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I'm curious if curiosity has started to drive backwards, at least occasionally, as part of an effort to reduce the rate of damage to the wheels.

This has come up after writing this answer to the question Does the Curiosity rover really have a chance of driving to the top of Mt. Sharp?

Emily Lakdawalla's 2014 blogpost Curiosity wheel damage: The problem and solutions says in part (it's very long and very thoughtful post):

4. How can they prolong the life of the wheels?

They can't go to Mars and switch out the wheels. Fortunately, they have identified several ways to reduce the rate at which the wheels accumulate damage.

Driving more judiciously.

Rover drivers are avoiding every pointy rock they can steer around. This only helps in the first 10 or 20 meters of a drive, where they can see smaller potentially hazardous rocks. On hazardous terrain, performing shorter drives allows them to avoid many potentially wheel-damaging rocks.

Driving backwards.

When they turn the rover around, the rover's middle and front wheels are dragged behind their supporting arms rather than being shoved forward. And the angle of the bogie arm that holds the rover's rear wheel is such that it does not experience the same kind of downward forces that the front and middle wheels do when the rover is driving forwards. Heverly showed a video, taken in the JPL Mars Yard, of a test wheel being driven over the sharpened metal spike with the rover driving backwards, and the wheel was only dented, not punctured.

There is a cost to driving backwards. At the end of each drive, they have to face forwards in order to acquire images of the path ahead for planning. They can't take those images while facing backwards, because the RTG and antennas on the rover's rear deck obscure the view from the cameras on the mast. So to drive backwards, they have to turn in place, then drive, then turn in place again. Each turn in place puts about 6 meters on the rover's wheels, or 12 meters for the drive. For short drives (which is what they do in bad terrain), this can swiftly add up. The drivers have to weigh the cost of increasing drive distance against the potential savings to the wheels of driving backwards. Driving backwards therefore is most valuable on long "blind" drives where the drivers aren't steering around smaller rocks.

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