Hi and thank you for your question!  Your caterpillar is not a caterpillar, it is a bristly rose slug (sawfly). There are 3 types of these creatures.  The other two only have one or two generations in a season but this one can have up to 6 generations each season. The best way to control these is by hand picking or using a stream of water to dislodge them from the leaves and plant. They are unable to crawl back up the plant to continue feeding.  Horticultural oils, insecticidal soap and neem are other options.  It is necessary to spray the upper and lower part of the leaf because these controls need direct contact in order to kill.  Bt will not work on this insect.  Also, if you decide to use a chemical, try to stay away from spraying the actual rose flower since pollinators can also be affected by chemical control.  Keep examining your plants and try to catch these critters when they are small.
 Hope this helps.  If you have more questions please contact us again.