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In the next step, considering the calculations in this section which have determined the rotations of gears No. 1, 3 and 5 of Fig 5, and the relationships of motion between the actuators and the internal mobile components in links No. 2, No. 3 and the base of robot, these were already calculated in section 2 and based on Figs. 2, 3 and 4 (equations No. | to 4). Because the rotational axes of those gears (gears No. 1, 3 and 5) and the rotational axes of joints No. 2 and 3 are the same and in one direction we can sum the amounts of those rotational angles so we can calculate the kinematic relationship between the mobile components and the actuators for all six joints of the 6-DOF robotic arm discussed in this paper, considering the positive direction of rotation for the actuators shown in Figure 8. This relationship can be expressed as shown below: * m: rotation related to the motor of joint; a: rotation related to the joint. In the kinematic simulation of this robot, it’s an important fact that we can calculate the related equations for the angular velocities and the angular accelerations between the actuators and the internal mobile parts by calculating the derivative of both sides of Equation (5-3) with respect to time. These equations provide three sets of information that are necessary for kinematic simulation of the robotic arm that we have assessed in this paper.

Figure 5 In the next step, considering the calculations in this section which have determined the rotations of gears No. 1, 3 and 5 of Fig 5, and the relationships of motion between the actuators and the internal mobile components in links No. 2, No. 3 and the base of robot, these were already calculated in section 2 and based on Figs. 2, 3 and 4 (equations No. | to 4). Because the rotational axes of those gears (gears No. 1, 3 and 5) and the rotational axes of joints No. 2 and 3 are the same and in one direction we can sum the amounts of those rotational angles so we can calculate the kinematic relationship between the mobile components and the actuators for all six joints of the 6-DOF robotic arm discussed in this paper, considering the positive direction of rotation for the actuators shown in Figure 8. This relationship can be expressed as shown below: * m: rotation related to the motor of joint; a: rotation related to the joint. In the kinematic simulation of this robot, it’s an important fact that we can calculate the related equations for the angular velocities and the angular accelerations between the actuators and the internal mobile parts by calculating the derivative of both sides of Equation (5-3) with respect to time. These equations provide three sets of information that are necessary for kinematic simulation of the robotic arm that we have assessed in this paper.

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Figure 1. Usual mechanism for motion structure of 6 DOF Robotic arms: 1- Link 1; 2- Link 2; 3- Link 3; 4- Robot wrist including links 4, 5, and 6; 5- place of setup motor for joint 1; 6- motor of joint 2; 7- motor of joint 3; 8- motors of joints 4, 5, and 6. (The picture belongs to Robotic arm, product of KUKA Company) producing power (such as hydraulic and pneumatic mechanisms) to the robot’s base, thereby transferrir motion from these motors to the rotational axes of the related joints.
Figure 1. Usual mechanism for motion structure of 6 DOF Robotic arms: 1- Link 1; 2- Link 2; 3- Link 3; 4- Robot wrist including links 4, 5, and 6; 5- place of setup motor for joint 1; 6- motor of joint 2; 7- motor of joint 3; 8- motors of joints 4, 5, and 6. (The picture belongs to Robotic arm, product of KUKA Company) producing power (such as hydraulic and pneumatic mechanisms) to the robot’s base, thereby transferrir motion from these motors to the rotational axes of the related joints.
Figure 2. Right: Transition procedure of rotational motion by the 4-bar linkage mechanism (Crank — Crank). Left: An equivalent functional structure with this mechanism was also provided on the right side, at the base of the robot’s motion mechanism. It’s a linkage mechanism of joint 3 and from the motor of that joint up to the joint 2 axis.
Figure 2. Right: Transition procedure of rotational motion by the 4-bar linkage mechanism (Crank — Crank). Left: An equivalent functional structure with this mechanism was also provided on the right side, at the base of the robot’s motion mechanism. It’s a linkage mechanism of joint 3 and from the motor of that joint up to the joint 2 axis.
Figure 3. Right: Procedure of influencing the rotation of joint 2 on the performance of the motion transfer linkage mechanism of joint 3 (and similarly for joints 4, 5, and 6): Mode 1 is the initial mode of the motor of joint 3 (green); Mode 2 is the motor of joint 3 after rotating +a degrees (blue); Mode 3 is the motor of joint 3 after rotating + (a+B) degrees (red). Left: An equivalent functional structure with this mechanism was also provided on the right side, the mechanism was applied from the base up to arm (Link) No. 3 of the robot’s motion mechanism. Thus, the procedure of interaction and the effect of rotational motion of a joint on the performance of the linkage mechanism make it possible to transfer rotational motion to the next joint or to the rectifier mechanism. Figure 4 shows an example of this in the connection between joints 2 and 3. In the figure (right), ‘wo aligned arms are visible in three different conditions in which we can define the orientation of link No. 3 Soncerning the rotation of joint No. 2 and by the rotation of the motor of joint 3.
Figure 3. Right: Procedure of influencing the rotation of joint 2 on the performance of the motion transfer linkage mechanism of joint 3 (and similarly for joints 4, 5, and 6): Mode 1 is the initial mode of the motor of joint 3 (green); Mode 2 is the motor of joint 3 after rotating +a degrees (blue); Mode 3 is the motor of joint 3 after rotating + (a+B) degrees (red). Left: An equivalent functional structure with this mechanism was also provided on the right side, the mechanism was applied from the base up to arm (Link) No. 3 of the robot’s motion mechanism. Thus, the procedure of interaction and the effect of rotational motion of a joint on the performance of the linkage mechanism make it possible to transfer rotational motion to the next joint or to the rectifier mechanism. Figure 4 shows an example of this in the connection between joints 2 and 3. In the figure (right), ‘wo aligned arms are visible in three different conditions in which we can define the orientation of link No. 3 Soncerning the rotation of joint No. 2 and by the rotation of the motor of joint 3.
Figure 5. Structure of the rectifier mechanism for motion transfer of joints 4, 5, and 6, which lies in arm 3: 1) motive gear of joint 6 connected to motion transfer link; 2) follower gear of joint 6 connected to motion transfer shaft; 3) motive gear of joint 5 connected to motion transfer link; 4) follower gear of joint 5 connected to the motion transfer shaft; 5) motive gear of joint 4 connected to the motion transfer link; 6) follower gear of joint 4 connected to the motion transfer shaft; 7) motion transfer shaft of joint 6; 8) motion transfer shaft of joint 5; 9) motion transfer shaft of joint 4; 10) related motion transfer link of linkage mechanism of joint 6; 11) related motion transfer link of linkage mechanism of joint 5; 12) related motion transfer link of linkage mechanism of joint 4. (The rotational directions drawn in this figure are to create rotation in the positive direction for joints 4, 5, and 6.)
Figure 5. Structure of the rectifier mechanism for motion transfer of joints 4, 5, and 6, which lies in arm 3: 1) motive gear of joint 6 connected to motion transfer link; 2) follower gear of joint 6 connected to motion transfer shaft; 3) motive gear of joint 5 connected to motion transfer link; 4) follower gear of joint 5 connected to the motion transfer shaft; 5) motive gear of joint 4 connected to the motion transfer link; 6) follower gear of joint 4 connected to the motion transfer shaft; 7) motion transfer shaft of joint 6; 8) motion transfer shaft of joint 5; 9) motion transfer shaft of joint 4; 10) related motion transfer link of linkage mechanism of joint 6; 11) related motion transfer link of linkage mechanism of joint 5; 12) related motion transfer link of linkage mechanism of joint 4. (The rotational directions drawn in this figure are to create rotation in the positive direction for joints 4, 5, and 6.)
Figure 6 [19]. Structure of Robot’s wrist mechanism to impose desired motion of joints 5 and 6: 1) motive gear connected to the main shaft of joint 6; 2) middle gear and waste rotation of joint 6; 3) gear connected to motion transfer shaft of joint 6; 4) fixed gear without any rotation for load support of gear No. 5; 5) motive gear connected to main shaft of joint 5; 6) gear connected to motion transfer shaft of joint 5. (The circular sides illustrated in this figure are to create rotation in the positive direction for joints 5 and 6.) After this step, the motion transfer shaft related to joint 4 from the rectifier mechanism is connected directly to the link’s arm and to the motion transfer shafts for rotational motion of joints 5 and 6 by means of three couples of bevel gears where the ratios between them are pairwise equal to | and all are formed at 90° angles. These bevel gears form the structure of the robot’s wrist and rotate the joints No. 5 and 6 axes [19]. Figure 6 [19] shows a view of the structure of the components of this part (Robot’s wrist) as a CAD model.
Figure 6 [19]. Structure of Robot’s wrist mechanism to impose desired motion of joints 5 and 6: 1) motive gear connected to the main shaft of joint 6; 2) middle gear and waste rotation of joint 6; 3) gear connected to motion transfer shaft of joint 6; 4) fixed gear without any rotation for load support of gear No. 5; 5) motive gear connected to main shaft of joint 5; 6) gear connected to motion transfer shaft of joint 5. (The circular sides illustrated in this figure are to create rotation in the positive direction for joints 5 and 6.) After this step, the motion transfer shaft related to joint 4 from the rectifier mechanism is connected directly to the link’s arm and to the motion transfer shafts for rotational motion of joints 5 and 6 by means of three couples of bevel gears where the ratios between them are pairwise equal to | and all are formed at 90° angles. These bevel gears form the structure of the robot’s wrist and rotate the joints No. 5 and 6 axes [19]. Figure 6 [19] shows a view of the structure of the components of this part (Robot’s wrist) as a CAD model.
Figure 7. Overall view of the outer structure of the CAD model of the 6 DOF Robotic arm: 1) Robot’s base (location of motors of joints); 2) link No. 1; 3) link No. 2; 4) link No. 3; 5) link No. 4; 6) link No. 5; 7) link No. 6 (end effector); 8) joint 2; 9) joint 3; 10) motor of joint 2. Based on the discussion of the linkage mechanism and its use to create 6 degrees of freedom in the notion of robotic arms with rotational joints, we attempted to design a CAD model of a 6 DOF robotic arm with rotational joints and a wrist with intersecting axes. Figs. 7, 8 and 9 provide pictures of the CAD model ‘or this robotic arm and associated information about the main parts that form it. For this purpose, this robot was checked by means of CATIA DMU Kinematic software for considering this fact that none of internal or >xternal components should conflict with each other. We have used data from the textbook entitled ‘Tabellenbuch Metall” (reference No. [20]), to choose models and specifications and to calculate mechanical art sizes (for example: screws, nuts, bearings and etc.). These specifications were used in the design of the “AD model for this operational robotic arm. Robot’s component materials include: (1) steel 304, (2) aluminum (Al 2219- T87) and (3) titanium (Tit / Ti-8Al-1Mo-1V) and their mechanical properties are resented respectively in reference No. [21].
Figure 7. Overall view of the outer structure of the CAD model of the 6 DOF Robotic arm: 1) Robot’s base (location of motors of joints); 2) link No. 1; 3) link No. 2; 4) link No. 3; 5) link No. 4; 6) link No. 5; 7) link No. 6 (end effector); 8) joint 2; 9) joint 3; 10) motor of joint 2. Based on the discussion of the linkage mechanism and its use to create 6 degrees of freedom in the notion of robotic arms with rotational joints, we attempted to design a CAD model of a 6 DOF robotic arm with rotational joints and a wrist with intersecting axes. Figs. 7, 8 and 9 provide pictures of the CAD model ‘or this robotic arm and associated information about the main parts that form it. For this purpose, this robot was checked by means of CATIA DMU Kinematic software for considering this fact that none of internal or >xternal components should conflict with each other. We have used data from the textbook entitled ‘Tabellenbuch Metall” (reference No. [20]), to choose models and specifications and to calculate mechanical art sizes (for example: screws, nuts, bearings and etc.). These specifications were used in the design of the “AD model for this operational robotic arm. Robot’s component materials include: (1) steel 304, (2) aluminum (Al 2219- T87) and (3) titanium (Tit / Ti-8Al-1Mo-1V) and their mechanical properties are resented respectively in reference No. [21].
Figure 9. a view of the inner structure of link No. 2,until the end effector of robot where the arrangement of motive components of mechanism is observed: 1) arm of joint No. 2 and the links in it; 2) joint 3; 3) link No. 3 and set of gears of rectifier mechanism in it; 4) set of gears of mechanism of robot wrist; 5) link No. 4; 6) link No. 5; 7) link No. 6 (end effector) Figure 8. view of the inner structure of the robot’ base on which the motors for the joints are placed: | container of motors and the links connected to ther in robot base; 2) link No. | (arm of joint 1); 3) joir 2; 4) link No. 2; 5) motor of joint 1; 6) motor of joit 2; 7) motor of joint 3; 8) motor of joint 4; 9) motor c joint 5; 10) motor of joint 6.
Figure 9. a view of the inner structure of link No. 2,until the end effector of robot where the arrangement of motive components of mechanism is observed: 1) arm of joint No. 2 and the links in it; 2) joint 3; 3) link No. 3 and set of gears of rectifier mechanism in it; 4) set of gears of mechanism of robot wrist; 5) link No. 4; 6) link No. 5; 7) link No. 6 (end effector) Figure 8. view of the inner structure of the robot’ base on which the motors for the joints are placed: | container of motors and the links connected to ther in robot base; 2) link No. | (arm of joint 1); 3) joir 2; 4) link No. 2; 5) motor of joint 1; 6) motor of joit 2; 7) motor of joint 3; 8) motor of joint 4; 9) motor c joint 5; 10) motor of joint 6.
Table 1. Denavit-Hartenberg notation parameters of the robot: 61 to 86 in this table shows the angle of rotation for the motors of each joint. 3.2. Tool Used in the Simulation
Table 1. Denavit-Hartenberg notation parameters of the robot: 61 to 86 in this table shows the angle of rotation for the motors of each joint. 3.2. Tool Used in the Simulation
Table 2. Set of Angles Used to Generate Frames for the Origin, Destination and Four Target Points
Table 2. Set of Angles Used to Generate Frames for the Origin, Destination and Four Target Points
Figure 11. Assignment of Frames Between Each of the Manipulators.
Figure 11. Assignment of Frames Between Each of the Manipulators.
In the following and in Fig. 12, the output graphical images of MATLAB SimMechanics software can be observed from the simulation model of the robot’s motion as well as the manner of positioning the tool’s frame on the goals frames. Figure 13 shows the 3D drawing of the motion of the robot’s end effector (frame connected to joint No. 6 of the robot), both theoretically and output of simulation, where the origin and destination points and the points related to the target frames are named and identified in accordance with the order given in Table 2.
In the following and in Fig. 12, the output graphical images of MATLAB SimMechanics software can be observed from the simulation model of the robot’s motion as well as the manner of positioning the tool’s frame on the goals frames. Figure 13 shows the 3D drawing of the motion of the robot’s end effector (frame connected to joint No. 6 of the robot), both theoretically and output of simulation, where the origin and destination points and the points related to the target frames are named and identified in accordance with the order given in Table 2.
Figure 13. In This Image the Black Continuous Lines Indicate the Desirable Path, While the Dashed Red Lines Show the Output Path in the Simulation for the Position of the Robot’s End Effector.
Figure 13. In This Image the Black Continuous Lines Indicate the Desirable Path, While the Dashed Red Lines Show the Output Path in the Simulation for the Position of the Robot’s End Effector.
* 0: output of simulation values, d: desired values
* 0: output of simulation values, d: desired values
(b) Left: The trajectories of desired and simulation output values of rotation angle of joint No. 3 Right: The trajectories of desired and simulation output values of rotation angle of joint No. 4
(b) Left: The trajectories of desired and simulation output values of rotation angle of joint No. 3 Right: The trajectories of desired and simulation output values of rotation angle of joint No. 4
Fig A 1. The scheme of SimMechanics model of kinematic simulation of this robotic arm You can find the scheme of this SimMechanics model on below and in Figure A. 1.
Fig A 1. The scheme of SimMechanics model of kinematic simulation of this robotic arm You can find the scheme of this SimMechanics model on below and in Figure A. 1.
Related topics:
EngineeringComputer ScienceRobot kinematicsParallel MechanismMechanism in BiologyKinematic SimulationLinkage MechanismRobotic Arms
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