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Inverse Optimal Feedback Stabilization of 3D Nonholonomic Vehicles in Spherical Coordinates

Kwang Hak Kim, Mamadou Diagne, Miroslav Krstic

arXiv:2609.36212v1math.OCcs.SYeess.SY

TL;DR

Inverse optimal stabilization of 3D nonholonomic vehicles has been precluded by the lack of strict CLFs. Building on spherical-coordinate CLFs, the paper develops damping LgV feedback that minimizes meaningful costs without solving the HJB equation. The design stabilizes on the largest domain permitted by the transformation and demonstrates quadratic-cost and input-constrained choices.

  • Problem

    Inverse optimal stabilization for 3D nonholonomic vehicles was precluded by the lack of strict CLFs, although inverse optimal control can avoid solving the HJB equation while retaining robustness margins.

    Source: §I · p. 1

  • Method

    The paper builds damping LgV inverse optimal feedback from strict CLFs in spherical coordinates for a vehicle actuated by surge velocity, pitch rate, and yaw rate.

    Source: §I · p. 1, §II · p. 2

  • Results

    The feedback renders the origin GAS on the transformation’s largest permitted domain, and the examples demonstrate quadratic running-cost and user-defined input-bound choices.

    Source: §I · p. 1, §II · p. 2, A. Families of inverse optimal… · p. 3

  • Takeaways & Limitations

    Penalty-function choice sets the cost structure and input constraints within the inverse optimal design, while bounded feedback addresses the nominal design’s unbounded effort near the excluded z-axis set.

    Source: §V · p. 6

    The spherical-coordinate design excludes the z-axis, where the coordinate map is undefined.

    Source: §II · p. 2

Abstract

from arXiv · show

Inverse optimal control provides optimality and robustness guarantees without solving the Hamilton--Jacobi--Bellman (HJB) equation, but its application to 3D nonholonomic vehicles has been precluded by the absence of strict control Lyapunov functions (CLFs). Recently, such strict CLFs were constructed for the 3D nonholonomic vehicle in spherical coordinates, which circumvent Brockett's obstruction, on the largest possible domain. Building on this construction, this paper develops a general inverse optimal stabilizing controller for the 3D nonholonomic vehicle actuated by surge velocity, pitch rate, and yaw rate. The design is a damping $L_gV$ feedback that minimizes a meaningful cost and inherits stability margins. Its flexibility is demonstrated through two choices of penalty function: a quadratic running cost on the state, recovering a near-classical optimal control formulation, and stabilization under user-defined input constraints. Numerical simulations illustrate stabilization performance, control-effort tradeoffs, and the effect of input constraints.

I. INTRODUCTION

The paper addresses why inverse optimal stabilization has been unavailable for 3D nonholonomic vehicles and builds on strict spherical-coordinate CLFs to develop inverse optimal stabilizers. The design minimizes meaningful costs without solving the HJB equation and allows user-defined input bounds.

I. INTRODUCTION

  • Point stabilization is harder than path following and tracking because it lacks the moving reference’s persistent excitation.Many underwater and aerial vehicles share longitudinal motion with pitch and yaw steering but lack direct sway or heave actuation.Source: §I · p. 1
  • Existing 3D approaches use time-varying, discontinuous, or hybrid feedback, with reported drawbacks including slow oscillatory convergence, chattering, or unpredictable maneuvers.A cited discontinuous logarithmic design’s exponential guarantee also depends on a state-dependent gain condition that is difficult to verify and may fail for large initial displacements.Source: §I · p. 1
  • The absence of strict CLFs had precluded inverse optimal control for 3D nonholonomic vehicles, despite inverse optimality avoiding the HJB equation while retaining robustness margins.Existing underactuated results were limited to practical stability, stochastic subsystem-level optimality, or the planar case.Source: §I · p. 1
  • The paper combines recently established strict CLFs in spherical coordinates with damping LgV feedback to obtain inverse optimal stabilizers without solving the HJB equation.Penalty choices demonstrate quadratic running costs and user-defined input bounds; the feedback retains stability margins.Source: §I · p. 1

II. 3D MODEL IN SPHERICAL COORDINATES

The model is transformed into spherical coordinates, whose singularity provides a route around Brockett’s obstruction. The resulting state domain excludes the z-axis, and a strict CLF on that domain supports the subsequent inverse optimal redesign.

II. 3D MODEL IN SPHERICAL COORDINATES

  • The vehicle model uses position, yaw, and pitch states with forward velocity (surge), pitch rate, and yaw rate as inputs.At a vertical orientation, horizontal velocity vanishes and yaw has no effect on motion.Source: §II · p. 2
  • The spherical transformation’s singularity is the mechanism that evades Brockett’s obstruction to continuous time-invariant stabilization.The transformation and its interpretation are given in Table I.Source: §II · p. 2, Table I · p. 2
  • The natural domain excludes the z-axis, where the spherical map is undefined; removing this measure-zero set leaves a connected state space.The paper identifies this as the largest open set on which a continuous time-invariant design is possible, and defines a metric that grows unbounded near the coordinate singularity.Source: §II · p. 2
  • A strict CLF derived from [21, Thm. 1] supplies the foundation for the inverse optimal redesign.The section defines GAS on the domain using a class KL bound and introduces the strict CLF conditions.Source: §II · pp. 2–3

III. GENERAL INVERSE OPTIMAL REDESIGN

The redesign uses the Legendre–Fenchel transform to formulate feedback laws that are optimal for a meaningful cost while stabilizing the system.

III. GENERAL INVERSE OPTIMAL REDESIGN

  • The redesign starts from the strict CLF and recalls the Legendre–Fenchel transform before specifying feedback laws.The transform is defined for a class K∞ penalty whose derivative is also class K∞.Source: §III · p. 3
  • The redesign’s purpose is to make stabilization optimal with respect to a meaningful cost, building on the strict CLF.Source: §III · p. 3
  • The transform uses the inverse of the derivative of µ.The paper denotes this inverse by (µ′)−1.Source: §III · p. 3

A. Families of inverse optimal stabilizers

Building on the strict CLF, the paper constructs damping L_gV feedback laws that minimize a meaningful cost and render the origin GAS on D. The design allows the cost to vary through penalty functions while retaining a continuous stabilizing feedback.

A. Families of inverse optimal stabilizers

B. Stability margins

The inverse-optimal controller retains stability under broad actuator distortions. Its margins include arbitrary positive constant gains, sign-preserving static nonlinearities, and strictly passive dynamic distortions.

B. Stability margins

  • Any positive constant scaling of each commanded input preserves negative-definite Vdot, yielding an infinite gain margin.Source: B. Stability margins · p. 4
  • Sign-preserving static actuator nonlinearities preserve GAS on D.Source: B. Stability margins · p. 4
  • Strictly passive dynamic distortion preserves stabilization for any positive unknown scale, unlike the one-half lower bound stated for systems with drift.Source: B. Stability margins · p. 4

IV. DESIGN FLEXIBILITY VIA PENALTY SELECTION

The section demonstrates penalty-selection flexibility through a quadratic running cost and a user-defined input-constrained design. Simulations contrast fast convergence with very large control effort against bounded inputs with slower convergence.

IV. DESIGN FLEXIBILITY VIA PENALTY SELECTION

  • The examples use equal penalties μ1 = μ2 = μ3 for clarity, although Theorem 1 permits different penalties for each input.Example 1 uses the uniform quadratic penalty; Example 2 uses a penalty of class K∞[0, π/2).Source: §IV · pp. 4–5

Example 1 (Near classical direct optimal control)

  • The quadratic penalty yields a classical quadratic-form running cost that approaches the classical state-quadratic cost as ˆε → 0.The examples use the same CLF gains, (0.5, 0.8, 1.2, 1.7, 1), and three initial conditions.Source: §IV · pp. 4–5
  • With Q = I5×5 and ˆε = 10−4, the quadratic-penalty design converges fast but requires very large control effort.The high effort can be impractical for platforms with actuator limits, motivating the input-constrained example.Source: §IV · p. 5, Figure 3 · p. 5

Example 2 (User-defined input constrained stabilization)

  • The constrained penalty produces feedback that is nearly linear for |ν_i| ≪ c_i/ε_i and approaches a saturated signum law as c_i → 0.The limiting law is −π/2 ε_i sgn(ν_i), within the imposed bounds.Source: §IV · p. 5
  • The input-constrained example keeps surge and steering rates within their specified bounds, but convergence is visibly slower than in Example 1.The bounds are |v(t)| ≤ 2 and |q(t)|, |r(t)| ≤ π/2; the example uses c_i = 0.5.Source: Figure 4 · p. 5, §IV · p. 6

V. CONCLUSION

The paper develops inverse optimal controllers for the 3D nonholonomic vehicle using strict CLFs in spherical coordinates. The controllers render the origin GAS on the transformation’s largest permitted domain while minimizing a meaningful cost without solving an HJB equation.

V. CONCLUSION

  • The controller family renders the origin GAS on the largest domain permitted by the spherical-coordinate transformation while minimizing a meaningful cost without solving an HJB equation.The quadratic penalty recovers a near-classical running cost, while bounded feedback addresses the nominal design’s unbounded effort near the excluded z-axis set.Source: §V · p. 6
  • The bounded feedback laws address the nominal design’s unbounded control effort near the excluded z-axis set, but require bidirectional surge that some marine and aerial vehicles cannot execute.Future work will study forward-surge-only stabilization and extension to vehicles with roll dynamics.Source: §V · p. 6
  • Penalty selection sets cost structure and input constraints within the framework, extending to 3D design freedom previously available only for the unicycle..Source: §V · p. 6
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