Underactuated Systems Research Articles

Geometric Extended State Observer on [formula omitted] with Fast Finite-Time Stability: Theory and Validation on a Multi-rotor Vehicle

This article presents an extended state observer for a vehicle modeled as a rigid body in three-dimensional translational and rotational motions. The extended state observer is applicable to a multi-rotor aerial vehicle with a fixed plane of rotors, modeled as an under-actuated system on the state-space TSE(3), the tangent bundle of the six-dimensional Lie group SE(3). This state-space representation globally represents rigid body motions without singularities. The extended state observer is designed to estimate the resultant external disturbance force and disturbance torque acting on the vehicle. It guarantees stable convergence of disturbance estimation errors in finite time when the disturbances are constant, and finite time convergence to a bounded neighborhood of zero errors for time-varying disturbances. This extended state observer design is based on a Hölder-continuous fast finite time stable differentiator that is similar to the super-twisting algorithm, to obtain fast convergence. Numerical simulations are conducted to validate the proposed extended state observer. The proposed extended state observer is compared with other existing research to show its advantages. A set of experimental results implementing disturbance rejection control using feedback of disturbance estimates from this extended state observer is also presented.

Simplified internal models in human control of complex objects

Humans are skillful at manipulating objects that possess nonlinear underactuated dynamics, such as clothes or containers filled with liquids. Several studies suggested that humans implement a predictive model-based strategy to control such objects. However, these studies only considered unconstrained reaching without any object involved or, at most, linear mass-spring systems with relatively simple dynamics. It is not clear what internal model humans develop of more complex objects, and what level of granularity is represented. To answer these questions, this study examined a task where participants physically interacted with a nonlinear underactuated system mimicking a cup of sloshing coffee: a cup with a ball rolling inside. The cup and ball were simulated in a virtual environment and subjects interacted with the system via a haptic robotic interface. Participants were instructed to move the system and arrive at a target region with both cup and ball at rest, ’zeroing out’ residual oscillations of the ball. This challenging task affords a solution known as ‘input shaping’, whereby a series of pulses moves the dynamic object to the target leaving no residual oscillations. Since the timing and amplitude of these pulses depend on the controller’s internal model of the object, input shaping served as a tool to identify the subjects’ internal representation of the cup-and-ball. Five simulations with different internal models were compared against the human data. Results showed that the features in the data were correctly predicted by a simple internal model that represented the cup-and-ball as a single rigid mass coupled to the hand impedance. These findings provide evidence that humans use simplified internal models along with mechanical impedance to manipulate complex objects.

Mathematical modeling of an articulated flying vehicle for precise control analysis in grasping highly oscillatory objects

In this research work, a detailed mathematical modeling approach has been taken in order to develop a complete package for the investigation of phenomena associated with the flight dynamics and control of an articulated aerial robot in grasping high weight oscillatory objects such as living creatures. To this end, the origin of forces and moments generated by a moving target is analyzed and the nonlinear differential equations of the system are derived. Then, the model is verified and the effect of target oscillations on the whole system dynamics is investigated. In the next step, due to the nonlinear nature of this under-actuated system and the presence of the force-generating target with unknown dynamics and model parametric and non-parametric uncertainties, two versions of adaptive sliding mode control (A-SMC) are designed to enable the system follows the nominal trajectories in a desirable way without needing to know the upper bounds of the time-varying forces and uncertainties. In this regard, the stability of closed-loop systems is proved based on Lyapunov theory. The performance and robustness of the designed controllers are evaluated and compared through numerical and Monte Carlo simulations with some standard criteria. At the end, the conclusion of this work is presented.

Optimal Periodic Impulsive Control for Orbital Stabilization of Underactuated Systems

Optimal Periodic Impulsive Control for Orbital Stabilization of Underactuated Systems

Constraint-following control design for nonlinear underactuated belt conveyors with uncertainty

There are many mismatched uncertainties in the nonlinear underactuated belt conveyor systems which leads to challenges for designing the controllers. This paper addresses this issue based on the concept of constraint-following by establishing the dynamic model and treating the desired trajectory as a constraint. First, the nominal control portion is studied in the absence of the deviation of initial condition and uncertainties. Second, the uncertainties are decomposed into matched and mismatched parts according to the match condition. Then, the robust constraint-following control portion that is solely based on matched uncertainty is designed. By utilizing the Lyapunov method, the proposed control with two portions is able to ensure the uniform boundedness and uniform ultimate boundedness of the underactuated systems with uncertainties. Simulations are additionally carried out for the purpose of verification.

Adaptive direct RBFNN consensus control for a class of unknown nonlinear underactuated systems

The consensus tracking control of a class of nonlinear underactuated multi-agent systems with uncertainties is studied in this paper. A radial basis function neural network (RBFNN)-based direct adaptive control algorithm is designed. Unlike many previous articles in which a neural network is employed to identify the unknown nonlinear functions in the system model or controller, this method directly approximates the ideal control law by a neural network, making the control law simpler. Based on the neural network direct control algorithm, a controller with a single-parameter learning scheme is designed. This new control method reduces the computational burden by reducing the amount of computational data. Finally, the closed-loop system is proved to be ultimately uniformly bounded stable; the application examples of the controllers are given by simulation.

Discrete time stability of augmented Lagrangian formalism based underactuated inverse dynamics control method

Stability problems of robotic systems arise sometimes suddenly, seemingly for no reason. The digital time sampling is often the main cause of these instabilities. Discrete models, which are capable of the prediction of stability, are available for low-degree-of-freedom template models of linear position and force control. However, for the inverse dynamics control of underactuated systems, the literature has a lack of generally applicable results related to the effect of time discretization. Several control approaches are available in the literature out of which a widely used one, the augmented Lagrangian formalism and its stability properties are analysed in this work. Theoretical stability properties are obtained for a generally usable, linear, underactuated, two-degree-of-freedom constrained template model. The actuator dynamics, the finite difference approximation of the feedback velocity and the filtering of the feedback data are considered in the model. These phenomena strongly affects the stability properties. The theoretically obtained stability maps are experimentally validated on an underactuated crane-like indoor robot. The position and orientation accuracy of the robot were assessed: the absolute position error was below 30 mm and the orientation error was below 3°.

Anti-swing position control of super-twisting sliding mode for underactuated tower crane based on nonlinear marine predator algorithm

Due to the nonlinear, underactuated, and strongly coupled characteristics of tower cranes, issues such as inaccurate load positioning and severe swinging are prone to occur during transportation, making the control problem extremely complex. To address these issues, this paper has designed a super-twisting robust sliding mode controller based on the nonlinear marine predator algorithm (NMPA) to ensure the accurate positioning of the desired location during the transportation process of the tower crane while reducing the oscillation of the load. First, a nonlinear dynamic model of a four-degree-of-freedom tower crane considering friction is constructed. Second, to address the difficulty of effectively controlling the underactuated tower crane system model due to its nonlinearity and strong coupling, the super-twisting algorithm is introduced, and a super-twisting sliding mode controller for anti-swing positioning during the crane transportation process is designed. Then, the parameters of the super-twisting sliding mode controller are optimized using the NMPA to suppress sliding mode chattering and quickly achieve the system’s steady state. Finally, the stability of the nonlinear tower crane system is analyzed using Lyapunov’s theory and LaSalle’s invariance principle to ensure the closed-loop stability of the error dynamics. The simulation results show that the proposed control method exhibits excellent load anti-sway performance and robustness.

TRAJECTORY TRACKING CONTROL OF A TWO WHEELED SELF-BALANCING ROBOT BY USING SLIDING MODE CONTROL

Two-Wheeled Self-Balancing Robots are widely used in various fields today. These systems have a highly unstable nature due to their underactuated structures. On the other hand, parameter uncertainties and external disturbances significantly affect their control performance. The best way to deal with parameter uncertainties that can easily lead controllers to instability is to use robust control methods. Dealing with these uncertainties is particularly crucial in control of underactuated and unstable systems such as Two-Wheeled Self-Balancing Robots. In this study, trajectory tracking control of a two wheeled self-balancing robot by using Sliding Mode Control (SMC) was realized. The chattering problem inherent in the SMC method was eliminated by employing tangent hyperbolic (tanh) switching function instead of signum function. The performance of the SMC controller has been examined under five different cases including external disturbance and various parameter uncertainties and compared with PID and LQR methods. The results showed that the SMC method is much more insensitive to parameter changes than the PID and LQR methods. It has also been observed that all three controllers maintain their stability against disturbance inputs, but the SMC method offers a better control performance.

Multiswitching Surface Based Sliding Mode Controller for a Class of Underactuated Nonlinear Systems: With Application to Ball and Plate System

Multiswitching Surface Based Sliding Mode Controller for a Class of Underactuated Nonlinear Systems: With Application to Ball and Plate System

Finite-Time Neural Network-Based Hierarchical Sliding Mode Antiswing Control for Underactuated Dual Ship-Mounted Cranes With Unmatched Sea Wave Disturbances Suppression.

As typical mechanical transportation equipment, cooperative dual ship-mounted cranes are widely used to transport large goods or containers in the marine environment. However, the control problem of the dual ship-mounted crane system is much more complex due to its underactuated characteristic and persistent unmatched disturbances. To solve these problems, we propose a novel neural network (NN)-based hierarchical sliding mode adaptive (HSMA) control method in this article. More specifically, an appropriate hierarchical sliding mode surface is first designed to connect the actuated and underactuated system state variables effectively. At the same time, the NNs are constructed to compensate for the unmatched interference of ship motions induced by sea waves simultaneously. Not only can the booms and the rope lengths reach their desired positions in finite time, but also the synchronous swing angles of the payload can be effectively eliminated. The asymptotic convergence of the closed-loop system's equilibrium points is achieved through rigorous mathematical proofs. Furthermore, the stability of each sliding mode surface is also analyzed utilizing the Lyapunov technique and Barbalat's lemma. Finally, numerous groups of compared numerical simulation results are investigated to further show the effectiveness and strong robustness of the proposed NN-based HSMA controller.

Barrier Function-Based Adaptive Composite Sliding Mode Control for a Class of MIMO Underactuated Systems Subject to Disturbances

Barrier Function-Based Adaptive Composite Sliding Mode Control for a Class of MIMO Underactuated Systems Subject to Disturbances

Motion planning in underactuated systems with impulsive phenomenon via dynamic shaping of virtual holonomic constraints

Rhythmic motions are traditionally achieved by developing predetermined paths for the states of the space system to follow. Since these paths are obtained offline, the dynamic behavior fails to adapt to changes in environmental conditions or user command desires. The solution we propose is a new strategy called dynamic shaping, in which the paths are formed online to allow the system to create an orbit with the characteristics we need. Hereupon, this paper focuses on applying this strategy to Dynamics with One Degree of Under-actuation and Impulsive Phenomenon (DSODUIP) to adapt the characteristics of outcomes to be in line with the demands.This research was conducted by leveraging the advantages of virtual holonomic constraints (VHCs) to establish these paths. Therefore, a novel two-level hierarchical control method is designed considering a stability criterion to avoid divergence. At the Low-Level, the controllers stabilize the output of system to follow the VHCs on the system. At the High-Level, the VHCs are modified to shape an orbit with our desired characteristic in the motion. As an illustrative example, the algorithm is implemented to adjust the average angular velocity of a devil stick and the hip velocity of a Three-Link biped robot. Their results vividly demonstrate smooth adjustments and efficient performance in achieving our desired outcomes.

Hierarchical attitude stabilization controller design and analysis for underactuated spacecraft on SO(3)

In this paper, the complete attitude stabilization of underactuated spacecraft with two parallel single gimbal control moment gyroscopes (SGCMGs) is explored on 3-dimensional special orthogonal group (SO(3)) and its corresponding Lie algebra (so(3)). Instead of assumption on zero total angular momentum in existing article, a less conservative criterion is adopted to ensure system controllability, which does not simplify system dynamics. On the kinematic level, an ideal controller is proposed to globally stabilize attitude without singular initial condition and stagnation behavior. Besides, the ideal kinematic controller is developed on so(3), which avoids singularity or unwinding phenomenon caused by other attitude parameters. On the dynamic level, a hierarchical controller consisting of two parts is proposed. The high-level sliding mode part enforces finite time stabilization of angular velocity about underactuated axis while the low-level part tracks ideal angular velocity commands about actuated axes. A rigorous proof of the whole dynamic-kinematic system that has not been explored before is given. Analysis of hierarchical control from the perspective of linear algebra provides a novel insight into controller design for underactuated systems. Furthermore, the paradigm of controller design which is applicable to other underactuated systems is derived. Simulation results validate the effectiveness of the proposed control logic.

A Gradient Dynamics-Based Singularity Avoidance Method for Backstepping Control of Underactuated TORA Systems.

In this paper, a gradient dynamics-based control method is proposed to directly tackle the singularity problem in the backstepping control design of the TORA system. This method is founded upon the construction of an energy-like positive function, which includes an auxiliary variable in terms of the intermediate virtual control law. On this basis, a gradient dynamics is created to obtain a new virtual control command, which is capable of making the auxiliary variable gradually approach zero, thereby mitigating the issue of division by zero. The core innovation is the integration of the gradient dynamics into the recursive backstepping design to overcome the singularity problem and stabilize the system at the equilibrium quickly. In addition, it rigorously proves that all the signals in the closed-loop control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero through a Lyapunov-based stability analysis. Comparative simulations demonstrate that the proposed approach not only avoids the singularity issue, but also achieves a better transient performance over other methods.

Output-feedback stabilization control for a class of underactuated systems via high-order sliding modes identification and compensation

We present a robust output-feedback control method for stabilizing a class of underactuated systems despite the effect of external perturbations and using only output information. Towards this aim, first, an extended high-order sliding-mode observer estimates the states and identifies the perturbations, theoretically, in finite time. The controller design relies on partial feedback linearization, compensating the coupled effect of the perturbations while stabilizing the internal dynamics. As a result, the closed-loop system is robust against external perturbations, and the system trajectories converge to a region around the origin. Depending on the system structure and control goal, two different formulations are proposed: robust non-collocated and robust collocated partial feedback linearization. The controller synthesis avoids nonlinear system transformations. The stability of the observer–controller scheme is done using the Lyapunov and input-to-state stability theories. Experimental results on a cart–pendulum system show the approach’s feasibility.

State Feedback Via Judicious Pole Placement and Linear Quadratic Regulator – Application to Rotary Inverted Pendulum

In the control system regulatory concept, placing closed loop poles too far from the origin in the stability region produces fast regulation time but requires huge forcing energy as a trade-off. As such, stabilizing an unstable system with minimum energy is needed, though this presents a challenge to the designer. At the design phase, the designer may ponder the optimized energy while compromising the possibility of catastrophic stabilization phenomena due to minimal forcing thrust towards the poles. In this manuscript, a simple Linear Quadratic Regulator (LQR) is proposed as an alternative to full state feedback (FSF) with judicious pole placement. The efficacy of both approaches was observed by exploiting a Rotary Inverted Pendulum (RIP) as a testbed. Beforehand, the RIP system dynamics were developed in the time domain. RIP is an under-actuated mechanical system that is inherently nonlinear and unstable. The main control objectives of RIP are swing-up control, stabilization control, switching control, and trajectory control. The methodology involved the appearance of weighted matrices that were necessary for the minimum cost function. The Riccati and Lyapunov criteria are also exploited to facilitate design. The result shows the comparative transient performances of the two approaches, where the LQR outperforms the FSF in many aspects.

Robust adaptive optimal trajectory tracking control for underactuated AUVs with position and velocity constraints in three‐dimensional space

AbstractThe safety and optimality of underactuated autonomous underwater vehicles (AUVs) during operations are essential factors to consider. In this context, a three‐dimensional robust adaptive optimal trajectory tracking control method under position and velocity constraints, unknown dynamics, and environmental disturbances is proposed. The main features of the method are: (1) The outputs of an underactuated AUV system are redefined to handle the underactuation problem. (2) The system with position and velocity constraints is transformed into an unconstrained system by a nonlinear state‐dependent transformation. (3) A critic‐identifier architecture is constructed using adaptive dynamic programming and neural networks in a backstepping framework. Specifically, critic networks and weight update laws without requiring initial stability control are designed to solve Hamilton‐Jacobi‐Bellman equations in kinematic and dynamic subsystems, and optimal virtual and actual control laws are obtained. (4) A neural network identifier is developed to estimate unknown dynamics. Disturbances are overcome by improving the cost function and solving for optimal control of the nominal dynamic subsystem. By stability analysis, tracking errors in the AUV closed‐loop system can converge to an arbitrarily small compact set of the origin, and the other signals are uniformly ultimately bounded. Simulation comparisons demonstrate the effectiveness and superiority of the proposed 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

Tilt-X: Development of a Pitch-Axis Tiltrotor Quadcopter for Maximizing Horizontal Pulling Force and Yaw Moment

In recent years, there has been a significant amount of research on tiltrotor multicopter unmanned aerial vehicles (TM-UAVs) in aerial robotics. Despite the varying frame types of TM-UAVs, they all still aim to decouple the propeller from the body, which means that the propeller’s attitude control is independent of the body’s attitude control. On the one hand, this solves the issue of multicopter unmanned aerial vehicles (M-UAVs) being limited by small roll and pitch angles, thereby improving flight performance. On the other hand, it addresses the drawbacks of M-UAVs as typical underactuated systems. However, the fact still remains that it cannot significantly change thrust direction, thus providing the necessary wrench direction for aerial manipulation. This paper presents a pitch-axis tiltrotor quadcopter unmanned aerial vehicle (UAV) design named Tilt-X, which can maximize horizontal pulling force and yaw moment when used as an aerial manipulator. This design contributes to tasks such as pushing, pulling, and twisting. The reliability of the design has been demonstrated through dynamic modeling and experimental validation.