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Towards safe and optimal flight: Viability Kernel MPC for Fully Actuated Multirotor
Massimiliano Bertoni, Alberto Piccina, Gianni Lunardi, Elias Fontanari, Andrea Del Prete, Angelo Cenedese, Giulia Michieletto
TL;DR
Real-world multirotor deployment remains limited by insufficient safety guarantees, especially in cluttered environments and near people or other robots. This paper integrates viability theory, neural-network approximation, and dynamically updated obstacle-free boxes into terminally constrained MPC, with simulations showing successful navigation and 1.00 ms average iteration time.
Problem
Existing approaches often provide short-term collision avoidance without long-term viability or depend on expensive offline reachability analysis, limiting safe multirotor deployment in cluttered environments.
Method
The method approximates a numerical viability kernel for a fully actuated multirotor and integrates it into terminally constrained MPC with dynamically computed axis-aligned bounding boxes and a certified safe fallback state.
Results
The tilted hexarotor successfully reached its target while dynamically updating the obstacle-free box, with average iteration time of 1.00 ms and maximum iteration time of 11.72 ms.
Takeaways & Limitations
The proof-of-concept combines viability theory with receding-horizon control to provide safety guarantees under state and input constraints without exhaustive offline reachability analysis.
Takeaways & Limitations
The current box-based approach can be conservative and the greedy AABB strategy may produce sub-optimal configurations in geometrically complex environments.
Abstract
from arXiv · showhide
Industrial aerial robotics demands safety guarantees for navigation in unstructured environments while optimizing performance and computational efficiency. This paper presents a method for generating safe pose trajectories for fully actuated multirotors within a Model Predictive Control (MPC) framework, leveraging both viability theory and data-driven methods. Obstacle avoidance is enforced through dynamically computed axis-aligned bounding boxes, providing formal safety guarantees without exhaustive offline reachability analysis. Numerical simulations on a fully actuated tilted hexarotor validate the approach, demonstrating successful navigation in cluttered environments with real-time computational performance.
I. INTRODUCTION
The paper targets safe pose-trajectory generation for fully actuated multirotors in cluttered environments, addressing limitations in long-term viability guarantees and scalable safety computation.
- Related work: Existing multirotor safety research spans platform fault tolerance and obstacle-constrained control strategies.The former addresses hardware damage and degradation, while the latter incorporates obstacle constraints into control formulations.
- Research gap: Most existing approaches provide only short-term collision avoidance or depend on computationally expensive offline reachability analysis.These limitations are problematic for high-dimensional nonlinear systems and dynamically changing environments.
- Approach: The proposed framework uses viability theory to characterize states from which the multirotor can safely reach an equilibrium under state and input constraints.It builds on a viability-based safe MPC framework and the Viability-Boundary Optimal Control approach.
- Contributions: The work approximates the viability kernel numerically and integrates it into terminally constrained MPC with a certified safe fallback state.The fallback is intended for cases of control infeasibility.
- Paper organization: The paper presents the multirotor model, viability-kernel computation, terminally constrained MPC, simulation validation, and concluding future directions.The notation defines R and N, vector and matrix conventions, and component-wise inequality notation.
II. MULTIROTOR MODEL
The multirotor is modeled as a 12-dimensional fully actuated dynamical system with rotor-rate-squared inputs and continuous-time dynamics discretized for control and viability computations.
- State representation: The state comprises position, Euler angles, linear velocity, and angular velocity, giving x ∈ X ⊂ R12.Position and linear velocity are expressed in the world frame, while angular velocity is expressed in the body frame.
- State representation: The twist component consists of linear and angular velocities and has dimension six.The velocity variables are grouped as v and ω.
- Control input: The control input stacks the squared, limited spinning rates of the n ≥ 4 rotors.The platform dynamics are represented compactly as ẋ = f(x,u).
- Dynamics: The explicit dynamics include Euler kinematics, gravity, force allocation, rotational coupling, and moment allocation.The allocation matrices map rotor inputs to wrench components at the multirotor center of mass.
- Discretization: A discrete-time approximation x_k+1 = f(x_k,u_k) is used for both MPC and viability-kernel computation.This matches the implementation of embedded digital control systems.
III. VIABILITY KERNEL COMPUTATION
The method computes a conservative viability-kernel approximation using VBOC, encodes it with a neural network, and incorporates dynamically defined obstacle-free boxes and safety margins.
- Viability theory: The viability kernel is the largest subset of safe states from which the system can remain safe indefinitely under suitable control.For discrete-time systems, it consists of states admitting an input sequence that keeps every future state within the constraint set.
- VBOC computation: VBOC reformulates the nonlinear viability problem through equilibrium reachability, defining viable states as those that can remain safe and reach an equilibrium.The refined kernel is a backward reachable set of the equilibrium set and is contained in the general viability kernel.
- VBOC computation: VBOC computes a conservative approximation ˆV by estimating the viability-kernel boundary through a finite-horizon optimal control problem.The approximation may not be control-invariant in general but can still support safety guarantees within MPC.
- Obstacle representation: Obstacle-free regions are modeled as axis-aligned bounding boxes around the multirotor position, with constraints determined by obstacle placement.The box faces are parallel to the world-frame coordinate planes and are parameterized by distances to those faces.
- Obstacle representation: Eroding each AABB by safety margin r yields a collision-free region containing the platform within an inscribed sphere.The margin represents the radius of a sphere within which the platform is fully inscribed.
- Data-driven encoding: A neural network maps Euler angles, twist direction, and AABB parameters to the maximum viable twist norm γ.This encoding embeds the viability check into each MPC iteration efficiently.
- Safety interpretation: The OCP horizon M bounds the number of steps needed to drive the system safely to equilibrium from ˆV under MPC.The initial constraint parameters include a twist direction and limits associated with rotor saturation and obstacles.
IV. TERMINAL-CONSTRAINT MPC WITH SAFE ABORT
The safe MPC framework optimizes tracking from an initial state while enforcing discrete dynamics and a terminal viability condition, with state viability serving as its safety mechanism.
- Safe MPC: State-viability verification is the key concept underlying the safe MPC framework for the fully actuated multirotor.The approach is investigated as a method for maintaining safety during receding-horizon control.
- Optimization problem: The MPC problem is formulated for a target state x_r and initial condition x_0.The optimization is posed over the system trajectory and control sequence.
- Optimization problem: The optimization enforces the discrete dynamics x_k+1 = f(x_k,u_k) over the prediction horizon.The dynamics constraint applies for k = 0,…,N−1.
- Cost function: The stage cost penalizes tracking error and control effort, while the terminal cost penalizes terminal deviation from the target.The supplied formulation defines the stage and terminal costs using norms relative to x_r.
A. MPC Terminal Constraint
The MPC dynamically defines collision-free AABB state constraints through greedy shrinking and uses the resulting box in its terminal viability formulation.
- A. MPC Terminal Constraint: At each MPC iteration, the state constraint is the set of states whose position lies within a collision-free AABB B⋆.B⋆ is computed from the current obstacle geometry.
- A. MPC Terminal Constraint: The greedy procedure identifies box faces intersecting obstacles and evaluates inward displacements that remove the associated collisions.Each candidate box is formed by shifting one intersecting face inward by its minimum required displacement.
- A. MPC Terminal Constraint: The algorithm selects the displaced-face box with the largest volume and repeats until no box face intersects an obstacle.The final box is the collision-free AABB B⋆ with parameters b⋆.
- A. MPC Terminal Constraint: Greedy shrinking prioritizes computational efficiency for real-time MPC replanning over globally optimal box configurations.The resulting box may be sub-optimal in geometrically complex environments.
B. Safe Abort Procedure
The terminal constraint places the predicted terminal state in a viability-based set so consecutive MPC infeasibilities can transition to a predefined safe abort maneuver.
- B. Safe Abort Procedure: The terminal constraint requires x_N to belong to X_N, thereby enforcing viability-kernel membership at every MPC iteration.The terminal set is defined using the viability-kernel approximation and an additional condition on the terminal state.
- B. Safe Abort Procedure: If MPC becomes infeasible, the controller applies the last feasible control sequence and starts a counter measuring steps since the last feasible solution.The counter also tracks the remaining steps until the planned viable terminal state is reached.
- B. Safe Abort Procedure: A newly feasible MPC solution before c reaches N resets the counter and returns the controller to normal operation.The reset occurs when feasibility is recovered with c < N.
- B. Safe Abort Procedure: When c = N, the viable terminal state is reached and the predefined safe abort maneuver is triggered.The maneuver is formulated as an optimal control problem initialized at the viable terminal state.
A. Simulation setup
The simulations evaluate the proposed controller on a fully actuated tilted hexarotor navigating among static obstacles, using neural-network viability evaluation and real-time MPC settings.
- A. Simulation setup: The scenario commands the multirotor to reach a target pose with zero velocity from a specified initial state while avoiding static obstacles.Obstacle centers and dimensions are defined in Table I.
- A. Simulation setup: Validation uses a coplanar star-shaped, interdependent cant-tilted α-Ted6R hexarotor with six propellers.Adjacent propeller axes are tilted in opposite directions, enabling independent control of all six degrees of freedom.
- A. Simulation setup: The multirotor dynamics are integrated with a 0.1 ms time step using fourth-order Runge–Kutta integration.The platform parameters are listed in Table II.
- A. Simulation setup: The viability constraint is evaluated by a two-hidden-layer GeLu neural network with 512 units per layer trained on 200,000 samples.The MPC uses CasADi and ACADOS with horizon N = 60 and sampling period T_s = 20 ms.
B. Results
The learned viability approximation agrees closely with the optimal-control solutions, while terminal-constrained MPC reaches the target and operates within the sampling period in cluttered simulation.
- B. Results: The neural network’s predicted viable region closely matches the OCP solutions for motion restricted to the y-axis.The comparison uses uniformly sampled displacement bounds d_y, d̄_y ∈ [−2, 2] m and OCP horizon M = 15.
- B. Results: The α-Ted6R successfully reaches the target pose even though the target is initially outside the obstacle-free box.The box is initially limited to ±2 m per axis and is dynamically updated as new regions are discovered.
- B. Results: During [0.8, 1.4] s, the z-position temporarily plateaus when the safety sphere becomes tangent to an AABB face.The MPC compensates during this interval by increasing velocity along the x-axis.
VI. CONCLUSIONS
The architecture combines viability theory with receding-horizon control to provide safety guarantees for fully actuated multirotor pose tracking without exhaustive offline reachability analysis. It is presented as a methodological proof of concept, with future work targeting less conservative obstacle representations and more complex environments.
- The proposed architecture combines viability theory with receding-horizon control to provide safety guarantees under state and input constraints.
- It supports safe pose trajectory tracking for fully actuated multirotors navigating in cluttered environments.
- The approach provides these guarantees without relying on exhaustive offline reachability analysis.
- The study is a foundational proof of concept, with future work extending the framework to dynamical obstacles and advanced perception or learning-based components.
- Future work also seeks more accurate obstacle-free-region representations to reduce the conservativeness of the current box-based approach.