Tham khảo tài liệu 'control of redundant robot manipulators - . patel and f. shadpey part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Schemes for Compliant and Force Control of Redundant Manipulators 111 T Ya -KDs JT - JT Ff H q qr C q q qr G q f q -JTeFex - jJ1 F where H C G f a are calculated based on estimated values of H C G Le f and a respectively. Fx is the measured end-effector interaction force with the environment Kj is a positive-definite matrix and s q qr . The last term on the right-hand side of the equation is only needed if another point of the manipulator other than the end-effector is in contact with the environment Fe denotes the measured reaction force corresponding to a second constraint surface and JC1 is the Jacobian of the contact point. We use the same Lyapunov candidate function as in 41 V t 2 sTHs a Tr a where r is a constant positive-definite matrix and a a a . Differentiating V t along the trajectory of the system leads to V t - sTKDs sTYa sTJTFx sTjT1 FZ where F F - F denotes force measurement error. This suggests that the adaptation law should be selected as a -nTs With this adaptation law equation leads to V t - STKjS sFTFX JT1 - - ijisi2 M UeilLII Jh Ifí and V t - ijsll2 811 si where ij is the minimum eigenvalue value of the matrix Kj and s satisfies the following inequality 112 4 Contact Force and Compliant Motion Control . II II . . II II wltell Jc1 l tel We also assume that Je a and Jc1 p. Now we consider two different cases precise and imprecise force measurements. Precise force measurements F 0 In this case inequality reduces to V t -kD s 2 which implies a s e. L or boundedness of a and s. Moreover it can be shown that 1 s 2dt -1dV t kD l s 2dt -1 fdV t J kDJ 0 0 _L V 0 - V kD a b which implies that s e L2 and consequently Jes Jcs e. IK. In order to establish a link between S and the tracking errors of ACT trajectories we assume that the tracking errors of the damped least-squares solution are negligible. Therefore multiplying both sides of equation by the .
