Control Dynamics of Robotic Manipulators by J. M. Skowronski

By J. M. Skowronski

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The threshold projected orthogonally into A^ determines A . The projection of the said family of levels behaves identically in A , and the m a p is isometric (distance preserving). 1 we may pick up other stable equilibria. Moving the origin of R by a simple transformation of variables, we may repeat our construction of the local cup about any other Dirichlet stable equilibrium. We then obtain a sequence of Z , k = 0, l , . . , m < oo, finite since A is bounded, with v = 0 referring to the basic cup.

See the two-dimensional illustration in Fig. 1. As stated by Lagrange and proved by Dirichlet, the isolated minima of V are the stable equilibria. We call them Dirichlet stable and use this term in the plain meaning of elementary physics in high school. 2 CONSERVATIVE REFERENCE FRAME. ENERGY SURFACE i 51 THRESHOLD 1 Fig. 1 bring back memories of a ball returning to a stable position at the b o t t o m of a convex two-dimensional V. The matrix \_ — dQ ldqj\ is called the functional coefficient of restitution of the system.

The matrix \_ — dQ ldqj\ is called the functional coefficient of restitution of the system. T h e restitution will be said to be positive if the matrix is positive definite, that is, all its principal minors are positive, which suffices for a local minimum. 1. An equilibrium is Dirichlet stable if the restitution in its neighborhood is positive. N o t e that the matrix \_ — dQ /dqf\ may be immediately replaced by [dUi/dqj], without a change of conditions. The sections i (q) = const = v of V are the potential energy levels Z , with v e [0, v ), o = sup i^(q) | q e A.

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