Design of a Three-Axis Rotary Platform

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1 Design of a Three-Axis Rotary Platform William Mendez, Yuniesky Rodriguez, Lee Brady, Sabri Tosunoglu Mechanics and Materials Engineering, Florida International University W Flagler Street, Miami, Florida United States wmend002@fiu.edu, yrodr025@fiu.edu, lbrad001@fiu.edu, tosun@fiu.edu ABSTRACT A three-degree-of-freedom (3-DOF) rotary platform (table or presenter) manipulator is a robotic system that brings desired rotational movements with high precision. For the design of this mechanism an extensive literature review was done to find the most suitable design for the application in the crystallography fielded. A kinematical analysis and dynamic equations are presented in this paper. The dynamic equations are derived from the implementation of the Newton s second law of rotational motion. The wobble error is neglected and a more synthesized set of equations are obtained. The final design presented was modelled in SolidWorks CAD software, and kinematical space constrains are satisfied. Finally, software integration with the corresponding servo-motors implementation is done. A small prototype was built to ensure the accuracy and the efficiency between the interface and the corresponding software. Keywords Rotary Platform, Presenter, 3-DOF, Crystallography, Gimbal, Mechanism, Kinematic Analysis, Dynamic Analysis. INTRODUCTION The purpose of this study is to design a 3-axis rotating platform for a laboratory that performs crystallography studies. The crystal samples are placed on an oscillating platform in order to apply a laser beam at different angles. Several types of manipulators were studied for the construction of the table. The first mechanism taken into consideration was the multi-axial parallel manipulator also known as Stewart mechanism. This particular mechanism is highly used for high precision motion such as flight simulators. One of the main designs parameters given by the crystallography process was the need for the bed to rotate a full 360 o at least in two of the axis. For this reason a variation of the parallel manipulators was included in the literature review; the three degree of freedom spherical manipulators. The second mechanism studied was the 3-DOF rotary table manipulator. The 3-DOF rotary table manipulator is also known as a 3 axis rotation table. The mechanism is formed of three rotary gimbals. The gimbals are generally denominated as inner, middle and outer. This particular robot is generally used to make a movement scenario in order to test guidance and navigation systems of moving vehicles. The second mechanism was chosen over the first one due to the complexity and amount of elements presented by the parallel manipulators mechanism. Due to limited space the rotary table is better suited for this application. For this particular project the first prototype will be presented, including three main design aspects, kinematical and dynamic analysis, software integration and construction. The construction will be done in order to test the software interface of the system. The final design of the prototype is presented in drawings and its fabrication will be part of the future work. PROBLEM STATEMENT The purpose of this project is to design a rotating platform in order to conduct crystallography experiments on various types of crystals. There is a limitation of workspace given by the relative location of the laser to the rotating manipulator. For a better analysis of the crystal, the platform should be able to rotate 360 o around two axis and plus or minus 30 o around the third one. For this reason a search of similar robotics configuration needs to be done in order to explore all the alternatives prior to design execution. A prototype needs to be built in order to test the chosen interface and programming requirements of the servo-motors in order to produce the desired motion. DESIGN ANALYSIS In this section of the report a kinematical and dynamic analysis will be presented. The analysis will include the derivation of the governing equations and conclusions regarding the behavior of the system in terms of the input variables. As it was mentioned, a 3-DOF rotary table manipulator is a mechanism consisting of three gimbals 1, these gimbals are classified as: outer, middle and inner. The main structural characteristic of these gimbals is their perpendicular rotation respecting each other. Figure 1. Two Design Configurations of the 3-DOF Platform A. Kinematic Analysis For the kinematical analysis and layouts of the 3-DOF rotary table manipulator, two main configurations are considered. There are 12 different possible considerations, but they are all a symmetric representation of these two [5]. The 1 Gimbals: A gimbal is a pivoted support that allows the rotation of an object about a single axis.

2 following figures describe the different layout, in which the outer gimbal is positioned differently respecting the middle and the inner gimbals. For the first layout, the outer gimbal is rotating around the pitch axis of the system, the middle and inner gimbals are rotating around the yaw and roll axis respectively. On the other hand, for the second layout the outer gimbal is rotating around the yaw axis and the middle and inner axis are rotating the pit and roll axis respectively. There are some advantages on the first layout over second one [4]. The main advantage of the first layout is that the outer gimbal, which has the highest moment of inertia and weight, can be moved with two actuators; on the other hand, for the second layout the outer gimbal can only be moved with one actuator. The second advantage is related with the reduction of mechanical problems such as wobbling effect. For this reason many authors chose the first layout for the derivation of the dynamic equation of the system. The following is the layout of the table manipulator with its defined frames. For the selected layout the following transformation matrix represents the kinematics of each link and it will be used to derive the dynamics equations of them. Note that c stands for cosine and s for sine and the values 1, 2 and 3 correspond to the angles φ, ψ and θ indicated in Fig 2. B. Dynamic Analysis The main formula applied for a 3-DOF rotary table equation, is derived from Newton s second law: (3) (2) This equation is applied for every point denoted for the k symbol, and the following is the notation used for the equation:, external torques around k point is the rotational momentum of the k point in proportion to the inertia point O is the k point moment of in proportion to the inertia point. is the rotational velocity of k point in relation of the inertia point defined in the inertia coordinate frame. is differential of any point in the inertia coordinate frame. Equation 3 can be represented as follows: Figure 2. Layout of the 3-DOF Platform Manipulator From the above figure, several coordinates frame can be described as follows [4]: 1. Inner coordinate frame connected to inner gimbal. 2. Middle coordinate frame which is connected to the middle gimbal. 3. Outer coordinate frame connected to the outer gimbal. 4. Inertia coordinate frame respecting to earth. In order to conduct an analysis to the kinematic of the spatial open chain mechanism [5], a link frame must be constructed from base to end (for i=0 to i= n). Then using the Denavit and Hartenberg s (D-H) method, a set of homogeneous transformation matrix can be obtained from the corresponding D-H parameters for each link. The general transformation matrix is obtained by using the following multiplication of transformation matrices: (1) (4) where means that u is defined in body coordinate frame. represents the entire external torques applied to a complete rotational body in proportion to A point. defines as the entire moment of inertia of the body. is the velocity of the body with respect to the inertial reference frame. is defined as a vector from any given point A to point K. represents a vector from point O to point A. Applying equation (4) to each of the gimbals, the dynamic equations are obtained in the following form: Dynamic equation of the inner gimbal is given by (5)

3 where several of the terms above are defined as follows: and are errors introduced in the dynamic equations due to wobble errors [8]. Dynamic equation of the middle gimbal: (12) The second velocity to be derived is the one from the middle gimbal, which is described as follows: Using equation (11), (13) (6) (14) Dynamic equation of the outer gimbal: (15) Lastly, the angular velocity of the outer gamble is described as, (7) Derivation of Rotational Velocities: The rotational velocity of the inner gimbal is described as follows: Derivation of the External Torques: The middle, outer and inner torques are described as follows: (16) (8) Each of these angular velocities can be represented as: (17) ; ; (9) (18) (10) (19) (11) Derivation of moments of inertia terms is briefly outlined in the following section.

4 By choosing the coordinate frames in such a way that their axes are aligned with the axes of the gimbals, the following matrix will be obtained [4]: (20), is the rotational torque that has apparent acceleration of the outer gimbal., represents the Coriolis acceleration-related torque component due to velocities and. (21) (22), is the rotational torque due to apparent acceleration of the middle gimbal with respect to the outer gimbal., is Coriolis acceleration-based system torque because of velocities and. is the rotational torque of the outer gimbal due to acceleration on the rotating table. Neglecting the unbalanced and wobble errors and using the terms introduced above, the dynamic equations previously described are rewritten as follows:, represents a rotational torque that contains acceleration of the inner gimbal. where (23) (24) (25) and, are terms representing Coriolis affect torques due to the velocities each gimbal experiences. PARTS DESIGNED AND ASSEMBLY The design of a three-axis rotary table requires a total of four bodies. These bodies include the base, the inner, middle and outer gimbals. To reduce significant effects of torque on the actuators responsible for turning these bodies, the material selection for the bodies should be lightweight; preferably plastic. The base, shown in Figure 3, is designed to provide a firm foundation to the three gimbals that will be rotating around their respected axis. A firm base is responsible for preventing vibrations that will cause inaccurate movement and yield the dynamic responses unpredictable. The vertical section of the base is hollow and allows the continuous rotational actuator to be housed securely inside. (11) In equations (8), (9) and (10), the coefficient represents the electromagnetic torques of the inner (I), middle (M) and outer (O). The coefficients of friction between every gimbal are denoted by. C. Physical Meaning of Mathematical Terms in the Dynamic Equations, represents the rotational torque due to acceleration of the inner gimbal with respect to the middle one. Figure 3. The Base Unit of the 3-DOF Platform The outer gimbal, shown in Figure 4, controls the tables yaw. The outer gimbal will be directly attached to the actuator mounted to the base. When the actuator is activated it will allow the outer gimbal to rotate 360. One side of the outer

5 gimbal houses the second actuator in the top most section. The actuator is firmly attached inside and is enclosed by a cap. Figure 4. Outer Gimbal The middle gimbal, shown in Figure 5, controls the tables pitch. The middle gimbal is rotated by the actuator housed in the outer gimbal. The middle gimbal allows the table to rotate 360 without interfering with the outer gimbal. The third and final actuator is housed in the middle gimbal in a similar fashion as the outer gimbal. Figure 5. Middle Gimbal The inner most gimbal seen in Figure 6, is responsible for the table s roll. The inner table does not include housing for an actuator as there are no additional gimbals to rotate. Figure 7. Final Assembly of the 3-DOF Platform SOFTWARE INTEGRATION The user interface for this platform will allow the user to input the degree values to rotate the sample base with respect to each axis with extreme exactitude. The user may choose the direction to rotate the sample base, which is a square thin surface in the center of the inner gimbal previously described. Using the DEBUGIN command from the Basic Stamp software several actions can be performed. First the user will enter the desired degrees on one axis and the program will activate the corresponding servo motor to perform the necessary rotation. Then the user can input another value for another axis and later another value for the third axis. This interface is configured in a way that when the value of 125 is entered for either axis the servo motors will not rotate at all. A reset value will be implemented to reset the platform to its original position. This option can be implemented for the three axes together or independently. Another desired action goal for this platform is to record a series of rotation values to repeat them several times as needed for different material samples. Through the use of recording desired actions/configurations it will save time when performing additional crystallographic analysis. Figure 8 shows the interface for each servo. Figure 6. Inner Gimbal Through the use of a lightweight body and proper foundation the 3 axis rotary table is able to rotate on all three axes: pitch, roll and yaw. The final assembly is illustrated in Figure 7. Figure 8. Initial System Interface (Basic Stamp Servo Control Default Interface by Parallax Inc.)

6 Figure 8 shows that the servo can be selected from the right column of available servos (labelled as right center and left) to activate them. The amount of rotation is entered via the left column entry to control the position of the servo. The main inconvenience of using the Basic Stamp interface, from Parallax Inc., is that it does not allow the user to define the degree of rotation. Instead it allows the entry of an arbitrary numerical value. Therefore, we have implemented a window that allows the user to specify the rotation in degrees. CONCLUSION The current study has yielded several conclusions as summarized below: A 3-DOF rotary platform was designed to be implemented into the crystallography experiments. Kinematics analysis was carried out to optimize the mechanism linkages. Dynamic equations have been derived and all design parameters, including possible wobble errors, the physical meaning of the equations are identified to be taken into consideration for the construction of the 3-axis rotation bed. A SolidWorks model was developed showing final prototype stage of the design. A prototype was built for integration with the computer through software interface. Through the use of a lightweight body and proper foundation the 3 axis rotary platform is able to rotate on all three axes: pitch, roll and yaw. The user interface will be implemented to allow recording the rotation steps for analysis as well as to accurately reproduce the motion. REFERENCES [1] K. Liu, J. M. Fitzgerald, and F. L. Lewis, Kinematic Analysis of a Stewart Platform Manipulator, IEEE Transactions On Industrial Electronics, Vol. 40, No. 2, April [2] T. Li, and S. Payandeh, Design of Spherical Parallel Mechanisms for Application to Laparoscopic Surgery, Robotica (2002) volume 20, pp , Cambridge University Press DOI: /S , United Kingdom, [3] S. Bai, Optimum Design of Spherical Parallel Manipulators for a Prescribed Workspace, Mechanism and Machine Theory, Vol. 45, No. 2, pp , February [4] M. Dorosti, and J. H. Nobari, Kinematic and Dynamic Analysis of 3-DOF Rotary Table Manipulator, IEEE Explorer, [5] A. Alasti, and H. Abedi, Kinematic and Dynamic Sensitivity Analysis of a Three-Axis Rotary Table, proceeding of IEEE Conference on Control Application, CCA 2003, and Paper Identity code: CD , [6] M. Dorosti, and H. Nobari, Extracting Full Dynamic Equations of 3-DOF Rotary Table, Proceeding of IEEE Conference on Computer, Control and Communication, IC4 2008, [7] Z. Qu, and Y. Yao, Analysis and Measurement of Wobble Error on Simulation Turntable, Harbin Institute of Technology, Harbin. [8] T. Seo, W. In, and J. Kim, A New Planar 3-DOF Parallel Mechanism with Continuous 360-degree Rotational Capability, Journal of Mechanical Science and Technology, 23, pp , [9] J. Kim, Y. Cho, Frank C. Park, and J. Lee, Design of a Parallel Mechanism Platform for Simulating Six Degrees-of freedom General Motion Including Continuous 360-degree Spin, CIRP Annals Manufacturing Technology, Vol. 52, No. 1, pp , [10] C. M. Gosselin, and E. Lavoie, On the Kinematic Design of Spherical Three-Degree-of- Freedom Parallel Manipulators, The International Journal of Robotics Research, Vol. 12, pp , 1993.

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