Udwadia Kalaba Research Articles

Formation control of networked mobile robots with weighted directed topology using the Udwadia–Kalaba approach

Formation control of networked mobile robots with weighted directed topology using the Udwadia–Kalaba approach

Multi-stage trajectory tracking control design under comprehensive constraints based on Generalized Udwadia–Kalaba theory

A novel control design based on Generalized Udwadia–Kalaba (GUK) theory is proposed to achieve the trajectory tracking under comprehensive constraints in this paper. To describe the complex task requirements, a constraint framework including equality and inequality constraints is constructed. Thus, the complex trajectory tracking problem is converted into a constraint-following problem. Based on the framework above, GUK equation is introduced to consider the equality and inequality constraints individually. Specifically, the equality constraints are transformed into second-order matrix form, while the inequality constraints are addressed via a diffeomorphism. Then, a novel control method including servo control term and high-order control term is proposed. Through these two control terms, the constraint forces for equality and inequality can be solved separately. The effectiveness of this control method is verified by the stability analysis based on Lyapunov minimax apprach and illustrative application of unmanned surface vehicle (USV). Meanwhile, the superiority of this method is demonstrated through a comparative analysis with Udwadia–Kalaba (UK) method, Linear Quadratic Regulator (LQR) method and sliding mode control (SMC) method.

An improved Udwadia–Kalaba approach for controller design in underactuated mechanical systems

An improved Udwadia–Kalaba approach for controller design in underactuated mechanical systems

Dynamical modeling of non-proportionally damped multibody systems using a modal Udwadia–Kalaba formulation based on complex modes of the dissipative subsystems

This paper presents a modal Udwadia–Kalaba formulation based on complex modal analysis. This alternative formulation is developed in the general case of vibro-acoustic systems with an internal cavity. In a similar manner to the original formulation, the global response of a constrained multibody system is expressed as a sum of its unconstrained response and a corrective term allowing the enforcement of constraints. The use of complex modes of the dissipative substructures has the advantage of leading to a set of ordinary differential equations, regarding the unconstrained response, even in the case of non-classically damped substructures. Moreover, in the frame of experimental substructuring, the estimation of complex modal parameters of a state-space representation is more straightforward than for real ones of the equivalent second order model, hence this alternative formulation is of practical interest in this context. An academic example with non-proportional damping allows to validate the proposed modal formulation. Then two experimentally identified models of a guitar and a harp from Central Africa are coupled to a stiff string whose modal parameters are known from the analytic theory. The guitar model contains an acoustic degree of freedom and allows to illustrate the suitability of the Udwadia–Kalaba formulation to model coupled vibro-acoustic substructures. The harp model includes highly complex modes which allows to highlight the advantage of expressing the coupled response of the global system in terms of its complex modal coordinates.

A generalized Udwadia–Kalaba control design for uncertain belt conveyor systems with inequality constraints

Uncertainty can lead to jitter or overshoot in mechanical systems, necessitating the design of multiple constraints to stabilize them. This paper proposes a control structure based on the generalized Udwadia–Kalaba equation to address these constraints simultaneously. An uncertain dynamical model is developed, incorporating both equality and inequality constraints. By integrating diffeomorphism theory, a robust control strategy is designed to ensure compliance with these constraints. Utilizing the Lyapunov approach, the uniform boundedness and uniform ultimate boundedness of the dynamical system are demonstrated. Finally, the feasibility of the proposed control method is validated through its application to a belt conveyor system.

A Separation Modeling Method for Morphing QUAV: Analytical Solutions for Constraint Forces Under Deformation

Abstract A morphing quadrotor unmanned aerial vehicle (QUAV) possesses the remarkable ability to alter its shape, enabling it to navigate through gaps smaller than its wingspan. However, these deformations result in changes to the system's center of gravity and moment of inertia, necessitating real-time computation of each state's variations. To address this challenge, we propose a dynamic modeling approach based on the Udwadia−Kalaba (U-K) method. The morphing QUAV is divided into three separate subsystems, with the dynamic modeling for each subsystem conducted independently. Subsequently, the QUAV's deformation states and inherent structure are introduced in the form of constraints, and the constrained forces are derived using the U-K equation. By combining these analytical solutions, the model of the QUAV under continuous deformation is obtained. This approach effectively simplifies the modeling computations caused by changes in the system's center of gravity and moment of inertia during deformation. A control approach is proposed to achieve attitude stabilization and altitude control for the morphing QUAV. Ultimately, the stable motion of the morphing QUAV is validated through numerical simulations.

Constraint‐following servo control for uncertain flexible transmission system with damping

AbstractDue to the flexible components, damping, underactuation, and uncertainties, the control for flexible transmission system (FTS) becomes very difficult. A constraint‐following servo control is designed to address this issue. The damping and applied load of system are integrated into the constraints, so that the desired performance could be achieved by accomplishing the constraints with Udwadia–Kalaba (U–K) equation. The system is divided into two subsystems by introducing a designed fictitious control to the load end, and the controllers of subsystems are designed, respectively. Since the accurate value of uncertainty is usually unknown, a corresponding leakage type of adaptive law is designed to estimate the size of its boundary without requiring the uncertainty itself. The effectiveness and superiority of proposed controller are verified by simulations.

Adaptive robust constraint-following control for morphing quadrotor UAV with uncertainty: A segmented modeling approach

During morphing, changes in a quadrotor UAV’s center of gravity and inertia contribute to unstable flight. Various uncertainties also impact its performance. In modeling, considering inertia and length variations due to morphing increases complexity. In control, detailed knowledge of uncertainties, such as parameter variability, is often needed. We propose a segmented modeling approach to obtain the quadrotor UAV model under morphing condition, along with an adaptive robust control to handle uncertainties within the system. The quadrotor UAV is innovatively segmented into three subsystems. Using the Udwadia–Kalaba (U–K) equation’s cascading and clustering features, we transform desired trajectories into constraints and integrate them with structural morphing constraints. This provides a direct analytical solution for constraint forces, which is immediately applicable to subsequent control design. This control consists of a nominal control part and an adaptive robust control part. Additionally, we only require that the uncertainty is bounded. The designed adaptation law is leaky, and adaptive parameters are adjusted based on the system’s performance to compensate for uncertainties within the system. The efficacy of the proposed modeling and control approaches is validated through simulation results.

Distributed Formation-Containment Control of Networked Mobile Robots Using the Udwadia–Kalaba Approach

Distributed Formation-Containment Control of Networked Mobile Robots Using the Udwadia–Kalaba Approach

Robust approximate constraint-following control design based on Udwadia–Kalaba approach and experimental validation for the joint module of cooperative robot

Robust approximate constraint-following control design based on Udwadia–Kalaba approach and experimental validation for the joint module of cooperative robot

Human–Machine Shared Steering Control Under High-Speed Emergency Obstacle Avoidance Scenarios

The allocation of controlling and driving authority is an important and difficult aspect of human–machine shared steering control (SSC). This paper addresses the SSC problem in high-speed emergency obstacle avoidance scenarios. A parallel SSC framework containing a dynamic driving authority allocation model and a path tracking controller is developed, where the human driver and controller can control the vehicle simultaneously. First, fuzzy logic is adopted in the SSC framework to actively adjust the driving authority between the human driver and the controller. The driver steering state and path tracking error are considered to reduce the negative impact of a driver’s mis-operation and weaken any human–machine conflict. Subsequently, the path tracking controller in the proposed SSC system is designed based on a nonlinear vehicle lateral model to improve the accuracy of the controller, particularly when the vehicle is facing large lateral acceleration. To address the nonlinear control problem, the Udwadia–Kalaba approach is employed and the Lyapunov stability of the controller is proved. Finally, the effectiveness of the proposed SSC system is proved through simulation results, which show that the vehicle has excellent path tracking performance in high-speed obstacle avoidance scenarios. In addition, the system can resolve the human–machine conflict problem.

Dynamic Modeling and Control of a Flexible Planar Three-Link Mechanism Using Joint and Piezoelectric Actuators

Abstract This paper presents a new method of controlling the end effector position and orientation of a flexible planar three-link mechanism. The coupled dynamic model was formulated using a method based on the Udwadia–Kalaba equations of motion for constrained systems. Lyapunov's theory was used to develop a nonlinear control law using piezoelectric actuators and the unconstrained link dynamic models. Numerical simulation was used to demonstrate the system tracking performance of the end effector using a reduced order controller applied to a higher order truth dynamic model.

Numerical and experimental investigation on the synthesis of extended Kalman filters for cable-driven parallel robots modeled through DAEs

Cable-driven parallel robots are parallel robots where light-weight cables replace rigid bodies to move an end-effector. Their peculiar design allows obtaining large workspaces, high-dynamic handlings, ease of reconfigurability and, in general, low-cost architecture. Knowing the full state variables of a cable robot may be essential to implement advanced control and monitoring strategies and imposes the development of state observers. In this work a general approach to develop nonlinear state observers based on an extended Kalman filter (EKF) is proposed and validated both numerically and experimentally by referring to a cable-suspended parallel robot. The state observer is based on a system model obtained by converting a set of differential algebraic equations into ordinary differential equations through different formulations: the penalty formulation, the Udwadia–Kalaba formulation, and the Udwadia–Kalaba–Phohomsiri formulation, which have been chosen since they can handle the presence of redundant constraints as often happens in cable-driven parallel robots. In the numerical investigation, the EKF is validated simulating encoders heavily affected by quantization errors to demonstrate the filtering capabilities of EKF. In the experimental investigation, a very challenging validation is proposed: only two sensors measuring the rotations of two motors are used to estimate the actual position and velocity of the end-effector. This result cannot be achieved by sole forward kinematics and clearly proves the effectiveness of the proposed observer.

An Evenly Partition Approach to the Modeling and Constraint-Following Control for the Spatial Cooperative Dual-Robot-System

Abstract Due to the breadth and complexity of spatial motion, and the nonlinear coupling caused by multiple degree-of-freedom, the control difficulties of spatial cooperative multirobot system are two mainly aspects, the modeling method and the dispose of uncertainty. For modeling, the process of directly using ordinary methods such as Lagrange mechanics are very complex. For control, the external disturbances and imperceptible errors will lead to the instability of dynamic system, these factors are collectively referred to as uncertainty. In this paper, a universal modeling method based on spatial cooperative multirobot system was proposed, and an adaptive robust controller was designed to solve the control problems. Simplify the connection link into a load and divide the system into two symmetrical subsystems by disconnecting the load center. The inherent physical position and given trajectories were unified as constraints. The constraints applied on system are creatively converted into consistent standard form through the Udwadia–Kalaba (U–K) method, and the corresponding constraints can be generated easily from the Udwadia–Kalaba basic equations. In addition, a nominal term to stable the system and an adaptive term was added to the controller. The adaptive term works through an adaptive parameter, the parameter estimates the uncertainty bound, thus, only the boundedness of uncertainty should be known, and more accurate control input can be gained. Eventually, the correctness of the model and controller has been confirmed through numerical simulation.

Cooperative-theoretic optimal adaptive robust control for fuzzy multiple manipulator systems

This article presents the study of the trajectory tracking issue of multiple manipulators from the viewpoint of servo constraint-following. First, on the framework of Udwadia-Kalaba (U-K) theory, the tasks of (holonomic or nonholonomic) trajectory tracking control are transformed into a series of servo constraints, and a nominal control is presented to realize the servo constraint-following for the system without uncertainty. Second, consider that there exists the time-varying uncertainty in the system, which is bounded and characterized by the fuzzy set theory. An adaptive robust control (ARC) is presented to achieve tracking errors uniform boundedness (UB) and uniform ultimate boundedness (UUB). Third, cooperative game theory is employed to obtain the dual optimal control parameters of the presented controller, and the mathematical derivation is performed to verify that there exists the Pareto optimal solution. Finally, simulation results show the effectiveness of the proposed control for the trajectory tracking control of multiple manipulators.

Distributed formation control of networked mobile robots from the Udwadia–Kalaba approach

AbstractIn this article, formation control of networked mobile robots is studied. Two control strategies for the leaderless case and the leader‐following case are developed from the Udwadia–Kalaba approach, respectively. Differing from the existing results about formation control, which focus on kinematic models of robots, this article considers dynamic models and obtained a lower‐cost control force. To improve the robustness of the communication network, two distributed formation controllers are designed by establishing distributed estimators. Different from the centralized controllers, the distributed controllers established here only require local information, thus save communication resources and improve system robustness. Finally, some numerical simulations are provided to demonstrate the proposed methods are effective and have lower control cost.

Robust approximate constraint following control for SCARA robots system with uncertainty and experimental validation

In the context of mechanical systems characterized by nonlinearity, uncertainty, and unknown external disturbances, this paper presents a novel and practical robust control algorithm based on the Udwadia–Kalaba (UK) equation that incorporates the ideal system trajectory as a servo constraint. Firstly, the constraint force of the system under servo constraints can be written in analytic form, which is the first part of the proposed control algorithm. The remaining parts deal with the initial condition incompatibility and the uncertainty within the system, respectively. Secondly, it is proved that the controlled system with uniform boundedness (UB) and uniform ultimate boundedness (UUB) by using the Lyapunov method. Finally, based on the SCARA robot platform, numerical simulations and experiments are performed using the rapid control prototype CSPACE. The results indicate that the proposed robust control method provides excellent dynamic performance under the constraints of the ideal trajectory. The effectiveness and practicality of the control method are further demonstrated.

Adaptive Robust Control for Uncertain Underactuated Mechanical Systems

AbstractAn adaptive robust control method is proposed for underactuated mechanical systems (UMSs) with uncertainty in the paper. UMSs are supposed to follow prescribed constraints when there is uncertainty. Inspired by the Udwadia–Kalaba modeling approach, those constraints are considered as control objectives realized by robust control. The uncertainty considered in UMSs is characterized as time‐varying and bounded, albeit with unknown bounds. In order to estimate the boundary information, a novel adaptive law is developed. By using the Lyapunov function, the UMS is guaranteed to be uniform boundedness and uniform ultimate boundedness. The simulation is validated by using a two‐wheeled bicycle as an example, compared to conventional PD control methods. The simulation comparison results show that the proposed control method exhibits notable characteristics, including fast transient response, small overshoots, and good robust performance.

Foot trajectory following control of hexapod robot based on Udwadia–Kalaba theory

This paper provides an adaptive robust control strategy for foot trajectory following control of hexapod robot on basis of the Udwadia–Kalaba theory. In this paper, the foot trajectory following control problem of the hexapod robot is transformed into the problem of solving the system control constraint force on basis of the Udwadia–Kalaba theory. Compared with the traditional control strategy, linearization or approximations are not required by using the Udwadia–Kalaba theory for nonlinear system such as the hexapod robot. Due to modeling error, measurement error and the change of working state, the system may have non-ideal initial conditions, vibration interference and other uncertain factors during operation, which affect the control accuracy. An adaptive robust controller is designed for solving uncertainties. Meanwhile, the stability is analyzed by using the second method of Lyapunov function. Finally, the accuracy and stability of the control method proposed are verified by establishing the leg model of the hexapod robot and conducting simulation analysis. The simulation results show that the provided adaptive control process has faster error convergence speed and response speed compared with the sliding mode control method.

Constrained Dynamics of an Aerial Manipulator Interacting With Flexible Cantilever Beams

Aerial manipulation poses complex challenges for aircraft design, modeling, and control, especially when dealing with nonlinear continuous systems. This article explores the coupled dynamics of a fully actuated aerial vehicle when interacting with a cantilever beam undergoing large deflections, focusing on achieving a more accurate model and identifying the challenges involved in this task. The analysis can be applied to prolonged interactions, such as tree branch deflection or long reach manipulation. Based on the Udwadia–Kalaba formulation, the presented model follows a relatively simple and systematic approach for imposing constraints without Lagrange multipliers. The resulting constrained equation of motion is obtained in closed form and captures the essential characteristics of the interaction: kinematic constraints and geometric nonlinearities. Furthermore, simulation and experiments demonstrate the feasibility and performance limitations of using a linear motion controller and how end-effector configuration can affect stability.