By Sadao Kawamura; Mikhail Svinin

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0) m. 5 2 x [m] Fig. 2. Natural motion through a regular point singularity. 0) m — a stationary point of the dynamical system. Simulation data are shown in Fig. 3. The exponential decay of all vector ﬁeld components is apparent from the speed graphs. We repeat the same simulation, but instead of natural motion, constant twist magnitude is required (q˙∗ = 1 rad/s). The results are shown in Fig. 4. It is seen that motion through the stationary point has been achieved. The lower left plot shows the graphs of the three vector ﬁeld components.

2 In addition, we may assume that V(¯ q 0 ) = U(q 0 ). Then, the integration constant is uniquely determined from Eq. (37) as: E = −T (¯ q 0 ). (38) It should be apparent now that natural motion of a manipulator with a prescribed end-eﬀector path is nothing else than nondissipative motion with prespeciﬁed initial energy. Note also that energy conserving systems are called natural [35]. Thus, the term natural motion, which was introduced in Section 3 to describe a pure geometrical phenomenon, is reaﬃrmed in a convincing way also from the viewpoint of energy conservation.

8) we obtain: ˆ ¯q˙ = λ˙ n ¯ (¯ q) (15) ˆ is the tangent vector of unit length at q ¯ , and the constant b has been where n ¯ set to one, without loss of generality. Deﬁnition (Natural motion) Manipulator motion with generalized velocities in proportion to the natural speed λ˙ is called natural motion. Corollary 1 Under natural motion, the magnitude of the end-eﬀector twist is in proportion to | det J |. Proof: Follows directly from Eqs. (12) and (15). Corollary 2 Natural motion requires nonzero initial conditions.