A New Approach To Estimate The Collision Probability For Automotive Applications

We revisit the computation of probability of collision in the context of automotive collision avoidance (the estimation of a potential collision is also referred to as conflict detection in other contexts). After reviewing existing approaches to the definition and computation of a collision probability we argue that the question "What is the probability of collision within the next three seconds?" can be answered on the basis of a collision probability rate. Using results on level crossings for vector stochastic processes we derive a general expression for the upper bound of the distribution of the collision probability rate. This expression is valid for arbitrary prediction models including process noise. We demonstrate in several examples that distributions obtained by large-scale Monte-Carlo simulations obey this bound and in many cases approximately saturate the bound. We derive an approximation for the distribution of the collision probability rate that can be computed on an embedded platform. In order to efficiently sample this probability rate distribution for determination of its characteristic shape an adaptive method to obtain the sampling points is proposed. An upper bound of the probability of collision is then obtained by one-dimensional numerical integration over the time period of interest. A straightforward application of this method applies to the collision of an extended object with a second point-like object. Using an abstraction of the second object by salient points of its boundary we propose an application of this method to two extended objects with arbitrary orientation. Finally, the distribution of the collision probability rate is identified as the distribution of the time-to-collision.Comment: Revised and restructured version, discussion of extended vehicles expanded, section on TTC expanded, references added, other minor changes, 17 pages, 18 figure

Partial Supersymmetry Breaking from Five Dimensions

Theories of partial supersymmetry breaking N=2 -> N=1 in four dimensions are derived by coupling the N=2 massless gravitino multiplet to N=2 supergravity in five dimensions and performing a generalized dimensional reduction on S^1/Z_2 with the Scherk-Schwarz mechanism. These theories agree with results that were previously derived from four dimensions.Comment: 15 pages, Latex; introduction slightly changed, one reference adde

Stability Analysis of Legged Locomotion Models by Symmetry-Factored Return Maps

We present a new stability analysis for hybrid legged locomotion systems based on the “symmetric” factorization of return maps.We apply this analysis to two-degrees-of-freedom (2DoF) and threedegrees- of-freedom (3DoF) models of the spring loaded inverted pendulum (SLIP) with different leg recirculation strategies. Despite the non-integrability of the SLIP dynamics, we obtain a necessary condition for asymptotic stability (and a sufficient condition for instability) at a fixed point, formulated as an exact algebraic expression in the physical parameters. We use this expression to characterize analytically the sensory cost and stabilizing benefit of various feedback schemes previously proposed for the 2DoF SLIP model, posited as a low-dimensional representation of running.We apply the result as well to a 3DoF SLIP model that will be treated at greater length in a companion paper as a descriptive model for the robot RHex

Dual Supersymmetry Algebras from Partial Supersymmetry Breaking

The partial breaking of supersymmetry in flat space can be accomplished using any one of three dual representations for the massive N=1 spin-3/2 multiplet. Each of the representations can be ``unHiggsed'', which gives rise to a set of dual N=2 supergravities and supersymmetry algebras.Comment: 12 pages, 1 figure, cosmetic change

Self-Stability Mechanisms for Sensor-Cheap Legged Locomotion

It is now well established that running animals’ mass centers exhibit the characteristics of a Spring Loaded Inverted Pendulum (SLIP) in the sagittal plane (Blickhan and Full, 1993). What control policy accomplishes this collapse of dimension by which animals solve the “degrees of freedom problem” (Bernstein, 1967)? How valuable might this policy be to gait control in legged robots

Supersymmetric Randall-Sundrum Scenario

We present the supersymmetric version of the minimal Randall-Sundrum model with two opposite tension branes.Comment: Latex, 9 pages. Published versio

Einstein equations for an asymmetric brane-world

We consider a brane-world of co-dimension one without the reflection symmetry that is commonly imposed between the two sides of the brane. Using the coordinate-free formalism of the Gauss-Codacci equations, we derive the effective Einstein equations by relating the local curvature to the matter on the brane in the case when its bare tension is much larger than the localized matter, and hence show that Einstein gravity is a natural consequence of such models in the weak field limit. We find agreement with the recently derived cosmological case, which can be solved exactly, and point out that such models can be realized naturally in the case where there is a minimally coupled form field in the bulk.Comment: 14 pages, Revte