three axis rotary platform

Florida Conference on Recent Advances in Robotics, FCRAR 2010 - Jacksonville, Florida, May 20-21, 2010

Design of a Three-Axis Rotary Platform

William Mendez, Yuniesky Rodriguez, Lee Brady, Sabri Tosunoglu

Mechanics and Materials Engineering, Florida International University 10555 W Flagler Street, Miami, Florida 33174 United States

[email protected], [email protected], [email protected], [email protected]

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 360o

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 360o 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 gimbals1,

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.


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.

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)

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.

(2)

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)

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:

(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)


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:

(6)

Dynamic equation of the outer gimbal:

(7)

Derivation of Rotational Velocities:

The rotational velocity of the inner gimbal is described as

follows:

(8)

Each of these angular velocities can be represented as:

; ; (9)

(10)

(11)

(12)

The second velocity to be derived is the one from the middle

gimbal, which is described as follows:

(13)

Using equation (11),

(14)

(15)

Lastly, the angular velocity of the outer gamble is described

as,

(16)

Derivation of the External Torques:

The middle, outer and inner torques are described as follows:

(17)

(18)

(19)

Derivation of moments of inertia terms is briefly outlined in

the following section.


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)

(21)

(22)

Neglecting the unbalanced and wobble errors and using the

terms introduced above, the dynamic equations previously

described are rewritten as follows:

(23)

(24)

(25)

where

(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.

, is the rotational torque that has apparent

acceleration of the outer gimbal.

, represents the Coriolis acceleration-related

torque component due to velocities and .

, 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.

, represents a rotational torque that contains

acceleration of the inner gimbal.

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.

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


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 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 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 8. Initial System Interface (Basic Stamp Servo Control Default

Interface by Parallax Inc.)


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.

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Analysis of a Stewart Platform Manipulator, IEEE

Transactions On Industrial Electronics, Vol. 40, No. 2,

April 1993.

[2] T. Li, and S. Payandeh, Design of Spherical Parallel

Mechanisms for Application to Laparoscopic Surgery,

Robotica (2002) volume 20, pp. 133–138, Cambridge

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United Kingdom, 2002.

[3] S. Bai, Optimum Design of Spherical Parallel

Manipulators for a Prescribed Workspace, Mechanism

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February 2010.

[4] M. Dorosti, and J. H. Nobari, Kinematic and Dynamic

Analysis of 3-DOF Rotary Table Manipulator, IEEE

Explorer, 2009.

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IC4 2008, 2008.

[7] Z. Qu, and Y. Yao, Analysis and Measurement of

Wobble Error on Simulation Turntable, Harbin Institute

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Parallel Mechanism with Continuous 360-degree

Rotational Capability, Journal of Mechanical Science

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Continuous 360-degree Spin, CIRP Annals –

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three axis rotary platform

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