Collision detection for rigid superellipsoids using the normal parameterization

The normal parameterization as an approach to describe geometries is introduced. The advantages of this description – as compared to other parameterizations or implicit functions – in the context of collision detection are: the possibility to explicitly calculate axis aligned bounding boxes for any convex geometry, an efficient iterative algorithm for collision detection between objects with arbitrary geometry that does not require any (analytical) derivatives. A system of several rigid superellipsoids is used to demonstrate the application and performance of the proposed approach in a multibody simulation

A Brief Survey on Non-standard Constraints: Simulation and Optimal Control

In terms of simulation and control holonomic constraints are well documented and thus termed standard. As non-standard constraints, we understand non-holonomic and unilateral constraints. We limit this survey to mechanical systems with a finite number of degrees of freedom. The long-term behavior of non- holonomic integrators as compared to structure-preserving integrators for holonomically constrained systems is briefly discussed. Some recent research regarding the treatment of unilaterally constrained systems by event-driven or time-stepping schemes for time integration and in the context of optimal control problems is outlined

Failing parametrizations: what can go wrong when approximating spectral submanifolds

Invariant manifolds provide useful insights into the behavior of nonlinear dynamical systems. For conservative vibration problems, Lyapunov subcenter manifolds constitute the nonlinear extension of spectral subspaces consisting of one or more modes of the linearized system. Conversely, spectral submanifolds represent the spectral dynamics of non-conservative, nonlinear problems. While finding global invariant manifolds remains a challenge, approximations thereof can be simple to acquire and still provide an effective framework for analyzing a wide variety of problems near equilibrium solutions. This approach has been successfully employed to study both the behavior of autonomous systems and the effects of non-autonomous forcing. The current computation strategies rely on a parametrization of the invariant manifold and the reduced dynamics thereon via truncated power series. While this leads to efficient recursive algorithms, the problem itself is ambiguous, since it permits the use of various approaches for constructing the reduced system to which the invariant manifold is conjugated. Although this ambiguity is well known, it is rarely discussed and usually resolved by an ad hoc choice of method, the effects of which are mostly neglected. In this contribution, we first analyze the performance of three popular approaches for constructing the conjugate system: the graph style parametrization, the normal form parametrization, and the normal form parametrization for “near resonances.” We then show that none of them is always superior to the others and discuss the potential benefits of tailoring the parametrization to the analyzed system. As a means for illustrating the latter, we introduce an alternative strategy for constructing the reduced dynamics and apply it to two examples from the literature, which results in a significantly improved approximation quality

The normal parameterization and its application to collision detection

Collision detection is a central task in the simulation of multibody systems. Depending on the description of the geometry, there are many efficient algorithms to address this need. A widespread approach is the common normal concept: potential contact points on opposing surfaces have antiparallel normal vectors. However, this approach leads to implicit equations that require iterative solutions when the geometries are described by implicit functions or the common parameterizations. We introduce the normal parameterization to describe the boundary of a strictly convex object as a function of the orientation of its normal vector. This parameterization depends on a scalar function, the so-called generating potential from which all properties are derived: points on the boundary, continuity/differentiability of the boundary, curvature, offset curves or surfaces. An explicit solution for collisions with a planar counterpart is derived and four iterative algorithms for collision detection between two arbitrary objects with the normal parametrization are compared. The application of this approach for collision detection in multibody models is illustrated in a case study with two ellipsoids and several planes

Explicit analytical solutions for two-dimensional contact detection problems between almost arbitrary geometries and straight or circular counterparts

Contact between complex bodies and simple counterparts like straight lines or circles occur in many two-dimensional mechanical models. The corresponding contact detection problems are complicated and thus far, no explicit formulas have been available. In this paper, we address the contact detection problem between two planar bodies: one being either a straight line or a circle and the other an almost arbitrary geometry—the only requirement is a unique contact point for all possible contact situations. To solve this general problem, a novel procedure is applied which provides necessary conditions for the description of the geometry based on the special case of a rolling contact. This results in a parameterization of the geometry which gives the potential contact point depending on the relative orientation between the two bodies. Although the derivation is based on a rolling contact, the result is valid in general and can also be used for efficient contact detection when the bodies are separated. The derived equations are simple and easy to implement, which is demonstrated for two examples: a foot-ground contact model and a cam-follower mechanism

The Influence of ground inclination on the energy efficiency of a bipedal walking robot

One of the major tasks in developing bipedal walking robots is to improve energy efficiency of their locomotion. In this paper a method for extending this research by considering ground inclinations is introduced. The investigated robot is driven by electric motors in its revolute joints. Robot\u27s gaits for different walking speeds are generated via numerical optimization, minimizing energy consumption during locomotion. Energy efficiency can be increased for differently inclined grounds where the robot utilizes its natural dynamics and gravity

Simultaneous optimization of gait and design parameters for bipedal robots

A walking bipedal robot’s energy efficiency depends on its gait as well as its design, whereas design changes affect the optimal gaits. We propose a method to take these interdependencies into account via simultaneous optimization of gait as well as design parameters. The method is applied to a planar robot with hybrid zero dynamics control and a torsion spring between its thighs. Periodic gaits are simulated by means of the hybrid zero dynamics. The implementation of the simultaneous optimization of gait parameters and spring stiffness via sequential quadratic programming is presented. Subsequently, an error analysis is performed to gain good convergence and short computation times of the optimization. The evaluation of gradients is identified as crucial for the algorithm’s convergence and therefore performed via complex step derivative approximations. The resulting implementation exhibits good convergence behavior and is provided as supplement to this paper. At 2.3 m/s, the simultaneous optimization results in savings in energy expenditure of up to 55%. A consecutive optimization of first gait and then stiffness yields only 11%, demonstrating the advantage of the presented method