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We have come a long way since the first space missions, and now we have state of the art avionics systems on board our spacecrafts. The more deep we go into space, the more autonomy our robotic spacecrafts need.

By autonomy, I mean to elaborate my question on two main conditions,

1) Autonomous Control over unfamiliar conditions (Self Repair, GNC without human interference)

2) Automatic scientific inference and hypothesis and decisions based on the inference

Are these problems already solved ?

If not, then

My question then comes to the following -:

What are some of the unsolved problems in robotics and avionics that need to be solved for such kinds of missions ?

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As far as autonomous control is concerned, much of this has already been developed. It had to be given the long time delays between Earth and even "nearby" destinations such as Mars. The Mars rovers, for example, are able to navigate to nearby locations on their own (Curiosity article). This is extremely limited in comparison to the autonomous vehicles entered into the DARPA grand challenges. The biggest road blocks in bringing that tech into space is the amount of computation needed and senors required. It can be done, but the DARPA challenge vehicles had a lot more power and mass available. For more information on the research side of this, look for SLAM (simultaneous localization and mapping).

This gives you a system that can move from point A to point B. That's great, but your question gets at the more difficult problem of how to decide where you want to go in the first place. This is starting to get into the domain of Artificial Intelligence.

There are many examples of artificial intelligence that work very well in specific domains. For example, NASA's Deep Space 1 mission demonstrated Remote Agent. Remote Agent was a software system that could do planning (for example, what sequences of actions are required to perform a task) as well as diagnose faults within the spacecraft (such as, why didn't I get a response after transmitting my data?). For this, Remote Agent has a model of the underlying hardware, and maintains "knowledge" of it such as sensor readings and actuation commands. It works well, however the system is limited. The underlying mathematical model is a POMDP, which can be solved for an optimal action given the state of the system (even given that some of this state is not know). But solving a POMDP becomes intractable for large / complex systems. So it does not scale well.

Making inferences from scientific measurements that direct future actions falls into the intractable domain. In fact, I would say that it is getting very close to the goal of build a general purpose AI.

So, for the first part of your question: these problems have demonstrate-able solutions. It is not easy, but with enough resources and effort robots that can navigate autonomously are possible.

For the the second part: unless the domain can be greatly restricted, these problems are generally intractable. However, approximate solutions that are probably "good enough" may be a viable.

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