Hard Science Answer With Current Technology
As a researcher who's familiar with some of this work (my last project was in brain computer interfaces and a colleague coded a system that used computer vision to evaluate human emotions), I am going to go ahead and give you a hard science answer based on current research even though you didn't ask for one.
Can robots sense human emotion? Yes, they already can (to a limited degree)
First, its helpful to realize that humans actively broadcast (communicate) many internal emotional states in ways that can be seen and heard - and recognized by computers. Many other emotional states can also be partially discerned by other physiological measures which are less obvious, but can be recorded by non-invasive means (IE, no need to install a neuro-jack into someone's skull or implant chips).
There are a number of advances in understanding how humans recognize emotional states, such as a test called Reading the Mind Through the Eyes, which shows that humans can detect a wide variety of emotions (36 distinct emotions in one analysis, ranging from "playful", "skeptical", and "hostile") based only on looking at pictures of the eyes and near-eye areas. In computer vision there are projects to detect and convey human emotions in computer animations using only facial expressions, and this similarly has been shown to successfully recognize many emotions - and to be able to re-broadcast them in a way that humans recognize.
Computers can already recognize many human emotional states through simple, cheap web cameras and live computation. Accuracy is not presently perfect ( either in humans or in computers), but if we already have this now than naturally advances can only improve the range of detected emotions and the accuracy of detection. Currently most technology uses only a single freeze-frame (or just a few frames) to make this detection.
Without even looking at the meaning of words, there are computerized methods of real-time detection of human emotion using only audio voice recordings. Just one example of this is EmoVoice (check out their project for more information) - and this included emotionally appropriate animated facial response to human speech...in 2005.
Again, more work can only advance the accuracy and range of emotions that can be detected.
Beyond Human Limitations - Brain Activity, Heart Rate, Skin Conductance, etc
The above methods are already currently useful and have full potential for going beyond human levels of accuracy in some limited-context scenarios - but what if we used information that normally was not available to humans?
Using wearable brain computer interfaces, such as EEG headsets (which detect electrical activity from the brain on the surface of the skin), we can already use a computer to identify a variety of emotional-cognitive states, such as attentiveness (is a driver paying attention?), sleepiness, positive/negative emotional reactions, etc. Having worked on areas related to this I can assure you that this is, in fact, hard to do reliably with current technology, yet there are many methods that have been shown to actually work! Emotional recognition is hard, but many advances are being made.
In one talk I attended just this year, Conceptual Priming for In-game BCI Training, an experiment showed that using only EEG a computer can even detect mentally visualized images - such as distinguishing between whether a person is thinking about a flashlight or a gun or neither, with about 60%+ accuracy based only on a single implicit training presentation lasting a few seconds. This is not an emotion, but if a wider range of images can be recognized then this could also be used to infer emotional states - or just obviate the need to detect a lie at all.
Theoretically you could ask someone "where did you hide the body?" and they would implicitly think of where they hid the body - and you could capture that snapshot directly from their brain. We are a long way off this being practical in the next 10+ years, but given 100-200+ years then this is extremely likely to be possible to do based on our current science - especially if methods similar to fMRI can be expanded to better detect human visual activation in the brain.
Other methods of emotion detection include things like examining skin conductance and heart rate. Alone these methods have proven more limited in the ability to detect emotion - but it has been shown to have connections to detecting stress levels, excitement, emotional valence, and emotional responsiveness. This information is emotional in nature, and some current research looks at the viability of combining these measures to detect a wider range of emotional states with greater accuracy - but the results aren't really "in" on this yet.
fMRI has also shown promise in detecting and evaluating emotions, but the technology is less practical currently as the machines are large and expensive and require a great deal of experimental control. In the future if this technology can be shrunk and made more resilient and easier to use, it may be a new field of computerized emotional state detection.
Put It All Together
I am not presently aware of research that has successfully combined multiple modes of emotion detection together - such as face expression, audio, brain activity, skin conductance, heart rate, etc. It would extremely likely be able to greatly improve the ability to detect a far larger range of human emotional states and responses to stimuli, and given enough computational power could do this all live within a single robot in the not-distant future.
Finally, I will leave you with one more set of keywords. There is a presently emerging field specifically dedicated to using computers to detect, respond to, and even "have" emotions: affective computing. To quote from the MIT Media Lab's Affective Computing Group:
Affective Computing is computing that relates to, arises from, or
deliberately influences emotion or other affective phenomena (Picard,
MIT Press 1997).
Emotion is fundamental to human experience, influencing cognition,
perception, and everyday tasks such as learning, communication, and
even rational decision-making. However, technologists have largely
ignored emotion and created an often frustrating experience for
people, in part because affect has been misunderstood and hard to
measure. Our research develops new technologies and theories that
advance basic understanding of affect and its role in human
experience. We aim to restore a proper balance between emotion and
cognition in the design of technologies for addressing human needs.
Our research has contributed to: (1) Designing new ways for people to
communicate affective-cognitive states, especially through creation of
novel wearable sensors and new machine learning algorithms that
jointly analyze multimodal channels of information; (2) Creating new
techniques to assess frustration, stress, and mood indirectly, through
natural interaction and conversation; (3) Showing how computers can be
more emotionally intelligent, especially responding to a person's
frustration in a way that reduces negative feelings; (4) Inventing
personal technologies for improving self-awareness of affective state
and its selective communication to others; (5) Increasing
understanding of how affect influences personal health; and (6)
Pioneering studies examining ethical issues in affective computing.
Affective Computing research combines engineering and computer science
with psychology, cognitive science, neuroscience, sociology,
education, psychophysiology, value-centered design, ethics, and more.
We bring together individuals with a diversity of technical, artistic,
and human abilities in a collaborative spirit to push the boundaries
of what can be achieved to improve human affective experience with
TLDR; Current technology already allows a wide range of human emotional detection by computers/robots using many available measures, from the human-like (facial expressions, sound of voice) to the beyond-human (brain activity, etc.). Additional research can only extend the range of emotional detection, improve accuracy, and expand our understanding of human emotions. There is no question that robots can possess various degrees of this ability in the future, because they already can right now.