Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition

Abstract

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Configuration-based compliance control of kinematically redundant robot arm Part I: Theoretical framework

“…In accordance to [10], [19], the complementary projector used in (13) is defined by the following relations:…”

“…while the Moore-Penrose generalized inverse satisfying the least norm condition is given by the following relation, [19]:…”

“…The cost function (19) generates a scalar potential field over the robot hyperdimensional configuration space. Since this potential field is nonlinear, continuous, and therefore differentiable function, the gradient optimization method, [9], [10] can be effectively applied to find the optimal nullspace joint vector q* which locally minimizes influence of the cross-joint members of the robot jointspace stiffness matrix.…”

Singular Factorization of an Arbitrary Matrix

“…This approach of Lanczos is similar to the methods in [15,16,27,28]. It can be considered that Jordan [15,16], Sylvester [30,31] and Beltrami [2] are the founders of the SVD [29], and there is abundant literature [4,6,7,11,30,34] on this matrix factorization and its applications.…”

Singular Value Decomposition

“…This approach of Lanczos is similar [7] to Schmidt [8] and Jordan [9,10] methods; we can consider that Jordan, Sylvester [3] and Beltrami [11] are the founders of the SVD [12], and there is abundant literature [13][14][15][16][17][18][19][20][21][22][23][24][25] on this matrix factorization and its applications.…”