This paper presents a new systematic method for identifying the values of the machine-tool settings required to obtain flank form modifications in hypoid gears. The problem is given a nonlinear least-squares formulation, and it is solved by the Levenberg–Marquardt method with a trust-region strategy. To test the method, the same ease-off topography was obtained by means of very different sets of machine-tool settings, including a set of only kinematic parameters and a highly redundant set of 17 parameters. In all cases, the goal was achieved in a few iterations, with residual errors well below machining tolerances and, even more importantly, with realistic values of all parameters. Therefore, significant improvements in practical gear design can be achieved by employing the overall proposed procedure.

Microgeometry optimization has become an important phase of gear design that can remarkably enhance gear performance. For spiral bevel and hypoid gears, microgeometry is typically represented by ease-off topography. The optimal ease-off shape can be defined as the outcome of a process where generally conflicting objective functions are simultaneously minimized (or maximized), in the presence of constraints. This matter naturally lends itself to be framed as a multi-objective optimization problem. This paper proposes a general algorithmic framework for ease-off multi-objective optimization, with special attention given to computational efficiency. Its implementation is fully detailed. A simulation model for loaded tooth contact analysis is assumed to be available. The proposed method is demonstrated on a face-hobbed hypoid gear set. Three objectives are defined: maximization of gear mesh mechanical efficiency, minimization of loaded transmission error, minimization of maximum contact pressure. Bound constraints on the design variables are imposed, as well as a nonlinear constraint aimed at keeping the loaded contact pattern inside a predefined allowable contact region. The results show that the proposed method can obtain optimal ease-off topographies that significantly improve the basic design performances. It is also evident that the method is general enough to handle geometry optimization of any gear type.

This paper presents an automatic procedure to optimize the loaded tooth contact pattern of face-milled hypoid gears with misalignments varying within prescribed ranges. A two-step approach is proposed to solve the problem: in the first step, the pinion tooth microtopography is automatically modified to bring the perturbed contact patterns (as the assembly errors are varied within the tolerance limits) match a target area of the tooth while keeping them off the edges; in the second step, a subset of the machine-tool settings is identified to obtain the required topography modifications. Both steps are formulated and solved as unconstrained nonlinear optimization problems. While the general methodology is similar to the one recently proposed by the same authors for the optimization at nominal conditions, here, the robustness issues with respect to misalignment variations are considered and directly included in the optimization procedure: no a posteriori check for robustness is therefore required. Numerical tests show that nominally satisfactory and globally robust hypoid pairs can be designed by a direct process and within a unified framework, thus avoiding tiresome trial-and-error loops.

Systematic optimization of the tooth contact pattern under load is an open problem in the design of spiral bevel and hypoid gears. In order to enhance its shape and position, gear engineers have been assisted by numerical tools based on trial-and-error approaches, and/or they have been relying on the expertise of skilled operators. The present paper proposes a fully automatic procedure to optimize the loaded tooth contact pattern, with the advantage of eventually reducing design time and cost. The main problem was split into two identification subproblems: first, to identify the ease-off topography capable of optimizing the contact pattern; second, to identify the machine-tool setting variations required to obtain such ease-off modifications. Both of them were formulated and solved as unconstrained nonlinear optimization problems. In addition, an original strategy to quickly approximate the tooth contact pattern under load was conceived. The results obtained were very satisfactory in terms of accuracy, robustness, and computational speed. They also suggest that the time required to optimize the contact pattern can be significantly reduced compared with typical time frames. A sound mathematical framework ensures results independent of the practitioner’s subjective decision-making process. By defining a proper objective function, the proposed method can also be applied to affect other contact properties, such as to improve the motion graph or to decrease the sensitivity of the transmission to assembly errors. Furthermore, it can be easily adapted to any gear drive by virtue of its systematic and versatile nature.

In this letter, we propose a two-stage strategy for optimal control problems of robotic mechanical systems that proves to be more robust, and yet more efficient, than straightforward solution strategies. Specifically, we focus on a simplified humanoid model, represented as a two-dimensional articulated serial chain of rigid bodies, in the tasks of getting up (sitting down) from (to) the supine and prone postures. Interactions with the environment are integral parts of these motions, and a priori unscheduled contact sequences are discovered by the solver itself, opportunistically making or breaking contacts with the ground through feet, knees, hips, elbows, and hands. The present investigation analyzes the effects on the computational performance of: 1) the explicit introduction of contact forces among the optimization variables, 2) the substitution of undesired contact forces with geometric constraints that prevent interpenetrations, and 3) the splitting of the planning problem into two consecutive phases of increasing complexity. To the best of our knowledge, these tests represent the only quantitative analysis of the performances achievable with different solution strategies for optimization-based, whole-body dynamic motion planning in the presence of contacts.