N-dimensional Convex Obstacle Avoidance using Hybrid Feedback Control (Extended version)

Technical Summary: N-dimensional Convex Obstacle Avoidance using Hybrid Feedback Control

Problem Statement
This paper addresses the problem of autonomous robot navigation in $n$-dimensional environments containing a priori unknown, disjoint convex obstacles of arbitrary shapes and sizes. The primary challenge is to design a control strategy that guarantees both safety (collision avoidance) and global asymptotic convergence to a predefined target location. Existing approaches face significant limitations:

• Artificial Potential Fields (APF) and Navigation Functions (NF) often suffer from undesired local minima or require global knowledge of the environment (e.g., diffeomorphic mappings).
• Control Lyapunov Function (CLF) and Control Barrier Function (CBF) methods, while effective in specific cases, generally provide only "almost global" stability guarantees due to the topological constraints of continuous time-invariant vector fields.
• Previous Hybrid Methods have been largely restricted to two-dimensional spaces or specific obstacle geometries (e.g., ellipsoids or strongly convex obstacles) and often require prior knowledge of obstacle shapes.

The specific constraints of this work include environments where obstacles are convex but may have non-smooth boundaries, and the robot possesses only local range-sensor data (no global map).

Methodology
The authors propose a hybrid feedback control scheme that operates via a switching mechanism between two distinct modes:

1. Move-to-Target Mode ($m=0$): The robot moves in a straight line toward the target.
2. Obstacle-Avoidance Mode ($m=1$): The robot navigates around an obstacle.

Key Technical Components:

• Hybrid System Framework: The system is modeled using differential and difference inclusions. The state vector $\xi$ includes the robot's position $x$, a "hit point" $h$ (where avoidance began), a unit vector $a$, a mode indicator $m$, and a timer variable $s$.
• Switching Logic:
  • The transition from move-to-target to obstacle-avoidance occurs when the robot enters a $\gamma$-neighborhood of an obstacle and the straight-line path to the target is obstructed (detected via a sensing cylinder intersection test).
  • The transition back to move-to-target occurs when the robot enters an "exit region" where the path to the target is clear, and the robot is closer to the target than the initial hit point $h$ (plus a hysteresis margin $\epsilon$).
• Obstacle-Avoidance Strategy:
  • Unlike 2D methods that force clockwise or counter-clockwise motion, the $n$-dimensional strategy confines the robot's motion to a specific two-dimensional plane $P(h, a)$. This plane is defined by the target location (origin), the hit point $h$, and a unit vector $a$ orthogonal to $h$.
  • Within this plane, the control vector is constructed using a rotation matrix and a parallel projection operator to guide the robot along the obstacle boundary without retracing its path.
  • A scalar function $\eta(x)$ ensures the robot remains within a safe neighborhood of the obstacle during the avoidance phase.
• Sensor-Based Implementation: The algorithm relies solely on range sensors (e.g., LiDAR) to identify the set of visible obstacle boundary points within a sensing radius $R_s$. It does not require a priori global knowledge of the environment.
• Smoothing Mechanism: To address the discontinuity of control inputs during mode switching (which can cause abrupt trajectory changes), the authors introduce a continuous scalar function $\lambda$ that smoothly interpolates between the two control vectors over a time interval $t_s$.

Key Contributions
The paper claims four primary contributions:

1. Global Asymptotic Stability: The proposed solution guarantees global asymptotic stability of the target location in unknown environments with convex obstacles of arbitrary shapes. This contrasts with existing hybrid results that are often limited to ellipsoidal obstacles.
2. $n$-Dimensional Applicability: The controller is designed for $n$-dimensional workspaces, whereas many existing sensor-based or hybrid approaches are restricted to 2D.
3. Arbitrary Obstacle Arrangements: The method imposes no restrictions on the relative arrangement of obstacles (other than a minimum separation distance to ensure the robot can pass between them), unlike methods requiring specific curvature conditions or non-flatness.
4. A Priori Unknown Environments: The approach is sensor-based and does not require global knowledge of obstacle geometries, unlike NF-based approaches that rely on diffeomorphic mappings.

Results and Validation

• Simulations:
  • 3D Environments: Simulations with 3D convex obstacles demonstrate the robot converging to the target from various initial positions while avoiding collisions.
  • 2D Environments: Tests with 12 convex obstacles show successful convergence from 24 different starting points.
  • Sensing Radius Analysis: The paper analyzes the effect of varying the sensing radius $R_s$, noting that smaller radii may lead to premature switching but the system remains robust.
  • Sphere Worlds: A modified version of the controller is applied to spherical obstacles, proving that the distance to the target decreases monotonically in this specific geometry.
  • Comparison: The proposed method is compared against a separating hyperplane approach [15]. The paper demonstrates that the hyperplane approach fails (converging to local minima) when obstacles do not satisfy specific curvature conditions, whereas the proposed hybrid controller succeeds.
• Experimental Validation:
  • The smoothed control law was implemented on a TurtleBot 4 Standard mobile robot in a physical workspace with five convex obstacles.
  • The robot successfully navigated from an initial position to a target, switching between modes as detected by its LiDAR sensor. The experimental results confirmed safe navigation and convergence.

Significance and Claims
The paper positions this work as a significant advancement over previous hybrid control strategies by extending global stability guarantees to $n$-dimensional spaces with arbitrary convex obstacles without requiring global environmental maps. The authors emphasize that the use of a switching mechanism overcomes the topological obstructions that prevent continuous vector fields from achieving global stability.

The authors conclude modestly, noting that while the current work handles convex obstacles, extending the approach to non-convex obstacles and robots with second-order dynamics (inertia) in $n$-dimensional environments remains a direction for future research. The paper does not claim to solve non-convex navigation problems but highlights the robustness of the convex case as a foundational step.