Alekhnovich, G. N.

Arebi, M. A.

Dainiak, I. V.

Repository of Belarusian National Technical University (BNTU)

 301 Belarusian State University of Informatics and Radioelectronics, Minsk  COMPUTER SIMULATION OF 6-DOF PARALLEL MECHANISM  Dainiak I. V., Arebi M. A., Alekhnovich G. N.  The six degree of freedom spatial movement system has been examined. Algorithms and simulation programs for direct and inverse kinematic problems solving of spatial mechanism are presented. As a result of computer simulation the movement system’s actuator spatial motion capabilities are obtained and investigated.  INTRODUCTION The modern phase of science and technology evolution in the field of automatic and robotic systems is characterized by the use of multi-degree-of-freedom movement systems, realizing almost any spatial movements of an executive or machining tool. Additional requirements for simultaneous implementation of all translational and angular motions are in increasing frequency demanded for new developed movement position devices. The multi-degree-of-freedom systems constructed on the basis of 6-DOF parallel mechanisms are among the most perspective solutions for such motions implementation. However, the task of rapid kinematic and dynamic analysis of such parallel mechanisms is one of the main difficulties in the sphere of robotics. Therefore the paper examines the simulation and analysis of the 6-DOF parallel mechanism in MATLAB/Simulink development environment to present a productive and up-to-date way of similar robotic manipulators engineering.  KINEMATIC STRUCTURE OF THE MECHANISM Kinematic structure of 6-DOF (degree-of-freedom) spatial parallel mechanism is shown on Fig. 1. It is composed of 6 independent legs connecting the mobile platform P13 with the base. Each of these legs is a serial kinematic chain that is controlled by one motor which actuates one of the joints.   Fig. 1. Mechanism structure  The parallel mechanism, presented on Fig. 1, is intended to the realizing of spatial 3D movements of mobile platform [1]. The points 1C , 2C , 3C  mobile platform P13 is directly connected to direct drives А1, В1, А2, В2, А3, В3 by shanks 7l , 8l ,…, 12l  (7, 8, 9,10,11,12) with spherical kinematic joints. The 6-DOF spatial parallel mechanism carry  302 out three coordinate axial displacements ( x , y , z ) and three angular rotations (ψ  − yaw, θ  − pitch, ϕ  − roll) of the mobile platform. The number of DOF was calculated by Somov-Malyshev’s formula [2]: 6651=∑ ν−⋅−==kkpknW , where 13=n ; 65 =p ; 04 =p ; 123 =p ; 021 == pp ; 6=ν . Thus, the mechanism has 6 degrees of freedom.  DIRECT AND INVERSE KINEMATIC PROBLEM The direct and inverse problems solution implies the implementation of algorithms in simulation software. These algorithms solve both direct and inverse problems for mechanism investigated. Solution direct and inverse problem kinematics for executive mechanism present important stages of system motion design as a whole Fig.2.   1C133C2C1A1B2A2B3A3B 134561098721112YXZOyxzrR Fig. 2. Geometric representation of the mechanism  DIRECT KINEMATIC PROBLEM Direct kinematics problem [2] is to determine the mechanism for a given structure and geometry of the moving parts, the laws of movement of executive-level relative to the fixed coordinate system according to the known laws of movement of the input or important links. Input parameters (as shown on Fig. 2) are the x and y coordinates of point’s pairs 11, BA , 22 , BA  and 33 , BA , which correspond to drives; coordinate z of these points are supposed to equal to zero. We accepted coordinates of points 321 ,, CCC  as output parameters Fig. 3. For the point iC  we’ll have: ,sin)(;sincos)(;coscos)(11iiiCiiNCiiiiNCCNzCNyyCNxxiiiiiϕ=αϕ−=αϕ−= where 3,2,1=i ; −iN  point between 11 BandA ; −ϕi −α i  angles; (as shown on Fig. 3) and the distance iiCN  is calculated by formula  303 222 )()(iiii ABABii yyxxlCN −−−−= .    Fig. 3  As we accepted, the distances 21CC , 32CC  and 13CC  are equal, then we can write const133221 ==== aCCCCCC . Finally, we’ll write equation system, which contain geometric condition for coordinates of points 321 ,, CCC : =−+−−−=−+−+−=−+−+−222222222222)()()()()()()()()(131313232323121212azzyyxxazzyyxxazzyyxxCCCCCCCCCCCCCCCCCC Solving these expressions together, we can get needed coordinates of platform points 321 ,, CCC . Unfortunately, it’s necessary to use numerical methods for the solving of equation system due to non-linearity of their equations. For simulation of mechanism we developed   the module in MATLAB environment, which solves the direct problem.  INVERSE PROBLEM Inverse kinematics problem [2] of the mechanism (Fig. 1) can be formulated as follows. It is necessary for the position and orientation of the movable element of mechanism, which is defined in movable coordinate system oxyz, specified by a discrete or parametrical function in relation to a stationary coordinate system OXYZ, to find the position of segments 1, 2, ..., 6 (see Fig. 1), defined in the coordinate system OXYZ coordinates of points iA  and iB , respectively, (i = 1, 2, 3). In the simulation software position and orientation of the movable coordinate system oxyz in relation to the fixed coordinate system OXYZ are characterized by, respectively, the Cartesian coordinates of platform center O( 0x , 0y , 0z ) and Euler angles ϕ , θ , ψ . The position and orientation matrix is given by .coscossincossincossinsinsincossinsinsincoscoscossincossincossinsinsinsincoscossincoscos,,ψθψθθ−ψθϕ+ψϕ−ψθϕ+ψϕθϕψθϕ+ψϕψθϕ+ψϕ−θϕ==⋅⋅= ψθϕψθϕ MMMM In the uniform coordinates matrix ψθϕ ,,M  shall transformed to the fourth-order matrix:  304 [ ]=⋅′=′⋅= ×ψθϕψθϕψθϕ10000033,,00,,,,0zyMxMMMMM , which are used for calculating of a platform point coordinates. Given matrix fully describes orientation and position of movable platform of parallel mechanism. Converting point coordinates 1C , 2C , 3C  in stationary coordinate system OXYZ can be realized by formula ⋅==11)()()()(iiiiiiiCCCCCCCzyxMzyxR . To find the points iA  and iB  appropriate to the location of the point 1C : ( ) ( ) ( )=+−=−+−.,2222)(22)(2)( 111Ryxzlyyxx CCC  22 xRy −±= , ( ) ( ) 22)()(22)(2)()(2 11111 22)( xRyxxlzyxR CCCCC −⋅±=⋅⋅−−+++ . When we denote ( ) ( ) AlzyxR CCC =−+++ 22)(2)()(2 111 )( , the previous expression takes a form 22)()( 11 22 xRyxxA CC −⋅±=⋅⋅− . Now we raise both parts of this expression into a second power: 022 =+− cbxax , where ( ) ( )( )2)(2)( 114 CC yxa += , )( 12 CAxb = , ( ) 22)(2 14 RyAc C−= .  From last quadratic equation we can find coordinates of points 332211 ,,,,, BABABA . Thus results solution the inverse kinematic problem, that implies the determination of the input variables (motors shafts rotation angles) out of the output variables (platform position x , y , z  and orientation ψ , θ , ϕ ) [2]. We developed the module in MATLAB environment, which solves the inverse problem.  HARDWARE-IN-THE-LOOP CONTROL MODEL SIMULATION IN MATLAB/SIMULINK The 6-DOF spatial parallel mechanism control system computer simulation is implemented as the hardware-in-the-loop control model simulation in MATLAB/Simulink modeling environment. MATLAB/Simulink in conjunction with Real-Time Workshop can automatically generate, package, and compile source code from Simulink models to create real-time software applications that can immediately be executed on a variety of systems and hardware platforms thus enabling the 6-DOF spatial parallel mechanism control system hardware-in-the-loop simulation as well. [3] The 6-DOF spatial parallel mechanism control system hardware-in-the-loop simulation structure is presented on Fig. 4. As the result of hardware-in-the loop simulation (Fig. 4), the 6-DOF spatial parallel mechanism  305 control system hardware-in-the-loop simulation model has been developed in MATLAB/Simulink for simulation on TI TMS320C2000 DSP debug hardware platform (target board).   Fig. 4. Structure of mechanism model  The developed hardware-in-the-loop simulation model enables the execution of the MATLAB/Simulink control system model directly on physical hardware debug platform, giving that way the possibility of rapid control prototyping for the 6-DOF parallel mechanism control system.  Hardware-in-the-loop simulation approach permits testing hardware components and debugging of controller software by means of connecting hardware to the program models which were developed and simulated within modeling environment.  REFERENCES  1. Litvinau Y., Jarski V. and Arebi M. The Simulation of Parallel Mechanisms in MATLAB/Simulink Development Environment // Proceeding of 53rd Internationals Wissenschaftliches Kolloquium Technische Universität Ilmenau, 08–12 September 2008. – P. 73. 2. Karpovich S., Jarski V. and Dainiak I. “Motion systems on the base of direct drive” // Minsk BSUIR – 2008 – 238 P. [in Russian] Litvinau Y., Jarski V., Dainiak I. and Arebi M. “Hardware-software modeling of a control system of multicoordinate system of moving” // BSUIR Reports.− 2007.−№ 6.−P. 50–55 

Computer simulation of 6-DOF parallel mechanism

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Computer simulation of 6-DOF parallel mechanism

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