Assuming you are looking for a near-future tech level solution, why not just have the artificial limbs integrate with your metabolic system for power? If you can feed blood into the prosthetic kind of like a dialysis machine, then you should be able to extract glucose and oxygen directly from the body. A sudden shortage of oxygen & glucose and spike in CO2 in your bloodstream will trigger all the same physical and neurological reactions as using an organic limb. This would allow your human parts to continue to function as normal without concerns about changing your diet or exercise routine. You also go the other way around and put hormone detectors in the limb so that it can respond to spikes in adrenaline or other hormones to scale its power settings accordingly.

Granted... it still means you will run out of breath when you work out too hard, but you could build in a reserve power system if you want that would fail over to battery control when you can't get enough air or something like that instead of your body's anaerobic solution. This way you can work your way up to a healthy level of exorcise, and then push yourself WAY beyond if the need arises until your battery is depleted.

Then all you need to do is handwave away the issues of discomfort, subpar performance, rejection, and infection which you seem to already have planned to do anyway.

At this point you have a more or less perfect and improved limb leaving all the problems purely psychological:

But, body image psychological disorders are no minor side-effect. Some people strongly associate thier appearance or natural ability with thier identity. We see this most pronounced in gender identity disorders where a person is born into a body that "is not theirs", now imagine how much worse that could get if you already had the body you want, and then you were forced to change it. Most people would choose thier augmentations like tattoos, plastic surgery, or cochlear implants. They would love them, they would consider it a part of who they are, but some people would have these changes forced on them, like someone who loses and arm in a car crash. Here they could be disturbing reminders of trauma. Being surrounded by people who are happily augmented could also make being unhappy with your augmentation even worse as there would be an element of shaming in admitting that you are not who you want to be. Various forms of augmentation identity disorders would be pretty rare, but they could become quite severe leading to depression or even suicide.

Will Smith's character in iRobot is a good example of how even a perfect prosthetic can have deep psychological ramifications.

I think most of the potential issues you list are non-issues, mostly because the technology required to augment people to the degree that you're suggesting solves those problems too. Let's take a closer look:

• People not changing their diet get fat(ter).

Simply augment them with an artificial stomach or other augment to speed up metabolism. Alternatively, you could force a change in diet by making people less hungry: after all, feelings like "hunger" are just chemicals. There's no rule that states you can't have an artificial gland to regulate one's sense of hunger.

• Lack of movement can cause heart failure. Increased further by possible fatness.

Besides obesity being engineered away (not difficult to do if you have augmentation tech), why not simply have artificial hearts? Furthermore, engineering a base biological fitness also wouldn't be to difficult. There are plenty of sci-fi solutions to making people who don't exercise fit--the reason that humans loose muscle mass when not exercising is because this has evolutionary advantages (a lot of biology is use-it-or-lose-it). Just as an example, you could implant thin wires inside biological muscles that artificially stimulate them and give them a "workout" while the person is asleep.

• Lack of movement can cause depression, tiredness and related mental problems.

Again. Depression, tiredness, and other mental problems are just chemicals in the brain. Exercise, for example, can release dopamine but we could also simply have an artificial organ which does this on command. Sure, solving mental health issues through copious use of drugs might not seem like an appropriate solution today, but in a future where these augmentations are ubiquitous, safe, and functional, I don't see why it wouldn't be possible to create a safe and effective cure for depression.

Additionally, the chemicals that are released into the brain through exercise can also be released in other human pursuits. Achieving a "feeling of accomplishment" is also possible through intellectual, creative, or social activities.

• No happiness or getting ‘high' from sport (runners high) when you do practice it.

See previous point. This can be solved with an artificial runner's high gland.

• Achievements like climbing a mountain can seem as ‘not your own'.

This is already an issue today, particularly in competitive environments where money directly translates to success. Today there's already great controversy about rich people who pay sherpas to "carry" them to the tip of Everest. The people that don't do this accuse those rich folks of "cheating".

I think that in a society where these augments are ubiquitous, physical achievement simply won't be very important to the people. Instead, skill-based accomplishments will be more highly regarded. For example, anyone can buy 'superlegs' but just because you have them, doesn't mean you're suddenly an expert surfer or rock-climber. Yes, you won't be bound by physical strength limits, but there's still a major skill component to the challenge.

• Body identity disorder or a form thereof could increase dramatically in augmented people.

This is an interesting point. Right now, BID is very rare, and mostly resigned to those who (often sexually) fetishize being an amputee of some sort. In fact, it's so rare that there is very little research on the topic or specific treatment outside of "standard" mental health treatment involving therapy and antidepressants.

Now, I'm not a medical professional, but I can tell you that the human brain has amazing neuroplasticity--the ability to adapt to extreme or unique change. This neuroplasticity is higher in children, but can probably also be induced through drugs. Regardless of how it's achieved, it allows users to adjust their own body image. Because of this, amputees with prostheses very quickly begin to see them as "part of myself". It's not uncommon for these people to begin feeling "phantom sensations", and that's with the relatively static and basic prosthesis we have today. I think the human mind would very quickly accept the augmented leg (or whatever), particularly if it moves and feels just like a real one.

Also, if someone's uncomfortable with their body, they can just get augments to change it.

Nefarious persons commandeer the prosthetics for nefarious ends.

https://www.imdb.com/title/tt0108598/mediaindex?ref_=tt_pv_mi_sm

The creators of these advanced prosthetics retain control over them. They can monitor and record what the prosthetics do and where they go. These nefarious creators can take complete control of the prosthetics, as the unfortunate Wallace learned when he unknowingly tried to commit a robbery in his sleep because his robot trousers had been hijacked.
These sorts of events will be commonplace in your world.

There is a condition that causes (partially) lost and deformed limbs, Phocomelia (1). On the wiki nothing is really said about the long-term general health problems suffered by people with Phocomelia.

A Study checking on various other studies is also very undefinitive (2). Most of the problems named, such as the mental health problems or musculoskeletal problems, are the cause of having no limbs at all, lowered mobility, having to use other limbs more often and generally having to compensate for their disability. Only one mention of obesity is done, which can be a direct result of not being as mobile and having less social/health/work engagements causing less healthy lifestyles.

The body of an amputee will in general think it still needs the same amount of food as before, but you can reduce food intake to lower the chance of obesity (3). There is the question of what powers your mechanical wonders. It might be prudent to have the augmentations work on the body's bloodsupply for day-to-day functions and have power supplies for enhanced augmentation usage. That would mean the augmentations are partially a synthetic organism living in symbiosis with the augmented body. It would also allow the person less checkups or having to rely on external power to keep going.

I wanted to draw a comparison to comatose patients as an analogy for the heart and lungs barely being in use but sustained, but found that the problems of infection, pneumonia and other defects were all the cause of being bedridden, and cardiovascular or lung problems were not on their list. So it looks like there would be relatively little problems with having your limbs replaced with augmentations.

The following links are basically some of the first links found on Google, use grains of salt where necessary:

(1): https://en.m.wikipedia.org/wiki/Phocomelia

(2): https://www.sciencedirect.com/science/article/pii/S1936657417301693

(3): https://www.hprc-online.org/nutrition/unique-nutrition-needs/healthy-eating-amputees#:~:text=Fill%20half%20your%20plate%20with,grains%20are%20your%20best%20bet.

Physical detriments

Depending on how the mechanical extremities are joined to the torso / stumps, various problems may arise.

The least invasive solution is using fitted cups and/or straps to hold everything in place.

Potential problems:

• chafing and/or pressure sores inside the cup (needs quite a bit of practice to figure out how to pad the stump to minimize those issues)
• quite a bit less resistant to torsion or tugging forces compared to a whole flesh-and-blood limb

Surgically attaching the limb to the torso: That one is best done by surgically attaching some kind of metal port to the torso / stump, where the mechanical limb can then be attached (à la Edward Elric of Fullmetal Alchemist). Reason: mechanical limbs should be easily detachable for repair / maintenance, since they don't contain any self-healing like real flesh-and-blood limbs would. Or do they?

Problems:

• risk of infection. To anchor the port properly, it will have to be screwed into one or more bones. However, another part of the port needs to be outside the body to connect to the prosthesis. Wherever the port exits the body will always be a potential locations for bacteria and viruses to invade.
• transmission of force to parts of the body not intended for it. Sure, the port is now anchored to a bone. It sure as fudge won't come off. But when you lift a load, normal flesh-and-blood muscles and sinew stabilize the arm and so take most of the torsion stress off the bone that way. With the screwed-in port, you don't have the muscle/sinew stabilizers holding your port in place - meaning all the torsion stress of the mechanical arm gets transferred directly to the bone. And bones really are not made for that...

Exo-skeleton

I can see only one way of getting around the 'joining' problems I see with all limb prothesises: strap the recipient into a mechanical exo-skeleton that hugs the torso and ends in all-mechanical limbs after the stumps. That gives you following advantages:

• any stress on the mechanical limbs can be buffered by the exoskeleton instead of stump skin and/or bone
• the limbs can be screwed into the exoskeleton without risking any infection
• torsion and pull resistance
• augmenting the mechanical limbs beyond human parameters can be buffered by a similarly augmented exo-skeleton. (It's nice if your new HammerArm Mark 9000 can lift a whooping 300 kg - if you don't have the torso strength to keep your spine in place you'll wrench your back / shoulder / hips / ... when trying to actually lift 300kg)
• can house some of your Infinite Energy Packs (TM)

Problems:

• if exo-skeleton takes over all / most muscle work of the torso, the typical problems of too little excercise will appear (obesity, loss of bone density, loss of stamina, heart problems, etc.)
• the exo-skeleton might also chafe / cause pressure sores

Nervous System

To make the limb replacement palatable, you need to make sure they feel / react at least as well as regular flesh-and-blood limbs. For that, you don't only need to catch the electro-magnetic signals going towards the limb (for muscle movement), but also simulate those leading away from the limb (temperature, touch, pain, proprioception, etc.). This is something we are only just starting to branch into with a single hand, and even there feedback is very rudimentary. It requires several implants to interface with the nerves, because there is more than one of them per limb...

Problems:

• finding and interfacing with all the nerves needed for a 'just like natural' feeling and level of control
• transmitting the signals to the limb: either you have a wire leading out of the body (infection risk), or you have wireless (very vulnerable to strong electro-magnetic fields)

Mental detriments

I think those depend a lot (and I really mean A LOT) on

• how long the time between amputation and fully fitted limb is (the less time spent limbless or with a 'dead' / badly functioning prosthesis the better)
• how much pain the limb causes (pressure sores, healing amputation, fitting)
• how much time the user is forced to spend "as a cripple" per day (do they need to take off the limbs for sleeping, can they wear them for only x hours per day, can they sleep with them)
• how much help the user needs with the limbs (visit a repairman once a week? Have a live-in maid to take them off every evening? Give them a tune-up oneself?)
• how often and seriously the limbs malfunction (daily wear and tear, misfired / misinterpreted nerve signals, etc.)
• how similar to flesh-and-blood limbs they look (do they have an artificial skin? are they metallic? do they 'bleed'? etc.)
• does the user perceive an advantage in artificial over natural limbs?
• is there discrimination against artificial limbs?
• how expensive is limb replacement?

Generally: the less problems there are with the artificial limbs and the longer they can be worn, the greater the acceptance both conscious and unconscious will be. (Disregarding all potential traumatic circumstances leading to the loss of limbs in the first place).

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I know several wheelchair users (age: 20-40 years) who consider their wheelchair their legs (= artificial limb). Yet most of them would swap the wheelchair for being able to walk in a heartbeat because

• there are so many places that are not wheelchair-accessible
• they still have their legs, they just don't work
• they have several other illnesses / injuries related to the cause that placed them in the wheelchair, and they think if they could walk it would be like the cause never happened ( = the other illnesses / injuries would miraculously vanish as well)
• they don't have the upper body strength to be as independent as they were before
• they get stared at by people and get comments like "you're just lazy - if you really wanted to walk, you could".
• getting a good wheelchair is a real pain (money, finding a fitting one, getting all the necessary modifications done, learning the ins and outs of that wheelchair, etc.)

Exception: an 83 year old grandma who is in an electric wheelchair. She loves her new legs because they gave her back most of her life - she can go shopping again, can visit church and the library and her friends because she doesn't need to fear falling down and cracking yet another hip / thigh bone.