Page - 213 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

SinceJNJ,M≡0, the transformation leads to the redundantcoordinates formulation M˜(q)q¨+ C˜(q, q˙)q˙+Q˜(q, q˙)= A˜ T (q)c (17) with NJ,M= ( P˜ A˜ ) , A˜∈Rm,n, P˜∈Rn−m,n (18) M˜ :=NTJ,MMNJ,M, C˜ :=N T J,M(CNJ,M+MN˙J,M), Q˜ :=N T J,MQ. (19) Unlikebefore, this formulationconsistsofnequations,whereδonesare independent. 4. Model-BasedControlwithanAugmentedPD-Controller Model-based control is very important for parallelmechanismswith actuation redundancy, because of the antagonistic forces. As the name, redundant actuation, implies, there aremore driving forces thandegreesof freedomm>δloc to control themechanism. However,with this feature it is possible to increase the internal preload and thus, e.g. to annihilate backlashdue tomanufacturingormanipulate theEEstiffness [9], [5], [6]. Thegeneralized force of an augmentedPDController consists of three parts. Thefirst part is a feed forward termcalculatedwith the inversedynamics,which releases the feedbackcontroller. Thus, the joint angle error ismuch smaller. The second one is a feedback term,withweighted error position andvelocity of the joint angles. Andfinally the third onehas nodynamic effect i.e. it increases the internal forces. For further information, see [2], [4], [8], [7]. 4.1. InverseDynamics The inversedynamics solution is thebasis for anaugmentedPDoracomputed torquecontroller. 4.1.1. MinimalCoordinatesFormulation Thesolutionof the inversedynamics isgivenbyminimizationof (c−c0)TW(c−c0)as c= ( AT (q) )+ W ( M(q) q¨i+C(q, q˙) q˙i+Q(q, q˙) )︸ ︷︷ ︸ 1 +NAT,W(q)c 0︸ ︷︷ ︸ 3 (20) with theweightingmatrixW and an arbitrary preload parameter vector c0. Furthermore, (AT)+W= W−1A(ATW−1A)−1 is the rightpseudoinverseandNAT ,W= Im−(AT) + WA T is anull spaceprojector ofmatrixAT. 4.1.2. RedundantCoordinatesFormulation Thenumberofequationsof the redundant coordinates formulation ishigher, than thenumberof free parametersc∈Rm (n<m). Furthermore, there areδ independent equations, i.e. onlyδ columnsof A˜ T are linear independent. ThereforeEq. 17mustbe rewrittenas y˜ = A˜ T c= A˜ T 1c1+ A˜ T 2c2, c1∈Rδ, c2∈Rm−δ, (21) c1 = ( A˜ T 1 )+( y˜− A˜T2c2 ) , ( A˜ T 1 )+ = ( A˜1A˜ T 1 )−1 A˜1, (22) 213