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

Julian Golembiewski, Timm Faulwasser

TL;DR

This work addresses the challenge of solving optimal control problems with nonsmooth dynamics, such as impacts and state discontinuities, where standard smooth methods fail. It proposes two tutorial problems—a ski jump without control and a pump-track bicycle model with pumping as the sole acceleration mechanism—and adopts a time-freezing approach to handle state jumps by embedding them in a Filippov differential inclusion within an extended state space. The methods are demonstrated via simulations (using direct discretization and the time-freezing framework), with scalability paths discussed for both problems, including extensions to friction and multi-contact scenarios. The authors provide a public codebase for reproducibility, highlighting practical implications for education and benchmarking of numerical methods in nonsmooth optimal control.

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.

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

TL;DR

This work addresses the challenge of solving optimal control problems with nonsmooth dynamics, such as impacts and state discontinuities, where standard smooth methods fail. It proposes two tutorial problems—a ski jump without control and a pump-track bicycle model with pumping as the sole acceleration mechanism—and adopts a time-freezing approach to handle state jumps by embedding them in a Filippov differential inclusion within an extended state space. The methods are demonstrated via simulations (using direct discretization and the time-freezing framework), with scalability paths discussed for both problems, including extensions to friction and multi-contact scenarios. The authors provide a public codebase for reproducibility, highlighting practical implications for education and benchmarking of numerical methods in nonsmooth optimal control.

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.

Paper Structure

This paper contains 13 sections, 14 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM SETTING AND PRELIMINARIES
  3. Class of nonsmooth dynamical systems
  4. Optimal control formulation
  5. NONSMOOTH OPTIMAL CONTROL
  6. Discretization of nonsmooth dynamical systems
  7. Time-freezing approach for systems with state jumps
  8. TWO TUTORIAL PROBLEMS
  9. Ski jump
  10. Accelerating a bike without pedaling
  11. Scalability of the proposed problems
  12. SIMULATION RESULTS
  13. CONCLUSION

Figures (5)

  • Figure 1: Model of a ski jumping venue.
  • Figure 2: Bicycle model on a pump track.
  • Figure 3: Ski jumping -- results for two different initial heights.
  • Figure 4: Bicycle on pump track -- simulation results for three different scenarios. The first row shows the system without input, and the second row shows the system with the optimal input $u^\star$.
  • Figure 5: Bicycle on pump track -- solution trajectories over time. The optimal input $u^\star$, the pumping motion, accelerates the system to reach the goal faster and to gain velocity.