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Control Problems in Robotics and Automation - B. Siciliano and K.P. Valavanis (Eds) Part 6

Tham khảo tài liệu 'control problems in robotics and automation - b. siciliano and k.p. valavanis (eds) part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Dynamics and Control of Bipedal Robots 109 of motion single support phase double support phase and instances where both lower limbs are above the ground surface. Accordingly the resulting motion is classified under two categories. If only the two former modes are present the motion will be classified as walking. Otherwise we have running or another form of non-locomotive action such as jumping or hopping. Equations of motion during the continuous phase can be written in the following general form X f x b x u 2.1 where X is the n 2 dimensional state vector f is an n 2 dimensional vector field b x is an n 2 dimensional vector function and u is the n dimensional control vector. Equation 2.1 is subject to m constraints of the form 0 x 0 2.2 depending on the number of feet contacting the walking surface. 2.2 Impact and Switching Equations During locomotion when the swing limb i.e. the limb that is not on the ground contacts the ground surface heel strike the generalized velocities will be subject to jump discontinuities resulting from the impact event. Also the roles of the swing and the stance limbs will be exchanged resulting in additional discontinuities in the generalized coordinates and velocities 15 . The individual joint rotations and velocities do not actually change as the result of switching. Yet from biped s point of view there is a sudden exchange in the role of the swing and stance side members. This leads to a discontinuity in the mathematical model. The overall effect of the switching can be written as the follows X sw X 2.3 where the superscripts X is the state immediately after switching and the matrix sw is the switch matrix with entries equal to 0 or 1. Using the principles of linear and angular impulse and momentum we derive the impact equations containing the impulsive forces experienced by the system. However applying these principles require some prior assumptions about the impulsive forces acting on the system during the instant of impact. .