Tham khảo tài liệu 'mobile robots -towards new applications 2008 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Biped Without Feet in Single Support Stabilization of the Vertical Posture with Internal Torques 31 constraint imposed on each torque rj U i 1 . n-1 U const We deduce from 6 the state model with x V V T x Ax br Here A 0nxn -D-1G b nx n-1 D-1B 7 8 9 Introducing a nondegenerate linear transformation x Sy with a constant matrix S we are able to obtain the well-known Jordan form Ogata 1990 of the matrix equation 8 y Ay dr 10 where Í 1 L S -1AS 0 Ì d S -1b l2n 7 11 Here X1 . X2nare the eigenvalues of the matrix A d is 2n x n-1 matrix. Let us consider that the positive eigenvalues have the smaller subscript. For the two-link biped we will obtain X1 0 Re Xi 0 i 2 3 4 for the three-link biped X1 0 X2 0 Re Xi 0 i 3 - 6 and for the five-link biped Xi 0 i 1 2 3 ReXj 0 j 4 -10 . 4. Problem Statement Let x 0 here 0 is a 2n x 1 zero-column be the desired equilibrium state of the system 8 . Let us design the feedback control to stabilize this equilibrium state x 0 under the constraint 7 . In other words we want to design an admissible satisfying the inequality 7 feedback control to ensure the asymptotic stability of the desired state x 0 . Let W be the set of the vector-functions r t such that their components ri t i 1 . n -1 are piecewise continuous functions of time satisfying the inequalities 7 . Let Q be the set of the initial states x 0 from which the origin x 0 can be reached using an admissible control vector-function. Thus the system 8 can reach the origin x 0 with the control r t e W only when starting from the initial states x 0 e Q . The set Q is called controllability domain. If the matrix A has eigenvalues with positive real parts and the control torques ri t i 1 . n -1 are limited then the controllability domain Q is an open subset of the phase space X for the system 8 see Formal sky 1974 . 32 Mobile Robots Towards New Applications For any admissible feedback control r r x L x U i 1 . n-1 the corresponding basin of attraction V belongs to the controllability domain
