Udwadia Kalaba Research Articles

Optimal robust path tracking control for multi-constrained underactuated vehicles based on uncertainty orthogonal decomposition

In this article, an optimal robust constraint-following control is proposed for the underactuated vehicle path tracking problem. First, an underactuated dynamic model is established according to Udwadia–Kalaba (UK) equation, and the system uncertainty is decomposed into matching and mismatching portions based on the system geometric characteristics. The mismatching uncertainty is orthogonal to task space; thus, its influence on system stability is eliminated. Second, the diffeomorphism method is used to creatively set inequality and equality constraints into new equality constraints, and a robust control method for front steering vehicles with parameter-tunable is proposed. Third, an optimal design scheme is proposed for the tunable parameters to minimize the comprehensive index of system performance and control cost. Carsim-Simulink co-simulation shows the effectiveness of the proposed optimal robust control. This article creatively solves the problem of optimal robust control for path tracking of multi-constrained underactuated vehicles.

Motion control of connected and automated vehicles in a lane‐changing scenario with guaranteed tracking performance

AbstractThe coupled, nonlinear vehicle dynamics and multi‐vehicle cooperative strategies make the lane‐changing maneuver control of vehicle platoons a challenging task. This paper presents the dynamic modeling and motion control of connected and automated vehicles (CAVs) for lane‐changing maneuver, utilizing the ideology of constraint‐following servo control. To guarantee a transient and steady tracking performance, a series of equality constraints and bilateral inequality constraints bounded by decaying functions or scalar constants are imposed on the dynamic state to formulate a constrained dynamical system. Since the original Udwadia‐Kalaba (UK) approach cannot handle inequality constraints and underactuated systems, a creative diffeomorphism method is applied to transform the bounded state into an unbounded one. Then, based on the UK approach and Lyapunov stability theory, a novel constraint‐following controller is designed to render the transformed states asymptotically stable in a full actuated system, and to achieve uniform stability performance in an underactuated system. Finally, numerical simulations are used to validate the effectiveness of the proposed control methodology.

Optimal design of Udwadia–Kalaba theory-based adaptive robust control for permanent magnet linear synchronous motor

The control performance of permanent magnet linear synchronous motor (PMLSM) is limited by uncertainty. The uncertainty is bounded by an unknown bound and may be time-varying. The boundary information of uncertainty is unknown but can be described by fuzzy set theory. To mitigate the impact of uncertainty on control performance, this paper proposes a Udwadia–Kalaba theory-based adaptive robust control (UKBARC). The proposed controller contains a Udwadia–Kalaba theory-based controller and an adaptive robust controller. In the Udwadia–Kalaba theory-based controller, the expected trajectory is taken as the expected constraint, and then the control task is converted to make the expected constraint be followed. Additionally, the adaptive robust controller is introduced to handle the system uncertainty without requiring the boundary information of uncertainty. The Lyapunov method is used to show that the performance of uniform boundedness (UB) and uniform ultimate boundedness (UUB) for the PMLSM system is guaranteed. The objective function used to characterize the trade-off between performance and control cost can be derived based on the fuzzy description of uncertainty. Thus, the optimal design of UKBARC (OUKBARC) can be achieved by minimizing the objective function. Simulation and experiment results together verify that high precision motion control can be ensured by using the proposed approach.

Robust Constraint-Following Control for Uncertain Mechanical Systems With Generalized Udwadia-Kalaba Approach

In this letter, we propose a robust constraint-following control approach for uncertain mechanical systems under both equality and inequality constraints. Particularly, both the global and local inequality constraints are systematically incorporated into the Udwadia-Kalaba (U-K) equation leveraging the diffeomorphism technique, wherein a novel smooth approximation of the local inequality constraints is proposed to address the non-differentiability resulting from its spatiotemporal dependence nature. Based on this development, the generalized U-K equation is mathematically established. With this, we develop a robust constraint-following control strategy to ensure satisfying system performance in the presence of uncertainties and various constraints. Moreover, by Lyapunov minimax approach, the proposed control strategy guarantees both uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the system. Finally, numerical simulations on the lateral motion control of an autonomous vehicle demonstrate the effectiveness of the proposed approach.

Research on Constraint Following Control of Flexible Joint Manipulators Based on Singular Perturbation

For the control of 2-link flexible joint manipulators, a control method based on the singular perturbation theory and the Udwadia-Kalaba (U-K) method was proposed. The control design was implemented by 2 steps. First, the system order was reduced based on the singular perturbation method and the system was divided into fast and slow sub-systems, to simplify the solution process and overcome the system flexibility. Second, the state feedback constraint following control law for the fast and slow sub-systems was designed with the U-K method, to make the constraint following errors of the fast and slow sub-systems converge to zero, even if the system can't initially satisfy the constraints. The proposed method can deal with holonomic and nonholonomic servo constraints at the same time without the auxiliary variables of the Lagrange multiplier and the pseudo generalized velocity. The method was applied to 2-link flexible joint manipulator systems to solve the flexible oscillation and constraint following problems. Through simulations on MATLAB, and was compared with the traditional PID control, to verify the effectiveness and superiority.

Whistling in the clavichord.

Sympathetic string vibration plays an essential role in the clavichord's sound quality and tonal identity. Sympathetic vibration comes from the undamped string segments between the bridge and tuning pins. Under some conditions, a specific note, a whistling tone, stands out of the reverberation halo due to sympathetic vibration. It is hypothesized that this whistling tone comes from resonance between played and sympathetic segments of strings that are coupled through the bridge. Vibratory measurements for three pairs of excited and sympathetic strings are conducted on a copy of a historical instrument built by Hubert in 1784. The influences of bridge mobility and tuning on sympathetic string frequency and damping are studied. The results show a significant increase in vibratory amplitude, frequency veering, and damping increase in the string segments when tuning approaches frequency coincidence. Numerical simulations of a reduced clavichord model corresponding to the experiments are conducted using the modal Udwadia-Kalaba formulation. Simulation gives a more accurate picture of the veering phenomenon. Simulation and experimental results are in good agreement, showing that whistling in the clavichord comes from string resonance. It is favored by frequency coincidence between excited and sympathetic string segments and by higher bridge mobility.

Modeling Flexible Multi-Body Systems Within the Udwadia–Kalaba Framework, a Lumped Parameter Approach

Abstract It is common practice within multi-body dynamic (MBD) modeling to assume that individual bodies are rigid however this can be an oversimplification especially when slender bodies are present resulting in inaccurate estimations of the system's natural frequencies and overall behavior. To address this shortfall, we extend the Udwadia–Kalaba (U–K) MBD formulation in this paper to model flexible multibody systems for purposes of exploring system natural frequencies. To model the flexibility, a lumped parameter approach is proposed, which in this work idealizes a flexible beam as a series of discrete rigid elements connected by torsional springs. In the U–K formulation, a mechanical system can also be discretized into rigid elements and adapted. This is viewed as a benefit for incorporating a lumped parameter approach within the U–K formulation to model flexible multibody systems. A flexible crank-slider mechanism is introduced and modeled within the Udwadia–Kalaba formulation to capture the dynamics of flexibility through linkage compliance. The model is validated against an alternatively formulated MBD model and system natural frequencies and mode shapes numerically predicted. Results of the study show the effectiveness and potential of extending the application of the Udwadia–Kalaba formulation by using a lumped parameter approach to dynamically model flexible multibody systems.

Formation tracking control design for uncertain artificial swarm systems under inequality constraints: Leakage and dead-zone type

The paper proposes a formation tracking control method for the uncertain artificial swarm systems under the inequality constraints. Not only can the agents perform swarm behaviors (e.g., convergence, formation and avoidance of collision), but they can also track the fixed targets in a constrained area (which is formulated as the inequality constraints, such as unilateral constraint and bilateral constraint.). The swarm behaviors are creatively considered as the servo constraints or the control objectives for the swarm agents. Based on the Udwadia–Kalaba (U–K) equation, those prescribed behaviors are realized by a model-based control design (that is the servo constraints force model-based feedforward control). To deal with the inequality constraints in the formation tracking process, a differential homeomorphism transformation is used to relieve the environmental constraints for the swarm agents. Moreover, the uncertainty of the swarm agents (i.e., the parameter uncertainty in modeling and the external disturbances) is considered, which is time-varying and unknown (but bounded). An uncertainty estimation method with dead-zone and leakage term is designed to calculate the possible upper bound of the uncertainty. In virtue of the estimated upper bound of the uncertainty, a robust control is designed for the uncertain swarm agents to obey the prescribed swarm behaviors in the formation tracking task. The system performances of the artificial swarm systems under the proposed control are theoretically guaranteed by a range of rigorous theorems and numerically verified by the simulations of three agents.

The adaptive neural network fuzzy sliding mode control for the 3-RRS parallel manipulator

In order to improve the high-precision tracking control of angle variables and the sliding mode equivalent control part for the 3-RRS parallel manipulator, an adaptive neural network fuzzy sliding mode control algorithm with self-adjusting switching gain is proposed. Firstly, considering the uncertainty of the constrained force between derived links and the moving platform, the complete dynamic model including ideal and non-ideal constrained force is established by combining with the Udwadia-Kalaba(U-K) equation and Lagrange method. Secondly, the neural network sliding mode controller is designed to realize the approximate solution of the sliding mode equivalent control part. At the same time, in order to reduce the chattering phenomenon of the neural network sliding mode controller, a fuzzy adjustment rule of switching gain is designed to better compensate for the uncertain terms. And then the stability of the control system is proved by the Lyapunov method. Finally, the proposed control algorithm is simulated on the 3-RRS parallel manipulator. The simulation results show that the chattering phenomenon is overcome. The high-precision control of angle variables and the sliding mode equivalent control part is realized.

Precise torques and sliding mode compensation for trajectory tracking of manipulator with uncertainty

The control based on dynamic model could improve the dynamic performance of manipulators and obtain better control effects than the control based on kinematic model. As manipulators are complex online multivariable systems, there are various uncertainties in different environments and working conditions. Accurate dynamic parameters are difficult to obtain in practical engineering applications. In this article, a controller is designed by combining the precise control torques obtained by the analytical dynamics method with the compensation control torques obtained by sliding mode method for trajectory tracking of the manipulator with bounded uncertainty. Precise control torques obtained from the Udwadia–Kalaba modeling method could be applied for the control of the ideal manipulator tracking the desired trajectory. Compensation control torques obtained from the sliding mode concept and the Lyapunov stability theory could be applied to compensate the uncertainties of parameters and external disturbances, thus enhancing the robustness of the system. By combining precise control torques with compensation control torques, the end point of the manipulator with uncertainty could track the end of the ideal manipulator and then track the desired trajectory. The simulation results of the three-link manipulator with uncertainty show that the control method can make the controlled target approach the desired trajectory in relatively small torque ranges and obtain high stability accuracy, and the chattering is effectively reduced at the stage of approaching the sliding mode surface.

Udwadia-Kalaba approach based distributed consensus control for multi-mobile robot systems with communication delays

In this paper, a distributed consensus algorithm for multi-mobile robot systems (MMRSs) with communication delays is proposed based on the Udwadia-Kalaba (UK) approach. The key feature of the proposed algorithm is that the consensus requirement is configured as a second-order constraint, and then a concise and explicit equation of motion for the constrained mechanical systems is formulated. Furthermore, the necessary and sufficient conditions for achieving the consensus of MMRSs with or without communication delays are developed under the network topology possessing a directed spanning tree. Finally, some numerical simulations are performed to verify the validity of the proposed consensus algorithm.

Optimal robust constraints-following control for rail vehicle virtual coupling

This paper puts forward the robust control problem of rail vehicle virtual coupling platoon that is subject to nonlinear and time-varying uncertainties. The uncertainties are caused by external disturbances and parameter variations. To this end, this paper designs a robust controller based on the Udwadia-Kalaba (UK) equation. First, we introduce the concept of virtual coupling and establish the longitudinal dynamics model of the rail vehicle. In order to ensure the control objective of the virtual coupling process, the spatial constraint on position and bidirectional inequality constraints for the spacing error between the marshalling trains are established. After changing the bounded state into the unbounded state through the state transformation, the UK approach can be used. Then, the constraint-following robust controller design is based on the UK approach and Lyapunov stability theory. The controller can render the unbounded state eventual uniform boundedness and uniform ultimate boundedness, which also satisfies the spatial constraint and the bidirectional restriction for the spacing error. Moreover, an optimized design scheme for the tunable parameter of this controller is proposed, which minimizes the control cost and comprehensive index of system performance. Finally, numerical simulations are given to illustrate the effectiveness of the proposed methods.

Constraint-following control design for the position tracking of a permanent magnet linear motor with inequality constraints

Abstract. In this paper, we mainly solve the problem of the trajectory following the control and boundary transfinite control of the permanent magnet linear motor (PMLM). Using the Udwadia–Kalaba (U–K) method, the explicit equation of the complete nonholonomic constraint equation is first established, and then the new input constraint equation is obtained by integrating the inequality constraint and the original equation constraint through tangent-state transformation mapping. This constraint equation can make the motor move along the ideal trajectory in a limited range, thus solving the control problem of equality and inequality creatively. The simulation results and PMLM experiment results, based on control Signal Processing And Control Engineering (cSPACE), show that the proposed control method can obtain better motion performance, and the motion displacement does not exceed the boundary while satisfying the trajectory tracking control performance, which proves the effectiveness and feasibility of the proposed method.

The neural network terminal sliding mode control for the 3-RRC parallel robot

In this paper, the 3 degrees of freedom (3-DOF) parallel robot (3-RRC) is taken as the research object. The Lagrange method is used to establish the reduced order dynamic equations of three branch chains. On the basis of the U-K (Udwadia-Kalaba) equation, the analytical expressions of ideal and non-ideal constrained forces are obtained. Then the complete dynamic model of 3-RRC parallel robot is established. In order to achieve high precision control of 3-RRC parallel robot, and fully considering the uncertainty of non-ideal constrained force and chattering problem in terminal sliding mode control algorithm, the neural network is used to adaptively adjust the gain of switching function and achieve universal approximation of the unknown non-ideal constrained force. The neural network terminal sliding mode control algorithm is proposed for the complete dynamic model of 3-RRC parallel robot, and the stability of the control system is proved by Lyapunov theorem. Finally, the simulation research is conducted on the 3-RRC parallel robot. Simulation results show that the tracking precision of angle positions and non-ideal constrained forces are all reached 10−2 order, which realize the high precision control of the 3-RRC parallel robot, weaken the chattering phenomenon, and verify the correctness and effectiveness of the proposed dynamic model and control algorithm.

Constrained control methods for lower extremity rehabilitation exoskeleton robot considering unknown perturbations

In this paper, trajectory tracking control is investigated for lower extremity rehabilitation exoskeleton robot. Unknown perturbations are considered in the system which are inevitable in reality. The trajectory tracking control is constructively treated as constrained control issue. To obtain the explicit equation of motion and analytical solution of lower extremity rehabilitation exoskeleton robot, Udwadia–Kalaba theory is introduced. Lagrange multipliers and pseudo-variables are not needed in Udwadia–Kalaba theory, which is more superior than Lagrange method. On the basis of Udwadia–Kalaba theory, two constrained control methods including trajectory stabilization control and adaptive robust control are proposed. Trajectory stabilization control applies Lyapunov stability theory to modify the desired trajectory constraint equations. A leakage type of adaptive law is designed to compensate unknown perturbations in adaptive robust control. Finally, comparing with nominal control and approximate constraint-following control, simulation results demonstrate the superiority of trajectory stabilization control and adaptive robust control in trajectory tracking control.

Constraint-Following Approach for Platoon Control Strategy of Connected Autonomous Vehicles

The current platoon control strategies of connected autonomous vehicles (CAVs) focus on controlling the fixed intervehicle distance, i.e., the string stability of the platoon system. Here, we aimed to design a CAV platoon control strategy based on a constraint-following approach to solve the problem of platoon starting. As the resistance of the vehicle during driving varies with time, this study regarded the CAV platoon system as a changing dynamic system and introduced the Udwadia–Kalaba (U–K) approach to simplify the solution. Apart from adding an equality constraint, unlike most other studies, this study imposed a bilateral inequality constraint on the intervehicle distance between successive CAVs to prevent collisions. Meanwhile, a diffeomorphism method was introduced to transform the bounded state into an unbounded state. The proposed control strategy could render each CAV compliant with both the original imposed bilateral inequality constraint and the equality constraint. The former avoids collisions, and the latter indicates the string stability of the designed CAV platoon system. The effectiveness of the proposed controller was verified by numerical experiments. The gap errors tend to converge to zero, which is not amplified by the propagation of traffic flow.

A Novel Theoretical Model Development and Simulation of Melt‐Electrospinning Using Kane's and Udwadia–Kalaba Methods

AbstractA novel analytical model development and simulation of melt‐electrospinning process is presented. Unconstrained equations of motion for the description of a discretized melt‐electrospun fiber are formulated using Kane's method. The motions of the spinneret and the collector plate are also incorporated into the kinematics formulation to simulate direct writing and three‐dimensional printing scenarios. Constraints describing viscoelastic joints in the system and the phenomenon of the melt‐electrospun fiber adhering to the collector plate are implemented using the Udwadia Kalaba method. Rotational viscoelastic elements are introduced to mimic the dampening of the whipping motion observed in the unstable region. A novel algorithm is devised to continuously run the simulation with no limit on the total duration and phase transitions. System responses such as fiber diameter, collection size, fiber elongations, and jet speeds are monitored for changes in control parameters including applied voltage, collector distance, and flowrate. The results demonstrate close agreement with experimental observations in the literature. The jet starts relatively straight on the onset of its trajectory and develops subtle whipping motion as it advances toward the collector plate. Reduction in the amplitude of the whipping motion is observed once the free end of the fiber adhered to the collector plate.

Pose Tracking Control for Spacecraft Proximity Operations Using the Udwadia–Kalaba Framework

This paper develops an analytical dynamics-based approach for simultaneous position and orientation tracking control of a chaser spacecraft with respect to an uncontrolled target. The control requirements are formulated as holonomic or nonholonomic constraints, which are expressed as linear and angular acceleration constraints. The complete six-degree-of-freedom formulation of the Udwadia–Kalaba-based pose tracking controller generating exact real-time control forces and torques is presented. For design purposes, the method assumes a precise knowledge of the system parameters and states, along with perfect control action by the actuators. The exponential convergence of the constraint dynamics is proven using the Lyapunov stability theory. Simulation results demonstrate exponentially stable position and orientation tracking for close-proximity operations in perturbed low Earth orbits. Finally, the controller is experimentally validated using the Spacecraft Proximity Operations Testbed at Carleton University.

Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles

<p style='text-indent:20px;'>Presented herein are a class of methodologies for conducting constrained motion analysis of rigid bodies within the Udwadia-Kalaba (U-K) formulation. The U-K formulation, primarily devised for systems of particles, is advanced to rigid body dynamics in the geometric mechanics framework and a novel development of U-K formulation for use on nonlinear manifolds, namely the special Euclidean group <inline-formula><tex-math id="M1">\begin{document}$ {\mathsf{SE}(3)}$\end{document}</tex-math></inline-formula> and its second order tangent bundle <inline-formula><tex-math id="M2">\begin{document}${\mathsf{T}^2\mathsf{SE}(3)} $\end{document}</tex-math></inline-formula>, is proposed in addition to the formulation development on Euclidean spaces. Then, a Morse-Lyapunov based tracking controller using backstepping is applied to capture disturbed initial conditions that the U-K formulation cannot account for. This theoretical development is then applied to fully-constrained and underconstrained scenarios of rigid-body spacecraft motion in a lunar orbit, and the translational and rotational motions of the spacecraft and the control inputs obtained using the proposed methodologies to achieve and maintain those constrained motions are studied.</p>

Iterative feedback tuning for optimal repetitive constraint‐following control of uncertain mechanical systems using Udwadia–Kalaba theory

AbstractThis article presents a novel optimal constraint‐following controller for uncertain mechanical systems (MSs). The MS's uncertainty is unknown (possibly time‐varying) and bounded, but the bound is unspecified. Employing the frame of Udwadia–Kalaba theory, we design a robust controller with two tunable control gains for MSs, which guarantees some deterministic performances. Assuming the constraints to be holonomic and periodic, we transform the controlled MSs and introduce an iterative feedback tuning method for optimizing the proposed controller. After optimization, the final control scheme can achieve the equilibrium between the system performance and control cost for an arbitrary single constraint period. Simulations on a two‐link rotational manipulator are provided at last to demonstrate the proposed approach.