conditional merge or left join two dataframes in R - r

Question

I am trying to add additional data from a reference table onto my primary dataframe. I see similar questions have been asked about this however cant find anything for my specific case.
An example of my data frame is set up like this
df <- data.frame("participant" = rep(1:3,9), "time" = rep(1:9, each = 3))
lookup <- data.frame("start.time" = c(1,5,8), "end.time" = c(3,6,10), "var1" = c("A","B","A"),
"var2" = c(8,12,3), "var3"= c("fast","fast","slow"))
print(df)
participant time
1 1 1
2 2 1
3 3 1
4 1 2
5 2 2
6 3 2
7 1 3
8 2 3
9 3 3
10 1 4
11 2 4
12 3 4
13 1 5
14 2 5
15 3 5
16 1 6
17 2 6
18 3 6
19 1 7
20 2 7
21 3 7
22 1 8
23 2 8
24 3 8
25 1 9
26 2 9
27 3 9
> print(lookup)
start.time end.time var1 var2 var3
1 1 3 A 8 fast
2 5 6 B 12 fast
3 8 10 A 3 slow
What I want to do is merge or join these two dataframes in a way which also includes the times in between both the start and end time of the look up data frame. So the columns var1, var2 and var3 are added onto the df at each instance where the time lies between the start time and end time.
for example, in the above case - the look up value in the first row has a start time of 1, an end time of 3, so for times 1, 2 and 3 for each participant, the first row data should be added.
the output should look something like this.
print(output)
participant time var1 var2 var3
1 1 1 A 8 fast
2 2 1 A 8 fast
3 3 1 A 8 fast
4 1 2 A 8 fast
5 2 2 A 8 fast
6 3 2 A 8 fast
7 1 3 A 8 fast
8 2 3 A 8 fast
9 3 3 A 8 fast
10 1 4 <NA> NA <NA>
11 2 4 <NA> NA <NA>
12 3 4 <NA> NA <NA>
13 1 5 B 12 fast
14 2 5 B 12 fast
15 3 5 B 12 fast
16 1 6 B 12 fast
17 2 6 B 12 fast
18 3 6 B 12 fast
19 1 7 <NA> NA <NA>
20 2 7 <NA> NA <NA>
21 3 7 <NA> NA <NA>
22 1 8 A 3 slow
23 2 8 A 3 slow
24 3 8 A 3 slow
25 1 9 A 3 slow
26 2 9 A 3 slow
27 3 9 A 3 slow
I realise that column names don't match and they should for merging data sets.

Answer 1

One option would be to use the sqldf package, and phrase your problem as a SQL left join:
sql <- "SELECT t1.participant, t1.time, t2.var1, t2.var2, t2.var3
FROM df t1
LEFT JOIN lookup t2
ON t1.time BETWEEN t2.\"start.time\" AND t2.\"end.time\""
output <- sqldf(sql)

Answer 2

A dplyr solution:
output <- df %>%
# Create an id for the join
mutate(merge_id=1) %>%
# Use full join to create all the combinations between the two datasets
full_join(lookup %>% mutate(merge_id=1), by="merge_id") %>%
# Keep only the rows that we want
filter(time >= start.time, time <= end.time) %>%
# Select the relevant variables
select(participant,time,var1:var3) %>%
# Right join with initial dataset to get the missing rows
right_join(df, by = c("participant","time")) %>%
# Sort to match the formatting asked by OP
arrange(time, participant)
This produces the output asked by OP, but it will only work for data of reasonable size, as the full join produces a data frame with number of rows equal to the product of the number of rows of both initial datasets.

Answer 3

Using tidyverse and creating an auxiliary table:
df <- data.frame("participant" = rep(1:3,9), "time" = rep(1:9, each = 3))
lookup <- data.frame("start.time" = c(1,5,8), "end.time" = c(3,6,10), "var1" = c("A","B","A"),
"var2" = c(8,12,3), "var3"= c("fast","fast","slow"))
lookup_extended <- lookup %>%
mutate(time = map2(start.time, end.time, ~ c(.x:.y))) %>%
unnest(time) %>%
select(-start.time, -end.time)
df2 <- df %>%
left_join(lookup_extended, by = "time")