[Continued from Getting embedded: a special connection (Part 2)]

My next real-time job was among the most fun, mainly because I got to recommend all the parts of our development system and tools. It was for yet another satellite tracking antenna, only this one was in an airplane, and therefore bopping around instead of bolted to the ground. We chose the Motorola 68332 chip. We used the Intermetrics C compiler, which included a very nice source-level debugger. For once, I didn't need floating-point software, but I still needed—guess what—functions f......

[Continued from Getting embedded: a special connection (Part 1)]

Changes in the winds
Fast forward to 1970. I was still programming an unseen mainframe in FORTRAN. That particular mainframe wasn't even in our building; it belonged to NASA. Our only contact with it was a courier, who made twice-daily runs to pick up our card decks and return printouts. Turnaround time was 24 hours. To keep the pump primed, each time we got a run back, we'd pore through the printout with a red pen in hand, marking it up for the next cycle.

But big changes were on the horizon: I'd......

At this moment, you have most likely heard that Embedded Systems Design magazine had its last print issue. That occasion is especially poignant for me, because so much—20+ years—of my career has been tangled up with the magazine in general, and the Programmer's Toolbox column in particular.

Some folks have been blessed (or cursed) by careers that are "linear." They start one job, stay with it, move up the ladder, and retire happy. Mine hasn't been that way. It's taken some sometimes-unexpected twists and turns—some more pleasant t......

[Continued from The evolving face of Mathcad (Part 1)]

Generating print for textbooks is one thing. For a single scientist or small group, generating it for published papers was quite another. I well remember serious technical papers written on a typewriter, with all the math equations written by hand. Or, not much better, generated by a typist skilled in typing subscripts, superscripts, and Greek letters. It was only natural that we average computer users would yearn to be able to generate book-quality print and equations, from a computer. In other words, let the tex......

There's some exciting news about a product I've lived with—and sometimes praised, sometimes cursed—for decades: Mathcad, PTC's leading math analysis tool.

Regular readers know that most of my columns are math-oriented, as is most of my work. For that reason, I'm a heavy user of Mathcad. For more than 15 years, I've maintained a love-hate relationship with Mathcad. I've mentioned it many times in my columns, in terms ranging from wholehearted endorsements to heated, curmudgeonly rants. Although I often curse Mathcad as I use it, the r......

Physicists know that many problems can be treated as though the dynamical event happened instantaneously. Examples might include the impact of a hammer and nail, two billiard balls colliding, or a baseball bat hitting the ball. Deep down, we know that there is a complex interaction involve. Materials get deformed, forces get exerted, and velocities get changed. But we don't need to know the details or the time history of the collision. From our perspective, the scale of time is so short that it might as well be zero.

We call such collisions impulsive. From our perspe......

Read the previous part here.

This is not an unreasonable approach, as long as you don't mind the 90 W part. A vertical descent is what we used for Surveyor and other unmanned missions. It simplifies the automated landing quite a bit, and you can change that landing longitude by making only the slightest tweaks to the outbound trajectory.

The vertical descent is simple to automate, but it's not very efficient. When you're landing vertically, your rocket motor expends a lot of fuel fighting gravity. With a grazing, nearly horizontal approa......

Click here to view the previous part.

This approach is, in fact, the same one adopted by Euler, Lagrange, and company. In their formulation of the RTBP, they wrote the equations of motion in the rotating system, which meant that they had to add centrifugal and Coriolis terms to the equations. We don't really have to do that. It's easier to compute things in an inertial system; we only need to use the rotating one during input and output transformations.

In 1960, I was using a simulation of the RTBP to study the circumlunar trajectory. I found it pre......

Read the first part here.

The three-body problem

One look at figure 1, though, tells you that this ain't your grandfathers ellipse. It's a strange and complicated trajectory, bending left, then right, then around the moon, in a sinuous path perhaps more familiar to a figure skater than an astronaut.

The trajectory is not an ellipse because this isn't a two-body problem, it's a three-body problem. As soon as the early astronomers—which included such mathematical giants as Gauss, Euler, Lagrange, and Poincaré......

We have a new topic this month, and it has absolutely nothing—well, almost nothing—to do with the topics we've been discussing lately—software development, testing, and the like. On the other hand, it certainly has to do with embedded systems and software. Systems that tend to be rather remote. Like 240,000 miles remote.

&nb......