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## Robust Controller Design for Uncer-tain Linear Systems with Circular Pole Constraints

### BibTeX

@MISC{Garcia_robustcontroller,

author = {G Garcia and J Bernussou and W M Haddad and D S Bernstein and D Mustafa and W M Haddad and D S Bernstein and S O . ; Reza Moheimaini and I R Petersen and Z Wang and G Tang and Chen and X Xie and L},

title = {Robust Controller Design for Uncer-tain Linear Systems with Circular Pole Constraints},

year = {}

}

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### Abstract

This paper documents the development, solution, and application of a computational model for the dynamic response of a small diameter, pneumatic tool used for boring horizontal tunnels in the soil. The model consists of (i) the tool component kinematics and kinetics, (ii) mechanics of the tool and soil interactions, (Hi) the compressible air dynamics, and (iv) the pressure control valve switching logic. The resulting model is represented by a set of coupled, sixth-order nonlinear differential equations. The boring tool design has several unique features, including dual pistonheads and a pilot-pressure actuated spool valve used to control the oscillatory piston. Implementation of these and other tool features in the computational model is discussed at length. The dynamic simulation and associated parametric studies establish the feasibility of the design for small diameter (25.4 mm) horizontal boring tools. Results for this design predict tunneling rates of about 60 meters/hr in a medium clay soil. Nomenclature and Nominal Values Used in Analysis = area, piston 467 X 10"** m^ (0.723 in^) = area, supply valve throat 23.7 X 10"'' m^ (0.0368 in^) = area, exhaust port throat = area, advance exhaust port throat 20.7 X 10 m^ (0.032 in') = area, return exhaust port throat 19.4 X lO""" m' (0.030 in") = elastic stiffness factor, (17) 2.214 dimensionless c = damping coefficient, piston/body 36.8 X 10' N s/m (0.00021 lb s/in.) Csoii = damping coefficient, body/soil 3.50 X 10' N s/m (20 lb s/in.) e = coefficient of restitution, body/soil 0.00 E = soil modulus (medium/hard clay) 6 x lO*" N/ m' (870 lb/in') piston/body cylinder contact force normal force on body from soil 716N (161 lb) resistance Force on body (sides) resistance force on body (tip) universal mass constant 1 kg m/N s' (386.1 Ibm in/Ms") elastic stiffness, piston/body impact elastic stiffness factor Introduction Trenching and tunneling are two methods commonly used to install underground cables and pipes. Several methods for boring horizontal holes in soil have been documented (Milligan, 1990; Transactions of the ASME Copyright © 1998 by ASME derground for installation of small (approximately 25 mm diameter) natural gas pipelines. The tunneling is accomplished by a pneumatically driven horizontal boring tool (HBT) attached to the PVC pipe which is suppUed with high pressure air from a standard 100 psi compressor stationed above ground at the installation site. Some contemporary designs of the pneumatic (air-hammer) HBT utilize a single piston (hammer, rammer, etc.) and a simple porting arrangement for supply (compressed) and exhaust air Utility companies have used the pneumatic HBT for several years to bore tunnels of 50 mm diameter and larger for the installation of service lines. For smaller, residential type lines, these large HBTs produce unnecessarily large diameter tunnels. The natural gas industry, for example, uses PVC pipe of 25 mm diameter or less for residential gas distribution lines. Excessively large holes increase soil displacement and require larger launch and recovery pits, additional operator handling, and ultimately increases the final installation cost. These factors make it is desirable to develop a small-diameter HBT. The primary goal of this applied research project was to provide an effective, efficient and economical device capable of boring 25 mm horizontal tunnels at a nominal rate of 50 m/hr. The design utilized a commercially available dynamic simulation computer software package. Equations necessary to model air compressibility in the HBT chambers were developed by the authors and adapted to the model utilizing a user input feature of the software package. Design features including the dual chamber/piston, the switching logic and control valve are evaluated. Major differences from the contemporary, large diameter designs include: (;) the use of a control valve for switching supply air to appropriate chambers for optimal oscillatory response, (ii) the location of the supply and exhaust channels and their activation ports, (Hi) the dual piston concept to increase the piston motive force and (iv) consideration of piston/ body sealing tolerance. As this paper is being written, a patent disclosure has been completed and three patents are pending for the system methodology and various design features. Hardware Configuration A typical geometric configuration of the HBT is presented in The soil forces acting on the HBT body are assumed sufficient to prevent body cylinder motion during piston advance. The control valve mechanism is designed to insure full impact of the leading pistonhead at the termination of the advance stroke. During the return stroke, sufficient impact reduction must occur to assure a net forward displacement. Mathematical Model The analysis of the prototype design is separated into two categories: rigid body dynamics of the HBT body and piston, and pneumatics of the air supplied with basic control valve logic. The governing equations were written for each of these processes and the necessary data was supplied to a dynamic simulation software package. Rigid Body Dynamics of the HBT Body and Piston. The motion of the piston mass, Mp, along its one-dimensional path is easily described using absolute the coordinate Xp and Newton's 2nd Law: Xp = "LFIMp (see Xp = [-c{Xp -XB) + AP{PL -PR) -Fimp!,a]/Mp (1) Air pressure in the left and right chambers, P^ and PR provide the piston motive force, ApiP^ -PR) and Fi^p represents the piston/body force at impact. The piston/body friction force is approximated by the viscous damping term, c{xp -Xg). Equation In this equation, the reaction forces at the piston/body interface (3rd law) and the force of the body/soil interaction are governing the motion. Assuming perfectly elastic contact (steel on steel), the piston/body impact force is given by Hooke's law (linear spring) as where 6 is the contact surface deformation at the piston/body interface during impact. The value of the equivalent impact stiffness, k, is discussed in the computer implementation section. A generalized body/soil force is very complicated to model The HBT side force resisting motion at the body/soil interface generated during the entire cycle is simulated by the Coulomb friction relation. where F^ is the soil normal force on the body and ix, is the effective body/soil friction coefficient. This force is normally of sufficient magnitude to constrain the HBT body from motion as the piston moves. Appropriate logic is required to compute proper values for this force (typical to dry friction,' 'slip/stick'' motion). The simulation software provided subroutines necessary for these calculations. The body/soil interface force at the HBT tip generated during the piston impact, was found to be reasonably simulated by the commonly used combination of a linear spring and viscous damping as The simulation software uses a proprietary computation to approximate the value of ^soii used in Eq. (4b). The software requires input values for the soil modulus, E, the coefficient of restitution, e, and the damping coefficient, Csoii. E was determined from Whitlow (1988) (also see Nomenclature). The body/soil impact tests produced only a small amount of soil restitution, or elastic deformation, for the relatively stiff clay tested. For simplicity, the coefficient of restitution, e, was given a value of 0, representing a perfectly plastic impact (no restitu- This definition is used in the analysis to (mathematically) connect the left and right side cylinder chambers to the supply and exhaust air ports. Using the value 1 or 0 for Y, depending on whether the piston is in the advance or return stroke allows simplification of the equations needed to describe the air flows entering and exiting the chambers, the air pressures developed in the chambers, and the exhaust port throat area. The value for 7 (1 or 0) depends on the position of the piston relative to the cylinder. The parameters /switchA and /switch/? are defined input to represent the positions along the cylinder where the control valve is actuated to change the air flow from advance mode to return mode (I switch A) and from return mode to advance mode (/switch/?). Physically these are positions where the supply air ports are located on the body and/or the piston rod where u( ) is the unit step function. Symbols x and x'dis are defined in the Nomenclature. When the piston is in the advance mode, high pressure supply air flows through the control valve and transmission lines and into the left chambers behind the pistonheads as shown schematically in The air is assumed to be characterized by the ideal gas law and to have negligible temperature changes. The ideal gas law can be written as: PV = mRT, for each chamber. Differentiating with time and solving for the time derivative of pressure gives the necessary differential equation for pressure change in the chambers. For this study the temperature change was ignored, so 294 / Vol. 120, JUNE 1998 Transactions of the ASME that T is constant and only P, V, and m change. The isothermic assumption produces only a small error, insignificant in the computation for piston motion. The change in volume, V, is due to the change in piston position relative to the body and of course the mass of air, m, in the chambers varies as air flows into or out of the chambers. From these considerations, the time rate of change of the pressure in the chambers as a function of the relative piston speed and air mass in the chambers is given by The X subscript, in Eq. with the volume flow rate given by the time derivatives of Eqs. (la) and (lb) where the initial (f = 0) volumes are VOL and VOR. Equations (6), where the valve throat Mach number M" is related to the supply chamber/port throat pressure ratio where P, and P" represent the static pressures at the supply (reservoir) and the inlet port (throat). The supply pressure is the fixed, inlet compressor pressure. The supply chamber is the left chamber during the advance mode and the right chamber during the return mode, hence the supply side chamber pressure is given in terms of the switching function Y as The model approximation assumes that the inlet valve throat static pressure, P", is related to the downstream supply chamber pressure, Pcsw, as follows: •"amb The first condition of Eq. (12) sets the port throat pressure to the "choke" or critical pressure simulating choked flow so that any further reduction of downstream pressure will not increase the mass flow rate. The second condition sets the port throat pressure to the chamber pressure simulating unchoked flow. The third condition sets the port throat pressure to the supply pressure so that there is no flow through the valve, simulating a physical check valve. Equations The mass flow rate through the exhaust port is given by where the exhaust port throat Mach number, Me, is calculated using and the exhaust chamber pressure is given by The design allows the exhaust throat area to differ in the advance (Aea) and return (Aer) modes. The exhaust throat port area is therefore written in terms of the switching function Y as: The exhaust throat pressure, Pg, is found using if 0 < PcEXH s P""b then P, = PCEXH if Pamb S PCEXH S 1.893Pa,"b then Pe = Pa,"b if 1.893P"",, s PCEXH then P, = 0.5283PCEXH Computer Implementation The information supplied above, along with the appropriate initial conditions on the integrated variables, is sufficient to program the simulation. modes from advance to return by alternating the supply and exhaust air paths. Equations Results A series of simulations were used to evaluate the feasibility of early design configurations. Typical design parameter values are listed in the table of Nomenclature. For these parameters, the piston position and speed increase monotonically to a maximum value at the time of impact with the body cylinder as illustrated in During the piston stroke, the body is held in place prior to piston-body impact by the soil resistance force on the body sides, /^soiii • The body and piston motion at the end of the advance stroke is shown in The impact forces, generated by this model, are shown in Transactions of the ASME the small initial chamber volume while the right chamber is at the ambient pressure to which it is ported. As the piston accelerates to the right (advances), P,. decreases, which in turn increases the air flow rate into the left chamber as indicated in Concluding Remarks Dynamic simulations of the small diameter boring tool indicated that the proposed design will provide significant boring rates. The simulation reported in this paper produced an ideal tunneling rate of about 60 meter/hr for operation in a medium/ stiff clay soil without excessive obstructions (large rocks, etc.). The target rate established at the beginning of this study was 50 meter/hr. Operation of the proposed small diameter device is enhanced by the dual piston and control valve switching operation. Further computer simulations and hardware prototype tests are required to produce an efficient and optimal working tool. Major improvements are expected from changes in control valve sensor locations, piston stroke length and mass, and exhaust port "snubbing." Acknowledgments This work was supported by the Columbia Gas Company of Columbus, Ohio. Special thanks are due Mr. Arthur C. Eberle, Director of Research at Columbia Gas during these studies, for sharing his valuable knowledge of soil boring, tunneling systems, and general manufacturing process technology. Engineering and Management, Vol. 117, No. 3, Sept., pp. 65-75. John, J. E. A., 1969, Gas Dynamics, Allyn and Bacon, pp. 40-42. Lee S. L., et al., 1988, "Rational Wave Equation Model for Pile Diiving Analysis," ASCE, Journal of Geotechnical Engineering, Vol. 114, No. 3, Mar, Milhgan, G., 1990, "Trenchless Construction Techniques," Engineering Digest, Vol. 36, No. 2, pp. 26-28. Whitlow, R., 1988, Basic Soil Mechanics, Longman Scientific and Technical, Essex, England. Fast Mapping of Obstacles Into Configuration Space ^ Wei Li,^ Chenyu Ma,^ Zushun Chen,^ Qi Cao,^ and Jingnan Ye^ This paper presents a practical approach to real-time path planning for a robot, based on fast mapping of obstacles into configuration space (C-space). Its basic idea is to define some specific points in workspace (W-space) as fundamental obstacles and to construct a database of fundamental obstacles' images in C-space (Li, 1990). In our previous works (Li, 1991; Li and Zhang, 1993), main fundamental obstacles were introduced to reduce the memory of the database; an analytical model for mapping fundamental obstacles was presented according to a robot's kinematics as well as geometry; and slice C-space obstacles were proposed to build a free subspace. Since mapping a Cartesian obstacle based on fundamental obstacles is superimposing images of fundamental obstacles on its borders, the study of the paper is to further speed up the time-consuming procedure of C-space construction by selecting the critical points. Finally, the computational time required by the existing approaches above is compared with that of the proposed one. References Off-Line Mapping of Fundamental Obstacles Since most robot's W-spaces are symmetric, they can be formed by moving an area around their symmetric axes. Such an area is known as a robot's fundamental area.