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0 0 ... 22) General Static Model 39 Obviously, the coeﬃcient of friction μ1 of the ﬁrst phalanx does not appear since this phalanx cannot slide on the object due to kinematic constraints. Coeﬃcients ηi are usually small and will hereafter be neglected. 6. Note that the expression of matrix J is not speciﬁc to underactuated ﬁngers and still holds in more general cases such as fully actuated ﬁngers performing enveloping grasps, provided that abduction/adduction is not considered. One also has to establish matrix T which relates vector ω a to the time derivatives of the phalanx joint coordinates as deﬁned in eq.

66) which corresponds simply to locking every even-numbered joint from the original single-stage design to zero degree. The expression of the matrix is straightforward and can be used to design a particular ﬁnger with a speciﬁc desired grasp force distribution. By using two stages instead of one, one can obtain speciﬁc ratios without requiring large values of ai and ci (link lengths). Thus, this mechanism will indeed generally lead to more compact designs. However, such designs will also have some drawbacks such as the limited space available for sensors, and a more complex architecture.

28) where ψi can be expressed as a function of ψi−1 and θi , namely −ci sin(θi+1 − ψi+1 ) li + ci cos(θi+1 − ψi+1 ) l2 + a2i + c2i − b2i + 2ci li cos(θi+1 − ψi+1 ) + arccos i . 29) Thus, one can obtain the expressions of angles ψi recursively. Eq. : used recursively to obtain θ˙a1 as a function of vector θ, θ˙a1= θ˙1 + = θ˙1 + = ... 30) Computation of the Transmission Matrix 41 For the last phalanx, one has θ˙an−1 = θ˙n−1 + hn θ˙n . 31) Therefore, one obtains n θ˙a1 = θ˙1 + hi h2 ˙ θ2 + . . + θ˙n h2 + l 1 h − li−1 i=2 i and, by observation of eqs.