Synergy–based Optimal Design of Hand Pose ... - Centro E Piaggio

gradient flow (10) onto the subspace tangent to the constraint,obtaining the search directions = ∇‖P p ‖ 2 FW . (11)Having the search direction for the constrained problem,the gradient flow is given byḢ = −4 [ P 2 pP o H T Σ(H) ] TW (12)where Σ(H) = (HP o H T +R) −1 . The gradient flow (10) guaranteesthat the optimal solution H ∗ has unit vectors as rows,if H(0) satisfies the latter condition.In [15] authors define a function V 2 (P) with P ∈ IR n×nthat forces the entries of P to be as “positive” as possible.In this section, for the second gradient, we extend thisfunction to measurement matrices H ∈ IR m×n with m < n,thus considering function V 2 : O m×n → IR, given byV 2 (H) = 2 3 tr[ H T (H − (H ◦ H)) ] , (13)where A ◦ B denotes the Hadamard or elementwise productof the matrices A = (a i j ) and B = (b i j ), i.e. A ◦ B = (a i j b i j ).The gradient flow of V 2 (H) isḢ = −H [ (H ◦ H) T H − H T (H ◦ H) ] , (14)which minimizes V 2 (H) converging to a matrix in P m×n ifH(0) ∈ O m×n (H(0) is the starting point at t = 0 for (14)).By combining (12) and (14), we can build the followinggradient flowḢ c,d = 4(1 − k) [ P 2 pP o H T c,d Σ(H c,d) ] TW++ k ¯H d[( ¯H d ◦ ¯H d ) T ¯H d − ¯H T d ( ¯H d ◦ ¯H d ) ] , (15)where k ∈ [0, 1] is a positive constant, Σ(H c,d ) =(H c,d P o Hc,d T + R)−1 , W = I n − Hc,d T (H c,dHc,d T )−1 H c,d is theprojecting matrix onto the set of all matrices whose rows areunit vectors, and, finally, ¯H d assumed the following form:[ ]0mc ×n¯H d = .H dThe gradient flow defined in (15) converges towards ahybrid sensing device, i.e. with both continuous and discretesensors, if H c,d (0) ∈ O m×n , minimizing the squaredFrobenius norm of the a posteriori covariance matrix. Noticethat, since this problem is not convex, we can only assurethat the proposed algorithm converges to a local minimum.To overcome this common problem in gradient methods aclassic solution can be provided by performing a multi–startsearch.IV. RESULTSFigure 1 shows the values of the squared norm of thea posteriori covariance matrix for increasing number m ofmeasures. In particular, values of V 1 for matrices Hc ∗ and Hd∗are reported, for both noise–free and noisy measures. Noticethat, in case of noise–free measures, V 1 values decreasewith the number of measures, tending to assume nearlyzero values in case of both continuous and discrete sensing.This fact is trivial because increasing the measurements wereduce the uncertainty of the measured variables. When allV 1(P o,H,R)0.140.120.10.080.060.040.02H * c , R=0H d* , R=0H c* , R≠ 0H d* , R≠ 001 2 3 4 5 6 7 8 9 10 11 12 13 14mFig. 1: Squared Frobenius norm of the a posteriori matrixwith noise–free and noisy measures (with noise covariancematrix R = diag(12) [ ◦ ]) for both H ∗ c and H ∗ d .the measured information is available V 1 assumes zero valuewith perfectly accurate measures. In case of noisy measures,V 1 values decrease with the number of measures but, in thiscase, it tends to a value which is larger, depending on thelevel of noise.In case of free-noise measures, if we analyze how muchV 1 reduces with the number of measurements w.r.t. the valueit assumes for only one measure, reduction percentage withthree measures is greater than 80% for both H ∗ c and H ∗ d .This result suggests that with only three measurements theoptimal matrix can furnish more than 80% of uncertaintyreduction. This is equivalent to say that a reduced number ofmeasurements is sufficient to guarantee a good hand postureestimation. In [4], [13], under the controllability point ofview, authors state that three postural synergies are crucial ingrasp pre–shaping as well as in grasping force optimizationsince they take into account for more than 80% of variance ingrasp poses. Here, the same result can be obtained in termsof the measurement process, i.e. from the observability pointof view: a reduced number of measures coinciding with thefirst three principal components enables for more than 80%reduction of the squared Frobenius norm of the a posterioricovariance matrix.V. EXPERIMENTSIn this work, we deal with the problem of hand posereconstructions considering noisy measures in the discretesensing case. We consider an additional zero–mean randomGaussian noise with standard deviation of 7 ◦ on each measureand hence a noise covariance matrix R = diag(7) [ ◦ ].This value is chosen in a cautionary manner, based on dataabout common technologies and tools used to measure handjoint positions [17]. Without loss of generality we haveadopted the 15 DoF model also in used [4], [13], [5] andreported in figure 3. As described in [5], an optical motioncapture system (Phase Space, San Leandro, CA - USA)with 19 active markers is used to collect a large number ofstatic grasp positions, see figure 4. Subject AT (M,26) hasperformed all the grasps of the 57 imagined objects describedin [4]; these data have been acquired twice to define a set of114 a priori data.