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Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 14

Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A Mathematical support 387 aịj a - o.ịj y n max k zo dotijiz dzk II33 -y Z Za From the latter expression and from A.3 we conclude the statement contained in A.2 . ộộộ Truncated Taylor Representation of a Function We present now a result well known from calculus and optimization. In the first case it comes from the theorem of Taylor and in the second it comes from what is known as Lagrange s residual formula . Giventheimportance t this lemma in the study of p lt deltaite tamtams Append X B the proof is presented in its complete form. Lemma A.4. Let f Rn - E be a continuOUS function with continuous partial derivatives up to at least the second one. Then for each X G Rn there existsa realnumber a s h t such Chat f f 1 f x 0 g WTx WTH ax x where Htax ts the HeesianmattacSehot tt ztssecond partial derivative of f x evaluated at ax. PreofLe X G llỉ bee conetianthector. Considerthetímhderivativh of tx Is ef e ds s tx- c Integratingfrom t 0 to t l f dWx J X Jũ yp- tx TX dt dxK J o x -nta Jo Ệ to Txdg . ox ih. The integral on the right-hand side above may be written as y t Tx dt A.5 where 388 A Mathematical Support y f A-6 Defining u ytffx V t 1 and consequently du . . 7 ytTz at dt 1 the integral A.5 may be solved by parts1 cl cl _ I y t Txdt t-ỉ ỹ t Tx dt y t Tx t-ỉ Jo Jo i 1 - y t Txdt y tí Tx. A.7 0 Now using the mean-value theorem for integrals2 and noting that 1 t Oforall tbetween 0 and 1 the integral on the right-hand side of Equation A.7 may be written as cl cl 1 t ỹ t Txdt y a Tx 1 i dt JO JO t1 1 x for TnZnoratinZhxX lA 1 we Incorporating this in A.7 TO get ------ We recall here the formula dv f1 du Jo e .1 l O iTcallOhat fcrrui ioos l i and 9 i continuous on the closed Interval a l l . and where 9 t 0 fcr e h tar the ruterval. there always ex Sts a uuurbe c such that a c and hitigit di h g0i di. A Mathematical support 389 fL 1 y f Tx dt y Q T X Jo and therefore A.4 may be written as a - 0 ý a Ta y 0 Ta . A.8 On the other hand using the definition of y i given in .