This paper presents an efficient optimization framework that solves trajectory optimization problems efficiently by decoupling state variables from timing variables, thereby decomposing a challenging nonlinear programming (NLP) problem into two easier subproblems. With timing fixed, the state variables can be optimized efficiently using convex optimization, and so the time variables can be optimized using a separate outer optimization. This is a bilevel optimization in which the outer objective function itself requires an optimization to compute. The challenge is that gradient optimization methods require the gradient of the objective function with respect to the time variables, which is not available. Whereas the finite difference method must solve many optimization problems to compute a gradient, this paper proposes a more efficient method: the dual solution (Lagrange multipliers) of the convex optimization problem is exploited to calculate the analytical gradient. Since the dual solution is a by-product of the convex optimization problem, the gradient can be obtained for free' with high accuracy. The framework is demonstrated on solving minimum-jerk trajectory optimization problems in safety corridors for unmanned aerial vehicles (UAVs). Experiments demonstrate that bilevel optimization improves performance over a standard NLP solver, and analytical gradients outperforms finite differences. With a 40 ,ms cutoff time, our approach achieves over 8 times better suboptimality than the current state-of-the-art.
Another strategy to determine a time allocation scheme is to use sampling @cite_1 . This approach randomly samples the duration of each segment until the corresponding QP can be solved, and is applicable to problems with a rather small feasible set, for example the problem of humanoid locomotion where a small change in time can make the optimization problem infeasible. However it may not scale well to generating trajectories with a large number of segments. Also, the work described in @cite_1 only seeks for a feasible time allocation and does not improve it.
@cite_1
2789286310
We tackle the transition feasibility problem, that is the issue of determining whether there exists a feasible motion connecting two configurations of a legged robot. To achieve this we introduce CROC, a novel method for computing centroidal dynamics trajectories in multi-contact planning contexts. Our approach is based on a conservative and convex reformulation of the problem, where we represent the center of mass trajectory as a Bezier curve comprising a single free control point as a variable. Under this formulation, the transition problem is solved efficiently with a Linear Program (LP)of low dimension. We use this LP as a feasibility criterion, incorporated in a sampling-based contact planner, to discard efficiently unfeasible contact plans. We are thus able to produce robust contact sequences, likely to define feasible motion synthesis problems. We illustrate this application on various multi-contact scenarios featuring HRP2 and HyQ. We also show that we can use CROC to compute valuable initial guesses, used to warm-start non-linear solvers for motion generation methods. This method could also be used for the 0 and 1-Step capturability problem. The source code of CROC is available under an open source BSD-2 License.