International Journal of Hybrid Information echnology, pp.4-4 http://dx.doi.org/.457/ijhit.4.7.5.37 Development of a MALAB oolbox for 3-PRS Parallel Robot Guoqiang Chen and Jianli Kang * Henan Polytechnic University, China jz97cgq@sina.com, kangjl@hpu.edu.cn Abstract Aiming at one kind of 3-PRS parallel robot, the study develops a toolbox in MALAB. he toolbox includes functions for forward kinematics, inverse kinematics, velocity kinematics, error analysis, schematic representation, and so on. he architecture of the 3-PRS robot is introduced firstly. he instructions of the functions, developing procedure and main algorithms are presented secondly. he toolbox encapsulates complicated mathematical formulas into the single function and provides standard inputs and outputs, which improves the reliability and makes it easy to use. Finally, an example calls the toolbox function, and verifies its correctness, reliability and convenience. Keywords: 3-PRS parallel robot, toolbox, forward kinematics, inverse kinematics. Introduction As an important branch of the industrial robot, the parallel robot has many advantages, such as high stiffness, high accuracy, little cumulate error, large load carrying capacity and compact structure [-7]. It has gained wide applications in all kinds of fields and attracted plenty of study. he 3-PRS parallel mechanism is very typical in the parallel robot family [4]. here are many complicated mathematical formulas in analysis on degrees of freedom, working envelope, kinematics, speed, acceleration, accuracy, and so on [8-8]. Especially, the trigonometric function computation, nonlinear equations and complex matrix make design and analysis tedious and discouraging [, 3-5, 7-8]. he engineering software MALAB has powerful performance in numerical computation, symbolic operation and graphical representation, so it is an efficient tool for the robot research [9]. he aim of this paper is to present a computer-aided analysis toolbox developed in MALAB for one kind of 3-PRS parallel robot, and give key algorithms.. Architecture of 3-PRS Parallel Robot he schematic representation of the 3-PRS parallel robot is shown in Figure [, 3-5,7-9]. he robot is composed of a moving platform, three limbs, three vertical rails and a fixed base. hree vertical rails vertically link to the base B B B. Moreover, B B B form an equilateral 3 3 triangle that lies on a circle with the radius R.he axis of the revolute pair R i for i, and 3 is perpendicular to the prismatic pair. Each limb L i for i, and 3 with the length of l i connects the corresponding rail by a prismatic pair R i. he moving platform and three limbs are connected by three spherical pairs b, b and b 3. hree spherical pairs form an equilateral triangle that lies on a circle with the radius r. he cutter with the length of h is placed at the *Corresponding Author: Jianli Kang is the corresponding author. Email: kangjl@hpu.edu.cn his work is supported by Scientific and echnological Project of Henan Province of China with grant No. 343. ISSN: 738-9968 IJHI Copyright c 4 SERSC
International Journal of Hybrid Information echnology center of the moving platform. he feed of the prismatic pair is given as H. Angles for i i i, and 3 are defined from the vertical rail to its corresponding limb L i. As shown in Figure, a fixed Cartesian reference coordinate system OXYZ is located at the center O of the base B B B. X-axis and Y-axis are in the base plane B 3 B B, X-axis points in the direction of O 3 B, and Z-axis is normal to the base plane and points upward. A moving coordinate frame o xyz is located at the cutter point o. he xy plane is parallel to the moving platform b b b, x- 3 axis points in the direction of C b, and z-axis is normal to the moving platform. he position and orientation of the cutter can be described using the coordinates x, y, z of the cutter o o l o o l o o l point and Euler angles, and rotating about Z, Y and X axes of the fixed system. he 3- PRS parallel robot possesses 3- DOF that are rotation about the Z-axis and about the Y- axis, and a translational motion z along the Z-axis [9]. hree parasitic motions include one rotation, one translational motion x about the X-axis, and one translational motion y about the Y-axis. he three parasitic motions, x and y can be expressed using the three independent motions, and z. R 3 H 3 3 B 3 L 3 R L B b 3 b o z C Z O y Y b h L x B R H X Figure. Schematic Representation of the 3-PRS Parallel Robot 3. oolbox Development in MALAB All toolbox function names begin with the prefixion 'PRS_'. In the toolbox functions, the parameters A, B, G, R, r, h, X and Y represent,,, R, r, h, x and y, respectively. H is a -by-3 vector that represents [ H, H, H ]. H is a -by-3 vector that represents [ 3,, ]. B, B, B3, R, R, R3, b, b and b3 are -by-3 vectors of the corresponding points 3 coordinates, respectively. R R 3 b b 3 Y O r b R X R 4 Copyright c 4 SERSC
International Journal of Hybrid Information echnology 3.. Function for Homogeneous ransformation Matrix = PRS_ran (A, B, G, P) returns a 3-by-4 array that represents the homogeneous transformation matrix computed using Equation () from o xyz to OXYZ. he input parameter P is the vector x y z. o o l o o l o o l C C C S S x S S C C S S S S C C S C y C S C S S C S S S C C C z where S, S, S, C, C and C represent s in, sin, sin, co s, co s and cos respectively. 3.. Function for Euler Angle G = PRS_ABG (A, B) returns the Euler angle that can be expressed by the other two angles and using Equation (). sin sin arctan c o s c o s 3.3. Function for Parasitic Motions x and y [X, Y]= PRS_XY (A, B, G, r, h) returns the parasitic motions x computed using Equation (3). r x c o s c o s sin sin sin c o s c o s h sin y h sin c o s r sin sin c o s r c o s sin 3.4. Function for Coordinates of B, B and B3 o o l o o l o o l () () and y that can be [B, B, B3]= PRS_BiXYZ (R) returns the coordinate vector of B, B and B3 using the following equation system. B = R 3 B = R R 3 B = R - R 3 (3) (4) 3.5. Function for Coordinates of R, R and R3 [R, R, R3]= PRS_RiXYZ (H, R) returns the coordinate vector of R, R and R3 using the following equation. Copyright c 4 SERSC 43
International Journal of Hybrid Information echnology R R H 3 R R R H 3 R R R H 3 3 (5) 3.6. Function for Coordinates of b, b and b3 [b, b, b3] = PRS_ForKinbi(H, H,PofPRS) returns the coordinate vector of b, b and b3. PofPRS is a vector including 4 elements that represent R, l, l and l 3. he coordinate vector of b, b and b3 in the system o xyz can be computed using Equation (6). Combined with the transformation matrix, the coordinate vector in the system OXYZ can be gotten. b r h 3 b r r h 3 b r r h 3 3.7. Function for Angles between Vertical Rails and Corresponding Limbs [H, Flag]= PRS_ForKinH (H, PofPRS, H) computes three feeds and angles using the optimal iteration algorithm based on based on Equation (7). PofPRS is a -by-5 vector that represent r, R, l, l and l 3. he initial iteration value H is a vector including 3 elements, and has heavy effect on the rationality of the solution. 3.8. Function for the Direction Vector of the z-axis b b b b b b 3 r (7) 3 3 Vz= PRS_zVectorFrombbb3 (b, b, b3) returns the direction vector of the z-axis. he direction vector can be computed using the following formula. e 3 b b b b b b b b b b b b 3 3 3 3 b b b b b b b b b b b b 3 3 3 3 (6) (8) 44 Copyright c 4 SERSC
International Journal of Hybrid Information echnology 3.9. Function for the Coordinates of the Cutter Point [Xool, Yool, Zool]= PRS_XYZFrombbb3(b,b,b3,h,dx,dy) returns the cutter point coordinates x, y, z. dx and dy are the offsets x and y along x-axis and y-axis o o l o o l o o l between the center C of the equilateral triangle and the perpendicular projection point C of the cutter point in the equilateral triangle plane, shown in Figure. he algorithm can be expressed as follows. b3 D b x E C F y C y x x, y, z Figure. Algorithm Diagram for Function PRS_XYZFrombbb3 he coordinates x, y, z C C C of the center C of the equilateral triangle bbb3 can be expressed as x y z C C C x x x 3 b b b 3 y y y 3 b b b 3 z z z 3 b b b 3 Based on the coordinates of b and b, the direction vector computed. he parameter equation for the y-axis line is he corresponding t of the point E is x x m t C y y n t C z z p t C t y m n p b m, n, p (9) of the y-axis can be () () he coordinates x, y, z E E E of E can be computed. Likewise, the coordinates of bb3 center D can be gotten, the direction vector of the x-axis can be computed, and the coordinates Copyright c 4 SERSC 45
International Journal of Hybrid Information echnology x y z of the point F can be gotten. Based on coordinates of points E and F, the center,, F F F coordinates of the parallelogram C F C E can be computed, and coordinates the point C are x, y, z C C C of x x x x y y y y z z z z C E F C C E F C C E F C he function PRS_zVectorFrombbb3 is called to compute the direction vector of the line that the cutter resides. Likewise, x, y, z can be computed. o o l o o l o o l 3.. Function for hree Euler Angles from the Direction Vector of the Cutter [A, B, G]= PRS_VectorOfoolABG(VectorOfool) returns Euler angles. he input parameter VectorOfool is a vector including three elements that represents the direction of the cutter. he 9 upper-left elements in the transform matrix form the direction cosine matrix from o xyz to OXYZ. he unit vectors of the three coordinate axes of OXYZ are E [,, ], E [,, ] and E 3 [,,], while the three unit vectors of o xyz are e, e and e 3. he direction cosine matrix D can be expressed as E e E e E e 3 D E e E e E e 3 E e E e E e 3 3 3 3 he unit vector e 3 is the utilization one of VectorOfool. he matrix elements in the third row and third row of D can be computed. he Euler angles and are computed using Equation () and (), then the Euler angle is gotten by calling the function PRS_ABG. 3.. Function for the Inverse Kinematics [H, H] = PRS_BacKin (A, B, Z, PofPRS) returns three feeds and three angles. PofPRS is a vector including 6 elements that represent r, R, l, l, l 3 and h. he initial iteration value H is a vector including 3 elements, and has heavy effect on the rationality of the solution. he Euler angle is computed using PRS_ABG, and the transformation matrix is computed using PRS_ran. he coordinates of b, b and b3 in the system o xyz can be expressed as [ x, y, z,] b i b i b i i,, 3, and the corresponding coordinates in the system OXYZ can be computed using the transformation matrix. he feeds of the three prismatic pairs H i,, 3 are i () (3) (i=,, 3) (4) H Z Z l X X Y Y i R i b i i R i b i R i b i where Z is the Z coordinate of R R i i, l i is the length of limb L i, X and X R i b i are the X coordinates of R i and b i respectively, and Y R i and Y b i are Y coordinates of R i and b i respectively. 46 Copyright c 4 SERSC
International Journal of Hybrid Information echnology 3.. Function for all Solutions of Angles Using Monte Carlo Simulation [H,N]= PRS_ForKinAllH (H, PofPRS, HMin, HMax, K, M, Error) returns all solutions of three angles between vertical rails and corresponding limbs using Monte Carlo simulation. he return value N is the number of the solutions. he input parameter PofPRS has the same meaning as the function PRS_BacKin, [HMin, HMax] is the interval where the solution resides, K is the number of Monte Carlo sampling, and M is the number of the possible solutions. he solutions are regarded as one if the absolute value of their difference is no more than the limiting error Error. he return number of the solutions N is no more than the set number M. 3.3. Function for the Position and Orientation Error Error= PRS_ManError (A, B, Z, PofPRS, EPofPRS) returns the position and orientation error. he return value Error is a vector including 6 elements that represent the errors of the three Euler angles and position coordinates. he input parameter PofPRS has the same meaning as the function PRS_BacKin. EPofPRS is a vector including elements that represent moving platform radius error r, fixing base radius error R, three limb-length-errors l, l, l, cutter length error h, three feeds errors H, H, H, offsets 3 3 x and y. he computation procedure is as follows. Step : hree feeds H and three angles H are computed using [H, H]= PRS_BacKin (A, B, Z, PofPRS). Step : he three angles are computed again using [H, Flag]= PRS_ForKinH (H+EPofPRS(7:9), PofRobot(:5)+EPofPRS(:5), H). Step 3: he three-dimensional coordinates of b, b and b3 are computed using [b, b, b3]= PRS_ForKinbi(H+EPofPRS(7:9), H, PofPRS(:5)+ EPofPRS(:5)). Step 4: he direction vector of the z-axis is computed using Vz= PRS_zVectorFrombbb3 (b, b, b3). Step 5: hree Euler angles A, B and G are computed using [A, B, G]= PRS_VectorOfoolABG (-Vz). Step 6: he coordinates x, y, z o o l o o l o o l are computed using [Xool, Yool, Zool] = PRS_XYZFrombbb3 (b, b, b3, EPofPRS(6), EPofPRS(), EPofPRS()). Step 7: he error Error is computed using the actual and nominal values. 3.4. Function for the Jacobian Matrix Jacobian = PRS_ Jacobian (A, B, Z, PofPRS, Flag) computes the Jacobian matrix according to the position and orientation, and the structure parameter. he input parameter PofPRS has the same meaning as the function PRS_BacKin. If Flag is, the function returns the Jacobian matrix; else, the inverse Jacobian matrix. he return value Jacobian is a 3-by-3 matrix. he 3-PRS parallel robot possesses 3- DOF, and z. Based on the Equation () and (4), three feeds are functions of, and z. he, and z are functions of the time t, and can be expressed as ( t ), ( t ) and Z ( t ). So Equation (4) can be expressed as So (5) H ( t ) F ( t ), ( t ), Z ( t ) i i Copyright c 4 SERSC 47
International Journal of Hybrid Information echnology where the Jacobian matrix J is dh dh d H d d d 3 Z J d t d t d t d t d t d t (6) H H H Z H H H J Z H H H 3 3 3 Z If J has full rank, the inverse Jacobian matrix can be computed. Because of the complexity of Equations (5) to (7), the Jacobian matrix function jacobian (f, v) can be used directly. 3.5. Function for the Acceleration Computation ah = PRS_ Acceleration (A, B, Z,VABX, aabx, PofPRS) computes the acceleration of three feeds. PofPRS has the same meaning as the function PRS_BacKin. VABX and aabx are the velocity and acceleration vectors of the cutter direction and position, and each includes 3 elements. he return value ah is a vector including 3 elements representing the acceleration of three feeds. he transmission relationship can be gotten through differentiation using Equation (6). 3.6. Function for the Structure Diagram Drawing PRS_ Plot(B,B,B3,R,R,R3,b,b,b3) plots the scheme of the 3-PRS robot. he function PRS_XYZFrombbb3 is called to compute the coordinates of the cutter point, and the MALAB function plot3 is used to draw the scheme. 4. Example Result and Discussion he structure parameters of a 3-PRS parallel robot are as follows. l l l 3 7, R 35 and h 8. he trajectory path of the cutter point is the intersection of the spherical surface and the plane z z, and the cutter is always perpendicular to the spherical surface. he trajectory path of the cutter point can also be expressed as x R s R z s c o s t y R s R z s s in t z z ( t 3 6 ) (7) r, he direction vector of the cutter is x, y, z, and the direction vector of the z-axis is x, y, z. he toolbox developed can be used to analyze the characteristics. If R s, z 6 and t 6, the computation procedure and key codes in MALAB are as follows. >> drsz=sqrt(*-(-6)*(-6)); 48 Copyright c 4 SERSC
International Journal of Hybrid Information echnology >> x=drsz*cos(pi/3); y=drsz*sin(pi/3); % Prepare parameters >> [A,B,G]= PRS_VectorOfoolABG(-[x,y,6]) ; % Compute three Euler angles >> [H,H]= PRS_BacKin(A,B,6,[,35,7,7,7,8]); % Compute feeds and angles >> [R,R,R3]= PRS_RiXYZ(H,35); % Compute coordinates >> [b,b,b3]= PRS_ForKinbi(H, H,[35,7,7,7]); % Compute coordinates >> [B,B,B3]= PRS_BiXYZ(35); %Compute coordinates >> PRS_Plot(B,B,B3,R,R,R3,b,b,b3,8); % Plots the scheme he feed vector of the three prismatic pairs is H=[46.5857, 46.5857,59.87333], the three angles vector between vertical rails and corresponding limbs is H =[.359,.359,.649], and the scheme of the robot is is shown in Figure 3. 5 Z 5 Y - - 3 Figure 3. Scheme of the Robot Position and Orientation for Position and Orientation he parameter t is made discrete over the interval [, 36 ). Each discretization value is computed, and the result is shown in Figure 3. PRS_BacKin is used for Figure 4 (a), PRS_XY for Figure 4 (b), and PRS_ManError for Figure 4(c) and (d). he error vector of the parameters is [.68,.77,.798,.5, -.76, -.566, -.636, -.98, -.788,.3,.878] in Figure 4(c) and (d). he three position-coordinates errors and three Euler angles errors differ greatly in value although the error source is same, which demonstrates that the position and orientation has tremendous effect on the error sensitivity. he translational motion of the 3-PRS robot possesses is only one z along the Z-axis, and it has two parasitic motions x and y, as shown in Figure 4(b). An X-Y table is needed to compensate the parasitic motions x and y, and the compensations Xp and Yp are shown in Figure 4 (b). X Copyright c 4 SERSC 49
International Journal of Hybrid Information echnology H 6 55 5 H H H3 4 X Y Xp Yp 45 - -5 4-4 3 4 3 4 t t a) Feeds of hree Prismatic Pairs b) Parasitic Motions and Compensations 5 x -4 - -. 3 4 3 4 t t c) hree Euler Angles Errors d) Position Errors 5. Conclusions Figure 4. Computing Results of the Example A toolbox for the 3-PRS parallel robot is developed in MALAB and key algorithms are given. he toolbox includes 6 functions for forward kinematics, inverse kinematics, velocity kinematics, error analysis, schematic representation of the robot, and so on. Finally, an example calls the toolbox function, and verifies its correctness, reliability and convenience. he toolbox is very useful for design and analysis of the 3-PRS robot characteristic. he developed toolbox and its application have several advantages. he toolbox encapsulates complicated mathematical formulas into the single function and provides standard inputs and outputs, which improves the reliability and makes it easy to use. he functions are saved as m-files and all file names end with the extension '.m', so it is helpful and almost necessary for the user to modify the codes for expansion. However, it should be noticed that the 3-PRS robot can be classified into four categories including seven kinds according to limb arrangements as discussed in [9]. Based on the toolbox here, the function for the other kinds can be extended. Acknowledgements his work is supported by Scientific and echnological Project of Henan Province of China with grant No. 343. he authors would like to thank the anonymous reviewers for their valuable work...5 -.5 Xool Yool Zool 4 Copyright c 4 SERSC
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