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Globally convergent homotopies for discrete-time optimal control

Willem Esterhuizen, Kathrin Flaßkamp, Matthias Hoffmann, Karl Worthmann

TL;DR

The paper introduces globally convergent homotopies for discrete-time optimal control by embedding a λ-parameterized path that connects an easy relaxed problem to a difficult original problem. By translating KKT conditions into a system of equations and modifying the homotopy to a transversal form ρ, it provides sufficient conditions (including a new assumption that nonconvex constraints are inactive at λ=0) to guarantee a probability-one, non-looping zero-curve that reaches a local NLP solution as λ→1. The method is then translated to OCPs by perturbing state constraints (e.g., obstacles) with λ, enabling a curve-tracking algorithm to solve challenging path-planning problems for linear dynamics and nonlinear Dubins vehicles. Numerical experiments illustrate a robust curve-tracking procedure, the competitive performance relative to sampling-based planners, and insights into computational trade-offs and potential online MPC applications. The work offers a principled convergence guarantee for homotopy-based NLP/OCP solvers in robotics-style path planning contexts and points to fruitful directions for scalability and integration with real-time control systems.

Abstract

Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle).

Globally convergent homotopies for discrete-time optimal control

TL;DR

The paper introduces globally convergent homotopies for discrete-time optimal control by embedding a λ-parameterized path that connects an easy relaxed problem to a difficult original problem. By translating KKT conditions into a system of equations and modifying the homotopy to a transversal form ρ, it provides sufficient conditions (including a new assumption that nonconvex constraints are inactive at λ=0) to guarantee a probability-one, non-looping zero-curve that reaches a local NLP solution as λ→1. The method is then translated to OCPs by perturbing state constraints (e.g., obstacles) with λ, enabling a curve-tracking algorithm to solve challenging path-planning problems for linear dynamics and nonlinear Dubins vehicles. Numerical experiments illustrate a robust curve-tracking procedure, the competitive performance relative to sampling-based planners, and insights into computational trade-offs and potential online MPC applications. The work offers a principled convergence guarantee for homotopy-based NLP/OCP solvers in robotics-style path planning contexts and points to fruitful directions for scalability and integration with real-time control systems.

Abstract

Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle).
Paper Structure (15 sections, 6 theorems, 55 equations, 10 figures, 1 algorithm)

This paper contains 15 sections, 6 theorems, 55 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline
  3. Related work on homotopies for nonlinear programs
  4. Preliminaries: Globally convergent homotopies for Systems of equations
  5. Globally convergent homotopies: Nonlinear programs
  6. From the KKT conditions to a system of equations
  7. Modifications to $\alpha$ to obtain a globally convergent homotopy
  8. A new condition to render $\rho$ globally convergent
  9. Globally convergent homotopies: Discrete-time optimal control
  10. Numerical examples
  11. Curve tracking algorithm
  12. Linear system
  13. Nonlinear example: Dubins' vehicle
  14. Comparison with sampling-based motion planners
  15. Conclusion

Key Result

Theorem 2.2

\newlabelthm:trans0 Let $\rho:\mathbb{R}^m \times (0,1) \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $C^2$-mapping, which is transversal to zero. Then the mapping $\rho_a:(0,1) \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ is transversal to zero for almost every $a\in\mathbb{R}^m$ with respe

Figures (10)

  • Figure 1: Under the conditions of Theorem \ref{['thm:trans']} components of $\gamma_a$ are $C^1$ curves that are diffeomorphic to an interval (curves 1-4) or a circle (curve 5) and that may be unbounded (curve 1); have infinite arc length (curve 2); have (resp. not have) limit points $(0,z_0)$ and $(1,z_1)$ (curve 3, resp. curve 4). They cannot be nondifferentiable (curve 6); bifurcate (curve 7); be discontinuous (curve 8); or self-intersect (curve 9).
  • Figure 1: Intuitive idea of a predictor-corrector algorithm. The predictor generates points (red dots), following the tangents to the curve at the current iterate. The corrector determines the next point on the curve (blue dots) with a root finding algorithm (like Newton's method) initiating from the predicted point.
  • Figure 2: Results of tracking the zero curve of the parametrised homotopy, $\rho_a$, see \ref{['def:para_homo']}, using Algorithm \ref{['alg:1']}, for the linear problem, \ref{['eq:linear_example_1']}-\ref{['eq:linear_example_6']}. Initial paths, associated with guesses $\mathbf{u}^0$, are indicated by the blue crosses (curves 1,2 and 3). The local solutions the algorithm converges to are indicated by red circles (curves 1', 2', 3').
  • Figure 3: Homotopy parameter over iterations for the problem in Figure \ref{['fig:Linear_2_obs_no_overlap_a']}. Numbers indicate which initial guess, $\mathbf{u}^0$, the sequence is associated with.
  • Figure 4: Result of tracking the zero curve for the linear problem, \ref{['eq:linear_example_1']}-\ref{['eq:linear_example_6']} with the obstacles moved so they overlap. Blue crosses: initial guess, red circles: solution.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Lemma 3.2
  • Proof 1
  • Proposition 3.3
  • Proof 2
  • Proposition 4.1
  • Proof 3