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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: [email protected] (D. Grzelczyk), bartosz.stanczyk
dokt.p.lodz.pl (B. Sta ́nczyk), [email protected] (J. Awrejcewicz).
s
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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 1 266 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
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c
s
s
w
s
o
t
p
s
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a
t
s
T
a
i
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p
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i
a
r
A
P
2
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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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