Minimum-time Path Research Articles

Minimum-time path planning for robot arms and their dynamics

Despite its close relationship to robot arm dynamics, conventional path planning does not take it into account, leading to the possible underutilization of the robot's capabilities. The authors have developed a minimum-time path-planning method in joint space taking into consideration robot arm dynamics as well as other realistic constraints. The main differences between this method and others are: (1) an absolute tolerance in the path deviation at each corner point can be specified; (2) local upper bounds on joint accelerations are derived from the arm dynamics so as to nearly fully utilize robot's capabilities; and (3) a set of local optimization problems, one at every local corner point, replaces the global minimum-time problem, thus making the minimum-time path-planning problem simpler and easier to solve. The method is applied to the path planning of the first three joints of the Unimation PUMA 600 series robot arm, using its simulator on a DEC VAX-11/780. The results show significant improvements in the total traveling time in addition to the ease and simplicity obtained from the decomposition of the global problem into a set of local optimization problems.

Minimum-time control of a two-link robot arm

Minimum time paths for a two-link robot arm are determined so that the tip moves a specified distance, starting from rest and ending at rest. Two torquers are used, one at the shoulder and one at the elbow, each having bounded torque. The optimal locations of the two end points in the robot coordinate system are determined as part of the solution. The paths are found to be “bang-bang” in both controls. For distances that are small compared to the reach of the arm, the minimum time paths are asymmetric, changing to symmetric paths when the distance is just slightly less than the maximum reach of the arm. A new gradient algorithm for systems with multiple bounded controls was developed to solve the problem.

Rapid solution of ray tracing problems in heterogeneous media

abstract The determination of local earthquake hypocenters and orgin times from first-P-arrival times by Geiger's method requires a technique for finding the minimum travel time (and derivatives) between the source and the station. Sophisticated ray tracing techniques have been developed for this purpose for use in complex velocity structures. Unfortunately, the two common techniques, shooting and bending, are generally prohibitively expensive for routine use in data analysis. The bending method is also particularly vulnerable to the problem of local minima in travel time. A method has been developed known as the ray initializer, which can be used to circumvent these problems in many cases. First, the technique can find a reasonable estimate of the minimum-time ray path in a quick and efficient manner. The velocity in a region local to the source and receiver is laterally averaged to yield an approximate layered velocity model. One-dimensional ray tracing techniques are used to find the minimum-time path for this layered structure. The ray path estimate can then be used as the starting path in a bending routine, a procedure resulting in more rapid convergence and the avoidance of local minima. Second, the travel time found by numerical integration along the estimated ray path is an excellent approximation to the actual travel time. Thus, in many cases, the ray initializer can be substituted for a three-dimensional ray tracing routine with a tremendous increase in efficiency and only a small loss in accuracy. It is found that the location of an explosion, derived using the ray initializer, is nearly identical to a complete ray tracing solution, even for a highly complex velocity structure.

The use of cutsets in Monte Carlo analysis of stochastic networks

Monte Carlo methods utilizing a new network concept, Uniformly Directed Cutsets (UDCs), are presented for analyzing directed, acyclic networks with probabilistic arc durations. The procedures involve sampling arc values for arcs not on a UDC and utilizing known probability information for arcs on a UDC. This approach results in less sampling effort and less associated variance than a straightforward simulation approach. A proof of this variance reduction is offered. The procedures provide estimates for project completion time distributions, criticality indices, minimum time distributions and path optimality indices. All of these network performance measures are useful to decision makers in project planning. Application areas include PERT-type network planning, equipment replacement analysis, reliability modeling, stochastic dynamic programming problems and maximal flow problems.

Long-range energy-state maneuvers for minimum time to specified terminal conditions

Some three-dimensional minimum-time paths to a specified final line (or point), heading, and energy are presented for an example supersonic aircraft. The optimum maneuvers have been determined using the calculus of variations and the energy-state approximation. These are compared with suboptimal solutions obtained with the additional assumption that only three discrete values of bank angle (−φmax, 0, +φmax) are available. Constraints on thrust, Mach number, angle-of-attack, dynamic pressure, and load factor are included. For ranges long enough that maximum velocity is attained en route, the initial and final ares can be determined separately, which greatly simplifies the solutions.

Application of the epsilon technique to a realistic optimal pursuit-evasion problem

The Balakrishnan epsilon technique is applied to a pursuit--evasion problem in which both the pursuer and evader act optimally. The problem consists of determining the control functions for both the pursuer and the evader that result in the time of interception being minimized and maximized, respectively. The pursuer is considered to be a point mass confined to a horizontal plane and subjected to realistic aerodynamic and propulsive forces which are nonlinear functions of the state and the control. The evader has a constant speed and has a limit on the rate with which it can change direction. The Balakrishnan epsilon technique involves the insertion of a penalty term for not satisfying the dynamic equations directly. A modified Newton-Raphson method is used to solve the coefficients of a functional expansion of the state variables which minimizes the cost; a simple search is used to find the control at each time point which minimizes the integrand of the cost function. Repeated application of both the Newton-Raphson method and the search results in rapid convergence to the minimum-time solution to a fixed point. Six to eight iterations are typically required. A complete family of minimum-time flight paths to points in several directions are computed and subsequently used to determine the intercept point for virtually any evader flight path. Interpolation between solutions yields both the optimal path and corresponding control for the interception. A similar set of solutions are generated for the evader. The general solution of the evader is superimposed on that of the pursuer in a way that is consistent with the problem's initial conditions. The intercept point for the max-min solution corresponds to the maximum of the minimum time required for both the pursuer and evader to reach that point. Once the intercept point is determined, the corresponding trajectories and control functions for both the pursuer and evader can be obtained by interpolating between adjacent solutions. The approach used is particularly efficient when a large number of optimal pursuit-evasion solutions are needed.

Minimum Time and Minimum Fuel Flight Path Sensitivity

Minimum Time and Minimum Fuel Flight Path Sensitivity

Computer Algorithm for Routing Oversized Trucks

A computer algorithm for routing oversized trucks through a network was developed. The basic input to the algorithm consists of the usual network description in terms of speed and distance on constituent links, and in addition, the height and weight limits on each link, and the height and weight of the truck to be routed between specified origins and destinations. The algorithm computes the minimum time path on the basic network or on the residual network subsequent to removal of all links having an allowable height and weight less than the corresponding parameters for the truck to be routed. The output also includes the distance along the minimum time path and the corresponding overweight fee computed on the basis of a certain rate per excess ton-mile traveled. These outputs can then be compared to determine the increase in user costs resulting from routing a truck over a minimum time path that avoids restricted links.

Urban Velocity Fields

The city is not an undifferentiated terrain and travel does not occur along straight-line paths at constant velocities. Variations in travel velocities at different locations bend the minimum time paths away from regions of high congestion. This paper discusses a transformation of the urban plane into a time surface on which distance corresponds to travel time, and describes the construction of minimum paths and isochrones for various velocity fields. This view of the urban transportation system allows us to discover some of the important features which are often hidden in a network description of the system.

Turn Penalty Effects on Minimum Time Paths

The transportation planning process is dependent upon the minimum path algorithm to develop the more logical route from the origin zone to the destination zone using the existing or proposed street system. Current philosophy assumes that an intersection turn penalty is necessary in order to prevent the algorithm from selecting illogical, stair-step paths through a grid street network. Minimum path trees for three levels of network detail were constructed with a turn penalty of 0.20 min and without a turn penalty. The differences resulting in the minimum path trees were examined to determine which path was the more logical. From these comparisons, it is concluded that the use of a turn penalty tends to produce a greater number of less logical routes, and the general practic of using turn penalties should be discontinued when level-of-service speeds are used in coding the representation of the street network.

Base Year Trip Tables for Statewide Traffic Models

Combining origin-destination data from scattered roadside interview stations results in extensive duplication since many movements are theoretically intercepted at more than a single station. The essence of the paper is in the method used to eliminate this duplication for a study covering a large area in West Virginia. Standard network building, tree tracing and tree skimming assignment programs were utilized to develop a matrix showing, for each origin-destination movement, the number of stations on the minimum time path between the two zones. The base year trip table was then produced by dividing each element of the composite trip table by the corresponding element of this matrix. The paper considers other methods that have been used in statewide planning studies to develop base year trip tables for traffic model development.

Conjugate gradient methods with an application to V/STOL flight-pathoptimization.

Conjugate gradient methods for optimization problems with terminal constraints, noting minimum time paths for V/STOL aircraft climb phase

Complete Elementary Solution to Some Optimum Trajectory Problems

A complete elementary and direct method is given for solving a class of optimum trajectory and design problems which arise in rocketry. The solution of problems which involve an optimum rela­ tion among final fuel consumption, position and time is reduced to elementary algebra. It is as­ sumed that aerodynamic forces are negligible and gravity is constant or can be expressed as a func­ tion of time. The problem of intercepting a ballistic missile with minimum fuel consumption is studied in some detail as a typical problem. For this problem a hyperbola can be plotted on a pre­ pared graph and the answers read. Solutions for the lower bound of fuel consumption, corre­ sponding to infinite thrust can be read also. It is assumed that enough of the initial conditions are given so that the problem is properly posed. A more general class including the Mayer reciprocal problems can also be handled, provided the velocity at one end is not involved in the end conditions. The method can be applied to initial design problems wherein it is desired to make an estimate of thrust and fuel requirements for a rocket which is to effect some specified mission, such as to gain a given altitude or intercept a specified type of target. A COMPLETE and elementary method is given in this paper for solving a class of problems which arises in the study of optimum performance of a rocket and in associated design problems. The method reduces the solution of prob­ lems, such as that of intercepting a target following a known path above the atmosphere, in a uniform gravitational field, to elementary algebra. For many problems a prepared graph can be used to reduce the work of solving to the plot­ ting of a simple curve and reading off values associated with the solution. The method is based on the following simplifying assump­ tions: The gravitational field is a known function of time, aerodynamic forces are negligible, the rocket is treated as a point mass, and the path of the target point is a known function of time. The method is carried through in some detail for the case of a rocket whose maximum thrust is constant, and for a ballistic target. The general type of problems which it treats are: Initial conditions are assumed given; it is desired to intercept a target following a known path in minimum time, or with minimum fuel consumption, or to maximize the range attained by a rocket with a given amount of fuel. Most of the problems which are Ma}^er reciprocal [(1) p. 574] 2 can be solved more or less directly, and some problems wherein the initial values are not all given. It is well known [see (4) or (8) for elementary proofs ] that under these assumptions, thrust must be fixed in direction and as large as possible during an initial period for maximum performance. With these assumptions, a single grid for dis­ placement can be drawn up in dimensionless form for all rock­ ets whose thrust is similar (defined in the text) during its period of application. On this graph the displacement of the target point may be plotted in the proper coordinate set, and from the relation between the two curves it can be deter­ mined at what times, if any, interception is possible, in­ cluding the earliest and latest times, and the time which cor­ responds to minimum fuel consumption. For a ballistic

The disturbance due to a line source in a semi-infinite elastic medium

When a cylindrical pulse is emitted from a line source buried in a semi-infinite homogeneous elastic medium, the subsequent disturbance at any point near the surface is much more complex than for an incident plane pulse. The curvature of the wave-fronts produces diffraction effects, of which the Rayleigh-pulse is the most important. In this paper the exact formal solution is given in terms of double integrals. These are evaluated approximately for the case when the depths of source and point of reception are small compared with their distance apart, allowing a description of the sequence of pulses which arrive at the point of reception. When that point is at the surface and distant from the epicentre, the disturbance there can be regarded as made up of the following pulses, in order of arrival: ( a ) for initial P -pulse at source: P -pulse, surface. S -pulse and Rayleigh-pulse; ( b ) for initial S -pulse: surface P -pulse, S - pulse and Rayleigh-pulse. If the initial pulse has the form of a jerk in displacement, the P - and S - pulses arrive as similar jerks, whereas the Rayleigh-pulse is blunted, having no definite beginning or end. The surface P-pulse takes a minimum-time path and arrives with a jerk in velocity. The surface S -pulse, on the other hand, is confined to the neighbourhood of the surface and arrives as a blunted pulse. Moreover, part of the S -pulse arrives before the time at which it would be expected on geometrical theory. Although derived on very restricting hypotheses, these results may throw some light on seismological problems. In particular, it is shown that when the sharp S -pulse of ray theory is converted by the presence of the surface S -pulse and the spreading of S into a blunted pulse, the duration of this composite pulse is of the same order of magnitude as the observed scatter of readings of Sg and other distortional pulses from near earthquakes.