SIMULATION ENVIRONMENT PROPOSAL, ANALYSIS AND CONTROL OF A STEWART PLATFORM MANIPULATOR Fabian Andres Lara Molina, Joao Mauricio Rosario, Oscar Fernando Aviles Sanchez UNICAMP (DPM-FEM), Campinas-SP, Brazil, rosario@fem.unicamp.br Abstract: The Stewart Platform is a six-axis parallel manipulator with position accuracy, stiffness, and payloadto-weight ratio exceeding conventional serial manipulators. It is proposed here a simulation environment to analyze the kinematics, dynamics and control of this manipulator. It is reviewed the manipulator geometry, the inverse kinematics, Jacobean and inverse Jacobean matrix and it is proposed a path generator. It is discussed the dynamic equation of the manipulator. It is present control position simulation of Stewart Platform manipulator for low velocities. Keywords: Stewart Platform Manipulator, Dynamic equation, Position Control. 1. INTRODUCTION The Stewart Platform manipulator is a parallel kinematic linkage system with six degrees of freedom proposed by Stewart [9]. The main advantages of this kind of manipulator are: (1) higher stiffness and payload-to-weight ratio due to major component of actuation forces are usually additive; (2) low position error because actuators position are not additive; although its workspace is shorter than conventional serial manipulators. In virtue of the mechanical advantages, Stewart Platform manipulator has been used in many applications: flight simulators, machine tools, biped locomotion system, surgery- manipulators. Its high position accuracy enabled its use for precision surgery: neuroenroscopy [5]. As this kind of parallel manipulator has high payload-to-weight ratio and position accuracy, it was used in flight simulators [11], locomotion system (biped locomotor to carry a human) [10], it was also used in CNC machine tool [14]. Kinematics of Stewart Platform manipulator has been studied deeply; it presents an extremely difficult problem for forward kinematics analyzes, the solution is obtained through analytical approaches or numerical schemes [6]; instead inverse kinematics is easy to obtain, it is obtained i.e, by means of a direct relation. There are few works related to dynamic analysis of Stewart Platform manipulator considering that it can be used in several ways: simulation of robotic systems, development of control strategies, sizing actuators and links [4]. The research on control of the Stewart platform has not yet been carried out thoroughly, complicate control schemes has been proposed for high performance: tracking control for Stewart Platform Manipulator [3] and adaptive control [1]. Through this work the mathematical model of the Stewart-Gough platform is implemented for creating simulation software that allows changing different parameters such as geometry, actuators length and controller setup. The results after testing various configurations are shown at the end of this paper. The organization of this paper is as follows: First the kinematic model is introduced in section 2 including manipulator geometry, inverse kinematic equations, Jacobean matrix and path generator and dynamic equations using Lagrange formulation. Section 4 presents the simulation results on the proposed environment for different geometric parameters, paths and control position for desired low velocity paths. Finally, in the last section, the conclusions and further work are discussed. 2. KINEMATICS ANALYSIS The Stewart-Gough platform is a 6 DOF parallel mechanism that consists of a rigid body top plate or mobile plate, connected to a fixed base plate through six independent kinematic legs. These legs are identical kinematic chains, composed of a universal joint, a linear electrical actuator and a spherical joint. 2.1. Manipulator Geometry The geometrical model of a platform expresses the position of the links due to a fixed coordinate system linked at the base and the platform of the manipulator. (Fig. 1) The bottom base geometry is designated by the B1 to B6 points, and the upper base by the P1 to P6 points [13]. Fig. 1. Platform Geometry 670
The links of the platform are defined by: (1) 0 3 2 1,3,5 2,4,6 And the links of the base by: (2) 0 3 2 1,3,5 2,4,6 2.2 Inverse Kinematics The inverse kinematics model of the manipulator expresses the joints linear movements as function of position and orientation due to a fixed coordinate system linked at the base of the platform (Fig. 2), that is: q fp (3) Where,,,,, represents the linear position of the joints, and the position-orientation vector of a platform s point. The coordinate systems A (u, v, w) and B (x, y, z) (figure 2) are attached to the mobile platform and the base.. (6) And so the equation for each actuator is obtained, 2.. 2. 2.. 2.. 1,2,3,4,5,6 (7) The previous equation describes the motion of the platform respect to the base. The parameters and are parameters that define the geometry of the manipulator. 2.3 Differential Kinematics Differentiating equation (7) respecting to time, it is obtained: q J P (8) And each term J ik is: 2. cos. cos. cos. sin. sin. sin. cos. 2. 2. (10) 2coscos. sin cos. sin. sin. 2. 2. (11) 2cos.sin. sin 2. (12) (9) 2cos cos sin cos sin sin 2cos cos sin sin sin cos (13) Fig. 2. Vector representation of the Manipulator The transformation form the mobile platform s centroid to the base is described with the position vector and the rotation matrix, where, 11 12 13 21 22 23 (4) 31 32 And when replacing terms, 33 (5) It can be write the vector-loop equation for the ith actuator s of manipulator: 2cos cos sin cos sin 2sin cos sin sin 2cos sin sin (14) 2 cos sin cos sin sin 2cossin sin sin cos 2coscos (15) 2.4 Actuator In the simulation environment was used an electric linear electric motor, which output is a linear motion. For each kinematic chain is used an actuator. Each actuator is composed by: spindle, gear head, motor and encoder. For electric and mechanic part of the actuator is: (16) (17). (18) 6712
.. (19) Where is the torque, the angular position of the motor axes, the current, and the inductance and resistance, and the inertia and friction of the axis load calculated on the motor side, P the pitch os spindle and N the gear head reduction. 2.5 Path Generator The objective of the path generator is to deliver the reference trajectory for each join according to a desired movement. A linear trajectory is used as a practical example, the following flux diagram shows how the path generator operates. Fig. 3. Path generator Flux diagram To express dynamic lumped equation it is need to find the kinematic and potential energy of mobile platform and legs respectively and to derive these expressions to find,, terms. 3. PROPOSAL ENVIRONMENT FOR ANALYSIS AND SIMULATION The simulation software was implemented using MATLAB, it contains the equations of the Stewart Platform manipulator modeling. The developing process was divided into to two main parts, the first one dealt with the kinematics modeling and the second one with the dynamics modeling, for this the MATLAB s GUI tool and the SIMULINK were used respectively on each case. The motion simulation for each joint is made from the reference vector, which is generated from a defined trajectory for the manipulator. The actuator s, mechanism s and controller s dynamics effects are considered over the six linear joints. Each q joint output enters the direct kinematic model, which delivers the manipulator s orientation and position; this is done with the objective of determining the error between the reference position and the manipulator s position after all the dynamics effects have been taken into account. It is showed the schematic diagram of the tool developed (Fig 4): 2.6 Dynamics Equations There are different methods to model the dynamics of Stewart Platform: Newton-Euler [2], Lagrange formulation [3], principle of virtual work [13], and kane s method [7]. The dynamic equation derived by Lagrange formulation [3] has the following lumped form:,. (20) Where is the inertia matrix., is the Coriolis (centrifugal acceleration) coefficient matrix given by:, 1 2 (21) is the gravity vector, which is derived by differentiating the potential energy as (22) The relationship between the output force vector F of the six actuators corresponding to the τ which is the force and torque applied on the gravity center of the platform is given by: (23) And for each term: (24), (25) (26) Fig. 4 Control diagram for the developed tool There are different parameters for the position and orientation, these are: for the reference one, for the error between the reference position and the exit and for the output Manipulator Position and orientation output. For the joint vector reference and is the joints output vector after the dynamics effects act over the manipulator. 3.1 Manipulator s Geometry Case Study The purpose of changing the manipulator s geometry is to enable simulation of different configurations; the developed tool allows varying it through these parameters: 6723
Table 1. Geometry Parameters Parameters Symbol Unit Platform Radius m Base Radius m Separation angle (Platform) Degrees Separation angle (Base) Degrees The defining parameters for the geometry are shown in the following table: Table 2. Study Cases Case 1 0.1 0.15 90⁰ 30⁰ 2 0.15 0.15 90⁰ 30⁰ 3 0.15 0.15 90⁰ 30⁰ 4 0.1 0.15 90⁰ 90⁰ 3.2 Inverse Kinematics Three different positions are analyzed, the working units are meters and degrees for length and angle, and so the results are: Table 3. Positions for inverse kinematics Case X Y Z α β γ 1 0 0.052 0 0⁰ 0⁰ 0⁰ 2 0.052 0.052 0.1 0⁰ 0⁰ 0⁰ 3 0 0 0.1 45.7⁰ 0⁰ 0⁰ a. case 1 a. Case 1 b. case 2 b. Case 2 c. case 3 Fig. 6 Inverse kinematics Cases c. Case 3 The simulator allows changing the initial position and orientation due to the total reference system R(x,y,z) in order for establishing a position and orientation difference between it and the base s reference system B(x,y,z), defined by. For the following example the Oi vector is 0 0 0.2 0 180 0 d. Case 4 Fig. 5 Geometry s Case Studies Fig. 7 Inverse kinematics Results 6734
3.3 Movement s Simulation Based on the Path Generator In the following example is presented a circular path trajectory on xy plane, the initial point is: 0 0 0.003 0 0 0, radius 30mm. Table 4. Actuator Parameters Parameter Value Unit Symbol Terminal resistance 0.605 Ω Terminal inductance 191x10 H Torque constant 29.2x10 Nm/A Speed constant 34.34 rad/s Mechanical constant 5.6x10 s Rotor inertia 78.7x10 Kg.m² Gear head Reduction 1/51 - Spindle pitch 1x10 m Fig. 8 Path Generator Results a. Actuator 1 b. Actuator 2 3.4 Dynamic Model and Control For the present work it is assumed that Stewart platform manipulator operates at lows speed, the non linear terms, on dynamic equation are negligible; the system in this specific case has a linear behavior. It is acceptable to use a PID controller in this case with high position performance [8]. The dynamic model of the manipulator was developed with MATLAB s SIMULINK, which allows simulating the dynamical effects over the platform. Table 5. Stewart Platform Manipulator Parameters Parameter Value Unit Platform Radius 0.1 m Base Radius 0.15 m Separation angle (Platform) 90 Degrees Separation angle (Base) 30 Degrees Mass mobile platform and load 5 Kg c. Actuator 3 d. Actuator 4 f. Actuator 5 g. Actuator 6 Fig. 9 Path Generator Results 3.5 Actuator In the figure is showed the output for a degree reference position input, 0.1. 4. CONCLUSION AND FURTHER WORK This paper has presented the kinematics, dynamics and control study of the Stewart Platform as a reconfigurable architecture. The simulator is a useful tool for designing and studying the dynamics response when changing different parameters of the platform. This simulation platform constitutes a powerful tool for future research activities such as controller s validation, kinematics analyses and generation of new models by varying the manipulator s parameters. Through the software the geometry parameters, actuators, inertias and gravity centers among others can be configured rapidly. Some further work is: Analyze the complete dynamic and implement a complete control strategy for position control of Stewart Platform manipulator. Make the possible the modification of the control s strategy in order to compare the manipulator s response with different controllers. REFERENCES Fig. 10 Path Generator Results Table 4 shows actuator parameters used in the simulation [1] Amir Ghobakhloo, Mohammad Eghtesad Mohammad Azadi Ghobakhloo, M. E. M. A. G. M. E. M. A. (2006), 'Adaptative-Robust Contol of the Stewart-Gough 6745
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