Reachable Region Research Articles

Omnidirectional Static Walking of a Quadruped Robot on a Slope

We propose successive gait transition with arbitrary body posture to enable a quadruped robot to walk statically and omnidirectionally on a slope. Body posture is determined by rotation around 3 axes, roll, pitch, and yaw. Successive gait transition with a minimum number of steps on a slope is realizable using common foot position before and after gait transition. The time required to transit between gaits is reduced by carefully designing foot position in crawling and rotating while limiting foot reachable region on a slope. The robot thus walks into any direction with arbitrary body postures. In this study, we also verify a tradeoff relation between motion speed and body posture. Computer simulation and experiments verified the feasibility of our proposed method and the stability of gait transition based on static stability margin.

Computing finitely reachable containable region for switching systems

In this paper the practical stability issue of a switching system is discussed in terms of solving a containability problem and an attraction problem. A novel computational procedure based on nonlinear programming is presented to compute a containable region, in which each trajectory from inside cannot move out under a given single-step-lookahead control policy. How to decide whether the obtained containable region is finitely reachable from a point outside the region is then examined. The proposed approach is demonstrated on a two-tank system.

Reachability of particle size distribution in semibatch emulsion polymerization

AbstractReachability analysis is critical in the determination of achievable end‐point products for complex process systems. In emulsion polymerization, an economically important method of producing various products, particle size distribution (PSD) is an important property of the latex produced. However, the population balance modeling framework does not yield direct solutions to reachability analysis for the resulting nonlinear distributed systems. In this article, a programming‐based approach is introduced for the reachability analysis of the final PSD for general particulate systems described by population balance models. In the application to a semibatch styrene homopolymerization system, the reachable region is defined for both the nominal and perturbed cases, with final PSDs characterized as multi‐Gaussian distributions. Sensitivity studies reveal the limiting factors in determining the achievable final products, and quantitative criteria are developed. The evolution of the reachable region in the case of midcourse correction reveals the difficulties encountered during in‐batch control, and shows the best control performance achievable given the limited on‐line measurements. © 2004 American Institute of Chemical Engineers AIChE J, 50: 3049–3059, 2004

Nonlinear model order reduction and control of particle size distribution in a semibatch vinyl acetate/butyl acrylate emulsion copolymerization reactor

This paper addresses the control of the full particle size distribution (PSD) in a semibatch emulsion copolymerization reactor. The numerical approximation of a fundamental population balance model results in a high order system to accurately describe the distribution of particle size; therefore, model order reduction is required. Pseudo random input signals are input to the mechanistic model to generate a data set which covers the reachable region of the system, on the basis of which the transformation matrices are calculated by principal component analysis (PCA). A linear time varying model with reduced order obtained from the transformation matrices is augmented in the prediction equation of linear model predictive control. The performance of the controller is evaluated to drive the particle size distribution at the final time of the batch to the desired distribution in the presence of disturbances.

Sectional and overall reachability for systems with S-shaped distillation lines

Total and minimum reflux bounds serve as limiting case scenarios for determining feasibility in azeotropic distillation columns. Specifically, all reachable feed compositions in a staged rectifying or stripping section have been reported as the union of points enclosed by the distillation and pinch-point curves through a desired product. By eigenvalue considerations of pinch bifurcations and geometric analysis of fundamental region structures, the region of all reachable compositions has seemingly been identified. Additional insights are needed to fully characterize the feasible compositions in a subset of azeotropic mixtures. This article describes how sectional profiles can actually cross pinch-point curves and generate a region of additional feasibility outside the one previously reported. It also demonstrates how a substantially larger region between the distillation line and pinch-point curve after the intersection is only accessible by composition profiles going through the extended region. This extended reachable region is explained by evaluating the conditions for a fixed point in composition space. The effect on sectional profiles is illustrated for both an artificially parametric VLE manifold and a nonideal azeotropic system with more than one simple distillation region. Sectional profiles extended beyond the traditional bounds affect coupled sections in a distillation cascade.

Feasibility of a Reactive Distillation Column with Ternary Mixtures

We develop feasibility criteria for reactive distillation systems in terms of phase and reaction equilibria. A reactive distillation column with the liquid-phase reaction R1 + R2 ↔ P1 can be feasible if the feed composition, the pseudo-feed composition, and the reaction difference point lie on a single straight line. In addition to this material balance constraint, the liquid composition vectors starting from the given top product composition should lie within or move toward the forward reaction region as well as lie within or move toward the reachable region from the bottom product of the nonreactive stripping section. We also perform feasibility studies on reactive distillation with the reaction 2R1 ↔ P1 + P2 in terms of the normalized product and reactant coefficient vectors. For these ternary reaction systems, the larger the reflux, the greater the reaction conversion. We finally propose several feasible flowsheets for the methyl tert-butyl ether production system.

差分方程式拘束を伴う離散時間非ホロノミック系の運動計画

The concept of discrete-time nonholonomic systems, in which the constraints cannot be represented as algebraic equations of generalized coordinates, is introduced. In particular, the constraints formulated as differece equations of generalized coordinates are considered. Such systems can be seen in the digital control of continuous-time nonholonomic systems, and in mechanical systems with repetitive and discontinuous constraints. A two-wheeled mobile robot and pivoting manipulation of a polyhedral object are described as simple examples. The k-step reachable region is defined as the set of the k-th state which the system can reach from the initial state, and the reachability of such systems is discussed. A motion planning method using the Jacobian matrix of the state with regard to the input series is proposed.

Linearization of Bilinear Models with Bounded Inputs

Abstract The use of bilinear models to control nonlinear plants with bounded inputs is discussed. It is shown that by redefinition of the problem as a feedback loop with a saturating complementary function and looking for extreme values on the boundary of the physically reachable region leads to a reasonable estimate of the stability conditions in presence of bounds.

A basic study on a pass operation of an object. Position change with two rotary arms.

This paper discusses the suitable positions of two wholly turnable arms and an efficient procedure of rotations and passes for changing the position of an object. The position has three degrees of freedom. Firstly, this paper deals with the reachable region of the direction of the object face that has two degrees of freedom by means of the two-arm operations under the conditions that the relative angle of arm directions is fixed and that the object is passed N times and rotated N+1 times. The results are that the angle of 90 degrees is suitable as the relative angle of arm directions and that three rotations are sufficient for an arbitrary changing of the direction of the object face. Secondly, this paper shows that three rotations are also sufficient for an arbitrary change of the object position, and it also shows the procedures.

Variable structure control of a robotic arm in the presence of uncertainty

AbstractBased on the theory of variable structure systems (VSS), control system design for the trajectory control of robotic systems is presented. It is assumed that the parameters of the system are uncertain and unknown frictional torques are acting at the various joints of the arm. For simplicity, the control of a three‐link PUMA arm is considered. For trajectory following, two control laws, CH and Cθ, based on the choice of coordinates of the end effector or joint angles as the controlled outputs, respectively, are derived. It is seen that whereas control Cθ has no singularity, certain singular surfaces arise where feedback elements of CH become infinity. These singular surfaces describe the boundary of the reachable region (workspace). Digital simulation results are presented to show that accurate trajectory following can be accomplished using the control CH or Cθ for large maneuvers of the arm in spite of the uncertainty in the system.

On the Movement of Robot Arms in 2-Dimensional Bounded Regions

The mover’s problem is the following: can an object in 3-dimensional space be moved from one given position to another while avoiding obstacles? It is known that the general version of this problem involving objects with movable joints is PSPACE hard, even for a simple tree-like structure moving in a 3-dimensional region. In this paper, we investigate a 2-dimensional mover’s problem in which the object is a robot arm with an arbitrary number of joints. In particular, we give a polynomial time algorithm for moving an arm confined within a circle from one given configuration to another. We also give a polynomial time algorithm for moving the arm from its initial position to a position in which the end of the arm reaches a given point within the circle. Finally, we show that 148 circles suffice to cover the boundary of the reachable region of a joint in an arm enclosed in a circle and that the boundary can be computed in polynomial time.

On Optimal Control for Distributed Parameter Systems

The optimal oontrol problem of the time-varying distributed parameter systems in a Banaoh space are discussed. The olosure of the reachable region of the system is proved to be a convex set even when the control domain is not convex. For the time optimal control a formula for computing the optimal time is obtained and the necessary and sufficient oonditions for holding the maximum principle are derived. The boundary control problem of parabolio systems is discussed for application. The maximum principle of the quadratio optimal control with non-convex constraint is proved.

A geometrical approach to problems of pursuit-evasion games

In this paper it is shown that a class of pursuit-evasion games can be treated more simply by the geometrical approach with the topological properties of the reachable region and by the method of functional analysis. Conditions for capture and optimal strategies are derived. A numerical example is given to show both the optimal open-loop strategies of players and the optimal trajectories of the game.

Minimal-time control of linear systems with energy constraints on the input components

There is a nonlinear minimizing problem inherent in the functional analysis solution to the minimal-time control problem. For systems with energy constraints in the input components an iterative method is developed for solving this nonlinear minimizing problem. The reachable region of a system so constrained is shown to be an ellipsoid only for certain special systems. Furthermore, normality is shown to be important if a system's inputs are separately constrained.

The minimal time control of multi-system problems with energy constrained inputs†

This paper presents an. application, of functional analysis to the minimal∗time control problem where two or more energy constrained systems of the same order must be taken to an initially unknown common point in state space. Two different types of problems with different forms of the energy constraint are described in detail. The development depends on the fact that the reachable region of a system with an energy constraint on the input vector is an ellipsoid.

Contributions to the theory of time-optimal control

A unified theory for the time-optimal control of general linear plants with generalized constraints on the control variable is developed. The geometrical approach taken provides new insight and interpretation for some results obtained by methods of functional analysis. Theorems stating necessary and sufficient conditions for the existence and uniqueness of the optimal control function, u 0, and explicit formulae for u 0, are derived by exploiting the topological properties of the reachable region—the set of the output points ( n-vectors) that can be reached using constrained controls. The detailed study of the conditions for the occurrence of corners and flat portions in the reachable region clarifies some “degenerate” cases for amplitude and area constraints, which have hitherto remained obscure. The synthesis problem is briefly discussed.