Digital Differential Analyzer Research Articles

Buffered DDA command generation in a CNC

In a computerized numerical controller (CNC), interpolating more than one block in a sampling interval, increases the feed rate. Some commands skipped by the generator are pre-saved in a circular buffer, to provide faster operation than that of a conventional digital differential analyzer. The feed rate can be increased when programmed distances are short. The high feed rate is confirmed by installing the buffered command generation algorithm in the motion board that includes a digital signal processor. The feed rate can reach 11.6 m/min, when minimal (1 μm) distance is programmed in all blocks.

Three-dimensional extension of Bresenham's algorithm and its application in straight-line interpolation

Conventional straight-line generating algorithms, such as the digital differential analyser (DDA), Bresenham's algorithm and the mid-point algorithm, are suitable only for planer straight lines on the coordinate planes, of which Bresenham's algorithm is the most efficient. In this paper, the authors have extended Bresenham's algorithm to spatial straight lines. Given a spatial straight-line segment with two end-points, the authors have applied Bresenham's algorithm to the projections of the line segment on two of the three coordinate planes, which is determined by the largest of the coordinate lengths of the line segment, thereby obtaining a three-dimensional extension of the algorithm. In a case study, the authors calculated the distance between each sample position and the given line segment. The result reveals that the actual error at each sample position is smaller than the maximum theoretical error, and the performance of the three-dimensional extension of Bresenham's algorithm is as good as that of Bresenham's original planer algorithm. One of its potential applications is the three-dimensional step straight-line interpolation used in computer numerical control (CNC) systems of machine tools and rapid prototyping machines. Application of the algorithm is contrasted with that of the traditional DDA step straight-line interpolation algorithm. The result confirms that the three-dimensional extension of Bresenham's algorithm is much better than the DDA straight-line interpolation algorithm.

Fixed-point digital differential analyser with antialiasing (FDDAA)

This paper presents a quick, efficient and simple method for drawing straight lines with antialiasing on a discrete display using the DDA algorithm with fixed-point arithmetic. The algorithm permits the drawing of grey tones lines by using a 3-pixels-high and 1-pixel-wide brush. The computing cost is low and only integer arithmetic is used. If a line is understood as a 1-pixel-width ( W) and n-pixels-long ( L) rectangle, its surface is given by W× L. The real length ( L) increases (hypotenuse of a rectangle triangle) as the line slope does, and so, the total radiation emitted that is proportional to its real surface. This algorithm takes into account this intensity increment. The algorithm maintains the width of the line perpendicular to its longitudinal axis independent of the slope.

Parallel Fixed-Point Digital Differential Analyser with Antialiasing. PFDDAA.

This paper introduces a quick, efficient and simple algorithm for drawing straight lines with antialiasing on discrete devices like printers or CRTs. It is based on the DDA algorithm. It uses 32 bits fixed-point arithmetic. This is a pipelined, parallelized and scalable SIMD version. It may be hardware implemented. It is suitable for MMX or Streaming SIMD Pentium (Pentium is a trade mark from Intel Corporation) III instructions. The algorithm allows drawing true colour lines using a brush 3 pixels high and 1 pixel wide in parallel. The computing cost is very low and only integer arithmetic is used. This algorithm takes into account that the real width of the line remains constant always and it does not depend on its slope. Given that the total radiation emitted by a line is proportional to its real surface, total emitted light also increases with its slope.

Development of New NC Controller for Multi-Axes Driving Machine.

Numerical control technique is widely used for controlling 2 or 3 axes driving machines such as machine tools or industrial robots by giving many pulse series to the industrial machine. NC languages for numerical control has been already prepared for driving many industrial machines and applying commands of the NC language the interpolation of the straight line or circular curve about trajectory line of tools can be realized easily. But the multi-axes machines or articulated industrial robots don't have the perpendicular coordinates, so DDA (Digital Differential Analyzer) technique that is a technique for the interpolation of straight line or circular curve cannot be applied to control multi-axes driving machines like 6-articulaled robot arms for the trajectory line of tools or robot hands. In this study, new NC controller that involves the forward kinematics of robot, its inverse Jacobian matrix and V-F converter is proposed by the computer software. The new NC controller that is proposed in this study is made by a feedback loop in the computer software and can generate on-line pulse series for realizing a desired posture and orientation of robot hand.

Design and performance evaluation of a pixel cache implemented within application-specific integrated circuits

The application-specific integrated circuit (ASIC) design and the performance of a graphics processor that uses a pipelined-cache with FIFO memory to transfer a 3D pixel array and its z values to the frame buffer in one cycle are described in detail. The functional modules in the graphics processor include: (1) a video refresh converter, (2) a module that combines texture-mapped patterns onto Phong-shaded surfaces, and (3) a bidircctional parallel link between external devices and the frame-buffer modules. Digital differential analyzer (DDA) algorithms and the size of the pixel cache relative to the frame-buffer bandwidth, have been selected for good overall performance. A drawing speed of 8 ns/pixel (32 bits/pixel) or 1.2 million Phong-shaded polygons/s (100-pixel polygons, texture mapped with hidden surface removal) was achieved when 60-ns accesstime single port DRAMs and synchronous DRAMs were used.

An interpolation scheme for tool-radius compensated parabolic paths for CNC

In general, NC systems provide only linear and circular interpolations. Complex shapes to be machined are first approximated with linear and circular segments to some predetermined tolerance. Parabolic curves are more suitable for piecewise approximation of higher-order curves because less segments than lines or circular arcs are required. In machining operations, the tool motion controlled by the interpolator does not move along the profile of the job. The tool path is offset from the profile by the tool radius. Tool-radius compensations for linear and circular tool paths involve simple offsets of the initial and final points. The tool paths remain linear or circular. For parabolic profiles, the offset tool paths are not parabolic; hence, parabolic interpolations cannot be used to generate the offset paths. This paper presents an algorithm to perform interpolations for the tool to follow parabolic curves as well as the offset paths passing through two user-defined end points. The parabolic interpolation is based on the principle of operation of the Digital Differential Analyzer (DDA) integrator and is capable of constant tangential velocity.

A low-cost raster engine for video game, multimedia PC and interactive TV

A low-cost Raster Engine (RE) is designed and implemented to improve the performance of 3D computer graphics and image composition application for video games, multimedia PCs and interactive TVs. Three operation modes: Gouraud and Phong shading algorithms, and image composition are incorporated in this chip. Modified digital differential analyzer (DDA), 2-level pipeline, and constant execution time for calculating cos/sup n/ /spl theta/ are proposed as the features of this design. The accelerator is implemented by 0.8 /spl mu/m SPDM CMOS VLSI technology and able to release more then 50% CPU loads.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A DDA parabolic interpolator for computer numerical control of machine tools

This paper is primarily concerned with the development of a parabolic interpolation algorithm for implementation in a computer numerical control (CNC) system to perform tool path control with constant tangential velocity (feedrate). Less segments of parabolic curves are required than of line or circular arcs for piece-wise approximation of higher-order curves. Hence, there is a need to develop an efficient algorithm for real-time control of a tool to follow a specified parabolic curve. A software approach based on the digital differential analyzer (DDA) method is presented. This integer computation method simplifies real-time implementation using assembly language or digital hardware. The developed parabolic interpolator has been implemented on a multiprocessor CNC system. Results of tests of the implemented algorithm are compared with those obtained from a hardware interpolator with parabolic interpolation capability. Results of fitting techniques using linear, circular and parabolic segments are also presented and plotted. Considerations in the parabolic interpolation include the accuracy in following the theoretical parabolic curve and the ability to maintain a constant tangential velocity. Word address codes for NC programming to generate the parabolic curves are also proposed.

Hardware realization of inverse kinematics for robot manipulators

For real-time processing of kinematic information required for intelligent robotic applications, a hardware realization of an inverse kinematics algorithm is a challenging task. This paper adopts an incremental unit computation method to accomplish the inverse kinematics of a three-axis articulated robot. This method starts from defining incremental units in joint and Cartesian spaces, which represent the position resolutions in each space. With this approach, calculation of the inverse Jacobian matrix can be realized through a simple combinational logic gate circuit. Furthermore, the incremental direct kinematics can be solved by using a digital differential analyzer (DDA) integrator. The hardware architecture to implement the algorithm is also described. Applying the hardware implemented by an erasable programmable logic device (EPLD) to the straight-line trajectory of an experimental robot, the authors have obtained an end-effector's maximum speed of 12.6 m/s.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Error-Free Interpolation of Parametric Surfaces

A Digital Differential analyzer (DDA) is a cost effective implementation for interpolating parametric curves and surfaces. However, the problems of register overflow and integer round-off have prevented it from being adopted by industry. Contrary to intuition, an increase in register capacity is shown to have no bearing on the register overflow problem. It is shown that there is a minimum number of interpolation steps Kl below which register overflow occurs. Correspondingly, it is also shown that there exists a maximum number of interpolation steps Ku above which the accumulation of integer round-off errors exceeds the acceptable limit. But, when Kl&amp;gt;Ku, conflict arises. A solution is given to resolve the possible conflict and to yield an error-free interpolation of curves and surfaces.

Circular interpolators for numerical control: A comparison of the modified DDA techniques and an LSI interpolator

In numerical control of machine tools, the control of the tool path to follow a specific curve is accomplished by interpolation. A well-known interpolation technique is based on the functional operation of the digital differential analyzer (DDA) integrator. This method of NC path control can be implemented on-line using digital hardware or software. This paper discusses modifications of the DDA interpolation technique for improved accuracy and computation speed, whilst retaining the suitability of the technique for integer computation or implementation using digital hardware. Results obtained from the best modified DDA technique are compared with those obtained from a widely-reported LSI hardware interpolator.

A speedy voxel-tracing method for cutting solid display of 3-D images

This paper describes a high-speed image-generation method which arbitrarily cuts out a three-dimensional (3-D) digital object (3-D image) in a 3-D frame memory and three-dimensionally displays its internal shape realistically. The idea of ray-tracing is applied to a 3-D memory space so that a high-speed ray-tracing is carried out in it by using a 3-D digital differential analyzer (DDA) called the reversible step type. To obtain cross points of a ray (straight line) and an object in a 3-D frame memory, an algorithm indicating address orders of all the voxels through which the ray passes is shown. A viewpoint of obtaining a realistic 3-D representation with a smooth surface from an irregular digital surface, calculation method of normal vector is important. A new factor, “existential density,” which can be obtained by a simple calculation around the surface of the digital object, is introduced; and a method to obtain a normal vector from this factor using the distribution of density gradient also is described. Since this calculation requires 3 × 3 × 3 operators, addition/ subtraction of integers and shift operations, the operation is high speed and suitable for hardware. The proposed algorithm is confirmed by a computer simulation which produces 3-D images with smooth shading.

Digital circles with non-lattice point centers

Digitization of a real circle is defined as the set of closest pixels (integer lattice points) to the circle. These pixels form a digital circle. First, this paper defines the term “closest pixels” and provides an algorithm to digitize an arbitrary circle, i.e., one whose radius is not necessarily an integer, and whose center is not necessarily a lattice point. This digitizing algorithm is simplified to the case of integer radius and integer-coordinate center so that it can be compared to two well-known algorithms: the Bresenham algorithm and an algorithm using the Digital Differential Analyzer (DDA) technique. As a result, the new algorithm consumes only 77% and 40% of the time required for Bresenham and DDA algorithms, respectively. Second, we investigate the problem of finding, for a given digital circle and center, the set of possible real radii that generate that digital circle. For some integerr, we look at all digital circles centered at the origin with real radius in the interval [r−0.5,r+0.5) and determine the paritions of this interval into sub-intervals of real radii generating the same digital circles. Third, we study fixing the digital circle and radius, and finding the set of possible centers. For some pixel (xc,yc), we find the partitions of the unit square [x c −0.5,x c +0.5)×[y c −0.5,y c +0.5) such that for each partition each center in that partition generates the same digital circle. The partition boundaries can be constructed by compass and ruler. Finally, the paper uses these results to determine whether or not a given finite set of pixels is a digital circle segment and if so, shows how to construct its region of possible centers by ruler and compass. We also present an algorithm to detect whether a closed digital are is a digital circle.

全腕式電動義手(TDU ys-86)のメカニズムと工学的評価

Many handicapped persons have had both arms amputated from the shoulder due to accidents or malignant tumors; these individuals need help in their daily life, especially for taking meals, and such needs can some-times be filled by a high-performance shoulder prosthesis. About 20 years ago electrical prostheses were developed for thalidomide victims in many countries; however, insufficient degree of freedom, problems in controlling, increase in weight in proportion to freedom degree improvement, and other problems gradually put them out of use. We therefore have developed over the past several years a new type of total-arm electrical prosthesis (right arm) with DC motors controlled by 3 single-chip microcomputers, and have developed technical support system for the arms using a CAD system. The arm consists of a space linkage shoulder mechanism capable of giving the arm 6 degrees of freedom, with a modular gripper including forward and side raising, rotation of upper arm and elbow bending, inside and outside direction forearm turning, and grippers opening and closing. A battery and control printed circuit board was made in order to keep the weight light. The arm is basically operated by chin switches. The main movement is divided into two kinds, i. e., manually controlled and combined movement by DDA (digital differential analyzer). With the cooperation of 3 disabled persons, we are currently undertaking training in wearing and operating the arm using a personal computer.

ＤＤＡ方式を用いたカッタの刃先方向制御

This paper describes a generalized method of cutter-direction control using DDA (Digital Differential Analyzer) algorithms. The method is applicable to cutting out two dimensional arbitrary shapes with high accuracy. The cutter path generation process requires a method for moving the cutter along the lines of the figure and for keeping the cutter tip tangent to lines of the figure. The algorithm is capable of generating equations for lines, arcs, ellipses, and arbitrary shapes. For complex shapes, the DDA algorithm generates the desired functions which represent the lines of figure and the tangent lines are generated from points input by a user. From this information, the motor control parameters for cutting these shapes may be easily obtained.

デジタル微分解析器とデジタル制御型機械式加力装置を用いた多自由度振動系のオンライン地震応答載荷実験 : 第2報 多層せん断型架構への拡張

In this paper, the authors have presented a generalized method for solving a non-linear response of multi-degrees of freedom structure based on a newly developted ON-LINE simulation system, which is composed of digital actuators and a digital differential analyzer. In order to examine the efficiency of this system, a series of on-line experiments have been conducted using a two and a three-storied steel building model. The test results have shown that the response obtained by the proposed technique agrees fairly well with a numerical analysis using a simple elasto-plastic hysteretic model.

デジタル微分解析器とデジタルアクチュエータを用いた多自由度振動系のオンライン地震応答載荷実験法 : 第1報 システムの提案

The authors have developed a method of analysis to find the non-linear earthquake response of multi-degrees of freedom structures by using a Digital Differential Analyzer(DDA) and the Digital actuators. Details of the system and the computing method are described herein. Based on this system, the non-linear seismic behavior of the two-degrees of freedom structures subjected to strong earthquakes were experimentally simulated.

On synchronized proportional motion of multi-robot systems with stepper motors and shared controller

SUMMARYThe synchronized action of robots is important in many applications, especially in the manufacturing processes. An algorithm to obtain a smooth synchronized movement of the multiple-robot system is presented for the case when only one processor is available to control all of the robots that incorporate stepper motors. The origin of this algorithm stems from the digital differential analyzer used in machine tool control.

An application of Bertrand curves and surfaces to CADCAM

A method is proposed for generating tool centre paths with an accuracy equal to the resolution of the machine tool. The method exploits the properties of Bertrand curves and surfaces, to generate tool motion which secures the desired degree of continuity in the machined contour or surface. It is shown that for cubic splines or bicubic patches, accurate tool motion can be generated by a fast DDA-like (Digital Differential Analyser) digital integration scheme, which reduces integration to successive additions. Accuracy is achieved by eliminating the integration error inherent in DDA.