M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Aug. 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....since the product is always larger than the lattice volume, with equality if and only if the basis is orthogonal. 2.3 Complexity results We refer to Cai [30, 31] for an up to date survey of complexity results. Ajtai [4] recently proved that SVP is NP hard under randomized reductions. Micciancio [98, 97] simplified and improved the result by showing that approximating SVP to within a factor 2 is also NP hard under randomized reductions. The NP hardness of SVP under deterministic (Karp) reductions remains an open problem. CVP seems to be a more difficult problem. Goldreich et al. 62] ....

....different than 1 in (v; 1) It is hoped that a shortest vector of that lattice is of the form (v Gamma u; 1) where u is a closest vector (in the original lattice) to v, whenever the distance to the lattice is smaller than the lattice first minimum. This heuristic may fail (see for instance [97] for some simple counterexamples) but it can also sometimes be proved, notably in the case of lattices arising from low density knapsacks. For exact SVP, the best algorithm known (in theory) is the recent randomized O(d) time algorithm by Ajtai et al. 6] which improved Kannan s ....

D. Micciancio. On the Hardness of the Shortest Vector Problem. PhD thesis, Massachusetts Institute of Technology, 1998.

....(see [52] For historical reasons, Hermite s constant refers to max1(L) and not max1(L) vol(L) 2.3 Complexity results We refer to Cai [24, 25] for an up to date survey of complexity results. Ajtai [4] recently proved that SVP is NP hard under randomized reductions. Micciancio [82, 81] simplified and improved the result by showing that approximating SVP to within a factor 2 is also NP hard under randomized reductions. The NP hardness of SVP under deterministic (Karp) reductions remains an open problem. CVP seems to be a more difficult problem. Goldreich et al. 50] ....

....(1 ) for any constant 0. However, Goldreich and Hastad noticed about a year ago that one can choose some = o(1) and still have polynomial running time, for instance using the blocksize k = log d= log log d in [101] Note that there exist simple counter examples (see for instance [81]) implement and can attain very high encryption decryption rates. But basically, all knapsack cryptosystems have been broken (for a survey, see [99] either by specific (often lattice based) attacks or by the low density attacks. The last significant candidate to survive was the Chor Rivest ....

D. Micciancio. On the Hardness of the Shortest Vector Problem. PhD thesis, Massachusetts Institute of Technology, 1998.

....4 Motion Uncertainty 4.1 Involved Uncertainties In previous sections, we assumed that the force fields and the motions of the parts can be perfectly controlled. However, for any real implementation, uncertainty should be taken into account. A lot of research has addressed the uncertainty issue [5, 9]. For part assembly, even though we only use sensing to find the initial orientations of the assembled parts, we still need to consider the following uncertainties mainly because of the force field control. Position uncertainty, see Figure 4 (a) Orientation uncertainty, see Figure 4 (b) ....

....push itself. Furthermore, we only need to take into account the uncertainties of because the uncertainties of can be transformed to relative uncertainties of with respect to . The problem becomes similar with a classical uncertainty problem, namely the peg in hole problem [5, 9]. Figure 5: a) is a rectangular part. b) shows a box to contain the part with all possible orientations when the uncertainty nty 33 . In order to assemble two parts, there should exist a goal region of and this goal region has to be larger than . If the goal region is much larger ....

[Article contains additional citation context not shown here]

M. A. Erdmann. On motion planning with uncertainty. Master 's thesis, Massachusetts Institute of Technology, Cambridge, MA, 1984.

....05A15, 20C33, 22E45. Key words and phrases. Oscillating tableaux, nonintersecting lattice paths. 1 defined by: if and only if i i for all i. Equivalently, if and only if the Ferrers diagram of is contained in the Ferrers diagram of . A (generalized) up down tableau from to [13, 31, 32, 37] is a sequence T = 0 ; 1 ; 2n Gamma1 ; 2n ) of partitions (Ferrers diagrams) for some n, such that = 0 1 2 Delta Delta Delta 2n Gamma1 2n = and i Gamma1 and i differ by a horizontal strip for each i = 1; 2; 2n. A horizontal strip (cf. 25, p. 4] is ....

S. Sundaram,On the combinatorics of representations of Sp(2n; C ), Ph. D. Thesis, Massachusetts Institute of Technology, 1986.

....similar. Also (1.3) and (1.5) for generic ff; fi, have interesting interpretations in terms of tableaux generating functions. For example, we can prove the following theorem. Theorem 4. The generating function P w( over all oscillating semistandard tableaux = i) fi ff (cf. [30, 11, 26]) with at most r rows equals (1.5) The weight w( is defined by Q x j (2i Gamma1) Gamma (2i) j j (2i Gamma1) Gamma (2i Gamma2) j i . 3. Norm generating functions for tableaux. Given a tableau we define the norm, n( of to be the sum of all the entries of . In this section ....

S. Sundaram, On the combinatorics of representations of Sp(2n; C ), Ph. D. Thesis, Massachusetts Institute of Technology, 1986.

....of contact points is small, for instance one to four, the effects of friction are easily computed. However, as the number of contact points grows, the problem becomes considerably more challenging. Simulation algorithms with exponential (in the number of contact points) running times are known[5] but are impractical for problems involving as few as 10 to 15 contact points. In order to make rigid body simulations with friction practical for computer graphics, efficient, polynomial time algorithms are needed. This paper considers the problems of computing friction forces for configurations ....

....a thin rod A and a fixed base B. By choosing A s angular velocity w and the magnitude g of the gravity force mgn acting on A, an indeterminate and an inconsistent configuration can be produced. This particular example can be found in a number of papers; for example, Lotstedt[11] Erdmann[5], or Mason and Wang[13] For a given value of w , the linear velocity of A is chosen such that the point p a on A has a non zero velocity tangent to B, and zero velocity normal to B. The unit vector n is normal to the surface of B. The unit vector t is tangent to the surface of B, and is ....

Erdmann, M.A., On Motion Planning with Uncertainty, M.S. Thesis, Massachusetts Institute of Technology, 1984.

....2 For historical reasons, Hermite s constant refers to max1(L) 2 =vol(L) 2=d and not max1(L) vol(L) 1=d . 5 2.3 Complexity results We refer to Cai [24, 25] for an up to date survey of complexity results. Ajtai [4] recently proved that SVP is NP hard under randomized reductions. Micciancio [82, 81] simplified and improved the result by showing that approximating SVP to within a factor p 2 is also NP hard under randomized reductions. The NP hardness of SVP under deterministic (Karp) reductions remains an open problem. CVP seems to be a more difficult problem. Goldreich et al. 50] ....

....(1 ) n for any constant 0. However, Goldreich and Hastad noticed about a year ago that one can choose some = o(1) and still have polynomial running time, for instance using the blocksize k = log d= log log d in [101] 4 Note that there exist simple counter examples (see for instance [81]) 7 implement and can attain very high encryption decryption rates. But basically, all knapsack cryptosystems have been broken (for a survey, see [99] either by specific (often lattice based) attacks or by the low density attacks. The last significant candidate to survive was the Chor Rivest ....

D. Micciancio. On the Hardness of the Shortest Vector Problem. PhD thesis, Massachusetts Institute of Technology, 1998.

.... following the rod) or apply a nonzero force. The three dimensional submanifold of contact configurations is fq 0 2 C 0 j F (q 0 ) yw Gamma y m Gamma r sin OE w = 0g. The conditions that O remain in contact with M are dF (q 0 (t) dt = 0 d 2 F (q 0 (t) dt 2 = 0: Erdmann [5, 6] refers to these constraints as the First and Second Variation Constraints, respectively. These constraints state that the velocity and acceleration of the system 3 Sign errors were propagated through equations in this section in (Lynch [11] normal to the constraint surface must be zero. For ....

M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Aug. 1984.

....Rivest and Tarjan [BFP 73] shows that any quantile of a data set of size N can be computed with at most 5:43N comparisons. The paper also establishes a lower bound of 1:5N comparisons for the problem. For an account of progress since then, see the survey by Mike Paterson [Pat97] Frances Yao [Yao74] showed that any deterministic algorithm that computes an approximate quantile requires Omega Gamma N) comparisons. Curiously, this lower bound is easily beaten by employing randomization. The folklore algorithm that outputs the median of a random sample of size O(ffl Gamma2 log ffi ....

F. F. Yao. On Lower Bounds for Selection Problems. Technical Report MAC TR-121, Massachusetts Institute of Technology, 1974. 12

....Building on results from classical mechanics, he identified a fundamental rule for predicting the sense of rotation (CW or CCW) of a part as it is pushed in the presence of Coulomb friction. Other geometric methods for predicting part motion in the presence of frictional contacts were described in [5] , 6] 16] 3] and [2] Mason s rule provided the basis for Brost s [1] push diagram, which represents all possible motions of a part as it is grasped by a parallel jaw gripper. When there is greater uncertainty in the initial orientation of a part, more than one action may be required to ....

Erdmann, M. A. On motion planning with uncertainty. MS Thesis, Massachusetts Institute of Technology (1984)

....primitives will be superior to any purely arithmetic (meaning linear combination of attributes) approach. The LISP expression database has to be built by hand through observation of the program s play. 2.3. 5 Elwyn Berlekamp A theory of games demonstrated in [3] can be applied to the endgame of go[45, 12]. It has the drawbacks (in its present form) of only being applicable to the late endgame of go (in fact, the very late endgame , in which who gets the very last move is being contested) and requiring a slight modification of the rules to be applicable. Presently there is no generalization of the ....

Yedwab, L., On playing well in a sum of games, PhD. thesis, Massachusetts Institute of Technology, 1985.

....f N is unique. All of the above work took place before the discovery of the importance of P and NP timecomplexities. As a result, the computational complexity of computing multiple contact forces was not of concern in any of the above work. In work done subsequent to this discovery, Erdmann[13] discusses the problem of computing constraint forces without friction, and notes that the problem is simply solved by an exhaustive search method, requiring exponential time. The earliest description of a simulator using exact methods to calculate constraint forces (by quadratic programming) ....

....accelerations of the objects. We call the first possibility, nonexistence of solution, inconsistency. The second problem, nonuniqueness of motion behavior, is called indeterminacy. Configurations with one contact point that exhibit inconsistency and indeterminacy have been discussed by Erdmann[13], L otstedt[28] and Mason and Wang[32] L otstedt[30] realizing that indeterminacy and inconsistency present major difficulties for a simulation process, proposes a modification of the Coulomb friction law that eliminates both inconsistency and indeterminacy, and further causes the LCP to always ....

M.A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, 1984.

....feedback loop at a high frequency, and does not garbage collect. Termination predicates have been introduced in the pre image motion planning framework of [LPMT84] An extensive discussion of the relative power of these termination predicates in the context of motion planning may be found in [Erd84] In the [LPMT84] framework, actions are pairs of the form (v; tp) where v is a velocity vector, and tp is a termination predicate. The motion continues in the direction v until the termination predicate returns true. The predicate, then, is a function of all of the information available to ....

M. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, 1984.

.... motion, and the direction of the slider motion is normal to the surface in three dimensional velocity space (two translational components and one angular component) Thus, for a particular contact mode, if we construct the convex hull of the pusher slider contact forces in forcemoment space [6] and project it to the limit surface, the set of normals to the intersected surface are the slider motions which are labeled P. All other slider motions are labeled I. This method of determining the force status of slider motions is impractical, as it assumes a known p( x) Instead, we can use the ....

M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Aug. 1984.

....an expanded foundation that is built on previous geometric concepts, while characterizing and unifying a broader class of problems. In the area of motion planning under uncertainty in sensing and prediction, many interesting concepts have been developed, such as preimages and forward projections [41, 67, 80, 83]) however, they are often tied to a particular set of uncertainty models that are based on crisp, geometric constructions (e.g. uncertainty cones and disks) In [73] it is shown how these concepts can be generalized to a broad class of problems that involve a variety of models and assumptions, ....

.... k ) y i k = h i (x k ) Multiple robots C 1 Theta Delta Delta Delta Theta C N x i = f i (x i (t) u i (t) y i (t) x(t) fl i : X U i [19, 85, 101] x i k 1 = f i (x i k ; u i k ) y i k = x k CP uncertainty C x = f(x(t) u(t) a (t) y(t) x(t) fl : X U [17, 31, 41] x k 1 = f(x k ; u k ; a k ) y k = x k EP uncertainty C Theta E x = f(x(t) u(t) a (t) y(t) x(t) fl : X U [27, 76, 107] x k 1 = f(x k ; u k ; a k ) y k = x k CS and CP unc. C x = f(x(t) u(t) a (t) y(t) h(x(t) s (t) fl : I k U [41, 67, 80] x k 1 = f(x k ; u k ....

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984.

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984.

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984.

....k ; n) on k is at most quadratic. Indeed, he proved a somewhat stronger result. The length of a path is the number of its edges. Let P k denote the class of all non crossing paths of length k. Clearly, every non crossing path of length 2k 1 has k pairwise disjoint edges. J anos Pach Theorem 5. 4 [75]. Let ex (P k ; n) denote the maximum number of edges that a geometric graph of n vertices can have without containing a non crossing path of length k. Then for every k and n, we have ex (P k ; n) Ck n; for a suitable constant C. It is possible, however, that the true order of magnitude of ....

G. Toth, Note on geometric graphs, preprint, Massachusetts Institute of Technology, 1998. Janos Pach

....from the motion planning community. Two common representations of uncertainty have been applied to motion planning problems. One representation restricts parameter uncertainties to lie within a specified set. A motion plan is then generated that is based on worst case analysis (e.g. 32] [62], 110] 124] We refer to this representation as nondeterministic uncertainty. The other popular representation expresses uncertainty in the form of a probability density function. This often leads to the construction of motion plans through average case or expected case analysis (e.g. 26] ....

....subset Q ae C free . For 8 Configuration Space Configuration Space Environment Space Environment Space Configuration Sensing Configuration Predictability Environment Sensing Environment Predictability Figure 1.2 Four sources of uncertainty in the motion planning problem. example, in [32] [62], 110] 124] this representation of uncertainty is used to guarantee that the robot recognizably terminates in a goal region. With a probabilistic model, the robot might infer a posterior probability density over configurations, p(q) that is conditioned on sensor observations, initial ....

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984. 260

....by the robot to make an inference about its configuration. This information could come from sensor measurements or motion history. With a nondeterministic uncertainty model, the robot might have sufficient information to infer that q lies in some subset Q ae C free . For example, in [9] [17], 38] 43] this representation of uncertainty is used to guarantee that the robot recognizably terminates in a goal region. With a probabilistic model, the robot might infer a posterior probability density over configurations, p(q) that is conditioned on sensor observations, initial ....

....future configuration will belong to a subset Q ae C free . The method of preimage backchaining constitutes of large body of work in which bounded uncertainties are propagated and combined with configuration sensing uncertainty, to guarantee that the robot will achieve a goal (e.g. 9] 13] [17], 20] 38] 43] With a probabilistic model, future configurations can be described by a posterior density over configurations, p(q) that is conditioned on the initial configuration and the executed motion command. Examples of work that include a probabilistic representation of ....

M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984.

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Aug. 1984.

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Washizu K. On the variational principles of elasticity and plasticity. Technical Report 25-18, Aeroelastic and Structures Research Laboratory, Massachusetts Institute of Technology, Cambridge, USA, 1955.

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984.

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1984.

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M. A. Erdmann. On motion planning with uncertainty. Master's thesis, Massachusetts Institute of Technology, Aug. 1984.

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L. Yedwab, On playing well in a sum of games, Technical Report MIT/LCS/TR-348, Massachusetts Institute of Technology, Cambridge, MA, 1985.

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On-Chip Prepreocessing. PhD thesis, Massachusetts Institute of Technology, 1983.

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