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# Rprp 3 Rpr

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RPRP SERIAL CHAIN MANIPULATOR & 3RPR PLANAR PARALLEL MANIPULATOR

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### Rprp 3 Rpr

1. 1. RPRP SERIAL CHAIN MANIPULATOR 3RPR PLANAR PARALLEL MANIPULATOR Sumit Tripathi Saket Kansara
2. 2. <ul><li>RPRP serial chain manipulator </li></ul><ul><li>Forward kinematics </li></ul><ul><li>Inverse kinematics </li></ul><ul><li>Trajectory tracking </li></ul><ul><ul><li>Circle </li></ul></ul><ul><ul><li>Ellipse </li></ul></ul><ul><li>Task space control </li></ul>
3. 3. RPRP GUI
4. 4. USE OF HOMOGENEOUS TRANSFORMATION TO DRAW LINK POLYGONS <ul><li>matlab_th1= theta1_g*pi/180; </li></ul><ul><li>matlab_th2 = matlab_th1+theta2_g*pi/180; </li></ul><ul><li>x1_t=x1_g*cos(matlab_th1); % So that prismatic joint 1 is always on top of revolute joint 1(circle) </li></ul><ul><li>y1_t=y1_g+ r_g*sin(matlab_th1); </li></ul><ul><li>Tm1 = [ cos(matlab_th1) -sin(matlab_th1) 0 x1_t % Transform to frame '1' </li></ul><ul><li>sin(matlab_th1) cos(matlab_th1) 0 y1_t </li></ul><ul><li>0 0 1 0 </li></ul><ul><li>0 0 0 1 ]; </li></ul>
5. 5. HOW DO WE APPLY CONTROL ?? <ul><li>xd=(xc+R*cos(w*t)); </li></ul><ul><li>yd=(yc+R*sin(w*t)); </li></ul><ul><li>ErrorX=[ (xd-x); (yd-y) ]; </li></ul><ul><li>K=[g 0; 0 g]; </li></ul><ul><li>error_correction = jacobian_i*K*ErrorX; </li></ul><ul><li>qdot=jacobian_i*Xdot + error_correction ; </li></ul>
6. 6. TASK SPACE CONTROL FOR TRACING CIRCLE <ul><li>No control </li></ul><ul><li>With control </li></ul>
7. 7. VARIATION OF RADIUS <ul><li>No control </li></ul><ul><li>With control </li></ul>
8. 8. TASK SPACE CONTROL FOR TRACING ELLIPSE <ul><li>No control </li></ul><ul><li>With control </li></ul>
9. 9. MANIPULATOR LIMIT REACHED !!! <ul><li>When any of the joint variables exceeds the maximum limit or goes below minimum limit, then it shows manipulator limit reached. </li></ul>
10. 10. 3RPR PARALLEL MANIPULATOR
11. 11. 3RPR PARALLEL MANIPULATOR <ul><li>Forward kinematics </li></ul><ul><li>Inverse kinematics </li></ul><ul><li>Trajectory tracing </li></ul><ul><li>GUI </li></ul>
12. 12. 3RPR PLANAR PARALLEL MANIPULATOR
13. 13. 3RPR GUI
14. 14. MANIPULATOR LIMIT REACHED
15. 15. FORWARD KINEMATICS EQUATIONS <ul><li>m = (k*sin(theta2 - pi)*cos(theta1)*sin(theta3) )/(sin(theta2-theta1)) - (k*sin(theta2- pi)*sin(theta1)*cos(theta3))/(sin(theta2-theta1)) - k*cos(2*pi/3)*sin(theta3) - k*sin(2*pi/3)*cos(theta3); </li></ul><ul><li>n = (-k*cos(theta2- pi)*cos(theta1)*sin(theta3))/(sin(theta2-theta1)) + (k*cos(theta2- pi)*sin(theta1)*cos(theta3))/(sin(theta2-theta1)) + k*sin(2*pi/3)*sin(theta3) - k*cos(2*pi/3)*cos(theta3); </li></ul><ul><li>p = (-a*sin(theta2 - 2*pi/3)*cos(theta1)*sin(theta3))/(sin(theta2-theta1)) + (a*sin(theta2 - pi/3)*sin(theta1)*cos(theta3))/(sin(theta2-theta1)) + a*sin(theta3-2*pi/3); </li></ul><ul><li>phi = 2*(atan2(-n + sqrt(n^2 -p^2 + m^2),(p-n))); </li></ul><ul><li>x = (k*sin(theta2 - phi-pi) + a*sin(theta2-pi/3))*cos(theta1)/(sin(theta2-theta1)); </li></ul><ul><li>y = (k*sin(theta2 - phi-pi) + a*sin(theta2-pi/3))*sin(theta1)/(sin(theta2-theta1)); </li></ul><ul><li>d1 = sqrt(x^2 + y^2); </li></ul><ul><li>d2 = sqrt((x- k*cos(phi+pi) - a*cos(pi/3))^2 + (y- k*sin(phi+pi)- a*sin(pi/3)) ^2) ; </li></ul><ul><li>d3 = sqrt((x - k*cos(-2*pi/3-phi) + a*cos(2*pi/3))^2 + (y - k*sin(-2*pi/3-phi) + a*sin(2*pi/3))^2); </li></ul>
16. 16. INVERSE KINEMATIC EQUATIONS <ul><li>theta1 = atan2(y,x); </li></ul><ul><li>theta2 = atan2((y - k*sin(phi+pi) - a*sin(pi/3)),(x - k*cos(phi+pi) - a*cos(pi/3))); </li></ul><ul><li>theta3 = atan2((y - k*sin(-2*pi/3-phi) - a*sin(2*pi/3)),(x - k*cos(-2*pi/3-phi) - a*cos(2*pi/3))); </li></ul><ul><li>d1 = sqrt(x^2 + y^2); </li></ul><ul><li>d2 = (sqrt((x- k*cos(phi+pi) - a*cos(pi/3))^2 + (y- k*sin(phi+pi)- a*sin(pi/3)) ^2)) ; </li></ul><ul><li>d3 = (sqrt((x - k*cos(-2*pi/3-phi) + a*cos(2*pi/3))^2 + (y - k*sin(-2*pi/3-phi) + a*sin(2*pi/3))^2)); </li></ul>
17. 17. HOW TO DO TRAJECTORY TRACING ?? <ul><li>J1 = [ -d1*sin(theta1) cos(theta1) -k*sin(theta4); </li></ul><ul><li>d1*cos(theta1) sin(theta1) k*cos(theta4); </li></ul><ul><li>1 0 1]; </li></ul><ul><li>Similarly calculate J2 and J3; </li></ul><ul><li>M = [ J1, -J2, zeros(3); </li></ul><ul><li>J1, zeros(3), -J3]; </li></ul><ul><li>theta_dependant_dot = -inv(M2)*M1*theta_independant_dot; </li></ul><ul><li>M1 = matrix of rates of independent variables </li></ul><ul><li>M2 = matrix of rates of dependent variables </li></ul><ul><li>theta_dot = [ theta1_dot, d1_dot, theta4_dot, theta2_dot, d2_dot, theta5_dot, theta3_dot, d3_dot, theta6_dot ]'; </li></ul><ul><li>Integrate theta_dot to get all the joint variables. Then use forward kinematics to plot. </li></ul>
18. 18. <ul><li>Thank You </li></ul>