By J. Baillieul, D. P. Martin, Roger W. Brockett, Bruce R. Donald
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Similarly, #2 may be computed using the numerical values of S2 and C2. Finally, substituting values for #1, #2, #3, #4, and #5 in the (1,1) and (2,1) elements of the following equation IT = f T' l 5rF-J-4rF4X 3 1 T - l o ^ - l handrp X zT 1 ' (45) MADHUSUDAN RAGHAVAN 40 yields two linear equations in s& and c$. After solving for 56 and CQ we may use their values to determine a unique value for 6Q. 6. Brief history of manipulator kinematics. The use of coordinate systems and transformation matrices to model linkages was introduced by DENAVIT and HARTENBERG[1955].
32 MADHUSUDAN RAGHAVAN For such an arrangement the coordinate transformation matrix *+ T is / cos Oi — sin 9i cos c^ sin Oi sin c^ sin 6i cos 6^ cos ai — cos ^ sin ai 0 sinc^ cosaj \ 0 0 0 ai cos Oi \ ai sin Oi d« 1 / where #, a, a, and d are shown in Figure 9. A formal derivation of - + 1 T may be Join t n + l Join t n- I Lin k n + l Lin k n - 2 Figure 9: Kinematic Parameters found in PAUL[1981], Chapter 2, page 53. For a revolute-jointed manipulator, Oi is the angle of rotation of link % + 1 relative to link z, about joint i.
RaPJ_ (49) (49) may be verified very easily by seeing that y T ' - ^ T is equal to / . 3. 7 LEMMA. Let v i , v 2 be arbitrary vectors. JR(Vlxv2), Vi,j. 8 P R O O F . Let v x = ( vn J R v i = ( j-1 j m v\2 ^13 ) Then (JRvi) x (*Rv 2 ) = , v 2 = ( v2\ j n ) vi = )lvn + )mvl2 v22 + jm; 1 3 . v23 ) . (50) Similarly J R v 2 = JIV21 + )mv22 + j nv23. (51) From (50) and (51), it follows that ( J R V l ) x (JRv 2 ) = {)\xij\)vnv21 + ()\xijm)vnv22 + ()l>