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The Kinematics of Final Point of the Holder Robot Manipulators
Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 102 (2016) 414 – 417
12th International Conference on Application of Fuzzy Systems and Soft Computing, ICAFS 2016, 29-30 August 2016, Vienna, Austria
The kinematics of final point of the holder robot manipulators S.Y. Agayevaa* a
Azerbaijan State Oil and Industry University, Baku, Azerbaijan
Abstract In this paper we investigate velocity and acceleration of the center of claw arm which consists of three links. The analytical formulas are obtained. These formulas can be used in engineering calculations of trajectories of robots. © byby Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016. Peer-review under responsibility of the Organizing Committee of ICAFS 2016 Keywords:Robot; claw arm; velocity; acceleration; transportation motion.
1. Introduction Robotics is very frequently used in different industries for performing different technological processes. Application of smart robots in oil mining field is very important area of research. From this point of view, investigations of kinematics of robots and manipulators are a challenging problem. In this paper we investigate kinematics of center of claw arms. In the existing works, deterministic models for description of motional manipulators are used which is related to strong axiomatic assumptions. In real situation, investigation of motion of multidimensional manipulators is related to uncertainty, mainly to fuzzy uncertainty. 2. Preliminaries Definition 1.Sum of fuzzy numbers Let A and ȼ be fuzzy numbers. The sum of fuzzy numbers A+B is defined as follows.
* Corresponding author, Tel.: +99 412 493 45 38 E mail address : [email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016 doi:10.1016/j.procs.2016.09.420
415
S.Y. Agayeva / Procedia Computer Science 102 (2016) 414 – 417
sup min > P A ( x), PB ( y)@
P( A B ) ( z )
z x y
Definition 2. The square of fuzzy number A denoted by P A2 is a continuous fuzzy number defined as
P A ( y) 2
y , P y
max P A
A
Definition 3. The square root of fuzzy number A denoted by P
P A ( y)
A
, is a continuous fuzzy number defined as
PA y2
3.Robot manipulator consists of elements AB, BC and CK. AB element has vertical axis, angle of rotation is I BC and CK rotate in vertical plane with angles of rotation T1 and T 2 . The velocity of the holder and acceleration are the final point which requires to define Wk to denote this velocity as V and acceleration as W . 4. Determination of velocity Rotation radius of point C around vertical axis is denoted as EC. EC
l1 sin T1
The length of elements BC and CK is denoted as l1 and l2 . The rotation radius of holder K around vertical axis is OE. OE OC CE l2 sin T1 T2 l1 sin T1 . The holder executes the compound motion. Its velocity consists of geometrical sum of 2 velocities
V
Vr Vc
(1)
Here is a relative velocity Vr located on drawing plane. The velocity Vr is equal to geometrical sum of Vr1 and Vr2 . Vr
Vr1 Vr2
(2)
Velocity Vr1 directed on the point C perpendicularly to BC to rotation T1
Vr1
T1l1
Vr2 directed on point CK perpendicularly to Ck to rotation T 2
(3)
416
S.Y. Agayeva / Procedia Computer Science 102 (2016) 414 – 417
Vr2
T T l 1
2
(4)
2
Transitional velocity is: Vc
IOE I ª¬l1 sin T1 l2 sin T1 T 2 º¼
(5)
OE=OC+CE Implementing the theorem of cosines to (2) Vr2 Vr12 Vr22 2Vr1 Vr2 cos Vr1 ^ Vr2 Vr12 Vr22 2V1V2 cos T 2 .
IP we take into account (1)-(5); cos VeVr
cos 900
0.
The velocity of holder’s final point is. V
^ª¬T l
2 2 1 1
1/ 2
`
(T1 T2 ) 2 l22 2l1l2T1 (T1 T2 ) cos T 2 º¼ ª¬l1 sin T1 l2 sin T1 T 2 º¼ I
Let us determine velocity by the projection methods V iV x jV y kVz or V V z directed perpendicularly to drawing plane: Vz Here Vx V1x V2 x V3 x V1 cos T1 V2 cos T1 T 2
Vy
V1 y V2 y V3 y
V1 sin T1 V2 sin T1 T 2
Vz
V1z V2 z V3 z
V3
(6)
V1 V2 V3
V3
ª¬l1 sin T1 l2 sin T1 T 2 º¼ I
There fore
V
Vx2 Vy2 Vz2
^»l
2 1
1/ 2
`
T12 l22 T1 T2 2V1V2 cos T 2 ¼º ª¬l1 sin T1 l2 sin T1 T 2 º¼ I
(7)
Because of (6) and (7) are equal o the task was right solved. 3. Determination of acceleration
Due to acceleration value has composite motion; absolute acceleration of this point is described as
W
Wr We Wc
Wr relative acceleration, We translational acceleration We Wc
(8) 2 Ze u Vr Coriolis acceleration and is equal to
2ZrVr sin Z ^ Vr . Summands (8) are described as W
Wer Wen Wr1r Wr1n Wr2r Wr2n Wc
(9)
S.Y. Agayeva / Procedia Computer Science 102 (2016) 414 – 417
We
WeW Wen
WeW
H e OE I l1 sin T1 l2 cos D ; D
Wen
Ze2 l1 sin T1 l2 cos D I2 l1 sin T1 l2 cos D
417
T1 T 2 900
The relative acceleration W2 consists of geometrical sum Wr1 and Wr2 Wr
Wr1 Wr2
Wr1
Wr12 Wr2n ;
Wr2
WrW2 Wr2n
On modules: Wr1W
T1l1 ;
Wr1m
T12 l1 ;
WrW2
Tr l2 ;
Wr2n
T22 l2 .
Acceleration of the holder manipulator can be determined as. W
Wx2 Wy2 Wz2
For determination Wx ,Wy ,Wz let’s project (9) to axis x, y, z . 4. Conclusion
In this paper we have investigated of three dimensional manipulators usually used in oil mining industry. We obtained analytical expressions for velocity and acceleration. We consider that in case of projections of acceleration are given as fuzzy numbers; it is possible on base of our analysis to compute value of acceleration.
References 1. Aliev RA . Intelligent robots. Baku , 2011 2. Gurbanov R, Ibrahimov I. Manual on theoretical mechanics. Baku, 1999. 3. Fu.K , Gonsales R, Li K. Robotics. Moscow , 1986.
ScienceDirect Procedia Computer Science 102 (2016) 414 – 417
12th International Conference on Application of Fuzzy Systems and Soft Computing, ICAFS 2016, 29-30 August 2016, Vienna, Austria
The kinematics of final point of the holder robot manipulators S.Y. Agayevaa* a
Azerbaijan State Oil and Industry University, Baku, Azerbaijan
Abstract In this paper we investigate velocity and acceleration of the center of claw arm which consists of three links. The analytical formulas are obtained. These formulas can be used in engineering calculations of trajectories of robots. © byby Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016. Peer-review under responsibility of the Organizing Committee of ICAFS 2016 Keywords:Robot; claw arm; velocity; acceleration; transportation motion.
1. Introduction Robotics is very frequently used in different industries for performing different technological processes. Application of smart robots in oil mining field is very important area of research. From this point of view, investigations of kinematics of robots and manipulators are a challenging problem. In this paper we investigate kinematics of center of claw arms. In the existing works, deterministic models for description of motional manipulators are used which is related to strong axiomatic assumptions. In real situation, investigation of motion of multidimensional manipulators is related to uncertainty, mainly to fuzzy uncertainty. 2. Preliminaries Definition 1.Sum of fuzzy numbers Let A and ȼ be fuzzy numbers. The sum of fuzzy numbers A+B is defined as follows.
* Corresponding author, Tel.: +99 412 493 45 38 E mail address : [email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016 doi:10.1016/j.procs.2016.09.420
415
S.Y. Agayeva / Procedia Computer Science 102 (2016) 414 – 417
sup min > P A ( x), PB ( y)@
P( A B ) ( z )
z x y
Definition 2. The square of fuzzy number A denoted by P A2 is a continuous fuzzy number defined as
P A ( y) 2
y , P y
max P A
A
Definition 3. The square root of fuzzy number A denoted by P
P A ( y)
A
, is a continuous fuzzy number defined as
PA y2
3.Robot manipulator consists of elements AB, BC and CK. AB element has vertical axis, angle of rotation is I BC and CK rotate in vertical plane with angles of rotation T1 and T 2 . The velocity of the holder and acceleration are the final point which requires to define Wk to denote this velocity as V and acceleration as W . 4. Determination of velocity Rotation radius of point C around vertical axis is denoted as EC. EC
l1 sin T1
The length of elements BC and CK is denoted as l1 and l2 . The rotation radius of holder K around vertical axis is OE. OE OC CE l2 sin T1 T2 l1 sin T1 . The holder executes the compound motion. Its velocity consists of geometrical sum of 2 velocities
V
Vr Vc
(1)
Here is a relative velocity Vr located on drawing plane. The velocity Vr is equal to geometrical sum of Vr1 and Vr2 . Vr
Vr1 Vr2
(2)
Velocity Vr1 directed on the point C perpendicularly to BC to rotation T1
Vr1
T1l1
Vr2 directed on point CK perpendicularly to Ck to rotation T 2
(3)
416
S.Y. Agayeva / Procedia Computer Science 102 (2016) 414 – 417
Vr2
T T l 1
2
(4)
2
Transitional velocity is: Vc
IOE I ª¬l1 sin T1 l2 sin T1 T 2 º¼
(5)
OE=OC+CE Implementing the theorem of cosines to (2) Vr2 Vr12 Vr22 2Vr1 Vr2 cos Vr1 ^ Vr2 Vr12 Vr22 2V1V2 cos T 2 .
IP we take into account (1)-(5); cos VeVr
cos 900
0.
The velocity of holder’s final point is. V
^ª¬T l
2 2 1 1
1/ 2
`
(T1 T2 ) 2 l22 2l1l2T1 (T1 T2 ) cos T 2 º¼ ª¬l1 sin T1 l2 sin T1 T 2 º¼ I
Let us determine velocity by the projection methods V iV x jV y kVz or V V z directed perpendicularly to drawing plane: Vz Here Vx V1x V2 x V3 x V1 cos T1 V2 cos T1 T 2
Vy
V1 y V2 y V3 y
V1 sin T1 V2 sin T1 T 2
Vz
V1z V2 z V3 z
V3
(6)
V1 V2 V3
V3
ª¬l1 sin T1 l2 sin T1 T 2 º¼ I
There fore
V
Vx2 Vy2 Vz2
^»l
2 1
1/ 2
`
T12 l22 T1 T2 2V1V2 cos T 2 ¼º ª¬l1 sin T1 l2 sin T1 T 2 º¼ I
(7)
Because of (6) and (7) are equal o the task was right solved. 3. Determination of acceleration
Due to acceleration value has composite motion; absolute acceleration of this point is described as
W
Wr We Wc
Wr relative acceleration, We translational acceleration We Wc
(8) 2 Ze u Vr Coriolis acceleration and is equal to
2ZrVr sin Z ^ Vr . Summands (8) are described as W
Wer Wen Wr1r Wr1n Wr2r Wr2n Wc
(9)
S.Y. Agayeva / Procedia Computer Science 102 (2016) 414 – 417
We
WeW Wen
WeW
H e OE I l1 sin T1 l2 cos D ; D
Wen
Ze2 l1 sin T1 l2 cos D I2 l1 sin T1 l2 cos D
417
T1 T 2 900
The relative acceleration W2 consists of geometrical sum Wr1 and Wr2 Wr
Wr1 Wr2
Wr1
Wr12 Wr2n ;
Wr2
WrW2 Wr2n
On modules: Wr1W
T1l1 ;
Wr1m
T12 l1 ;
WrW2
Tr l2 ;
Wr2n
T22 l2 .
Acceleration of the holder manipulator can be determined as. W
Wx2 Wy2 Wz2
For determination Wx ,Wy ,Wz let’s project (9) to axis x, y, z . 4. Conclusion
In this paper we have investigated of three dimensional manipulators usually used in oil mining industry. We obtained analytical expressions for velocity and acceleration. We consider that in case of projections of acceleration are given as fuzzy numbers; it is possible on base of our analysis to compute value of acceleration.
References 1. Aliev RA . Intelligent robots. Baku , 2011 2. Gurbanov R, Ibrahimov I. Manual on theoretical mechanics. Baku, 1999. 3. Fu.K , Gonsales R, Li K. Robotics. Moscow , 1986.