Edward B. Magrab

A general procedure to evaluate robot positioning errors

A new, explicit, and complete formulation that describes the geometric errors due to both origin translation and misalign ments of axes in the positioning of an open-loop robot ma nipulator has been presented. The formulation does not use the usual Denavit-Hartenberg approach. The results clearly display the role of each quantity involved and allow easy physical interpretation of each error term. First, a general kinematic formulation for an ideal robot with an arbitrary number of links is developed. The geometric errors in axes locations and orientations are then shown to be skew coordi nate transformations with origin translations and are incor porated into the analysis using general tensor algebra. The final forms of forward and backward transformations contain up to 9 (N + 2) error parameters for a robot with N physical links. This theoretical formulation is applied to a specific six-degrees-of-freedom robot. It is observed that of the possi ble 72 errors in this robot, only 53 can independently contrib ute to the error at the end effector and are the only ones that can be determined by measurements made at the end effec tor. It is shown how a correlation matrix analysis can be used to isolate these 53 parameters. In addition. it is shown how, using a successive orthogonalization technique, one can eliminate from consideration error parameters (eight in the present case) that are too small to be significant but contrib ute to the ill-conditioning of the coefficient matrix. Applica tion of the procedure to computer-simulated data shows that the formulation and the numerical techniques developed here combine to form a powerful method to calibrate a robot.

Radiation from the Forced Harmonic Vibrations of a Clamped Circular Plate in an Acoustic Fluid

The forced harmonic vibrations of a solid circular plate clamped to a rigid infinite baffle and bounded on one side of an inviscid fluid of infinite extent is considered. The motion in the plate is described by the Mindlin‐Timoshenko theory, which includes the effects of transverse shear and rotatory inertia. The external force, which is applied from the in vacuo side of the plate, can have any spatial distribution. The solution is obtained with the use of orthogonal functions which result from transforming the polar cylindrical coordinates describing the fluid motion into oblate spheroidal coordinates and the solution to the in vacuo vibrations of the clamped plate. This method of solution removes all previous restrictions regarding the solution of this problem, namely, boundary conditions of the plate, symmetry of loading, applicable frequency range, and extensive numerical calculations for the determination of the complete nearfield pressure. Expressions are obtained for the near‐ and farfield pressure in the fluid and the radiated power from the plate‐fluid surface. A numerical case for a concentrated load at the center of the plate is presented.

Farfield radiation of internally generated transient acoustic pulses transmitted through a spherical shell

The effects of a fluid‐loaded spherical shell on the propagation to the farfield of an internally generated, transient acoustic pulse is obtained. The pulse is generated by a spherical transducer that is located concentrically within the shell and has a specified spatial and temporal velocity distribution. A shell theory that incorporates the effects of transverse shear and rotary inertia is used. The fluid between the transducer and the shell, and the fluid external to the shell, is described by the linear acoustic wave equation. The spatial dependence of the transmitted pulse is expanded in a series of spherical harmonics. The temporal dependence is determined by numerical inversion of the Fourier transformed solution. Computations are performed for axisymmetric transducer velocity pulses of realistically short duration. It is found that a steady‐state analysis is incapable of predicting the nature of the farfield pulse, since the applied waveform and the farfield waveform were dissimilar. The shell mem...

Studies on the spatial variation of decaying sound fields

The spatial variation of the reverberation time was measured in the NBS reverberation room in the 1/3‐octave bands from 80 to 10 000 Hz to determine the following: (1) the effects on the precision of the spatially averaged reverberation time due to (i) vane speed and vane orientation, (ii) loudspeaker location, and (iii) the area and location of an absorbing panel and its absorption coefficient; (2) the selection of the parameters in (1) above such that the measurement uncertainty of the reverberation time is minimized; and (3) the overall measurement uncertainty for this optimum configuration as a function of the number of microphone locations and the number of decay curves recorded at each microphone location. For an 11‐m2 panel with relatively little low frequency absorption and with the vanes oriented at 22.5° from the vertical and rotating at 7.5 rpm, an analysis of variance indicated that the total uncertainty of the measured average reverberation time (one standard deviation from the mean) was less...

Forced Harmonic and Random Vibrations of Concentric Cylindrical Shells Immersed in Acoustic Fluids

The forced harmonic and random vibrations of an elastic cylindrical shell surrounded by an inviscid fluid and concentrically contained by another thin elastic shell which itself is immersed in another inviscid fluid of infinite extent is considered. The motion of the shells and fluids is assumed independent of the axial coordinate. The motion of the shells is described by a theory which accounts for transverse shear and rotatory inertia. The motion of the fluids is described by the classical wave equation. Expressions for the acoustic pressure at the outer surface of the inner shell and the inner and outer surfaces of the outer shell are obtained along with the displacements (velocity) at these surfaces. Numerical results for the near‐ and farfield acoustic pressure are given for the case wherein the interior of the cylinder is subjected to diametrically opposed point forces which vary either harmonically or randomly with time. The random excitation is assumed to be spatially uncorrelated, broad‐band whit...

Reply to ’’Comments on ’Environmental effects on microphones of various constructions’ ’’ [J. Acoust. Soc. Am. 61, 1647 (1977)]

To answer the question as to whether or not the microphones were given sufficient time to ’’recover’’ from a particular humidity we elaborate further on the experimental procedure.

Comparison of various techniques for the determination of the random incidence response of microphones and sound level meters

The measurements necessary for the determination of the random incidence responses of microphones and sound level meters were made from 500–20 000 Hz in an anechoic chamber using both single‐frequency tenes and 1/3‐octave bands of noise. The random incidence responses were computed from these data using formulas given in the relevant IEC and ANSI standards documents and according to a Gaussian‐type numerical integration formula. This Gaussian formula appears to introduce less numerical error due to the approximation procedure used to compute the random incidence response than either the IEC or ANSI formulas. The random incidence responses of the microphones and sound level meters tested in the anechoic room were also measured in two reverberation rooms using both discrete sampling of the reverberant field and two types of continuous sampling: linear and circular. The results from the various methods and procedures are compared to the exact values (determined by a numerical procedure that used at least 30 ...

Measurement of reverberation time in the National Bureau of Standards reverberation room

The spatial variation of the reverberation time in the 1/3‐octave bands from 80 to 10 000 Hz was studied as a function of: (1) the surface area of the absorbing material ranging from 0 to 13.4 m2; (2) the orientation and rotational speed (0–10 rpm) of the vanes; (3) loudspeaker placement and type; (4) discrete microphone versus high‐speed multiplexed data acquisition; (5) the number of microphones; and (6) the number of decay times averaged at a given microphone location. For the case of the vanes rotating at 7.5 rpm it was found that the standard deviation of spatial variation of the reverberation time from 12 microphones ranged from 1% to 12% of the mean value in the frequency bands below 400 Hz, was less than 0.5% from 400 to 5000 Hz and increased to 1.5% at 10 000 Hz. The values were virtually independent of the surface area of the absorbing material placed in the room. With the vanes stationary these percentages increased by as much as a factor of 5. In addition to the decrease in the spatial varianc...

Vibrations of prolate spheroidal shells of constant thickness

The general displacement‐equilibrium equations, which include the effects of transverse shear and rotary inertia, have been derived for a prolate spheroidal shell of constant thickness. The solution is formulated for a shell that is immersed in an inviscid fluid of infinite extent and subjected to an harmonically time‐varying, arbitrarily spatially distributed force normal to the shell surface. The approximate formal solutions for the three displacements of the shell surface and the two rotations of the shell cross section are obtained using an extension of Galerkins variational method developed by Chi and Magrab [Proceedings of the International Conference on Variational Methods in Engineering (University of Southampton, 1974)]. Numerical results are presented for the lowest seven axisymmetric natural frequencies of the shell in vacuo. Using 15‐term solutions for both thick and thin shells, which have eccentricities that vary from 0.46 to 0.99, the approximate natural frequencies are found to converge t...

Effects of ground cover and reflecting surfaces on sound‐level meter measurements

The effects of soft (grass) and hard (asphalt) horizontal reflecting planes on the measurement of the sound pressure level have been determined as a function of source‐receiver distance for 15 one‐third octave bands of noise from 250 to 6300 Hz and for several combinations of heights of the source and receiver relative to the reflecting surfaces. Comparisons are made between these measurements (semifree field) and measurements made in a free field (anechoic chamber). It is found that interference effects, characterized by the ratio of the difference between the distances of the direct and bounce paths to the wavelength, tend to dominate the semifree‐field data and show strong noise‐band dependency. Computed A‐weighted sound pressure levels are compared for the various semifree‐field and free‐field source‐receiver combinations. A second and more complicated configuration consisting of two orthogonal intersecting planes has also been studied. Again, comparisons to the free‐field data have been made.