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Constructive Safety-Critical Control for a Class of Underactuated Systems: A Hierarchical Approach

Massimiliano de Sa, Aaron D. Ames

arXiv:2610.10987v1eess.SYcs.SYmath.OC

TL;DR

Constructive safety-critical control for underactuated systems with nontrivial geometry remains an open challenge. This paper studies shape maps and symmetries to build natural layered architectures and hierarchical CBFs for a class of quadrotor-like systems. It gives conditions for local and global shape maps and constructs a safe hierarchical controller under stated conditions.

  • Problem

    For underactuated systems, it is unclear how to construct suitable reduced-order models and tracking certificates for hierarchical CBF synthesis, particularly when system geometry constrains tracking.

    Source: §I · p. 1, §III · p. 3

  • Method

    The paper uses shape maps and system symmetries to characterize natural layered architectures and develop a constructive hierarchical CBF synthesis procedure.

    Source: §I · pp. 1–2

  • Results

    Theorem 1 establishes local shape maps near each F_g outside Z*(G), and a global shape map when π has a global section.

    Source: B. Shape Maps · p. 5

  • Takeaways & Limitations

    For the studied class of underactuated systems, shape maps link virtual-system input directions to configurations and support layered control with hierarchical CBF synthesis.

    Source: §I · p. 2, §V · p. 8

    The results concern a class of underactuated systems; generalization to a broader class is left for future work.

    Source: §V · p. 8

Abstract

from arXiv · show

Underactuated systems with nontrivial geometry are abundant in practical robotic problems. However, little is known about constructive safety-critical control design in the presence of underactuation. In this work, we provide a constructive framework for synthesizing safe control architectures and control barrier functions for a class of quadrotor-like underactuated systems. By studying the geometry of their underactuation, we characterize when these systems have natural layered architectures useful for safety-critical control design. Using these architectures, we devise a constructive, hierarchical CBF synthesis procedure and provide conditions under which it succeeds.

I. INTRODUCTION

The introduction frames constructive safety-critical control for underactuated systems with nontrivial geometry as an open problem. It studies when symmetries and shape maps yield natural layered architectures and closed-form CBFs for a class of quadrotor-like systems.

I. INTRODUCTION

  • Constructive CBF synthesis for underactuated systems remains less understood than for fully actuated systems, despite practical safety needs such as quadrotor collision avoidance.Existing hierarchical methods assume a reduced-order CBF, a tracking controller, and a Lyapunov tracking certificate; how to construct these pieces, or establish their existence, is unknown.Source: §I · p. 1
  • The paper studies how symmetries and shape maps can establish natural layered architectures and closed-form safety certificates for quadrotor-like geometric mechanical systems.Its two questions concern when a layered architecture exists and when a full-system safety certificate can be assembled in closed form.Source: §I · p. 1
  • The contributions are shape maps linking virtual-system inputs to references, symmetry-based layered architectures, and a constructive hierarchical CBF synthesis procedure.The work focuses on a class of underactuated systems rather than general underactuated systems.Source: §I · pp. 1–2

B. Simple Mechanical Control Systems

The paper represents a simple mechanical control system geometrically and specifies how controllers act through its input codistribution. Its motion equation uses the metric-induced Levi-Civita connection, and locally Lipschitz controllers yield unique maximal trajectories.

B. Simple Mechanical Control Systems

  • A simple mechanical control system consists of a configuration manifold, kinetic-energy metric, potential function, and input codistribution.The input codistribution has rank m on the n-dimensional configuration manifold.Source: B. Simple Mechanical Control Systems · p. 2
  • A controller maps tangent-bundle states to forces in the input codistribution at the corresponding state.Source: B. Simple Mechanical Control Systems · p. 2
  • The Levi-Civita connection enables directional derivatives on the configuration manifold, with ∇_qdot qdot interpreted as acceleration.For locally Lipschitz controllers, the motion equation has a unique maximal trajectory from each initial tangent vector.Source: B. Simple Mechanical Control Systems · p. 2

C. Error Functions & Transport Maps

Configuration error functions quantify mismatch between configurations, while transport maps enable comparison of velocities in different tangent spaces. The paper gives their compatibility condition and an SO(3) example.

C. Error Functions & Transport Maps

  • A configuration error function is symmetric, vanishes exactly when its configurations agree, is proper in its first argument, and has a positive-definite Hessian at agreement.The paper also defines an L-uniformly quadratic condition for such functions.Source: C. Error Functions & Transport Maps · p. 2
  • A transport map compares vectors from different tangent spaces, and compatibility with a configuration error function is expressed through its second differential.The map is linear, equals the identity when its two configurations coincide, and transports a fixed vector to a smooth vector field.Source: C. Error Functions & Transport Maps · p. 2
  • On SO(3), the paper gives a trace-based configuration error and a compatible transport map for rotation-matrix tangent vectors.Source: C. Error Functions & Transport Maps · p. 2

D. Control Barrier Functions

A CBF certifies a safe set through a control condition on its derivative. The CBF-QP filter minimally modifies a nominal input, and the resulting trajectories remain in the safe set under the stated regularity conditions.

D. Control Barrier Functions

  • A CBF for a simple mechanical control system on D defines the safe set as the nonnegative level set of h and requires a control condition involving its Lie derivative.The condition uses a function α in K∞.Source: D. Control Barrier Functions · pp. 2–3
  • The CBF-QP safety filter minimally modifies a nominal controller to produce a safe input.For locally Lipschitz nominal controller and α, the resulting control law is locally Lipschitz on C.Source: D. Control Barrier Functions · p. 3
  • From any initial state in C, trajectories under the filtered controller remain in C for all positive times in their maximal interval of existence.The paper also notes that smooth closed-form approximations of the filter can be constructed.Source: D. Control Barrier Functions · p. 3

III. ARCHITECTURES & ACTUATION GEOMETRY

Hierarchical CBF design combines safety for a reduced-order model with certified tracking by the full system, but constructive design for underactuated systems remains less understood. The section frames the tracking and synthesis questions and examines the quadrotor architecture to identify the structure that makes it possible.

III. ARCHITECTURES & ACTUATION GEOMETRY

  • A safe vector field on G and a certifiable tracking controller can yield a full-system CBF by combining tracking and safety certificates,,.The supplied passage contrasts this established process for fully actuated systems with the less-understood underactuated case.Source: §III · p. 3
  • The motivating problem asks when a κ-tracking controller has a closed-form certificate and when that certificate yields a CBF for π^-1(C0).Source: §III · p. 3
  • For the quadrotor, a virtual point-mass controller generates fd, which is converted to an orientation aligning Rde3 with fd before the quadrotor tracks it.Sufficiently fast orientation tracking enables stable κ-tracking with a closed-form certificate [10].Source: §III · p. 3, Figure 2 · p. 3
  • The section reverses the quadrotor example to identify the structural features that make its κ-tracking architecture possible.Source: §III · p. 3

A. Trivial Simple Mechanical Control Systems

Trivial SMCS form a quadrotor-like class whose product geometry and symmetry isolate underactuation in the group component. This structure supports a decomposition into shape and group subsystems and a fully actuated virtual model.

A. Trivial Simple Mechanical Control Systems

  • For the quadrotor, translations preserve both the kinetic-energy metric and input codistribution, giving the system translational symmetry.The example identifies S = SO(3) and G = R3, with the virtual model a fully actuated point mass with gravity.Source: A. Trivial Simple Mechanical Control… · pp. 3–4
  • A trivial SMCS has configuration space Q = S × G, a product metric, a potential depending on G, and an input codistribution invariant under the group action.Its shape component is fully actuated, while underactuation is concentrated in G.Source: A. Trivial Simple Mechanical Control… · p. 3
  • The equations of motion decompose into G- and S-subsystems, with underactuation in G and full actuation in S.The force codistribution structure governs the coupling between the subsystems.Source: A. Trivial Simple Mechanical Control… · pp. 3–4
  • The quadrotor illustrates the trivial-SMCS form, with its rotational and translational configuration components and thrust-direction force codistribution.Source: A. Trivial Simple Mechanical Control… · p. 4

B. Shape Maps

A shape map assigns a shape to a nonzero force covector so that the force lies in the system’s input codistribution. The section characterizes when local or global shape maps exist and illustrates the construction for a quadrotor.

B. Shape Maps

  • A shape map takes a nonzero force covector to a shape where that force belongs to the input codistribution; L-compatibility bounds force-projection error using configuration error.Source: B. Shape Maps · p. 4
  • Shape-map existence reduces to local Lie shape maps, which correspond to local sections of the bundle π : A → Sg∗.The section also characterizes the submersion condition for π in terms of condition (*).Source: B. Shape Maps · pp. 4–5
  • Theorem 1 guarantees local shape maps near every nonzero covector when condition (*) holds and π is surjective, and guarantees a global map when π has a global section.The local existence argument obtains local sections and transfers them to shape maps.Source: B. Shape Maps · p. 5
  • Global shape maps can face topological obstructions, although they often exist on large domains.Source: B. Shape Maps · p. 5
  • For the quadrotor, the example maps nonzero force directions other than the downward direction to orientations whose body thrust axis aligns with the force.Source: B. Shape Maps · p. 5

IV. HIERARCHICAL CBF SYNTHESIS

The section applies the shape-map result to constructive safety-critical control of trivial SMCS. It outlines a layered design in which controllers and tracking certificates are combined to produce a CBF.

IV. HIERARCHICAL CBF SYNTHESIS

  • Theorem 1 supplies a shape map linking virtual-system inputs to true-system configurations, an ingredient used in the quadrotor architecture and extended here to trivial SMCS.Source: §IV · p. 5
  • The section assumes a safe configuration set defined as a compact zero superlevel set of h0.Source: §IV · p. 5
  • The outlined procedure designs controllers and performance certificates for each architecture layer, then combines them into a CBF for Σ.Source: §IV · p. 6

A. Layered Architecture

The architecture layers a safe kinematic vector field on G, a virtual model that tracks it, and control inputs for the full system. A CBF is then constructed for this architecture.

A. Layered Architecture

  • The full-system input combines a PD controller that tracks the shape reference with a projection of the virtual-model input onto the G component.The reference is induced by the virtual-model controller and the safe vector field on G.Source: A. Layered Architecture · p. 6
  • The architecture proceeds from a kinematic reduced-order model on G to a virtual mechanical model, then to the true system.The virtual model tracks the safe vector field; the true system tracks the virtual input.Source: A. Layered Architecture · p. 6
  • The resulting layered controller is used to construct a CBF for the full system.The section describes a certificate of the form h, with positive parameters σ and λ.Source: A. Layered Architecture · p. 6

B. Constructing a Safe Hierarchical Controller

The section constructs controllers and tracking certificates for each layer, then combines their bounds to establish a CBF for the full system. The guarantee depends on the tracking layers’ performance and constrains how aggressively the safe kinematic field can be chosen.

B. Constructing a Safe Hierarchical Controller

  • The kinematic field κ must satisfy a safety condition parameterized by α and a robustness condition parameterized by δ.Outside these requirements, κ can be chosen to satisfy other performance specifications.Source: B. Constructing a Safe Hierarchical… · p. 6
  • The virtual-model controller tracks the safe vector field’s velocities, with the derivative of its tracking-error energy equal to −k~d||ė~||².The controller is designed for the virtual model on G.Source: B. Constructing a Safe Hierarchical… · p. 6
  • When grad V_G is identically zero, verifying the nonzero-controller assumption requires a more careful topological analysis that the authors leave for future work.When grad V_G is nonzero, the assumption is often straightforward to verify.Source: B. Constructing a Safe Hierarchical… · p. 6
  • A geometric PD controller uses the full actuation of the S-subsystem to track the reference induced by the virtual-model input.Its tracking performance is certified using two energy-like functions, Wϵ and W.Source: B. Constructing a Safe Hierarchical… · p. 6
  • The barrier construction uses Wϵ to form h and W to ensure L-regularity conditions and the safety constraint on h’s safe set.The construction bounds the tracking energies and combines the resulting controller-layer estimates.Source: B. Constructing a Safe Hierarchical… · pp. 6–7
  • The constructed safe set C is contained in C0, so configuration-safety, L-regularity, and bounds established on C0 also hold on C.This inclusion holds for the stated ranges of σ, λ, and ϵ.Source: B. Constructing a Safe Hierarchical… · p. 7
  • Under the stated parameter choices, the closed-loop system satisfies dot h > −αh on the safe set, and h is a CBF for the full system.The theorem establishes this conclusion on C = {h ≥ 0}, with h defined on TS × O.Source: B. Constructing a Safe Hierarchical… · pp. 7–8
  • The bound on α limits the aggression of κ according to the performance of the tracking controllers; h can certify safety even when used hierarchically rather than in an end-to-end filter.The safe velocity controller can be designed first and tracked by the low-level controller.Source: B. Constructing a Safe Hierarchical… · p. 8

V. CONCLUSION

The paper presents a constructive, hierarchical study of safety-critical control for underactuated systems, using shape maps to characterize when layered control architectures exist and to develop hierarchical CBF synthesis.

V. CONCLUSION

  • Shape maps provide precise conditions for correspondence between input directions in G and shapes in S, supporting a layered architecture and hierarchical CBF synthesis procedure.The paper identifies generalization to a broader class of underactuated systems as future work.Source: §V · p. 8
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