# Planning under Uncertainties with Closed-Loop Sensitivity: Recent Results and Perspectives

Tommaso Belvedere, Amr Afifi, Simon Wasiela, Andrea Pupa

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Tommaso Belvedere, Amr Afifi, Simon Wasiela, Andrea Pupa. Planning under Uncertainties with Closed-Loop Sensitivity: Recent Results and Perspectives. ERF 2025 - European Robotics Forum, Mar 2025, Stuttgart, Germany. pp.1-5, 10.1007/978-3-031-89471-8_9. hal-05021252

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# Planning under Uncertainties with Closed-Loop Sensitivity: Recent Results and Perspectives

Tommaso Belvedere<fnref n="1" />, Amr Afifi<fnref n="2" />, Simon Wasiela<fnref n="3" />, and Andrea Pupa<fnref n="4" />

**Abstract.** This paper presents a comprehensive summary of recent advancements in motion planning under parametric uncertainties, focusing on the application of closed-loop state sensitivity. This concept provides a framework for quantifying how deviations in model parameters affect the behavior of a system in closed-loop, facilitating the generation of robust trajectories. Various methods have been proposed to improve the resilience of robotic systems to model inaccuracies. However, these approaches often face challenges such as computational complexity and limitations in real-time applications. This paper synthesizes key results from several recent works, highlighting the development of techniques that optimize trajectory robustness while reducing computational overhead. Additionally, we outline the practical applications of these methods, discussing their validation through simulations and experiments on robotic systems subject to non-negligible uncertainties in their models.

## 1 Introduction

Robust planning in the presence of uncertainties is a critical challenge in robotics, especially as robotic systems are increasingly deployed in real-world tasks such as autonomous navigation, manipulation, and interaction with humans. Uncertainties in robot dynamics, sensor noise, and environmental conditions can significantly degrade performance, leading to deviations from the planned trajectory and, in some cases, failure to complete tasks. To address this, numerous strategies have been developed that focus on generating robust trajectories or designing controllers capable of mitigating the effects of these uncertainties.

One of the viable methods among these approaches utilizes the concept of closed-loop state sensitivity. This notion quantifies how model parameter variations affect the trajectory of a system in closed-loop, offering a way to assess and mitigate the impact of uncertainties while accounting for the controller's performance. Early research focused on minimizing the sensitivity at key points along

<footnote n="1">¹ CNRS, Université de Rennes, Inria, IRISA, Campus de Beaulieu, Rennes, France, tommaso.belvedere@irisa.fr</footnote>

<footnote n="2">² Robotics and Mechatronics Lab, Faculty of Electrical Engineering, Mathematics & Computer Science, University of Twente, Enschede, The Netherlands. a.n.m.g.afifi@utwente.nl</footnote>

<footnote n="3">³ LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France swasiela@laas.fr</footnote>

<footnote n="4">⁴ Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Italy andrea.pupa@unimore.it</footnote>
a trajectory to increase accuracy, while more recent methods incorporate uncertainty tubes to bound these deviations, enhancing safety. Despite these advances, issues related to computational complexity, especially in real-time scenarios, remain a key challenge.

The goal of this paper is to provide a concise summary of recent research that tackles motion planning under parametric uncertainties, with a focus on techniques leveraging closed-loop state sensitivity. Specifically, we aim to: (i) synthesize results from recent works on trajectory optimization methods, uncertainty tubes, and sensitivity-aware planners; (ii) highlight advances in reducing computational complexity while maintaining robust performance in the presence of model uncertainties; (iii) discuss real-world applications and validation of these methods through simulations and experiments on robotic systems.

By summarizing these contributions, this paper provides a clear overview of the current state-of-the-art in planning under uncertainties and outlines potential directions for future research in this area.

## 2 Closed-loop Sensitivity of Nonlinear Systems

Consider a dynamical system $\dot{x} = f(x, u, p)$, where $x \in \mathbb{R}^{n_x}$ and $u \in \mathbb{R}^{n_u}$ are the state and input vectors, $p \in \mathbb{R}^{n_p}$ represents uncertain model parameters, which have nominal value $p = p_c$. Let $\varphi: \mathbb{R}^{n_x} \times \mathbb{R} \to \mathbb{R}^{n_x}$ be the flow of the dynamical system under the feedback control action

$$ \begin{aligned} u &= \mu(x, \xi, y_d, p_c) \\ \dot{\xi} &= g(x, \xi, y_d, p_c), \end{aligned} \quad (1) $$

where $\xi \in \mathbb{R}^{n_\xi}$ are the controller internal states, with known initial condition $\xi(0) = \xi_0$, and $y_d$ is a generic reference output that the controller is designed to track. Then, for an initial state $x(0) = x_0$, the closed-loop state trajectory is $x(t) = \varphi(x_0, t)$. Following [1], we use the notion of closed-loop sensitivity to describe how changes in the parameters affect the system trajectories under the action of (1). Define the state sensitivity $\Pi \in \mathbb{R}^{n_x \times n_p}$ as

$$ \Pi(t) = \frac{d\varphi(x_0, t)}{dp}\Big|_{p=p_c}. \quad (2) $$

This quantity is evaluated around the nominal trajectory (i.e., the one obtained when $p = p_c$) and represents a first-order approximation of the effects of a parameter deviation on the closed-loop state trajectory. For instance, a system that is structurally less affected by a particular parameter, or for which the controller is already able to compensate for parameter inaccuracies, will present a smaller (in some norm) closed-loop state sensitivity.
A closed-form expression for (2) is, in general, not available but, as shown in [1], it is possible to obtain the following update rule:

$$ \begin{aligned} \dot{\Pi} &= \frac{\partial f}{\partial x} \Pi + \frac{\partial f}{\partial u} \left( \frac{\partial \mu}{\partial x} \Pi + \frac{\partial \mu}{\partial \xi} \Pi_{\xi} \right) + \frac{\partial f}{\partial p} \quad \Pi(0) = 0, \\ \dot{\Pi}_{\xi} &= \frac{\partial g}{\partial x} \Pi + \frac{\partial g}{\partial \xi} \Pi_{\xi} \quad \Pi_{\xi}(0) = 0, \end{aligned} \quad (3) $$

having defined $\varphi_{\xi}: \mathbb{R}^{n_{\xi}} \times \mathbb{R} \to \mathbb{R}^{n_{\xi}}$ and $\Pi_{\xi}(t) = \frac{d}{dp}\varphi_{\xi}(\xi_0, t)$ to denote the sensitivity of the controller states. In general, we might be interested in describing the sensitivity of any function of the closed-loop state and inputs. Due to its inherent importance in feedback control we define as well the input sensitivity $\Pi_u = \frac{\partial \mu}{\partial x} \Pi + \frac{\partial \mu}{\partial \xi} \Pi_{\xi}$, which describes how perturbed closed-loop trajectories result in a perturbed input evolution due to the feedback action [2]. If also the initial state $x_0$ is considered uncertain, it is possible to define state/input sensitivities such as $\Psi(t) = \frac{d\varphi}{dx_0}$ in the same fashion to the just described sensitivity to parameters [3].

# 3 Applications

## 3.1 Minimum-sensitivity planning and control adaptation

A first possible use of the closed-loop sensitivity metric is that of generating reference trajectories that are inherently robust to parametric uncertainties, improving their performance in closed-loop [1]. This is particularly useful when task requirements allow for some redundancy in the trajectory optimization, such as in collision-free navigation or regulation tasks. The main advantage of this approach is that, once the controller is defined, it is possible to robustify the closed-loop system behavior by only modifying the trajectory, as demonstrated in [4] using the standard commercially available PX4 flight controller.

This can be achieved by solving an optimization problem of this form [2]:

$$ \begin{aligned} \min_{a \in \mathcal{A}} & \quad w_1 \|\Pi(t_f)\|^2 + w_2 \int_{t_0}^{t_f} \|\Pi(\tau)\|^2 d\tau \\ \text{s.t.} & \quad A_{eq}a = b_{eq}, \end{aligned} \quad (4) $$

where $\|\cdot\|$ represents some norm that should be minimized, e.g. the Frobenius norm. $a$ represent the coefficient that parametrize the trajectory (e.g., polynomial, Bézier curve) and $\mathcal{A}$ is the set of its possible values. $w_1$ and $w_2$ are the weights associated with the sensitivity only at the end of the trajectory and the one over the entire trajectory, respectively $A_{eq}$ and $b_{eq}$ represents the boundary conditions of the trajectory. The goal of the optimization problem in (4) is to build a trajectory that minimizes a combination of the sensitivity at the end and during the execution. Figure 1 shows the effects of reducing the sensitivity at the end, i.e., $w_2 = 0$, on a unicycle. Of course, optimization problem in (4) can be modified to minimize any sensitivity matrices, e.g. $\Pi_u$. Moreover, it can be extended to compute the optimal gains of a specific controller, ensuring that the closed-loop system becomes less sensitive to parameter uncertainties [5].
<figure><img src="image_2.png" /><figcaption>Fig. 1: Difference between optimized and non-optimized trajectories with perturbed model parameters [1, 5]. In both examples, the goal is to precisely reach a target configuration.</figcaption></figure>

## 3.2 Safe trajectory optimization with sensitivity-based tubes

While minimizing the norm of the sensitivity matrix can be beneficial, it may not always be the optimal strategy to ensure robustness. In some cases, the goal is to achieve robust guarantees while tracking the reference trajectory, ensuring safe execution. In [3], the concept of state sensitivity is used to build an envelope, called tube, centered on the nominal state trajectory, which contains all possible perturbed system state evolutions. This tube can then be optimized to enforce constraints even in the event of perturbations, ensuring the system's safety.

Assume that the uncertain parameters $p$ have a maximum variation $\Delta p_{\max} = (\Delta p_{\max,1}, \dots, \Delta p_{\max,n_p})$ w.r.t. their nominal values $p_c$. Define a direction $n \in \mathbb{R}^{n_x}$ (typically one of the canonical basis) along which one is interested in finding the maximum deviation the system suffers. The tube radius at time $t$ is:

$$ \rho_n(t) = \sqrt{n^T K_{\Pi}(t)n}, \qquad (5) $$

where $K_{\Pi} = \Pi W \Pi^T$, with $W = \text{diag}\{\Delta p_{\max,i}^2\}$, is the kernel matrix of an ellipsoid in the state space centered at the nominal state $x_c$.

Starting from the radius of the tube in (5), it is possible to compute the envelope for the entire trajectory and use it inside the optimization problem in (4) to robustify constraints. In particular, the tube formulation has been successfully exploited to ensure a collision-free trajectory [3, 4], avoiding intersections between the tube around the nominal state and the obstacles. And to prevent the depletion of the energy tank in a manipulation task [6], guaranteeing that the tank state remains above a predefined lower bound. Figure 2 shows applications to an hexarotor UAV and a robot arm.

## 3.3 Robust sampling-based planning and learned sensitivity dynamics

While ensuring robust trajectory generation and execution, the optimization methods presented so far rely on local techniques that can be susceptible to local minima and/or deadlocks in complex environments. To address this issue, sensitivity-based tubes were encoded within sampling-based algorithms in [7].
<figure><img src="image_3.png" /><figcaption>Fig. 2: Results of the optimization with the tube constraints. On the left, a hexarotor avoids a virtual obstacle [3], while on the right, a UR10e robot performs a manipulation task without depleting the energy tank [6].</figcaption></figure>

These sampling-based methods are global planning techniques that typically involve randomly generating tens of thousands of robot states and attempting to connect them using a steering method, which generates a local trajectory between two states. The result is commonly referred to as a tree [8]. In order to ensure the generation of a robust global trajectory, the sensitivity-based tubes must be computed for each local trajectory of the tree, which involves propagating the sensitivity dynamics over a large number of trajectories.

Although the results of [3] demonstrated that propagating the sensitivity dynamics and computing the tubes only took a few tens of milliseconds on trajectories with a hundred points for complex systems such as the hexarotor, the results of [7] have demonstrated that this process becomes a bottleneck for sampling-based applications.

To overcome this limitation, the work of [9] exploited the similarity in structure between differential equations, required to propagate the sensitivity (see Sect. 2), and recurrent neural networks, allowing for direct prediction of the tubes. The results have shown that using recurrent neural networks provides a time gain of one order of magnitude.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Experimental validation of the robust sensitivity-aware sampling-based method. On the left, drone navigation through a narrow window. On the right, in-flight ring-catching task, where uncertainty in the perch end-effector position is minimized.</figcaption></figure>
## 4 Conclusion and Future Directions

This paper provides a concise overview of recent research developments addressing the challenge of motion planning in the presence of parametric uncertainties. First, a mathematical overview of the concept of closed-loop sensitivity is presented. Following this, a range of techniques and approaches that exploit state sensitivity are discussed, showing how these approaches are suitable not only for simulation but also for real-world applications.

Future work will aim at embedding closed-loop sensitivity into online control techniques, such as Model Predictive Control, and at performing a comparison between the notion of closed-loop sensitivity and other methods used for robust planning, such as contraction theory. Moreover, further studies should explore the interplay between closed-loop sensitivity and standard methods for online parameter estimation.

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