A.1 Rotations in 2 dimensions In 2-dimensional settings, there exists a single scalar angular position, the yaw angle θ. In order to perform the transformation, we have to express the angular positions in a format suitable for linear transformations; we do so by transforming them to rotation matrices, perform a matrix multiplication, and then transform the angular positions back to angle format. In 2 dimensions, we use eq. After the rotation, we can convert them back to angle format using the 2-argument arc-tangent function: θ = atan2(sinθ,cosθ) (14) Simplified rotations In 2 dimensions, the computations can be simplified since rotations commute. First, we show that chained rotations result in angle addition/subtraction, that is: Q(θi) Q(θj) = cosθi sinθi sinθicosθi cosθj sinθj sinθjcosθj (15) = cosθicosθj sinθisinθj cosθisinθj sinθicosθj sinθicosθj +cosθisinθj sinθisinθj +cosθicosθj (16) = cos(θi +θj) sin(θi +θj) sin(θi +θj) cos(θi +θj) (17) = Q(θi +θj) (18) Following the same approach, we compute the inverse rotation: Q (θi) Q(θj) = Q( θi) Q(θj) = Q(θj θi) (19) Thus, instead of rotating the angular positions (expressed in rotation matrix form) using the rotation matrix Q, in practice we perform the transformation directly to the angles via addition/subtraction, and replace the matrix Qwith the identity matrix I1 1.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

The rapid advancement in physical human-robot interaction (HRI) has accelerated the development of soft robot designs and controllers. Controlling soft robots, especially soft hand grasping, is challenging due to their continuous deformation, motivating the use of reduced model-based controllers for real-time dynamic performance. Most existing models, however, suffer from computational inefficiency and complex parameter identification, limiting their real-time applicability. To address this, we propose a paradigm coupling Pseudo-Rigid Body Modeling with the Logarithmic Decrement Method for parameter estimation (PRBM plus LDM). Using a soft robotic hand test bed, we validate PRBM plus LDM for predicting position and force output from pressure input and benchmark its performance. We then implement PRBM plus LDM as the basis for closed-loop position and force controllers. Compared to a simple PID controller, the PRBM plus LDM position controller achieves lower error (average maximum error across all fingers: 4.37 degrees versus 20.38 degrees). For force control, PRBM plus LDM outperforms constant pressure grasping in pinching tasks on delicate objects: potato chip 86 versus 82.5, screwdriver 74.42 versus 70, brass coin 64.75 versus 35. These results demonstrate PRBM plus LDM as a computationally efficient and accurate modeling technique for soft actuators, enabling stable and flexible grasping with precise force regulation.

This paper presents the Coordinate Heart System (CHS), a geometric framework for emotion representation in artificial intelligence applications. We position eight core emotions as coordinates on a unit circle, enabling mathematical computation of complex emotional states through coordinate mixing and vector operations. Our initial five-emotion model revealed significant coverage gaps in the emotion space, leading to the development of an eight-emotion system that provides complete geometric coverage with mathematical guarantees. The framework converts natural language input to emotion coordinates and supports real-time emotion interpolation through computational algorithms. The system introduces a re-calibrated stability parameter S in [0,1], which dynamically integrates emotional load, conflict resolution, and contextual drain factors. This stability model leverages advanced Large Language Model interpretation of textual cues and incorporates hybrid temporal tracking mechanisms to provide nuanced assessment of psychological well-being states. Our key contributions include: (i) mathematical proof demonstrating why five emotions are insufficient for complete geometric coverage, (ii) an eight-coordinate system that eliminates representational blind spots, (iii) novel algorithms for emotion mixing, conflict resolution, and distance calculation in emotion space, and (iv) a comprehensive computational framework for AI emotion recognition with enhanced multi-dimensional stability modeling. Experimental validation through case studies demonstrates the system's capability to handle emotionally conflicted states, contextual distress factors, and complex psychological scenarios that traditional categorical emotion models cannot adequately represent. This work establishes a new mathematical foundation for emotion modeling in artificial intelligence systems.

Additionally, some digitizers capture other function-based parameeters, such as the vertical pressure exerted by the pen tip, azimuthal and altitude angles of the pen, and even the pen's in-air trajectory. As a physiological biometric trait, a signature is used in various applications, including access control, commercial transactions, document forgery detection, and the provision of evidence in legal scenarios such as the verification of last wills [9]. In biometrics, where impostors may attempt to forge signatures with varying degrees of skill, robust verification methods are crucial. Since the execution of a signature inherently involves movements of the hand, arm, and forearm, it is hypothesized that these motions may contain kinematic and dynamic unique characteristic of the signer [7]. From a kinematic perspective, this action can be characterized by the arm's angular position, θ(t), and angular velocity, ω(t). Dynamically, these movements are facilitated by force torques, τ(t), applied at the joints. One method used to obtain this valuable biomechanical information involves a physical robot programmed to mimic the act of signing. While a robot's ability to accurately replicate these movements depends on its configuration, working area, and degrees of freedom, it can effectively capture kinematic and dynamic features during the process. However, accessing these robots is costly and cumbersome.

Empowering embodied agents, such as robots, with Artificial Intelligence (AI) has become increasingly important in recent years. A major challenge is task open-endedness. In practice, robots often need to perform tasks with novel goals that are multifaceted, dynamic, lack a definitive "end-state", and were not encountered during training. To tackle this problem, this paper introduces \textit{Diffusion for Open-ended Goals} (DOG), a novel framework designed to enable embodied AI to plan and act flexibly and dynamically for open-ended task goals. DOG synergizes the generative prowess of diffusion models with state-of-the-art, training-free guidance techniques to adaptively perform online planning and control. Our evaluations demonstrate that DOG can handle various kinds of novel task goals not seen during training, in both maze navigation and robot control problems. Our work sheds light on enhancing embodied AI's adaptability and competency in tackling open-ended goals.

A low-cost measurement system using filtering of measurements for two-wheeled balancing robot stabilisation purposes has been addressed in this paper. In particular, a measurement system based on gyroscope, accelerometer, and encoder has been considered. The measurements have been corrected for deterministic disturbances and then filtered with Kalman, $\alpha$-$\beta$ type, and complementary filters. A quantitative assessment of selected filters has been given. As a result, the complete structure of a measurement system has been obtained. The performance of the proposed measurement system has been validated experimentally by using a dedicated research rig.

The safety-critical nature of vehicle steering is one of the main motivations for exploring the space of possible cyber-physical attacks against the steering systems of modern vehicles. This paper investigates the adversarial capabilities for destabilizing the interaction dynamics between human drivers and vehicle haptic shared control (HSC) steering systems. In contrast to the conventional robotics literature, where the main objective is to render the human-automation interaction dynamics stable by ensuring passivity, this paper takes the exact opposite route. In particular, to investigate the damaging capabilities of a successful cyber-physical attack, this paper demonstrates that an attacker who targets the HSC steering system can destabilize the interaction dynamics between the human driver and the vehicle HSC steering system through synthesis of time-varying impedance profiles. Specifically, it is shown that the adversary can utilize a properly designed non-passive and time-varying adversarial impedance target dynamics, which are fed with a linear combination of the human driver and the steering column torques. Using these target dynamics, it is possible for the adversary to generate in real-time a reference angular command for the driver input device and the directional control steering assembly of the vehicle. Furthermore, it is shown that the adversary can make the steering wheel and the vehicle steering column angular positions to follow the reference command generated by the time-varying impedance target dynamics using proper adaptive control strategies. Numerical simulations demonstrate the effectiveness of such time-varying impedance attacks, which result in a non-passive and inherently unstable interaction between the driver and the HSC steering system.

Integrating sensors into rotational mechanisms could make it possible for engineers to build smart hinges that know when a door has been opened, or gears inside a motor that tell a mechanic how fast they are rotating. MIT engineers have now developed a way to easily integrate sensors into these types of mechanisms, with 3D printing. Even though advances in 3D printing enable rapid fabrication of rotational mechanisms, integrating sensors into the designs is still notoriously difficult. Due to the complexity of the rotating parts, sensors are typically embedded manually, after the device has already been produced. However, manually integrating sensors is no easy task.