The majority of the words that you have written in bold are a type of adverbial (a type of adverb) which is called an disjunct. (whether they are all disjuncts could be argued.).
.Disjuncts allow the speaker/writer to influence the hearer or reader.. They represent the speaker's attitude about what he is going to say.
Here are some examples where the adjunct is written in bold:
Naturally, you are going to go you. = the speaker is certain you will go.
Obviously,I agree with the president. = Its obvious that the speaker will agree.
Of course, she is late. = I think it is to be expected.
Regrettably, he didn't attend the meeting.= I think its sad or too bad that he didn't come.
Surely, you are going to go to school today. = a strong sense of persuasion.
Frankly, I couldn't care less about comma rules. = the speaker really doesn't care.
In each case we use a comma to separate the emotion or attitude of the speaker from the rest of the sentence. Some authorities believe that a comma should be used if the disjunct doesn't flow with the rest of the sentence and in most cases disjuncts are separated by a comma.
There are no hard and fast rules about comma use. and convention plays a role in determining some rules. You learn from experience and even English teachers can get it wrong.
Source Grammar 33 manual U of Saskatchewan, Longman's Dictionary, Guide to Grammar Writing by Charles Darling.
You do need a comma here. There is a systematic answer for this.
Whenever a sentence begins with an adverbial phrase, use a comma.
(source: https://owl.purdue.edu/owl/general_writing/punctuation/commas/extended_rules_for_commas.html)
An adverbial phrase explains things like: Manner, timing, reason, purpose among others. In this case, you have an infinitive of purpose as an adverbial phrase.
In your case:
To further our understanding of this phenomenon, (Adverbial of purpose)
we conducted yet another experiment. (main clause)
Other similar examples:
To help him start his car, we pushed it until it started rolling.
To better understand what they were doing, we started spying on them.
Usage:
These phrases use infinitives of purposes as adverbial phrases. It is much more common to place them at the end of a sentence like so:
We pushed his car until it started rolling to help him start it.
One might be tempted to think that it makes no difference, but it does. The reason one would put this at the beginning is to emphasize the purpose. Consider this context.
To help him start his car, we pushed it until it started rolling. To help him repair it, we called a mechanic. To convince him to leave, we offered him some of the stale food we had.
Also consider this phrasing in which the first purpose is logical, whereas the second oen is somewhat surprising:
We started spying on them to better understand what they were doing. To join them, we confessed that had been spying on them.
Best Answer
Your examples are of clauses that begin with adjuncts†.
In most cases, this is a matter of style. A writer with a heavy punctuation style will almost always insert these commas, while a writer with a light punctuation style is less likely to. That is to say, these commas are generally optional. As you can see, I favor a heavier style and nearly always insert these commas. In my opinion, most sentences are clearer and easier to read when you do so.
There are a few cases where a comma is (more or less) required, because without the comma, you'll most likely misread the sentence. For example:
If I remove the comma, we get a much scarier sentence:
In a sentence like this, even a writer with a light punctuation style would insert the comma, because leaving it out changes the meaning completely!
But if I take some examples from my second paragraph above and remove the commas, the sentences are still acceptable (and some would say preferable):
I've removed the comma after in most cases, and that's fine. So is this:
So it really is a matter of style--except in a few sentences, where the comma can change the meaning altogether!
I should add that this is the subject of a prescriptive rule: some students are taught to always insert these commas. It's a good rule, actually--you'll rarely go wrong if you follow it. However, it should be noted that this practice is far from universal in published writing, so it's not accurate as a descriptive rule.
†The linked Wikipedia article, which is incoherent, defines phrase as two or more words, while in modern linguistics, a phrase is one or more words; it defines the term adverbial in a quasi-functional, quasi-semantic sense (quasi-functional in that it says adverbial phrases "do the work of an adverb", and quasi-semantic in that it limits the term to adjuncts of time, place, and manner, which are not syntactically distinct or well-defined categories, and of course it excludes many types of adjuncts which are syntactically alike). Confusingly, the top half of the article uses the term adverb phrase in the same way; even if you admit adverbial as a functional category, adverb phrase (on the pattern of noun phrase and verb phrase) should probably refer to a phrase with an adverb as a head (as in very quickly). And the bottom half (under the heading "Adverbial Phrase Distribution") and the sole citation are completely unrelated to the rest of the article, since these talk about actual adverb phrases, but of course the article makes no mention of the fact that it describes two completely different things with the same terminology! This is all very confusing, and I recommend you ignore it completely. I've chosen to talk instead about clauses that begin with adjuncts, which is a more accurate description of your examples.