This study develops a ccoc framework capable of handling rare event probabilities. Therefore, the framework uses the gpc method to calculate the probability of fulfilling rare event constraints under uncertainties. Here, the resulting cc evaluation is based on the efficient sampling provided by the gpc expansion. The subsim method is used to estimate the actual probability of the rare event. Additionally, the discontinuous cc is approximated by a differentiable function that is iteratively sharpened using a homotopy strategy. Furthermore, the subsim problem is also iteratively adapted using another homotopy strategy to improve the convergence of the Newton-type optimization algorithm. The applicability of the framework is shown in case studies regarding battery charging and discharging. The results show that the proposed method is indeed capable of incorporating very general cc within an ocp at a low computational cost to calculate optimal results with rare failure probability cc.

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To satisfy an increasing demand to reconstruct an athlete’s motion for performance analysis, this paper proposes a new method for reconstructing the position and velocity in the context of ski jumping trajectories. Therefore, state-of-the-art wearable sensors, including an inertial measurement unit, a magnetometer, and a GPS logger are used. The method employs an extended Rauch-Tung-Striebel smoother with state constraints to estimate state information offline from recorded raw measurements. In comparison to the classic inertial navigation system and GPS integration solution, the proposed method includes additional geometric shape information of the ski jumping hill, which are modeled as soft constraints and embedded into the estimation framework to improve the position and velocity estimation accuracy. Results for both simulated measurement data and real measurement data demonstrate the effectiveness of the proposed method. Moreover, a comparison between jump lengths obtained from the proposed method and video recordings shows the relative root-mean-square error of the reconstructed jump length is below 1.5 m depicting the accuracy of the algorithm.

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This paper presents an approach to model a ski jumper as a multi-body system for an optimal control application. The modeling is based on the constrained Newton-Euler-Equations. Within this paper the complete multi-body modeling methodology as well as the musculoskeletal modeling is considered. For the musculoskeletal modeling and its incorporation in the optimization model, we choose a nonlinear dynamic inversion control approach. This approach uses the muscle models as nonlinear reference models and links them to the ski jumper movement by a control law. This strategy yields a linearized input-output behavior, which makes the optimal control problem easier to solve. The resulting model of the ski jumper can then be used for trajectory optimization whose results are compared to literature jumps. Ultimately, this enables the jumper to get a very detailed feedback of the flight. To achieve the maximal jump length, exact positioning of his body with respect to the air can be displayed.