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Advanced Robotics - Control of Interactive Robotic Interfaces Volume 29 Part 5

Tham khảo tài liệu 'advanced robotics - control of interactive robotic interfaces volume 29 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 66 2 Control of Port-Hamiltonian Systems a Energy Fig. 2.7. Energy and force of non saturated regular and saturated bold spring. b Force amount of stored energy increases too much then the force generated by the springs that is the force that the actuators should apply can be greater than the physical limits of the actuators themselves. If the robot is controlled by means of the PD g q controller this situation can happen e.g. if the initial error is sufficiently high. Consider K diag fci . fc that is the spring energy in 2.117 can be written as E Et -r 2Í 1 1 i l Then suppose that each actuator is limited i.e. jjTO fi fi M Ỉ Consider XM and xm x-L m . xn m such that fi M kiXi M and jjTO kịXitm. The saturation of each actuator can be taken into account if the following energy function is introduced CÍ.Ẹ Ei s En s 2.118 where fi m 2 i m Ikixl xijm Xi xijM 2.119 fi M 2 Note that the passivity properties of the spring are preserved since the proposed energy function is c1 and bounded from below. The energy function of a 1-dimensional spring and the relative force in function of the state are represented in Fig. 2.7 both for the non-saturated and saturated case. Clearly the energy functions are different in the saturation 2.5 A Variable Structure Approach to Energy-Based Control 67 zone when the stored energy becomes infinite the force generated by a nonsaturated spring increases to infinity while it is limited for the saturated case. The saturation of each actuator can be taken into account in the passive control of a robot if instead of the energy function 2.115 we consider it is assumed n Hc p Eps qi qpd 2.120 i l where EjjS - is defined as in 2.119 kị 0 can be freely assigned and fi m fi M depend on the characteristics of the ị-th actuator. Since Hc is characterized by a global minimum in qd the control action 2.114 still assures the global stability of this configuration. From Eq. 2.116 it follows that D Kd 0 2.121 at dp dp for q Ạ 0. Since HC1 is bounded from .