Page - 212 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

3.3. DifferentFormulationsofMotionEquations Equations (8) and (9) are the point of departure, for many formulations. These formulations are necessary,becausesolving this systemofequations (Eq. 8,9),which iscalledadifferential algebraic equation(DAE), isverycomplex.Moreover, it isnotappropriate for the inversedynamics. Toreduce it to an ordinary differential equation (ODE), the constraint forcesmust be eliminated. This paper presents theminimaland redundantcoordinates formulation [2], [4], [8], [7]. 3.3.1. MinimalCoordinatesFormulation There are six independent joint angles,without thegeometrical constraints.While introducing these fourconstraints, thenumberof independentangleswillbe reduced formsix to two. Thus, thecoordi- nates canbesplit independentqd and independentqiones q= ( qd qi ) , qd∈Rn−δ, qi∈Rδ. (10) Moreover, thekinematicconstraints (Eq. 7) canbedivided too, toexpress thedependent jointveloc- itiesexplicitly Jq˙=Jqdq˙d+Jqiq˙i=0, q˙=Fq˙i, F= (−J−1qdJqi Iδ ) , F∈Rn,δ (11) with the identitymatrix I. MatrixF is therefore anorthogonal complementof the JacobianmatrixJ, i.e. theproduct ofbothvanishes identically (JF≡ 0). Since theconstraint forcesvanishwithmatrix F, it is anappropriateprojectorand leads to theminimalcoordinates formulation M(q)q¨i+C(q, q˙)q˙i+Q(q, q˙)=A T(q)c (12) with F= ( P A ) , A∈Rm,δ, P∈Rn−m,δ (13) M :=FTMF, C :=FT(CF+MF˙), Q :=FTQ. (14) Thisformulationconsistsofδ independentequations.Adrawbackistheselectionoftwoindependent, local appropriate coordinates. Therefore parametrization singularities canoccur. Amethod to avoid this is to switchbetweenmotionequationswithdifferent independentcoordinates selection [4]. 3.3.2. RedundantCoordinatesFormulation Theproblemof the latter formulation(Eq. 12)are theparametrizationsingularities,due to thechoice of independent coordinates. There are two possibilities to avoid this. The first way, the switching method,hasbeenmentionedbefore. Theotherway is touseanother formulationwithoutanycoordi- nates selection,bymeansofanull-spaceprojector NJ,M := In−J+MJ, NJ,M∈Rnn (15) with the rightpseudoinverse J+M=M −1JT(JM−1JT)−1. (16) 212