Natural Orthogonal Complement Constraint equations twist Shape matrix -R Joint Equations(6.63)and(6. 64) pertaining to the first link E141=0 (665a) 81+R1w1=0 6.65b K1t1=0 6 K;-1t-1+Kt;=0,t=1,:7 666b) Er O K R11 6.67a K;;1≡ ei o (667b) E; O R (6.67
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint Equations (6.63) and (6.64) pertaining to the first link:
Natural Orthogonal Complement Constraint equations twist shape matrix -R Joint Kll O6 O6 6 6 K21K22O6 O K (6.68) O606O Kn-1 ) Oc O6 O6 K K 6n xin matrix O6 denoting the 6 x 6 zero matrix
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint 6n 6n matrix
Natural Orthogonal Complement Constraint equations twist shape matrix -R Joint Define partial Jacobian j.6=t (6.69) 6x n matrix with its element defined as ifj≤ ei xr 0 (670) otherwise Mapping the first i joint rates to ti of the ith link
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint Define partial Jacobian 6 n matrix with its element defined as Mapping the first i joint rates to ti of the ith link
Natural Orthogonal Complement Constraint equations twist shape matrix -R Joint +1 +1 aj a;+a+1+…+a-1+p2ifj< r;;≡〈p ifj=勾 (6.71) 03 otherw
Natural Orthogonal Complement • Constraint Equations & Twist Shape Matrix – R Joint
Natural Orthogonal Complement Constraint Equation and twist Shape matrix-R Joint t;=61ti+b2t2+…+6;tn;i=1,…,7 (672) 10 0 0 T三 (6.73) 77 Easy to verify KT=o -11 Oe o O t110 0 K21K22O6 O 0 O6 Oa O6 K O OG O6 O K nn-1 KnnI Tmi t K t 11U1 0 Recall K;-1ta-1+Kt=0,=1,…:7
Natural Orthogonal Complement • Constraint Equation and Twist Shape Matrix – R Joint Easy to verify Recall