Implementation of PID, Bang-bang and Backstep-

ping controllers on 3D Printed Ambidextrous Robot

Hand

Mashood Mukhtar, Emre Akyürek, Tatiana Kalganova, Nicolas Lesne

Abstract Robot hands have attracted increasing research interest in recent years

due to their high demand in industry and wide scope in number of applications.

Almost all researches done on the robot hands were aimed at improving

mechanical design, clever grasping at different angles, lifting and sensing of

different objects. In this chapter, we presented the detail classification of control

systems and reviewed the related work that has been done in the past. In

particular, our focus was on control algorithms implemented on pneumatic

systems using PID controller, Bang-bang controller and Backstepping controller.

These controllers were tested on our uniquely designed ambidextrous robotic

hand structure and results were compared to find the best controller to drive such

devices. The five finger ambidextrous robot hand offers total of 13 degrees of

freedom (DOFs) and it can bend its fingers in both ways left and right offering

full ambidextrous functionality by using only 18 pneumatic artificial muscles

(PAMs).

__________________________

Mashood Mukhtar, Emre Akyürek, Tatiana Kalganova

Department of Electronic and Computer Engineering, Brunel University, Kingston Lane,

Uxbridge, London, UK, e-mail: Mashood.Mukhtar@brunel.ac.uk

Nicolas Lesne

Department of System Engineering, ESIEE Paris, Noisy-le-Grand Cedex, France

1 Introduction

Robots play a key role in our world from providing daily services to industrial

use. Their role is in high demand due to the fact they offer more productivity,

safety at workplace and time reduction in task completion. Number of control

schemes are found in the literature [1] to control such robot such as adaptive con-

trol [2], direct continuous-time adaptive control [3], neurofuzzy PID control [4]

hierarchical control [5], slave-side control [6], intelligent controls [7], neural

networks [8], just-in-time control [9] ,Bayesian probability [10], equilibrium-

point control [11], fuzzy logic control [12], machine learning [13], evolutionary

computation [14] and genetic algorithms [15], nonlinear optimal predictive con-

trol [16],optimal control [17], stochastic control [18], variable structure control

[19], chattering-free robust variable structure control [20] and energy shaping

control [21], gain scheduling model-based controller [22], sliding mode control

[23], proxy sliding mode control [24], neuro-fuzzy/genetic control [25] but there

are five types of control systems (Fig.1) mainly used on pneumatic muscles.

Fig.1. Classification of control algorithm implemented on pneumatic systems.

Focus of our research was on the feedback control system and non-linear con-

trol systems. Feedback is a control system in which an output is used as a feed-

back to adjust the performance of a system to meet expected output. Control sys-

tems with at least one non-linearity present in the system are called nonlinear

control systems. In order to reach a desired output value, output of a processing

unit (system to be controlled) is compared with the desired target and then feed-

back is provided to the processing unit to make necessary changes to reach closer

to desire output. The purpose of designing such system is to stabilise the system

at certain target.

Tactical sensors are employed to investigate the grasping abilities of the ambi-

dextrous hand. These sensors have been used several times in the past on the ro-

bot hands driven by motors [26], [27] and robot hands driven by pneumatic artifi-

cial muscles [28] and [29]. The automation of the ambidextrous robot hand and

its interaction with objects are also augmented by implementing vision sensors on

each side of the palm. Similar systems have already been developed in the past.

For instance, the two-fingered robot hand discussed in [30] receives vision feed-

back from an omnidirectional camera that provides a visual hull of the object and

allows the fingers to automatically adapt to its shape. The three-fingered robot

hand developed by Ishikawa Watanabe Laboratory [31] is connected to a visual

feedback at a rate of 1 KHz. Combined with its high-speed motorized system that

allows a joint to rotate by 180 degrees in 0.1 seconds, it allows the hand to inter-

act dynamically with its environment, such as catching falling objects. A laser

displacement sensor is also used for the two-fingered robot hand introduced in

[32]. It measures the vertical slip displacement of the grasped object and allows

the hand to adjust its grasp. In our research, the aim of the vision sensor is to de-

tect objects close to the palms and to automatically trigger grasping algorithms.

Once objects are detected by one of the vision sensors, grasping features of the

ambidextrous robot hand are investigated using three different algorithms, which

are proportional-integrative-derivative (PID), bang-bang and Backstepping con-

trollers. Despite the nonlinear behavior of PAMs actuators [33], previous re-

searches indicates that these three algorithms are suitable to control pneumatic

systems.

The work presented in this chapter aim to validate the possibility of controlling

a uniquely designed ambidextrous robot hand using PID controller, Bang-bang

controller and Backstepping controller. The ambidextrous robot hand is a robotic

device for which the specificity is to imitate either the movements of a right hand

or a left hand. As it can be seen in Fig.2, its fingers can bend in one way or an-

other to include the mechanical behavior of two opposite hands in a single device.

The Ambidextrous Robot Hand has a total of 13 degrees of freedom (DOFs) and

is actuated by 18 pneumatic artificial muscles (PAMs) [58].

2 Implementation of Controllers

2.1 PID Controller

PID controller is a combination of proportional, integral and derivative controller

[34].Proportional controller provides corrective force proportional to error pre-

sent. Although it is useful for improving the response of stable systems but it

comes with a steady-state error problem which is eliminated by adding integral

control. Integral control restores the force that is proportional to the sum of all

past errors, multiplied by time [35]

Fig.2. Ambidextrous Robot Hand (Ambidextrous mode)

On one hand integral controller solves a steady-state error problem created by

proportional controller but on the other hand integral feedback makes system

overshoot. To overcome overshooting problem, derivative controller is used [36].

It slows the controller variable just before it reaches its destination. Derivatives

controller is always used in combination with other controller and has no influ-

ence on the accuracy of a system. All these three control have their strengths and

weakness and when combine together offers the maximum strength whilst mini-

mizing weakness.

te

dt

dKdtteKteKOutput d

t

iPPID

0

2.1.1 PID controller adapted to the Ambidextrous Hand

PID control loops were implemented on an ambidextrous robot hand using the

parallel form of a PID controller, for which the equation is as follows:

𝑢(𝑡) = 𝐾𝑝𝑒(𝑡) + 𝐾𝑖 ∫ 𝑒(𝜏)𝑑𝜏𝑡

0

+ 𝐾𝑑

𝑑

𝑑𝑡𝑒(𝑡) (1)

Output 𝑢 (t) should be calculated exactly in the same way as described in

equation (1) but this is not the case due to asymmetrical tendon routing of

ambidextrous robot hand [37].Mechanical specifications are taken into

consideration before calculating the 𝑢 (t). It is divided into three different outputs

for the three PAMs driving each finger. These three outputs are 𝑢𝑝𝑙(𝑡), 𝑢𝑚𝑟(𝑡)

and 𝑢𝑚𝑙(𝑡), respectively attributed to the proximal left, medial right and medial

left PAMs. The same notations are used for the gain constants. The adapted PID

equation is defined as:

[

𝑢𝑝𝑙(𝑡)

𝑢𝑚𝑟(𝑡)

𝑢𝑚𝑙(𝑡)

] = [

𝐾𝑝𝑝𝑙𝐾𝑖𝑝𝑙

𝐾𝑑

𝐾𝑝𝑚𝑟𝐾𝑖𝑚𝑟

𝐾𝑑

𝐾𝑝𝑚𝑙𝐾𝑖𝑚𝑙

𝐾𝑑

]

[

𝑒(𝑡)

∫ 𝑒(𝜏)𝑑𝜏𝑡

0

𝑑

𝑑𝑡𝑒(𝑡) ]

(2)

No. of position Proximal left PAM

(bars)

Medial left PAM

(bars)

Medial right PAM

(bars)

1 2.497 1.173 0.017

2 4.028 0.250 0.782

3 3.129 0.175 1.481

4 1.972 1.406 0.017

5 2.954 1.115 2.363

6 1.643 0.357 2.617

7 0.038 2.725 1.227

8 0.022 1.411 2.039

9 0.008 0.491 2.990

Table 1. Pressure at ambidextrous robot hand’s extreme positions.

To imitate the behaviour of human finger correctly, some of the pneumatic arti-

ficial muscles must contract slower than others. This prevents having medial and

distal phalanges totally close when the proximal phalange is bending. Thus, ex-

periments are done to collect PAMs’ pressure variation according to the fingers’

position. As the finger is made of one proximal phalange and two other phalanges

for which the movement is coupled, the finger has nine extreme positions. These

positions are numbered from 1 to 9. Position 1 refers to the proximal and medi-

al/distal phalanges reaching both the maximum range on their left side. Position 2

refers to the proximal phalange reaching its maximum range on its left side,

whereas the medial/distal phalanges are straight. Position 3 refers to the proximal

phalange being on its maximum range on its left side, whereas the medial/distal

phalanges are on their maximum range on the right side. Position 4 to 6 refers to

the finger when the proximal phalange is straight. Last position, position 9, is

when the proximal and medial/distal phalanges reach both the maximum range on

their right side. The experiment is summarized in Table 1 and Fig.3.

Fig.3. Pressure variations at extreme finger positions.

When the fingers’ close from vertical to left position (position 5 to the average

of positions 4 and 2), it is observed that the pressure of medial left PAM varies

thrice less than the pressure of the proximal left PAM, whereas the pressure of

the two medial PAMs vary from the same ratios but in opposite ways. Therefore,

the proportional constant gains are consequently defined as:

[𝐾𝑝𝑚𝑙

𝐾𝑖𝑚𝑙

] = [𝐾𝑝𝑝𝑙

/3

𝐾𝑖𝑝𝑙/3

] = [−𝐾𝑝𝑚𝑟

−𝐾𝑖𝑚𝑟

] (3)

When the fingers’ close from vertical to right position (position 5 to the average

of positions 6 and 8), it is observed that the pressure of medial left PAM varies

0.4 times slower than the pressure of the medial right PAM. Moreover, the pres-

sure of the medial right PAM varies in an opposite way as the one of the proximal

left PAM. Therefore, when the object interacts on the left side of the hand, equa-

tion can be written as follows:

[𝐾𝑝𝑚𝑟

𝐾𝑖𝑚𝑟

] = [0.4 𝑋 𝐾𝑝𝑚𝑙

0.4 𝑋 𝐾𝑖𝑚𝑙

] = [−𝐾𝑝𝑝𝑙

−𝐾𝑖𝑝𝑙

] (4)

Same ratios are applied to the derivative gain constants when object interact on right side of the hand.

0

1

2

3

4

1 3 5 7 9

Pre

ssu

re v

aria

tio

n (

bar

s)

Fingers' position number

Proximal leftPAM

Medial leftPAM

Medial rightPAM

2.1.2 Results obtained with the PID controller

Using equation (2), PID control loops with identical gain constants are sent to the four fingers with a target of 1 N and an error margin of 0.05 N, whereas the thumb is assigned to a target of 12 N with an error margin of 0.5 N. A can of soft drink is brought close to the hand and data is collected every 0.05 sec. The experiment re-sult is illustrated in Fig.4

Fig.4. The Ambidextrous Robot Hand holding a can (a grasping movement implemented with

PID controllers) and a graph representing data collected after every 0.05 seconds.

From Fig.4, it can be seen that grasping of a can of soft drink began at 0.15 ap-

proximately and it became more stable after 0.2 sec. An overshoot has occurred on

all fingers but stayed in limit of 0.05N where system has automatically adjusted at

the next collection. These small overshoots occurred because different parts of the

fingers get into contact with the object before the object actually gets into contact

with the force sensor. This results bending of fingers slower when phalanges

touches the object. Since the can of soft drink does not deform itself, it can be de-

duced that the grasping control is both fast and accurate when the Ambidextrous

Hand is driven by PID loops.

2.1.3 Comparison with other grasping algorithms using PID

Mechanical and control features of the ambidextrous hand are compared with other

hands in table 2. The grasping algorithm designed by J.Y. Nagase et al. in [38] is

the most accurate and the slowest engineered after 2010. As for fuzzy logic, it is

observed that the movements are more precise but slower than the ones driven by

PID and PD controls. It is also noticed that PD control is implemented more often

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0.1 0.15 0.2 0.25 0.3

Forc

e (N

)

Time (sec)

Forefinger

MiddlefingerRingfingerLittlefinger

than PID control to grab objects. This is explained because the grasping time is

usually reached in a shorter delay than the full rotation of fingers from open to

close positions. The Ambidextrous Hand has a grasping time and a maximum

overshoot close to the best ones obtained with other robot hands. It can therefore

successively grab objects despite its limited number of DOFs, and it is the only ro-

botic model that can grab objects with an ambidextrous behavior, either as a left or

a right hand, equally performing well.

Table 2. Comparison of various robot hands with our ambidextrous robot hand.

3.1 Bang-bang Controller

Bang-bang controllers are used to switch between two states abruptly. They are

widely used in systems that accept binary inputs. Bang-bang controller is not

Ro

bot

han

d

# f

ing

ers

Ty

pe

of

actu

ators

# D

OF

s

Gra

spin

g a

lgo

rith

ms

An

thro

po

mo

rph

ic p

o-

siti

on

ing

of

fin

ger

s T

hu

mb o

ppo

site

to

oth

er f

ing

ers

duri

ng

gra

spin

g

Gra

spin

g t

ime

(sec

)

Max

. ov

ersh

oo

t or

er-

ror

(%)

Am

bid

exte

rity

Gifu Hand II [39] 5 Motors 16 PID 0.55ab

16%ab

High-speed hand [40] 3 Motors 8 PD 0.05a N/A

ITU Hand [41] 2 SMAs 1a N/A 3.76 N/A

I. Yamano and T. Maeno

[42] 5 SMAs 20 N/A N/A 6%

a

L. Zollo et al. [43] 3 Motors 10a PD N/A N/A

S. Nishino et al. [29] 5 PAMs 13a PID, Cascade 1.0

a 6%

a

D. Gunji et al. [32] 2 Motors 1 PD 0.6a N/A

T. Yoshikawa [44] 2 Motors 4a PID 0.55

a 22%

a

J.Y. Nagase et al. [45] 4 PAMs N/

A Fuzzy logic 0.9

a 9%

ad

DLR Hand II [46] 4 Motors 16a N/A 28.7 N/A

DLR Hand [47] 5 Motors 19 Cascade 0.1a N/A

Shadow Hand [26] 5 Motors 20 PID 0.15a N/A

ACT Hand [48] 5 Motors 23 N/A N/A N/A

T. Nuchkrua et al. [49] 3 PAMs 3 N/A N/A N/A

TU Bionic Hand [50] 5 Motors 15 PID 0.208 N/A

DEXMART Hand [51] 5 Motors 20 NN 0.25a 25%

ad

Ambidextrous Hand [52] 5 PAMs 13 PID 0.20 8% a Estimations made from curves, pictures or videos of the robot hands

b Experiments do not concern grasping but contact tasks

c Only for grasping

d For a target of 1 N ± 10%, whereas the error can exceed 30% for lower force

targets

popular in robotics due to its shooting functions that are not smooth [53] and need

regularisation [54]. Fuzzy logic [55], [56] and PID are usually combined with

bang-bang controller to add flexibility and determine switching time of Bang-bang

inputs [37]. Bram Vanderborght et al. in [57] controlled a bipedal robot actuated

by pleated pneumatic artificial muscles using bang-bang pressure controller. Ste-

phen M Cain et al. in [58], studied locomotor adaptation to powered ankle-foot or-

thoses using two different orthosis control methods. Dongjun Shin et al. in [59],

proposed the concept of hybrid actuation for human friendly robot systems by us-

ing distributed macro-Mini control approach [60]. The hybrid actuation controller

employ modified bang-bang controller to adjust the flow direction of pressure reg-

ulator. Zhang, Jia-Fan et al. in [61], presented a novel pneumatic muscle based ac-

tuator for wearable elbow exoskeleton. Hybrid fuzzy controller composed of bang-

bang controller and fuzzy controller is used for torque control.

3.1.1 Bang-bang control adapted to the Ambidextrous Hand

The bang-bang controller allows all fingers to grasp the object without taking any

temporal parameters into account. A block diagram of Bang-bang controller is

shown in Fig.5. Execution of algorithm automatically stops when the target set

against force is achieved. The controller also looks after any overshooting issue if

it may arise. To compensate the absence of backward control, a further condition is

implemented in addition to initial requirements. Since the force applied by the four

fingers is controlled with less accuracy than with PID loops, the thumb must offset

the possible excess of force to balance the grasping of the object. Therefore, a bal-

ancing equation is defined as:

𝐹𝑡𝑚𝑖𝑛 = 𝑊𝑜 + ∑ 𝐹𝑓

4

𝑓=1

(5)

Fig.5. Bang-bang loops driven by proportional controllers.

3.1.2 Results obtained with bang-bang control

The phalanges must close with appropriate speed’s ratios to tighten around objects when interacting with object. Results obtained from experiment using the bang-bang controller are provided in Fig.6. This time, it is noticeable that the can of soft drink deforms itself when it is grasped on the left hand side. The graph obtained from the data collection of the left mode is provided in Fig. 6.

Fig.6. Graph shows force against time of the four fingers while grasping a drink can with bang-

bang controllers. The ambidextrous robot hand holding a can with a grasping movement imple-

mented with bang-bang controllers. It is seen the can has deformed itself.

It is noted during experiment that speed of fingers does not vary when it touches the objects. That explains why we got higher curves in Fig.6 than the previously obtained in Fig.4. This makes bang-bang controller faster than PID loops. The bang-bang controllers also stop when the value of 1 N is overreached but, without predicting the approach to the setpoint, the process variables have huge over-shoots. The overshoot is mainly visible for the middle finger, which overreaches the setpoint by more than 50%. Even though backward control is not implemented in the bang-bang controller, it is seen the force applied by some fingers decreases after 0.2 sec. This is due to the deformation of the can, which reduces the force ap-plied on the fingers. It is also noticed that the force applied by some fingers in-crease after 0.25 sec, whereas the force was decreasing between 0.2 and 0.25 sec. This is due to thumb’s adduction that varies from 7.45 to 15.30 N from 0.1 to 0.25 sec. Even though the fingers do not tighten anymore around the object at this point, the adduction of the thumb applies an opposite force that increases the forc-es collected by the sensors. The increase is mainly visible for the forefinger and the middle finger, which are the closest ones from the thumb. Contrary to PID loops, it is seen in Fig.6 that the force applied by some fingers may not change be-tween 0.15 and 0.2 sec, which indicates the grasping stability is reached faster with

0.9

1

1.1

1.2

1.3

1.4

1.5

0.1 0.15 0.2 0.25 0.3

Forc

e (N

)

Time (sec)

Forefinger

Middle finger

Ring finger

Little finger

bang-bang controllers. The bang-bang controllers can consequently be applied for heavy objects, changing the setpoint of 𝐹𝑓 defined in (5).

3.1.3 Comparison with other bang-bang controls

The features related to the bang-bang control implemented on the ambidextrous

hands are compared to others in Table 3. For example, the bipedal walking robot

discussed in [57] implemented bang-bang controller on robotic structure driven by

pneumatic artificial muscles. and hand designed by Z. Xu et al. [62] is actuated by

air cylinders instead of PAMs, its architecture is much closer to the one of the

Ambidextrous Hand. Table 3 shows that bang-bang control is not usually imple-

mented on complex structures, as the ambidextrous hand and the two legs dis-

cussed in [63] are the only architectures that exceed ten actuators. The ambidex-

trous hand is also the only robotic structure in Table 3 that has more than ten

DOFs. The execution times are not very significant, given that the aims of the

bang-bang controls are totally different from one project to other; it is nevertheless

observed that the execution times never exceed 0.4 sec, as bang-bang controls aim

at making a system switch from one state to another as fast as possible. The system

proves its efficiency for the walking robot introduced in [57], but provides a huge

overshoot of 53% when it is implemented on the Ambidextrous Hand. Conse-

quently, despite the originality of the bang-bang control to grab objects, its imple-

mentation on an ambidextrous device and its grasping time of 0.15 sec (25%

shorter than the one obtained with the PID control), the bang-bang control is not

accurate enough to control the fingertips’ force of the Ambidextrous Hand.

Ban

g-b

ang

co

n-

trol

Ro

boti

c st

ruct

ure

Ty

pe

of

actu

ators

Aim

of

the

ban

g-

ban

g c

ontr

ol

Co

ntr

oll

er u

sed

as a

n o

ute

r lo

op

# D

OF

s

# a

ctu

ators

Ex

ecu

tion

tim

e

(sec

)

Err

or

max

. (%

)

Am

bid

exte

rity

R. Van Ham et

al. [57]

Modular

part of a

leg

PAMs Pressure control PID N/

A 2 0.2

a N/A

B. Vanderbor-

ght et al. [64] Two legs PAMs

Generate a joint tra-

jectory PI 3 6 0.4

a

4% for pressurea

0.27% for anglea

B. Vanderbor-

ght et al. [65] Two legs PAMs

Generate a joint tra-

jectory PID 3 6 0.2

a

3% for pressure

4.5% for anglea

B. Vanderbor-

ght et al. [63] Two legs PAMs

Generate a joint tra-

jectory Delta-p 6 12 0.15

a

17% for pressurea

17.5% for posi-

tiona

Z. Xu et al.

[62]

Index of a

hand

Air cylin-

ders

Evaluation of speed

capabilities None 2

b 4

b 0.33 N/A

Ambidextrous

Hand [37] Hand PAMs Force control Proportional 9

b 14

b 0.15

53% for

the force

a Estimations are made from curves b These numbers do not correspond with the actuators of the robotic structure but with the ones used for the bang-bang control

Table 3. Comparison of bang-bang controls’ characteristics between the Ambidextrous Hand and

other robotic models.

4.1 Backstepping Controller

Backstepping was offered as a mean to offer stabilise control with a recursive

structure based on derivative control and it was limited to nonlinear dynamical

systems. Mohamed et al. presented a synthesis of a nonlinear controller to an elec-

tro pneumatic system in [66]. First, the nonlinear model of the electro pneumatic

system was presented. It was transformed to be a nonlinear affine model and a co-

ordinate transformation was then made possible by the implementation of the non-

linear controller. Two kinds of nonlinear control laws were developed to track the

desired position and desired pressure. Experimental results were also presented

and discussed. P. Carbonell et al. compared two techniques namely sliding mode

control (SMC) and Backstepping control (BSC) in [67] and found out BSC is

somewhere better than SMC in controlling a device. He further applied BSC cou-

pled with fuzzy logic in [68]. In [69], a paralleled robot project is discussed which

is controlled by BSC. Since literature review revealed no multifinger robot hand

(actuated by PAMs) is ever driven using BSC, research presented in this chapter

validates the possibility of driving multifinger robot hand using BSC.

4.1.1 BSC adapted to the Ambidextrous Hand

BSC consists in comparing the system’s evolution to stabilizing functions. Deriva-tive control is recursively applied until the fingers reach the conditions implement-ed in the control loops. First, the tracking error 𝑒1(𝑡) of the BS approach is defined as:

[𝑒1(𝑡)

�̇�1(𝑡)] = [

𝐹𝑡 − 𝐹𝑓(𝑡)

−�̇�𝑓(𝑡)] (6)

then evaluated using a first Lyapunov function defined as:

𝑉1(𝑒1) = 1

2𝑒1

2(𝑡) < 𝐹𝑔𝑚𝑖𝑛 (7)

�̇�1(𝑒1) = 𝑒1(𝑡) ∗ �̇�1(𝑡) = −�̇�𝑓(𝑡) ∗ 𝑒1(𝑡) (8)

The force provided by the hand is assumed not being strong enough as long as 𝑒1

2/2 exceeds a minimum grasping force defined as 𝐹𝑔𝑚𝑖𝑛 . In (8), it is noted

that �̇�1(𝑡) ≠ 0 as long as 𝐹𝑓(𝑡) keeps varying. Therefore, �̇�1(𝑒1) cannot be stabi-

lized until the system stops moving. Thus, a stabilizing function is introduced. This stabilizing function is noted as a second error 𝑒2(𝑡):

𝑒2(𝑡) = 𝑘 ∗ �̇�1(𝑡) (9)

Fig.7. Basic Backstepping controller is shown.

with 𝑘 a constant > 1. 𝑒2(𝑡) indirectly depends on the speed, as the system cannot stabilized itself as long as the speed is varying. Consequently, both the speed of the system and 𝑒2(𝑡) are equal to zero when one finger reaches a stable position, even if 𝐹𝑓(𝑡) ≠ 𝐹𝑡. 𝑘 aims at increasing �̇�1(𝑡) to anticipate the kinematic moment

when �̇�1(𝑡) becomes too low. In that case, the BSC must stop running as 𝐹𝑓(𝑡) is

close to 𝐹𝑡. Both of the errors are considered in a second Lyapunov function:

𝑉2(𝑒1, 𝑒2) = 1

2(𝑒1

2(𝑡) + 𝑒22(𝑡)) < 𝐹𝑠 (10)

�̇�2(𝑒1, 𝑒2) = 𝑒1(𝑡) ∗ �̇�1(𝑡) + 𝑒2(𝑡) ∗ �̇�2(𝑡) (11)

where 𝐹𝑠 refers to a stable force applied on the object. This second step allows sta-bilizing the system using derivative control. Using (9), (11) can be simplified as:

�̇�2(𝑒1, 𝑒2) = �̇�1(𝑡) ∗ (𝑒1(𝑡) + 𝑘�̈�1(𝑡)) (12)

The block diagram of backstepping process is illustrated in Fig.7. According to the force feedback 𝐹𝑓(𝑡), the fingers’ positions adapt themselves until the condi-

tions of the Lyapunov functions (𝑉1,�̇�1) and (𝑉2,�̇�2) are reached.

4.1.2 Results obtained with the BSC

The grasping features and the force against time graph is shown in Fig.8. During the experiment, it was noticed that speed of finger tightening using backstepping control was much slower compared to PID controller and bang-bang controller. Since the backstepping controller’s target is based on force feedback and speed stability, it offers greater flexibility than PID and bang-bang but takes longer to stablise. Finger provided enough force to grab the can of soft drink at 0.30 sec, but it is seen the system carries on moving until 0.40 sec. Therefore it was deduced that BSC takes longer to stablise. The force collected for the thumb at the end of the experiment is 13.10 N, which is a value close to the one obtained with the PID control. It can also be noted that the fingers’ speed is slower using BSC, as none of

the sensors collect more than 0.80 N after 0.15 sec. The higher speeds of the PID and bang-bang controllers are respectively explained because of the integrative term and the lack of derivative control.

Fig.8. Graph shows force against time of the four fingers while ambidextrous hand grasping a

can of soft drinks using backstepping controllers.

4.1.3 Comparison with other BSCs

Backstepping control implemented on various hands is compared with the ambi-

dextrous hand in Table 4. Some robotic structures actuated by motors are also in-

cluded in the table as their number of DOFs is closer to the one of the ambidex-

trous hand and some of their BSCs are related to force control. BSCs are usually

implemented on structures with less than five DOFs, such as manipulators, arms

or parallel robots. The ambidextrous hand is the only robotic structure in table 4

that has more than ten DOFs (BSC is only implemented on nine of them). It is

quite clear that the number of DOFs does not exceed two when the BSCs are im-

plemented on structures driven by PAMs. The BSC of the ambidextrous hand has

an execution time of about 0.37 sec, which is one of the shortest of Table 4, with

[70]and [71] that have execution times estimated to 0.3 sec, as well as [72] with

an execution time of 0.45 sec. However, the maximum error of 4% for the ambi-

dextrous hand’s BSC is much higher than the ones obtained with the practical re-

sults introduced in [68], [72] and [73]. The control algorithms introduced in [72]

appears to be ones of the most efficient, as the arm has 6 DOFs and the BSC is

both among the fastest and the most accurate. Nevertheless, the BSC of the ambi-

dextrous hand is the only one that is implemented on an ambidextrous structure

and which is used to grab objects. In conclusion, the implementation of the BSC

0.6

0.7

0.8

0.9

1

1.1

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Forc

e (N

)

Time (s)

Forefinger

Middlefinger

Ringfinger

Littlefinger

on the ambidextrous hand is quite successful, as the obtained results are among

the best in Table 4. Nevertheless, its grasping time of 0.37 sec is 85% slower than

the grasping time obtained with PID control in, and also much slower than the

grasping times of most of robot hands summarized in Table 2. Consequently, the

BSC is not the best option to grab objects with the Ambidextrous Hand.

BS

C

Robo

tic

stru

ctu

re

Typ

e o

f

actu

ato

rs

Aim

of

the

BS

C

Alg

ori

thm

s

to w

hic

h

it i

s

com

bin

ed

# D

OF

s

# a

ctu

ato

rs

Ex

ecuti

on

tim

e (s

ec)

Err

or

max

.

(in

%)

Am

bid

ex-

teri

ty

C.-Y Su and Y. Ste-

panenko [74] Manipulator Motors

Trajectory

control None N/A N/A 0.3a 10%a

P. Carbonell et al. [67] N/A, single

PAM PAM

PAM’s

length control

None N/A 1 2.4a 8%a

P. Carbonell et al. [68] N/A, single

PAM PAM

PAM’s

length control

Fuzzy logic N/A 1 7a 1%a

D. Nganya-Kouya et al. [75]

Manipulator N/A

Force and

position

control

None 4 N/A 9a N/A

Lotfazar et al. A. [76] Manipulator Motors Trajectory

control None 5 5 2a N/A

S.-H. Wen S.-H. Wen

and B. Mao. [77] Manipulators N/A

Force and

position control

NN N/A N/A 1.9ab 9%ab

H. Aschemann and D.

Schindele [71]

Parallel ro-

bot PAMs

Position

control None 2 4 0.3a 4%a

M.R. Soltanpour and M.M. Fateh [78]

Manipulator Motors Trajectory control

SMC 2 N/A 7.6ab <10-3%b

X. Liu and A. Liadis [79] Parallel ro-

bot Motors

Position

control Fuzzy logic 2 2 N/A N/A

L. Qin et al. [72] Arm N/A Trajectory

control SMC 6 N/A 0.45a 2%a

H. Aschemann and D.

Schindele [73] Axis PAMs

Position

control None 1 2 4a 1.4%a

Ambidextrous Hand,

2014 [80] Hand PAMs

Force control

None 9c 14c 0.37 4%

a Estimations are made from curve

b

Results are obtained through a simulation c Four DOFs and four actuators are unused for the BSC

Table 4. Comparison of BSCs’ characteristics between the Ambidextrous Hand and other robot-

ic mode

CONCLUSION

PID, Bang-bang and BSC are tested and their performance in controlling an ambi-dextrous robotic hand is analyzed. PID controller was found the best when applied as compare to Bang-bang and Backstepping control. Backstepping control tech-

nique was validated for the first time on an ambidextrous robot hand. By combining PID controllers and force sensors, this research proposes one of the cheapest solu-tions possible. In future, PID controller could be used with artificial intelligent con-trollers to further improve the controls.

ACKNOWLEDGMENT

The authors would like to cordially thank Anthony Huynh, Luke Steele, Michal

Simko, Luke Kavanagh and Alisdair Nimmo for their contributions in design of

the mechanical structure of a hand, and without whom the research introduced in

this paper would not have been possible.

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