TABLE II.

Nomenclature Used in This Paper

• {A} - A right handed frame with {x̂a, ŷa, ẑ a} as its axes and point a as its origin.

• Apa/b position vector from point b to a expressed in {A}.

• ARB - orientation of frame {B} relative to {A}

• Cv A/B, CωA/B - linear and angular velocity of frame {A} with respect to frame {B} expressed in frame {C}.

• CtA/B = [CvA/B, CωA/B]-twist of frame {A} with respect to {B} expressed in {C}. Unless otherwise stated, all twists are defined in base frame {B0}.

• [p^] - the skew-symmetric cross product matrix of vector p.

• θi - the angle describing the bending of the ith segment from its straight configuration θi = π/2, Fig. 4(c).

• δi - the angle describing the plane in which the ith segment (i=l, 2) bends. This angle is defined from the bending plane to the first backbone. The direction of δi is defined by the right-hand rule about ẑbi.

• ξ = (b1z, blx, θ1, δ1, θ2, δ2, q7)T - augmented configuration variables vector. The variables b1z and bx represent the coordinates of point b1 in frame {B0},. Variable q7 is the wrist rotation angle measured according to the right-hand rule about ẑt. All joint values are defined with respect to a home configuration in which the dexterous arm is straight and q7=0 as defined in Fig. 4(b).

• Li - Length of the primary backbone of the ith segment.

• Parallelogram lengths di: d1 = ||p5 − p2|| = ||p6 − p3||, d2 = ||p7 − p2||, d3 = (p4−p7)Tx̂p2, d4 = ||p4 − p1||, d5 = (b1 − p6)Tẑb0, d6 = (bl − p6)T x̂b0, d7= (p3−b0)T x̂b0, d8 = (p3 − p2)Tẑb0, d9 = (p1 − b0)T x̂b0, dl0 = (p1 − p2)T x̂b0