# Why Every GPS Overestimates Distance Traveled

## Math proves that GPS distance, on average, exceeds the actual distance traveled

Illustration: University of Salzburg

Runners, mariners, airmen, and wilderness trekkers beware: Your global positioning system (GPS) is flattering you, telling you that you have run, sailed, flown, or walked significantly farther than you actually have. And it’s not the GPS’s fault, or yours.

Blame the statistics of measurement. Researchers at the University of Salzburg (UoS), Salzburg Forschungsgesellchaft (SFG), and the Delft University of Technology have done the math to prove that the distance measured by GPS over a straight line will, on average, exceed the actual distance traveled. They also derive a formula for predicting how big the error will be. The open-access paper was published in the International Journal of Geographical Information Science; an earlier version is available on Arxiv.

GPS course calculations are subject to both interpolation error (a function of the sampling interval) and measurement error (the everyday orneriness of real-world physical systems). The Salzburg team—including first author Peter Ranacher of UoS and senior author Siegfried Reich of SFG—discovered a systematic bias in distance measurement errors.

Measurement errors have many causes. The paper specifically cites: propagation delay (atmospheric fluctuations affect the speed of the GPS signal); ephemeris error (uncertainty in the precise position of the GPS satellite); satellite clock drift; hardware error (the shortcomings of the terrestrial GPS unit); signal reflections (which can increase the length of the signal path); and unfavorable satellite geometry (available GPS satellites are too low in the sky or too close together or too few, for example).

Put them together and you have readings that scatter around the true position. The Salzburg researchers found that distances derived from position measurements with randomly distributed errors will, on average, come up longer than the actual separation between two points. There are three components to their calculation:

• The reference distance (d0): the actual Euclidean distance between two points
• The variance (Vargps): the “mean of the square minus the square of the mean” of the position error, an index of how accurate the position measurement is. Variance is the square of the standard deviation, σ2
• The autocorrelation (C, perhaps more properly the autocovariance) of the measurement error. This can vary from a maximum of Vargps (if the errors are closely covariant) to 0 (if they are random) to -Vargps (if there is an inverse correlation).

The Salzburg formula for the average Overestimation of Distance (OED) is then,

OED = (d02 + Vargps - C)1/2 - d0

The variance is always positive, so if the autocorrelation is lower than the variance, the overestimation of distance will always be positive. And the autocorrelation is generally lower than the variance.

The problem becomes particularly acute when the user (or the GPS) calculates the total distance traveled by adding together the lengths of multiple segments. The differences between the true and measured distance will fluctuate—sometimes short, but more often long. Because the GPS-measured distance skews long, though, the total GPS distance error will tend to grow with each added segment.

Not content with mere calculation, Ranacher, Reich, and their colleagues went on to test their findings experimentally. In an empty parking lot, they staked out a square course 10 m on a side, reference-marked each side at precise 1-m intervals, and set a GPS-equipped pedestrian (a volunteer, one hopes) to walk the perimeter 25 times, taking a position reading at each reference mark.

The researchers analyzed the data for segment lengths of 1 meter and 5 meters. They found that the mean GPS measurement for the 1-m reference distance was 1.2 m (σ2 = 0.3) and the mean GPS measurement for the 5-m reference distance was 5.6 m (σ2 = 2.0).  They also ran a similar experiment with automobiles on a longer course, with similar results.

Now, that pedestrian-course error of 10 to 20 percent is exaggerated because of the low-cost GPS receiver used and the short reference distances. But it is big enough that your GPS watch could tell you you’re crossing the finish line of a 42,195-meter marathon while the real terminus is more than 400 meters ahead.

That’s not a hypothetical example. For years, runners have complained that their GPS watches and other devices have mismeasured the distances they’ve run over supposedly verified courses, or suddenly finding that they set personal record times the first time they use a GPS to measure their course. There have been a number of confident explanations. Most involved either interpolation error (measuring the distance between successive plots as a straight line, which will likely report a shorter-than-actual distance over a twisty course) or the runner’s non-optimal choice of routes (adding to the verified distance on each leg, and reporting a longer actual distance traveled). Maybe they’ll like this explanation better.

The Ranacher team’s results do not mean that measuring the lengths of complex courses by GPS is futile. They point out that moment-by-moment GPS velocity measurement is not subject to the same sources of error, so that calculating distance traveled by integrating velocity should yield reasonably accurate results.

This post was edited on 6 November 2015 to correct the reported overestimation of distance reported from the GPS meaasurements on a 10-meter-square course. The correct average overestmates were 1.2 m on a 1-m reference distance and 5.6 m on a 5-m reference distance. (The incorrect values in the original post were 1.02 and 5.06.)

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## When Gamers Get Nasty

### His pivot from defense helped a tiny tuning-fork prevent SUV rollovers and plane crashes

Vertical

In 1992, Asad M. Madni sat at the helm of BEI Sensors and Controls, overseeing a product line that included a variety of sensor and inertial-navigation devices, but its customers were less varied—mainly, the aerospace and defense electronics industries.

The Cold War had ended, crashing the U.S. defense industry. And business wasn’t going to come back anytime soon. BEI needed to identify and capture new customers—and quickly.

Getting those customers would require abandoning the company’s mechanical inertial-sensor systems in favor of a new, unproven quartz technology, miniaturizing the quartz sensors, and turning a manufacturer of tens of thousands of expensive sensors a year into a manufacturer of millions of cheaper ones.

Madni led an all-hands push to make that happen—and succeeded beyond what anyone could have imagined with the GyroChip. This inexpensive inertial-measurement sensor was the first such device to be incorporated into automobiles, enabling electronic stability-control (ESC) systems to detect skidding and operate the brakes to prevent rollover accidents. According to the U.S. National Highway Traffic Safety Administration, in the five-year period spanning 2011 to 2015, with ESCs being built into all new cars, the systems saved 7,000 lives in the United States alone.

The device went on to serve as the heart of stability-control systems in countless commercial and private aircraft and U.S. missile guidance systems, too. It even traveled to Mars as part of the Pathfinder Sojourner rover.

### Vital Statistics

Current job: Distinguished adjunct professor, University of California, Los Angeles; retired president, COO, and CTO, BEI Technologies

Date of birth: 8 September 1947

Birthplace: Mumbai, India

Family: Wife (Taj), son (Jamal)

Education: 1968 graduate, RCA Institutes; B.S., 1969, and M.S., 1972, University of California, Los Angeles, both in electrical engineering; Ph.D., California Coast University, 1987

Patents: 39 issued, others pending

Hero: My father, overall, for teaching me how to learn, how to be a human being, and the meaning of love, compassion, and empathy; in art, Michelangelo; in science, Albert Einstein; in engineering, Claude Shannon

Most recent book read:Origin by Dan Brown

Favorite books:The Prophet and The Garden of the Prophet, by Kahlil Gibran

Favorite music: In Western music, the Beatles, the Rolling Stones, Elvis Presley; in Eastern music, Ghazals

Favorite movies: Contact, Good Will Hunting

Favorite cities: Los Angeles; London; Cambridge, U.K.; Rome

Leisure activities: Reading, hiking, listening to music

Most meaningful awards:IEEE Medal of Honor: “For pioneering contributions to the development and commercialization of innovative sensing and systems technologies, and for distinguished research leadership”; UCLA Engineering Alumnus of the Year 2004

For pioneering the GyroChip, and for other contributions in technology development and research leadership, Madni received the 2022 IEEE Medal of Honor.

Engineering wasn’t Madni’s first choice of profession. He wanted to be a fine artist—a painter. But his family’s economic situation in Mumbai, India (then Bombay) in the 1950s and 1960s steered him to engineering—specifically electronics, thanks to his interest in recent innovations embodied in the pocket-size transistor radio. In 1966 he moved to the United States to study electronics at the RCA Institutes in New York City, a school created in the early 1900s to train wireless operators and technicians.

“I wanted to be an engineer who would invent things,” Madni says, “one who would do things that would eventually affect humanity. Because if I couldn’t affect humanity, I felt that I would have an unfulfilling career.”

After two years completing the electronics technology program at the RCA Institutes, Madni went on to the University of California, Los Angeles (UCLA), receiving a B.S. in electrical engineering in 1969. He continued on to a master’s and a Ph.D., using digital signal processing along with frequency-domain reflectometry to analyze telecommunications systems for his dissertation research. While studying, he also worked variously at Pacific States University as an instructor, at Beverly Hills retailer David Orgell in inventory management, and at Pertec as an engineer designing computer peripherals.

Then, in 1975, newly engaged and at the insistence of a former classmate, he applied for a job in Systron Donner’s microwave division.

Madni’s started at Systron Donner by designing the world’s first spectrum analyzer with digital storage. He had never actually used a spectrum analyzer before—these were very expensive instruments at the time—but he knew enough about the theory to talk himself into the job. He then spent six months working in testing, picking up practical experience with the instruments before attempting to redesign one.

The project took two years and, Madni reports, led to three significant patents that started his climb “to bigger and better things.” It also taught him, he says, an appreciation for the difference between “what it is to have theoretical knowledge and what it is to commercialize technology that can be helpful to others.”

He went on to develop numerous RF and microwave systems and instrumentation for the U.S. military, including an analyzer for communications lines and attached antennas built for the Navy, which became the basis for his doctoral research.

Though he moved quickly into the management ranks, eventually climbing to chairman, president, and CEO of Systron Donner, former colleagues say he never entirely left the lab behind. His technical mark was on every project he became involved in, including the groundbreaking work that led to the GyroChip.

Before we talkabout the little quartz sensor that became the heart of the GyroChip, here’s a little background on the inertial-measurement units of the 1990s. An IMU measures several properties of an object: its specific force (the acceleration that’s not due to gravity); its angular rate of rotation around an axis; and, sometimes, its orientation in three-dimensional space.

The GyroChip enabled electronic stability-control systems in automobiles to detect skidding and prevented countless rollover accidents. Peter Adams

In the early 1990s, the typical IMU used mechanical gyroscopes for angular-rate sensing. A package with three highly accurate spinning mass gyroscopes was about the size of a toaster oven and weighed about a kilogram. Versions that used ring-laser gyroscopes or fiber-optic gyroscopes were somewhat smaller, but all high-accuracy optical and mechanical gyros of the time cost thousands of dollars.

So that was the IMU in 1990, when Systron Donner sold its defense-electronics businesses to BEI Technologies, a publicly traded spinoff of BEI Electronics, itself a spinoff of the venerable Baldwin Piano Co. The device was big, heavy, expensive, and held moving mechanical parts that suffered from wear and tear, affecting reliability.

Shortly before the sale, Systron Donner had licensed a patent for a completely different type of rate sensor from a group of U.S. inventors. It was little more than a paper design at the time, Madni says, but the company had started investing some of its R&D budget in implementing the technology.

The design centered on a tiny, dual-ended vibrating tuning fork carved out of quartz using standard silicon-wafer-processing techniques. The tines of the fork would be deflected by the Coriolis effect, the inertial force acting on an object as it resists being pulled from its plane of rotation. Because quartz has piezoelectric properties, changes in forces acting upon it cause changes in electric charge. These changes could be converted into measurements of angular velocity.

The project continued after Systron Donner’s divisions became part of BEI, and in the early 1990s BEI was manufacturing some 10,000 quartz gyroscopic sensors annually for a classified defense project. But with the fall of the Soviet Union and ensuing rapid contraction of the U.S. defense industry, Madni worried that there would be no more customers—at least for a long time—for these tiny new sensors or even for the traditional mechanical sensors that were the main part of the division’s business.

“We had two options,” Madni recalls. “We stick out in the sands and peacefully die, which would be a shame, because nobody else has this technology. Or we find somewhere else we can use it.”

“If I couldn’t affect humanity, I felt that I would have an unfulfilling career.”

The hunt was on. Madni says he and members of his research and marketing teams went to every sensors conference they could find, talking to anyone who used inertial sensors, regardless of whether the applications were industrial, commercial, or space. They showed the quartz angular-rate sensors the company had developed, touting their price, precision, and reliability, and laid out a path in which the devices became smaller and cheaper in just a few years. NASA was interested—and eventually used the devices in the Mars Pathfinder Sojourner rover and the systems that allowed astronauts to move about in space untethered. Boeing and other aircraft and avionics-system manufacturers began adopting the devices.

But the automotive industry clearly represented the biggest potential market. In the late 1980s, car companies had begun introducing basic traction-control systems in their high-end vehicles. These systems monitored steering-wheel position, throttle position, and individual wheel speeds, and could adjust engine speed and braking when they detected a problem, such as one wheel turning faster than another. They couldn’t, however, detect when the direction of a car’s turn on the road didn’t match the turn of the steering wheel, a key indicator of an unstable skid that could turn into a rollover.

This quartz tuning fork responds to inertial forces and forms the heart of the GyroChip. Peter Adams

The industry was aware this was a deficiency, and that rollover accidents were a significant cause of deaths from auto accidents. Automotive-electronics suppliers like Bosch were working to develop small, reliable angular-rate sensors, mostly out of silicon, to improve traction control and rollover prevention, but none were ready for prime time.

Madni thought this was a market BEI could win. In partnership with Continental Teves of Frankfurt, Germany, BEI set out to reduce the size and cost of the quartz devices and manufacture them in quantities unheard of in the defense industry, planning to ramp up to millions annually.

This major pivot—from defense to one of the most competitive mass-market industries—would require big changes for the company and for its engineers. Madni took the leap.

“I told the guys, ‘We are going to have to miniaturize it. We are going to have to bring the price down—from \$1,200 to \$1,800 per axis to \$100, then to \$50, and then to \$25. We are going to have to sell it in hundreds of thousands of units a month and then a million and more a month.’”

To do all that, he knew that the design for a quartz-based rate sensor couldn’t have one extra component, he says. And that the manufacturing, supply chain, and even sales management had to be changed dramatically.

“I told the engineers that we can’t have anything in there other than what is absolutely needed,” Madni recalls. “And some balked—too used to working on complex designs, they weren’t interested in doing a simple design. I tried to explain to them that what I was asking them to do was more difficult than the complex things they’ve done,” he says. But he still lost some high-level design engineers.

“The board of directors asked me what I was doing, [saying] that those were some of our best people. I told them that it wasn’t a question of the best people; if people are not going to adapt to the current needs, what good do they do?”

Others were willing to adapt, and he sent some of those engineers to visit watch manufacturers in Switzerland to learn about handling quartz; the watch industry had been using the material for decades. And he offered others training by experts in the automotive industry, to learn about its operations and requirements.

The changes needed were not easy, Madni remembers. “We have a lot of scars on our back. We went through a hell of a process. But during my tenure, BEI became the world’s largest supplier of sensors for automotive stability and rollover prevention.”

In the late 1990s, Madni says, the market for electronic stability-control systems exploded, as a result of an incident in 1997. An automotive journalist, testing a new Mercedes on a test track, was performing the so-called elchtest, often referred to as the “elk test”: He swerved at normal speed, intending to simulate avoiding a moose crossing the road, and the car rolled over. Mercedes and competitors responded to the bad publicity by embracing stability-control systems, and GyroChip demand skyrocketed.

Thanks to the deal with Continental Teves, BEI held a large piece of the automotive market for many years. BEI wasn’t the only game in town at that point—Germany’s Bosch had begun producing silicon-based MEMS rate sensors in 1998—but the California company was the only manufacturer using quartz sensors, which at the time performed better than silicon. Today, most manufacturers of automotive-grade rate sensors use silicon, for that technology has matured and such sensors are cheaper to produce.

While manufacturing for the auto market ramped up, Madni continued to look for other markets. He found another big one in the aircraft industry.

The Boeing 737 in the early and mid-’90s had been involved in a series of crashes and incidents that stemmed from unexpected rudder movement. Some of the failures were traced to the aircraft’s power control unit, which incorporated yaw-damping technology. While the yaw sensors weren’t specifically implicated, the company did need to redesign its PCUs. Madni and BEI convinced Boeing to use BEI’s quartz sensors in all of its 737s going forward, as well as retrofitting existing aircraft with the devices. Manufacturers of aircraft for private aviation soon embraced the sensor as well. And eventually the defense business came back.

Asad Madni explains a problem in electron ballistics to a classmate at the RCA Institutes in 1966 [top]. In 1977, Madni [seated, center] discusses the communications-line analyzer he developed for the U.S. Navy. Asad Madni

Today, electronic angular-rate sensors are in just about every vehicle—land, air, or sea. And Madni’s effort to miniaturize them and reduce their cost blazed the trail.

By 2005, BEI’s portfolio of technologies had made it an attractive target for acquisition. Besides the rate sensors, it had earned acclaim for its development of the unprecedentedly accurate pointing system created for the Hubble Space Telescope. The sensors and control group had expanded into BEI Sensors & Systems Co., of which Madni was CEO and CTO.

“We weren’t looking for a buyer; we were progressing extremely well and looking to still grow. But several people wanted to buy us, and one, Schneider Electric, was relentless. They wouldn’t give up, and we had to present the deal to the board.”

The sale went through in mid-2005 and, after a brief transition period and turning down a leadership position with Schneider Electric, Madni officially retired in 2006.

While Madni says he’s been retired since 2006, he actually retired only from industry, crossing over into a busy life in academia. He has served as an honorary professor at six universities, including the Technical University of Crete, the University of Texas at San Antonio, and the University of Waikato, in New Zealand. In 2011, he joined the faculty of UCLA’s electrical and computer-engineering department as a distinguished scientist and distinguished adjunct professor and considers that his home institution. He is on campus weekly to meet with his advisees, who are working in sensing, signal processing, AI for sensor design, and ultrawideband high-speed instrumentation. Madni has advised 25 graduate students to date.

One of his former UCLA students, Cejo K. Lonappan, now principal systems engineer at SILC Technologies, says Madni cares a lot about the impact of what his advisees are doing, asking them to write an executive summary of every research project that goes beyond the technology to talk about the bigger picture.

“Many times in academic research, it is easy to get lost in details, in minor things that seem impressive to the person doing the research,” Lonappan says. But Madni “cares a lot about the impact of what we are doing beyond the engineering and scientific community—the applications, the new frontiers it opens.”

S.K. Ramesh, a professor and former dean of electrical engineering and computer science at California State University, Northridge, has also seen Madni the advisor in action.

“For him,” Ramesh says, “it’s not just about engineering. It’s about engineering the future, showing how to make a difference in people’s lives. And he’s not discouraged by challenges.”

“We had a group of students who wanted to take a headset used in gaming and use it to create a brain-control interface for wheelchair users,” Ramesh says. “We spoke to a neurologist, and he laughed at us, said you couldn’t do that, to monitor brain waves with a headset and instantaneously transfer that to a motion command. But Prof. Madni looked at it as how do we solve the problem, and even if we can’t solve it, along the way we will learn something by trying.”

Says Yannis Phillis, a professor at the Technical University of Crete: “This man knows a lot about engineering, but he has a wide range of interests. When we met on Crete for the first time, for example, I danced a solo Zeibekiko; it has roots from ancient Greece. He asked me questions left and right about it, why this, why that. He is curious about society, about human behavior, about the environment—and, broadly speaking, the survival of our civilization.”

Madni went into engineering hoping to affect humanity with his work. He is satisfied that, in at least some ways, he has done so.

“The space applications have enhanced the understanding of our universe, and I was fortunate to play a part of that,” he says. “My contributions [to automotive safety] in their own humble way have been responsible for saving millions of lives around the world. And my technologies have played a role in the defense and security of our nation. It’s been the most gratifying career.”