matlab_RRR运动控制机器人

5/13/2009

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Table of Equations .................................................................................................................................................. 3

Table of Figures ....................................................................................................................................................... 4

1. INTRODUCTION ................................................................................................................................................... 5

1.1. THE SYSTEM ................................................................................................................................................. 5

1.1.1. General Overview ................................................................................................................................. 5

1.1.2. Parts and Specification ...................................................................................................................... 6

1.2. THE RELEVANCE ........................................................................................................................................... 6

1.3.2. Identifying key variables ................................................................................................................. 10

2. LITERATURE REVIEW ..................................................................................................................................... 11

3. ANALYTICAL FORMULATION ......................................................................................................................... 12

3.1. KINEMATIC ANALYSIS ............................................................................................................................. 12

3.2. DYNAMIC ANALYSIS FORMULATION ...................................................................................................... 16

3.2.1. FBD of Platform: .............................................................................................................................. 17

3.2.2. FBD of Legs: ..................................................................................................................................... 17

3.2.3. FBD- Sliding Block:........................................................................................................................... 18

3.3. STATIC ANALYSIS .................................................................................................................................... 18

4. MAPLESIM MODELING .................................................................................................................................. 19

7. CONCLUSION ................................................................................................................................................. 27

8. REFERENCES .................................................................................................................................................. 28

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Equation 1: Angular velocities and accelerations ................................................................................................. 10

Equation 2: Position level vector loop closure equation ...................................................................................... 12

Equation 3: Velocity level loop closure equation ................................................................................................. 12

Equation 4: Inverse kinematic relation for first loop ............................................................................................ 13

Equation 5: Inverse kinematics (velocity level) relating each actuator velocity and twist vector ...................... 13

Equation 6: Inverse kinematics (velocity level) for the system ............................................................................ 13

Equation 7: Inverse kinematics (acceleration level) for the system ..................................................................... 14

Equation 14: Dynamic equations (Force, Moment) for end-effector ................................................................... 17

Equation 15: Relation for constraint force between the 3 legs and the end-effector ......................................... 17

Equation 16: Dynamic equation of each leg ......................................................................................................... 17

Equation 17: Relation for constraint forces between 3 legs and corresponding 3 moving slider blocks ............ 18

Equation 18: Dynamic equation for each slider block .......................................................................................... 18

Equation 19: Relation for the actuator force of each link .................................................................................... 18

Equation 20: Inverse dynamic relating actuator forces to external loads ........................................................... 18

Equation 21 - Reference Frames and Paramaters ................................................................................................ 19

Equation 22 - Kinematic Positional EOM - Point1 ................................................................................................ 21

Equation 23 - Kinematic Positional EOM - Point2 ................................................................................................ 21

Equation 30 - Overall Constraints ......................................................................................................................... 22

Equation 31 - DOF wise Constraints ..................................................................................................................... 22

Equation 32 - Independent DOF calculation ......................................................................................................... 22

Equation 33 - Null Space Component Calculaton ................................................................................................. 22

Equation 34- Dynamic Parameters ....................................................................................................................... 24

Equation 35-Position and Velocity Level equations ............................................................................................. 24

Equation 36-Lagrangian Formulation ................................................................................................................... 25

Equation 37-EOM1 ................................................................................................................................................ 25

Equation 38-EOM2 ................................................................................................................................................ 26

Equation 39-EOM3 ................................................................................................................................................ 26

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Figure 1: Hexapod model ........................................................................................................................................ 5

Figure 2: Simplified 2-dimensional model diagram ................................................................................................ 8

Figure 3 - MapleSim Model - Top View ................................................................................................................. 19

Figure 4 - MapleSim Model - PRR Sub-System ..................................................................................................... 20

Figure 5 - MapleSim Model - Link ......................................................................................................................... 20

Figure 6 - MATLAB Simulation Result ................................................................................................................... 23

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The system analyzed in this work is a 6 Degrees-of-Freedom (DOF) Parallel Platform Manipulator by

Quanser Inc. called Hexapod.

The Hexapod has been created as a collaborative R&D venture between ARM Lab at SUNY-Buffalo led

by Prof. Venkat Krovi and the company Quanser Inc. The device is capable of moving high payload at

high acceleration within a compact workspace. It is a 6-DOF of Parallel Manipulators with a rigid

platform. The platform is joined to rigid fixed length arms at three points that form an equilateral

triangle. The arms are linked to the platform in pairs, each pair coming together at one of their ends

and subsequently attached to the manipulator though a spherical joint. The free ends of each of the six

arms are linked to the base by a universal joint which can slide on the base. The bases of the arms are

along prismatic joints. The same pair of links that join together at the platform occupy one of the

straight line axes, thus there are three straight horizontal slots on the platform, each with two link