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By J. M. Skowronski

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6). In turn, the functions Qf(q), Qf(q) have special shapes due to coupling of the particular spring forces that may act on m . 21)]. The gravity forces are in general trigonometric functions (sine or cosine) in rotary joint coordinates and linear functions in prismatic joint coordinates, coupled by the joint variables added together. 1). The spring forces are quite often analytic functions as well and thus are also power series developable. Frequently, they m a y be represented as the sum of terms with direct action between m and the base 6o;(4;) d those between links representing spring couplings Qfj{q — qj), i j .

It corresponds to A by the fact that the boundary 8A = H of A is an isometric image of Z . Since the thresholds of V a n d coincide, H is that H which passes through the boundary of A and since both H and A are defined, the H is defined as well, a n d so is A . The set H is called the separating set (generalized separatrix) of A . Hence, if the potential threshold does not appear: A — A (linear or hard Q \ the energy threshold Z never appears either and A = A. c c c 9 l9 l9 n n c c c H c CH CH v H H H H CH c qv qv CH c H H P qv CH CH CH H CH 56 2 F O R C E CHARACTERISTICS, E N E R G Y , A N D P O W E R It is now easy to describe the interior of Z .

Then, since the basic level is a double minimum of both V a n d 7, there is a neighborhood A a A of the basic equilibrium on which H increases with k i l > • • • > knl> \4n\- Since T has n o extremal values except 0, it is only the potential energy threshold which m a y break down the increase. T o any potential energy level i^(q) = const, there corresponds some Z in Jt. Thus the potential energy threshold becomes the energy threshold Z :H(q,q) = h . Correspondingly, the basic potential energy cup Z becomes the basic energy cup Z .

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