Continuum of Motion Equations and Control Laws for the Inverted Pendulum Cart and Rotary Pendulum

2020 ◽  
Author(s):  
Constance Lare ◽  
Warren N. White
Keyword(s):  
Inverted Pendulum ◽  
Equations Of Motion ◽  
Control Law ◽  
Control Laws ◽  
Motion Equations ◽  
Base Radius ◽  
Limiting Condition ◽  
And Control ◽  
The Given ◽  
Insight Into

Abstract This paper questions whether the controller properties for a given rigid body mechanical system still apply as the given system is changed. As a first attempt in this investigation, the controller for the underactuated rotary pendulum is investigated as the system morphs into an underactuated inverted pendulum cart. As the limiting condition of the inverted pendulum cart is approached, the investigation allows the controller to also morph. The authors show that, as the pendulum base radius grows, the rotary pendulum equations of motion morph into the inverted pendulum cart dynamics. The paper presents necessary conditions for the successful morphing of the dynamic equations. The morphing process for the controller tests the idea whether the control law also satisfies the same continuum basis as the motion equations. The paper presents a framework for the class of controllers investigated for providing insight into when the controller morphing may be successful. This paper presents dimensionless quantities that render the equations of motion and controller for the inverted pendulum cart and rotary pendulum into dimensionless form. These dimensionless quantities allow comparison of controllers and systems that are not possible through simple inspection. This comparison ability is especially useful for quantifying the nonlinearities of a given system and controller compared to another system and controller having different parameter sizes, a comparison rarely seen in the control literature.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

Robust Trajectory Following Control of Robotic Systems

1985 ◽  
Vol 107 (4) ◽  
pp. 308-315 ◽  
Author(s):  
S. N. Singh ◽  
A. A. Schy
Keyword(s):  
Degrees Of Freedom ◽  
Digital Simulation ◽  
Robotic Systems ◽  
Control Law ◽  
Robot Arm ◽  
Control Laws ◽  
Payload Uncertainty ◽  
Trajectory Following ◽  
And Control ◽  
Three Degrees Of Freedom

Using an inversion approach we derive a control law for trajectory following of robotic systems. A servocompensator is used around the inner decoupled loop for robustness to uncertainty in the system. These results are applied to trajectory control of a three-degrees-of-freedom robot arm and control laws Cθ and CH for joint angle and position trajectory following, respectively, are derived. Digital simulation results are presented to show the rapid trajectory following capability of the controller in spite of payload uncertainty.

On Aero Control Law Joined Compensatory PID for Glide down to Fixed Low Altitude System and Simulation Researches

2012 ◽  
Vol 198-199 ◽  
pp. 1021-1024
Author(s):  
De Hai Yu ◽  
Dong Cai Qu ◽  
Jian Hua Lu ◽  
Bin Wen Lu
Keyword(s):  
Control Law ◽  
Simple Analysis ◽  
Control Laws ◽  
Control Scheme ◽  
Improve Accuracy ◽  
Working Principle ◽  
Simulation Results ◽  
Low Altitude ◽  
Constant Altitude ◽  
And Control

In order to improve accuracy of constant altitude fly at low altitude and fly track of glide down to fixed altitude, aeroplane’s control scheme of glide down to fixed low altitude with PID compensatory link were designed. At the same time, the corresponding control laws had been designed. After simple analysis about working principle of the aeroplane’s control system, simulation researches were done to optimize designed control laws, so that achieving expectant requirement. Simulation results show that designed control scheme and control law were accurate and effective.

Application of the Nonlinear Tschauner-Hempel Equations to Satellite Relative Position Estimation and Control

2017 ◽  
Vol 71 (1) ◽  
pp. 44-64 ◽  
Author(s):  
Ranjan Vepa
Keyword(s):  
Relative Motion ◽  
Position Estimation ◽  
True Anomaly ◽  
Linear Quadratic ◽  
Control Laws ◽  
Motion Equations ◽  
Motion Models ◽  
Estimation And Control ◽  
Oblateness Effect ◽  
And Control

In this paper we develop the nonlinear motion equations in terms of the true anomaly varying Tschauner–Hempel equations relative to a notional orbiting particle in a Keplerian orbit, relatively close to an orbiting primary satellite to estimate the position of a spacecraft. A second orbiting body in Earth orbit relatively close to the first is similarly modelled. The dynamic relative motion models of the satellite and the second orbiting body, both of which are modelled in terms of independent relative motion equations, include several perturbing effects, such as the asymmetry of the Earth gravitational field resulting in the Earth's oblateness effect and the third body accelerations due to the Moon and the Sun. Linear control laws are synthesised for the primary satellite using the transition matrix so it can rendezvous with the second orbiting body. The control laws are then implemented using the state estimates obtained earlier to validate the feedback controller. Thus, we demonstrate the application of a Linear Quadratic Nonlinear Gaussian (LQNG) design methodology to the satellite rendezvous control design problem and validate it.

Reciprocal Variable Feedback: Induced Sensing for Nonlinear Systems Design and Control

1998 ◽  
Vol 120 (2) ◽  
pp. 157-163
Author(s):  
Aristides Gogoussis ◽  
Max Donath
Keyword(s):  
Control Theory ◽  
Feedback Linearization ◽  
Systems Design ◽  
Physical Reality ◽  
Physical Processes ◽  
Control Laws ◽  
Physical Systems ◽  
Broad Variety ◽  
And Control ◽  
Insight Into

System performance can be significantly improved when both the design of the plant and of the controller are considered concurrently. Control theory can be applied to a broad variety of systems, including those that are physical in nature and many that are not. Despite the generality of control theory, there are many situations in which opportunities are missed for using less conservative control laws and simpler overall implementations. This is due to the use of formulations that do not explicitly reveal the existence of intrinsic information pertaining to the particular domain of application. Such is the case with many physical systems. However, the various constraints associated with physical reality (in the form of principles, laws, etc.) open up several possibilities which can be exploited for system design and control. In this paper, we propose the Reciprocal Variable Feedback principle as a means for facilitating the control of plants with complicated nonlinear dynamics in the presence of parameter and/or structural uncertainty. The RVF principle exploits the effort-flow relationships associated with power interactions in order to assist in the design and control of physical processes. This is accomplished by using appropriate sensors instead of computation based on models (e.g., feedback linearization) and can be implemented within many physical domains. A motion control example is used to provide insight into the nature of the principle. It is expected that in the future, additional principles will be identified and introduced for integrating design with the control of dynamical systems.

Human Step Rehabilitation Using a Robot Attached to the Pelvis

2004 ◽  
Author(s):  
D. Aoyagi ◽  
W. E. Inchinose ◽  
D. J. Reinkensmeyer ◽  
J. E. Bobrow
Keyword(s):  
Force Control ◽  
Control Algorithm ◽  
Control Law ◽  
Task Space ◽  
Control Laws ◽  
Position Feedback ◽  
Pelvic Motion ◽  
Linear Force ◽  
And Control ◽  
Piston Chamber

This paper describes a new robot capable of manipulating pelvic motion during human step training on a treadmill. The robot, PAM (Pelvic Assist Manipulator), uses two pneumatically actuated subsystems arranged in a tripod configuration to measure and control the pelvis of a person during body weight supported stepping on a treadmill. The device can be used in a back-drivable mode to record pelvic trajectories, either specified manually by a therapist or pre-recorded from unimpaired subjects, then replay these trajectories using a PD position feedback control law in the task space and a non-linear force control algorithm for each piston chamber. The control laws are presented, along with data that demonstrate the ability of the device to record and replay the pelvic motions that occur during normal walking.

Model-Based Sliding Mode Control for a Robot With SMA Actuators

2005 ◽  
Author(s):  
Hashem Ashrafiuon ◽  
Jala Vijay Reddy
Keyword(s):  
Sliding Mode Control ◽  
Sliding Mode ◽  
Equations Of Motion ◽  
Constitutive Law ◽  
Control Law ◽  
Robot Arm ◽  
Control Laws ◽  
Sma Actuators ◽  
Model Based ◽  
Mode Control

This paper presents a model-based sliding mode control law for a planar three-degree-of-freedom robot arm actuated by two rotary Shape Memory Alloy (SMA) actuators and a servomotor. The SMA actuators use a combination of SMA wires and pulleys to produce rotational motion. A model of the robot is developed which combines robot equations of motion with the SMA wire heat convection, constitutive law, and phase transformation equations. Two second-order sliding surfaces are defined leading to derivation of asymptotically stable control laws within the actuation region of the SMA wires. Outside the actuation region, constant inputs are used based on the one-way nature of the SMA actuators. The control law is shown to be effective in several simulations for both set point and trajectory tracking of the robot.

scholarly journals Implementation of automation in vibration-isolating supports of new type for engines of sea and river vessels

2021 ◽  
Vol 2142 (1) ◽  
pp. 012004
Author(s):  
P A Fomichev ◽  
E V Fomicheva
Keyword(s):  
Power Plants ◽  
Vibration Isolation ◽  
Modern Technology ◽  
Control Law ◽  
Control Laws ◽  
Active Systems ◽  
Single Process ◽  
New Type ◽  
Almost All ◽  
And Control

Abstract In almost all areas of modern technology, the problem of vibration isolation arises, which is associated with the goal of improving the quality, reliability and productivity of various technical objects. One of the important tasks of shipbuilding is to reduce vibration levels of ship power equipment. This article proposes a non-linear control law for an electromagnetic hydraulic vibration-insulating support (EGVO) for ship power plants (SEP), which allows you to compensate for external harmonic disturbances at the SEP, which leads to an increase in their technological safety. The proposed control law of EGVO has advantages over various control laws of vibration-insulating systems of an active type for SEP. The proposed method can be used to create vibration-isolating active systems for various technical objects. The essential novelty of this method is a consistent synthesis of control laws and the creation of a single process of technological self-organization and control.

scholarly journals Stabilizing a Rotary Inverted Pendulum Based on Logarithmic Lyapunov Function

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Jie Wen ◽  
Yuanhao Shi ◽  
Xiaonong Lu
Keyword(s):  
Lyapunov Function ◽  
Inverted Pendulum ◽  
Control Method ◽  
Quadratic Function ◽  
Control Law ◽  
Logarithmic Function ◽  
Control Laws ◽  
Lyapunov Control ◽  
Comparative Results ◽  
Rotary Inverted Pendulum

The stabilization of a Rotary Inverted Pendulum based on Lyapunov stability theorem is investigated in this paper. The key of designing control laws by Lyapunov control method is the construction of Lyapunov function. A logarithmic function is constructed as the Lyapunov function and is compared with the usual quadratic function theoretically. The comparative results show that the constructed logarithmic function has higher numerical accuracy and faster convergence speed than the usual quadratic function. On this basis, the control law of stabilizing Rotary Inverted Pendulum is designed based on the constructed logarithmic function by Lyapunov control method. The effectiveness of the designed control law is verified by experiments and is compared with LQR controller and the control law designed based on the quadratic function. Moreover, the system robustness is analyzed when the system parameters contain uncertainties under the designed control law.

scholarly journals Applying Eigenstructure Assignment to Inner-Loop Flight Control Laws for a Multibody Aircraft

2021 ◽  
Author(s):  
Alexander Köthe ◽  
Robert Luckner
Keyword(s):  
Rigid Body ◽  
Flight Control ◽  
Reference Model ◽  
Equations Of Motion ◽  
Unmanned Aircraft ◽  
Control Law ◽  
Eigenstructure Assignment ◽  
Control Laws ◽  
Target Model ◽  
Alternative Technologies

AbstractUnmanned aircraft used as high-altitude platform system has been studied in research and industry as alternative technologies to satellites. Regarding actual operation and flight performance of such systems, multibody aircraft seems to be a promising aircraft configuration. In terms of flight dynamics, this aircraft strongly differs from classical rigid-body and flexible aircraft, because a strong interference between flight mechanic and formation modes occurs. For unmanned operation in the stratosphere, flight control laws are required. While control theory generally provides a number of approaches, the specific flight physics characteristics can be only partially considered. This paper addresses a flight control law approach based on a physically exact target model of the multibody aircraft dynamics rather than conventionally considering the system dynamics only. In the target model, hypothetical spring and damping elements at the joints are included into the equations of motion to transfer the configuration of a highly flexible multibody aircraft into one similar to a classical rigid-body aircraft. The differences between both types of aircraft are reflected in the eigenvalues and eigenvectors. Using the eigenstructure assignment, the desired damping and stiffness are established by the inner-loop flight control law. In contrast to other methods, this procedure allows a straightforward control law design for a multibody aircraft based on a physical reference model.

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