AN
ALGORITHM TO SOLVE THE INVERSE KINEMATICS TO A STEWART PLATFORM
Florian
Ion Tiberiu Petrescu
Bucharest
Polytechnic University, Romania
E-mail: petrescuflorian@yahoo.com
Relly
Victoria Virgil Petrescu
Bucharest
Polytechnic University, Faculty of
Transport, Romania
E-mail: rvvpetrescu@gmail.com
Submission: 2/15/2018
Revision: 3/22/2018
Accept: 2/27/2019
ABSTRACT
Mechanical systems in motion type parallel
structures are solid, fast and accurate. Between mobile systems parallel the
best known and used system is that of a Stewart platform, as being and the
oldest system, quickly, solid and accurate. The paper presents a few main
elements of the Stewart platforms. In the case where a motto element consists
of a structure composed of two elements in a relative movement from the point
of view of the train of propulsion and especially in the dynamic calculations,
it is more convenient to represent the motto element as a single moving item.
The paper presents an exact, original analytical geometry method for
determining the kinematic and dynamic parameters of a parallel mobile
structure. Compared with other methods already known, the presented method has
the great advantage of being an exact analytical method of calculation and not
one iterative-approximately.
Keywords: An algorithm, Mechatronics, Robotics, Parallel moving mechanical systems, A Stewart structure
1.
INTRODUCTION
The humanoids robots are used now as
a tool for research in several scientific fields.
Researchers need to understand the
structure of the human body and behavior (biomechanics) to build and to study
robots humanoids. On the other hand, the attempt simulation of the human body
leads to a greater understanding of it. Human knowledge is a field of study,
which is focused on the way in that people learn from sensory information in
order to acquire the skills and insightful motor. Such knowledge is used to
develop models for the calculation of human behavior and has been improved in
time.
It has been suggested that robotics
highly advanced will facilitate its increase even in ordinary people.
With all that the original purpose
of humanoid research has been to build a better orthosis and prosthesis for
human beings, knowledge has been transferred between the two disciplines. Some
examples are Prosthesis footswitch with electrical adjustment for impaired
neuromuscular, orthosis ankle-foot, biological realistic prosthesis leg and
forearm prosthesis (AVERSA et al., 2017a; AVERSA et al., 2017b; AVERSA et al.,
2016a; AVERSA et al., 2016b; AVERSA et al., 2016c; AVERSA et al., 2016d; CAO et
al., 2013; DONG et al., 2013; GARCIA et al., 2007; GARCIA-MURILLO et al., 2013;
GOUGH-STEWART PLATFORM; HE et al., 2013; LEE, 2013; LIN et al., 2013; LIU et
al., 2013; MELO et al., 2012; MIRSAYAR et al., 2016; PADULA; PERDEREAU, 2013;
PERUMAL; JAWAHAR, 2013; PETRESCU; PETRESCU, 2011, ; PETRESCU; PETRESCU, 2012, ;
PETRESCU; PETRESCU, 2014, ; PETRESCU; PETRESCU, 2016a; PETRESCU; PETRESCU,
2016b; PETRESCU et al., 2016a; PETRESCU et al., 2016b; PETRESCU, 2012; REDDY et
al., 2012; TABAKOVIĆ et al., 2013; TANG et al., 2013; TONG et al., 2013;
WANG et al., 2013; WEN et al., 2012).
In addition to the research, robots
humanoids are developed to perform human activities, such as personal assistance,
where they would be able to help places of work diseased and the elderly and
dirty or dangerous. Workplaces ordinary, such as to be a yacht or a worker of a
production line of cars are also suitable for the humanoids." In essence,
as they can use tools and operate the equipment and vehicles designed to human
form, those humanoids could carry out, theoretically, any load a human being
may, as long as they have the software itself. However, the complexity to do
this is deceptively big.
They are also more popular for the
provision of entertainment. For example, Ursula, food Sex Female, sing, play
music, dances and speaks to the public her from Universal Studios. More
highlights Disney hire the use of animatrons, robots that look, move and speak
in the same way as human beings, in some thematic shows.
These animatrons look so realistic
that it can be difficult to decipher the remote whether or not they are in fact
they are human. Though they look realistic, they do not have yet any cognitive
autonomy or natural. Various robots humanoids and possible their applications
in everyday life are presented in a documentary film independently, called Plug
and Pray, which has been launched in 2010.
Robots humanoids, in particular with
the algorithms of artificial intelligence, could be useful for future dangerous
mission and/or at a high distance for the spatial scan without the need to turn
around again and to get back on the ground once the mission is completed.
A sensor is a device which measures
some attribute of the world. As one of the three primitives of robotics (apart
from the planning and control), detection plays an important role in the fault
finding sequential paradigms.
Sensors can be classified on the
basis of the physical process which works with or, depending on the type of
metering information which they give that output.
Proprioceptive
sensors sense the position, the orientation and speed of the rubber body of
humanoid.
In addition, people do not use their
own proprioceptive sensors (e.g., to the touch, muscular extension, limb
position) to help with robots Humanoid orientation. Their uses accelerometers
to measure the acceleration, from which the speed can be calculated by means of
the integration; tilt sensors to measure the tilt; sensor of force placed on
her arms and legs to measure the force of contact with the robot environment;
position sensors, which indicates the actual robot position (from which the
speed can be calculated by the derivation of the movement laws) or even the
speed sensors.
The arrays tactels can be used to
provide data on what has been reached. The shadow of the hand uses an array of
34 tactile arranged under the skin of polyurethane on each finger. Touch
sensors also provide information about the forces and the torques transferred
between the robot and the other objects.
The vision (view) refers to the
processing of data in any way that uses the electromagnetic spectrum to produce
an image. In the robots, humanoids are used to recognize the objects and
determine their properties (They put the sensors to the works at more than in a
similar way the eyes of human beings). Most robots humanoids use CCD cameras
that the sensors.
Sensors allow sound robots humanoids
to hear the speech and the sounds of the environment and to carry out the
functions that the ears of the human being. Microphones are usually used for
this task.
Actuators are the motors responsible
for the movement in and of the robot.
Robots humanoids are constructed in
such a way that they mimic the human body so that they can use the actuators
which carried out the movements such as the muscles and joints, though with a
different structure. To obtain the same effect as the human movement, robots
humanoids use actuator in rotating main. They may be either electrical wiring,
pneumatic, hydraulics, piezoelectric, ultrasound.
Actuators hydraulic and electrical
have a behavior very rigid and may be made only to act in a manner consistent
with the, through the use of strategies relatively complex for the control of the
feedback. While the electrical components of the motor actuation using cored
are more suited for high speed and low load, hydraulic works well at low speed
and high load.
Elements of the piezoelectric
actuator generate a movement with a large capacity of force when it is applied
to the voltage. They can be used for positioning the ultra-fine and for
generating and handling large forces or pressure in situations static and
dynamic.
Elements of the actuator with
ultrasound are designed to produce movements in an order micrometer at
frequencies ultra-sound (over 20 kHz). They are useful for vibration control
applications, positioning, and fast switching.
Elements of the pneumatic actuator
operate based on the compressibility of the gas. As they are inflated, extend
along the axis and how to deflate, contracts. In the case where an end is
fixed, the other will move in a linear trajectory. These components are
intended for low speed and low load/average. Between the components of the
pneumatic actuator are cylinders the gaiter, motors pneumatic, stepper motors
gauge and of the artificial muscles pneumatic.
In the planning and control, the
essential difference between the humanoids and other types of robots (such as
industrial), is the fact that the robot move must be human consumption as it
may be, using locomotion with feet, in particular, lever biped. Planning the
ideal for the movements of the humanoids during the normal course should lead
to minimize power consumption, as it happens in the human body. For this
reason, the studies on the dynamics and control of these types of structures
are becoming increasingly important.
The problem of walking and of the
stabilization on the surface for the robots is of great importance. Maintenance
of the center of gravity of the robot over the center of the camp in order to
ensure a stable position can be chosen as an objective of the control. In order
to maintain the dynamic balance during their walk and a robot needs information
on the contact force and the movement to the actual and desired. The solution
to this problem is based on a major concept, Zero Point Time (ZMP).
Another feature of the robots
humanoids is that moves, gather information (using sensors) to "real
world" and to interact with her. They do not remain as other manipulators
robots who work in environments very structured. In order to enable the
humanoids to move in complex environments, planning and control must focus on
the detection of self-collision, planning and the way of avoiding obstacles.
The humanoids have not yet some
features of the human body. These include structures with the variable
flexibility to provide a fuse (to the robot in itself and for the people) and
redundancy movements, i.e., more degrees of freedom and availability task, therefore,
at the level. With all that these features are desirable for the robots
humanoids, they will bring more complexity and new problems of planning and
control. The field of dealing with the control of the whole body with these
problems and to address proper coordination of many degrees of freedom, for
example in order to carry out more tasks simultaneously control, while in the
following an order given priority.
Robotic screwing unit with automatic
feeding of screws are automatic machines with anthropomorphic arms: extremely
flexible in all aspects; they allow to screw on different planes and have a
high reconversion factor. In case of change of product or mode of production,
the arm can be used in the most diverse applications.
Anthropomorphic industrial robots
have become the most prevalent and most used. They are most prevalent on the
planet because they were very well put in place and are more easily designed,
manufactured and implemented, compared to other types of robots and
manipulators. The most common is the structure of with a base made up of three
rotating elements, 3R. It is a mechanical structure, furniture, with three
degrees of mobility, easy designed, with a high mobility and a large workspace.
They are big advantages it has established itself in the world of industrial
robots and was generalized.
Like all industrial robots and this
anthropomorphic structure, it was launched in the auto industry, which
commissioned and produced almost all modern industrial robots. The main
advantages of a structure of this kind are great mobility, a wider working
space, a good dynamic, fast-moving and acceptable accuracy for industrial
operations daily conjunction with most common.
When it comes to reliability and
stability excessive anthropomorphic structure can’t cope the tasks, she
successfully being replaced by parallel structures.
Today the moving mechanical systems
are utilized in almost all vital sectors of humanity. The robots are able to
process integrated circuits sizes micro and nano, on which the man they can be
seen only with electron microscopy. Dyeing parts in toxic environments, working
in chemical and radioactive environments or at depths and pressures at the deep
bottom of huge oceans, or conquest of cosmic space and visiting some new
exoplanets (PETRESCU et al., 2017a; PETRESCU et al., 2017b; ; PETRESCU et al.,
2017c; PETRESCU et al., 2017d; PETRESCU et al., 2017e; PETRESCU et al., 2017f;
PETRESCU et al., 2017g; PETRESCU et al., 2017h; PETRESCU et al., 2017i;
PETRESCU et al., 2017j; PETRESCU et al., 2017k; PETRESCU et al., 2017l;
PETRESCU et al., 2017lm; PETRESCU et al., 2017n; PETRESCU et al., 2017o;
PETRESCU et al., 2017p; PETRESCU et al., 2017q; PETRESCU et al., 2017r;
PETRESCU et al., 2017s; PETRESCU et al., 2017t; PETRESCU et al., 2017u;
PETRESCU et al., 2017v; PETRESCU et al., 2017w; PETRESCU et al., 2017x;
PETRESCU et al., 2017y; PETRESCU et al., 2017z; PETRESCU et al., 2017ab;
PETRESCU et al., 2017ac; PETRESCU et al., 2017ad; PETRESCU et al., 2017ae), are
with robots systems possible and were turned into from the dream in reality
because of use of mechanical platforms sequential gearbox.
The man will be able to carry out
its mission supreme, conqueror of new galaxies, because of mechanical systems
sequential gear-box (robotics systems). Robots were developed and diversified,
different aspects, but today, they start to be directed on two major
categories: systems serial and parallel systems. Parallel systems are more
solid but more difficult to designed and handled, and for this reason, the
serial systems were those which have developed the most. In medical operations
or radioactive environments are preferred mobile systems parallel, because of
their high accuracy positioning.
Moving mechanical systems parallel
structures are solid, fast and accurate. Mechanical systems in motion type
parallel structures are solid, fast and accurate. Between mobile systems
parallel the best known and used system is that of a Stewart platform, as being
and the oldest system, quickly, solid and accurate.
A platform Gough-Stewart is a type
of parallel robot which has six actuators prismatic, frequently winches electric
or hydraulic actuators attached in pairs at three positions on the base plate
of the platform, passing over the three mounting points on a top plate. The
devices placed on the top plate can be moved in the six degrees of freedom in
which it is possible that a body free-suspended to move. These are the three
movements linear x, y, z (lateral, Longitudinal and vertical) and the three
revolutions step, roller, & yaw sensor. The terms "six axes" or
"6-DOF" (degrees of freedom), the platform is also used, also
"Synergy" (see below).
This specialized aspect of six Jack
has been used for the first time by VE (Eric Gough) in the United Kingdom and
has been operational in 1954, design later being made public in a document 1965
D Stewart on British Institute Engineers mechanics. Although the short title
Stewart Platform is now used for this look Jack, it would be more appropriate
to Eric Gough to call it a platform Gough/Stewart. To be more specific, the
original platform Stewart has had a design slightly different. See references
for more detailed at the end of this Article.
To ensure that movements are
produced by a combination of movements of several collars, such a device is
sometimes called a movement synergistic platform due to the synergy (reciprocal
interaction between the manner in which the sockets are programmed. Because the
device has six jacks, is often, also known as a hexapod (six feet). Trademark
the name "hexapod" (through geodesic technology) was originally for
platforms of Stewart used in machine tools. However, the term is now used for
platforms of 6-jack outside the machine tool, since this simply means "six
feet".
The paper presents a few main
elements of the Stewart platforms. Begin with the study of geometric, kinematic
elements of the system and then shall be presented and some elements of
dynamics.
In the event that a structural motto
element consists of two elements in a relative movement from a structural point
of view, the drive train and especially the dynamic it is more convenient to
represent the motto element as a single component mobile. In this way remain
seven elements in movement (the six motto elements or feet, to which shall be
added the mobile platform 7) and a fixed component. Kinematics of positions
shall be determined by an original method of analytical geometry (Fig. 1). The
study of mechanical solids is achieved by means of specific calculations.
Figure 1:
The structure and geometry of a Stewart platform
2.
MATERIALS AND METHODS: STRUCTURE AND
GEOMETRY OF A STEWART PLATFORM
An
equilateral triangle in the lower and upper platform circles is used to
simplify calculations. The base is the ABC (fixed) triangle with the xOyz
fixed, rectangular axle system, and for the mobile (upper) platform, the DEF
(mobile platform) triangle is adopted. The center of the fixed triangle is O,
and the mobile triangle is S.
The
reverse kinematics is much easier to determine, but it will still be studied
for rational reasons, making it more logical to impose certain successive
positions of the mobile platform (which it must occupy in turn) and, on their
basis, determine the length of the six arms or legs corresponding to each
position imposed in part.
In
Figure 2 we determine the position parameters (spatial Cartesian coordinates)
for fixed points A, B, C. For point A we obtain x = r and y = z = 0.
For
point B the relations (1) are used, and the system (2) is considered for the
determination of the coordinates of point C.
Figure 2:
Base geometry (fixed plane) ABC
(1)
(2)
For the DEF mobile platform (see figure 3) the equations
(3) can be written.
(3)
Figure 3:
DEF mobile plan, geometry
It is determined the spatial coordinates of the point D,
when the height of this point is known, h, ie the coordinate of the zD,
the radius R is known, all the coordinates of the central point S of the upper
platform are known. We mention that all the coordinates of the points A, B and
C have already been determined and are already known.
The coordinates x and y of point D must be determined
because z coordinate is already known. It solves the system formed by the first
two equations and ultimately obtains the last two relationships that generate
the solutions yD and xD. We basically use the equation of
the sphere having the center in S and the radius R to reach point D. However,
we need the equation of the center circle S and the radius R that is inscribed
in the plane of the mobile triangle. In order to obtain from the sphere a
circle we intersect the sphere with the plane of the mobile triangle (PETRESCU;
PETRESCU, 2014).
Two equations with three unknowns appear instead of an
equation, but an unknown one disappears by intersecting our circle with the
horizontal plane of height h known, h being the height at which the point D is
to be found. In this way the two written equations will remain with only two
unknowns, xD and yD, since zD becomes known
being identical to the height h. The system of the two equations with two
unknowns is solved and the values yD and xD are obtained
respectively. For yD, two different situations are required, for
which two different equations are used.
The general case when alpha is different from zero is
solved with the equation obtained from the system, and the particular case in
which alpha is equal to zero is solved using the same equation to which the
boundaries have been applied and so the equation has changed its shape
initially, losing the alpha value from the denominator. In the program, an if
logical counter was used for these distinct situations. Next, the coordinates
of points F and E are easily determined by an original rotation method (PETRESCU;
PETRESCU, 2014) using equations 4 and 5.
With the known coordinates of points D, E, F imposed by
the position of the DEF plane and the choice of point D, the necessary lengths
of the legs (motor elements) are determined (see relations 6).
(4)
(5)
(6)
The computing program used (written in excel) will be
presented in Appendix 1.
3.
RESULTS AND DISCUSSION
One applied the computational
relationships for some possible situations and the program worked correctly. If
the input parameters are not correct, in the sense that they cannot be met by
the Stewart platform, then the program will not work (relays in relationships
will vehemently oppose unrealistic situations described by inappropriate input
parameters). Table 1 shows two different cases.
Table 1: Two different cases
|
Calculation example 1. |
Calculation example 2. |
||||
|
|
A |
B |
|
A |
B |
|
7 |
h[m] |
1.3 |
7 |
h[m] |
1.276 |
|
8 |
R[m] |
0.1 |
8 |
R[m] |
0.1 |
|
9 |
xS[m] |
0 |
9 |
xS[m] |
0 |
|
10 |
yS[m] |
0 |
10 |
yS[m] |
0 |
|
11 |
zS[m] |
1.3 |
11 |
zS[m] |
1.3 |
|
12 |
a[] |
0 |
12 |
a[] |
0.707 |
|
13 |
b[] |
0 |
13 |
b[] |
0 |
|
14 |
g[] |
1 |
14 |
g[] |
0.707 |
|
15 |
zD[m] |
1.3 |
15 |
zD[m] |
1.276 |
|
16 |
yD[m]a=0 |
0.1 |
16 |
yD[m]a=0 |
0.116667 |
|
17 |
yD[m]aǂ0 |
#DIV/0! |
17 |
yD[m]aǂ0 |
0.094281 |
|
18 |
yD[m] |
0.1 |
18 |
yD[m] |
0.094281 |
|
19 |
xD[m] |
0 |
19 |
xD[m] |
0.02357 |
|
20 |
xE[m] |
-0.0866 |
20 |
xE[m] |
-0.09343 |
|
21 |
yE[m] |
0.05 |
21 |
yE[m] |
0.004466 |
|
22 |
zE[m] |
1.3 |
22 |
zE[m] |
1.264645 |
|
23 |
xF[m] |
0.086603 |
23 |
xF[m] |
0.069865 |
|
24 |
yF[m] |
0.05 |
24 |
yF[m] |
0.004466 |
|
25 |
zF[m] |
1.3 |
25 |
zF[m] |
1.264645 |
|
26 |
xA[m] |
0.1 |
26 |
xA[m] |
0.1 |
|
27 |
yA[m] |
0 |
27 |
yA[m] |
0 |
|
28 |
zA[m] |
0 |
28 |
zA[m] |
0 |
|
29 |
xB[m] |
-0.05 |
29 |
xB[m] |
-0.05 |
|
30 |
yB[m] |
-0.086 |
30 |
yB[m] |
-0.086 |
|
31 |
zB[m] |
0 |
31 |
zB[m] |
0 |
|
32 |
xC[m] |
-0.05 |
32 |
xC[m] |
-0.05 |
|
33 |
yC[m] |
0.086 |
33 |
yC[m] |
0.086 |
|
34 |
zC[m] |
0 |
34 |
zC[m] |
0 |
|
35 |
l1[m] |
1.307 |
35 |
l1[m] |
1.282 |
|
36 |
l2[m] |
1.314 |
36 |
l2[m] |
1.291 |
|
37 |
l3[m] |
1.307 |
37 |
l3[m] |
1.268 |
|
38 |
l4[m] |
1.301 |
38 |
l4[m] |
1.268 |
|
39 |
l5[m] |
1.307 |
39 |
l5[m] |
1.272 |
|
40 |
l6[m] |
1.301 |
40 |
l6[m] |
1.265 |
This original algorithm and computing program (presented in the appendix) manages to greatly ease the designer's work on such difficult systems.
4.
APPLICATIONS
In
the 1800s, Augustin Louis Cauchy, a pioneer in mathematical analysis, studied
the stiffness of an "articulated octahedron" which is the ancestor of
the hexapod. In 1949, V. E. Gough advanced in research and built a parallel
mechanism to test tires under different loads.
A
few years later, in 1965, D. Stewart began using a variant of the hexapod for
flight simulators. The robot he built will be renamed on his behalf the
"Stewart Platform". Over the years, the hexapod has been improved by
sever-al engineers such as K. Cappel, Mc Callion etc.
A
platform Gough-Stewart is a type of parallel robot which has six actuators
prismatic, frequently winches electric or hydraulic actuators attached in pairs
at three positions on the base plate of the platform, passing over the three
mounting points on a top plate (Gough-Stewart platform,
from Wikipedia).
The
devices placed on the top plate can be moved in the six degrees of freedom in
which it is possible that a body free-suspended to move.
These
are the three movements linear x, y, z (lateral, Longitudinal and vertical) and
the three revolutions step, roller, & yaw sensor. The terms "six
axes" or "6-DOF" (degrees of freedom), the platform is also
used, also "Synergy".
This
specialized aspect of six Jack has been used for the first time by VE (Eric
Gough) in the United Kingdom and has been operational in 1954, design later
being made public in a document 1965 D Stewart on British Institute Engineers
mechanics. Although the short title Stewart Platform is now used for this look
Jack, it would be more appropriate to Eric Gough to call it a platform
Gough/Stewart. To be more specific, the original platform Stewart has had a
design slightly different.
To
ensure that movements are produced by a combination of movements of several
collars, such a device is sometimes called a movement synergistic platform due
to the synergy (reciprocal interaction between the manner in which the sockets
are programmed.
Because
the device has six jacks, is often, also known as a hexapod (six feet).
Trademark the name "hexapod" (through geodesic Technology) was
originally for platforms of Stewart used in machine tools. However, the term is
now used for platforms of 6-jack outside the machine tool, since this simply
means "six feet".
The
presented system may be useful in particular to the surgical robots which
operate patients; those systems require a very high accuracy of positioning.
Such
systems of high precision of positioning may be useful in particular for the
future operations on the brain, heart, liver, kidneys, but also to prosthesis
miscellaneous.
These
platforms can position very accurately even some very large weights, such as to
the modern telescope stationary.
The
design of the Stewart platform is widely used in the simulation of the flight,
in particular in the so-called flight simulator full for which there is a need
for all 6 degrees of freedom. This application has been developed by Redifon,
whose simulators offering has become available for Boeing 707, Douglas DC-8,
South Aviation Caravelle, Canadair CL-44, Boeing 727, the Comet, Vickers
Viscount, Vickers Vanguard, Convair CV-990, Lockheed C130 Hercules, Vickers
VC10 and Fokker F-27 1962.
In
this role, the payload is the pilot reply and a system of visual display,
normally in the order of several channels, in order to show the visual scene
out of the world the crew of the aircraft, which are trained. Weights in the
case of the payload of a flight simulator full for an airplane of large
transport may be up to about 15,000 kilograms.
Similar
platforms are used in simulators, mounted can usually be found on the large
meals x-y driving position in order to simulate the acceleration on a
short-term basis acceleration in the long term, can be simulated by tilting the
platform and an area of active research is how to mix the two.
Eric
Gough has been an engineer auto and has worked at the Redoubt Dunlop, factory
Dunlop tires of the Birmingham, England. He developed or "Universal
Tir-Testare Machine" (also called "Universal Rig") and in 1950
and the platform was operational in 1954. The device has been able to
mechanically tires tested in accordance with the combined tasks. Dr. Gough died
in 1972, but the testing of its platform continued to be used up to the end of
1980 when the factory was closed and then demolished. His rig has been saved
and transported to the marginal storage Science Museum (London), at Wrought on
near Swindon.
The
AMiBA radio telescope, a Cosmic Microwave Background experiment, is mounted on
a 6 m carbon fiber hexapod. A hexapod robot is a walker robot whose locomotion
is based on three pairs of legs. The study of the progress of insects is of
particular interest to present an alternative to the use of wheels. The term
thus refers to robots of biological inspiration imitating in the present case
hexapod animals such as insects.
Hexapod
robots are considered more stable than biped robots because in most cases
hexapods are statically stable. Because of this, they do not depend on
real-time controllers to stand or walk. However, it has been shown that at high
displacement rates, insects are dependent on dynamic factors.
Insects
were chosen as models because their nervous system is simpler than that of
other animal species.
In
addition, complex behaviors can be attributed to only a few neurons and the
path between sensory inputs and motor outputs is relatively short.
The walking behavior of
the insect and the neural architecture are used to improve the locomotion of
the robot. Conversely, biologists use hexapod robots to test different
hypotheses.
5.
CONCLUSIONS
The paper presents an exact, original analytical geometry method for determining the kinematic and dynamic parameters of a parallel mobile structure.
Compared with other methods already known, the presented method has the great advantage of being an exact analytical method of calculation and not one iterative-approximately.
6.
FUNDING INFORMATION
Research contract: Contract number 36-5-4D/1986 from 24IV1985, beneficiary CNST RO (Romanian National Center for Science and Technology) Improving dynamic mechanisms.
Contract research integration. 19-91-3 from 29.03.1991; Beneficiary: MIS; TOPIC: Research on designing mechanisms with bars, cams, and gears, with application in industrial robots.
Contract research. GR 69/10.05.2007: NURC in 2762; theme 8: Dynamic analysis of mechanisms and manipulators with bars and gears.
4-Labor contract, no. 35/22.01.2013, the UPB, "Stand for reading performance parameters of kinematics and dynamic mechanisms, using inductive and incremental encoders, to a Mitsubishi Mechatronic System" "PN-II-IN-CI-2012-1-0389".
All these matters are copyrighted! Copyrights: 394-qodGnhhtej, from 17-02-2010 13:42:18; 463-vpstuCGsiy, from 20-03-2010 12:45:30; 631-sqfsgqvutm, from 24-05-2010 16:15:22; 933-CrDztEfqow, from 07-01-2011 13:37:52.
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Appendix
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|
|
Calculation program |
|
|
A |
B |
|
|
|
|
|
7 |
h[m] |
=1.3 |
|
8 |
R[m] |
0.1 |
|
9 |
xS[m] |
0 |
|
10 |
yS[m] |
0 |
|
11 |
zS[m] |
1.3 |
|
12 |
[] |
0 |
|
13 |
[] |
0 |
|
14 |
[] |
1 |
|
15 |
zD[m] |
=B7 |
|
16 |
yD[m]a=0 |
=B10+(B11-B15)*B14+B8 |
|
17 |
yD[m]aǂ0 |
=B10+((B11-B15)*B13*B14+B12*SQRT(B8^2*(B12^2+B13^2)-(B11-B15)^2*(B12^2+B13^2+B14^2)))/(B12^2+B13^2) |
|
18 |
yD[m] |
=IF(B12=0,B16,B17) |
|
19 |
xD[m] |
=B9+SQRT(B8^2-(B15-B11)^2-(B18-B10)^2) |
|
20 |
xE[m] |
=B9-1/2*(B19-B9)-SQRT(3)/2*(B18-B10) |
|
21 |
yE[m] |
=B10-1/2*(B13*(B15-B11)-B14*(B18-B10))-SQRT(3)/2*(B14*(B19-B9)-B12*(B15-B11)) |
|
22 |
zE[m] |
=B11-1/2*B8*B12-SQRT(3)/2*B8*B13 |
|
23 |
xF[m] |
=B9-1/2*(B19-B9)+SQRT(3)/2*(B18-B10) |
|
24 |
yF[m] |
=B10-1/2*(B13*(B15-B11)-B14*(B18-B10))-SQRT(3)/2*(B14*(B19-B9)-B12*(B15-B11)) |
|
25 |
zF[m] |
=B11-1/2*B8*B12+SQRT(3)/2*B8*B13 |
|
26 |
xA[m] |
=B8 |
|
27 |
yA[m] |
=0 |
|
28 |
zA[m] |
=0 |
|
29 |
xB[m] |
=-1/2*B8 |
|
30 |
yB[m] |
=-SQRT(3)/2*B8 |
|
31 |
zB[m] |
=0 |
|
32 |
xC[m] |
=-1/2*B8 |
|
33 |
yC[m] |
=SQRT(3)/2*B8 |
|
34 |
zC[m] |
=0 |
|
35 |
l1[m] |
=SQRT((B19-B26)^2+(B18-B27)^2+(B15-B28)^2) |
|
36 |
l2[m] |
=SQRT((B19-B29)^2+(B18-B30)^2+(B15-B31)^2) |
|
37 |
l3[m] |
=SQRT((B20-B29)^2+(B21-B30)^2+(B22-B31)^2) |
|
38 |
l4[m] |
=SQRT((B20-B32)^2+(B21-B33)^2+(B22-B34)^2) |
|
39 |
l5[m] |
=SQRT((B23-B32)^2+(B24-B33)^2+(B25-B34)^2) |
|
40 |
l6[m] |
=SQRT((B23-B26)^2+(B24-B27)^2+(B25-B28)^2) |
|
|
|
|