Tutorial Problems for Nonsmooth Dynamics and Optimal Control: Ski Jumping and Accelerating a Bike Without Pedaling

TL;DR

This paper introduces two accessible tutorial problems involving nonsmooth dynamics: a ski jump model with impacts and a bicycle control problem inspired by BMX techniques, aimed at aiding research and education.

Contribution

It proposes two novel, scalable tutorial problems that address nonsmooth phenomena in dynamics and optimal control, filling gaps in existing test problems.

Findings

The ski jump model effectively simulates state-dependent jumps and impacts.

The bicycle control problem demonstrates practical acceleration techniques without pedaling.

Abstract

Nonsmooth phenomena, such as abrupt changes, impacts, and switching behaviors, frequently arise in real-world systems and present significant challenges for traditional optimal control methods, which typically assume smoothness and differentiability. These phenomena introduce numerical challenges in both simulation and optimization, highlighting the need for specialized solution methods. Although various applications and test problems have been documented in the literature, many are either overly simplified, excessively complex, or narrowly focused on specific domains. On this canvas, this paper proposes two novel tutorial problems that are both conceptually accessible and allow for further scaling of problem difficulty. The first problem features a simple ski jump model, characterized by state-dependent jumps and sliding motion on impact surfaces. This system does not involve control inputs and serves as a testbed for simulating nonsmooth dynamics. The second problem considers optimal control of a special type of bicycle model. This problem is inspired by practical techniques observed in BMX riding and mountain biking, where riders accelerate their bike without pedaling by strategically shifting their center of mass in response to the track's slope.