This paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

This paper presents a new formulation of computing three-dimensional (3-D) frictional form-closure grasps of n robotic fingers. As 3-D form-closure grasps involve 6-D wrench space, we first propose a recursive reduction technique to transform the complicated problem in the 6-D space into a simpler 3-D one. Next, we rewrite the sufficient and necessary condition for form-closure grasps into its equivalent form of two sets of linear inequalities. Then, according to the linear inequality theory, the problem is transformed to searching for a set of points which ensure the inconsistency of each of the two linear inequality systems. To search for such points, we proposed two methods: The first one is based on testing whether the convex region formed by each linear inequality system is empty, while the second one relies on the potential field method. We have implemented the algorithm and confirmed their efficiency for the synthesis of 3-D form-closure grasps.