CHAPTER 5
 SOLVING 4-DOF MANIPULATORS Reduced System of Equations


 For 4-DOF robot arms, the inverse kinematic problem is solved when 4 joint values are found that satisfy the equation


 A1 A2 A3 A4 = P. (5.1)


Equation (5.1) decouples into a position equation


 R1 (R2 (R3 14 + 13) + 12) + 11 = p, (5.2)


and an orientation equation given by


 R1 R2 R3 R4 = R. (5.3)


When the fourth joint .is revolute 14=0 is obtained by proper choice of frame F4 and Eq. (5.2) simplifies to


 R1 (R2 13 + 12) + 11 = p. (5.4)


 A reduced system of four equations in the sines and cosines of joint angles e1 and e3 can be derived by considering the quantities tz, pz and the inner products t.p and p.p expressed in frame F1

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