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.

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.

An optimal prescribed performance control with fixed-time for fuzzy mechanical systems is explored. Specifically, a novel bounded performance function is proposed, which pre-assigns the convergence time, transient convergence trend (dynamic, variable and gentle initial stage) and steady-state tracking accuracy. More possibilities for transient performance of fuzzy systems are developed. Then, fuzzy set theory is introduced to describe system uncertainties. A fuzzy mechanical system with prescribed performance is constructed in the homeomorphism mapping space. An adaptive control method with finite-time stability is proposed. Thus, the preassigned performance is met through a two-layer collaborative convergence characteristic, rather than relying solely on performance constraint. High-order adaptive laws reduce control costs and avoid overcompensation. Control design is always deterministic rather than based on fuzzy rules. A more effective way is reflected in fuzzy-based optimization. Based on fuzzy sets to measure uncertainty, a cooperative game optimization strategy is designed to obtain the optimal decision for multiple objectives and control parameters. The best combination of performance and cost is solved. The effectiveness of the proposed method is verified via the steer-by-wire system.

This work investigates a game-theoretic optimization approach towards robust control of uncertain dynamical systems with state and input constraints. The uncertainty involved is possibly rapidly time-varying but bounded within a prescribed fuzzy set. For this, the associated fuzzy dynamical system is appropriately established and constructed based on fuzzy set theory. To cope with the bounded state and input constraints, a novel state-and-input diffeomorphism technique is proposed, where a transformed system is formulated such that the prescribed inequality constraints are innovatively merged into the stabilization and trajectory tracking problems. Furthermore, a diffeomorphism-based robust control (DBRC) strategy is developed to ensure the uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the transformed system. Under this proposed control architecture, the constraint satisfaction of the original system is thus always analytically ensured based on the rigorous properties of the diffeomorphism technique. The resulting control parameter optimization problem then has to take into account the multiple considerations (and compromise) amongst the factors of the steady-state performance; the finite convergence time; and the control effort. For this, a two-player Nash game is formulated and solved in an effective manner. The Nash equilibrium is obtained and the existence of the solution is also proved theoretically. With this methodology, and with the resulting attainment of the desired Nash equilibrium, the attendant outcome of superior system performance is achieved. Finally, numerical simulations on a steer-by-wire (SBW) system demonstrate the effectiveness of the proposed approach.

A Stackelberg game-based design for an adaptive robust control for the fuzzy uncertain underactuated mechanical systems (UMSs) is proposed. The emphasis is on fuzzy-based uncertainty and inequality constraint. The uncertainty is time varying and bounded within a prescribed fuzzy set. For the inequality constraint, we creatively have it merge into constraint-following performance by a diffeomorphism technique. An adaptive robust control strategy is then proposed. Deterministic performance is guaranteed provided the control design parameters are within feasible regions. To further enhance the performance, we introduce a two-player Stackelberg game setting. The optimal choice of design parameters can be solved. The feasibility of this design is demonstrated on an autonomous wheeled mobile robot (AWMR), which is confined in a bounded space.

This article develops a diffeomorphism-based robust bounded control (DRBC) for the permanent magnet linear synchronous motor system subject to inequality constraints (i.e., bounded input and position constraints) and uncertainties. The uncertainties, including parameter uncertainties and external disturbances, are potentially nonlinear and fast time varying. The bound of the uncertainty is described by a fuzzy set. To overcome the bounded input constraint, a robust bounded control is proposed based on a novel input diffeomorphism scheme, which is in a deterministic form and not if–then rule based. Furthermore, to overcome the bounded position constraint, a transformed system is formulated by a state diffeomorphism scheme, which transforms the bounded state-constrained system to an unconstrained one. Thus, the output of the controlled system can be restricted to a prescribed range. The DRBC guarantees both the uniform boundedness and the uniform ultimate boundedness of the transformed system. A fuzzy performance index, which combines the steady-state performance (the average fuzzy performance) and control effort, is then established based on the fuzzy description of uncertainty. As a result, the design parameter optimization can be solved by minimizing the performance index. Experimental results demonstrate that the DRBC is of superior tracking performance and robustness without violating the prescribed inequality constraints.

This article investigates the constraint following control and cooperative game-based optimization of the spacecraft rendezvous system with parametric uncertainty and unknown external disturbances. For the nominal system, the constraint force with the analytical expression is implemented by applying Udwadia–Kalada approach. Aiming at uncertainties and initial condition deviations, a flexible robust constraint following control with a self-adjusting term is proposed. The deterministic performance is guaranteed: uniform boundedness and uniform ultimate boundedness. Meanwhile, the control input is always stable in a small range. Further, for improving the system performance with lower control cost, a cooperative game-based optimization scheme is proposed. The Pareto optimality and Pareto frontier are obtained. The effectiveness of the above methods are verified by simulation of spacecraft rendezvous system with two servo constraints.

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 propose a high-order robust control for fuzzy dynamical systems. The time varying but bounded uncertainty in this system is described by the fuzzy set theory. The control is deterministic and is not based on IF-THEN fuzzy rules. By the Lyapunov approach, we prove that the control is able to guarantee uniform boundedness and uniform ultimate boundedness. In addition, the tunable parameters in the high-order control are regarded as two players in a cooperative game. Two cost functions are also proposed based on the two players. These two cost functions are related to system performance and control cost. Then, the optimal design problem is solved by finding the Pareto-optimality parameters. Numerical simulations are performed 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.

A novel class of high-order robust controls is presented for uncertain fuzzy systems in this article. The optimal tunable parameters are also obtained based on the Stackelberg game theory. First, a dynamical system structure is formulated with uncertainty. The uncertain portion in the system is bounded, nonlinear, and time-varying which is within prescribed fuzzy set. Thus, this is a fuzzy system. Then, the proof based on the Lyapunov minimax approach shows that the novel high-order robust controls are able to assure deterministic system performance, which is uniform boundedness and ultimate uniform boundedness. Furthermore, the optimal choice of the tunable parameters in the control is considered. We creatively apply the Stackelberg strategy to solving the optimization problem. Two parameters are regarded as leader and follower in a sequential game, respectively. Based on the Stackelberg game rules, we are able to design different cost functions for different parameters. The cost functions include both performance measures and control cost. Finally, the simulations of an electronic throttle system are presented for demonstration.

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 article, we are dedicated to coping with the control design for underactuated mechanical system (UMS) with system uncertainties, which is also subject to a set of servo constraints. For the nominal system without any initial condition deviations, this article provides a general model-based servo control with the idea of second-order constraints, which can be derived from both holonomic and nonholonomic ones. For the uncertain system, with the uncertainties being separated into matched portions and mismatched portions, this article provides an adaptive robust control scheme to guarantee the UMS to attain the deterministic performance on the basis of nominal control. The leakage-type adaptive law is to size up the unknown uncertainty bounds. The cart-pole system and the rotational-translational actuator system are used to verify the availability of the proposed adaptive robust control.

The optimal design problem of adaptive robust control for fuzzy mechanical systems with uncertainty is investigated in this paper. The uncertainty that may be nonlinear and (possibly fast) time-varying is assumed to be bounded, and the knowledge of the bound only lies within a prescribed fuzzy set. Based on the Udwadia and Kalaba's approach, an adaptive robust controller, which is deterministic and is not the usual if-then rules-based is proposed to render the system to follow a class of prespecified constraints approximately. The adaptive law is of leakage type that can adjust the magnitude of the adaptive parameter based on the nonlinear performance-dependent gain. The resulting controlled system is uniformly bounded and uniformly ultimately bounded, which is proved via the Lyapunov minimax approach. Furthermore, we propose a novel concept: fuzzy confidence to measure the expectation value of a fuzzy number. Then, a fuzzy-based system performance index that includes the expectation value of the uniform ultimate boundedness (the average fuzzy performance) and the control cost is formulated. The optimal design problem associated with the control can then be solved by minimizing the performance index. As a result, the performance of the fuzzy mechanical system is both deterministically guaranteed and fuzzily optimized under this control.

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.