Gravity balancing of a spatial serial 4-dof arm without auxiliary links using minimum number of springs

The principle of gravity balancing has been studied for a long time. It allows a system to be in indifferent equilibrium regardless of the configuration. In the literature, gravity balancing has often been achieved using appropriate combinations of springs and auxiliary links. Some paper address potential layouts without auxiliary links, but limited to planar mechanisms. This paper proposes a method to passively balance an anthropomorphic arm, with spatial kinematics, avoiding the use of auxiliary links. The approach used in this paper includes the analysis of all the contributions to the potential energy of the arm. It is shown that they are proportional (according to geometrical and inertial parameters) to scalar products between configuration-dependent unit vectors and/or configuration- independent unit vectors. Analysing the potential energy contributions for each combination of unit vectors, it is shown how to minimize the number of springs required to balance the mechanism without additional links. As a result, four possible layouts are developed, all of them using only two springs. Features and design issues of the four layouts are discussed. Finally, one of them is chosen for actual implementation

A probabilistic approach to the geometry of the ℓᵨⁿ-ball

This article investigates, by probabilistic methods, various geometric questions on Bᵨⁿ, the unit ball of ℓᵨⁿ. We propose realizations in terms of independent random variables of several distributions on Bᵨⁿ, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in Bᵨⁿ. As another application, we compute moments of linear functionals on Bᵨⁿ, which gives sharp constants in Khinchine’s inequalities on Bᵨⁿ and determines the ψ₂-constant of all directions on Bᵨⁿ. We also study the extremal values of several Gaussian averages on sections of Bᵨⁿ (including mean width and ℓ-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in ℓ₂ and to covering numbers of polyhedra complete the exposition

A probabilistic approach to the geometry of the \ell_p^n-ball

This article investigates, by probabilistic methods, various geometric questions on B_p^n, the unit ball of \ell_p^n. We propose realizations in terms of independent random variables of several distributions on B_p^n, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B_p^n. As another application, we compute moments of linear functionals on B_p^n, which gives sharp constants in Khinchine's inequalities on B_p^n and determines the \psi_2-constant of all directions on B_p^n. We also study the extremal values of several Gaussian averages on sections of B_p^n (including mean width and \ell-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in \ell_2 and to covering numbers of polyhedra complete the exposition.Comment: Published at http://dx.doi.org/10.1214/009117904000000874 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Balancing sums of random vectors

We study a higher-dimensional 'balls-into-bins' problem. An infinite sequence of i.i.d. random vectors is revealed to us one vector at a time, and we are required to partition these vectors into a fixed number of bins in such a way as to keep the sums of the vectors in the different bins close together; how close can we keep these sums almost surely? This question, our primary focus in this paper, is closely related to the classical problem of partitioning a sequence of vectors into balanced subsequences, in addition to having applications to some problems in computer science.Comment: 17 pages, Discrete Analysi

Extremal Problems in Minkowski Space related to Minimal Networks

We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in $n$ for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n and that equality holds iff the space is linearly isometric to \ell^n_\infty, the space with an n-cube as unit ball. We also remark on similar questions raised in [FLM] that arose out of the study of singularities in length-minimizing networks in Minkowski spaces.Comment: 6 pages. 11-year old paper. Implicit question in the last sentence has been answered in Discrete & Computational Geometry 21 (1999) 437-44

Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of $T$ unit vectors in $\mathbb{R}^d$, find $\pm$ signs for each of them such that the signed sum vector along any prefix has a small $\ell_\infty$-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Koml\'os problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an $O(\sqrt{\log d+ \log T})$ bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk's bound and consider natural generalizations of prefix discrepancy: We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in $T$ compared to Banaszczyk's bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of $O(\sqrt{\log d+ \log\!\log T})$ with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk's bound in the worst case. We also introduce a generalization of the prefix discrepancy problem where the discrepancy constraints correspond to paths on a DAG on $T$ vertices. We show that an analog of Banaszczyk's $O(\sqrt{\log d+ \log T})$ bound continues to hold in this setting for adversarially given unit vectors and that the $\sqrt{\log T}$ factor is unavoidable for DAGs. We also show that the dependence on $T$ cannot be improved significantly in the smoothed case for DAGs. We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.Comment: 22 pages. Appear in ITCS 202

Tradeoffs for nearest neighbors on the sphere

We consider tradeoffs between the query and update complexities for the (approximate) nearest neighbor problem on the sphere, extending the recent spherical filters to sparse regimes and generalizing the scheme and analysis to account for different tradeoffs. In a nutshell, for the sparse regime the tradeoff between the query complexity $n^{\rho_q}$ and update complexity $n^{\rho_u}$ for data sets of size $n$ is given by the following equation in terms of the approximation factor $c$ and the exponents $\rho_q$ and $\rho_u$: $$c^2\sqrt{\rho_q}+(c^2-1)\sqrt{\rho_u}=\sqrt{2c^2-1}.$$ For small $c=1+\epsilon$, minimizing the time for updates leads to a linear space complexity at the cost of a query time complexity $n^{1-4\epsilon^2}$. Balancing the query and update costs leads to optimal complexities $n^{1/(2c^2-1)}$, matching bounds from [Andoni-Razenshteyn, 2015] and [Dubiner, IEEE-TIT'10] and matching the asymptotic complexities of [Andoni-Razenshteyn, STOC'15] and [Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt, NIPS'15]. A subpolynomial query time complexity $n^{o(1)}$ can be achieved at the cost of a space complexity of the order $n^{1/(4\epsilon^2)}$, matching the bound $n^{\Omega(1/\epsilon^2)}$ of [Andoni-Indyk-Patrascu, FOCS'06] and [Panigrahy-Talwar-Wieder, FOCS'10] and improving upon results of [Indyk-Motwani, STOC'98] and [Kushilevitz-Ostrovsky-Rabani, STOC'98]. For large $c$, minimizing the update complexity results in a query complexity of $n^{2/c^2+O(1/c^4)}$, improving upon the related exponent for large $c$ of [Kapralov, PODS'15] by a factor $2$, and matching the bound $n^{\Omega(1/c^2)}$ of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal complexities $n^{1/(2c^2-1)}$, while a minimum query time complexity can be achieved with update complexity $n^{2/c^2+O(1/c^4)}$, improving upon the previous best exponents of Kapralov by a factor $2$.Comment: 16 pages, 1 table, 2 figures. Mostly subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1605.02701 [cs.DS]