Ternary Logic Encodings of Temporal Behavior Trees with Application to Control Synthesis

Behavior Trees (BTs) provide designers an intuitive graphical interface to construct long-horizon plans for autonomous systems. To ensure their correctness and safety, rigorous formal models and verification techniques are essential. Temporal BTs (TBTs) offer a promising approach by leveraging existing temporal logic formalisms to specify and verify the executions of BTs. However, this analysis is currently limited to offline post hoc analysis and trace repair. In this paper, we reformulate TBTs using a ternary-valued Signal Temporal Logic (STL) amenable for control synthesis. Ternary logic introduces a third truth value \textit{Unknown}, formally capturing cases where a trajectory has neither fully satisfied or dissatisfied a specification. We propose mixed-integer linear encodings for partial trajectory STL and TBTs over ternary logic allowing for correct-by-construction control strategies for linear dynamical systems via mixed-integer optimization. We demonstrate the utility of our framework by solving optimal control problems.

Risk-Constrained Belief-Space Optimization for Safe Control under Latent Uncertainty

Many safety-critical control systems must operate under latent uncertainty that sensors cannot directly resolve at decision time. Such uncertainty, arising from unknown physical properties, exogenous disturbances, or unobserved environment geometry, influences dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider partially observed dynamical systems whose dynamics, costs, and safety constraints depend on a latent parameter maintained as a belief distribution, and propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) control framework that plans under this belief while enforcing a Conditional Value-at-Risk (CVaR) constraint on a trajectory safety margin over the receding horizon. The resulting controller optimizes a risk-regularized performance objective while explicitly constraining the tail risk of safety violations induced by latent parameter variability. We establish three properties of the resulting risk-constrained controller: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the risk weight in the objective tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In physics-based simulations of a vision-guided dexterous stowing task in which a grasped object must be inserted into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared to 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces.

Long-Horizon Geometry-Aware Navigation among Polytopes via MILP-MPC and Minkowski-Based CBFs

Autonomous navigation in complex, non-convex environments remains challenging when robot dynamics, control limits, and exact robot geometry must all be taken into account. In this paper, we propose a hierarchical planning and control framework that bridges long-horizon guidance and geometry-aware safety guarantees for a polytopic robot navigating among polytopic obstacles. At the high level, Mixed-Integer Linear Programming (MILP) is embedded within a Model Predictive Control (MPC) framework to generate a nominal trajectory around polytopic obstacles while modeling the robot as a point mass for computational tractability. At the low level, we employ a control barrier function (CBF) based on the exact signed distance in the Minkowski-difference space as a safety filter to explicitly enforce the geometric constraints of the robot shape, and further extend its formulation to a high-order CBF (HOCBF). We demonstrate the proposed framework in U-shaped and maze-like environments under single- and double-integrator dynamics. The results show that the proposed architecture mitigates the topology-induced local-minimum behavior of purely reactive CBF-based navigation while enabling safe, real-time, geometry-aware navigation.

RRT$^η$: Sampling-based Motion Planning and Control from STL Specifications using Arithmetic-Geometric Mean Robustness

Sampling-based motion planning has emerged as a powerful approach for robotics, enabling exploration of complex, high-dimensional configuration spaces. When combined with Signal Temporal Logic (STL), a temporal logic widely used for formalizing interpretable robotic tasks, these methods can address complex spatiotemporal constraints. However, traditional approaches rely on min-max robustness measures that focus only on critical time points and subformulae, creating non-smooth optimization landscapes with sharp decision boundaries that hinder efficient tree exploration. We propose RRT$^η$, a sampling-based planning framework that integrates the Arithmetic-Geometric Mean (AGM) robustness measure to evaluate satisfaction across all time points and subformulae. Our key contributions include: (1) AGM robustness interval semantics for reasoning about partial trajectories during tree construction, (2) an efficient incremental monitoring algorithm computing these intervals, and (3) enhanced Direction of Increasing Satisfaction vectors leveraging Fulfillment Priority Logic (FPL) for principled objective composition. Our framework synthesizes dynamically feasible control sequences satisfying STL specifications with high robustness while maintaining the probabilistic completeness and asymptotic optimality of RRT$^\ast$. We validate our approach on three robotic systems. A double integrator point robot, a unicycle mobile robot, and a 7-DOF robot arm, demonstrating superior performance over traditional STL robustness-based planners in multi-constraint scenarios with limited guidance signals.

Control Barrier Functions via Minkowski Operations for Safe Navigation among Polytopic Sets

Safely navigating around obstacles while respecting the dynamics, control, and geometry of the underlying system is a key challenge in robotics. Control Barrier Functions (CBFs) generate safe control policies by considering system dynamics and geometry when calculating safe forward-invariant sets. Existing CBF-based methods often rely on conservative shape approximations, like spheres or ellipsoids, which have explicit and differentiable distance functions. In this paper, we propose an optimization-defined CBF that directly considers the exact Signed Distance Function (SDF) between a polytopic robot and polytopic obstacles. Inspired by the Gilbert-Johnson-Keerthi (GJK) algorithm, we formulate both (i) minimum distance and (ii) penetration depth between polytopic sets as convex optimization problems in the space of Minkowski difference operations (the MD-space). Convenient geometric properties of the MD-space enable the derivatives of implicit SDF between two polytopes to be computed via differentiable optimization. We demonstrate the proposed framework in three scenarios including pure translation, initialization inside an unsafe set, and multi-obstacle avoidance. These three scenarios highlight the generation of a non-conservative maneuver, a recovery after starting in collision, and the consideration of multiple obstacles via pairwise CBF constraint, respectively.

Accelerating Proximal Policy Optimization Learning Using Task Prediction for Solving Games with Delayed Rewards

In this paper, we tackle the challenging problem of delayed rewards in reinforcement learning (RL). While Proximal Policy Optimization (PPO) has emerged as a leading Policy Gradient method, its performance can degrade under delayed rewards. We introduce two key enhancements to PPO: a hybrid policy architecture that combines an offline policy (trained on expert demonstrations) with an online PPO policy, and a reward shaping mechanism using Time Window Temporal Logic (TWTL). The hybrid architecture leverages offline data throughout training while maintaining PPO's theoretical guarantees. Building on the monotonic improvement framework of Trust Region Policy Optimization (TRPO), we prove that our approach ensures improvement over both the offline policy and previous iterations, with a bounded performance gap of $(2\varsigma\gamma\alpha^2)/(1-\gamma)^2$, where $\alpha$ is the mixing parameter, $\gamma$ is the discount factor, and $\varsigma$ bounds the expected advantage. Additionally, we prove that our TWTL-based reward shaping preserves the optimal policy of the original problem. TWTL enables formal translation of temporal objectives into immediate feedback signals that guide learning. We demonstrate the effectiveness of our approach through extensive experiments on an inverted pendulum and a lunar lander environments, showing improvements in both learning speed and final performance compared to standard PPO and offline-only approaches.

Control of Microrobots Using Model Predictive Control and Gaussian Processes for Disturbance Estimation

This paper presents a control framework for magnetically actuated micron-scale robots ($\mu$bots) designed to mitigate disturbances and improve trajectory tracking. To address the challenges posed by unmodeled dynamics and environmental variability, we combine data-driven modeling with model-based control to accurately track desired trajectories using a relatively small amount of data. The system is represented with a simple linear model, and Gaussian Processes (GP) are employed to capture and estimate disturbances. This disturbance-enhanced model is then integrated into a Model Predictive Controller (MPC). Our approach demonstrates promising performance in both simulation and experimental setups, showcasing its potential for precise and reliable microrobot control in complex environments.

Interpretable Generative Adversarial Imitation Learning

Imitation learning methods have demonstrated considerable success in teaching autonomous systems complex tasks through expert demonstrations. However, a limitation of these methods is their lack of interpretability, particularly in understanding the specific task the learning agent aims to accomplish. In this paper, we propose a novel imitation learning method that combines Signal Temporal Logic (STL) inference and control synthesis, enabling the explicit representation of the task as an STL formula. This approach not only provides a clear understanding of the task but also allows for the incorporation of human knowledge and adaptation to new scenarios through manual adjustments of the STL formulae. Additionally, we employ a Generative Adversarial Network (GAN)-inspired training approach for both the inference and the control policy, effectively narrowing the gap between the expert and learned policies. The effectiveness of our algorithm is demonstrated through two case studies, showcasing its practical applicability and adaptability.

Learning Robust and Correct Controllers from Signal Temporal Logic Specifications Using BarrierNet

In this paper, we consider the problem of learning a neural network controller for a system required to satisfy a Signal Temporal Logic (STL) specification. We exploit STL quantitative semantics to define a notion of robust satisfaction. Guaranteeing the correctness of a neural network controller, i.e., ensuring the satisfaction of the specification by the controlled system, is a difficult problem that received a lot of attention recently. We provide a general procedure to construct a set of trainable High Order Control Barrier Functions (HOCBFs) enforcing the satisfaction of formulas in a fragment of STL. We use the BarrierNet, implemented by a differentiable Quadratic Program (dQP) with HOCBF constraints, as the last layer of the neural network controller, to guarantee the satisfaction of the STL formulas. We train the HOCBFs together with other neural network parameters to further improve the robustness of the controller. Simulation results demonstrate that our approach ensures satisfaction and outperforms existing algorithms.

Efficient LQR-CBF-RRT*: Safe and Optimal Motion Planning

Control Barrier Functions (CBF) are a powerful tool for designing safety-critical controllers and motion planners. The safety requirements are encoded as a continuously differentiable function that maps from state variables to a real value, in which the sign of its output determines whether safety is violated. In practice, the CBFs can be used to enforce safety by imposing itself as a constraint in a Quadratic Program (QP) solved point-wise in time. However, this approach costs computational resources and could lead to infeasibility in solving the QP. In this paper, we propose a novel motion planning framework that combines sampling-based methods with Linear Quadratic Regulator (LQR) and CBFs. Our approach does not require solving the QPs for control synthesis and avoids explicit collision checking during samplings. Instead, it uses LQR to generate optimal controls and CBF to reject unsafe trajectories. To improve sampling efficiency, we employ the Cross-Entropy Method (CEM) for importance sampling (IS) to sample configurations that will enhance the path with higher probability and store computed optimal gain matrices in a hash table to avoid re-computation during rewiring procedure. We demonstrate the effectiveness of our method on nonlinear control affine systems in simulation.