Endpoint Trajectory Research Articles

Inverse cartesian trajectory control and stabilization of a three‐axis flexible manipulator

AbstractThis article treats the question of end point trajectory control of a flexible manipulator based on the nonlinear inversion technique. The manipulator has two rigid links and the third link is elastic. A parameterization of the Cartesian coordinates of a point close to the end effector position is suggested. Using these coordinates as output variables, an inverse feedback control law is derived for tracking reference Cartesian trajectories. The stability of the zero dynamics associated with the end point motion control is examined. It is shown that inverse control of the end point causes divergent oscillatory flexible modes. In addition, for regulating the end point to a fixed position, a linear stabilizer is designed to damp the elastic vibration. Simulation results are presented to show that in the closed‐loop system, reference end point trajectories can be accurately followed in spite of the parameter uncertainty in the arm dynamic model. © 1994 John Wiley & Sons, Inc.

A receptor‐oriented methodology for determining source regions of particulate sulfate observed at Dorset, Ontario

A statistical receptor‐oriented model was developed for long‐range transport of atmospheric sulfate to Dorset (elevation 320 m, latitude 4513′26″ N and longitude of 7855′52″ W), Ontario. This model computes the potential for sources within 1 latitude by 1 longitude grid cells across North America that contribute to the airborne concentrations measured at the ground station at Dorset. Airborne concentration data and air parcel backward trajectories were incorporated explicitly in the model calculation to identify the geographical areas of potential contributing sources. The present model is qualitative in nature; however, it provides a reasonable receptor‐oriented approach to examine the long‐range transport of atmospheric species. In order to fully understand the methodology and in a hope to optimize it, several aspects of the PSCF methodology have been examined in detailed in this study. Results of this study are presented that suggest interpolation of trajectory endpoints to increase the counting statistics for the potential source contribution function (PSCF) values is not reliable. The average concentration provides a reasonable criterion value; however, using the fiftieth percentile value as the criterion point provides an opportunity for identifying source areas that cannot be previously found by using the average concentration. The fiftieth percentile value may be a better choice for the particulate sulfate data in this case since Dorset is a relatively clean background site. Using the seventy‐fifth percentile, which is generally larger than the average, may not be suitable because it reduces the number of degrees of freedom. This could render the model to behave like a regular trajectory analysis model that has been used commonly for analyzing episodic pollution events. Separation of data into summer and winter periods is useful to illustrate the effects of photochemistry and meteorology on the PSCF results. Invoking the total probability concept and examining the trajectory arrival at different heights directly above the sampling site, the total PSCF was computed. This resultant function thus provides a time‐integrated geographical map useful for identifying sources of airborne particulate sulfate in a receptor‐oriented manner.

Adaptive control of flexible multilink manipulators

An adaptive self-tuning control scheme is developed for end-point position control of flexible manipulators. The proposed scheme has three characteristics. First, it is based on a dynamic model of a flexible manipulator described in cartesian coordinates, which eliminates the burden and inaccuracy of translating a desired end-point trajectory to joint coordinates using inverse kinematic relations. Second, the effect of flexibility is included in the dynamic model by approximating flexible links with a number of rigid sublinks connected at fictitious joints. The relatively high stiffness of the fictitious joints is shown to result in a decomposition of the model into two subsystems operating at different rates. This allows for stabilization of the oscillatory modes associated with the flexible links by a fast feedback control in addition to a slower control for trajectory tracking. Third, the control is constructed from measurements of the end-point position and deformations of the flexible links, with the manipulator parameters required to form the control obtained using a recursive least-squares estimation algorithm, which is fast enough for on-line applications. Satisfactory results are obtained from digital simulation of a two-link flexible manipulator.

Experiments on the Tracking Control of a Flexible One-Link Manipulator

Experimental study on the tracking control of a flexible onelink manipulator is reported in this paper. A tracking control scheme is developed based on the system transfer function. In the proposed control scheme, the desired control input for a given end-point trajectory is obtained by using a command feedforward controller instead of solving the inverse dynamic equations of the system. The proposed control scheme requires small amount of computations and can be easily implemented for real time control. The experimental results are presented, which shows very good tracking performance.

Coordination of symmetrical and asymmetrical human gait.

Most human gait forms assume symmetrical, alternating patterns of interlimb coordination (e.g., crawling, walking, running). Human galloping is a notable exception. In contrast to extensive information on galloping in animals, little is known about this gait in humans. Therefore, kinematic and topographical analyses of running and galloping were undertaken to investigate the manner in which the lower limbs are uncoupled to produce this asymmetrical gait. Seven adult females were filmed while running and galloping at their preferred speed. Analysis of the gaits revealed differences in the following: (a) preferred speed, (b) coupling between upper- and lower-limb girdles, and (c) point of foot fall (end-point trajectories). In contrast to clear differences in interlimb coordination, intralimb coordination was remarkably similar across gaits, although when galloping was adopted, the rear leg did show more variable change than the front leg.

A Hierarchical Distributed Path Planning for Redundant Manipulators Based on Virtual Arm

This paper proposes a collision-free path planning algorithm in the task space based on virtual arms. The virtual arm has the same kinematic structure as the actual arm except that its end-point is located at the joint or link of the actual arm. Therefore, the configuration of the actual arm can be represented as a set of end-points of the virtual arms, and the path planning for multi-joint manipulators can be performed only in the task space. Our method adopts a hierarchical strategy which consists of the global level, the intermediate level, and the local level. The global level plans the collision-free endpoint trajectory of the actual arm based on the global representation of the task space. The intermediate level generates the subgoals for the actual and virtual endpoints based on the current positions and the actual endpoint trajectory specified by the global level. The local level moves each end-point to the corresponding subgoal, avoiding the close obstacles based on the local informations of the task space. The effectiveness of the method is verified by computer simulations using a planar manipulator with redundant joint degrees of freedom.

Control of Manipulator/Vehicle System on Water.

The manipulator/vehicle system floating on water consists of a vehicle and a manipulator attached to it. Similar to the space manipulator/vehicle system, the system on water is not fixed to an inertial coordinate system. Moreover, the external forces affect the motion of the system, and the interaction of motions among the vehicle, the manipulator and water causes vibration of the system. This paper discusses the control issues of a manipulator/vehicle system floating on water and proposes an algorithm to control both the endpoint trajectory of the manipulator and the vibration of the vehicle simultaneously using redundant actuators. We develop the basic kinematic equations and then derive the equation, which describes the dynamic relation between the system and the external forces/moments applied to it, then propose a simultaneous control algorithm of the manipulator endpoint trajectory and the vibration of the vehicle. Simulation results illustrate the effectiveness of the proposed control algorithm.

Climatic change in mixed-layer trajectories over large regions

Using a modified version of the NOAA ARL/ATAD model, a series of over 300,000 mixed-layer back trajectories were calculated in eighteen hour time steps through the period 1948–1985 for seventeen trajectory endpoints in and around the state of Virginia. Results of these trajectory runs were aggregated both seasonally and annually to a spatial grid system covering approximately the eastern two-thirds of the United States. From this were generated plots of trajectory frequencies, residence times and transport times. A principal components analysis was performed on the spatial distribution of the gridded trajectory frequency data. The first principal component is dominated by mixed-layer-flow from the Ohio Valley and the Upper Midwest. A time-series plot of this component indicates a dramatic increase in this northwesterly flow over the study period, especially in the 1950s and early 1960s. A possible link between this observed phenomenon and anthropogenic climate alteration is discussed.

On the Accuracy of End-Point Trajectory Tracking for Flexible Arms by Noncausal Inverse Dynamic Solutions

The problem of open-loop control of the end-point trajectory of a single-link flexible arm by an inverse dynamic solution is addressed in this paper. A finite element discretization of the system is used to obtain a set of ordinary differential equations describing the motion. Theoretical difficulties pertaining to the inverse problem for flexible structures are exposed, and it is shown that a noncausal solution for the actuating torque enables a tracking of an arbitrary tip displacement with any desired accuracy.

Composite pseudolink end-point control of flexible manipulators

A novel method of decomposing the dynamical equations of a three-link flexible manipulator is described. The concept of a 'pseudolink' is used to separate the full manipulator-system model into a pseudolink subsystem model and a flexible subsystem model. A composite controller is designed for achieving accurate end-point positioning and trajectory tracking of a predetermined trajectory described in terms of pseudolinks, and for stabilizing the flexible subsystem. Simulation results are presented to demonstrate the effectiveness of the control method. Practical steps for implementing this strategy in an actual system are also discussed. >

Robot Performance as Influenced by Mechanical System

The paper presents results of a study of 2D continuous path performance of an industrial robot. Based on comparison of simulation results with experimental position measurements, several causes of deviation from the programmed trajectory were identified. Path inaccuracies are due to both the electromechanical system (robot inertia, mechanical compliance, structural damping, actuator overload), and to the software (interpolation algorithm, control system). To experimentally measure the endpoint trajectory of the robot, an inexpensive but effective system has been developed.

An investigation of the dominant source regions of fine sulfur in the western United States and their areas of influenve

Two methods for analyzing source-receptor relationships are used to determine the dominant source regions of particulate sulfur in the western U.S. The first method, area of influence analysis, is based on the residence time of back trajectory endpoints. The other method, principal component analysis, involves examination of the gradients of the spatial eigenvectors of fine sulfur concentrations to determine source regions. Results of both methods are similar and show that southern California, northeastern Mexico, and the large coal-fired power plants in the Four Corners region contribute most strongly to the long-range transport of fine sulfur into remote areas of the western U.S. Several sources with smaller areas of influence are also identified.

An advective-interpolation method for estimating pollutant movement during anatex

A temporal interpolation technique has been developed that uses meteorological information to show the evolution of an average air concentration pattern into its next time period average. Forward and backward trajectories are constructed from every point at the desired time to the next and previous time period patterns. The concentration is then the weighted average of the concentrations at the respective trajectory endpoints. The technique can be used to smooth transitions between successive images for movie animation or video presentations. An example sequence from the ANATEX video is shown.

A study of the sources of acid precipitation in Ontario, Canada

Since 1980, the Acid Precipitation in Ontario Study (APIOS) Event Wet/Dry Deposition Network has collected a substantial precipitation chemical data base. Data from the three stations in Dorset, Kingston and London were selected for investigation of the sources of the constituents found in the collected precipitation. Factor analysis was applied to the chemical data sets. Three factors reproduce the concentrations observed in precipitation. These factors represent the acid gas sources (SO 2 and NO x sources), Ca and Mg sources, and Na and Cl sources (marine aerosol). The acid gas sources are responsible for the acidity of the precipitation. To locate these sources in terms of geographic region; air parcel back trajectories corresponding to the precipitation sample time periods are incorporated into factor analysis with the chemical data. This information is in the form of the geographic distribution of back trajectory endpoints. The region in which the trajectory endpoints fall is divided into subregions. New variables are defined as the number of the endpoints in each of the subregions. They reflect the residence time of the air parcels in the subregions. Factor analysis is applied to the combined chemical and endpoint variable data sets. The results show the relationship between the chemical nature of the sources and the geographic regions that the air parcels passed. The potential source contribution function (PSCF) is another approach to locate the sources. PSCF is a conditional probability function. It can be calculated for each 1° longitude by 1° latitude cell by dividing the number of trajectory endpoints that correspond to samples with factor scores or pollutant concentrations greater than specified values by the number of total endpoints in the cell. Source areas are indicated by high PSCF values. Both of the approaches suggest that the Ohio River Valley and some areas along the east coast are major source areas for the observed precipitation acidity in Ontario.

Intercomparison of three long-range trajectory models applied to Arctic haze

hree long-range trajectory models which have been applied to Arctic haze problems were compared. These models employ different parameterizations of vertical motions, and are based on different meteorological analyses. Median horizontal displacements between 5-day trajectory endpoints were mostly in the 800—1000 km range. Model sensitivity to the meteorological data was shown to contribute a substantial component to the overall uncertainty. Based on these comparisons, we estimate that the usefulness of trajectory models after 5 days is limited to the identification of distant source regions with horizontal dimensions on the order of 1000 km.

On Reachable Sets and Extremal Rearrangements of Control Functions

A characterization is provided for the set $\mathcal{A}(x^0 ;\mathcal{V})$ of states reachable from a given state $x^0$ along trajectories of a linear control system with controls in the set $\mathcal{V}$ of rearrangements of a prescribed function v. This reachable set is shown to be convex and compact. A necessary and sufficient condition (maximum principle) for extremal controls (that is, controls generating trajectory endpoints in thee boundary assl $\partial A(x^0 ;\mathcal{V})$ is provided. Finally, some results on existence of solutions to Mayer-type optimal control problems are deduced.

Deducing planning variables from experimental arm trajectories: pitfalls and possibilities.

This paper investigates whether endpoint Cartesian variables or joint variables better describe the planning of human arm movements. For each of the two sets of planning variables, a coordination strategy of linear interpolation is chosen to generate possible trajectories, which are to be compared against experimental trajectories for best match. Joint interpolation generates curved endpoint trajectories called N-leaved roses. Endpoint Cartesian interpolation generates curved joint trajectories, which however can be qualitatively characterized by joint reversal points. Though these two sets of planning variables ordinarily lead to distinct predictions under linear interpolation, three situations are pointed out where the two strategies may be confused. One is a straight line through the shoulder, where the joint trajectories are also straight. Another is any trajectory approaching the outer boundary of reach, where the joint rate ratio always appears to be approaching a constant. A third is a generalization to staggered joint interpolation, where endpoint trajectories virtually identical to straight lines can sometimes be produced. In examining two different sets of experiments, it is proposed that staggered joint interpolation is the underlying planning strategy.

Precipitation Growth Trajectories in a CCOPE Storm

Abstract The growth trajectories of precipitation particles that attain diameters from 0.5 to 2.0 cm are modeled within the wind field of a small, relatively steady-state, southeastern Montana thunderstorm. The trajectories are calculated backwards, from systematic arrays of particles of specified sizes at a level near cloud base. Using a simple set of criteria for rejecting the obviously impossible trajectories, the patterns of accepted trajectory end-points are compared with the radar echo patterns. Good agreement lends credence to the qualitative aspects of the trajectories. For a given size of precipitation particle, the method helps one to assign different trajectory types to specific regions within the horizontal plant on which the calculations were started. The relative importance of the different types of trajectories can thus be estimated. Particle origin mechanisms are discussed in terms of the regions in which the trajectories are found to start. The variety of successful trajectories leading t...

Calculation of A Velocity Distribution from Particle Trajectory End-Points

AbstractThe longitudinal component of the velocity of a particle at or near a glacier surface is considered, its position as a function of time being termed its trajectory. Functional relationships are derived for obtaining the trajectory from the spatial distribution of velocity and for obtaining the velocity distribution from the trajectory. It is established that the trajectory end-points impose only an integral condition on the velocity distribution, and that no individual point on the velocity distribution can be determined if only the end-points are known. An example is given of a deduced velocity distribution that is consistent with (although not uniquely determined by) the end-points of several trajectories on the lower reach of Columbia Glacier, Alaska. It is shown that constructing a velocity distribution by assigning the average trajectory velocity to the trajectory mid-point can be subject to errors of several per cent for velocity distribution features that are typical of actual glaciers. The error in this method is determined, and closed-form expressions for the trajectory are obtained, for linear velocity distributions and for two classes of second-degree distributions. The class of functions is identified to which the velocity distribution must belong for this error to be zero.

Calculation of A Velocity Distribution from Particle Trajectory End-Points

Abstract The longitudinal component of the velocity of a particle at or near a glacier surface is considered, its position as a function of time being termed its trajectory. Functional relationships are derived for obtaining the trajectory from the spatial distribution of velocity and for obtaining the velocity distribution from the trajectory. It is established that the trajectory end-points impose only an integral condition on the velocity distribution, and that no individual point on the velocity distribution can be determined if only the end-points are known. An example is given of a deduced velocity distribution that is consistent with (although not uniquely determined by) the end-points of several trajectories on the lower reach of Columbia Glacier, Alaska. It is shown that constructing a velocity distribution by assigning the average trajectory velocity to the trajectory mid-point can be subject to errors of several per cent for velocity distribution features that are typical of actual glaciers. The error in this method is determined, and closed-form expressions for the trajectory are obtained, for linear velocity distributions and for two classes of second-degree distributions. The class of functions is identified to which the velocity distribution must belong for this error to be zero.