Coriolis Matrix Research Articles

Adaptive passive admittance control of a compliant collaborative robot

Compliance of the collaborative robots plays an important role in human-robot interaction. Due to the complex nonlinearities caused by the compliance, and parameter uncertainties, good performance control method for the compliant collaborative robot is always a challenging task. Therefore, to overcome the challenge of the compliance, and uncertain dynamics, this paper develops an adaptive passive admittance control (APAC) approach for a novel compliant collaborative robot, whose compliant mechanism is constituted of the inerters, springs, and dampers. First, through force-current analogy between mechanical network and electrical network, the compliance of the proposed collaborative robot with the inerters, springs, and dampers, is represented by a novel and simplified admittance matrix to facilitate dynamics model of the proposed collaborative robot. Then, for the proposed compliant collaborative robot, an APAC strategy is presented to dispose of the uncertain parameters of the dynamics. The inertial matrix, Coriolis matrix, admittance matrix, and gravity vector of the compliant collaborative robot are expressed as linear representation of orthogonal basis functions by using function approximation technique (FAT). The boundedness of the robot dynamics is proved by detailed stability analysis which uses Lyapunov function. Finally, simulation results indicate that the proposed APAC of the compliant collaborative robot is effective.

Adaptive fuzzy control of underwater vehicle manipulator system with dead-zone band input nonlinearities via fuzzy performance and disturbance observers

In this paper, an adaptive fuzzy control design problem is investigated for an underwater vehicle manipulator system (UVMS) based on a fuzzy performance observer (FPOB) and fuzzy disturbance observer (FDO). The UVMS is considered as a nonlinear system comprising model uncertainties, external disturbances and dead-zone band input nonlinearities. To mitigate the combined effects of the dead-zone and the hysteresis, a novel pre-deadzone compensator is proposed. Then, fuzzy logic systems (FLSs) with online adaptations are utilized to evaluate the unknown components of the inertia matrix, the Coriolis matrix and the damping matrix. In addition, a fuzzy performance observer is constructed whose errors are used to estimate the external disturbances. Further, an H∞ fuzzy control technique is developed to reduce the errors in estimating the external disturbance. Then, the stability and tracking performance of the closed-loop system are analyzed using the Lyapunov stability theory. It is shown that all signals of the closed-loop system are uniformly ultimately bounded. Finally, simulations are performed to demonstrate the effectiveness of the proposed control scheme in addressing the tracking control problem of the UVMS in presence of the dead-zone band and disturbances.

Modelling and control of nonlinear negative imaginary systems

This article investigates mathematical modelling and control of a nonlinear negative imaginary system using an illustrative benchmark physical example of a quadrotor dynamic model. A generalized methodology based on Euler–Lagrange equation is applied to obtain nonlinear negative imaginary dynamic model for the quadrotor. In this method, the Kronecker product is employed to formulate the Coriolis matrix, which is then used to construct a mathematical model of a quadrotor. This article further presents nonlinear negative imaginary systems theory-based analysis and synthesis framework to find the control solution for quadrotor’s attitude stability problem while hovering. The multi-loop control scheme is applied to the quadrotor that directly uses the Euler angles (angular positions) instead of angular velocity measurements. Numerical simulation results of this paper show that the investigated control strategy ensures the asymptotic stabilization of quadrotor attitude system model in the presence of external disturbance.

Coriolis mixing of the K = 1 and K = 0 mixed symmetry states in the well deformed even–even nuclei

The Coriolis matrix elements responsible for mixing of the $$1^+ K=1$$ and $$1^+ K=0$$ states are calculated in the framework of the Quasiparticle Phonon Model for several Gd and Dy isotopes. In many considered cases these matrix elements are equal to several tens of keV and are comparable with energy distances between the mixed levels. The results obtained indicates that Gd isotopes could be more suitable for finding deviations from Alaga rules in M1 transitions from $$1^+$$ state to the states of the ground band.

Tracking Design of an Uncertain Autonomous Underwater Vehicle with Input Saturations by Adaptive Regression Matrix-Based Fixed-Time Control.

In this study, a simplified model of an autonomous underwater vehicle (AUV) with input saturation based on kinematic and dynamic equations was built. Subsequently, a simplified model of the AUV was used to represent its main dynamic features. In terms of trajectory tracking, only the system’s structure (i.e., the regression matrix, which is flexible and non-unique) from the nominal model of the transformed system was required to design the proposed adaptive regression matrix-based fixed-time controller (ARM-FTC). A nonlinear auxiliary sliding surface was contained in the control design to shape the system’s frequency response. When the operating point was in the neighborhood of the zero auxiliary sliding surface, nonlinear filtering gains were increased to accelerate its tracking ability. Furthermore, the skew-symmetric property condition of the time-derivative of the inertia matrix and the Coriolis and centrifugal force matrices was not necessitated for the controller design. Under an appropriate condition for lumped uncertainties, the fixed-time convergence of the auxiliary sliding surface and the corresponding tracking error is guaranteed to go to zero by the Lyapunov stability theory. Finally, a comparative study was conducted through simulations for the AUV with external disturbance and input saturation among the known parameters, learning parameters reflecting a regression matrix, and another asymptotical robust tracking control scheme. The results validate the fast tracking ability of a desired time-varying trajectory of the proposed control scheme.

Modelling and analysis of a bladed drum subject to the Coriolis and mistuning effects

The interaction between the Coriolis and mistuning effects on a finite-element model (FEM) of a simplified bladed drum compressor has been investigated in this paper. The Subset of Nominal Modes (SNM), the Component Mode Mistuning (CMM) and the Integral Mode Mistuning (IMM) techniques have been chosen and adapted to consider the reduction of the Coriolis matrix. Both free and forced responses of the ROMs have been validated using the FEM solutions as the benchmark. A new analysis using Monte Carlo simulations has been performed to consider disc and blade mistuning independently, and to study the domination of one over the other. The impact of the Coriolis effect has been highlighted through observation of the model with and without the Coriolis matrix. The evolution of the global dynamic behaviour with rotational speed has been investigated for the first time for different excitation frequencies and engine orders (EO). This in-depth investigation of a blisk allowed a better understanding of the interaction between the Coriolis and mistuning effects. The blade-dominated responses and the modes which remain close to the veering regions have strong mistuning effects, with significant localisation and amplitude magnification. The results mainly highlight that the disc-dominated and the blade–disc responses tend towards a tuned behaviour with the appearance of travelling waves.

Parameter estimation and adaptive control of Euler–Lagrange systems using the power balance equation parameterisation

It is widely recognised that the existing parameter estimators and adaptive controllers for robot manipulators are extremely complicated, stymieing their practical use – in particular, for robots with many degrees of freedom. This is mainly due to the fact that the existing parameterisation includes the complicated signal and parameter relations introduced by the Coriolis and centrifugal forces matrix. In an insightful remark of their seminal paper, Slotine and Li suggested to use the parameterisation of the power balance equation, which avoids these terms – yielding significantly simpler designs. To the best of our knowledge, such an approach was never actually pursued in on-line implementations, because the excitation requirements for the consistent estimation of the parameters are ‘very high’. In this paper, we use a recent technique of generation of ‘exciting’ regressors developed by the authors to overcome this fundamental problem. The result is applied to general Euler–Lagrange systems and the fundamental advantages of the new parameterisation are illustrated with comprehensive simulations of a 2 degrees-of-freedom robot manipulator.

Numerical Methods to Compute the Coriolis Matrix and Christoffel Symbols for Rigid-Body Systems

Abstract This article presents methods to efficiently compute the Coriolis matrix and underlying Christoffel symbols (of the first kind) for tree-structure rigid-body systems. The algorithms can be executed purely numerically, without requiring partial derivatives as in unscalable symbolic techniques. The computations share a recursive structure in common with classical methods such as the composite-rigid-body algorithm and are of the lowest possible order: O(Nd) for the Coriolis matrix and O(Nd2) for the Christoffel symbols, where N is the number of bodies and d is the depth of the kinematic tree. Implementation in C/C++ shows computation times of the order of 10–20 μs for the Coriolis matrix and 40–120 μs for the Christoffel symbols on systems with 20-degrees-of-freedom (DoF). The results demonstrate feasibility for the adoption of these algorithms within high-rate (>1 kHz) loops for model-based control applications.

Non-singular terminal sliding-mode control for a manipulator robot using a barrier Lyapunov function

This study introduces a design of robust finite-time controllers that aims to solve the trajectory tracking of robot manipulators with full-state constraints. The control design is based on the construction of a distributed state constraint non-singular terminal sliding mode (CNTSM). The CNTSM design includes the gain self-adapting tuning method, which can ensure finite-time convergence to the sliding surface aside from the states to its corresponding reference trajectories. The implementation of the time-varying gain ensures the fulfillment of the accurate tracking for the references while the position and velocity constraints are satisfied permanently. A barrier Lyapunov function is proposed to develop the finite-time stability analysis of the designed controllers. The CNTSM realization uses the tracking error as well as its estimated derivative, which is calculated using a variant of adaptive super-twisting algorithm operating as robust differentiator. The proposed CNTSM is numerically evaluated on a two-link RM with uncertain inertia and Coriolis matrices. Simulation and experimental results evidence the efficiency of the CNTSM controller demonstrating a better tracking performance while the full-state constraints are satisfied in counterpart with the classical non-singular terminal sliding mode which is not able to keep such restrictions.

Parameter Estimation and Adaptive Control of Euler-Lagrange Systems Using the Power Balance Equation Parameterization

It is widely recognized that the existing parameter estimators and adaptive controllers for robot manipulators are extremely complicated to be of practical use. This is mainly due to the fact that the existing parameterization includes the complicated signal and parameter relations introduced by the Coriolis and centrifugal forces matrix. In an insightful remark of their seminal paper Slotine and Li suggested to use the parameterization of the power balance equation, which avoids these terms—yielding significantly simpler designs.To the best of our knowledge, such an approach was never actually pursued, mainly because the excitation requirements for the consistent estimation of the parameters is “very high". In this paper we use a recent technique of generation of “exciting" regressors developed by the authors to overcome this fundamental problem. The result is applied to general Euler-Lagrange systems and the fundamental advantages of the new parameterization are illustrated with comprehensive simulations of a 2 degrees-of-freedom robot manipulator.

Adaptive sliding mode output feedback control for dynamic positioning ships with input saturation

In this paper, a finite-time observer-based adaptive sliding mode output feedback controller is developed for dynamic positioning ship while the Coriolis matrix, hydrodynamic nonlinear damping matrix, input saturation, unmeasured states, unknown time-varying environment disturbances and transient performance are considered simultaneously. A continue finite-time state observer is proposed to estimate the unmeasured states of dynamic positioning ship in finite time and an auxiliary system is constructed to handle with the control saturation. A terminal sliding mode surface with the signal generated by the auxiliary system is designed while the good transient performance is achieved under multiple constraints and the nominal output feedback controller proposed in this paper is chatter-free guaranteeing the steady performance. In additional, an adaptive update law is introduced that the prior knowledge of disturbance is no need and the finite-time convergence of loop-system is proved strictly. Finally, simulations are given to illustrate the effectiveness and advantages of the proposed control scheme.

Adaptive Finite-Time Saturated Tracking Control for a Class of Partially Known Robots

Only partial system knowledge for the generalized robotic dynamics with input saturation, i.e., the regression matrix for the specific robot, is required to design the proposed adaptive finite-time saturated tracking control (AFTSTC). It contains a nonlinear auxiliary tracking error, which can shape the system frequency response. As the operating point is in the neighborhood of the zero auxiliary tracking error, nonlinear filtering gains can be increased to accelerate its tracking ability. Moreover, the traditional skew-symmetric matrix's condition for the time derivative of inertia matrix and the Coriolis and centrifugal force matrix is not necessarily required such that the uncertainties and the finite-time convergence are reduced. The computational complexity as compared with the fuzzy neural network adaptive control is also addressed. The stability of the closed-loop system is verified by the Lyapunov stability theory. Finally, two practical robotic examples are given to validate the effectiveness and robustness of the proposed control.

Robust Regressor-Free Control of Rigid Robots Using Function Approximations

This paper develops a novel regressor-free robust controller for rigid robots whose dynamics can be described using the Euler–Lagrange equations of motion. The function approximation technique (FAT) is used to represent the robot’s inertia matrix, the Coriolis matrix, and the gravity vector as finite linear combinations of orthonormal basis functions. The proposed controller establishes a robust FAT control framework that uses a fixed control structure. The control objectives are to track reference trajectories in worst case scenarios where the robot dynamics are too costly to develop or otherwise unavailable. Detailed stability analysis via Lyapunov functions, the passivity property, and continuous switching laws shows uniform ultimate boundedness of the closed-loop dynamics. The simulation results of a three-degree-of-freedom (DOF) robot when the robot parameters are perturbed from their nominal values show good robustness of the proposed controller when compared with some well-established control methods. We also demonstrate success in the real-time experimental implementation of the proposed controller, which validates practicality for real-world robotic applications.

Dynamic Modeling and Inverse Optimal PID with Feed-forward Control in H∞ Framework for a Novel 3D Pantograph Manipulator

This paper affords dynamic modeling and control for a new 3D pantograph manipulator. The new manipulator possesses pure decoupled translational motions and it is characterized by large workspace to size ratio, high speed, rigidity, and accuracy. Euler-Lagrange first type method is used to get the dynamic model. However, the resulted dynamic model is too complex to be used in model-based control techniques. Therefore, a simplified nominal plant is proposed. It allows the inverse dynamic solution efficiently. However, an explicit form of the nominal Coriolis and centrifugal matrix cannot be obtained due to the complicated kinematic terms. Considering these dynamic characteristics as well as the required robust trajectory tracking performance of the manipulator, a new controller is proposed. The new controller is called inverse optimal PID with Feed-Forward Control which is designed in H∞ framework. The new controller has the following merits; robustness, optimality, simple implementation, and efficient execution without the need of explicit forms of dynamic matrices. The extended disturbance in the proposed controller is smaller than that in the inverse optimal PID control (IPID) and contains one type of error contrary to the nonlinear robust motion controller (NRIC). The performance of the proposed controller is compared with those of IPID and NRIC controllers for different trajectories and payloads. The dynamic simulation results via co-simulation of MSC-ADAMS® and MATLAB®/Simulink software prove the robustness of the proposed controller against speed/payload variations. The proposed controller is found to have higher performance compared with IPID and NRIC controllers. These results assure the feasibility of the 3D pantograph manipulator with the proposed controller for pure translational tracking applications.

Vibration localization for a rotating mistuning bladed disk with the Coriolis effect by a state-space decoupling method

Slight deviations of blades due to manufacturing tolerances can cause mistuning of bladed disk leading to localized vibration, which can accelerate fatigue. Moreover, the rotating blades are subjected to the Coriolis effect and the influence of the Coriolis force on the natural frequencies of high-speed rotational bladed disks such as those of an aero-engine become more apparent. In this paper, the effect of Coriolis force on the forced response localization of a mistuned bladed disk are investigated, for conditions where the natural frequency located in the first and second modal families of the bladed disk. Mistuning is introduced by varying the Young’s modulus of each blade. Due to the asymmetric Coriolis matrix, it is not possible to directly decouple the system. A state-space decoupling method is developed to decouple the system to effectively calculate the forced response of bladed disk with the consideration of the Coriolis effect. The results show that response localization factor is increased by 13.09% considering the Coriolis force compared to the system without considering the Coriolis effect, in the case of where the first modal family is considered. In addition, the response localization factor with the consideration of the Coriolis force is decreased by 30.85% compared to the system without considering the Coriolis force, when the second modal family is considered. It indicates that the forced response localization with the consideration of the Coriolis effect will be changed obviously with the rotational speed increasing, when the concerning natural frequency is located in the first and second modal families. Furthermore, the effect of Coriolis force causes changes in the resonant frequencies and resonant amplitude, but does not introduce additional resonant peaks for the case of the mistuned bladed disk.

Model Predictive Stabilization of Constrained Underactuated Autonomous Underwater Vehicles With Guaranteed Feasibility and Stability

This paper studies the model predictive stabilization problem of underactuated autonomous underwater vehicles (AUVs) with control input constraints, where the full dynamic and kinematic model including Coriolis and centripetal matrix, the hydrodynamic added mass, and damping is utilized, and a four-control-input configuration is considered. A novel model predictive control algorithm is proposed based on the homogeneity of system model and an existing time-varying control law. Theoretical results on ensuring algorithm feasibility and closed-loop stability are provided. It is shown that if the terminal set level and the cost weights are designed appropriately, the developed algorithm is iteratively feasible and the closed-loop system is asymptotically stabilized at the equilibrium point. Comparison studies verify the effectiveness of the developed results.

Characteristic model-based control of robotic manipulators with dynamic uncertainties

In this paper, we investigate the problem of the discrete-time adaptive control design of robotic manipulators by utilizing the characteristic model theory. Different from the traditional plants manipulated by the characteristic model-based control strategy, the Coriolis matrix in the robotic dynamics makes the bound of the characteristic model coefficients rely on the state information, proposing great challenge for the controller design and stability analysis. To conquer this problem, we develop a projection estimation algorithm depending on the state information, and propose an adaptive control law via introducing a discrete-time sliding vector. A new stability analysis framework is proposed by use of the recursive induction methodology, giving rise to the asymptotic convergence of joint positions. Simulations are performed to test the effectiveness of our methodology. Our design extends the characteristic model theory frame by proposing the control design and stability analysis methods for the the characteristic model with the state-relevance coefficients.

Математическое обеспечение конечно-элементного моделирования микромеханических датчиков инерциальной информации в рамках неклассической теории изгиба

A complete mathematical support of the 3D finite beam element for modeling of the micromechanical inertial measurement sensors and their components has been developed. The mathematical support includes mass matrix, stiffness matrix, Coriolis matrix, and centrifugal matrix. The mathematical support takes full account of the gyroscopic effect and the theory of Timoshenko. Employing of the Variational principles of mechanics and Lagrange's equation makes the process of derivation of the mathematical support clear, accurate and well-founded. The developed software was verified by a numerical simulation of the influence of the gyroscopic effect on the dynamics of the simplest model of the vibrating gyro. The results obtained due to the numerical simulation by using the developed mathematical support were compared with the results obtained in ANSYS, well-known engineering simulation software. The difference between these results was less than 5 %. This difference can be explained by the dissimilarity of the elements used in ANSYS and in the developed software. This paper shows that the developed mathematical support can be used for development of special software, which ensures, in contrast to the universal proprietary closed-source software such as ANSYS, a transparent implementation of the algorithms, a complete control of the progress of computing and significantly lower cost. Thus, the developed mathematical support for the three-dimensional finite element based on the theory of Timoshenko can be used to solve a wide range of problems of statics and dynamics, including the gyroscopic effect, e.g. in the area of research and development of the microelectromechanical sensors of the inertial information.

Reduced-order modeling for rotating rotor-bearing systems with cracked impellers using three-dimensional finite element models

A novel reduced-order modeling method is presented in this paper for dynamics analysis of rotating impeller–shaft-bearing assembly with cracked impellers. Based on three-dimensional finite element model, the complex component mode synthesis (CMS) method is employed to generate an efficient reduced-order model (ROM) for studying the effects of crack on the global vibration of the rotating assembly. First, a modeling framework for impeller–shaft-bearing systems in rotating frame is presented. Rotational effects, including Coriolis matrix and centrifugal softening, have been taken into account. Then, the governing equation of motion of the damped gyroscopic system is reduced by the complex CMS method. Finally, the obtained ROM is employed to study the effects of crack on assembly׳s vibration. During the steady-state response analysis, external excitations on the impeller due to rotor–stator interactions have been taken into account, which was however neglected in previous investigations on rotordynamics. Numerical results show that the lower-order eigenvalues and the unbalance response of the assembly are not sensitive to the local crack on impeller. Nevertheless, the flexible coupling between impeller and shaft becomes more complex when the air flow-induced excitations are considered. Under EO1 traveling wave excitations, a crack leads to slight changes in the assembly׳s response. In contrast, the effect of crack becomes significant when the assembly is excited by EO2 and higher EO excitations. Moreover, the nonlinear crack breathing effects affect the assembly׳s response obviously. Finally, a potential technique for detecting the crack on impeller during operation is discussed.

A Trajectory Tracking Robust Controller of Surface Vessels With Disturbance Uncertainties

This brief considers the problem of trajectory tracking control for marine surface vessels with unknown time-variant environmental disturbances. The adopted mathematical model of the surface ship movement includes the Coriolis and centripetal matrix and the nonlinear damping terms. An observer is constructed to provide an estimation of unknown disturbances and is applied to design a novel trajectory tracking robust controller through a vectorial backstepping technique. It is proved that the designed tracking controller can force the ship to track the arbitrary reference trajectory and guarantee that all the signals of the closed-loop trajectory tracking control system of ships are globally uniformly ultimately bounded. The simulation results and comparisons illustrate the effectiveness of the proposed controller and its robustness to external disturbances.