A Course of Study in the Simulation and Control of PRT Systems

J. Edward Anderson

Control of Personal Rapid Transit Systems

The problem of precise longitudinal control of vehicles so that they follow predetermined time-varying speeds and positions has been solved. To control vehicles to the required close headway of at least 0.5 sec, the control philosophy is different from but no less rigorous than that of railroad practice. The preferred control strategy is one that could be called an "asynchronous point follower." Such a strategy requires no clock synchronization, is flexible in all unusual conditions, permits the maximum possible throughput, requires a minimum of maneuvering and uses a minimum of software. Since wayside zone controllers have in their memory exactly the same maneuver equations as the on-board computers, accurate safety monitoring is practical. The paper discusses the functions of vehicle control; the control of station, merge, and diverge zones; and central control.

Simulation of the Operation of Personal Rapid Transit Systems

 A computer simulation program developed by the author to study the operation of personal rapid transit (PRT) systems of any size    and configuration. The control scheme is asynchronous with maneuvers commanded by wayside zone controllers. The simulation runs on a PC, is accurate in every detail, and can be used to run an operational system, which would use dual-redundant computers on the vehicles at wayside to manage specific zones, and in a central location to manage the flow of empty vehicles and to perform other system-wide functions. Some results are given.

Longitudinal Control of a Vehicle

Generally applicable formulae for the gain constants in a proportional plus integral controller required for stable control of the speed of any vehicle in terms of natural frequency, damping ratio, vehicle mass, and thruster time constant.  An example, based on a simulation of the controller and vehicle, is given.  The theory shows that only speed and position feedback are needed.  Acceleration feedback is unnecessary.

Synchronous or Clear-Path Control in Personal Rapid Transit

An equation is derived for the ratio of the maximum possible station flow to average line flow in a PRT or dual-mode system using fully synchronous control.  It is shown that such a system is impractical except in very small networks.

Failure Modes and Effects

A wide range of failure modes in PRT systems are treated with estimates the mean time to failure of each and the degree of redundancy needed to meet requirements of performance and safety.  In developing the results, many details of the control system required are explained.

The Geometry of a Vehicle Moving in 3-D Space

The Reference Frames and the Velocity Vector.  Components of Acceleration.  Maximum Speed based on Comfort Acceleration.  The components of Jerk.  The Differential Equations of the Spiral Transitions.

Plane Transition Curves at Constant Speed.  The Transition Curve with no Region of Constant Curvature.    The Transition Curve with a Region of Constant Curvature.  The Roll-Rate Limit.  Nonlinear Effects

Yaw-Pitch Coupling.  Large Yaw Angles.  Superelevation.

Equations needed to compute a direction change in a horizontal or vertical plane

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The Governing Differential Equations.  Calculation of the Slope of the Curve.  Calculation of the Coordinates of the Curve in the Region of Positive Jerk.  The Limit Condition for a Section of Constant Curvature.  Calculation of the Coordinates of the Curve in the Region of Constant Curvature.  Calculation of the Coordinates of the Curve in the Region of Negative Jerk.  A Program for Calculating the Curve in Local Coordinates

The Transition to an Off-Line Station

Generally applicable differential equation for the transition curve.  Solution with constant speed.  Equations for constant-speed transition.  The transition to an off-line station.  Limits.  Quarter and half point values.

Transition with variable speed.  The Curvature.  The Slope of the Transition Curve.  The Transition Curve.     The Length of the Transition.  The Station Speed.  The Maximum Slope of the Transition Curve.  Solution for large lateral displacement.  Collection of the Equations for the Transition.  Calculation of the Speed into a Station How does the Station Throughput change with Station Speed?  A Program to Compute the Transition.  Numerical Solution for the Transition for Arbitrary Speed Profile.

A process for developing a program that will simulate the operation of PRT vehicles in a network of any configuration

The step-by-step process required to develop a program to simulate the operation of any PRT network is discussed.

Layout of a PRT Network

Quantitative layout of a PRT network including properties needed for vehicles and passengers.

List of constant values for the system.  Programs to calculate and plot the system.

Setup of Control Zones in a PRT Network

Design criteria.  Hardware required for control.  Equation for minimum distance between branch points.

Control Strategy.  Explanation of Control Zones.

Speed & Position vs. Time

The equations needed to calculate speed and position vs. time for acceleration to line speed, stop in given distance, slip given amount, speed change, and emergency stop.  The results are useful both in the code of a real system and in a simulation program.

Description and code for a PRT Vehicle Controller

A vehicle controller designed to follow speed/and distance vs. time profiles.

Positioning of Vehicles and their Movement

Required number of vehicles.  Initial vehicle placement.  Vehicle movement.  When can a vehicle leave a station?  Resolving a merge conflict.  A diverge point.  Entering and moving through a station.

Additional Code needed to Operate a PRT Simulation

The demand matrix.  Generation and Processing of Passengers.

Equations for Command Point Positions

Switch, Deceleration, Diverge, and Merge Command Points

Structure of a Simulated PRT Control System

Functions of vehicle, zone, and central computers.  Description of the physical system to be simulated.

Stopping Distance vs. Transition Length

Derivation of the relationship between stopping distance and the transition length to an off-line station.

The Program of Calculations Required to provide data to Operate the PRT Network Simulation Program

Universal constants.  Apex data.  Station data.  Demand matrix.  Branch data.  Compute azimuth, direction change, curve properties, straight sections, start coordinates, station data, guideway coordinates, jump points, main arc lengths at jump points, branch-point apexes, distance of branch-command points from the branch points, station Command-Point distances from the branch-point ahead.  Load vehicles.  Compute station-to-station distances and the number of upstream station past branch points.  Definitions of the arrays used in the simulation program.

The PRT Network Simulation Program

The Command Points and Actions

Command Line Speed, Reset On Station Exit, Diverge Control, Merge Control, Switch At Station Switch Point, Decelerate to Berth, Advance In Station, Call Empty Vehicles, Speed Change, Emergency Stop.

Additional Routines needed in the Simulation

Calculate Maneuvers, Up-Date Times, Power and Energy

For further details, contact J.E. Anderson at PRT International, LLC