Adriana
Comanescu
IFToMM, Romania
E-mail: adrianacomanescu@yahoo.com
Alexandra Rotaru
IFTOMM, Romania
E-mail: alexandra.rotaru11@gmail.com
Liviu Marian
Ungureanu
IFTOMM, Romania
E-mail: ungureanu.liviu.marian@gmail.com
Florian Ion
Tiberiu Petrescu
IFToMM, Romania
E-mail: fitpetrescu@gmail.com
Submission: 1/26/2021
Accept: 3/8/2021
ABSTRACT
The positional modeling of the 2T6R robot mechanism is done for inverse kinematics, i.e. when the imposed positions of the end effector T, imposed, belonging to the final element 3, are known and the necessary positions and speeds of the two input motors, the two leading elements, are determined, 1 and 6. It is proposed to solve a simple algorithm in the program MathCad 2000, which uses for initiation the logical function If Log. The kinematic output parameters, i.e. the parameters of the foot and practically of the final effector, i.e. those of the point marked with T, will be determined for initiating the working algorithm using the logical functions, "If log (ical)", with the observation that here plays the role of parameters input; is positioned as already specified in reverse kinematics when the output is considered as input and input as output. The logical functions used, as well as the entire calculation program used, were written in Math Cad 2000.
Keywords: If Log; Math Cad; Robot; 2T6R
robot; Kinematics;
Inverse kinematics
1.
INTRODUCTION
The
industrial robot is an integrated mechanical electronic informational system,
used in the production process in order to achieve manipulation functions
analogous to those performed by human hands, giving the manipulated object any
freely programmed movement, within a technological process that takes place in a
specific environment.
Intelligent
robots represent the highest stage of development, at which the sensors are
much more numerous and more complex, there are specific blocks and subsystems
for moving and orienting their sensors, for measuring their movement, for
processing information.
1)
Trajectory generating mechanism (MGT): the mechanism
formed by those kinematic couplings that make possible the displacement of the
characteristic point M on the imposed trajectory. To generate the T trajectory,
3 degrees of freedom are also necessary: rotation around the Oz axis; vertical
displacement along the Oz axis, and a radial displacement along the x-axis.
2)
The orientation mechanism (MO) is the mechanism formed
by the kinematic couplings that ensure the spatial orientation of the object, ie the mechanism that rotates after x ', y', and z
'(palm-forearm of the human hand).
3)
The gripping mechanism (MP) ensures the gripping and
fixing of the manipulated object. If instead of handling we need welding,
painting, cutting, processing, measuring ..., then the end effector will no
longer be a gripper (gripping hand) but another corresponding final effector
element.
Classification
in terms of trajectory generation:
Robots
with continuous positioning (in which the trajectory is generated
continuously), which involves special blocks for correlating movements on 2 or
3 degrees of freedom, are called motion interpolators. The drive system and the
control system must be suitable for this mode of operation. There must always
be a well-defined bi-univocal correspondence between the command-movement.
The
control system must be able to manage the movements on each degree of freedom
and to correlate the movements with each other, in the sense of generating the
mathematically described trajectory. Controllers, sensors, motion limiters are
needed, in addition to the actuation system with actuators (electric or
hydraulic motors, rarely pneumatic), for actuation, command, and permanent
control of the realized movement.
The
controller is in fact a microchip, a microprocessor, which controls the whole
process of the robot, from head to tail, through system drivers some
specialized programs that control all the movements of the robot, these drivers
being in constant contact with the machine, system, and computer, direct and
reverse connections. The control drivers perform practically all the necessary
commands, in the sense that they will move from the microchip (central unit and
controller) to the robot, actuators (motors) effectively commanding the
necessary imposed or self-detected movements (to the latest generation
intelligent robots).
The
drivers also check the execution of the movement by the entire robot mechanism
and if the elements of the robot mechanism are in the indicated parameters
(prescribed by the controller and microprocessor). In particular, the permanent
positions of the end-effector end element, with the end-effector point T, its
trajectory, and the sequential positions occupied in time are checked, so that
they correspond to the commands given by the microprocessor, and the controller
verifies their accuracy within certain prescribed limits. Whether the final
element is within the prescribed limits or not is communicated by the sensors
that permanently check the system parameters.
There
are sensors for motion, speed, acceleration, shock, temperature, pressure ...
The sensors give the reverse signal showing what happens to the whole system
and especially to the final element end effect at any given time, and if the
parameters of a point, but especially those of the final element, of the T
point tracer, or effector, do not correspond at a given moment, the necessary
correction is made immediately, the movement to the next step being corrected
accordingly on each axis, more or less as the case may be.
The
role of the limiters is not to let certain moving elements exceed certain
limits. For example, they will stop the rotational movement reaching a certain
angle and will control the reverse movement, ie the
reverse rotation to the other end where the process will be reversed again.
Both motion sensors and limiters are built according to the principles of
transmitters, being generally very small.
There
are also robots with sequential positioning.
Mechanic
geometric parameters: Guide device: the set of all kinematic torques that
compete to achieve the trajectories and spatial orientation of the manipulated
objects within the imposed limits (MGT + MO). Final effector: clamping
mechanism (in case of handling robots) or device (in case of specific
operations). Load capacity: the maximum size of the mass that can be handled,
in conditions of total safety, for the most unfavorable position of the robot
and for the highest value of the acceleration that can develop it, in ascending
vertical movement.
Unfavorable
position: that position of the gripping mechanism, in which the manipulated
object is maintained and moved only under the effect of the frictional forces,
generated by the tightening action between the object and the ‘fingers’ of the
mechanism. Normalized load-bearing capacities: 0.250; 1; 2.5; 6.4; 10; 25; 64;
100 ... etc.
Classification
of robots according to the value of load-bearing capacity: Microrobots (tens of
grams); Minirobots (hundreds of grams); Medium robots
(of the order of kilograms); and Heavy robots (of the order of hundreds of kg);
(Antonescu & Petrescu, 1985;
1989; Antonescu et al., 1985a; 1985b; 1986;
1987; 1988; 1994; 1997; 2000a; 2000b; 2001; Atefi et
al., 2008; Avaei et al., 2008; Aversa et al.,
2017a; 2017b; 2017c; 2017d; 2017e; 2016a; 2016b; 2016c; 2016d; 2016e; 2016f;
2016g; 2016h; 2016i; 2016j; 2016k; 2016l; 2016m; 2016n; 2016o; Azaga & Othman, 2008; Cao et al., 2013; Dong et al.,
2013; El-Tous, 2008; Comanescu, 2010; Franklin, 1930;
He et al., 2013; Jolgaf et al., 2008; Kannappan et al., 2008; Lee, 2013; Lin et al., 2013;
Liu et al., 2013; Meena & Rittidech, 2008;
Meena et al., 2008; Mirsayar et al., 2017; Ng et
al., 2008; Padula, Perdereau
& Pannirselvam, 2008; 2013; Perumaal
& Jawahar, 2013; Petrescu,
2011; 2015a; 2015b; Petrescu & Petrescu, 1995a; 1995b; 1997a; 1997b; 1997c; 2000a; 2000b;
2002a; 2002b; 2003; 2005a; 2005b; 2005c; 2005d; 2005e; 2011a; 2011b; 2012a;
2012b; 2013a; 2013b; 2016a; 2016b; 2016c; Petrescu et
al., 2009; 2016; 2017a; 2017b; 2017c; 2017d; 2017e; 2017f; 2017g; 2017h;
2017i; 2017j; 2017k; 2017l; 2017m; 2017n; 2017o; 2017p; 2017q; 2017r; 2017s;
2017t; 2017u; 2017v; 2017w; 2017x; 2017y; 2017z; 2017aa; 2017ab; 2017ac;
2017ad; 2017ae; 2018a; 2018b; 2018c; 2018d; 2018e; 2018f; 2018g; 2018h; 2018i;
2018j; 2018k; 2018l; 2018m; 2018n; Pourmahmoud, 2008;
Rajasekaran et al., 2008; Shojaeefard
et al., 2008; Taher et al., 2008; Tavallaei & Tousi, 2008; Theansuwan & Triratanasirichai,
2008; Zahedi et al., 2008; Zulkifli
et al., 2008).
A special
character in the study of robots is the study of inverse kinematics, with the
help of which the map of the motor kinematic parameters necessary to obtain the
trajectories imposed on the effector can be made. For this reason, in the
proposed mechanism, we will present reverse kinematic modeling in this paper.
The kinematic output parameters, i.e. the parameters of the foot and
practically of the end effector, i.e. those of the point marked with T, will be
determined for initiating the working algorithm with the help of logical
functions, " If
Log(ical)",
with the observation that here they play the role of input parameters; it is
positioned as already specified in the inverse kinematics when the output is
considered as input and the input as output. The logical functions used, as
well as the entire calculation program used, were written in Math Cad 2000.
2.
METHODS AND MATERIALS
The positional modeling of the 2T6R
robot mechanism is done for inverse kinematics, ie
when the imposed positions of the end effector T, imposed, belonging to the
final element 3, are known and the necessary positions and speeds of the two
input motors, the two leading elements, are determined, 1 and 6. It is proposed
to solve a simple algorithm in the program MathCad
2000, which uses for initiation the logical function If Log.
The kinematic output parameters, i.e.
the parameters of the foot and practically of the final effector, i.e. those of
the point marked with T, will be determined for initiating the working
algorithm using the logical functions, "If log (ical)",
with the observation that here plays the role of parameters input; is
positioned as already specified in reverse kinematics when the output is
considered as input and input as output. The logical functions used, as well as
the entire calculation program used, were written in Math Cad 2000.
If it is desired that a point T of
the execution element realizes a sequence of sequences on a trajectory required
by a certain process it is necessary that through the inverse model to
establish the kinematic parameters of the active torques. The dynamic
parameters and consequently their actuation according to the process can be
determined using the directly associated model. In the following, these
characteristics for a 2T6R type manipulator robot are analyzed in detail.
The bimobile
mechanism of Figure 1 can be used in various handling applications or in technological
processes. There are six movable kinematic elements and eight kinematic
couplings, of which two active kinematic couplings of translation A and D. The
mechanism can achieve with the T end of the effector 3 any curve in a certain
plane domain. It is obtained from the bimobile and bicontour kinematic chain of Figure 2a from which derives
the direct structural model (Figure 2b) for which the base, the effector, as
well as the active kinematic torques are nominated.
The
kinematic scheme of the 2T6R plan robot presented with two cylindrical drive
motors is arranged in figure 1.
Figure 1: The kinematic scheme of the 2T6R plan robot presented with two
cylindrical drive motors
Figure 2: Structural scheme of the 2T6R mechanism
The connection of the modular groups
(Figure 3) corresponding to the direct structural model (Figure 2b) comprises
two initial active modular groups respectively GMAI (A, 1) and GMAI (D, 6), and
two passive modular groups of dyad type, i.e. GMP1 ( 2.5) and GMP1 (3.4).
Figure 3:
The connection of the modular groups corresponding to the direct structural
model (Figure 2b), comprises two initial active modular groups respectively
GMAI (A, 1) and GMAI (D, 6), and two passive modular groups of dyad type, i.e.
GMP1 ( 2.5) and GMP1 (3.4).
The structural model directly is
used to establish the algorithm for calculating the components of the reaction
torque in the kinematic torques based on the kinetic-static calculation
modules.
The inverse structural model (Figure
4a) of the mechanism allows the determination of the parameters of the active
torques A and D depending on the characteristics of the T point required by the
technological process in which the system is used, representing the theme
presented in this paper.
In the inverse structural model the
passive modular group GMP8 from Figure 4b. In the connection of the groups (Figure
5) this structure is noticeable.
Figure 4: The
inverse structural model (Figure 4a) of the mechanism allows the determination
of the parameters of the active torques A and D depending on the
characteristics of the T point. In the inverse structural model (Figure 4b) the
passive modular group GMP8.
Figure 5: The connection of the groups GMP8(1, 2, 3, 4, 5, 6)
3.
RESULTS AND DISCUSSION
The
constant geometric parameters of the mechanism of Figs. 1 are listed in Table
1.
Table 1: Constant geometric
parameters
TB |
1. |
TC |
0.8 |
AE |
0.3 |
EB |
0.8 |
AB |
1.1 |
DC |
0.8 |
DE |
0.3 |
XA |
0. |
YD |
0. |
The input parameters of the point T
corresponding to a trajectory, for example, rectilinear and alternative for a
certain interval are shown in Table 2 and plotted in FIGURE 6.
Table 2: Initial parameters of the T
point trajectory
Initial parameters
of the T point |
T0(-0.2, -0.8) |
The description of the input parameters for the
inverse model of the mechanism is presented in table 3. The trajectory of the T
point represented in figure 6 is thus described.
Table 3: Input parameters
Coordinates of
the current point T |
XTk:=if[k≤10,XT0+k.0.05,XT0+10.0.05-(k-10).0.05] YTk:=YT0 |
Figure 6: The trajectory of the point T
Figure 7 shows the dependent
positional parameters for the inverse model of the mechanism. The algorithm for
their determination is given in Table 4 and uses the connection of the modular
groups of the inverse model (Figure 5).
Figure 7: The dependent positional parameters for the inverse model of
the mechanism
The GMP 8 passive modular group (Figure
4b) has the following dependent parameters: YAk(XTk,YTk), XDk(XTk,YTk),
Φ2k(XTk,YTk), Φ3k(XTk,YTk),
Φ4k(XTk,YTk), Φ5k(XTk,YTk).
Table 4: passive modular group
The pattern |
Position-dependent
parameters |
GMP8(1,2,3,4,5,6) |
AB.cos(f2)=XTk+TB.cos(f3) YA+AB.sin(f2)=YTk+TB.sin(f3) XD+DC.cos(f4)=XTk+TC.cos(f3) YD+DC.sin(f4)=YTk+TC.sin(f3) AE.cos(f2)=XD+DE.cos(f5) YA+AE.sin(f2)=DE.sin(f5) |
BPT(C) |
XCk:=XTk+TC.cos(f3k) YCk:=YTk+TC.sin(f3k) |
BPT(B) |
XBk:=XTk+TB.cos(f3k) YBk:=YTk+TB.sin(f3k) |
BPT(E) |
XEk:=XAk+AE.cos(f2k) YEk:=YAk+AE.sin(f2k) |
Variation of the parameters of the active
kinematic translation couples YAk(XTk,YTk),
XDk(XTk,YTk) can be seen in
the Figure 8a and Figure 8b.
a
b
Figure 8: Variation of the parameters of the active kinematic
Similarly, the variation of the
angular parameters (expressed in degrees) noted Φ20k(XTk,YTk), Φ30k(XTk,YTk), Φ40k(XTk,YTk), Φ50k(XTk,YTk) will be represented graphically in figure 9.
Figure 9: The variation of the angular parameters
To perform the kinetostatic
analysis it is necessary to determine the parameters of the kinematic couples
marked with C, B, E, the trajectories being shown respectively in Figure 10a, FIGURE
10b and FIGURE 10c.
a
b
c
Figure 10: perform the kinetostatic analysis
4.
CONCLUSIONS
A special character in the study of
robots is the study of inverse kinematics, with the help of which the map of
the motor kinematic parameters necessary to obtain the trajectories imposed on
the effector can be made. For this reason, in the proposed mechanism, we will
present reverse kinematic modeling in this paper.
If it is desired that a point T of
the execution element realizes a sequence of sequences on a trajectory required
by a certain process it is necessary that through the inverse model to
establish the kinematic parameters of the active torques. The dynamic
parameters and consequently their actuation according to the process can be
determined using the directly associated model. In the following, these
characteristics for a 2T6R type manipulator robot are analyzed in detail.
The bimobile
mechanism of Figure 1 can be used in various handling applications or in
technological processes. There are six movable kinematic elements and eight
kinematic couplings, of which two active kinematic couplings of translation A
and D. The mechanism can achieve with the T end of the effector 3 any curve in
a certain plane domain. It is obtained from the bimobile
and bicontour kinematic chain of Figure 2a from which
derives the direct structural model (Figure 2b) for which the base, the
effector, as well as the active kinematic torques are nominated.
Inverse kinematic modeling is
generally the most sought after, as the most important, but in most situations,
it is also the most difficult to determine. In the presented paper, the
MathCad2000 software was used in order to facilitate the calculations, because
the software automatically solves the linear and nonlinear systems through its
internal procedures that must be called within the program.
As an important function, the "IfLog" logic function was used twice in the program to
initiate the calculations, by determining the input variables in the inverse
kinematics.
5.
ACKNOWLEDGEMENT
This
text was acknowledged and appreciated by Dr. Veturia CHIROIU Honorific member of Technical Sciences
Academy of Romania (ASTR)
PhD supervisor in Mechanical Engineering.
6.
FUNDING INFORMATION
a) 1-Research contract: 1-Research contract: Contract number 36-5-4D/1986 from 24IV1985, beneficiary CNST
RO (Romanian National
Center for Science and Technology) Improving dynamic mechanisms.
b) 2-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.
c) 3-Contract research. GR 69/10.05.2007: NURC in 2762; theme 8: Dynamic analysis of mechanisms and manipulators with bars and gears.
d) 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".
e) 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.
7.
ETHICS
Authors
should address any ethical issues
that may arise after the publication of this manuscript.
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