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Add to Cart. Applied Mechanics and Materials Volumes Main Theme:. Edited by:. Wen-Pei Sung and Ran Chen. Online since:. January Cited by. Related Articles. Paper Title Pages. Abstract: Accurate predictive modeling is an essential prerequisite for optimization and control of production in modern manufacturing environments. For slender bar turning operations, dimensional deviation is one of the most important product quality characteristics due to the low stiffness of part. The local operator in the present setting is. Unfortunately, the method presented here is only useful in spaces of a few dimensions.
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If , then it may be applied, for example, to the systems in Section If dynamics are considered, then in many circumstances the dimension is too high because the dimension of is usually twice that of. For example, if is a rigid body in the plane, then the dimension of is six, which is already at the usual limit of practical use. It is interesting to compare the use of dynamic programming here with that of Sections If Dijkstra's algorithm is used or even breadth-first search in the case of time optimality , then by the dynamic programming principle, the resulting solutions are approximately optimal.
Frontiers in interpolation and approximation (Boca Raton, ). - ОГЛАВЛЕНИЕ / CONTENTS
To ensure convergence, resolution completeness arguments were given based on Lipschitz conditions on. It was important to allow the resolution to improve as the search failed to find a solution. Instead of computing a search graph, value iteration is based on computing cost-to-go functions.
In the same way that both forward and backward versions of the tree-based approaches were possible, both forward and backward value iteration can be used here. Providing resolution completeness is more difficult, however, because is not fixed. It is therefore not known whether some resolution is good enough for the intended application. If is known, then can be used to generate a trajectory from using the system simulator. If the trajectory fails to reach , then the resolution can be improved by adding more samples to and or by reducing.
Under Lipschitz conditions on , the approach converges to the true optimal cost-to-go [ 92 , , ]. Therefore, value iteration can be considered resolution complete with respect to a given. The convergence even extends to computing optimal feedback plans with additional actions that are taken by nature, which is modeled nondeterministically or probabilistically. This extends the value iteration method of Section The relationship between the methods based on a search graph and on value iteration can be brought even closer by constructing Dijkstra-like versions of value iteration, as described at the end of Section 8.
These extend Dijkstra's algorithm , which was viewed for the finite case in Section 2.
The improvement to value iteration is made by recognizing that in most evaluations of This is caused by two factors: 1 From some states, no trajectory has yet been found that leads to ; therefore, the cost-to-go remains at infinity. A forward or backward version of a Dijkstra-like algorithm can be made. Consider the backward case. The notion of a backprojection was used in Section 8. This was used in 8.