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While self-driving cars won’t get distracted or drive drunk, that only accounts for a third of wrecks that occur, according to the insurance industry.

Self-driving cars likely have a long, long way to go.

In a blow to hopes for a future free of car crashes with the coming of self-driving cars, a study released Thursday by the Insurance Institute for Highway Safety shows totally driverless cars would have a difficult time achieving such a goal.

The IIHS looked at more than 5,000 police-reported crashes from the National Motor Vehicle Crash Causation Survey, which the insurance industry-funded group said represents vehicle crashes that resulted in one car towed and required emergency medical services.

Combing through the files, the IIHS then sorted the crashes into five categories: sensing and perception; predicting; planning and deciding; execution and performance; and incapacitation errors. Self-driving cars will be able to eliminate sensing and perception errors, or crashes that result in the driver’s distraction, and autonomous technologies won’t be subject to the influence of drugs or alcohol. So, that takes incapacitation errors out. From the sample, that accounts for 34% of crashes. Let’s note the figure is not an insignificant number of crashes automated cars could prevent — 2 million a year in the US alone.

“It’s likely that fully self-driving cars will eventually identify hazards better than people,” said Jessica Cicchino, IIHS vice president for research, “but we found that this alone would not prevent the bulk of crashes.”

As tech companies and automakers paint a picture of a future without a single crash, however, the IIHS found it will be a lot harder for self-driving cars to prevent all of them.

Where the trouble comes, the IIHS says, is errors of planning and deciding, and execution and performance. IIHS says self-driving cars will need to feature programs that can in no way conflict with safety and law protocols. This would mean a future where humans cannot instruct a self-driving car to go faster than a posted speed limit, or perform some sort of illegal maneuver like a U-turn.

“Our analysis shows that it will be crucial for designers to prioritize safety over rider preferences if autonomous vehicles are to live up to their promise to be safer than human drivers,” said the study’s lead author Alexandra Mueller, a research scientist at the IIHS.

Via CNet.com