For what 'rounding constant' exists in Round5?(NIST PQC Round 2 Algorithm)

I am reading a paper Round5.

This public key encryption scheme is based on Ring-LWR but I found it is a little bit different from typical LWR-based PKE scheme.

In the key generation algorithm of Round5 (Algorithm 1, Line 3 in the paper), they compute

$$b= \left< \Bigl \lfloor \frac{p}{q}\left(\left< as \right>_{\Phi_{n+1\ }(x)} +h_1\right) \Bigr\rceil \right>_p$$

where $$\left< \cdot \right>_f$$ means reduction modulo $$f$$.

As far as I know,

$$b= \left< \Bigl \lfloor \frac{p}{q}\left(\left< as \right>_{\Phi_{n+1\ }(x)} \right) \Bigr\rceil \right>_p$$

is the "typical" LWR-based public key, but in the Round5, there is an additional $$h_1$$ term.

What is the role of $$h_1$$?

What Algorithm 1 Line 3 actually has is $$b=\left\langle\left\lfloor\frac pq\left(\langle as\rangle_{\Phi_{n+1}(x)}+h_1\right)\right\rfloor\right\rangle_p$$ (note the use of the floor function rather than the rounding function).
Note that $$h_1=p/2q$$ (see the second complete paragraph on page 5) and $$\lfloor x\rceil=\lfloor x+\frac12\rfloor$$. Adding $$h_1$$ and multiplying by $$\frac pq$$ is equivalent to adding 1/2 and this is just to allow us to implement the rounding function as an integer part function (which is easier in code).