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Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4.4 The Residual Dynamics 103 and with an abuse of notation it may be written as h t q q Mfiqfi - M q qd C qfi - Cfiq q qd ỡ qd - gfij . This function has the characteristic that h t 0 0 0 for all t but more importantly the residual dynamics h t g. q has the virtue of not growing faster than IIqII and ộ . Moreover it may grow arbitrarily fast only when so does g independently of ộ . In order to make this statement formal we need to recall the definition and properties of a continuously differentiable monotonically increasing function the tangent hyperbolic. As a matter of fact the statement can be shown for a large class of monotonically increasing functions but for clarity of exposition here we restrict our discussion to ex e x tanh x x which is illustrated in Figured.1. Figure 4.1. Graph of tangent hyperbolic tanh x As it is clear from Figure 4.1 tanh x is coin ninons monotonically increasing. Also it has continuous derivat ives and it satisfies M tanh x and 1 tanh x for all .r é III All these observations are stated formally below. Definition 4.1. Vectorial tangent hyperbolic function I We define the vectorial tangent hyperbolic function as tanh aj tanh xi tanh x 4.13 104 4 Properties of the Dynamic Model where X G IRn The first partial derivative of tanh aj is given by ỡtanh dx a Seclific diag secỉĩ rCj 4.14 where sechfxi The vectorial tangent hyperbolic function satisfies the following properties. For any X X G IR tanh a 4 a tanh a 2 tanh a 2 o. tanh a Ta 0esh aj i O4 i where 4 O.With tanh a defined as in 4.13 the constants - 1 012 a Ũ 03 1 ds I. Property f.f. Residual dynamics Vector h t q q The vector of residual dynaomcs h t q q -f n x 1 depends on the position errors q velocity errors q and on the desired joint motion qd qd and qd thatissupposedto be bounded.Inthisrespect we denote by I sllei and IIsbIIM tlie supreme values over the norms of the desired velocity and acceleration. In addition h t q q has the following property There exist constants khi kh2