Deep Neural Networks (DNNs), used for perception, policy learning, or state estimation, abound in many contemporary AI-driven robot control systems. Being very opaque “Black boxes,” DNNs complicate greatly the formal verification of their behavior [8].

Problem: How can we mathematically prove that a neural network controller will always operate within specified safety boundaries, avoid collisions, or maintain stability under all permissible inputs and environmental conditions [9]?

Challenges: Conventional formal verification techniques are impractical given the non-linear, high-dimensional character of DNNs and their sensitivity to hostile attacks [10]. New mathematical methods are required to examine reachable sets of DNN-controlled systems, check Lipschitz continuity features, and formulate probabilistic safety promises [10]. The DNN-controlled system entails creating strong techniques for measuring uncertainty flow across neural networks and incorporating them into Control Lyapunov Functions (CLFs) or Control Barrier Functions (CBFs) [10], [11].

Sensor noise, model errors, unexpected disturbances, and limited knowledge all contribute to the inherent uncertainty of real-world situations. Reliable operation is required for AI-driven robots despite this variability [12], [13].

Problem: How can we accurately quantify and propagate uncertainties through complex AI models and control loops [14], ensuring that decision-making accounts for these uncertainties in a principled manner? Bayesian approaches present an intriguing path, but their computational difficulty is their scaling to high-dimensional robot states [15] and deep learning models. Still, an open area is developing tractable solutions for probabilistic inference [16], resilient state estimation (e.g., robust Kalman filters, particle filters for non-Gaussian uncertainties), and decision-making under extreme uncertainty (e.g., using robust optimization or minimax control). An open area covers mathematically, defining how downstream control activities are affected by perception inaccuracies [16].

Particularly deep learning models [17], AI systems are susceptible to adversarial attacks, in which case tiny, barely noticeable changes to inputs might produce very erroneous results. For robots, this might show up as misreading instructions, misidentifying things, or sensing phantom barriers [18].

Problem: How can we build AI-driven robot control systems that are certainly robust against adversarial perturbations in their sensor inputs or internal states [18]?

Challenges: Modern adversarial training approaches might produce few assurances and could lower performance on clean data [19]. Understanding the geometry of adversarial instances in high-dimensional state spaces, creating certified robustness solutions for robotic applications, and designing control laws that are automatically resistant to such assaults require novel mathematical frameworks [20]. A novel mathematical framework investigates relations between control theory, game theory, and adversarial machine learning [21].

Although RL, has had outstanding performance in simulated environments, its use on actual robots is sometimes hindered by great sample inefficiency [3], [4]. Training, a robot in the actual world is expensive, labor-intensive, and possibly dangerous [3], [4].

Problem: How can we create mathematically grounded RL systems that learn optimal or near-optimal control policies with much less real-world contact [22]?

Challenges: This covers research into model-based RL (Learning dynamics models), off-policy learning, transfer learning from simulation to reality (Sim-to-real), and efficient exploration strategies [23]. Mathematically means improving knowledge of the circumstances for successful policy transfer, developing tighter bounds on sample complexity [24], and creating best experimental design plans for robot learning. Active research fields include Bayesian optimization, meta-learning, and information-theoretic approaches to exploration.

Robots used in the real world need to constantly change to new circumstances, dynamic surroundings, and fresh jobs without forgetting earlier acquired abilities (Catastrophic forgetting) [25].

Problem: How can we mathematically model and solve the issue of lifelong learning for robot control [26], therefore allowing continuous adaptation and skill acquisition without performance drop on past tasks? Challenges: This calls for fresh mathematical models for knowledge representation, memory management [27], and transfer learning with incremental policy and model updating. Relevant are methods from biologically inspired neural architectures, concept drift adaptation [28], and online learning. It is essential to create metrics and theoretical guarantees for measuring and guaranteeing good transfer and to minimize catastrophic forgetting in robotic applications [29].

One of the main drawbacks of modern AI systems is their weak generalization to data or circumstances much different from their training distribution (Out-of-Distribution (OOD)) [8]. For robots, this means that a policy developed in one context might perform horrifically in another rather similar environment [30]. Problem: How can we mathematically characterize and enhance the generalization abilities of AI-driven robot controllers to fresh [30], unfamiliar settings and duties?

Challenges: This calls for a deeper knowledge of the inductive biases of learning algorithms [31], the intrinsic dimensionality of robotic tasks, and the creation of domain adaption methods with robust theoretical guarantees. Mathematical bases for developing more generalizable robot behaviors can be found in causal inference, invariant learning, and robust optimization [32].

Robots must correctly infer and project human intentions [34], goals, and future actions for efficient teamwork.

Problem: How can we enable proactive and cooperative robot behavior by means of strong mathematical models for real-time human intent inference [34], particularly in uncertain or partially visible situations? Challenges: Major mathematical problems are quantifying the uncertainty in intent predictions and creating control policies resistant to misinterpretations [34]. This also covers knowledge of cognitive states and human tastes [34].

Many HRI situations include shared control between the person and the robot [35], or the degree of robot autonomy changes depending on context.

Problem: How can we mathematically devise optimal control strategies for shared autonomy systems, guaranteeing safety [36], efficiency, and user satisfaction that flawlessly mix human input with robot autonomy?

Challenges: In this include human factors, optimal control, and dynamic system modeling. Mathematical models are required for human trust, cognitive load, and error propagation in shared control loops [18]. This also entails creating arbitration systems and hand-over procedures with official assurances [37].

Of first importance is guaranteeing honesty [38], responsibility, and fairness in robot behavior.

Problem: Directly into the mathematical formulation of robot control objectives and learning algorithms [38], how can we embed ethical principles and fairness limitations?

Challenges: This is a developing but essential field. It entails transforming abstract ethical ideas into measurable mathematical limits (e.g., ensuring non-discrimination, minimizing harm, maximizing societal benefit) [39]. Mathematical limits may include constrained optimization, multi-objective optimization, and the integration of social welfare functions into control design [39].

Often in a dispersed way without a centralized coordinator [40], many real-world projects involve teams of robots working together to meet shared objectives.

Problem: How can we create scalable and strong mathematical frameworks for decentralized control and coordination of big multi-robot systems [40], therefore guaranteeing emergent desirable behaviours and preventing undesirable ones?

Challenges: In this include swarm intelligence, graph theory, game theory, and distributed optimization [41]. A major mathematical difficulty is guaranteeing stability, convergence, and fault tolerance in decentralized learning and control techniques [42]. This also covers constrained control, task delegation, and resource distribution among communications [43].

Many robotic systems have hybrid dynamics, that is, a mix of continuous physical movement and discrete logical transitions (e.g., switching between modes, contact events) [44].

Problem: Especially when AI components control the discrete transitions [45], how can we create mathematically precise techniques for designing and validating controllers for hybrid robotic systems? Challenges: This calls for formal methods, discrete event systems [46], and continuous control theory. Key mathematical issues are guaranteeing Zeno behavior avoidance [45], stability across mode changes, and robustness to uncertainties in event detection. Event-triggered control [47] where control updates occur only when needed provides efficiency but complicates stability analysis.

Soft robots' innate compliance and infinite-dimensional state spaces call into question conventional rigid-body control approaches.

Problem: Particularly when AI is used to learn their intricate, non-linear dynamics [47], how can we create mathematical models and control techniques for very deformable soft robots?

Challenges: This combines continuum mechanics, functional analysis, and innovative techniques for state estimation and control of high-dimensional [48], non-linear systems. Significant mathematical difficulties arise in learning correct forward and inverse models for soft robots and developing controllers able to use their compliance for safe interaction.

To create a complete picture of their surroundings [50], robots use several sensors (Cameras, LiDAR, IMUs). AI-driven perception systems can extract high-level semantic information.

Problem: How can we mathematically fuse heterogeneous sensor data, including semantic information, in a robust and computationally efficient manner to provide accurate and reliable state estimates for control [51]? Problems: This entails robust estimation approaches, deep learning for feature extraction, and probabilistic graphical models. An open field is quantifying the uncertainty in semantic labels and integrating it into state estimation frameworks (e.g., semantic SLAM) [52]. Equally important is strong handling of sensor malfunctions, occlusions, and new objects.

Many robotic jobs require dealing with limited knowledge [53] regarding the surroundings or the robot's condition.

Question: How can we create mathematically correct techniques for optimal state estimation and control under high partial observability, especially when AI models are applied to predict missing data [53]?

Challenges: This entails Partially Observable Markov Decision Processes (POMDPs), but realistically robot applications would find scaling them impractical [54]. There are needed approximate inference techniques, active perception strategies, and information-theoretical approaches to sensing [55].

For debugging, certification, and human supervision, knowing why an AI-driven robot makes a specific choice is essential [57].

Problem: How can we create mathematical models to ensure that human operators can understand and explain the decision-making processes of AI-driven robot controllers [58]?

Challenges: This goes beyond just picturing neural network activations. It entails creating techniques to draw human-understandable rules or explanations from intricate policies [59], attribute control actions to certain inputs, and measure the "Reasonableness" of a robot's behavior. Intricate policies may entail symbolic AI integration, counterfactual explanations, or saliency maps [11].

A major difficulty is deploying sophisticated AI models on limited robot hardware (e.g., limited CPU, GPU, memory, power) [61].

Problem: While maintaining performance and safety guarantees, how can we create mathematically optimal techniques for compressing, quantizing, and optimizing AI models for effective run on edge robotic systems [32]?

Challenges include hardware-aware co-design, efficient architectures (e.g., MobileNets), quantization, and neural network pruning [62]. Important establishing theoretical limits on the performance loss caused by model compression and guaranteeing real-time performance.

Many control challenges entail real-time, under tight deadlines, resolution of sophisticated optimization issues [63].

Problem: Often with AI-driven parts, how can we create mathematically efficient algorithms for real-time optimal control and motion planning capable of handling high-dimensional state spaces and non-linear dynamics [64]?

Challenges: This calls for progress in approximate dynamic programming, Model Predictive Control (MPC), and numerical optimization [65]. Active areas include utilizing AI for warm-starting optimization issues [66], developing effective solvers, or directly learning control policies satisfying real-time limitations.