Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | P D Control with Gravity Compensation 293 A possible modification to the PD control law with gravity compensation consists in replacing the derivative part D which is proportional to the derivative of the position error i. e. to the velocity error q qd q by a term proportional to qd-0 where ứ e IR is as said above the result of filtering the position q by means of a dynamic system of first-order and of zero relative degree. Specifically the p D control law with gravity compensation is written as T Kpq Kv qd - tf ỡ q X Ax ABq ử X Bq where Ap Kntlinv areniagont positivedefinite matricer i4 diag ai andto rt diagtM rll g l l ii arereal airictlypositiveconstants but otherwise arbitraryfoo i 1 2 . Figure shows the block-diagram corresponding to the robot under P D control with gravity compensation. Notice that the measurement of the jointvelocity q es nol raqniredby1 hecoatrallet. Figure . Blowl-diagram P D gfaigol with gravity compensation Define ị X Bqd. The equation that describes the behavior in closed loop may be obtained by combining Equations III. 1 and - which may be written in terms of the state vector ịT qT qT as 294 P D Control r d ỉt ộ -q- Aị ABq Bqd q - qd - M q -1 Kpq Kv qd - Bq - C q q q . j r H1 zT1T o A sufficient condition for the origin q q Oe nr to be a unique equilibrium point of the closed-loop equation is that the desired joint position qd be a constant vector. In what is left of this section we assume that this is the case. Notice that in this scenario the control law may be expressed as 1- Kpq - li o iP ị q g q I p Li J Which is Close to the PD with gravity compensation control law 7 1 when the desired position qd is constant. Indeed the only difference is replacement of the velocity q by . i dip i diag Q Vet thereby avoiding the use of the velocity q in the control law. A wesliow inthe followingsubsections p D control with Ravity con1-peneationmeetsthe yoeitioncvntrolhbjective that is CT IH-X_it - where qd e IR
