Small steps for mankind: Towards a kinematically driven dynamic simulation of curved path walking

A simple approach for the simulation of bipedal locomotion is presented. It is based on a kinematically driven rigid body dynamics simulation. As inputs, our system uses a 14-DOF simplified human body model, and the instantaneous walking direction and the gait length. From these parameters, the trajectories of the left and right feet are computed over time, using a purely kinematical recipe. Optionally, trajectories of other body parts (head, shoulders, hips, hands, knees, etc.) may be derived from a simple kinematic body model. Using these additional trajectories provides for increased control over the motion, adding extra kinematic constraints to the dynamical system. The trajectories, together with optional force fields (gravity etc.), serve to drive the motion of the articulated human body model. This means that for those points in the body that have not been specified kinematically, a rigid body dynamics calculation is used to describe their motion over time. An implementation of these ideas turns out to allow a frame update rate of about 20 Hz on a personal IRIS^TM workstation, which is sufficiently fast for real-time interaction.     

N-body spacetime constraints

Animators frequently choreograph complex motions for multiple objects that interact through collision and obstruction. In such situations, the use of physically based dynamics to confer visual realism creates challenging computational problems. Typically forward simulation is well understood, but the inverse problem of motion synthesis—that of synthesizing motions consistent both with physical law and with the animator's requirements—is generally tedious and sometimes intractable. We show how N-body inverse problems can be formulated as optimization tasks. We present a simply stated, but combinatorially formidable example that exhibits all of the essential sources of complexity common to N-body motion synthesis, and show how it can be solved approximately using heuristic methods based on evolutionary computation.     

Effective spacetime from multidimensional gravity

We study the effective spacetimes in lower dimensions that can be extracted from a multidimensional generalization of the Schwarzschild-Tangherlini spacetimes derived by Fadeev, Ivashchuk, and Melnikov (Phys. Lett. A 161, 98 (1991)). The higher-dimensional spacetime has D = (4 + n + m) dimensions, where n and m are the number of “internal” and “external” extra dimensions, respectively. We analyze the effective (4 + n) spacetime obtained after dimensional reduction of the m external dimensions. We find that when the m extra dimensions are compact (i) the physics in lower dimensions is independent of m and the character of singularities in higher dimensions, and (ii) the total gravitational mass M of the effective matter distribution is less than the Schwarzschild mass. In contrast, when the m extra dimensions are large, this is not so; the physics in (4 + n) does explicitly depend on m as well as on the nature of singularities in high dimensions, and the mass of the effective matter distribution (with the exception of wormhole-like distributions) is larger than the Schwarzschild mass. These results may be relevant to observations for an experimental/observational test of the theory.     

Geodesic portrait of de Sitter-Schwarzschild spacetime

De Sitter-Schwarzschild space-time is a globally regular spherically symmetric spacetime which is asymptotically de Sitter as r → 0 and asymptotically Schwarzschild as r → ∞. A source term in the Einstein equations smoothly connects de Sitter vacuum at the origin with Minkowski vacuum at infinity and corresponds to an anisotropic vacuum fluid defined by symmetry of its stress-energy tensor which is invariant under radial boosts. In the range of the mass parameter M ≥ M _crit, de Sitter-Schwarzschild spacetime represents a vacuum nonsingular black hole, while M < M _crit corresponds to a compact gravitationally bound vacuum object without horizons, called a G-lump. Masses of objects are related to both de Sitter vacuum trapped inside and to smooth breaking of the spacetime symmetry from the de Sitter group at the origin to the Poincaré group at infinity. We here present a geodesic survey of de Sitter-Schwarzschild spacetime.     

Casimir effect for two spheres in a wormhole spacetime

We consider the Casimir effect for a static, spherically symmetric wormhole surrounded by two perfectly conducting spheres. We construct an expression for zero-point energy in this model. It is shown that the sign of the Casimir force depends on the nonminimal coupling constant ξ.     

Building spacetime meshes over arbitrary spatial domains

We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finite-element methods. Our method generalizes and improves the Tent Pitcher algorithm of Üngör and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T], our algorithm builds a simplicial mesh of the spacetime domain X × [0, T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of spacetime elements to the local quality and feature size of the underlying space mesh.This work was supported in part by The Center for Process Simulation and Design at the University of Illinois, Urbana-Champaign, under NSF ITR grant DMR-0121695. A preliminary version of this paper was presented at the 11th International Meshing Roundtable [9].     

Real-time rendering in curved spaces

A hypersphere surface provides a finite 3D world in which the user can fly freely without encountering boundaries, while hyperbolic space provides a spacious environment. The algorithm for rendering a scene in a hypersphere is identical to the standard algorithm for rendering a scene in ordinary flat 3D space. Indeed, the computations are so similar that off-the-shelf 3D graphics cards, when fed the correct matrices, will do real-time animations in a hypersphere just as easily and as quickly as they do in flat space.     

Separability and hidden symmetries of Kerr-Taub-NUT spacetime in Kaluza-Klein theory

The Kerr-Taub-NUT spacetime in the Kaluza-Klein theory represents a localized stationary and axisymmetric object in four dimensions from the Kaluza-Klein viewpoint. That is, it harbors companion electromagnetic and dilaton fields, thereby showing up the signature of the extra fifth dimension. We explore the separability structure of this spacetime and show that the Hamilton-Jacobi equation for geodesics admits complete separation of variables only for massless geodesics. This implies the existence of hidden symmetries in the spacetime, which are generated by a conformal Killing tensor. Using a simple trick built up on a conformally related metric (an “effective” metric) with the Killing tensor, we construct an explicit expression for the conformal Killing tensor.     

Thermodynamics of horizons in a globally regular spherically symmetric spacetime

We address the question of thermodynamics of horizons in a globally regular spherically symmetric spacetime which is asymptotically de Sitter as r ?? 0 and as r ?? ??. A source term in the Einstein equations smoothly connects two de Sitter vacua with different values of the cosmological constant and corresponds to an anisotropic vacuum fluid defined by symmetry of its stress-energy tensor, which is invariant under radial boosts. In the most general case, the spacetime has three horizons, an internal one, r _a , which is a cosmological horizon for an observer in the R-region 0 ?? r ?? r _a ; the horizon r _b > r _a which is the boundary of the T-region r _a < r < r _b seen as a black or white hole by an observer in the R-region r _b < r < r _c , where r _c is his cosmological horizon. We present a detailed analysis of the thermodynamics of horizons using the Padmanabhan approach relevant to the case of non-zero pressures.     

Einstein-Born-Infeld on Taub-NUT spacetime in 2k + 2 dimensions

We wish to construct solutions of Taub-NUT spacetime in Einstein-Born-Infeld gravity in even dimensions. Since the Born-Infeld theory is a nonlinear electrodynamics theory, in leads to nonlinear differential equations. However, a proper analytical solution was not obtained, we try to solve it numerically (by the Runge-Kutta method) with initial conditions coinciding with those of our previous work in Einstein-Maxwell gravity. We solve the equations for 4, 6 and 8 dimensions and do data fitting by the least-squares method. For N = l = b = 1, the metric turns to the NUT solution only in 8 dimensions, but in 4 and 6 dimensions the spacetime does not have any NUT solution. Talk given at the International Conference RUSGRAV-13, June 23–28, 2008, PFUR, Moscow.     

Hyper-reoriented walking in minimal space

Virtual Reality - We present a new reorientation technique, “hyper-reoriented walking,” which greatly reduces the amount of physical space required in virtual reality (VR) applications...     

Computational geometry in a curved world

We extend the results of straight-edged computational geometry into the curved world by defining a pair of new geometric objects, thesplinegon and thesplinehedron, as curved generalizations of the polygon and polyhedron. We identify three distinct techniques for extending polygon algorithms to splinegons: the carrier polygon approach, the bounding polygon approach, and the direct approach. By these methods, large groups of algorithms for polygons can be extended as a class to encompass these new objects. In general, if the original polygon algorithm has time complexityO(f(n)), the comparable splinegon algorithm has time complexity at worstO(Kf(n)) whereK represents a constant number of calls to members of a set of primitive procedures on individual curved edges. These techniques also apply to splinehedra. In addition to presenting the general methods, we state and prove a series of specific theorems. Problem areas include convex hull computation, diameter computation, intersection detection and computation, kernel computation, monotonicity testing, and monotone decomposition, among others.     

Generalized Bargmann-Wigner equations in curved space-time

The Bargmann-Wigner equations are formulated to represent bosonic fields in terms of fermionic fields in curved space-time. The bispinor field is explicitly studied including lower spin states. The correspondence among fermions and bosons through bispinors is applied to the superradiance problem in Kerr black-hole spacetime. Superradiance phenomena of the type of positive frequency (0 < ω) and negative momentum near the horizon (p _H < 0) are shown not to occur.     

Computational geometry in a curved world

We extend the results of straight-edged computational geometry into the curved world by defining a pair of new geometric objects, thesplinegon and thesplinehedron, as curved generalizations of the polygon and polyhedron. We identify three distinct techniques for extending polygon algorithms to splinegons: the carrier polygon approach, the bounding polygon approach, and the direct approach. By these methods, large groups of algorithms for polygons can be extended as a class to encompass these new objects. In general, if the original polygon algorithm has time complexityO(f(n)), the comparable splinegon algorithm has time complexity at worstO(Kf(n)) whereK represents a constant number of calls to members of a set of primitive procedures on individual curved edges. These techniques also apply to splinehedra. In addition to presenting the general methods, we state and prove a series of specific theorems. Problem areas include convex hull computation, diameter computation, intersection detection and computation, kernel computation, monotonicity testing, and monotone decomposition, among others.This research was partially supported by National Science Foundation Grants MCS 83-03926, DCR85-05517, and CCR87-00917.This author's research was also partially supported by an Exxon Foundation Fellowship, by a Henry Rutgers Research Fellowship, and by National Science Foundation Grant CCR88-03549.     

Spin-Based Quantum Dot Quantum Computing in Silicon

The spins of localized electrons in silicon are strong candidates for quantum information processing because of their extremely long coherence times and the integrability of Si within the present microelectronics infrastructure. This paper reviews a strategy for fabricating single electron spin qubits in gated quantum dots in Si/SiGe heterostructures. We discuss the pros and cons of using silicon, present recent advances, and outline challenges. PACS: 03.67.Pp, 03.67.Lx, 85.35.Be, 73.21.La     

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell-Cattaneo-Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An h-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.     

Consistent curved beam element

Curved beam finite elements with shear deformation have required the use of reduced integration to provide improved results for thin beams and arches due to the presence of a spurious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approximation of displacements, and a quadratic polynomial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor k as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allowing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consistent beam element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick beams and arches.     

Spin-Based Quantum Information Processing in Magnetic Quantum Dots

We define the qubit as a pair of singlet and triplet states of two electrons in a He-type quantum dot (QD) placed in a diluted magnetic semiconductor (DMS) medium. The molecular field is here essential as it removes the degeneracy of the triplet state and strongly enhances the Zeeman splitting. Methods of qubit rotation as well as two-qubit operations are suggested. The system of a QD in a DMS is described in a way which allows an analysis of the decoherence due to spin waves in the DMS subsystem.on leave from Institute of Physics, Odessa UniversityPresented at the 36th Symposium on Mathematical Physics, “Open Systems & Quantum Information”, Toruń, Poland, June 9–12, 2004.     

Neuromorphic walking gait control

We present a neuromorphic pattern generator for controlling the walking gaits of four-legged robots which is inspired by central pattern generators found in the nervous system and which is implemented as a very large scale integrated (VLSI) chip. The chip contains oscillator circuits that mimic the output of motor neurons in a strongly simplified way. We show that four coupled oscillators can produce rhythmic patterns with phase relationships that are appropriate to generate all four-legged animal walking gaits. These phase relationships together with frequency and duty cycle of the oscillators determine the walking behavior of a robot driven by the chip, and they depend on a small set of stationary bias voltages. We give analytic expressions for these dependencies. This chip reduces the complex, dynamic inter-leg control problem associated with walking gait generation to the problem of setting a few stationary parameters. It provides a compact and low power solution for walking gait control in robots.