Page - 218 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

For a signal energy optimal manipulationof the robot a cost functional is introduced, which consists of the quadratic input signals in every time step and of a so-called Scrap-function which defines the end pointdeviation. TheidentifiedmovementsweretestedonaPUMAsixaxisrobot. Withthemeasuredcontrolvariables the required energy was evaluated. Based on this test data a considerably energy reduction was detected. 2. Problem definition Atfirst, let usconsider anonlineardynamical system q˙=v M(q)v˙= f˜(q,v,u,t), (1) where q ∈ Rn is the vector of generalized coordinates and v ∈ Rn is the vector of generalized velocities. In addition,M is then×nmass matrix and f˜ ∈ Rn the force vector. The vectoru indicates thecontrolvariables inan openedorenclosed regionΓ⊆Rm. By introducing thevectorof statevariablesxT=(qTvT)wemayrewriteEquation(1) by x˙=f(x,u,t) x(t0)=x0. (2) In general the force vectorf is a continuous vector field which depends on the statesx, controlsu andontimet. In robotics, thepositionandvelocityof the toolcenterpoint (TCP)willbeofparticular interest instead of the joint angles and angular velocities. Hence, the system outputy∈Rl is given by y=g(x). In order tomeet apredefined end pointwehave tosatisfy theboundarycondition g(x(tf))= y¯. (3) However,wesubstitute theboundaryconditionofEquation (3)by theoptimalcontrolproblem x˙=f(x,u,t) J= ∫ tf t0 h(x,u,t) dt+S(tf,x(tf))−→Min. (4) where the integraldescribes theenergy consumptionand theScrap-functionS includes theend point error. If theclosed regionΓ is not empty the solutionof the optimalcontrol problem of Equation (4) leads toan energy optimalmanipulationof thedynamical systemofEquation (2). 3. Gradient computation To determine thegradient of thecost functional (4)wefirst add zero terms to it: J= ∫ tf t0 h(x,u,t)+pT [f(x,u,t)− x˙]︸ ︷︷ ︸ =0Eq. (2) dt+S(tf,x(tf)) (5) 218