Holonomic Constraints Research Articles

A Novel Trajectory Tracking Control of AGV Based on Udwadia-Kalaba Approach

Several control approaches have been used in order to achieve the trajectory tracking control of automated guided vehicle (AGV). But they are all based on the D'Alembert principle which requires to determine the Lagrangian multipliers. Unlike those control methods, in this paper, a novel approach of trajectory tracking control is presented, which is based on Udwadia-Kalaba approach. This approach presents a novel, concise and explicit equation of motion for constrained mechanical systems with holonomic and/or nonholonomic constraints as well as constraints that may be ideal or non-ideal. The Udwadia-Kalaba equation is uncoupled and free from the Lagrangian multipliers. In this paper, we classify constraints into structural constraints and performance constraints. Structural constraints are set up regardless of trajectory control and we use structural constraints to establish the dynamic model. Then the desired trajectory is set as performance constraints and constraint torques are solved by analyzing Udwadia-Kalaba equation. For AGV, a nonholonomic mechanical system, second-order constraints are used to get constraint forces and the numerical simulation is conducted by using Matlab. The results show that the AGV's movement meets the requirement and the tracking trajectory is exact and perfect. Compared with other approaches, this approach requires very little computation. Through two cases of tracking a circle and a line, how to use Udwadia-Kalaba equation to realize trajectory tracking control is shown in detail.

Time-switching patrolling controller for holonomic and nonholonomic multi-robot systems

In this paper, we propose a distributed controller for holonomic and nonholonomic multi-robot systems to patrol multiple targets. Distributed control is a control method that uses only information observed by the robot itself and gathered from its surrounding without relying on centralized control. In order for the robots to accomplish their patrols, we propose an objective function that changes switching weights depending on the time when the targets are not observed. As a result, each robot determines which targets to observe based on the time weights and the information from surrounding robots to achieve efficient patrols. We design a distributed controller based on the gradient-flow method for both holonomic and nonholonomic systems. Finally, simulation and experimental results show that the proposed controller is useful for the robots to realize patrolling.

An electric kickscooter multibody model: equations of motion and linear stability analysis

AbstractIn this work, a detailed multibody model of an electric kickscooter is presented. The model includes toroidal wheels as well as rear and front suspensions. The equations of motion are derived and linearized along the steady forward motion of the vehicle. Using an efficient linearization approach, suitable for complex multibody systems with holonomic and nonholonomic constraints, allows for obtaining the reduced linearized equations of motion as a function of the geometric, dynamic, wheels’, and suspensions’ parameters. The proposed electric kickscooter multibody model is validated with the stability results of a previously presented electric kickscooter benchmark. Since the resulting eigenvalues are parameterized regarding the design parameters, a detailed linear stability analysis of the system is performed. In particular, the influence on the stability of the toroidal geometry of the wheels, the elliptic cross-section of the toroidal wheels, the rider model, the steering axis inclination angle, the inertia tensor of the front frame, and the rear and front suspensions is analyzed. The model presented, together with the linearized equations of motion obtained in this work, enables a systematic analysis of the stability of these vehicles, which helps design new electric kickscooters with improved vehicle safety conditions and oriented to a wider range of potential users.

Adaptive region-reaching control for a non-holonomic mobile robot via system decomposition approach

This article focuses on the region-reaching control for a non-holonomic mobile robot system with two actuated wheels. The region-reaching control task consists of three basic control objectives: reaching an objective region, maintaining a rest state in the objective region, and stabilising at a desired posture. The constraints imposed on the motion of the mobile robot system are not integrable and have no inverse resolution, and thus increase the complexity of the controller design for non-holonomic systems. A system decomposition approach is proposed to convert the non-holonomic constraint control system to the cascade holonomic constraint control subsystems, and an adaptive control gain technique is presented to guarantee the stability of the non-holonomic system. Then, a novel torque controller is presented to implement the region-reaching control task. Specially, the objective region can be considered as a parking spot for a mobile robot.

Alternate Forms to Lagrange–D'Alembert Principle for Treatment of Rheonomic Constraints

Abstract Lagrange–D'Alembert principle is based on the concept of the nonactual and nonmeasurable virtual displacement and the assumption that the system-constraint forces are workless. Because time is not considered in defining the virtual displacements, virtual changes in prescribed displacements that characterize rheonomic constraints, referred to as driving constraints, are zero. Consequently, Lagrange–D'Alembert principle does not account systematically, from the outset, for rheonomic constraints, which are not workless and have power associated with them. In multibody system implementations, rheonomic-constraint forces are considered as constraint forces and not as applied forces. Consequently, the statement of the virtual-work principle that virtual work of the system-inertia forces is equal to the virtual work of the system-applied forces because the virtual work of system-constraint forces is zero omits inclusion of rheonomic constraints forces. This paper discusses using alternate forms to Lagrange–D'Alembert principle to account for rheonomic constraints from the outset by using actual and measurable motion variables to replace the virtual displacements. The analysis presented in this paper, which is applicable to both holonomic and nonholonomic systems, shows that the power of the system-inertia forces is equal to the power of the system-applied forces plus the power of rheonomic-constraint forces. It is demonstrated that the existence of Lagrange multipliers is not rooted in the definition of the virtual displacement or virtual work, but in the independence of the constraint functions. It is shown that when redundant coordinates are used, the effect of the rheonomic constraints appears explicitly in the constraint equations while this effect appears as generalized inertia forces when using the independent coordinates.

Nonlinear nonholonomic systems: a simple approach and various examples

The main theme of the article is the study of discrete systems of material points subjected to constraints not only of a geometric type (holonomic constraints) but also of a kinematical type (anolonomic constraints). The setting up of the equations of motion follows a simple principle which generalizes the holonomic case. Furthermore, attention is paid to the fact that the kinematical variables retain their velocity meaning, without resorting to the pseudo-velocity technique. Particular situations are examined in which the modeling of the constraints can be carried out in several ways to evaluate their effective equivalence. Numerous examples, many of which taken from the most recurring ones in the literature, are provided in order to illustrate the proposed theory.

Shaping the laser control landscape of a hydrogen transfer reaction by vibrational strong coupling. A direct optimal control approach.

Controlling molecular reactivity by shaped laser pulses is a long-standing goal in chemistry. Here, we suggest a direct optimal control approach that combines external pulse optimization with other control parameters arising in the upcoming field of vibro-polaritonic chemistry for enhanced controllability. The direct optimal control approach is characterized by a simultaneous simulation and optimization paradigm, meaning that the equations of motion are discretized and converted into a set of holonomic constraints for a nonlinear optimization problem given by the control functional. Compared with indirect optimal control, this procedure offers great flexibility, such as final time or Hamiltonian parameter optimization. A simultaneous direct optimal control theory will be applied to a model system describing H-atom transfer in a lossy Fabry-Pérot cavity under vibrational strong coupling conditions. Specifically, optimization of the cavity coupling strength and, thus, of the control landscape will be demonstrated.

The Bifurcations of Completely Integrable Holonomic Systems of First-Order Differential Equations

As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are R+-simple and stable. Using this result, the parametric differential equation models in electrical power systems and engineering can be studied.

KeyCLD: Learning constrained Lagrangian dynamics in keypoint coordinates from images

We present KeyCLD, a framework to learn Lagrangian dynamics from images. Learned keypoints represent semantic landmarks in images and can directly represent state dynamics. We show that interpreting this state as Cartesian coordinates, coupled with explicit holonomic constraints, allows expressing the dynamics with a constrained Lagrangian. KeyCLD is trained unsupervised end-to-end on sequences of images. Our method explicitly models the mass matrix, potential energy and the input matrix, thus allowing energy based control. We demonstrate learning of Lagrangian dynamics from images on the cl_bnmsqnk pendulum, cartpole and acrobot environments. KeyCLD can be learned on these systems, whether they are unactuated, underactuated or fully actuated. Trained models are able to produce long-term video predictions, showing that the dynamics are accurately learned. We compare with Lag-VAE, Lag-caVAE and HGN, and investigate the benefit of the Lagrangian prior and the constraint function. KeyCLD achieves the highest valid prediction time on all benchmarks. Additionally, a very straightforward energy shaping controller is successfully applied on the fully actuated systems.

Whole Body Control of an Autonomous Mobile Manipulator Using Series Elastic Actuators

The flexible joint robot has superior attractive features because of its high mobility, high load ratiohigh torque fidelity, robustness for external disturbance, task adaptability, and safety. In this paper, an autonomous mobile manipulator driven by the designed series elastic actuators (SEAs) is developed. A complete dynamic model of nonholonomic mobile manipulator including a mobile platform and manipulator with joint flexibility operating simultaneously is proposed. An integrated whole body trajectory control framework is proposed for such robot to perform mobile manipulation tasks. Considering the nonholonomic and holonomic constraints in the mobile manipulation, the whole-body dynamics is formulated and reduced. To address the highly nonlinear of the dynamics and model uncertainty, a novel integral Lyapunov function (ILF)-based adaptive neural network control for task tracking under uncertainties of the flexible joint robot model is proposed. Compared with existing methods, the proposed method provides an alternative for controlling flexible joint robots. The feasibility of the proposed method is verified by the extensive trajectory tracking experiment results in our developed flexible joint manipulator.

Consistent Lagrangians for Irreducible Interacting Higher-Spin Fields with Holonomic Constraints

Consistent Lagrangians for Irreducible Interacting Higher-Spin Fields with Holonomic Constraints

A unified analytical expression of the tangent stiffness matrix of holonomic constraints

While an accurate computation of the tangent stiffness matrix of multibody systems is usually of critical importance in various numerical analyses, one contribution often neglected in dynamics is represented by the tangent stiffness matrix of constraints. This term, resulting from the change of joints reactions with respect to the coordinates of the connected bodies, is indeed easily discarded for the sake of performance due to its slight effect on dynamic simulation results. On the contrary, it plays a critical role when static or eigenvalue analyses are required, especially for those cases featuring free motions. While the topic of holonomic constraint equations and their respective Jacobian matrix is not new in literature, this article aims to provide a general and unified formulation based on quaternion parametrization together with a consistent analytical expression of the tangent stiffness matrix derived through linearization. The article includes also the rheonomic contribution to holonomic constraints. The formulations presented in this work are built on a mixed-basis formulation, in use in many engineering applications, and allow to easily derive specialized versions (e.g. revolute, cylindrical, prismatic joints, etc.) from the same equation set. Examples demonstrate the doubly-positive effect of this additional stiffness term: first, static analyses are able to converge to the actual equilibrium position and second, the eigenvalues analyses proved to be more consistent. For this latter set of tests the non-linear dynamic results, observed in the frequency domain, are compared against those coming from eigenvalue analysis, in order to prove the augmented accuracy of the results.

Performance Comparison of Meta-Heuristics Applied to Optimal Signal Design for Parameter Identification.

This paper presents a comparative study that explores the performance of various meta-heuristics employed for Optimal Signal Design, specifically focusing on estimating parameters in nonlinear systems. The study introduces the Robust Sub-Optimal Excitation Signal Generation and Optimal Parameter Estimation (rSOESGOPE) methodology, which is originally derived from the well-known Particle Swarm Optimization (PSO) algorithm. Through a real-life case study involving an Autonomous Surface Vessel (ASV) equipped with three Degrees of Freedom (DoFs) and an aerial holonomic propulsion system, the effectiveness of different meta-heuristics is thoroughly evaluated. By conducting an in-depth analysis and comparison of the obtained results from the diverse meta-heuristics, this study offers valuable insights for selecting the most suitable optimization technique for parameter estimation in nonlinear systems. Researchers and experimental tests in the field can benefit from the comprehensive examination of these techniques, aiding them in making informed decisions about the optimal approach for optimizing parameter estimation in nonlinear systems.

Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics

Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE.

Necessary and Sufficient Optimality Conditions for Fuzzy Variational Problems of Several Dependent Variables in Terms of Fuzzy Granular Derivatives

This paper is intended to investigate fuzzy variational problems of several dependent variables. Firstly, we establish both necessary and sufficient optimality conditions for fundamental fuzzy variational problems and fuzzy variational problems with natural boundary conditions. Then, the necessary and sufficient optimality conditions for fuzzy variational problems with isoperimetric constraints and holonomic constraints are discussed. Some examples are given to illustrate our results.

Arbitrarily High-Order Energy-Conserving Methods for Hamiltonian Problems with Quadratic Holonomic Constraints

Arbitrarily High-Order Energy-Conserving Methods for Hamiltonian Problems with Quadratic Holonomic Constraints

Online regulation of walking gait speed for a five-link bipedal robot via adaptive deforming of virtual holonomic constraints

Online regulation of walking gait speed for a five-link bipedal robot via adaptive deforming of virtual holonomic constraints

Optimization of a coupled launching process considering hydrodynamic interaction effects

Crane-assisted launching and recovering of small boats from a large vessel is common in a wide range of operational scenarios and may be required during research missions or in emergency situations for rescue boats. As the mothership moves in natural irregular seas, the resulting accelerations and motions of the small boat can lead to dangerous situations, such as a collision between the mothership and the small boat or a slamming of the small boat onto the free water surface can occur, posing a great danger to the people involved. In the paper, the development of a coupled simulation method for the motions of the mothership and the small boat during the launching in the time-domain is described. The developed simulation method is based on the impulse response function. The implementation of the suspension on a crane is realised by a holonomic constraint and a Lagrange’s equation. The parameter influencing the launching operation include the current sea state, the stationary wave field induced by the motherships forward speed, at which the small boat is launched and the initial position of the small boat relative to the water surface. By varying the above mentioned parameters, the process can be optimised for different sea states in terms of forces and accelerations. In considering the forward speed of the mother boat during the launching process, it can be observed that in many cases the small boat is operated at high Froude numbers of greater than 0.4. This leads to an exceedance of the operational limits of the programmes used in this paper.

Second-order-optimal filtering on SE(2)×R2 for the Chaplygin sleigh

In the last decades many filters have been proposed to solve the problem of pose estimation that naturally arises in robotics, both for manipulators and mobile robots. The dynamics of mobile robots can be mathematically described by exploiting the theory of Lie groups embedding holonomic and nonholonomic constraints. In this paper we design a second-order-optimal filter for the so-called Chaplygin sleigh, that is a mechanical system with a nonholonomic constraint. In particular we examine the importance to know the exact dynamic equations, and to exploit the underline Lie group structure of the system. Moreover, we investigate the conditions that ensure the preservation of the nonholonomic constraint by properly choosing the affine connection which guarantees that the orthogonal velocity is equal to zero. In this work the sensing system consists of a GPS-like configuration (Global Positioning System) obtained by using two antennas attached to the planar rigid body and an INS-like unit (Inertial Navigation System) to measure the velocities.

Singularity analysis of spatial single-DOF mechanisms based on the locations of the instantaneous screw axes

The instantaneous kinematics of a mechanism can be fully described through the instantaneous screw axes (ISAs) associated to the relative motions between mechanism's links. Characterizing the instantaneous kinematics of a mechanism includes its singularity analysis, which is the identification of mechanism configurations (singularities) where the instantaneous degrees-of-freedom (DOFs) of the mechanism “somehow” change. In single-DOF mechanisms with time-independent either holonomic or first-order non-holonomic constraints, the locations of the instantaneous screw axes only depend on the mechanism configuration. This property is fully exploited in planar single-DOF mechanisms where all the singularity conditions have been associated to the coincidence of particular instant centers (ICs), thus making the singularity identification possible with a simple analysis of the mechanism configuration. Differently, for single-DOF spatial mechanisms, the geometric conditions on ISAs that identify a singular configuration have not been enunciated for the general case, yet. This paper fills this lack by providing the explicit expressions of all the possible instantaneous input-output relationships, the exhaustive enumeration of the geometric conditions on the ISAs that identify singular configurations and a geometric and analytic technique for the singularity analysis of spatial single-DOF mechanisms, which is the natural extension of the IC-based ones conceived for planar single-DOF mechanisms.