Page - 221 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

replacements x(t) y(t) z(t) x y z TCP θ1 θ2 θ3 Figure 1. Schematics of the six-axis PUMA robot Figure2. Imageofthesix-axisPUMArobot Note, that it is stronglyrecommended touseaquasi-Newtonmethodwhichdirectlyapproximates the inverseof theHessian. Otherwise, if theoriginal Hessian is computed, avery large and dense matrix mustbe inverted, since thenumberofcomponentsofJmightbecomelarge. The inverseof theHessian afterk+1 iterations isgivenby H˜ −1 k+1= ( I−pkq T k qTkpk ) H˜ −1 k ( I− qkp T k qTkpk ) + pkp T k qTkpk (13) whereI is the identitymatrix,pk is thegradient direction of thekth-iteration andqk is the change of thegradientduring the last iteration. 5. Application to the six-axis-robot The presented method is used to minimize the signal energy consumption of the robot which is de- picted in Figure 1. The reason why we have chosen this robot is that a lot of different parameters are available which are necessary for the evaluation and verification of the results. Afterwards, the simulationresults areverified at a real six-axis-robotwhich is showninFigure2. 5.1. Problem definition The system consists of three degrees of freedom, θ1, θ2 and θ3 which denote the relative rotation angles of the joints. Due to the complicated structure of the equations of motion and the minor influence on the energy consumption the three wrist joints are fixed. First of all the equations of motionarederivedandhave theform x˙=f(x,u,t)with the initialconditionx(t0)=x0 andwhere u=[M1,M2,M3] T contains the torquesof themotorsandx=[θ1,θ2,θ3, θ˙1, θ˙2, θ˙3]T is thevectorof states of the dynamical system. The system outputy= g(x) is a nonlinear function which depends on thestatesand describes thecoordinates of the toolcenterpointy=[x(t),y(t),z(t)]T. For the energy optimal manipulation of the robot from a start-pointx0 to a given end-point y, y˙ (c.f. Table1)withina predefined time tf wedefine thecost functional in the form J= ∫ tf t0 uTudt︸ ︷︷ ︸ signal-energy +S(tf,x(tf)). (14) 221