STUDY
OF FORCES IN A 2T9R ROBOT MECHANISM
Adriana
Comanescu
IFToMM, Romania
E-mail: adrianacomanescu@yahoo.com
Alexandra Rotaru
IFToMM, Romania
E-mail: alexandra.rotaru11@gmail.com
Florian
Ion Tiberiu Petrescu
IFToMM, Romania
E-mail: fitpetrescu@gmail.com
Submission: 1/18/2021
Accept: 1/20/2021
ABSTRACT
The paper presents in detail a method of calculating the forces acting on a 2T9R type robot. In order to determine the reactions (forces in the kinematic couples), one must first determine the inertial forces in the mechanism to which one or more useful loads of the robot can be added. The torsor of the inertia forces is calculated with the help of the masses of the machine elements and the accelerations from the centers of mass of the mechanism elements, so the positions, velocities, and accelerations acting on it will be determined, i.e. its complete kinematics. The calculation method applied by a MathCad program intelligently uses data entry through the IFLOG logic function so that the calculations can be automated. So the effective automation of the calculation program is done exclusively through the IFLOG functions originally used in the paper.
Keywords: IFLOG; Robot; 2T9R robot; Forces;
Kinematics; Geometric-analytical method; Direct kinematics; Inverse kinematics
1.
INTRODUCTION
Robots have always fascinated us,
but today we use them massively, in almost all industrial areas, especially
where they work hard, repetitive and tiring, in toxic, chemical, radioactive
environments, underwater, in the cosmos, in dangerous environments, on mined
lands, in hard to reach areas, etc. It can be said once again that, just as
software and microchips have helped us to quickly write various useful programs
and implement them directly, so robotics has made our daily work much easier.
Thanks to robots, automation is
almost perfect today, product quality is very high, the manufacturing price has
dropped a lot, you can work in continuous fire, people have escaped hard work,
tiring, repetitive, in toxic environments and now can treat other problems more
important, such as design, scientific research, to work only 5 days a week with
high income and, in the future, due to the massive implementation of
increasingly modern robots with increased capabilities, man will reach the work
week only 4 days.
An even greater increase is expected
in the number of specialized robots implemented in large factories and
factories around the world.
Due to
the massive use of industrial robots, the diversification in this field has gained high levels. For
this reason, we want to
study in this paper a new robot model, 2T9R, extremely complex in movements, useful in any type of
work, a versatile robot, which can weld, cut, process different
parts, to assemble them, or to manipulate
them from one working strip
to another, and in the same way,
he can also paint the different machined
components before their assembly.
The robot has various advantages due to its
complex mode arranged since the design and will be able to
easily adapt to any type
of automated manufacturing cell. For this reason,
and because it is an original one and has not been studied before,
we want that
in this paper we review its
study completely with the determination
of all the
forces that act it and that
appear within it, the one
that it also
requires a complete kinematic
calculation (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).
2.
METHODS AND MATERIALS
The
present study will start with a description of the 2T9R robot proposed to be
analyzed, in terms of the forces acting on it. The 2T9R mechanism (Figure 1)
has a constructive model based on a bimobile
kinematic chain having three independent contours (Figure 2a) obtained from the
bicontour chain of the 2T6R mechanism.
Figure 1: The mechanism 2T9R
The
direct structural model (Figure 2b) consists of two initial active modular
groups GMAI (A, 1) and GMAI (G, 8) which constitute the linear motors that
drive it and two passive modular groups, one of the type of the GMP2 triad (2,
3,4,6) and the other of the GMP1 dyad type (5,7). The connection of the modular
groups for the direct model is shown in Figure 3.
Figure 2: Structural scheme of the mechanism
Figure 3: Electronic or wiring diagram (block diagram) of the mechanism
The
direct structural model (Figure 2b) and the connection of the corresponding
modular groups (Figure 3) are used to determine the reaction torsor in each kinematic coupling using the kinetostatic principle.
To
study the main plane mechanism of the 2T9R robot, its kinematic elements, kinematic
torques, and positioning angles of the elements that also have rotation are
initially established (Figure 4).
Figure 4:
Determining the kinematic elements, the kinematic torques, and the angles that
position the elements that also have a rotation
For
the kinetostatic analysis (determination of the
forces in the mechanism) the centers of mass marked with the letter T (Figure
5) are positioned as follows: O ≡ T5 ≡ T4; B ≡ T2 ≡ T3;
E ≡ T6; F ≡ T7. Their placement does not influence the algorithm
for calculating the components of the reaction torsion in the kinematic
torques.
Figure 5: Positioning the centers of mass T of all the elements of the
mechanism
It
is considered a single external force RT acting on the system neglecting other
external forces (for example - gravitational forces). This simplification
brings some peculiarities in the form of terms from the calculation algorithm
without restricting its generality. The forces of weight are not recommended to
be introduced in the sizing calculations because their influence is sometimes
by addition and sometimes by decrease it being therefore opposite and having
negative effects on the sizing of a mechanism. On the other hand, in large
(large) robots, if they still work fast (at high speeds), the inertial forces
(internal forces, which arise even in the mechanism due to its masses) are
considerable and much higher than those weights that automatically become
negligible.
2.1.
Determination of Reactions in the
kinematic torques of the triad (2,3,4,6)
The study of forces is always
processed inversely to the kinematic one, ie not from
the motors to the final effector element, but inversely, from the modular group
furthest from the motors to them. For this reason, the force calculations start
on the triad (2,3,4,6) from Figure 6.
Figure 6:
The forces on the triad (2,3,4,6). The known forces are shown in blue; the
reactions (unknown forces in the kinematic couplings) are drawn in green.
To determine the unknown forces, the
reactions (from the kinematic couplings), the following calculation relations
are written (from 2 ROx is made explicit, from 3 RAx, which is introduced in relation 1 and I is obtained,
and in relation 4 and II is obtained, where I and II represent two linear
equations with two unknowns that make up a linear system that can be solved
immediately by Kramer III):
With
(IV) on determines ROx si
RAx:
From (5) results relation (V) which
determines REx, and from (6) results the
expression (VI) which generates REy:
In order to perform the triad
calculations (2,3,4,6) it is necessary to present briefly the expressions by
which the known inertial forces, inside the mechanism, due to the masses of the
component elements (16-20) are determined by calculations:
2.2.
Determination of Reactions in the
kinematic couplings of the dyad (5,7)
Dyad 5.7 has the following charges (Figure
7), where the already known forces are shown in blue, and the unknown ones in
green, i.e. the reactions in the kinematic torques of the dyad, which will be
determined.
Figure 7:
Forces of the dyad 5-7
Can
write the relations 21-22:
From relation (22) one explicitly
reaction RGy (24) which is introduced in
relation (21) obtaining directly the value RGx
(23), and then RGy (24):
Now, one write the relations
(25-30):
The torsor
of the inertial forces on dyad 5,7 is determined by the relations (31-32):
2.3.
Determination of the reactions in
the kinematic torques of the motor element 8 and calculation of the driving
force Fm8
Figure 8 shows all the forces acting
on the linear motor element 8, in the rotation torque G (between elements 8 and
7) and in the translation torque T8 (between elements 8 and 0) materialized by
the guideline between the motor piston 8 and its axis of vertical symmetry
coinciding with the guide 0, considering as the point of actuation of the
forces 08 the center of mass T8. The forces in the torque are the x-axis and
y-axis projections of the already known R78 reaction (thus shown in dark blue).
Also known the torsion of the
inertial forces on element 8, represented here only by an inertial force along
the guide axis y (its action being concentrated in the center of mass T8),
there is no movement on the x-axis acceleration and automatic and force
inertial on this x-axis is canceled, and the inertial moment is also canceled
permanently because there is no rotational motion, the angular and automatic
acceleration and the inertial moment being canceled.
Figure 8:
Forces acting on the engine element 8
The
driving force that moves the linear motor element 8 also acts in the center of
mass. Practically except for the reaction in coupling G all other forces act on
the center of mass T8. Relationships can be written (33-36):
It is specified here that if the
points G and T8 coincide the moment M08 is canceled together with the phase
shift (=0).
The
procedure is then repeated for engine 1 (Figure 9, relations 37-40).
2.4.
Determination of the reactions in
the kinematic torques of the motor element 1 and calculation of the driving
force Fm1
Figure 9:
Forces acting on the engine element 1
It is specified that if points A and
T1 coincide the moment M01 is canceled together with the phase shift (=0).
Remarks:
Any torque introduces a reaction that decomposes along the coordinate axes (in
the plane) into two components along the x and y axes, while each translation
torque introduces a reaction perpendicular to the torque guide axis and a
moment.
Any
reaction in any pair is easily determined by having the modulus (size) given by
the radical in the sum of the squares of the two scalar components of the
reaction, and its position (the direction of the vector defining it) is given
by an alpha angle measured from the horizontal which passes through the origin
of the reaction (the respective coupling) and which has the trigonometric
functions described by the two-component scalar and the vector of the
respective reaction.
2.5.
Determination of robot speeds and
accelerations
The
kinematic calculation of the robot's speeds and accelerations is done only by
direct kinematics as it is operated in reality, while the positions can be
determined in two distinct situations, by direct kinematics when we are
interested in the normal operation of the robot, finding the workspace. and the
trajectories described by the effector element (or other component kinematic
couplings), or by using inverse kinematics when the positions that the final
element (effector) must occupy successively are already imposed and the
successive positions of the driving elements must be determined, for this robot
the linear motors 1 and 8.
2.6.
Determination of robot speeds and
accelerations to the dyad 5,7
As
stated, only direct kinematics is used to determine speeds and accelerations,
so the calculations from dyad 5.7 are started (Figure 10).
Write
the calculation relationships in the system (41):
The
scalar coordinates, velocities, and accelerations of points G and O are known,
with the help of which, using the equations of the two circles formed, the
scalar coordinates of point F are determined. Then easily determine the angles
FI5 and FI7 with their derivatives, w5,e5, w7,e7.
(41)
Figure 10:
Direct kinematics on dyad 5.7: speeds and accelerations
2.7.
Determination of speeds and
accelerations in the triad 2,3,4,6
In figure 11 you can see the
positions with the sizes characteristic of triad 2,3,4,6 starting from which
the relations of positions, speeds, and accelerations are written.
Position
relations being considered already solved and all known position values (solved
separately by direct or inverse kinematics as required), derived directly twice
and thus obtaining triad speeds and accelerations (2,3,4,6), equations (42-52).
Figure 11:
Kinematics of the triad 2,3,4,6
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
3.
RESULTS AND DISCUSSION
Table
1 gives the input data, more precisely the known lengths of the mechanism (In
the calculation program used these lengths represent the constant geometric
parameters):
Table 1: Constant geometric parameters
|
XA |
0.1 |
ET |
1.35 |
|
XG |
-0.15 |
OF |
0.15 |
|
AB |
1.15 |
FG |
0.45 |
|
CD |
0.88 |
TD |
0.9 |
|
OE |
0.88 |
BD |
0.7 |
|
OC |
0.45 |
BC |
0.18 |
|
ED |
0.45 |
|
|
The
point T located on the effector 6 (Figure 1, 4-5) describes a rectangular
trajectory (Figure 12). Its characteristics are shown in Table 2.
Table 2: Initial parameters of the T
point trajectory
|
Initial
parameters of the T point |
T0( 1.5,-0.9 ) |
|
The step of
moving the T point horizontally - v |
-0.05 |
|
The step of
moving the T point vertically - v1 |
0.05 |
Figure 12:
The trajectory of the T-point, the end effector
The
trajectory of the point T in Figure 12 is described by the relationships in
Table 3.
The
coordinates represent the input parameters for the algorithm of the inverse
positional model in Table 3.
Table 3: The input parameters
|
Point T coordinates |
XTk =if [ k
≤10,XT0+ kv,if [10< k≤ 15,XT0+
10v,if[15<k ≤25,XT0+10v-(k-15)v,XT0]]] YTk=if [k
≤10,YT0,if[10<k≤15,YT0+(k-10)v1, if [15< k ≤25,YT0+
5v1,YT0 +5v1- (k - 25)v1]]] |
Going
through the connection of the modular groups for the inverse structural model (Figure
2b, 3) the algorithm presented in Tables 2-3 allows the successive calculation
of the dependent parameters (Figure 4), as follows: - for the dyad RRR(5,6) - Φ5k(XTk,YTk), Φ6k(XTk,YTk)
can be seen in Figure 13 [deg], as Φ50k(XTk,YTk),
Φ60k(XTk,YTk);
Figure 13:
Variation of angles FI5 and FI6 considered in [deg] depending on the
independent parameter k
· for the dyad RRR(3,4) – Φ3k(XTk,YTk), Φ4k(XTk,YTk)
can be seen in the Figure 14 [deg], as Φ30k(XTk,YTk),
Φ40k(XTk,YTk);
Figure 14: Variation of angles FI3
and FI4 considered in [deg] depending on the independent parameter k
· for dyad RRT(1,2) – YAk(XTk,YTk) and Φ2k(XTk,YTk) seen in Figure 15, where
· Φ2k(XTk,YTk) in [deg] is Φ20k(XTk,YTk);
Figure 15: The variation of the
parameter YA and the angle FI2 considered in [deg] depending on the independent
parameter k
· for dyad RRT(8,7) – YGk(XTk,YTk) and Φ7k(XTk,YTk) seen in Figure 16, where
· Φ7k(XTk,YTk) in [deg] is Φ70k(XTk,YTk).
Figure 16: Variation of parameter YG
and angle FI7 considered in [deg] depending on the independent parameter k
It
is considered a single external force (technological resistance) RTk that acts on the system neglecting other external
forces (for example - gravitational forces) and the system of inertial forces.
This simplification brings some peculiarities in the form of terms from the
calculation algorithm without restricting its generality.
The
external force RTk (Figure 17) is considered constant
on the initial and horizontal portion of the trajectory of the point T (Figure
12) and is described by the relation (53):
Figure 17: The external force RTk is considered constant on the initial and horizontal
portion of the trajectory of the point T
RTk := if (k≤ 10, 20, 0)
(53)
Using
the connection of the modular groups for the direct structural model (Figure 3)
the passive module GMP2 (2,3,4,6), a 6R triad (Figure 5, 6, 18) is analyzed in
a first stage, for which elaborated algorithm, relations (1-20).
Figure 18:
Passive module GMP2(2,3,4,6), the triad 6R
Applying
the calculation algorithm (1-20) for the GMP2 triad (2,3,4,6) is determined
reaction torsion components, as follows:
· in the kinematic torque of E →
X56k, Y56k from Figure 19;
Figure 19: Reaction torque in the
kinematic rotation coupling of E → X56k, Y56k on the GMP2 modular group
(2,3,4,6), triad type 6R
· in kinematic rotation couple from
the point A → X12k, Y12k from Figure 20;
Figure 20: Reaction torque in the
kinematic torque of A → X12k, Y12k on the GMP2 modular group (2,3,4,6),
6R triad type
· in the kinematic rotation couple
from the point B → X23k = -X32k, Y23k = -Y32k;
· in the kinematic rotation couple
from the point C → X43k =-X34k, Y43k =-Y34k;
· in the kinematic rotation couple
from the point D → X63k=-X36k, Y63k=-Y36k;
· in the kinematic rotation couple
from the point O → X04k, Y04k from Figure 21;
Figure 21: Reaction torsion in the
kinematic torque of O → X04k, Y04k on the GMP2 modular group (2,3,4,6),
triad type 6R
The
next module in the modular group connection of the direct structural model (Figure
7) is GMP1 (7.5) shown in Figure 22 a, b, an RRR dyad for which the kinetostatic model is rendered by the relations (21-32).
Figure 22:
Reaction torsor on the GMP1 dyad modular group (7.5)
In
this calculation stage it is determined:
· in the kinematic torque from E
→ X87k, Y87k from Figure 2. 3;
Figure 23:
Reaction torsor in kinematic coupling E, →
X87k, Y87k, on the GMP1 dyad modular group (7.5)
· in the kinematic rotation couple
from the point O → X05k, Y05k from Figure 24.
Figure 24:
Reaction torsor in kinematic coupling O, →
X05k, Y05k, from the GMP1 dyad modular group (7.5)
In
the following steps, the initial active modular groups GMAI (G, 8) and GMAI (A,
1) shown in Figs. 25 a, b.
Figure 25:
The reaction torsor of the initial active modular
groups GMAI (G, 8) a, and GMAI (A, 1) b
The
components (NO8k, T08k) of the active translation coupling G are shown in Figs.
26, and for the active coupling of A (NO1k, T01k) in Figs. 27.
Figure 26:
Reaction torsor from the initial active modular group
GMAI (G, 8)
Figure 27:
Reaction torsor from the initial active modular group
GMAI (A,1)
This
bimobile 2T9R mechanism (Figure 1) can be used by the
simultaneous action of active translation torques in A and G point T having a
chosen trajectory and law of motion. If one of these active couplings is
locked, the mechanism remains with only one degree of mobility. The connections
of the modular groups are given in both cases: respectively, for G blocked and
for A blocked in Figs. 28 a, b.
Figure 28:
The connections of the modular groups for the two distinct situations when G is
blocked and the case when A is blocked, respectively
Applying
the calculation modules it is possible to study the behavior of the mechanism
with a degree of mobility in the mentioned situations. Thus, if the active
coupling G is blocked, the variation of the dependent parameters of the
resulting mechanism is studied, with a degree of mobility (Figure 29) for the
extreme blocking positions Φ50 minimum and Φ50 maximum.
Figure 29:
The case in which the active coupling G is blocked when studying the variation
of the dependent parameters of the resulting mechanism, with a degree of
mobility for the extreme locking positions Φ50 minimum and Φ50
maximum.
4.
CONCLUSIONS
The
kinematic and kinetostatic modeling of a 2T9R robotic
mechanism is generally quite difficult and lucrative, but it has the advantages
of obtaining a well-developed theoretical model that can be used in practice to
design or use such robots, extremely interesting and useful, which have
increased maneuverability, a large workspace, a correct and fast dynamics of
movement, without vibrations or noises, the mechatronic module presented can be
designed and built-in various ways depending on the requirements and objectives
of the workplace in which it will be implemented.
The
paper presented the inverse and direct kinematic models, the kinetostatic (forces) model that is always studied
inversely, together with the related calculation relations.
In
the results and discussions section, the diagrams obtained by calculation using
the MathCad 2000 program were actually presented.
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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