Initial state iterative learning for final state control in motion systems

In this work, an initial state iterative learning control (ILC) approach is proposed for final state control of motion systems. ILC is applied to learn the desired initial states in the presence of system uncertainties. Four cases are considered where the initial position or speed is a manipulated variable and the final displacement or speed is a controlled variable. Since the control task is specified spatially in states, a state transformation is introduced such that the final state control problems are formulated in the phase plane to facilitate spatial ILC design and analysis. An illustrative example is provided to verify the validity of the proposed ILC algorithms.     

Directional control of the motion of a rolling disk by using a rotor fixed in the disk's plane

This work deals with the stabilization and control of a system which is composed of a disk rolling on a plane, and a circular rotor plate fixed in the disk's plane. The disk's motion is controlled by the above-mentioned rotor, a “tillting moment” and a pedalling moment. It is shown here that by applying a kind of inverse dynamics control, the motion of the disk is stabilized about a given angle, while simultaneously controlling its speed and direction in such a manner that the point of contact between the disk and the horizontal plane will be able, during a given time interval [0, t_f], to move from a given point. r_A to a another given point r_B, both of them fixed in the plane.     

Quasi-Static Motion of a Two-Link System along a Horizontal Plane

Quasi-static motion of a two-link system along a rough horizontal plane is considered. Dry friction acts between the links of the system and the plane. During the motion, the configuration of the linkage changes, whereas the points of contact of the linkage and the plane remain the same. The control torque applied at the joint of the linkage is chosen so as to provide equilibrium condition for each of the links. It is shown that the quasi-static motion of the two-link system is uncontrollable and the uniqueness of this motion is proved for given initial position of the linkage. The trajectories of the vertices are found, depending on the parameters of the system.     

Cyclic motion design and analysis for a passive object manipulation using an active plate

We propose a novel scheme for manipulating a passive object using an active plate. In previous studies, cyclic manipulations that transport objects on a plate, position control of an object on a plate, and juggling have been realized. In most manipulations using plates, object motions in the direction of the gravitational force are not considered. The objective of this study is to control an object’s orientation with respect to the gravitational force direction using an active plate for realizing hitherto unrealized object motion. In this context, a tumble doll, which is a planar rigid sphere, is defined as the object. Motions of the object and active plate are designed to be cyclic. A state vector composed of the object’s angle and angular velocity is defined, and the cyclic motion is expressed as a nonlinear discrete system. Fixed points of the state vector are searched for in the designed cyclic motion. A stability analysis around the fixed points is conducted using a Poincaré map. As a result, the fixed points are shown to be asymptotically stable. Finally, experimental results are used to verify that the object’s angle can be manipulated with the designed cyclic motion using the plate.     

Dynamic analysis of impact in swing-through crutch gait using impulsive and continuous contact models

The dynamics associated with the impact of the crutch with the ground is an important topic of research, since this is known to be the main cause of mechanical energy loss during swing-through gait. In this work, a multibody system representing a subject walking with crutches is used to investigate the behavior of two different contact models, impulsive and continuous, used for impact analysis. In the impulsive (discrete) approach, the impact interval is considered to be negligible and, therefore, the system configuration is constant. The postimpact state is directly obtained from the preimpact one through algebraic equations. In the continuous approach, the stiffness and dissipation characteristics of the contact surfaces are modeled through nonlinear springs and dampers. The equations of motion are integrated during the impact time interval to obtain the postimpact state, which, in principle, can differ from that obtained by means of the impulsive approach. Although both approaches have been widely used in the field of biomechanics, we have not found any comparative study in the existing literature justifying the model chosen for impact analysis. In this work, we present detailed numerical results and discussions to investigate several dynamic and energetic features associated with crutch impact. Based on the results, we compare the implications of using one contact model or the other.     

Optimal Capture of a Tumbling Object in Orbit Using a Space Manipulator

This paper introduces an optimal capture strategy for a manipulator based on a servicing spacecraft to approach an arbitrarily rotating object, such as a malfunctioning satellite or a piece of orbital debris, for capturing with minimal impact to the robot’s base spacecraft. The method consists of two steps. The first step is to determine an optimal future time and the target object’s corresponding motion state for the robot to capture the tumbling object, so that, at the time when the gripper of the robot intercepts the target the very first instant, the resulting impact or disturbance to the attitude of the base spacecraft will be minimal. The second step is to control the robot to reach the tumbling object at the predicted optimal time along an optimal trajectory. The optimal control problem is solved with random uncertainties in the initial and final boundary conditions. Uncertainties are introduced because sensor and estimation errors inevitably exist in the first step, i.e., determination process of the initial and final boundary conditions. The application of the method is demonstrated using a dynamics simulation example.     

MODELING FRACTAL DYNAMICAL SYSTEMS IN THE COMPLEX SPACE. FROM MACRO-OBSERVATION TO MICRO-OBSERVATION

The thesis presented here describes the dynamics of fractal systems subject to fractional Brownian motions (of order n ) with independent increments. One has to define the problem in the complex plane and consider the complexvalued state. After some preliminaries intended to support the paper it is described how dynamics with real-valued states can be continued in the complex plane, and three modeling axioms are proposed. A short background on fractional Brownian motion is displayed for the convenience of the reader, and, as an illustration, the approach is applied to stock market dynamics.     

A differential approach for modeling revolute clearance joints in planar rigid multibody systems

A non-penetration approach of frictional contact analysis is presented for modeling revolute clearance joints of planar rigid multibody systems. In the revolute clearance joint, the motion modes of the journal are divided into three categories, namely, the free motion, collision, and permanent contact modes. The switch between different contact modes is identified by the state of the journal and bearing, including the gap and the normal relative velocity. When impact in the revolute clearance joint is detected, the collision process is simulated by the impulse-based differential approach, where Stronge’s improved model for restitution is employed to determine the relative velocity after impact. Instead of algebraic equations, the impact process is described by a set of ordinary differential equations (ODEs), which avoids solving complementarity problems. Moreover, in the permanent contact mode, the constraint-based approach and modified Coulomb’s friction law are adopted. The permanent contact mode maintains for most of the time and the governing ODEs are non-stiff. There is general agreement that the constraint-based approach is more efficient than the force-based method. A slider–crank mechanism with a revolute clearance joint is considered as a demonstrative application example where the comparison with the continuous contact force model is investigated.     

Biped side step in the frontal plane

A three-link planar model of a biped is studied in the frontal plane. When this model is subjected to certain on-off constraints, the forces of constraint are derived as functions of the system state and the input, and can be shown in a simple block diagram. Linear state-variable feedback is designed to stabilize and decouple the system for small motions about an equilibrium state. The motion of this biped is implemented as a sequence of state transitions between stable equilibrium states. Ramp state and input references are used to guide the biped system from one equilibrium state to the next. Single-support postural stabilization and side step and return motions were investigated by computer simulation in order to evaluate the effectiveness of this control strategy.     

A model for deployment of a freefall lifeboat from a moving ramp into waves

The equations of motion of a freefall lifeboat are formulated using Kane’s method. Deployment from a moving ramp is assumed to occur in a known plane and a 2D model suffices. A segmented approach is used to model ramp contact. A separate 3D model is presented for the water entry phase. The hydrodynamic loads at water entry are also modeled using a segmented approach. The equations are solved numerically using a standard Runge–Kutta MATLAB routine.     

Reaching a consensus in networks of high-order integral agents under switching directed topologies

Consensus problem of high-order integral multi-agent systems under switching directed topology is considered in this study. Depending on whether the agent’s full state is available or not, two distributed protocols are proposed to ensure that states of all agents can be convergent to a same stationary value. In the proposed protocols, the gain vector associated with the agent’s (estimated) state and the gain vector associated with the relative (estimated) states between agents are designed in a sophisticated way. By this particular design, the high-order integral multi-agent system can be transformed into a first-order integral multi-agent system. Also, the convergence of the transformed first-order integral agent’s state indicates the convergence of the original high-order integral agent’s state, if and only if all roots of the polynomial, whose coefficients are the entries of the gain vector associated with the relative (estimated) states between agents, are in the open left-half complex plane. Therefore, many analysis techniques in the first-order integral multi-agent system can be directly borrowed to solve the problems in the high-order integral multi-agent system. Due to this property, it is proved that to reach a consensus, the switching directed topology of multi-agent system is only required to be ‘uniformly jointly quasi-strongly connected’, which seems the mildest connectivity condition in the literature. In addition, the consensus problem of discrete-time high-order integral multi-agent systems is studied. The corresponding consensus protocol and performance analysis are presented. Finally, three simulation examples are provided to show the effectiveness of the proposed approach.     

Prediction of part orientation error tolerance of a robotic gripper

This paper presents a model which predicts the part orientation error tolerance of a three-fingered robotic gripper. The concept of “self-alignment” is introduced, where the gripper uses the grasping process to bring the workpiece into its final state of orientation. The gripper and part are represented mathematically, and initial contact locations upon grasp closure determined. This information is used to solve for the contact forces present, and criteria are developed to determine if beneficial part motion resulting in self-alignment is expected. The results are visualized via a boundary projected on a reference plane below the part. The model is validated experimentally with a number of part configurations with favorable results. This method presents a useful tool by which the mechanical designer can quantitatively predict the performance of an intuitively designed gripping system.     

Optimal motion control for a system of two bodies on a straight line

The optimal control problem for motions of a system of two rigid bodies on an inclined straight line in a plane that are periodic in velocity is solved. The external body (frame) moves on a plane under the action of a force from the inner body in the course of its motions relative to the frame under dry friction between the frame and plane. The acceleration of the inner body relative to the outer one is the control whose absolute value is bounded. An optimal control that maximizes the average velocity of the system motion for a given period is found. It is shown that optimal relative acceleration of the inner body has three intervals of constancy on this period, and the outer body is in the state of rest on a part of the period (in the case of horizontal straight line, it is in a state of rest on half a period), and during the rest of the period, it moves in the desired direction and never performs a reversion. It is established that, for the found control law and under an additional constraint on the amplitude of oscillations of the inner body, it is possible to make the motion velocity of the system arbitrarily large under arbitrarily large accelerations of the inner body and an under arbitrarily large frequency of its oscillations simultaneously.     

Insectomorphic robot climbing a freely rolling ball

The problem of control of autonomous motion of a six-legged robot from a support horizontal plane to a ball that can freely move on this plane in an arbitrary direction is solved. Further robot motion aimed at acceleration or deceleration of the ball both in the direction of the longitudinal axis of its body and in the transverse direction ensuring dynamic stability of the robot on the ball is synthesized. Analytical conditions of implementability of the maneuver of climbing a ball are found. Formulas for estimating the maximum ball radius for which climbing of the robot is possible are obtained. Using the developed control algorithms, the robot can climb the ball and, staying on it, move the ball to the desired position in the plane. Robot motion is performed owing to the dry friction forces. Asymptotic stability of the programmed motion of the whole system is provided by a PD controller, which implements necessary step cycles of legs motion and the planned law of body motion. Results of 3D computer simulation of the controlled robot dynamics are discussed.     

An insectomorphic robot climbing over a freely rolling ball

An algorithm for the control of an insectomorphic robot climbing over a ball that rolls freely on a horizontal plane is developed and tested using computer simulation. The proposed motion involves three maneuvers. First, the robot climbs the ball at rest from the horizontal surface. At the end of this maneuver, the ball gains an angular velocity due to errors in the execution of the programmed motion. The further motion of the robot is designed so as to reduce the velocity gained in the course of climbing to an acceptable level. The motion is completed by the maneuver of getting down to the supporting horizontal plane from the almost motionless ball. The robot motion is implemented using the Coulomb friction without any special devices. The asymptotic stability of the programmed motion of the system as a whole is ensured by a PD controller that implements the step cycles of the leg motions and the planned motion of the body. Results of 3D computer simulation of the robot motion are discussed. The model of the mechanical robot-ball system is formed using the Universal Mechanism program package; this model is described by an automatically derived system of differential equations that take into account the dynamics of all solid elements.     

Limit cycle oscillations in a satellite attitude control system

Limit cycle oscillations in an attitude control system using an integral-pulse-frequency modulator are studied. A previous paper showed, through a study of some geometrical properties of the state transition equations, the necessary and sufficient relationships among the physical parameters for the ultimate behavior to be a two-pulse limit cycle for arbitrary initial states [1]. Here it is shown that two-pulse limit cycles can exist if those relationships are not satisfied provided the initial state lies in a certain region near the origin of the state space. The boundaries of this region are established analytically by geometrical means and are verified by simulator experiments. These experiments also show the existence of four-pulse, six-pulse and higher order limit cycles for certain initial states and combinations of physical parameter values.     

A differential-geometric approach for 2D and 3D object grasping and manipulation

This article presents an expository work on a differential-geometric treatment of fundamental problems of 2D and 3D object grasping and manipulation by a pair of robot fingers with multi-joints under holonomic or nonholonomic constraints. First, Lagrange’s equation of motion of a fingers-object system whose motion is confined to a vertical plane is derived under holonomic constraints when rolling contacts between finger-ends and object surfaces are permitted. Then, a class of control signals called “blind grasping” and constructed without knowing the object kinematics or using any external sensing like vision or tactile sensation is shown to realize stable object grasping in a dynamic sense. Stability of motion and its convergence to an equibrium manifold are treated on the basis of differential geometry of solution trajectories of the closed-loop dynamics on the constraint manifolds. Second, a mathematical model of 3D object grasping and manipulation by a pair of multi-joint robot fingers is derived under the assumption that spinning motion of rotation around the opposing axis between contact points does no more arise. It is shown that, differently from the 2D case, the instantaneous axis of rotation of the object is time-varying, which induces a nonholonomic constraint expressed as a linear differential equation of rotational motion of the pinched object. It is shown that there is a class of control signals constructed without knowing the object kinematics or using external sensings that can realize “blind grasping” in a dynamic sense. Finally, it is shown that the proposed differential geometric treatment of stability can naturally cope with redundancy resolution problems of surplus degrees-of-freedom (d.f.) of the overall fingers-object system, which is closely related to Bernstein’s d.f. problem.     

刚杆非定点碰撞的几何学及动力学模拟

两刚性杆件在运动过程中其接触碰撞点的位置是不确定的.本文给出了此模型接触碰撞时的两种接触碰撞模式,分析了寻找接触碰撞点位置的方法,由此得出判断接触碰撞模式的方法,并通过MATLAB对此模型进行了几何学及动力学模拟.