tailieunhanh - Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 7

Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Lyapunov Function for Global Asymptotic Stability 165 Time Derivative of the Lyapunov Function The time derivative of the Lyapunov function candidate along the trajectories of the closed-loop system may be written as q q qT Kpq - Kvq - C q q q qTM q q - Kpq T q 7qTSech2 q TM q q - ytanh ộ TM q q - ytanh q T Kpq - Kvq - C q q q where we used Equation in anh qj -Sech2 q q thatis Sech2 q g diag sech2 ẽ where sech on oi uSa and therefore Sech2 q is a diagonal matrix whose elements sech2 are positive and smaIler than 1. Using Property Mt which establishes that qT a M - O 0 and M q C q q ty q nY thetinw drrivativr of the ayahunov function candidate yields th n V q q -qTKvq yqTSech2 q TM q q - ctauhi q tyq ytanh q TKvq - tnnh q TC q q Tq . p now . by a negative of the states q and q. To that end it is convenient to find upper-bounds for each term of . The first term of may be trividlh bounded by qT Ì si q 2 . To upper-boundthe syCondtermyf we inn Iseeh2 x u ni Sech2 q q lliill . From this argument we also have yh i T q w M a n 7TMat M 11 q 2 . On the otherdond nobb fhrtinview vf ThufoTlowing inuqoalityal-u holds tpueoinve u p is a eiayono1gositipeeuWnitom7-rix 166 7 PD Control with Gravity Compensation 7tanh q TKpq 7Amin Kp tanh q 2 which in turn implies the key inequality -7tanh ộ TKpq -yXmin Kp tanh q 2 . A bound on 7tanh q TKvq that is obtained directly is 7tanh ộ TK ý 7ĂMax JQ q IItanh ộ II . The upper-bound on the term 7tanh ộ Tơ q q Tq must be carefully selected. Notice that 7tanh ộ Tơ q q Tq 7QTC q q tanh q 711411 II Cq q tanh q . Then considering Property but in its variant that establishes the existence of a constant c suchthat IIC qqa y kc a y for all q x y e IR we obtain -7tanh ộ Tơ q ý Tý 7fcơl ý 2 tanh ậ . Makingate oft heinequality tanh ât which tsys that tanh q Jn for all q e IR we obtain i -7tanh ậ Tơ q ý Tý y kC1 q 2 . Thwprevious Cotmds yield that