Path Feasibility Governor (PathFG)

• PathFG is a supervisory mechanism that regulates the progression along a geometric path, ensuring feasibility and safety under dynamic constraints.
• It integrates a dedicated path planner with nonlinear MPC by modulating references using certificates like terminal sets and invariant ellipsoids.
• PathFG expands the region of attraction and reduces computational burdens, enabling safe and adaptive control in non-convex environments.

Searching arXiv for Path Feasibility Governor and closely related reference-governor papers. arxiv_search(query="Path Feasibility Governor", max_results=10) Path Feasibility Governor (PathFG) denotes a governor-based mechanism that manipulates the reference passed to a controller so that progression along a path remains feasible and safe under the system dynamics and constraints. In the explicit 2025 formulation, PathFG integrates a path planner with nonlinear Model Predictive Control (MPC) by guiding the controller along a path while ensuring constraint satisfaction, stability, and recursive feasibility (Zhang et al., 12 Jul 2025). Earlier robotics papers implemented the same functional role through a reference governor or virtual governor, even when “PathFG” was not introduced as a formal term: the governor filters a path-following or goal-directed reference using motion prediction, invariant ellipsoids, or terminal-set certificates so that the resulting closed-loop motion remains collision-free and dynamically feasible (İşleyen et al., 2022, Li et al., 2020).

1. Conceptual definition

PathFG is best understood as a supervisory layer between planning and control. A planner provides a geometric path or reference field, while the governor admits only those reference updates that are certified feasible for the closed-loop system. In the nonlinear MPC formulation, the plant evolves as

$$x_{k+1} = f(x_k,u_k), \qquad x_k \in \mathcal X,\; u_k \in \mathcal U,$$

and the path is a continuous map

$p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$

where every point on the path corresponds to a strictly admissible steady-state reference (Zhang et al., 12 Jul 2025). The governor then regulates the progress parameter $s\in[0,1]$, rather than commanding the final target directly.

A central theme across the literature is that PathFG is not a full trajectory optimizer. It is a reference-modulation device. In linear MPC Feasibility Governor (FG) formulations, the governor computes an auxiliary reference $v_k$ as close as possible to the desired reference while keeping the underlying optimal control problem feasible: $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$ That construction is presented for setpoints, but the same papers describe a path interpretation obtained by applying the same feasibility-preserving modulation to a sequence of references or to a path parameterization (Skibik et al., 2021, Liao-McPherson et al., 2020).

2. Genealogy across reference-governor and MPC formulations

The PathFG idea emerged by convergence of two lines of work. One line comes from reference governors for motion planning and path following; the other comes from feasibility governors for MPC.

In the linear MPC line, the Feasibility Governor is an add-on unit that enlarges the region of attraction of MPC by manipulating the reference so that the optimal control problem remains feasible. Offline polyhedral projection algorithms compute the feasible state–reference set $\Gamma_N$, and online implementation solves a convex quadratic program in the auxiliary reference $v$ (Skibik et al., 2021). A closely related earlier formulation for linear MPC with piecewise-constant references emphasizes the same architecture, proving constraint satisfaction, asymptotic stability, zero-offset tracking, and finite-time convergence of the internal reference (Liao-McPherson et al., 2020).

In the nonlinear navigation line, a 2020 method for safe robot navigation in cluttered environments used a virtual governor that tracks the farthest feasible point along a path while invariant ellipsoids bound the closed-loop output trajectory under bounded disturbances. That paper does not define PathFG explicitly, but it implements the core PathFG function through a governor law driven by a local safe zone constructed from obstacle distance and an ellipsoidal output bound (Li et al., 2020). The 2022 framework for extending low-order motion planners to high-order robot models makes the same point even more directly: the paper does not explicitly define “PathFG” as a formal term, yet its reference governor filters a path-following planner through motion prediction and safety assessment, thereby ensuring that the commands are feasible and safe for high-order robot dynamics (İşleyen et al., 2022).

The 2025 paper “Integrating Planning and Predictive Control Using the Path Feasibility Governor” makes the concept explicit and names it PathFG. It frames PathFG as a modular mechanism for integrating path planners with nonlinear MPC in non-convex environments, with safety, recursive feasibility, asymptotic stability, and a significantly expanded region of attraction (Zhang et al., 12 Jul 2025).

Formulation	Governor variable	Feasibility certificate
Linear MPC FG	Auxiliary reference $v_k$	$(x_k,v)\in \Gamma_N$
Virtual governor for path following	Governor point $\mathbf g$	Invariant ellipsoid and local safe zone
High-order reference governor	Governor position $p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$0	Predicted robot motion range and safety level
Nonlinear MPC PathFG	Path progress $p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$1	Terminal-set membership $p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$2

3. Core mechanism and closed-loop architecture

Across these formulations, PathFG realizes a bidirectional interface between planning and control. Planning intent is transmitted downward through a governed reference, while feasibility or safety information is transmitted upward through a certificate computed from the current closed-loop state (İşleyen et al., 2022).

In the explicit nonlinear MPC formulation, the path is parameterized by $p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$3, and the governor uses the previous optimal terminal prediction $p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$4 to decide how far along the path the controller may safely progress. The feasibility set is

$p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$5

and the governor law is

$p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$6

This converts path following into repeated admissible advancement along a path, with each advancement certified by terminal invariance (Zhang et al., 12 Jul 2025).

The nonlinear path-following formulation based on invariant ellipsoids uses a virtual first-order governor system. Given a path $p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$7, the governor computes the largest feasible advancement along the path within the local safe zone,

$p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$8

and then applies

$p:[0,1]\to \mathcal R_\epsilon,\qquad p(1)=r,$9

If the free energy $s\in[0,1]$0, the governor stops; as the robot catches up, the safe zone expands and progression resumes (Li et al., 2020).

The high-order motion-planning formulation uses a continuous-time governor law driven by a safety level computed from the predicted robot motion range: $s\in[0,1]$1 This law scales the reference planner’s vector field by the available safety margin, so the governor moves only as much as the predicted motion range can remain in free space (İşleyen et al., 2022).

4. Feasibility and safety certificates

What distinguishes PathFG from a purely heuristic path-speed regulator is the presence of an explicit certificate of admissible progression. The certificate differs by formulation, but its role is constant: it bounds the future closed-loop motion tightly enough that the governor can decide whether further progression is allowed.

For feedback-linearizable nonlinear systems with bounded disturbances, the 2020 method computes invariant ellipsoidal bounds on the output tracking error. The relevant bound is the output peak bound

$s\in[0,1]$2

which is upper-bounded either by an SDP-based construction or by a Lyapunov-equation-based construction. The free energy

$s\in[0,1]$3

determines the local safe zone. When $s\in[0,1]$4, the governor pauses; otherwise it advances toward the farthest path point inside that zone (Li et al., 2020).

For high-order robots, the certificate is the predicted robot motion range $s\in[0,1]$5, which satisfies

$s\in[0,1]$6

The paper gives two concrete constructions: Lyapunov ellipsoids and Vandermonde simplexes. Safety is measured by the safety level, defined as the minimum distance from the predicted motion range to the free-space boundary. The governor then filters the planner reference according to that safety level, yielding recursive feasibility in the sense that safe at $s\in[0,1]$7 implies safe at $s\in[0,1]$8 (İşleyen et al., 2022).

In the nonlinear MPC PathFG, the certificate is terminal-set membership. Rather than predicting all future path progress explicitly, the governor checks whether the previous terminal prediction lies in the terminal slice corresponding to a candidate path parameter $s\in[0,1]$9. This is a shift-and-append-terminal-control certificate: if the previous terminal state belongs to the terminal invariant set at the new reference, then the optimal control problem remains feasible at the next step (Zhang et al., 12 Jul 2025).

5. Guarantees and region-of-attraction expansion

The chief theoretical guarantees associated with PathFG are recursive feasibility, safety or constraint satisfaction, and convergence.

In the nonlinear MPC formulation, if the planner produces a feasible path, then the PathFG+MPC closed loop is well defined and satisfies

$v_k$0

Under standard MPC assumptions, $v_k$1 is an asymptotically stable equilibrium. The paper emphasizes that ungovened MPC has region of attraction

$v_k$2

whereas PathFG expands it to

$v_k$3

that is, to all states from which a feasible path to $v_k$4 exists (Zhang et al., 12 Jul 2025).

In the linear MPC FG setting, the same logic appears in state–reference space. The closed-loop system remains in

$v_k$5

so recursive feasibility and output constraint satisfaction follow for all time. For constant desired reference, the papers also establish asymptotic stability and zero-offset tracking to $v_k$6 or to its closest admissible projection $v_k$7, with finite-time convergence of the internal governor reference $v_k$8 to $v_k$9 (Skibik et al., 2021, Liao-McPherson et al., 2020).

In the high-order motion-planning framework, safety means the robot state remains in free space for all $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$0, while stability means the robot and governor converge asymptotically to zero motion at the goal. The paper explicitly notes that overly conservative or inaccurate motion prediction slows or halts the governor: this preserves safety but can reduce performance (İşleyen et al., 2022).

6. Implementations, applications, and limitations

PathFG has been instantiated in several application settings. The 2020 work applies a virtual governor with invariant ellipsoids to safe robot navigation in unknown environments using local obstacle sensing, including an Ackermann-drive example based on feedback linearization and dynamic extension (Li et al., 2020). The 2022 framework targets second-, third-, and fourth-order robot dynamics and demonstrates that tighter motion prediction via Vandermonde simplexes yields faster, more adaptive motion than Lyapunov ellipsoids in cluttered environments and circular corridors (İşleyen et al., 2022). The 2025 paper validates PathFG on a simulated quadrotor in a cluttered environment, using obstacle half-space linearizations inside a convex MPC problem (Zhang et al., 12 Jul 2025).

A consistent practical benefit is reduction of horizon or replanning burdens. The 2022 paper contrasts the governor-based feedback planning approach with brittle trajectory optimization and significant replanning cycles (İşleyen et al., 2022). The 2025 quadrotor study reports that ungoverned MPC requires $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$1 for feasibility in the same environment, whereas PathFG maintains feasibility with $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$2; the reported average governor time is approximately $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$3 s and the full PathFG+MPC time is approximately $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$4 s per step under a $$v_k = g(x_k,r)=\arg\min_{v\in R_\epsilon}\|v-r\|_2^2 \quad \text{s.t.}\quad (x_k,v)\in \Gamma_N.$$5 s sampling period (Zhang et al., 12 Jul 2025).

The principal limitations are also common across formulations. PathFG requires a certificate that is both correct and sufficiently nonconservative: invariant ellipsoids, motion-range bounds, or terminal sets can stall progress if they are overly conservative (Li et al., 2020, İşleyen et al., 2022). The nonlinear MPC theory is nominal; robustness to disturbances and model uncertainties is identified as a direction for pairing PathFG with robust MPC, such as tube MPC (Zhang et al., 12 Jul 2025). The 2020 path-following guarantees assume static obstacles within the sensing horizon at each step, and rapidly moving obstacles require additional prediction or reactive safety layers (Li et al., 2020).

PathFG is therefore most precisely described as a governor architecture for safe progression along a path under closed-loop feasibility certificates. In reference-governor language, it separates path generation from admissible path advancement; in MPC language, it filters the reference so that the optimal control problem remains feasible; and in motion-planning language, it closes the kinodynamic gap between a geometric path and the actual constrained robot dynamics.