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Fig. 9. The generated angles by servo motors to make the MEE follows the triangular Fig. 8. The triangle path followed by 3RRR planar parallel robot

Figure 9 The generated angles by servo motors to make the MEE follows the triangular Fig. 8. The triangle path followed by 3RRR planar parallel robot

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where sp,1,9,1=||Pa1 — Qo1 \|? which is independent from the chosen reference frame. In most cases, the two sequences of points are similar, it will be convenient to abbreviate of sequences of points e.g. Pyi,...-Pan and Qp1,...... Opn. The geometric interpretation of those points represents an important element in distance geometry. This determinant is known as Cayley-Menger determinants [23],[24] and [25]. The main advantages of this modeling in relation to the classic formalisms are to introduce bi-direction movements and more precise positioning. D(Piut,--+;Pan;Qb1,---;Qpn) by D(Pi,..-,Pn) which is so called Cayley-Menger determinant. The D(P,,...,P,) is ((n —1)!)? times the squared hypervolume of the spanned by the point P,,...,P, in R’"-). Hence, for two points P,P, (i.e.n = 2),D(Pa,Ph) = d(Pu,P,)*, where d(P,,P,) is the Euclidean distance between P, and P,. For three points P,P)P,(i.e.n = 3),D(PaPhP:) = £5 \V/(D(PaPpP-)) is the sign area of the triangle P,,P,P. (shown in Fig. 1(a)),where the = sign accounts for the two mirror locations of P. respect to the line defined by P,P. The two vectors P,P,andP,P, can be ex- pressed by scalar product as: D(P,P,;PaP-) = (Pa — P,).(Pa — P.).This dot product is computed as d(Pa, P,)d(Pa, Pp) cos @ ,with 8 being the angle between the lines P,P, and P,P. The angle (8) between P,P, and P,P. can be calculated from the following formula:
where sp,1,9,1=||Pa1 — Qo1 \|? which is independent from the chosen reference frame. In most cases, the two sequences of points are similar, it will be convenient to abbreviate of sequences of points e.g. Pyi,...-Pan and Qp1,...... Opn. The geometric interpretation of those points represents an important element in distance geometry. This determinant is known as Cayley-Menger determinants [23],[24] and [25]. The main advantages of this modeling in relation to the classic formalisms are to introduce bi-direction movements and more precise positioning. D(Piut,--+;Pan;Qb1,---;Qpn) by D(Pi,..-,Pn) which is so called Cayley-Menger determinant. The D(P,,...,P,) is ((n —1)!)? times the squared hypervolume of the spanned by the point P,,...,P, in R’"-). Hence, for two points P,P, (i.e.n = 2),D(Pa,Ph) = d(Pu,P,)*, where d(P,,P,) is the Euclidean distance between P, and P,. For three points P,P)P,(i.e.n = 3),D(PaPhP:) = £5 \V/(D(PaPpP-)) is the sign area of the triangle P,,P,P. (shown in Fig. 1(a)),where the = sign accounts for the two mirror locations of P. respect to the line defined by P,P. The two vectors P,P,andP,P, can be ex- pressed by scalar product as: D(P,P,;PaP-) = (Pa — P,).(Pa — P.).This dot product is computed as d(Pa, P,)d(Pa, Pp) cos @ ,with 8 being the angle between the lines P,P, and P,P. The angle (8) between P,P, and P,P. can be calculated from the following formula:
Fig. 1. Bilateration representation (a) geometric problem (b) vectors projection.
Fig. 1. Bilateration representation (a) geometric problem (b) vectors projection.
Fig. 2. The structure of 3RRR planar parallel robotic platform. The platform structure of the 3RRR planar parallel robotic is shown in Fig. 2. The topology of the 3RRR is generally composed of single MEE and three fixed based points each point is called actuated (or active) point because it is attached to a servo motor. The active point (point A; i € (1 2 3)) is connected by three identical manipulator arms, each arm con- sisting of two free links (J); and /); ) that are connected by revolution joint at the point B; it is called a passive point.The axes of the three revolution joints intersect at a common point called the center of rotation (point E). The servo motors are used to change the angles of the parallel manipulator arms pro- vides three degrees of freedom rotational motion that produce changing the location and orientation of MEE. The kinemat- ics model for 3RRR parallel planar robot is classified into two models: the forward kinematic model and the inverse kinematic model. In this section, the forward and inverse kinematic are derived.
Fig. 2. The structure of 3RRR planar parallel robotic platform. The platform structure of the 3RRR planar parallel robotic is shown in Fig. 2. The topology of the 3RRR is generally composed of single MEE and three fixed based points each point is called actuated (or active) point because it is attached to a servo motor. The active point (point A; i € (1 2 3)) is connected by three identical manipulator arms, each arm con- sisting of two free links (J); and /); ) that are connected by revolution joint at the point B; it is called a passive point.The axes of the three revolution joints intersect at a common point called the center of rotation (point E). The servo motors are used to change the angles of the parallel manipulator arms pro- vides three degrees of freedom rotational motion that produce changing the location and orientation of MEE. The kinemat- ics model for 3RRR parallel planar robot is classified into two models: the forward kinematic model and the inverse kinematic model. In this section, the forward and inverse kinematic are derived.
The above set of equation can be rewritten as below:
The above set of equation can be rewritten as below:
The angle 6; and length A;E can be computed from the fol- lowing equations: From Egns. (16), (17) and (18) the angles a; can be found as:
The angle 6; and length A;E can be computed from the fol- lowing equations: From Egns. (16), (17) and (18) the angles a; can be found as:
Fig. 5. Angle arms of each base joint of 3RRR planar parallel robot. Fig. 4. The workspace of 3RRR planar parallel robot Trajectories can be divided as a continuous one such circu- ar or sinusoidal and as discontinuous trajectories such square and triangular. The main difference is the sharp angles that exist in the non-continuous trajectories which may cause a jump in the turning at sharp angles. Therefore, two trajecto- ries have been tested to evaluate the performance of the 3RRR planar parallel robot. The first trajectory is a circular path as shown in Fig. 6 and it has radius 0.3m and centered at the origin coordinates. Fig. 7 shows the angle arms of each base joint of the 3RRR platform that is generated by the servo motors to move the end-effector along this circular path. The second trajectory is the triangular one and it is equilateral and side length equals to 0.55m, which has sharp convex points as shown in Fig. 8 Similarly, Fig. 9 shows the angle arms of each base joint of the 3RRR platform that are generated to move
Fig. 5. Angle arms of each base joint of 3RRR planar parallel robot. Fig. 4. The workspace of 3RRR planar parallel robot Trajectories can be divided as a continuous one such circu- ar or sinusoidal and as discontinuous trajectories such square and triangular. The main difference is the sharp angles that exist in the non-continuous trajectories which may cause a jump in the turning at sharp angles. Therefore, two trajecto- ries have been tested to evaluate the performance of the 3RRR planar parallel robot. The first trajectory is a circular path as shown in Fig. 6 and it has radius 0.3m and centered at the origin coordinates. Fig. 7 shows the angle arms of each base joint of the 3RRR platform that is generated by the servo motors to move the end-effector along this circular path. The second trajectory is the triangular one and it is equilateral and side length equals to 0.55m, which has sharp convex points as shown in Fig. 8 Similarly, Fig. 9 shows the angle arms of each base joint of the 3RRR platform that are generated to move
Fig. 6. The circular path followed by 3RRR planar parallel robot the end-effector along the triangular path. It is observable that that the 3RRR planar parallel robot can follow variety of trajectories as demonstrated in the aforementioned figures. This has successfully proven the effectiveness of the proposed system in satisfying the performance smoothly and feasibly. The performance criterion can be understood by tracking the given reference trajectories as required.
Fig. 6. The circular path followed by 3RRR planar parallel robot the end-effector along the triangular path. It is observable that that the 3RRR planar parallel robot can follow variety of trajectories as demonstrated in the aforementioned figures. This has successfully proven the effectiveness of the proposed system in satisfying the performance smoothly and feasibly. The performance criterion can be understood by tracking the given reference trajectories as required.
Fig. 7. The generated angles by servo motors to make the MEE follows the circular path
Fig. 7. The generated angles by servo motors to make the MEE follows the circular path
Fig. 12. The serial communication between the PC and the Maestro controller. Fig. 11. The channel setting of servos
Fig. 12. The serial communication between the PC and the Maestro controller. Fig. 11. The channel setting of servos
Fig. 10. Controlling servos using Pololu Masestro
Fig. 10. Controlling servos using Pololu Masestro
as depicted in Fig.11. The serial communication used in the Polou Maestro Control Center is based on ’UART type as shown in the setting tab provided in in Fig. 12. The baud rate is set to 9600 to speed up the communication. The computer specifications used in our experiments are CPU is 1.6GHz, core i5 and RAM equals 4GB. After conducting of all afore-
as depicted in Fig.11. The serial communication used in the Polou Maestro Control Center is based on ’UART type as shown in the setting tab provided in in Fig. 12. The baud rate is set to 9600 to speed up the communication. The computer specifications used in our experiments are CPU is 1.6GHz, core i5 and RAM equals 4GB. After conducting of all afore-
Fig. 14. The movement of the 3RRR parallel planar robot in case of triangular trajectory.
Fig. 14. The movement of the 3RRR parallel planar robot in case of triangular trajectory.
Fig. 13. The practical implementation of the 3RRR parallel planar robot In this case study, a triangular movement will be tested to validate the effectiveness of the platform movement as demonstrated in Fig.14. The movements of the platform based on the time responses of both X and Y axes are shown in Figs. 15 (a) and (b), respectively. When the MEE moves, the coordinates of X and Y change accordingly within the area of workspace and this proves the design and satisfies the operating condition practically.
Fig. 13. The practical implementation of the 3RRR parallel planar robot In this case study, a triangular movement will be tested to validate the effectiveness of the platform movement as demonstrated in Fig.14. The movements of the platform based on the time responses of both X and Y axes are shown in Figs. 15 (a) and (b), respectively. When the MEE moves, the coordinates of X and Y change accordingly within the area of workspace and this proves the design and satisfies the operating condition practically.
Fig. 15. The coordinates response of the triangular trajectory
Fig. 15. The coordinates response of the triangular trajectory
Fig. 16. The movement of the 3RRR parallel planar robot in case of circular trajectory.
Fig. 16. The movement of the 3RRR parallel planar robot in case of circular trajectory.
Fig. 17. The coordinates response of the circular trajectory
Fig. 17. The coordinates response of the circular trajectory
Fig. 18. The movement of the 3RRR parallel planar robot in case of complex trajectory.
Fig. 18. The movement of the 3RRR parallel planar robot in case of complex trajectory.
Furthermore, a complex trajectory has been applied which combines both continuous and none continuous paths. In other words, sharp turning and smooth tracking are found to demonstrate and validate the movement the system.This case study is described as shown in Fig. 18. Moreover, the time response of X and Y coordinates are shown in Fig. 19. These graphs are changed continuously as observed in which the complex trajectory is obtained. In real life, such a trajectory can be used in computer numerically controlled machine.
Furthermore, a complex trajectory has been applied which combines both continuous and none continuous paths. In other words, sharp turning and smooth tracking are found to demonstrate and validate the movement the system.This case study is described as shown in Fig. 18. Moreover, the time response of X and Y coordinates are shown in Fig. 19. These graphs are changed continuously as observed in which the complex trajectory is obtained. In real life, such a trajectory can be used in computer numerically controlled machine.
Fig. 19. The coordinates response of the complex trajectory
Fig. 19. The coordinates response of the complex trajectory
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