.1 with a fixed K

Figure 3: Trajectory example 1. Left: base yaw angle of

not constrained, constrained (fixed Kc) and fuzzy-adaptive

trajectories. Right: calculated Kc adaptive value.

Figure 4: Trajectory example 2. Left: base yaw angle of

not constrained, constrained (fixed Kc) and fuzzy-adaptive

trajectories. Right: calculated Kc adaptive value.

Figure 5: Trajectory example 3. Left: base yaw angle of

not constrained, constrained (fixed Kc) and fuzzy-adaptive

trajectories. Right: calculated Kc adaptive value.

5 CONCLUSIONS

In this paper is proposed a fuzzy algorithm suitable

to control the yaw angle of a free-flying robot

operating in a space zero-g environment. The

performed simulations, which used parameters of a

real robotic platform tested by authors, has shown

that the yaw angle obtained with the fuzzy algorithm

is smaller than that obtained with a classical

approach, while the end-effector position error was

comparable. The gain matrix was incremented up to

the point in which the system was close to

instability; when this condition is reached, the

algorithm promptly reacts (possible oscillations may

be considered as an indication that the gain matrix is

too large) and the fuzzy controller decreases the

elements of the gain matrix. In fact, the proposed

algorithm acts as a stabilizer for the robot under

control. It detected oscillations and reacted to

stabilize the system; it behaved as a system control

tuner. The proposed method is potentially suitable

for solving a large class of control problems and

could in principle be applicable to any kind of

robot’s geometrical constraint.



constan

uzzy-adaptive



E pos.

err. (m)



Yaw max

d. (rad)



Yaw mean

d. (rad)

Oscill.

E pos. err.

(m)

Yaw max d.

(rad)

Yaw mean

d. (rad)



Oscil



2.5 10

-5



0.064



0.020

0.0014

2.6 10

-5

0.028



0.005



0.048

100



9.3 10

-5



0.041



0.011

0.0030

3.0 10

-5

0.030



0.006



0.016

500



2.0 10

-5



0.030



0.006

0.0096

2.7 10

-5

0.024



0.004



0.060

1000



0.0013



0.027



0.005

0.0587

2.8 10

-5

0.023



0.004



0.070

5000



0.56

0.233



0.019

0.5439

2.8 10

-5

0.010



0.001



0.110

A FUZZY LOGIC APPROACH USED IN THE INVERSE KINEMATIC ALGORITHM OF A SPACE ZERO-G FREE

FLYING ROBOT

433

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