Tham khảo tài liệu 'robot manipulators trends and development 2010 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 192 Robot Manipulators Trends and Development Coriolis and centripetal terms C x x properties The matrix xT AT x 2C x x x 0 is skew-symmetric so AT x C x x C x x T. 134 We need to keep in mind that the equality described in 134 can be written in the following form AT x C x x C x x T 0 135 Proof. Considering the definition on the inertia matrix M x equation 125 and the Coriolis and centripetal terms C x x equation 126 both in cartesian space we will verify the equation 134 is fulfilled. Therefore we initiated transposing the Coriolis matrix thus we have C x x T J q T C q q T J q 1 J q T J q T J q T M q J q 1 136 what it allows us to solve operation C x x C x ã T C x x C x x T J q T C q qf J q 1 J q TM q J q 1 J q J q 1 J q T C q q T J q 1 J q T J q T J q T M q J q 1 137 As is observed we can put together the following terms C x x C x x T J q T C q q J q 1 J q TM q J q 1 J q J q 1 J q T C q q T J q 1 J q T J q T J q T M q J q 1 138 Thus we have C x x C x x T J q T c q q C q q T J q 1 - J q TM q J q 1 J q J q 1 - J q T J q T J q TM q J q 1 139 Applying 46 we have C x x C x x T J q T M q J q 1 J q TM q J q 1 J q J q 1 140 J q T J q TJ q TM q J q 1 Now replacing 125 in 140 Cartesian Control for Robot Manipulators 193 C x x C x x T J q -TM q J q -1- J q -T M q J q -1 J q J q -1 y - - y M x 141 - J q -T J q TJ q -TM q J q -1 M x thus we have C x x C x x T J q -T M q J q -1 - M x J q J q -1 - J q -T J q T M x 142 Equation 142 represents the first part on the proof. The second step consists on deriving matrix M x defined in 125 thus we have M x J q -T M q J q -1 J q -T M q J q -1 J q -T M q J q -1 143 Using the equation 125 we can find M q as follows M x J q -T M q J q -1 J q TM x M q J q -1 J q T M x J q M q This allows us to replace M q expressed in 144 in 143 as follows M x J q -T J q TM x J q J q -1 J q -T M q J q -1 J q -T J q TM x J q J q -1 Some terms can be eliminated applying the identity matrix property M x J q -T J q TM x J q Hqy J q -TM q J q -1
