Controllability and stabilization of the programmed motions of a transport robot

A non-linear mathematical model for the motion of a transport robot (TR) with a caterpillar chassis and with drives based on DC motors, which is a non-holonomic electromechanical system, is considered. Non-linear canonical transformations of the coordinates of the state and control space are constructed, which reduce the initial equations of motion of the TR to a simpler canonical form, which is convenient for analysing and synthesizing control systems for the TR. The conditions for the TR to be controllable as a controlled object are found. Algorithms are given for constructing programmed motions (PMs) of the TR. Stabilizing control laws are synthesized under which the PMs of the TR are asymptotically stable and transients of a specified nature are ensured.     

Methods of stabilizing the motions of reversible dynamic systems using non-linear canonical transformations

Methods of synthesizing stabilizing and robust control laws for non-linear reversible systems which ensure asymptotic stability of programmed motions, specified figures of merit and decomposition of transients are considered. Non-linear canonical transformations of state space and the controls are obtained which simplify the synthesis and analysis of the laws of the stabilization of reversible dynamic systems.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Dynamics modeling and experiments of wave driven robot

Wave driven robots (WDRs) take ocean energies as the power sources and are often used for long-term monitoring of the marine environment. The unique multi-body joint structure and special operation mechanism of a WDR make the dynamics modeling problem unusual. The dynamic model of a WDR was put forward by taking the interconnection of forces and motions between the float body (float) and the submerged glider (glider) into account. Numerical simulation of longitudinal motion and the comparison between simulation and tank test of reciprocating steering motion of the "Ocean Rambler" WDR were carried out. The dynamic model proposed in this paper was consistent with the motion characteristics of "Ocean Rambler" WDR. Simulations of PID heading control demonstrated the unique control characteristics of the WDR, which proved the significance of the established dynamic model of the WDR in control algorithm design.     

Three-Dimensional Maneuvering and Control of Flexible Robots

This paper develops a general approach to the three-dimensional maneuver and vibration control of a robot in the form of a chain of flexible links. The equations for the rigid-body maneuvering motions are derived by means of Lagrange equations in terms of quasi-coordinates and the equations for the elastic deformations by means of ordinary Lagrange equations. The equations of motion are derived for the full system simultaneously, using recursive equations to relate the motions of a given link to the motions of the preceding links in the chain. The maneuver is carried out by means of joint torques and the vibration is suppressed by means of point actuators dispersed throughout the links. The controls are designed by the Liapunov direct method. A numerical example demonstrates the theoretical developments.     

保守双摆的不可积性和混沌

本文用Birkhoff级数正则变换方法求出保守双摆运动方程的近似积分,并把近似积分的等值曲线与数值仿真结果作了比较.由此清楚地看出.当能级提高时,系统由近可积的成为不可积的,即其运动情况由规则的转变为混沌的.本文还介绍了演示上述性态的一个保守双摆模型.     

路面方向扰动下重型汽车的横向稳定性及分岔、混沌运动

针对三轴重型汽车建立了二自由度非线性人-车-路闭环模型,考虑驾驶员控制和路面方向扰动,推导了系统动力学方程．在运用Hopf分岔理论进行分析的基础上,以临界车速为评价指标,通过数值模拟研究了轴距、预瞄距离、载重量、驾驶员控制时滞和轮胎侧偏刚度对转向稳定性的影响,并确定了转向系统的数值稳定范围．另外,还通过分岔图、时程曲线、相轨线、功率谱、Poincaré图和Lyapunov指数研究了不同车速下汽车的非线性动力学响应．结果表明,随着车速的增加汽车可能发生周期运动、拟周期运动及混沌运动,汽车的横向稳定性与车辆和驾驶员参数密切相关.     

Modelling of the controlled motion ofa point on a plane

A solution of the problem of feedback control of the motion of a point on a plane is presented. The equations of the controlprogramme (the objective) are set up as a system of differential equations with a given set of singular trajectories in the domain of admissible positions of the controlled point, as well as a given topological structure of the partition into trajectories. These equations define the vector field of velocities of the programmed motions of the point and are used to find the corresponding control forces.     

A velocity observer design for tracking task-based motions of unicycle type mobile robots

The paper presents a design of a nonlinear velocity observer and its application within a model-based tracking control strategy for tracking task-based motions of unicycle type mobile robots. The strategy is the model reference tracking control strategy for programmed motion and it enables switching between controllers employed in it to improve a tracking precision as well as switching between coordinates used for modeling based on a type of a nonholonomic system. The strategy benefits by adding the velocity observer to its architecture due to the reduction of a number of measurements needed for feedback tracking.     

Controllability and stabilization of programmed motions of reversible mechanical and electromechanical systems

Problems of controllability and methods of stabilizing programmed motions of a large class of mechanical and electromechanical systems which are reversible with respect to the control are considered. Criteria of the controllability and stabilizability of reversible systems are obtained. Programmed motions and algorithms of programmed control are designed in analytical form and algorithms of programmed motions for non-linear reversible systems are synthesized.     

Phase Space Transformations of Gaussian Diffusions

It is shown that for Gaussian diffusions, the transformation back to Brownian motion, usually accomplished via the Girsanov (or Feynman–Kac) formula and time-shift, can be obtained by a classical canonical, i.e. symplectic, transformation in phase space. The method is based on constants of motion, in this case the Wronskian. Similar transformations for general diffusions are briefly discussed.     

Modeling and identification of emotional aspects of locomotion

The studies of emotional facial expressions and emotional body language are currently receiving a lot of attention in the cognitive sciences. In this project, we study implicit bodily expression of emotions during standard motions, such as walking forwards.An underlying assumption of our work is that all human motion is optimal in some sense and that different emotions induce different objective functions, which result in different deformations of normal motion.We created a 3D rigid-body model of a human of which we use the forward dynamics simulation in an optimal control context. We performed two kinds of optimizations: (i) reconstruction of dynamic quantities, such as joint torques, of pre-recorded data of emotional walking motions and (ii) forward optimization that generates neutral and varied walking motions using different objective functions. Optimizations are performed with the software package MUSCOD-II, which uses a direct multiple-shooting discretization scheme. The results of this work form the foundation for further analysis of emotional motions using inverse optimal control methods.     

Chaotic motion in a micro-electro–mechanical system with non-linearity from capacitors

This investigation is to provide a possible prediction for design, manufacturing, testing and industrial applications of a simplified micro-electro–mechanical system (MEMS). The chaotic motion in a certain frequency band of such a MEMS device is investigated, and the corresponding equilibrium, natural frequency and responses are determined. Under alternating current (AC) voltage, the resonant condition for such a system is obtained. It is observed that the lower-order resonant motions can be easily converted to the mechanical force and sensed to the electrical signal. The chaotic motions in the vicinity of a specified resonant separatrix are investigated analytically and numerically. For given voltages, the AC frequency bands are obtained for chaotic motion in the specific resonant layers and resonant motions, and such chaotic motions can be very easily sensed by the output transducer in MEMS.     

Mathematical Models for the Triaxial Attitude Control Testbed

The Triaxial Attitude Control Testbed has been developed as part of a research program at the University of Michigan on multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models; these include fan actuators, reaction wheel actuators and proof mass actuators that are fixed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constrained to move relative to the triaxial base body. The models expose the dynamic coupling between the rotational motion of the triaxial base body, the relative or shape motion of the unactuated auxiliary degrees of freedom, and dynamics associated with actuation mechanisms. Many different model simplifications and approximations are developed. Control models for the triaxial attitude control testbed are formulated that reflect specific assumptions.     

Design and Control of a Pneumatic Hybrid Actuator

To simulate arbitrary force/displacement relationships, a hybrid actuator consisting of a .uidic muscle and a linear pressure spring is presented. Fluidic Muscles are interesting in their use as actuators in robotics, since they have a high power/weight ratio, a slip-stick free motion and a long durability. The operating point is de.ned as the half contracted-stroke of the muscle. The present paper describes a procedure to simulate virtual stiffness of a linear actuator by choosing an operating point of the pre-stressed muscle and applying PID Control to produce desired forces as function of state. The results are presented for a testbed. It is shown how the aforementioned control scheme produces a rapid and .exible stiffnes simulation. The device can be employed for later use in general environments such as motion simulations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Construction of 3D Conformal Motions

This paper exposes a very geometrical yet directly computational way of working with conformal motions in 3D. With the increased relevance of conformal structures in architectural geometry, and their traditional use in CAD, its results should be useful to designers and programmers. In brief, we exploit the fact that any 3D conformal motion is governed by two well-chosen point pairs: the motion is composed of (or decomposed into) two specific orthogonal circular motions in planes determined by those point pairs. The resulting orbit of a point is an equiangular spiral on a Dupin cyclide. These results are compactly expressed and programmed using conformal geometric algebra (CGA), and this paper can serve as an introduction to its usefulness. Although the point pairs come in different kinds (imaginary, real, tangent vector, direction vector, axis vector and ‘flat point’), causing the great variety of conformal motions, all are unified both algebraically and computationally as 2-blades in CGA, automatically producing properly parametrized simple rotors by exponentiation. An additional advantage of using CGA is its covariance: conformal motions for other primitives such as circles are computed using exactly the same formulas, and hence the same software operations, as motions of points. This generates an interesting class of easily generated shapes, like spatial circles moving conformally along a knot on a Dupin cyclide.     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

Bäcklund transformations for the Jacobi system on an ellipsoid

We consider analogues of auto- and hetero-Bäcklund transformations for the Jacobi system on a threeaxis ellipsoid. Using the results in a Weierstrass paper, where the change of times reduces integrating the equations of motion to inverting the Abel mapping, we construct the differential Abel equations and auto-Bäcklund transformations preserving the Poisson bracket with respect to which the equations of motion written in the Weierstrass form are Hamiltonian. Transforming this bracket to the canonical form, we can construct a new integrable system on the ellipsoid with a Hamiltonian of the natural form and with a fourth-degree integral of motion in momenta.     

Computed torque control of an under-actuated service robot platform modeled by natural coordinates

The paper investigates the motion planning of a suspended service robot platform equipped with ducted fan actuators. The platform consists of an RRT robot and a cable suspended swinging actuator that form a subsequent parallel kinematic chain and it is equipped with ducted fan actuators. In spite of the complementary ducted fan actuators, the system is under-actuated. The method of computed torques is applied to control the motion of the robot.The under-actuated systems have less control inputs than degrees of freedom. We assume that the investigated under-actuated system has desired outputs of the same number as inputs. In spite of the fact that the inverse dynamical calculation leads to the solution of a system of differential–algebraic equations (DAE), the desired control inputs can be determined uniquely by the method of computed torques.We use natural (Cartesian) coordinates to describe the configuration of the robot, while a set of algebraic equations represents the geometric constraints. In this modeling approach the mathematical model of the dynamical system itself is also a DAE.The paper discusses the inverse dynamics problem of the complex hybrid robotic system. The results include the desired actuator forces as well as the nominal coordinates corresponding to the desired motion of the carried payload. The method of computed torque control with a PD controller is applied to under-actuated systems described by natural coordinates, while the inverse dynamics is solved via the backward Euler discretization of the DAE system for which a general formalism is proposed. The results are compared with the closed form results obtained by simplified models of the system. Numerical simulation and experiments demonstrate the applicability of the presented concepts.