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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS13.070
Study of Caterpillar-like Motion of a Four-link Robot S.F. Jatsun, L.Yu. Vorochaeva, S.I. Savin, A.S. Yatsun
Department of Mechanics, Mechatronics and Robotics, South-West State University, Kursk, Russia
e-mail: [email protected]
Abstract: In this paper we study a four link robot that performs
caterpillar-like motion. The device moves on a rough horizontal
surface due to friction forces that are applied to the robot at the
points of contact. The specific feature of the robot is that it has
active supports that allow control over coefficients of dry friction.
The object’s mathematical model is developed, the stages of
motion singled out, an algorithm for realizing caterpillar-like
motion and the results of numerical modeling are presented.
Keywords: caterpillar-like robot, motion stages, active and
passive supports, control torques.
1 Introduction The design of bionic robots whose motion is based on
animals is one of the important areas of development in
modern mechatronics and robotics. Caterpillar-like robots
fall into one of the broad classes of such robots.
Multilink mechanisms that describe the more
important aspects of robots of this type are used as
mathematical approximations i.e. models. These
approximations allow us to study the most important
features of the robot’s dynamics. Each link of the
multilink mechanism is presented as an absolutely rigid
body with finite mass. This allows the links that form the
mechanism to rotate relative to each other, implementing
different kinds of gaits [1-13].
Paper [1] proposes a joint torque control method based
on the assumption that there is only one active joint in the
four-link mechanism executing the climbing gait. Besides
the active joint the other three joints are all considered as
passive joints whose torques tend to zero, although they
are driven by motors in reality. Article [13] presents the
application of developing and employing modular robots
for the research of caterpillar-like motion. First an
investigation on the locomotion kinematics adopted by
natural caterpillars is given systematically. Paper [4]
describes some of the biomechanics of caterpillar
locomotion and gripping. It then describes recent work to
build a multifunctional robotic climbing machine based on
the biomechanics and neural control system
(neuromechanics) of caterpillars, Manduca sexta. In [5] a
rope climbing robotic caterpillar was designed and
achieved by imitating the gait of a natural caterpillar. A
simple scalable sinusoidal oscillator is successfully
employed for implementing diverse bionic locomotion
patterns including caterpillar-like, millipede-like, and
earthworm-like motions as described in [6].
This paper is dedicated to studying the motion of four–
link caterpillar-like robot equipped with devices that
enable it to change the way it interacts with the supporting
surface.
2 Description of the caterpillar-like robot
A diagram of the caterpillar-like robot is presented on the
figure 1. The number of the device’s links is chosen equal
to four: the extreme among them are the “head” and “tail”,
while the middle ones form the "складывающуюся
section", that provides transverse pull-up of the “tail” to
the “head” and straightening of the links into one line.
This is the minimum possible number of links required to
execute such caterpillar-like motion during which the
“head” and “tail” can exchange places as a result of which
the object moves backwards and forwards. The object can
move on an absolutely solid rough surface with no elastic-
dissipative properties, for example on asphalt, concrete
and ice. Bodies 1 and 2, 2 and 3, 3 and 4 are connected to
each other via rotational motors 5, 6 and 7. The interaction
of the robot with the supporting surface occurs at four
points via supports 8-11 which are mounted on links 1 and
4. The difference between this robot and other known
designs is the possibility to control friction acting at
supports 9 and 10. This is possible by the use of special
motors that can change the properties of the contact
surfaces of the supports. [14, 15].
Figure 1 Diagram of the robot
Let’s consider the design of the supports. Supports 9
and 10 (figure 2) consist of frame 1 which is rigidly
connected to the lower part, 2 of the corresponding unit of
the four-link robot, springs 6 and 7, electromagnetic motor
3 and metal armature 4 with sharp tip point 5 mounted on
it.
Figure 2 Central support
When the coils of the electromagnets are powered a
magnetic flux directed perpendicular to the supporting
surface is induced which causes the metal armature move
downwards, so the points clings to the supporting surface
thereby considerably increasing the coefficient of dry
friction. If the design of the support meets certain
conditions it is possible to completely fixate the central
body onto the surface. This in particular depends on the
choice of the material of the point’s tip.
Figure 3 Passive support
Passive supports 8 and 11 (figure 3) consists of frame
1 which is rigidly connected to lower part 2 of the
corresponding link and spherical joint 3 that provides a
low coefficient of friction when the support is sliding
along the surface.
Thus it is possible to change friction coefficients at
supports 9 and 10 at different stages of the motion
depending on the chosen control method.
3 Mathematical model of the robot
The robot moves in such a way that one of its extreme
links is periodically fixed onto the surface by means of
special adhesion systems. Thus the robot at every point in
time is a three-link mechanism moving relative to the
currently fixed link. To describe the motion of the four-
link robot we introduce a stationary absolute coordinate
system, Oxy and relative coordinate systems, Oixiyi which
are rigidly linked with points О1, О2, О3 and О4 in such a
way that the Oixi axis is directed along the corresponding
links (as shown in figure 4). Angles φi describe the
rotations of the Oixiyi coordinate systems relative to Oxy.
Figure 4 Analytical diagram of the mechanism
We will assume that all of the robot’s links are
absolutely rigid bodies and rods of li and whose mass mi is
concentrated at their centers of symmetry, Сi (i=1-4).
Motors 5, 6 and 7 are located at the points О2, О3 and О4
and generate torques: М12(М21), М23(М32) and М34(М43).
Active supports 9 and 10 are also located at these same
points while passive supports 8 and 11 - at points О1 and
О5.
To write a generalized mathematical model of the
robot’s motion in the vertical plane, Оху we look at the
case when link 1 is fixed onto the horizontal surface and
that the coordinates of its center of mass and angle of
inclination to the horizontal plane are constant:
constxC 1 , constyC 1 and const1 . Stationary
points О1 and О2 in fig. 4 are fixed. Taking into
consideration the constraints imposed on the system the
generalized coordinates are the links’ angles of rotation, φi,
i=2-4:
T432 q (1)
Differential equations of the system’s motion obtained
using second order Lagrange equations have the following
form:
224223
2223212
42424
2
4
3232432
3
424244
3232433
2
24322
coscos
2/cos
2/)sin(
)sin()2(
2/)cos(
)cos()2/(
)(
glmglm
glmMM
llm
llmm
llm
llmm
lmmJ
(2)
3343334323
43432
2
2
233243
2
2
434344
2332432
2
34
2
3333
cos2/cos
2/)sin(
2/)sin()2(
2/)cos(
2/)cos()2(
)4/(
glmglmMM
llm
llmm
llm
llmm
lmlmJC
(3)
2/cos2/)sin(
2/)sin(2/)cos(
2/)cos(4/
4443434434
2
3
24424
2
2344343
244242
2
4444
glmMllm
llmllm
llmlmJC
(4)
where 3CJ , 4CJ are the central moments of inertia of the
links, 2J - the moment of inertia of link 2 relative to point
О2.
4 Stages of motion for caterpillar-like motion of the
robot
To implement caterpillar-like motion we will use the
following cyclogram for control inputs (see figure 5).
Figure 5 Cyclogram for control torques M12, M43 and
electromagnetic forces F2 and F4 at contact elements during
caterpillar-like motion
The principle of this type of motion lies in periodic
relative motion of one of the robot’s links with respect to
the others under the action of torques M12 and M43
generated by motors located at points О2 and О4 and also
in the successive connection-disconnection of contact
elements that enable friction control. In this work we
assume that fixation of the contact elements onto the
surface is implemented when they are connected ( af )
and that motion on a smooth surface ( 0pf ) is
executed when they interact with the surface, where af
and pf are coefficients of friction at active and passive
supports respectively.
There are two conditions that should be met before we
can fixate links 1 and 4 by means of frictional forces
controlled by electromagnets: links 2 and 3 should be
oriented in such way that their angles of rotation vary in
certain intervals (as shown by expressions (5) and (6));
links 2 and 3 should rotate in specified directions. Thus
kinematic constraints are periodically imposed on the
mechanical system which corresponds to the fact that the
velocities of points О2, О4 are equal to zero under the
action of forces F2 and F4:
0
0
0
,0
0,
222
222
22
22
0
2
2
kn
kn
kn
kn
if
ifF
F (5)
0
0
0
,0
0,
333
333
33
33
0
4
4
kn
kn
kn
kn
if
ifF
F (6)
where φ2n, φ3n are the initial values of the angles of
rotation of links 2 and 3, φ2k, φ3k -the final values of these
angles, F20, F4
0 – certain fixed values of friction.
Torques M12 and M43 can be calculated using the
following expressions:
0
0
0
,0
0,
222
222
22
22
0
12
12
kn
kn
kn
kn
if
ifM
M (7)
0
0
0
,0
0,
333
333
33
33
0
43
43
kn
kn
kn
kn
if
ifM
M (8)
where 0
12M , 0
43M are some constant values of torques.
The caterpillar-like motion can be represented as a
sequence of stages (see figures 6 and 7).
a
b
c
Figure 6 Sequence of the robot’s positions during the first
stage of motion: a) initial position; b) intermediate position;
c) final position
a
b
c
Figure 7 Sequence of the robot’s positions during the second
stage of motion: a) initial position; b) intermediate position,;
c) final position
In figures 6 and 7 the black triangles represent points
that are assumed to be fixated.
The four link mechanism is initially at rest with its
entire links parallel to the Ox axis. In this state we have
φ1=φ2=φ3=φ4.
Active support 10 that controls friction force at point
О4 is switched on during the first stage of motion and so
friction coefficient at this point has its maximum value.
Active support 9 is switched off and link 1 is in contact
with the supporting surface via passive support 8. Torque
М43 that forces link 3 to rotate around point О4 through
angle φ3=φ3k, where φ3k is the required value of this angle,
begins to act at point О4. Passive support 8 which is
mounted at point О1 slides along the surface and link 1
remains parallel to the Ox axis (see figure 8).
Figure 8 Schematic representation of the caterpillar-like
robot during the first stage of motion
During this stage the position of the mechanism can be
described using one generalized coordinate, 2 which can
be determined by one differential equation:
2/coscoscos
2/coscos
2/cos
2/)sin(
2/)cos()(
1122331
22332
33343
23322
2
2
323222
2
3233
lllgm
llgm
glmM
llm
llmlmJ
(9)
where J3 is the moment of inertia of the link 3 with respect
to point О3.
In the second stage connected and disconnected active
supports exchange places and now point 2 is fixated onto
the surface. Torque 43M ceases to act, while torque 12M
starts acting at point О2. It is necessary to point out that
the torques are switched on and off simultaneously and
that they are equal in magnitude and direction. In this
stage link 2 executes angular motion relative point О2
until the following condition holds: k22 , where k2
is the required value. Links 2, 3 and 4 form a crank-slide
mechanism, where link 2 is the crank, link 3 – the
connecting rod and link 4 –the slider. Link 4 interacts with
the surface at point О5 by means of a passive support. As a
result of this stage the robot’s links are aligned along the
Ох axis, occupying a position similar to the initial one (fig.
9).
Figure 9 Schematic representation of the caterpillar-like
robot during the second stage of motion
In this stage we calculate the generalized coordinate,
φ2 using the following differential equation:
2/coscoscos
2/coscos
2/cos
2/)sin(
2/)cos(
4433224
33223
22212
32323
2
3
323233
2
2322
lllgm
llgm
glmM
llm
llmlmJ
(10)
where J2 is the moment of inertia of the link 2 with
respect to point О2.
5 Numerical simulation of the robot’s caterpillar-like
motion
The algorithm for simulating the robot’s caterpillar-like
motion is shown in the figure 10. The simulation uses
iterative algorithm and at time t0=0 s the mechanism is in
its initial state. To determine the stage of the motion we
use counter n. When n=1 we have the first stage of motion
and when n=2 - the second. At each moment in time the
characteristics of the system are calculated by formulas
corresponding either to the first or second stage. In the
first stage the counter’s value is determined based on the
orientation of the link 3 i.e. angle 3
. As soon as n=2 the
motion progresses to the second stage where the counter
can assume values n=1 and n=2 depending on the
orientation of the link 2 which is determined by angle φ2.
Figure 10 Algorithm used to simulate robot’s movement
Masses of the links are given by expression mi=0.1 kg
and the lengths of the links - l1=l4=0.1 m, l2=l3=0.3 m. the
initial values for the generalized coordinates and their
derivatives are equal to zero. The required values of
angles: 18/732 kk radians and torques
generated by electric motor: 8,10
43
0
12 MM Nm. The
numerical solutions of the differential equations are
obtained using Mathcad software by implementing a
numerical integration algorithm that assumes the
generalized accelerations to be constant in intervals [ti,
ti+h], where h is a constant time step.
The simulation results are presented in the form of
time graphs of angular and linear displacements (see
figures 11-13).
In figure 11 we can see that the angular displacements
of links 2 and 3 change in antiphase by the same laws and
links 1 and 4 move parallel to the surface, so 041
rad.
Figure 11 Time graphs of the links’ angular
displacements 1 – φ1(t), 2 – φ2(t), 3 – φ3(t), 4 – φ4(t)
From the graphs in fig. 12 it can be seen that during
the robot’s motion links 1 and 4 are fixated to the surface
in turns. The centers of mass of links 2 and 3 move along
the horizontal axis on trajectories with the same form, but
shifted relative to each other by time equal to the duration
of one stage of the motion.
Figure 12 Time graphs of the x-coordinates of the
centers of mass of links 1-4: 1 – xC1(t), 2 – xC2(t), 3 – xC3(t), 4 –
xC4(t)
The graphs of the ordinates of the centers of mass of
links 2 and 3 are the same. Links 1 and 4 are not separated
from the surface: yC1=yC4=0 m (see figure 13).
Figure 13 Time graphs of the y-coordinates of centers of
mass of links 1-4: 1 – yC1(t), 2 – yC2(t), 3 – yC3(t), 4 – yC4(t)
The motion of the links’ extreme points along the Ох
axis is shown in figure 14.
Figure 14 Time graphs of the x-coordinates of the extreme
points of links 1-4: 1 – xО1(t), 2 – xО2(t), 3 – xО3(t), 4 –
xО4(t), 5 – xО5(t)
Point О3 at which links 2 and 3 are connected to each
other never ceases to move along the horizontal axis.
Besides this we observe its ascent along the Оу axis in the
first stage of motion and its descent in the second stage
until it touches the surface.
Points О1 and О2 move in an identical way and the
same goes for points О4 and О5. The graphs do not
overlay each other because of the different initial positions
of these points. In figure 14 we can see repeating pattern
of fixation states and motion of the points: points О1 and
О2 move during the first stage and points О4 and О5 -
during the second stage. Motion of the points mentioned
above in the vertical direction is absent as shown in figure
15.
Figure 15 Time graphs of the y-coordinates the extreme
points of links 1-4: 1 – уО1(t), 2 – уО2(t), 3 – уО3(t),
4 – уО4(t), 5 – уО5(t)
Figures 16 and 17 show the graph of the average
velocity of the robot as a function of control torques and
the value of angle k2 .
Figure 16 Graph of ),( 043
012 MMvsr
Average speed of the robot increases by a curvilinear
law as the magnitudes of torques 4312,MM increase. The
convexity of this curve is directed downwards. From 2
Nm onwards the graph becomes linear.
Figure 17 Graph of )( 2ksrv
The graph of the average velocity as a function of
angle φ2k is also curvilinear, but as the angle increases the
average speed falls at first, attains a minimum at φ2k=550
and then rises. For values greater than φ2k=550 average
speed growth with growing φ2k. Based on the analysis of
figure 17 we can conclude that in order to achieve
caterpillar-like motion with the highest average speed the
value of angle φ2k has to be minimal.
6 Conclusions
In this paper mathematical model that describes
motion of a four link mechanism was presented. The
mechanism interacts with the surface by means of passive
and active supports and the latter allow the robot to
control friction coefficients. The design of these contact
elements and an algorithm that implements caterpillar-like
motion was presented. The results of numerical simulation
were shown in the form of time graphs of linear an
angular displacements and graphs of the robot’s average
speed as a function of variable parameters.
Work is performed by Russian Science Foundation,
Project № 14-39-00008.
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