Mechatronics 37 (2016) 63–78

Contents lists available at ScienceDirect

Mechatronics

journal homepage: www.elsevier.com/locate/mechatronics

Prototype, control system architecture and controlling of the hexapod

legs with nonlinear stick-slip vibrations

Dariusz Grzelczyk

∗, Bartosz Sta ́nczyk , Jan Awrejcewicz

Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski Str., 90-924 Lodz, Poland

a r t i c l e i n f o

Article history:

Received 31 August 2015

Revised 31 December 2015

Accepted 6 January 2016

Available online 27 January 2016

Keywords:

Legged locomotion

Multi-legged robot

Hexapod robot

Inverse kinematics

Central pattern generator

Stick-slip vibrations

a b s t r a c t

The paper introduces the constructed prototype of the hexapod robot designed based on the biomechan-

ics of insects for inspection and operation applications as well as for different research investigations

related to the walking robots. A detailed discussion on the design and realization of mechanical con-

struction, electronic control system and devices installed on the robot body are presented. Moreover, the

control problem of the robot legs is studied in detail. In order to find the relationship between move-

ments commonly used by insects legs and stable trajectories of mechanical systems, first we analyze

different previous papers and leg movements of real insects. Next, we are focus on the control the robot

leg with several oscillators working as a so-called Central Pattern Generator (CPG) and we propose other

model of CPG based on the oscillator describing stick-slip induced vibrations. Some advantages of the pro-

posed model are presented and compared with other previous applied mechanical oscillators with help

of numerical simulations performed for both single robot leg and the whole robot. In order to confirm

the mentioned numerical simulations, the conducted real experiments are described and some interest-

ing results are reported. Both numerical and experimental results indicate some analogies between the

characteristics of the simulated walking robot and animals met in nature as well as the benefits of the

proposed stick-slip vibrations as a CPG are outlined. Our research work has been preceded by a biological

inspiration, scientific literature review devoted to the six-legged insects met in nature as well as various

prototypes and methods of control hexapod robots which can be found in engineering applications.

© 2016 Elsevier Ltd. All rights reserved.

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. Introduction

Legged locomotion is very popular in nature and lots of animal

pecies use this way for traveling on Earth. Probably for this rea-

on also multi-legged walking robots (inspired by walking animals)

elong to the important group of mobile robots found in engineer-

ng applications [1,2] . Walking robots are good mobile machines

apable of traversing over irregular and uneven terrain, includ-

ng obstacles and gaps, while providing many degrees-of-freedom

DOF) if manipulation tools are required [3] . It should be em-

hasized that though multi-legged walking robots require addi-

ional effort to control their locomotion, they (similar like crawl-

ng robots) can go, where the wheeled ones cannot. From a view-

oint of application this is why different types of a biologically

nspired multi-legged robots are required in engineering and can

e used for exploration of the highly broken and unstable land-

capes [4] . There are a lot of examples of biological inspirations

nd constructed robots in the scientific literature, and interesting

∗ Corresponding author. Tel.: +48 426312225; fax: +48 426312489.

E-mail addresses: dariusz.grzelczyk@p.lodz.pl (D. Grzelczyk), bartosz.stanczyk

dokt.p.lodz.pl (B. Sta ́nczyk), jan.awrejcewicz@p.lodz.pl (J. Awrejcewicz).

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ttp://dx.doi.org/10.1016/j.mechatronics.2016.01.003

957-4158/© 2016 Elsevier Ltd. All rights reserved.

nd compact state of the art in this area can be found, for instance,

n one of the recent paper [5] . Based on the latter reference and

any others cited therein important papers a brief summary de-

oted to biological paragon from nature, methods of investigations

f locomotion, as well as the constructed multi-legged robots, is

resented.

The first gait studies were based on the observation of animals

n nature. In 1899 Muybridge used 24 cameras for studying the

otion patterns of the running horse [6] , and at present the cited

aper is treated as classical one in the study of walking gaits [5] .

ince then, a number of experimental observations of other ani-

als (including reptiles, amphibians and insects) were conducted,

nd on the basis of the observation was attempted to use their

ovement in different walking machines. For instance, insects and

piders are relatively simple creatures, which are able to success-

ully operate by using many legs at once in order to navigate a

iversity of terrains and they served as an inspiration for numer-

us researches [7–12] . Studies conducted in papers [13,14] were in-

pired by the movements of the cockroaches. Paper [15] presents

oth simulation studies and physical results obtained on the im-

lementation of a model of praying mantis behavior on a robotic

exapod equipped with a real time color vision system. In turn,

64 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

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for instance in papers [5,16,17] the authors used crab gaits as a

biological inspiration for the constructed robots. Despite the pas-

sage of many years since the first observations made by Muybridge

this kind of experimental research is still used by contemporary

researchers, for instance in the mentioned earlier paper [15] . Us-

ing video camera also in one of the recent paper [5] the motion

features of the biologic crab are studied, including both transfer

and support phases, as well as transition between them. It should

be noted that in addition to experimental studies simultaneously

mathematical methods describing the legged locomotion system

were developed. First, a good mathematical foundation in this area

was presented by the authors of the paper [18] . McGhee [19] pro-

posed (new, as for those times) mathematical description of gaits

and demonstrated an enormous amount of possible gaits of multi-

legged animals (the so-called McGhee formula). Since that time,

still appear different and offered by many researchers mathemati-

cal models for the testing of the multi-legged robots, both in terms

of construction and control.

There are a lot of different types and prototypes of legged

robots which can be found in the literature, namely: bipeds,

tripods, quadrupeds, hexapod, or octopod [3] . An interesting liter-

ature review of the most important and the most known multi-

legged robots can be found in [5] . Studies of multi-legged robots

have been initiated in 1960s by McGhee and Frank [20] , who con-

structed the first four-legged robot named “Phoney Pony”. Since

that time a wide variety of four-legged (quadruped) robots have

been built, including: TITAN-VIII [21] —the robot driven by wire-

pulley system, BISAM [22] —the self-adaptive quadruped robot,

SIL04 [23] —the robot driven by DC servomotor with torus worm

gear, Tekken [24] —the robot controlled by the system including

CPG and reflection mechanism, ARAMIES [25] —the robot for work-

ing in non-structural environment, and BigDog [26] —the robot de-

signed for transport of military materials. In this paper we present

only the most interesting and well-known prototypes of the six-

legged robots, which are still very popular and most widely stud-

ied. First, an anti-torpedo, amphibian multi-legged crablike robot

called ALUV has been constructed in 1990s [27] . Other constructed

hexapod robot, which can walk like crab and also possesses the

ability of anti-overturn, is the robot called Ariel [28] . Next, impor-

tant hexapod robot inspired by crab and called Lemur was mod-

eled by imitating body similar to the body of octopus with crab

legs [16] . In order to detection and removal of the landmines, the

six-legged walking robot called COMET-III that ensures stable walk-

ing in the mine field has been constructed [29,30] . Later, a mili-

tary six-legged robot named as SILO-6 primarily for terrain mine-

clearing has been built [31,32] . In turn, in order to explore un-

known celestial body in the outer space, different prototypes of

the hexapod robots were also constructed, namely Genghis [33] ,

Hannibal [34] and Attila [35] . Yet another interesting type of six-

legged robot is the RHex robot—a biologically inspired hexapod

runner that travels better than one body length per second and

uses a clock excited alternating tripod gait to walk and run in a

highly maneuverable and robust manner [4] . In turn, SensoRHex is

a modified and functionally improved version of the original RHex

robot, which possesses six half circular rotary compliant legs, and

each of them is actuated by a gearbox DC motor controlled in a

position or current (torque) mode [36] . The paper [37] introduces

the manufactured by the authors hexapod robot called ROBOTURK

SA-3, which can be used as both the quadrupedal and hexapod

robot. In one of the recent paper [38] a novel hexapod robot called

HITCR-II, characterized by high-integration and control with multi-

sensors and suitable for walking on unstructured terrain, is pro-

posed. Agheli et al. [3] introduce SHeRo, a scalable hexapod robot

designed for maintenance, repair and operations within remote,

inaccessible, irregular and hazardous environments. On the other

hand, a novel Abigaille-III hexapod robot powered by 24 miniature

ear motors, which uses dual-layer dry adhesives to climb smooth

nd vertical surfaces, is presented by Henrey et al. [39] .

Recently, also eight-legged robots have become popular, and as

xamples we can list [5] : a biomimetic eight-legged robot SCOR-

ION [40] and searching and rescuing robot Halluc II [41] . The

CORPION robot consists of three main body parts and eight ho-

ogenous legs, and the joints of legs are actuated by standard DC

otors with high gear transmission ratio for sufficient lifting ca-

acity. The control system of the mentioned robot combines the

PG and the reflex. In turn, Halluc II is a robotic vehicle with eight

heels and legs designed to drive or walk over rugged terrain,

hich is also provided with wireless network capabilities and a

ystem of cameras and sensors that monitor the distance to poten-

ial obstacles. In result, the robot constantly assesses how best to

djust the position of its legs and wheels.

Though the mentioned eight-legged robots become very pop-

lar and each of them certainly possesses interesting original fea-

ures, however, it should be noted that eight-legged robots are usu-

lly studied based on six-legged robots to imitate some specific

nimals [5] . Hexapod robots due to their simplicity, static balance

reater than in case of four-legged robots, with various configura-

ions and leg designs, have engaged a number of researchers. Six-

egged robots imitating insects make their movements using six

egs and according to the McGhee formula [19] have an especially

reat spectrum of different types of gaits. However, it is difficult

o define and describe all gaits that insects or hexapod robots can

se. In all cases of the mentioned gaits some of the legs are per-

orming swing movements in the air, while the rest are support-

ng and propelling forward the body on the ground. Generally, six-

egged robots have superior walking performances in comparison

ith those having fewer legs, especially in terms of larger statical

nd dynamical stability, greater walking speeds or lower control

ethod complexity (a control of their legs still does not belong to

asy tasks). Since for keeping the stability of the robot only three

egs are sufficient, hexapod possesses the great flexibility in walk-

ng. To maintain a balance of the robot only three legs are enough,

ut to perform a movement four legs are required. Thus, in case of

ailure of two extremities, hexapod can still continue his motion.

robably this is why over the last 30 years an extensive research

as been conducted in this field, and different prototypes of hexa-

od walking robots have been constructed and investigated. Finally,

t should also be noted that analysis of gait algorithms used in six-

egged robots is similar to that regarding the case of four-legged

nes. Moreover, the mentioned gait analysis can be relatively sim-

ly extended and applied to eight-legged robots.

On the basis of presented above brief literature review, one may

an conclude that analysis of walking robots, especially hexapod

obots, still belong to challenging tasks of many researchers. Walk-

ng robots have a greater ability to adapt to various kinds of ter-

ains in comparison with wheeled or tracked ones [42] . New con-

tructions of walking robots still arise, and to control their move-

ent still CPGs are used. In addition, currently built structures are

sually equipped with various types of measuring or actuating de-

ices which expand their application possibilities. In addition to

he operational capacity of the constructed prototypes an impor-

ant issue concerns the energy consumption of these systems. Min-

mization of the mentioned energy consumption extends time of

he robot work, which often plays a key role to carry out the re-

lized mission. This paper issues and research have been inspired

y both current trends in mechatronics and the mentioned above

equirements for modern robots. As a result, we construct own

rototype of the hexapod robot, which can be used both for ex-

erimental studies as well as for inspection and operational ap-

lications. The robot contains many additional devices installed on

ts body and a system of wireless data transmission. In this pa-

er we are focused on the general presentation of the mechanical

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 65

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Fig. 1. A scheme of a typical stick insect leg with four functional segments: Coxa

(Cx), Femur (Fe), Tibia (Ti) and Tarsus (Ta).

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tructure and the control system of the constructed robot. More-

ver, we propose a new model of CPG and finally we compare it

ith other previously used CPGs. The advantages of the proposed

PG model are shown based on both simulation and experimental

tudies. Since the legged robots are the most commonly used and

onstructed walking machines, the proposed methods and results

an be adapted to a wide group of robots reported in the literature.

The rest of the paper is organized as follows. In Section 2 a

cheme of morphology of a typical stick insect leg is presented

nd described. Next, a kinematic model and Computer Aided De-

ign (CAD) of the three-joint leg mechanism of a robot leg are il-

ustrated and discussed. In Section 3 the CAD project, mechanical

onstruction, electronic control system and devices installed on the

obot body are presented. Section 4 is focused on the mathemati-

al modeling of the robot leg mechanism, including direct and in-

erse kinematics rules. Section 5 includes a brief introduction to

he CPGs. Three typical oscillators are considered, a new oscillator

orking as a CPG is proposed and some aspects of robot gaits are

iscussed. Numerical investigations of a single robot leg and the

hole robot are conducted in Section 6 . The obtained numerical

imulations are experimentally verified in Section 7 . In addition,

ther experimental results are presented and discussed in there.

inally, conclusions of our studies are given in the last Section 8 .

inally, we acknowledge that some ideas adopted in this paper are

otivated by the discussed references or have been shortly pre-

ented in our previous papers [43–45] .

. Biological inspiration of the hexapod robot and mechanism

f its legs

In this section, as a biological inspiration for the constructed

rototype of the hexapod robot, a scheme of morphology of a typ-

cal stick insect leg is presented and described in detail. Next, a

inematic model and CAD three-joint leg mechanism of the robot

eg is considered, as well as the definitions of the links and joints

re given.

.1. Biological inspiration

It is well known that the legs belong to the most challenging

arts of walking robots. During modeling of the hexapod robot legs

e have been inspired by a morphology of a typical stick insect

egs. Usually each insect leg is made up of five basic segments

onnected by joints, namely: Coxa, Trochanter, Femur, Tibia, and

arsus [12] . On the basis of this fact it can be concluded that the

ccurate representation of insect legs in the hexapod robots led to

he need to build mechanisms with 5-DOF. However, in the liter-

ture the mechanisms having 3-DOF modeling hexapod legs usu-

lly can be found, since the mentioned additional DOFs are rarely

sed in a typical walking process. Following the results reported

n other papers (for instance [9,46] ), similar schematic diagram of

morphology of a stick insect leg is taken into account ( Fig. 1 ).

oth structure and physiology of the considered insects are reason-

bly well known [14] . Coxa, Femur and Tibia lie approximately in

plane, further referred as a leg plane. In engineering prototypes

f the robots usually Tarsus segments are ignored.

TC describes the angle position of the Thorax–Coxa joint, CT

enotes the angle position of the Coxa–Trochanterofemur joint,

hereas FT is the angle position of the Femur–Tibia joint [9,46] .

C joint is responsible for forward and backward movements

Protractor–Retractor muscles), CT joint enables elevation and de-

ression of the leg (Levator–Depressor muscles), while the FT joint

nables extension and flexion of the Tibia (Flexor–Extensor mus-

les) [42] . The single stride cycle of the walking leg (the dashed

ines in Fig. 1 ) can be regarded as being in one of two functional

tates, either a swing movement (also referred as a return stroke)

r a stance movement (also referred as a power stroke). In the

wing movement the insect leg is lifted off the ground and moved

o a position where the next stance movement occurs. In turn, in

he stance movement the body of the insect is supported by the

egs and moved in the desired direction. During swing movement

nd stance movement the leg tip lies approximately in a plane.

he mentioned two phases of leg movement are mutually exclu-

ive, and it means that leg cannot be in swing state and in stance

tate at the same time. Moreover, the point PEP denotes the poste-

ior extreme position (the position at which the leg is lifted off the

round to start a swing movement), while the point AEP denotes

he anterior extreme position (the position, where the leg switches

rom swing to stance phase by touching the ground).

.2. Mechanism of the hexapod robot legs

Direct copying the five anatomical arrangement of insect leg in-

reases the mechanical construction of the leg and its control sys-

em, and this is why usually legs with only three segments (Coxa,

emur and Tibia) are developed in engineering applications (see

apers [12,14,40,42] and others). Also in this paper the additional

nd rarely used by insects DOFs are neglected, and therefore three-

oint leg mechanism of the constructed robot is considered. On the

asis of the schematic diagram of an insect leg presented above,

oth kinematic model and CAD view of the robot leg are consid-

red ( Fig. 2 ).

In Fig. 2 the angle ϕ 1 corresponds to the angular position of the

horax–Coxa joint (TC joint), the angle ϕ 2 corresponds to the an-

ular position of the Coxa–Trochanterofemur joint (CT joint), while

he angle ϕ 3 corresponds to the angular position of the Femur–

ibia joint (FT joint). In real insects the CT- and FT-joints belong

o simple hinge joints (1-DOF), while the TC-joint connected the

eg to the body is more complex. However, also its movement

an be modeled by rotation around a slanted axis [9] and this is

hy in the constructed robot we use the same servomotors in

ll joints of its legs. The lengths of the three straight line links

which corresponds to the Coxa, Femur and Tibia) are denoted by

1 , l 2 and l 3 , respectively, and changes in all leg stiffness are ne-

lected. In the considered construction the links l 2 and l 3 are the

ongest ones, since usually Femur and Tibia are the longest leg

egments of insects living in nature and such construction is op-

imal to overcome obstacles [12] . In the constructed prototype of

he robot l 1 = 27 mm, l 2 = 70 mm and l 3 =120 mm, whereas due

o real constraints the angle values change within the following

anges: ϕ 1 = 45 °… 135 °, ϕ 2 = −90 °… 90 ° and ϕ 3 = 0 °… 150 °. From

mechatronic point of view during walking of the robot each leg

an be modeled as a manipulator of three segments with lengths

, l and l connected through hinge joints with angles ϕ , ϕ

1 2 3 1 2

66 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

Fig. 2. The kinematic structure (a) and CAD design of a single robot leg (b).

Fig. 3. The project designed in CAD Inventor software (a) and the prototype of the constructed hexapod robot (b).

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and ϕ 3 . In result, the hexapod robot can be treated as vehicle that

walks on six independent legs.

3. Project and prototype of the hexapod robot

In this section the constructed hexapod robot is introduced.

First, CAD project and mechanical construction are presented. Sec-

ond, the main components and additional devices installed on the

robot body are described. Finally, the control system of the robot

is considered and discussed.

3.1. General presentation of the constructed hexapod robot

Because robot construction requires frequent design changes,

first we consider virtual model of the robot, which is well suited

for design optimization as well as for virtual experiments of its

motion. Fig. 3 shows the CAD project and general view of the pro-

totype of the constructed hexapod robot. The robot has originally

been designed for teaching purposes and to conduct research ex-

periments. However, in the future it may also be used to study its

ossibilities of inspection and operating applications, as well as for

urther research investigations. The size of the prototype is about

50 mm × 250 mm × 250 mm (length × width × height) and it has

n additional 80 mm ground clearance when standing. It weighs

early 4 kg and can moving at maximum speed 5 km/h.

The mechanical design of the robot focuses on two main com-

onents, i.e. the body and the legs. The robot body houses the

lectronic circuits, the installed equipment, the battery, and con-

ects all the legs together. The legs are appropriately distributed

o reduce the possibility of motion interference between legs and

o improve the walking stability. The robot has six identical legs

anufactured in aluminum alloy, and each leg is moved by three

lectric actuators. In result, the constructed robot is characterized

y a high stability, both during standing and walking events.

.2. Equipment of the robot

Fig. 4 shows the most important equipment (including sensors

nd actuators) and additional important electro-mechanical com-

onents installed and distributed on the robot body. Motivated

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 67

Fig. 4. The prototype of the constructed hexapod robot with equipment installed and distributed on the robot body.

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Section 5 .

lso by anatomical structure of real crab found in nature, we have

quipped the robot into two-jaw gripper, which movement is real-

zed via three actuators. Its function aims on the ability to perform

ifferent tasks related to inspection and operational applications.

he hexapod is equipped with the BTM-222 receiver–transmitter

llowing to send signals from distance up to 100 m. The applied

echnology yields communication with the robot via any device

ossessing a Bluetooth wireless network and the appropriate soft-

are. Technology of 18B20 temperature sensor allows for temper-

ture measurement of the surrounding environment with an ac-

uracy of 0.1 °C, and data from the ultrasonic measurement sys-

em HC-SR04 allow to estimate distances and to avoid obstacles. In

urn, the color camera with lighting and the microphone ANL-02-

W controlled via additional actuator send audio and video data

rom remote and inaccessible places. A choice of the ATmega162

icrocontroller has been motivated by a number of the generated

6-bit PWM (Pulse-Width Modulation) waves. The controller han-

les the high-level computation involving generation of the CPG

ignal and the inverse kinematic relations, handles user input data

nd sends the data read from the installed sensors via RS232 com-

unication, as well as can control independently up 32 servo mo-

ors in the same time. In order to keep the robot at an upright

osition and the need for continuous operation of all actuators

he robot requires a lot of battery power. However, despite using

he battery FlightPower EONX30, the developed prototype is able

o work only about one hour. The applied electric servo actuators

ower Pro Mg995 (DC motors with a high gear transmission ra-

io) are capable of providing a relatively high torque and have rela-

ively sufficient power to walk on rough terrain and carry a heavier

ayload.

.3. Control system of the robot

Walking of a hexapod robot is a complex task requiring the co-

rdination of all controlled legs on any type of walking surfaces

nd it does not belong to easy tasks [14,42] . In many biological

ystems of legged locomotion the number of DOF is larger than is

equired to perform the appropriate task. In result, the coordina-

ion of several legs during walking requires the control system to

elect one out of numerous alternative movements. The mentioned

ossibilities of different coordination of the robot legs (order of ad-

ustment of individual legs) generate different types of gaits. An-

ther issue is the single leg movement, particularly the trajectory

lotted by the leg tip. The control system architecture presented in

ig. 5 ensures control both choosing of the appropriate gait and the

hape of the trajectory plotted by the leg tips from the software

evel (however, this issue is not considered in detail). It consists

f a number of distinct modules which are responsible for solving

articular subtasks.

Data from the sensors installed on the robot are processed by

he microcontroller Atmega 162 and transmitted to the mobile

hone using wireless network (BTM-222 module). Similarly works

ommunication to the opposite direction, namely transmission of

ontrol signals (for instance different parameters of the robot gait,

ontrol signals for the gripper or orientation of the camera) from

he mobile phone (with the algorithm developed in Java-me) to the

icrocontroller.

In order to proper control of the robot motion it should be

uaranteed that the legs move along the desired trajectories as

losely as possible. Usually standard DC motors do not have suf-

ciently high accuracy and they require calibration in the con-

rol system. In this regard we use the mentioned actuators Tower

ro Mg995 possessing a servo feedback (the inner loop of posi-

ion control) and proportional controller, which based on the sig-

al reading from the sensor (potentiometer contained within each

ervo motor) provide measurements and controls of joint angles.

he control signals for the individual servos are absolute angu-

ar positions, to which the servos (in a given time depending on

alking speed) are moved as fast as possible. Control of the angu-

ar position of each servo is performed using the PWM technique

nd digital electronic circuits made in TTL technology, which al-

ow to supply intermediate amounts of power by varying the ra-

io of discretely switching on and off the power supply. The pro-

osed electronic control system is able to control the 32 servos

t the same time, however only 22 servos are used in the con-

tructed robot. Generation of the angular positions for the individ-

al servos installed in the robot legs using CPGs is discussed in

68 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

Fig. 5. The control system architecture of the constructed hexapod robot.

Fig. 6. The robot leg located in the body reference frame (a) and the workspace of the leg tip in the z − m leg plane.

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4. Kinematics of the hexapod leg

This section is focused on the mathematical modeling of the

mechanism depicted in Fig. 2 imitating a leg of a stick insect, in-

cluding direct and inverse kinematics. Legs of walking robot are

similar to industrial robotic manipulators and from dynamic and

mechatronic point of view can be treated as a multibody system

with 3-DOF [47] . This is why this approach is also used in this

paper.

4.1. Direct kinematics of the hexapod leg

Kinematic description of a multibody system is a recipe for

transformation describing the geometrical relationship between

the generalized coordinates q and coordinates x of the global base

coordinate system. As the global base coordinate system we take

the body reference frame which is considered as inertial reference

frame. Generally, the mentioned relation has the following non-

linear form

x = f (q ) , (1)

where q = [ ϕ 1 , ϕ 2 , ϕ 3 ] T and x = [ x, y, z] T . In mechatronic applica-

tions it is of particular importance to calculate the position of the

chosen point referred as the end-effector which plays a key role

for the realized mechatronic task. In the considered structure as

the end-effector we take the leg tip of the robot. The kinematic

scheme of the considered robot leg in the body reference frame

or arbitrary configuration is shown in Fig. 6 a. The direct kinemat-

cs can be determined by taking coordinates x, y and z (the leg tip

ositions in the body reference frame) as a function of the lengths

i and the angles ϕ i ( i = 1 , 2 , 3 ) in the form

x = cos ϕ 1 ( l 1 + l 2 cos ϕ 2 + l 3 cos ϕ 2 cos ϕ 3 + l 3 sin ϕ 2 sin ϕ 3 ) , y = sin ϕ 1 ( l 1 + l 2 cos ϕ 2 + l 3 cos ϕ 2 cos ϕ 3 + l 3 sin ϕ 2 sin ϕ 3 ) , z = l 2 sin ϕ 2 − l 3 cos ϕ 2 sin ϕ 3 + l 3 sin ϕ 2 cos ϕ 3 .

(2)

The workspace of the considered flat mechanism of the robot

eg (in the mechanism plane) is shown in Fig. 6 b. The presented

rea is bounded by the curves obtained after inserting of the ex-

reme angular positions of the appropriate joints and applying the

irect kinematics rules.

.2. Inverse kinematics of the hexapod leg

In the case of inverse kinematics, in order to obtain the cor-

ect configuration of the system, the appropriate generalized coor-

inates q are considered as a function of the global coordinates x .

n this way formally we obtain the inverse relationship

= f −1 (x ) , (3)

hich is usually strongly nonlinear, and analytically solutions can

e obtained only in special cases. Through the analyzing of the

cheme of the robot leg shown in Fig. 6 a it can be seen that the

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 69

Fig. 7. Projection of the considered robot leg in the leg plane z − m defined by the

mechanism of the leg.

a

r

ϕ

a

F

t

ϕ

c

α

o

n

N

t

p

i

t

ϕ

k

s

5

c

w

w

h

c

p

5

t

C

s

m

e

c

g

[

a

W

a

f

a

5

R

s

t

r

a

j

fi

i

t

t

i

t

a

t

i

a

b

p

j

n

[

b

r

t

[

o

a

i

t

r

a

m

e

t

t

t

5

c

ngle ϕ 1 is decoupled from angles ϕ 2 , ϕ 3 and it is governed by the

elations

1 =

⎧ ⎨

⎩

arctan

(y x

)if x > 0 ,

π2

if x = 0 ,

π − arctan

(y

−x

)if x < 0 .

(4)

In order to obtain relations for angles ϕ 2 and ϕ 3 we take into

ccount the mechanism of the robot leg lying in the leg plane (see

ig. 7 ).

The Pythagorean theorem for the right-angled triangle ACD and

he cosine theorem for the triangle ABC yield

3 = arccos

(c 2 − l 2 2 − l 2 3

2 l 2 l 3

), c 2 = z 2 +

(√

x 2 + y 2 − l 1

)2

. (5)

In order to get the relation for the angle ϕ 2 we employ the

osine theorem for the triangle ABC and we obtain

= arccos

(l 2 3 − l 2 2 − c 2

−2 l 2 c

). (6)

Due to real constraints of the leg mechanism and the definition

f the angle ϕ 3 the value of the mentioned angle ϕ 3 cannot be

egative. Accordingly, the angle α will always take positive values.

ext, taking the right-angled triangle ACE, we have

an (−β) =

−z √

x 2 + y 2 − l 1 ⇒ β = arctan

(

z √

x 2 + y 2 − l 1

)

. (7)

It should also be observed that values of angle β can be both

ositive or negative. If √

x 2 + y 2 − l 1 ≥ 0 , then ϕ 2 = α + β . In turn,

f √

x 2 + y 2 − l 1 < 0 , then ϕ 2 = α − (π − β) . Finally, the relation for

he angle ϕ 2 takes the following form

2 =

{α + β if

√

x 2 + y 2 − l 1 ≥ 0 ,

α − (π − β) if √

x 2 + y 2 − l 1 < 0 . . (8)

The obtained in this section relationships for direct and inverse

inematics are used in both numerical simulations and real control

ystem of the robot.

. Methods of hexapod control based on the CPGs

This section provides a brief introduction to the CPGs which can

ontrol the movement of legs of walking robots. Three typical and

ell known oscillators together with the proposed new oscillator

orking as a CPG are considered in detail. Finally, typical gait of

exapod robot chosen for the further studies and the method of

onversion of the CPG signal into joint space of the robot legs are

resented.

.1. Literature review of CPGs

The control approach of the walking robots usually combines

wo biological control principles, namely the CPG and the reflex.

PG is able to produce rhythmic motion without the need for sen-

ory feedback, while reflexes are realized based on the sensor-

otor-feedback [40] . Wave gaits generated by CPG method are

volutionarily proven to be successful for legged locomotion, be-

ause the movement characteristics of real insects is easy to be

enerated by CPG and they are also frequently observed in nature

47] . Reflex is generated based on a sensor in a feedback, and as

n example of a robot using reflex systems only, the robot named

alknet [7] can be given, which local sensory feedback at the legs

nd coupling between the control of neighbored legs are sufficient

or a stable walk.

A lot of scientific papers is devoted to the method of gait gener-

tion which includes the method based on CPGs (see papers [48–

0] ). The first CPG model was proposed by Cohen, Holmes and

and in 1980s through the study on the dissection of a lamprey

pinal cord [51] . Since then, many researchers have been applying

he CPG algorithms to control various bio-inspired prototypes of

obots. Not the best, but the simplest method is to calculate joint

ngles for the leg on AEP and on PEP points and then to change all

oint angles proportionally and simultaneously from their initial to

nal values [52] . This is called the mass-spring model [53] which,

n general, gives curvilinear trajectory of the leg tip. Next, oscilla-

ions of numerous linear/nonlinear oscillators with stable orbits in

he phase space have been usually used and applied as a CPGs. For

nstance, in [54] various oscillators and similarities/differences be-

ween them are discussed. In other paper [55] several oscillators

re coupled together to construct the CPG model and, as a result,

he robot controlled in this way can walk on the land and swim

n the water. In [42] CPG is constructed by isochronous oscillators

nd several first-order low-pass filters. Various gaits are obtained

y changing phase shift between the signals which control the ap-

ropriate robot legs. In order to predefine the robot leg swing tra-

ectory, also fixed curves or composite curves (polynomials and si-

usoids) have been utilized by some researchers [40,56] . The paper

49] shows that hexapod robot can perform various types of gait

y chaos control, however the proposed model is difficult to be di-

ectly applied. A brief state-of-the-art devoted to the implementa-

ion of CPG algorithms for control hexapod motion is presented in

42] or in the review paper [57] . It should be also noted that vari-

us types of oscillators used as CPGs give various leg movements,

nd transitions between them should have a smooth character. For

nstance, this problem is considered in [42] .

Our viewpoint is similar to the one presented by authors of

he mentioned paper [42] . Our assumptions are as follows: (i) the

obot will not be totally identical to real insects, especially in the

ctuators; (ii) the robot is usually driven by electric or hydraulic

otors, while real insects are driven by muscles which have higher

nergy efficiency. This is why we also use a relatively simple con-

rol system based on the CPG method. However, on the contrary to

he previous paper, we propose other CPG to control the joints of

he robot legs.

.2. Three typical oscillators as a CPGs

We consider three popular and well known nonlinear os-

illators working as a CPGs: Hopf oscillator, van der Pol

70 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

a) Hopf oscillator b) van der Pol oscillator c) Rayleigh oscillator

-15

-10

-5

0

5

10

15

-4 -3 -2 -1 0 1 2 3 4

Z

X-30

-20

-10

0

10

20

30

-4 -3 -2 -1 0 1 2 3 4

Z

X-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

Z

X

Fig. 8. Phase trajectories of the three typical nonlinear oscillators for ω = 2 , μ = 6 and various initial conditions. (For interpretation of the references to colour in the text,

the reader is referred to the web version of this article.)

a b

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Z

X-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Z

X

Fig. 9. Stable phase trajectories of the proposed CPG model for different initial conditions: for v dr = 0 . 5 (a) and for v dr = 0 . 8 (b). (For interpretation of the references to

colour in the text, the reader is referred to the web version of this article.)

5

o

c

o

p

o

t

a

p

o{

w

F

d

v

δ

m

t

s

w

m

s

i

t

v

t

c

l

f

oscillator and Rayleigh oscillator. Ordinary differential equations

(ODEs) describing the mentioned oscillators are presented in the

non-dimensional form. A simple, isochronous Hopf oscillator is

governed by the following nonlinear first order ODEs {˙ X = (μ − X

2 − Z 2 ) X + ωZ,

˙ Z = (μ − X

2 − Z 2 ) Z − ωX. (9)

Next, van der Pol oscillator is described by the first order ODEs{˙ X = Z,

˙ Z = μ(1 − X

2 ) Z − ω

2 X, (10)

while the Rayleigh oscillator is governed by the following nonlin-

ear first order ODEs {˙ X = Z,

˙ Z = μ(1 − Z 2 ) Z − ω

2 X. (11)

Stable orbits of the considered oscillators in the phase space

are shown in Fig. 8 . The presented numerical solutions have been

obtained for fixed values of the parameters ω = 2 , μ = 6 and var-

ious initial conditions (marked with squares). It is obvious that no

matter what initial value is, they will always converge to the same

limit cycle (bolded red curve).

The trajectories presented in Fig. 8 oscillate stably regardless of

the initial conditions and therefore they are often used to gener-

ate the trajectory of a leg tip of robot legs. The presented solu-

tions have been obtained for fixed parameters μ and ω. However

by changing the values of these parameters we can change the

shape of the stable orbit and its period. In result, we can control

the length and period of the single robot stride. It is clear that the

obtained in this way trajectories cannot be directly used to con-

trol the robot leg. First, they must be converted to the workspace

of the leg mechanism and then to the joints space of the leg

by using inverse kinematics rules. This problem is explained in

Section 5.4 .

.3. A novel CPG model

Motivated by the carried out study of a trajectory of a leg tip

f real insects and various biologically inspired robots, in order to

ontrol the hexapod robot leg other 1-DOF nonlinear mechanical

scillator with stick-slip induced vibrations is employed in this pa-

er as a novel CPG model. It can be observed that the trajectory

f stable orbit of the proposed CPG (after inverting with respect

o the coordinates axes) looks like the shape of the trajectory of

leg tip of a stick insect shown in Fig. 1 (see dashed curves). The

roposed model is governed by the following non-dimensional first

rder ODEs

˙ X = Z,

˙ Z = −d c Z − X + F f r ( v r ) , (12)

here d c denotes the non-dimensional coefficient of damping, and

f r ( v r ) =

F s 1+ δ| v r | sgn ( v r ) is the non-dimensional dry friction force

epending on the non-dimensional relative sliding velocity v r = dr − ˙ X with respect to the constant velocity v dr . Parameters F s and

characterize function F fr ( v r ). In further calculations we approxi-

ate non-smooth signum function sgn ( v r ) by the smooth and of-

en applied hyperbolic tangent function in the form

gn ( v r ) = tanh

(v r ε

)(13)

ith control parameter ɛ . The system parameters of the proposed

odel have a great impact on the frequency of vibrations and the

hape of the obtained stable orbits. In further studies we take

nto consideration stable trajectories of the proposed model ob-

ained for the following initial parameters: d c = 0 . 01 , F s = 1 , δ = 3 ,

dr = 0 . 5 , and the control parameter ε = 10 −4 . Fig. 9 shows the ob-

ained stable orbits for different parameter v dr and different initial

onditions (also marked with squares). In turn, Figs. 10 and 11 il-

ustrate similar stable orbits for different parameter F s and for dif-

erent parameter δ, respectively.

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 71

a b

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Z

X-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Z

X

Fig. 10. Stable phase trajectories of the proposed CPG model for different initial conditions: for F s = 0 . 8 (a) and for F s = 1 . 2 (b). (For interpretation of the references to colour

in the text, the reader is referred to the web version of this article.)

a b

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Z

X-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Z

X

Fig. 11. Stable phase trajectories of the proposed CPG model for different initial conditions: for δ = 1 (a) and for δ = 5 (b). (For interpretation of the references to colour in

the text, the reader is referred to the web version of this article.)

a b

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-0.75 -0.5 -0.25 -3E-15 0.25 0.5 0.75

Z

X

Hopf oscillatorvan der Pol oscillatorRayleigh oscillators�ck-slip oscillator

-110

-100

-90

-80

-70

-60

-50

-45 -30 -15 0 15 30 45

z [mm]

x [mm]

Hopf oscillatorvan der Pol oscillatorRayleigh oscillators�ck-slip oscillator

Fig. 12. Normalized periodic orbits of different CPGs (a) and the same orbits converted into workspace of the leg mechanism.

r

b

D

l

m

5

t

t

s

s

b

c

t

F

f

w

X

g

∈

s

t

s

c

a

s

s

r

i

s

l

o

c

c

t

t

t

t

The trajectories presented in Figs. 9 –11 oscillate stably (bolded

ed curves) regardless of the initial conditions and therefore can

e used to generate the trajectory of a leg tip of robot legs.

ifferent shapes and periods of the stable orbits can be regu-

ated by changing the system parameters of the proposed CPG

odel.

.4. Control structure of the robot leg movements

The stable trajectories of the presented oscillators can be used

o control the leg tip of the robot. The shape and size of such

rajectories have an impact on the length and height of a single

tride, whereas the orbit period corresponds to the period of the

ingle stride of the robot leg. We can change the gait parameters

y changing parameters describing the applied CPG, or convert the

onsidered trajectory of CPG into the workspace of the robot leg in

he appropriate way. In this paper we use the following method.

irst, the appropriate calculations of the stable orbits are started

rom initial condition which lies in the stable orbit and, as a result,

e omitted transient states. Second, the obtained orbits (variables

and Z ) are scaled by the normalization in such a way that the

enerated trajectories lie in the unit ranges: X ∈ [ −0 . 5 , 0 . 5] and Z

[0, 1]. Third, the period of the stable orbit is controlled by the

oftware implemented in the microcontroller using the regulated

ime delays. Patterns of the stable orbits obtained in this way are

hown in Fig. 12 a. Next, multiplying the variables X and Z of an os-

illator working as a CPG, we can directly change both the length

nd height of the robot stride. The example of the trajectory of CPG

caled in this way to the workspace of the robot leg mechanism is

hown in Fig. 12 b.

As it was mentioned earlier, periodic stable orbits cannot be di-

ectly used to control of the robot legs, they must be converted

nto joints space by using the inverse kinematics rules first. If a

traight propulsion is the major task of the walking robot, then the

eg tips ought to move along straight lines parallel to the course

f the robot and parallel to its longitudinal body axis [14] . In this

ase the body of the robot should translate itself without any os-

illations. This is why in the conducted simulations and real con-

rol of the robot we use variable X of CPG to control the leg tip in

he direction of the x -axis, variable ˙ X = Z to control the leg tip in

he z direction, while the coordinate y of the leg tip is constant (in

he considered robot we take y = 100 mm). Although it differs from

72 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

Fig. 13. Numeration of the robot legs (a) and diagram showing tripod gait (black color denotes stance movement, while the white color denotes swing movement).

Fig. 14. Schematics showing generation of angles values for each servo of the robot

legs.

t

f

a

s

t

a

(

C

a

o

v

X

j

o

e

s

o

6

i

s

N

reality, since real insects oscillate during straight walking [52] , this

concept is preserved during planning of the leg tips.

a) Hopf oscillator b) van der Pol oscillator c

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

Joint angles [rad]

t [s]

φ1

φ2

φ3

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

Joint angles [rad]

t [s]

φ1

φ2

φ3

Fig. 15. Time series of angles ϕ 1 (red curve), ϕ 2 (green curve) and ϕ 3 (blue curve) an

regarding the leg tip obtained for different CPGs. (For interpretation of the references to c

Considering legged locomotion we can list the following three

ypical and most known gaits of six-legged robots (from slow to

ast): wave gait, tetrapod gait and tripod gait [42] . Moreover, there

re many other gaits that are created through adjustable phase

hift between the signals used to control various robot legs. In fur-

her studies we take into consideration only tripod gait as a typical

nd fast, which is schematically presented in Fig. 13.

Fig. 14 schematically illustrates a method to generate signals

angle values) for each leg of the robot. The signal of the applied

PG is time-shifted by controlling the phase shift, which allows to

chieve an adequate gait realized by the appropriate coordination

f the different robot legs. The obtained periodic orbits are con-

erted to the workspace of the robot leg (by scaling the variables

and Z of the CPG). Next, the obtained signals are converted to the

oint spaces using the inverse kinematics relations. For the purpose

f all numerical simulations presented in Section 6 (and further

xperimental investigations presented in Section 7 ), we take the

tride length of 60 mm, the stride height of 30 mm, and the period

f the single robot stride equal to 2 s.

. Numerical investigations

In order to investigate the proposed CPG model and to compare

t with the other ones, application of the algorithms to control a

ingle leg with help of a numerical simulation is illustrated first.

ext, some interesting numerical results regarded to the whole

) Rayleigh oscillator d) stick-slip oscillator

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

Joint angles [rad]

t [s]

φ1

φ2

φ3

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

Joint angles [rad]

t [s]

φ1

φ2

φ3

d visualization of the robot leg configurations with the plotted stable trajectory

olour in this figure legend, the reader is referred to the web version of this article.)

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 73

Fig. 16. Comparison of the stable trajectory of the leg tip plotted by using the proposed stick-slip oscillator with a trajectory plotted by a leg tip of a real stick insect.

r

u

i

6

s

s

l

i

s

c

t

t

b

e

d

p

j

l

fi

w

p

s

p

s

s

“

t

l

c

i

m

r

6

i

e

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6

z [mm]

t [s]

Hopf oscillator

van der Pol oscillator

Rayleigh oscillator

stick-slip oscillator

Fig. 17. The distance of the robot leg tip to the center of the leg coordinate system

in z direction obtained numerically for different CPGs.

l

y

i

s

t

g

d

f

a

w

c

i

a

i

P

c

h

a

7

v

a

m

obot are presented and discussed. In numerical simulations we

sed the standard fourth order Runge–Kutta method implemented

n Scilab.

.1. Numerical simulations of a single robot leg

The presented numerical simulations refer to a single robot leg,

ince the results for the remaining ones are the same. Fig. 15

hows time series of angles ϕ 1 , ϕ 2 and ϕ 3 . It also presents robot

eg configurations with the stable trajectory plotted by the leg tip

n regular time intervals as a result of the application of the pre-

ented time series of angles to the appropriate leg joints.

The presented numerical results show how the considered os-

illators can be used to control the leg tip of the robot. In all cases,

he leg tip starts from initial configuration (which corresponds to

he initial condition of the applied CPG), finally plotting the sta-

le trajectory in the workspace of the leg mechanism. Moreover, as

xpected, the mentioned plotted trajectories are limited to a two-

imensional plane, which is parallel to the xz plane.

Fig. 16 shows comparison of the stable trajectory of the leg tip

lotted for the proposed stick-slip oscillator as a CPG and the tra-

ectory plotted by a leg tip of a real stick insect (presented ear-

ier in Fig. 1 ). The leg tip of the robot starts from the initial con-

guration, finally plotting the stable trajectory in the workspace,

hich lies in the plane parallel to the xz plane. The so-called “slip

hase” of the stick-slip vibrations (proposed CPG model) corre-

ponds to the swing movement of the leg, and the so-called “stick

hase” of the mentioned stick-slip vibrations corresponds to the

tance movement of the leg. Moreover, “transition from stick to

lip phase” corresponds to the posterior extreme position, whereas

transition from slip to stick phase” corresponds to the anterior ex-

reme position of the robot leg.

Fig. 17 illustrates the distance of the leg tip to the center of the

eg coordinate system (see Fig. 6 a) in z direction obtained numeri-

ally for all CPGs considered in this paper. As it can be seen, only

n the case of the proposed CPG model there are no changes of the

entioned distance in the stance movement during walking of the

obot.

.2. Numerical simulations of the whole robot

Fig. 18 shows configurations of all legs plotted in regular time

ntervals in four phases of single stride of the robot and for differ-

nt CPGs (the arrow indicates the forward direction).

The center of the coordinate system is placed at a constant

evel, at a height of 100 mm above the ground. Through the anal-

sis of configurations of individual legs it can be observed that,

n cases of Hopf, van der Pol and Rayleigh oscillators, there exist

uch phases of the robot motion in which the distance between

he center of the coordinate system and the leg tips touching the

round changes significantly. Moreover, it can be seen that for van

er Pol oscillator, some periods of the robot motion are realized

aster ( t = 0 … 0,5 s and t = 1 … 1,5 s), and other slower ( t = 0,5 … 1 s

nd t = 1,5 … 2 s). Only in case of the proposed stick-slip oscillator

orking as a CPG, there are almost no fluctuations of the gravity

enter of the robot in any phase of its movement. In order to better

llustrate this observation, also fluctuations of the gravity center as

function of time for all considered CPGs (also obtained numer-

cally) are shown in Fig. 19 . As it can be seen, for Hopf, van der

ol and Rayleigh oscillators working as CPGs, the gravity center is

yclically moved up and down. The proposed CPG model does not

ave these disadvantages, namely the gravity center of the robot is

t constant level in each phase of its motion.

. Experimental investigations

The obtained numerical simulations have been experimentally

erified using the constructed hexapod robot, and the appropri-

te results are presented in this section. The performed experi-

ental investigations are related to: (i) trajectories plotted by leg

74 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

a) Hopf oscillator

t = 0...0,5 s t = 0,5...1 s t = 1...1,5 s t = 1,5...2 s

b) van der Pol oscillator

t = 0...0,5 s t = 0,5...1 s t = 1...1,5 s t = 1,5...2 s

c) Rayleigh oscillator

t = 0...0,5 s t = 0,5...1 s t = 1...1,5 s t = 1,5...2 s

d) Stick-slip oscillator

t = 0...0,5 s t = 0,5...1 s t = 1...1,5 s t = 1,5...2 s

Fig. 18. Configurations of the robot legs (plotted in regular time intervals) in different equal phases of the single robot stride (from the left to the right) obtained numerically

for different CPGs.

7

e

s

p

i

F

m

t

a

i

tip of a single robot leg; (ii) fluctuations of the robot gravity cen-

ter; (iii) variations of the displacement and velocity of the robot in

forward direction; (iv) energetic cost estimation of the robot. The

mentioned experimental trajectories and fluctuations of the gravity

center of the robot have been obtained with the camera using the

point tracking method (the so-called capture motion method). The

displacement and velocity variation of the gravity center in mov-

ing direction have been obtained by using a potentiometer with

the appropriate rope fixed to the robot body during its walking. In

turn, an energetic investigations have been carried out based on

measurements of voltage and current data of individual servos in-

stalled in the robot legs.

.1. A single robot leg

First, for simplification of the presentation without loss of gen-

rality, we consider the situation in which only one leg performs

tance and swing movements and the body of the hexapod is sup-

orted by the rest of the legs like a stationary base. The exper-

mentally obtained trajectories plotted by the leg are depicted in

ig. 20 , and the obtained results coincide with the appropriate nu-

erical simulations (also presented in Section 6 ). Moreover, the

rajectory obtained for the proposed CPG indicates the greatest

nalogies with a trajectory of a leg tip of walking animals exist-

ng in nature.

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 75

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3 3.5 4

z [mm]

t [s]

Hopf oscillator

van der Pol oscillator

Rayleigh oscillator

stick-slip oscillator

Fig. 19. Variations of the robot gravity center in z direction obtained numerically

for different CPGs.

-110

-100

-90

-80

-70

-60

-50

-40 -30 -20 -10 0 10 20 30 40

z [mm]

x [mm]

Hopf - numerical Hopf - experimentalvan der Pol - numerical van der Pol - experimentalRayleigh - numerical Rayleigh - experimentalstick-slip - numerical stick-slip - experimental

Fig. 20. Stable phase trajectories of the leg tip obtained numerically and experi-

mentally for different CPGs.

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3 3.5 4

z [mm]

t [s]

Hopf oscillator

van der Pol oscillator

Rayleigh oscillator

stick - slip oscillator

Fig. 21. The fluctuation curves of the gravity center of the robot obtained experi-

mentally for different CPGs.

7

i

o

o

2

a

s

a

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12

x [mm]

t [s]

Hopf oscillator

van der Pol oscillator

Rayleigh oscillator

stick-slip oscillator

Fig. 22. Displacement curves along x direction (moving direction of the robot) ob-

tained experimentally for different CPGs.

-100

-50

0

50

100

150

200

250

300

0 2 4 6 8 10 12

v [mm/s]

t [s]

Hopf oscillator van der Pol oscillatorRayleigh oscillator stick-slip oscillator

Fig. 23. Velocity curves along x direction (in moving direction of the robot) for

different CPGs.

a

t

i

v

t

i

i

c

c

e

p

d

s

p

H

l

t

o

a

i

i

a

a

.2. The gravity center fluctuations

Fig. 21 shows the fluctuation curves in z direction of the grav-

ty center of the robot for different CPGs. When we use the typical

scillators considered in this paper (Hopf, van der Pol and Rayleigh

scillators), the robot gravity center varies in the range about 15–

0 mm (about a half of the stride height). By comparing the fluctu-

tion ranges of the gravity center presented in Fig. 21 it is clearly

een that, in case of proposed CPG, there are low changes observed

nd it agrees with numerical simulations presented in Section 6 .

In addition to the obtained numerical solutions of the fluctu-

tions of the robot gravity center and their experimental verifica-

ion, we also conducted experimental studies of the robot motion

n the forward direction. Figs. 22 and 23 depict the displacement

ariations and velocity variations (obtained by numerical differen-

iation of the displacement variations) of the robot gravity center

n moving direction, respectively.

As it can be seen, the choice of the applied CPG has a great

mpact on the presented experimental results, and the CPG model

onstructed based on the van der Pol oscillator is the worst. In this

ase we can see the greatest unnecessary accelerations and decel-

rations in the moving direction of the robot. As can be seen, the

resented in Fig. 23 velocity variations of the robot in the moving

irection indicate large changes. It should be noted that these re-

ults have been obtained by numerical differentiation of the dis-

lacement variations and may be slightly different from reality.

owever, it can be seen characteristic fluctuation of the robot ve-

ocity (two times per one robot stride), which are the lowest for

he proposed CPG. It should be also noted that both fluctuations

f the gravity center as well as unnecessary changes of speed and

cceleration in the moving direction of the robot can have a great

mpact on energy consumption during walking. This is why dur-

ng normal walking both the variation curves of the gravity center

nd acceleration of the robot in moving direction should be zero

s close to zero as possible.

76 D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78

Fig. 24. Comparison of the total energy cost of the robot obtained experimentally

for different CPGs.

w

l

i

t

n

m

i

r

p

r

s

t

e

s

c

b

w

a

a

i

a

t

p

t

q

v

t

a

t

p

7.3. Energy demand of the robot

As it is well known, power consumption belongs to one of the

main operational restrictions on autonomous walking robots [47] .

Power is a limited resource in the mentioned systems, and there-

fore the proper development of walking robots can be limited by

the problem of their high energy consumption. This is why over

the last three decades numerous researchers have been also ex-

ploring power consumption optimization techniques for these mo-

bile machines. A brief literature review presented below regard-

ing to power consumption optimization for walking robots can

be found in reference [47] . Based on the literature [58–64] re-

ferred in the mentioned paper [47] , some optimization problem

of power consumption and energy costs from other viewpoint are

summarized by optimizing the gait parameters in the form of the

following topics: (i) the protraction movement trajectory of the

robot leg [58] ; (ii) energy cost analysis with respect to the stride

and stance length [59,60] ; (iii) examination of the optimum stride

length changes at a given speed with different body masses [61] ;

(iv) testing various parameters used to define the leg tip trajecto-

ries of legged robots walking on irregular terrains [62] ; (v) analy-

sis of the torque contributions of different dynamic components in

real leg trajectories taking into account backlash, friction and elas-

ticity effects in the gear reduction system [63] ; (vi) distribution of

identical legs around the robot body in order to use smaller actu-

ators and for lower energy consumption [64] .

In this paper we also present some experimental investigations

devoted to the energy consumption of the robot during walking

when different CPGs are used. Power electric energy demand in all

servomechanisms of the robot during walking has been calculated

using computer program prepared in LabView. In all presented

cases the stride length of the robot and the number of stride

lengths are the same. During experimental measurement, the ob-

tained total length of the road of the robot was 80 cm and the time

was equal 22 s. Experimental results shown in Fig. 24 illustrate en-

ergy cost for used CPGs. As it was mentioned earlier, both unneces-

sary variations of the gravity center and acceleration/deceleration

of the robot during walking have a negative impact on energy cost,

because all servomechanisms installed in robot legs must perform

additional and unnecessary work. As demonstrated numerically in

the previous section and also experimentally in this one, the CPG

proposed in this paper does not possess these disadvantages and

this is why, for the mentioned CPG, energy demand cost during

walking of the robot is the smallest.

8. Conclusions and future studies

A biologically-inspired prototype of the hexapod robot is in-

troduced in this paper. Both mechanical construction of the robot

ith equipment installed on its body and its control system are il-

ustrated and presented. Mathematical description (both direct and

nverse kinematics) of the robot leg is reported. Three different

ypical and well known nonlinear oscillators are used and, finally,

on-linear stick-slip induced vibrations are introduced as a novel

odel of CPG controlling the leg movement of the robot. Some

nteresting numerical simulations of the robot leg and the whole

obot are obtained and experimentally verified. Finally, some as-

ects of an energy efficiency analysis and obtained experimental

esults are presented and discussed.

Considering the performed numerical and experimental analy-

is, the following main concluding remarks can be drawn:

1. The constructed prototype of the robot with proposed con-

trol system and installed equipment can be used for inspec-

tion and operation applications in inaccessible environments, as

well as for other different research investigations. For instance,

the robot equipped with wireless system helps us transmit au-

dio and video data in inspection applications, whereas the pro-

posed control system allows to operate all used servos indepen-

dently and to generate a wide range of different robot gaits.

2. There are two possible ways of accelerating the robot move-

ment, namely: (i) increasing the free frequency of oscillations of

the applied CPG; or (ii) enlarging the amplitude of the CPG sig-

nal during its conversion into the working space by scaling vari-

ables of GPG, in order to generate a larger stride for the robot.

What is more, different scale coefficients on different sides of

the hexapod body allow to change the direction of its move-

ment. The amplitude and frequency of the CPG signal can be

controlled independently. In consequence, also the length of the

stride and speed of walking can be controlled independently.

3. The presented numerical configurations of the robot leg and

trajectories plotted by the leg tip indicate good analogies be-

tween the movements of the simulated walking robot leg and

the animal leg movements met in nature. The proposed CPG

model can produce swing and stance trajectories, which closely

resemble movements observed in walking stick insects.

4. The numerically obtained and experimentally verified results of

the studies of the whole robot show that the proposed CPG has

some advantages in comparison to other designs presented in

this paper. Namely, the robot controlled by the mentioned CPG

does not have unnecessary fluctuations of the gravity center

and low acceleration/deceleration in moving direction.

5. The proposed model, being based on the stick-slip induced vi-

brations, can be more energy efficient from the point of view

of energy cost in comparison to others.

It should also be noted that if we use various CPG algorithms,

he precise description of the considered mechanical model is nec-

ssary. Besides, if a leg gets stuck by an obstacle, the robot should

top the leg, detect the current orientation and position, and recal-

ulate new trajectory. This is why some sensor information should

e employed, for instance piezoelectric sensors on the leg tip

orking in a feedback loop. In real situations, in nature, insects

cquire distribution of terrains surrounding them through eyes in

dvance when planning leg movements and strides. This problem

s especially important where the hexapod movement cannot be

chieved by employing simple contact information from reactions

o the ground. Therefore, the constructed prototype of the hexa-

od robot can also be equipped with a binocular stereo vision sys-

em to recognize external environment in the form of a series of

uadratic surface patches which are fit by an elevation map in the

ertical direction associated with a regular grid in horizontal direc-

ions. Based on this information, the range of tip motion in upper

nd forward directions may be respectively scaled at each stride of

he hexapod. The proposed CPG model is relatively simple in com-

arison with other control methods presented in literature and it is

D. Grzelczyk et al. / Mechatronics 37 (2016) 63–78 77

s

I

c

s

s

w

s

o

t

p

s

b

r

a

t

s

T

a

i

w

p

t

i

a

r

A

P

2

R

[

[

[

[

[

[

[

[

[

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ufficient especially when robot walks on the flat, regular surfaces.

n nature insects have considerable flexibility in their motion, be-

ause the path taken by a leg can easily be modified if the leg

trikes an obstacle. This problem is more difficult to solve in con-

tructed prototypes of the multi-legged robots. However, in cases

hen robot is walking on irregular surfaces, this problem can be

olved by the appropriate scaling of the CPG signal in each cycle

f the robot stride, as it was mentioned earlier. It is well known

hat the development of multi-legged robots is restricted by the

roblem of their high power consumption. For the mentioned rea-

ons the proposed movements of the legs of the hexapod robot can

e used to overcome long distances, particularly in the regular ter-

ains in a more efficient way. Optimization of energy efficiently is

n area without improving the power supply unit allow to increase

he mission time of the robot.

The obtained simulations are presented for a purely kinematic

ystem, and the constructed prototype has to act in the real world.

herefore, the robot has to deal with dynamics. Future work will

lso involve dynamic control schemes of legs, studies of optimal-

ty criteria proposed in literature, as well as various gaits for the

alking robot. Although the obtained knowledge gained from the

resented studies can be applied to improve the constructed proto-

ype of the robot leading to better performance, more future stud-

es are required in this field. Future work in this area will therefore

ddress the refinement of the considered prototype of the hexapod

obot.

cknowledgments

The work has been supported by the National Science Centre of

oland under the grant OPUS 9 no. 2015/17/B/ST8/01700 for years

016–2018.

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