A unified solution to swing-up control for n-link planar robot with single passive joint based on virtual composite links and passivity

This paper concerns the swing-up control of an n-link revolute planar robot with any one of the joints being passive. The goal is to design and analyze a swing-up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point, at which all the links are in the upright position. We present a unified solution based on the notion of virtual composite link (VCL), which is a virtual link made up of one or more active links. By using the angles of two series of VCLs separated by the passive joint and using the total mechanical energy of the robot, we design a swing-up controller and analyze the global motion of the robot under the controller. The main new results of this paper are: (1) we obtain a lower bound for each control gain related to the angle of each VCL such that the closed-loop system has only one undesired equilibrium point in addition to the upright equilibrium point, and we present an original proof of the conditions on the control gains for a class of n-link underactuation-degree-one planar robots with an active first joint; (2) we provide a bigger control gain region for achieving the control objective than those of previous results on three- and n-link robots with a passive first joint; (3) we validate the theoretical results via numerical simulations on a 4-link robot with the passive joint in each of the four positions. This paper gains an insight into the passivity-based control of underactuated multiple-DOF systems.     

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.     

A simple and quick control strategy for a class of first-order nonholonomic manipulator

This paper presents a simple and quick control strategy for a class of first-order nonholonomic manipulator: planar n-link manipulator with a passive first joint (no gravity). The control target is driving its endpoint from any initial position to any target position. First, we reduce the planar n-link manipulator to a planar three-link one by maintaining the states (angles and angular velocities) of n-3 active links in the initial value all the time. That is, we only adjust the states of the remaining two active links in the whole control process, and these two adjusted active links are chosen to guarantee that the target position is in the reachable region of the planar n-link manipulator by using the enumeration method. Then, we divide the whole control process into two stages for the reduced planar three-link manipulator. In each stage, the manipulator becomes a planar two-link one by maintaining the states of one adjusted active link to be the constant value. The state constraint existing between the passive first link and the adjusted active link is obtained by using the integral characteristics of a planar two-link manipulator. Meantime, the geometric constraint between the position of the endpoint and angles of all joints is obtained based on the homogeneous coordinate transformation method. According to the above two kinds of constraint, the target angles of the two adjusted active links are calculated by employing the particle swarm optimization algorithm. When the two adjusted active links are controlled to their target angles in turn, the control target of the planar n-link manipulator is completed. Finally, simulation results demonstrate that the proposed control strategy is valid and rapid.     

Stabilization of underactuated four-link gymnast robot using torque-coupled method

In this paper, we use an underactuated four-link gymnast robot (UFGR) with a passive first joint to model a gymnast on the bar, and consider the stabilization of the UFGR at the straight-up position. First, we introduce a coupled relationship between control torques for the UFGR. It decouples some state variables of the UFGR from others and changes the nonlinear UFGR system into a cascade-connected system. And then, we use an energy-based method to design a stabilizing controller for the zero dynamics of the cascade-connected system. And the triangle lemma guarantees the control objective of the UFGR to be achieved. The torque-coupled method transforms the stabilization of an underactuated four-link manipulator into that of an underactuated two-link acrobot. This makes the structure of the control system simple. Moreover, our proposed control strategy is easy to extend to the stabilization control of other multi-input nonlinear underactuated systems. Simulation results using the characteristics parameters of a gymnast demonstrate the validity of the proposed method.     

A proposed hybrid neural network for position control of a walking robot

The use of a proposed recurrent neural network control system to control a four-legged walking robot is presented in this paper. The control system consists of a neural controller, a standard PD controller, and the walking robot. The robot is a planar four-legged walking robot. The proposed Neural Network (NN) is employed as an inverse controller of the robot. The NN has three layers, which are input, hybrid hidden and output layers. In addition to feedforward connections from the input layer to the hidden layer and from the hidden layer to the output layer, there is also a feedback connection from the output layer to the hidden layer and from the hidden layer to itself. The reason to use a hybrid layer is that the robot’s dynamics consists of linear and nonlinear parts. The results show that the neural-network controller can efficiently control the prescribed positions of the stance and swing legs during the double stance phase of the gait cycle after sufficient training periods. The goal of the use of this proposed neural network is to increase the robustness of the control of the dynamic walking gait of this robot in the case of external disturbances. Also, the PD controller alone and Computed Torque Method (CTM) control system are used to control the walking robot’s position for comparison.     

Trajectory planning and tracking control for autonomous bicycle robot

Control of the autonomous bicycle robot offers considerable challenges to the field of robotics due to its nonholonomic, underactuated, and nonminimum-phase properties. Furthermore, instability and complex dynamic coupling make the trajectory planning of the bicycle robot even more challenging. In this paper, we consider both trajectory planning and tracking control of the autonomous bicycle robot. The desired motion trajectory of the contact point of the bicycle’s rear wheel is constructed using the parameterized polynomial curve that can connect two given endpoints with associated tangent angles. The parameters of the polynomial curve are determined by minimizing the maximum of the desired roll angle’s equilibrium of the bicycle, and this optimization problem is solved by the particle swarm optimization algorithm. Then, a control scheme that can achieve full-state trajectory tracking while maintaining the bicycle’s balance is proposed by combining a planar trajectory tracking controller with a roll angle balance controller. Simulation results are presented to demonstrate the effectiveness of the proposed method.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

索牵引并联机器人静态平衡位置的正解研究

研究了一种大跨度但运动缓慢的索牵引并联机器人的位置正解问题,此时可认为系统处于静平衡状态,但必须考虑柔索垂度的影响。索牵引并联机器人的静态平衡位置的正解就是确定机器人末端执行器的位置和姿态,使得系统能在给定的柔索长度下处于静平衡状态。这涉及到系统的非线性力学分析而不仅仅是一个几何问题。本文采用悬链线方程描述柔索的力学特性,结合索端位移与机器人末端执行器位移之间的关系推导出机器人的末端执行器位移与索长变化之间的增量关系,据此提出一种迭代策略完成正解计算。数值算例验证了本文方法的有效性。     

Finite-time output feedback control for a class of second-order nonlinear systems with application to DC–DC buck converters

The problem of output feedback control for a class of second-order nonlinear systems is investigated in this paper. Using the techniques of finite-time control and finite-time convergent observer, an observer-based finite-time output feedback controller is proposed which can guarantee that the system’s state converges to the equilibrium in a finite time. As an application of the proposed theoretical results, the problem of finite-time control without current signal for the DC–DC buck converters is solved. Simulation results are given to demonstrate the effectiveness of the proposed method.     

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

We study three-dimensional Westervelt model of a nonlinear hydroacoustics without dissipation. We received all of its invariant submodels. We studied all invariant submodels described by the invariant solutions of rank 0 and 1. All invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. With a help of these invariant solutions we researched: (1) a propagation of the intensive acoustic waves (self-similar, axisymmetric, planar and one-dimensional) for which the acoustic pressure and a speed of its change, or the acoustic pressure and its derivative in the direction of one of the axes are specified at the initial moment of the time at a fixed point , (2) a spherically symmetric ultrasonic field for which the acoustic pressure and a speed of its change, or the acoustic pressure and its radial derivative are specified at the initial moment of the time at a fixed point. Solving of the boundary value problems describing these processes is reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. We found all the conservation laws of the first order for the Westerveld equation written in dimensionless variables.     

Input torque sensitivity to uncertain parameters in biped robot

Input torque is the main power to maintain bipedal walking of robot, and can be calculated from trajectory planning and dynamic modeling on biped robot. During bipedal walking, the input torque is usually required to be adjusted due to some uncertain parameters arising from objective or subjective factors in the dynamical model to maintain the pre-planned stable trajectory. Here, a planar 5-link biped robot is used as an illustrating example to investigate the effects of uncertain parameters on the input torques. Kine-matic equations of the biped robot are firstly established by the third-order spline curves based on the trajectory planning method, and the dynamic modeling is accomplished by taking both the certain and uncertain parameters into account. Next, several evaluation indices on input torques are intro-duced to perform sensitivity analysis of the input torque with respect to the uncertain parameters. Finally, based on the Monte Carlo simulation, the values of evaluation indices on input torques are presented, from which all the robot param-eters are classified into three categories, i.e., strongly sensi-tive, sensitive and almost insensitive parameters.     

An asymptotically stable collision-avoidance system

Artificial potential fields, which are widely used in robotics for path planning and collision avoidance, are normally beset by difficulties arising from the existence of local minima. This article proposes a solution that involves an asymptotically stable point-mass system governed by differential equations. The system represents a planar point robot moving from its initial position to the desired goal whilst avoiding a static obstacle. Because the system is asymptotically stable, its Lyapunov function, which produces artificial potential fields around the goal and the obstacle, has no local minima other than the goal configuration in the pathwise-connected proper subset of free space which contains the goal configuration. As an application, we consider the point stabilization of a planar mobile car-like robot moving in the presence of a static obstacle.     

Invariant submodels and exact solutions of Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics with dissipation

We study three-dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) model of the nonlinear hydroacoustics with dissipation. This model is described by third order quasilinear partial differential equation of the (KZK). We obtained that the (KZK) equation admits an infinite Lie group of the transformations, depending on the three arbitrary functions. This is due to the fact that in the (KZK) model the main direction of the wave’s propagation is singled out. The submodels of the (KZK) model.are described by the invariant solutions of the (KZK) equation. We studied essentially distinct, not linked by means of the point transformations, invariant solutions of rank 0 and 1 of this equation. Also we considered the invariant solutions of rank 2 and 3. The invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the invariant solutions that we called by “Ultrasonic knife” and “Ultrasonic destroyer”. The submodel “Ultrasonic knife” have the following property: at each fixed moment of the time in the field of the existence of the solution near a some plane the pressure increases indefinitely and becomes infinite on this plane. The submodel “Ultrasonic destroyer” contains a countable number of “Ultrasonic knives”. The presence of the arbitrary constants in the integro-differential equations, that determine invariant solutions of rank 1 provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original (KZK) model. With a help of these invariant solutions we researched a propagation of the intensive acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change, or the acoustic pressure , speed and acceleration of its change in the radial direction, or the acoustic pressure, speed and acceleration of its change in the direction of one of the axes are specified at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solutions of boundary value problems, describing these wave processes. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. Application of the obtained formula generating the new solutions for the found solutions gives families of the solutions containing three arbitrary functions.     

Planning motion trace of robot by spinor method

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Orientation-error observer-based tracking control of nonholonomic mobile robots

In this paper, a novel trajectory tracking controller is proposed for mobile robots with unknown orientation angle by employing the orientation-error observer (OEO). In order to overcome the local stability resulted from linearization design methods, an asymptotically stable controller is designed using Lyapunov’s direct method. This method breaks down nonlinear systems into low-dimensional systems and simplifies the controller design using virtual auxiliary error function and partial Lyapunov functions. A state-feedback controller for the nonlinear error dynamics of the mobile robot is combined with an observer that estimates the orientation-error based on available trajectory information and measurement of the position coordinates. The stability of the system is easily proved via the Lyapunov theory. Abundant simulation and experiment results validate the effectiveness and superiority of the proposed control method.     

Adaptive neural network control of bilateral teleoperation with constant time delay

This paper proposes a novel approach for bilateral teleoperation systems with a multi degrees-of-freedom (DOF) nonlinear robotic system on the master and slave side with constant time delay in a communication channel. We extend the passivity based architecture to improve position and force tracking and consequently transparency in the face of offset in initial conditions, environmental contacts and unknown parameters such as friction coefficients. The proposed controller employs a stable neural network on each side to approximate unknown nonlinear functions in the robot dynamics, thereby overcoming some limitations of conventional controllers such as PD or adaptive controllers and guaranteeing good tracking performance. Moreover, we show that this new neural network controller preserves the control passivity of the system. Simulation results show that NN controller tracking performance is superior to that of conventional controllers.     

RECURRENT NEURAL NETWORK MODEL BASED ON PROJECTIVE OPERATOR AND ITS APPLICATION TO OPTIMIZATION PROBLEMS

The recurrent neural network (RNN) model based on projective operator was studied. Different from the former study, the value region of projective operator in the neural network in this paper is a general closed convex subset of n-dimensional Euclidean space and it is not a compact convex set in general, that is, the value region of projective operator is probably unbounded. It was proved that the network has a global solution and its solution trajectory converges to some equilibrium set whenever objective function satisfies some conditions. After that, the model was applied to continuously differentiable optimization and nonlinear or implicit complementarity problems. In addition, simulation experiments confirm the efficiency of the RNN.     

Lyapunov-Based Controller for the Inverted Pendulum Cart System

A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.Contributed by Prof. F. Pfeiffer.