This paper presents a novel method for reformulating non-differentiable collision avoidance constraints into smooth nonlinear constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid obstacles while moving in an n-dimensional space. The proposed reformulation does not introduce approximations, and applies to general obstacles and controlled objects that can be represented as the union of convex sets. We connect our results with the notion of signed distance, which is widely used in traditional trajectory generation algorithms. Our method can be applied to generic navigation and trajectory planning tasks, and the smoothness property allows the use of general-purpose gradient-and Hessian-based optimization algorithms. Finally, in case a collision cannot be avoided, our framework allows us to find "leastintrusive" trajectories, measured in terms of penetration. We demonstrate the efficacy of our framework on a quadcopter navigation and automated parking problem, and our numerical experiments suggest that the proposed methods enable real-time optimization-based trajectory planning problems in tight environments.Source code of our implementation is provided at https://github.com/XiaojingGeorgeZhang/OBCA.
A robust Learning Model Predictive Controller (LMPC) for uncertain systems performing iterative tasks is presented. At each iteration of the control task the closed-loop state, input, and cost are stored and used in the controller design. This paper first illustrates how to construct robust control invariant sets and safe control policies exploiting historical data. Then, we propose an iterative LMPC design procedure, where data generated by a robust controller at iteration j are used to design a robust LMPC at the next iteration j + 1. We show that this procedure allows us to iteratively enlarge the domain of the control policy and it guarantees recursive constraints satisfaction, input to state stability, and performance bounds for the certainty equivalent closed-loop system. The use of different feedback policies along the horizon is the key element of the proposed design. The effectiveness of the proposed control scheme is illustrated on a linear system subject to bounded additive disturbances.
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