This article proposes a Nash game-optimized adaptive robust control framework for bipedal parallel wheel-legged robots. Specifically, the framework targets a balance between tracking accuracy and control effort under underactuation, constraint coupling, and substantial uncertainties. To address these challenges, a constraint following adaptive robust controller is adopted, while gain selection is posed as a two-player Nash game between performance and cost objectives. Consequently, the resulting equilibrium yields controller gains without heuristic tuning. Furthermore, Lyapunov analysis establishes uniform ultimate boundedness of the closed-loop trajectories under bounded disturbances and model errors. Finally, numerical simulations verify that the proposed approach achieves a superior balance, concurrently enhancing tracking accuracy while significantly reducing control effort compared to conventional methods.

There are always multi-stage and multi-type precision requirements for robotic arms in industrial manufacturing. Due to uncertainties and disturbances, the tracking error is forced to exceed the time-varying precision boundary and the system is even broken. Therefore, an intelligent optimization framework of fuzzy adaptive prescribed time control for robotic arms with multi-stage precision is developed in this paper. Firstly, the multi-stage precision requirements are explained as time-varying piecewise inequality constraints of the tracking error. Based on homeomorphic mapping, a model reconstruction method is proposed by designing the transformation function to form a reconstructed system. Then, based on the constraint force analysis of the reconstructed system, a leaky adaptive prescribed time control method is proposed, which simultaneously handles uncertainties, disturbances and inequality constraints. The error convergence is achieved within a prescribed time by setting piecewise constraints. The overcompensation is avoided via the leaky-type adaptive law. The proposed method is proven to be uniformly bounded and uniformly ultimately bounded. Furthermore, an optimization strategy is designed based on fuzzy set theory to optimize system performance and control cost, forming a set of general trade-off rules. Finally, the effectiveness of the proposed method is verified. The results show that the time-varying piecewise inequality constraints are fully satisfied at low control cost.

Human-machine systems (HMSs) are dedicated to integrating intelligent human decisions with machine operations to achieve synergistic operational functionality. We focus on constraint-following control within the HMS, considering potential uncertainties, environmental disturbances, and limited operational space. A hierarchical hybrid control scheme is proposed, consisting of a preemption algorithm and a human decision algorithm. Specifically, the preemption algorithm relies on online state feedback from mechanical system signals, such as position and velocity, which can be implemented in hardware or software; the human decision algorithm takes inputs from electrophysiological signals or language commands. In this development, a Lagrangian density function is constructed that integrates optimal decision making with a uniformly bounded threshold. The intelligent decision-making problem in the HMS is creatively analyzed and mathematically solved leveraging variational calculus, resulting in the analytical expression of the optimal membership function associated with human decisions. Furthermore, a series of numerical simulation experiments are conducted using a bionic upper-limb prosthetic system as an example, and the comparison results demonstrate the superiority and effectiveness of the proposed method.

Trajectory tracking in bimanual humanoid robots, whose closed-chain kinematic structures inherently amplify the effects of modeling errors, external disturbances, and time-varying parameters, is a challenging task. To address this, we reformulate the dual-arm tracking task as a servo constraint-following problem and derive the system dynamics under approximate constraints using the Udwadia-Kalaba method. The humanoid system is modeled as a constrained mechanical structure subjected to fast-varying, bounded uncertainties with unknown limits. On this basis, we propose a robust control framework that guarantees both uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the tracking error, ensuring stability and performance even under severe parametric and dynamic uncertainties. To reconcile the trade-off between transient dynamics and steady-state accuracy—essential for service-oriented tasks such as door opening or coffee pouring—we integrate a Nash equilibrium-based optimization mechanism into the controller design. By formulating a two-player non-cooperative game over the controller's key tuning parameters, we analytically derive the existence, uniqueness, and closed-form solutions of the game, achieving an optimal balance between competing objectives. Comprehensive simulations on a reduced-order bimanual humanoid model validate the proposed approach, demonstrating superior tracking accuracy, disturbance rejection, and energy efficiency compared to benchmark methods. The proposed strategy offers a theoretically grounded and practically implementable solution for robust, constraint-compliant humanoid manipulation.

This article studies a prescribed performance adaptive robust control (PPARC) scheme for uncertain robotic manipulators. First, a state transformation is introduced to embed the predefined output constraints into the trajectory tracking servo constraints. Second, a PPARC is designed to fulfill the reference trajectory tracking and ensure the tracking errors within the predefined output constraints regardless of uncertainty and initial condition deviations. Then, Lyapunov stability analysis is conducted to prove the uniform boundedness and uniform ultimate boundedness of the tracking error. Moreover, the optimization of the control parameter is translated into an optimal design problem, and a fuzzy-based cost function is proposed for the optimal design problem. The existence and uniqueness of the solution to the optimal design problem are theoretically proved. Finally, the effectiveness and superiority of the proposed control scheme are demonstrated based on simulation studies.

A confined space is an area in an industrial facility that has limited access and allows only restricted movement due to physical constraints. Confined spaces often require special safety precautions and may be subject to specific regulations to ensure the well-being of workers. Flexible manufacturing cells typically work in confined spaces in order to increase efficiency and decrease cost. The operation can be further complicated if uncertainty is involved. We propose an adaptive robust controller for uncertain mechanical systems in a confined space to enhance system performance while ensuring system safety. The design procedure consists of five phases: constraint-following formulation, fuzzy uncertainty prescription, diffeomorphism transformation, adaptive robust control design, and Nash game-based optimization. The effectiveness of the control scheme is demonstrated by numerical simulation experiments for a humanoid robot arm.

The inequality constraints, system nonlinearities, parameter uncertainties, and external disturbances are always unavoidable in practical mechanical systems. This article proposes an adaptive robust control (ARC) algorithm from the view of servo constraint following to tackle the control problem of mechanical systems subject to the above factors. For the inequality constraints, a creative diffeomorphism which could convert the two-sided bounded state variables to the unbounded ones is explored, which could render the transformed nonlinear system free from inequality constraints. For the system uncertainties, a leakage-type ARC algorithm is developed, which could render the system the practical stability. The permanent magnet linear motor (PMLM) system is utilized as a typical application to verify the proposed state transformation and ARC approach. Numerical simulations show that the displacement of the PMLM system could well track the desired trajectory without violating the given bound line.

While designing control for a class of underactuated mechanical systems (UMSs), the uncertainty and the prescribed nonholonomic tracking trajectories should be taken into consideration. Uncertainty considered in this article is time varying and bounded, and the bound of uncertainty is described using the fuzzy set theory, namely, fuzzy UMSs. An analytical dynamics-based view is taken in which the prescribed tracking trajectories are viewed as servo constraints which can be linear, nonlinear, holonomic, and nonholonomic. Using this view, a novel closed form solution of adaptive robust control is found with leakage type adaptive law to guarantee deterministic system performance, including uniform boundedness and uniform ultimate boundedness. In order to find the optimal dual gain parameters of the designed control, a two-player cooperative game is proposed for which the Pareto optimality can always be guaranteed. The effectiveness of the proposed control is shown through numerical simulation of a two-wheeled inverted pendulum vehicle.

In this article, we apply a high-order control to a dynamical system with uncertainty. There are two characteristics. First, the uncertain part, which is time-varying but bounded, is described in a fuzzy aspect. Specifically, the uncertainty lies within a fuzzy set and the bound is regarded as a fuzzy number. Second, the systems are uniformly bounded and uniformly ultimately bounded with a deterministic controller based on the Lyapunov theory. To obtain better system performance and lower control input, we apply the noncooperative game theory to optimize the parameters by establishing a Nash game. Then, the D-operation is proposed for the uncertainty related to fuzzy numbers. Finally, we perform the numerical simulations of the steering-by-wire system for verification.

This article investigates the trajectory tracking problem of underactuated mechanical systems (UMSs) with companion nonholonomic servo constraints and uncertainties. For such motion tasks, the existing approaches in the literature attempt unrealistically to furnish a reliable closed-form solution, rendering it difficult to have high-quality tracking performance with theoretical support. In addition, the uncertainties typically pose substantial difficulty in the controller synthesis. Here, by invoking the methodology of fuzzy sets, the uncertainties in the UMSs are elegantly represented; and with this, the formulation becomes such that a closer link between the uncertain dynamical model of the UMSs and the real world is established. The reference trajectories are regarded appropriately as servo constraints, and subsequently an adaptive robust controller is designed to accomplish the trajectory tracking task from a specific viewpoint of servo constraint tracking. As supported by rigorous proofs, the closed-form solution to the proposed controller is obtained with guaranteed Lyapunov stability. Leveraging on the closed-form solution, the global optimizer to the controller gain parameter can be determined, which is shown to exhibit several important properties including existence and uniqueness. Finally, a numerical example is presented to demonstrate the effectiveness of the designed method.

This article addresses the problems of the adaptive robust control design and the parameters optimization for a class of underactuated mechanical systems. We describe the uncertainty by using the fuzzy set theory to form the fuzzy dynamical systems. A novel adaptive robust control with two tunable gain parameters is designed based on the formed system, and the nonholonomic servo constraints are also considered in the control design. By employing the principle of the Nash game theory, we establish an optimization method for the two gain parameters, and the Nash equilibrium orients the optimal values of the two gain parameters. Finally, the effectiveness of the designed optimal control is shown by a numerical simulation.

In this paper, we propose a fuzzy approach for optimal robust control design of an automotive electronic throttle (ET) system. Compared with the conventional ET control systems, we establish the fuzzy dynamical model of the ET system with parameter uncertainties, nonlinearities, and external disturbances, which may be nonlinear, (possibly fast) time varying. These uncertainties are assumed to be bounded, and the knowledge of the bound only lies within a prescribed fuzzy set. A robust control that is deterministic and is not the usual if-then rules-based control is presented to guarantee the controlled system to achieve the deterministic performance: uniform boundedness and uniform ultimate boundedness. Furthermore, a fuzzy-based system performance index including average fuzzy system performance and control cost is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the fuzzy-based performance index. With this optimal robust control, the performance of the fuzzy ET system is both deterministically guaranteed and fuzzily optimized.

We first investigate the fundamental properties of the mechanical system as related to the control design. Then a new robust control is proposed for mechanical systems with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty in the mechanical system. The desirable system performance is deterministic. The proposed control is deterministic and is not the usual if-then rules-based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the Lyapunov minimax approach. The resulting control design is systematic and is able to render the deterministic performance. A mechanical system is chosen for demonstration.

We first investigate the fundamental properties of the mechanical system related to the control design. Then, a new optimal robust control is proposed for mechanical systems with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty in the mechanical system. The desirable system performance is deterministic (assuring the bottom line) as well as fuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if-then rule based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the Lyapunov minimax approach. A performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. The resulting control design is systematic and is able to guarantee the deterministic performance, as well as minimizing the cost. In the end, a mechanical system is chosen for demonstration.