Jul 2026· IEEE Transactions on Neural Networks and Learning Systems· Vol PP, pp. 1-14· 0 citations
Medicine
TL;DR
A variable stiffness Euclidean flow is proposed, Using diffeomorphic mapping, complex demonstrated trajectories are transformed into straight-line trajectories, eliminating the need for the integration process typically required to obtain integral curves.
Abstract
Imitation learning based on dynamical systems (DSs) can generate real-time motion planning with intrinsic stability and robustness, providing significant advantages in highly uncertain dynamic environments. However, most DS approaches tend to deviate from their original integral curves when subjected to perturbations, limiting their applicability in scenarios that require tracking specific reference trajectories or passing through critical path points. To address this issue, this article proposes a variable stiffness Euclidean flow. Using diffeomorphic mapping, complex demonstrated trajectories are transformed into straight-line trajectories, eliminating the need for the integration process typically required to obtain integral curves. Furthermore, a gradient system based on these straight-line trajectories is designed to converge to the demonstrated trajectories, creating symmetric attractor-like behavior around the demonstrated trajectory. When applied to the original system, this approach preserves the desired velocity of the demonstrated trajectory. The effectiveness of the proposed method is demonstrated using standard datasets and real robot experiments. More videos and model details are available on the project website (https://anonymous.4open.science/w/EFVSDS-WebPage-5D06/).
Trajectory planning strongly influences tracking accuracy, actuator demand, and overall execution behavior in robotic manipulators. Classical planners such as cubic, quintic, and trapezoidal profiles are widely used for their simplicity and smoothness, yet they remain purely kinematic and ignore system dynamics and control effort during trajectory generation. As a result, nominally smooth trajectories can lead to inefficient nonlinear execution and increased corrective control action. This paper presents a control-aware optimal trajectory planning framework that explicitly incorporates manipulator dynamics and actuator effort within a finite-horizon formulation. A midpoint linearization strategy is introduced to improve approximation accuracy for large point-to-point motions. In contrast to prior comparisons, the proposed approach enables fair, isolated evaluation of trajectory generation effects under identical closed-loop nonlinear execution conditions. To this end, a unified evaluation framework is developed in which all planners are executed under identical nonlinear dynamics, controller structure, and actuator constraints. Simulations on a nonlinear simplified UR5 manipulator show that the proposed approach consistently reduces tracking error, corrective torque, and closed-loop execution cost compared to classical methods, achieving substantial reductions in actuator effort and execution cost across all evaluated scenarios, demonstrating that kinematic smoothness alone does not ensure dynamically efficient execution.
Controlling spherical robots is challenging due to their nonlinear dynamics, underactuated characteristics, and non-holonomic constraints. These challenges become more pronounced in the presence of parameter variations and external disturbances. To address these issues, this paper proposes an Adaptive Dynamic Programming (ADP)-based control framework for spherical robot dynamics. The stability properties of the proposed method are analyzed using Lyapunov theory. The kinematic control layer is designed based on the feedback linearization approach, while the dynamic controller employs ADP to compensate for uncertainties and disturbances. The effectiveness of the proposed method is evaluated through two simulation case studies involving trajectory-tracking tasks under parametric uncertainties and external disturbances. The simulation results show that the proposed controller is capable of achieving accurate trajectory tracking while maintaining stable system performance. To provide a comparative assessment, a Sliding Mode Control (SMC) scheme is also implemented under the same conditions. The obtained results indicate that the ADP-based controller can reduce tracking errors and power consumption compared with the considered SMC approach. In one case study, the tracking error integral achieved by the ADP controller is approximately 36% lower than that of SMC, while the corresponding power consumption is reduced by about 23%. These results demonstrate the potential of the proposed ADP framework for improving trajectory-tracking performance in spherical robots under uncertain operating conditions.
Hadi Sazgar, Ali Keymasi‐Khalaji, Aliakbar Ghasemzadeh· Journal of Vibration and Con...· 0 citations
Context—Snake-like robots are biomimetic systems that can move effectively in narrow, complex, and restricted environments thanks to their modular and flexible body structures composed of numerous serially connected joints. These characteristics offer significant advantages, particularly in areas such as pipeline inspection, search and rescue operations, industrial maintenance applications, and exploration missions. The multiple degrees of freedom distributed along the body enable the robot to achieve high maneuverability but also make the control problem quite complex. Due to the dynamic interactions between segments, friction-based motion characteristics, and nonlinear system behavior, achieving reliable and accurate trajectory tracking emerges as a significant engineering problem.Objective—In this study, a reinforcement learning (RL) based control method has been developed to solve the trajectory tracking problem for snake-like robots in a two-dimensional plane.Method—In the proposed approach, the robot’s dynamic model was created in the Webots simulation environment, an open-source simulation program, and all training and testing processes were carried out in this environment. During the learning process, policy- based RL algorithms from the Stable-Baselines library were used. In this context, Proximal Policy Optimization (PPO) and three different RL algorithms were used during the training process. To enable the robot to adapt to different orientation scenarios, seven different angles defined in the range of +45 to −45 and trajectories of varying lengths were used. Thus, the goal was for the agent to learn a generalizable control policy not only for a specific trajectory type but also for tracks with different slopes and orientations.Results—The results obtained show that the PPO algorithm produced a higher average reward compared to other methods and exhibited a more stable learning process. After training was completed, the developed method was tested both on trajectories used during the training phase and on previously unseen trajectories. For the 0 trajectory, maximum errors were recorded as 0.093 m and 0.040 m for the x and y axes, respectively. Furthermore, the system exhibited robust generalization capabilities on a +22.5 trajectory, not encountered during the training phase, yielding maximum errors of 0.099 m and 0.052 m.Conclusion—These findings demonstrate that the proposed RL-based control approach can effectively solve the two-dimensional trajectory tracking problem in snake robots. In future studies, the proposed method can be extended to the three-dimensional trajectory tracking problem, or it can be evaluated under more complex conditions, such as scenarios involving obstacles.
Furkan Mezgil, Mustafa Can Bingöl· Pamukkale Üniversitesi Mühen...· 0 citations
This paper presents a Neural Backstepping control strategy for trajectory tracking of a differential-drive mobile robot. The proposed approach combines a dynamic-level backstepping controller with a lightweight single-hidden-layer adaptive neural network to compensate uncertain nonlinear dynamics through online adaptation. The backstepping component provides a Lyapunov-based stabilizing structure, whereas the neural approximator improves tracking performance without requiring deep architectures, offline training stages, or computationally demanding optimization procedures. The adaptive law for the neural output weights is derived from the stability analysis, ensuring bounded closed-loop signals and uniformly ultimately bounded tracking errors in the presence of bounded approximation uncertainties. The controller is evaluated through simulations using four reference trajectories: circular, lemniscate, Lissajous, and waypoint-based paths. The same control gains and neural network configuration are used in all cases, showing that the proposed scheme can track different trajectory geometries without trajectory-specific retuning. The simulation results show satisfactory tracking performance, with position RMSE values below 0.04 m for all evaluated trajectories. These results indicate that the proposed Neural Backstepping controller provides a suitable balance between tracking accuracy, online adaptation capability, and implementation simplicity for differential-drive mobile robot trajectory tracking.
J. Zepeda-Hernández, I. Santos-Ruiz, G. Valencia‐Palomo et al.· Mathematics· 0 citations
Soft robotic snakes, characterized by continuous body deformation, offer a superior platform for investigating the fundamentals of efficient biological locomotion. This paper introduces a mathematical framework for modeling and analyzing the planar rolling gait of these soft robotic snakes. We leverage principles of continuous deformation and distributed contact dynamics to derive an analytical relationship between the joint parameters of the robot and its resulting rolling velocity. To validate the model and enable practical application, we develop a configuration-space closed-loop controller for accurate trajectory tracking. The stability of the proposed controller is formally confirmed using the input-to-state stability criteria. Simulations and experimental validation demonstrate that the controller achieves accurate trajectory tracking and confirm that the measured rolling speed adheres to the theoretical prediction. These results affirm the effectiveness and accuracy of our gait modeling and control strategy for enhancing the locomotion performance and fundamental understanding of soft robotic snakes. Note to Practitioners—Soft robotic snakes offer unique advantages for applications requiring navigation in cluttered or confined environments, yet their highly compliant bodies and non-linear pneumatic actuation have historically made reliable motion control a significant barrier to deployment. This work addresses that practical need by providing an engineering framework that allows practitioners to command the snake robot to roll at a predictable speed and in a predictable direction, overcoming the unreliability of open-loop actuation. The presented geometric model can be used to estimate forward speed based on desired snake parameters, while the core practical contribution is the closed-loop controller that actively compensates for real-world issues. This controller maintains consistent body curvature despite common challenges like sensor noise, fabrication imperfections, material stiffness variations, and pneumatic delays, all of which typically undermine performance on physical hardware. Validation confirms this approach delivers improved trajectory tracking and more repeatable rolling behavior than current methods. Engineers deploying this framework should note its assumption of planar rolling on relatively uniform terrain and reliance on standard IMU sensing and pneumatic actuation, which may limit performance on highly irregular surfaces or when extreme speed in pressure dynamics is required. This foundation is essential for extending soft snake capabilities toward future, complex behaviors like autonomous navigation and obstacle-aware gaits.
Dulanjana M. Perera, Dimuthu D. K. Arachchige, Nathan Byrd et al.· IEEE Transactions on Automat...· 0 citations
Quadrotors (UAVs) are widely used in intelligent inspection, environmental monitoring, and logistics due to their simple structure, strong maneuverability, and vertical take-off and landing capabilities. However, their highly nonlinear, strongly coupled, and highly constrained dynamic characteristics make trajectory tracking control a challenging task. To improve trajectory tracking accuracy and control robustness, this paper proposes a quadrotor trajectory tracking method based on model predictive control (MPC). First, a six-degree-of-freedom dynamic model of the quadrotor is established and linearized with small disturbances to transform it into a state-space model suitable for MPC design. An MPC optimization controller is then constructed, with an objective function that minimizes state error and imposes an input energy penalty, while explicitly considering the system's input and state constraints. Simulation results demonstrate that this method exhibits good tracking accuracy and control smoothness for typical trajectory tracking tasks (such as circular and spiral trajectory tracking). Compared with traditional PID and LQR controllers, the proposed method significantly improves maximum error, mean square error, and interference rejection. This study provides an engineering-feasible optimization control framework for UAV trajectory control.
Fei Peng, Zhong Tao, Qiang Gao et al.· SAE technical paper series· 0 citations