Kumudakriya is a janya of the 51st Melakartha Kamavardhani (or what is commonly referred to as Pantuvarali)
S R1 G3 M2 D1 S - S N3 D1 M2 G3 R1 S
The absence of the Pa (and Ni in arohana) gives it a unique flavour! There are only four compositions in this raga according to Lakshman's list. Dikshitar's arddhanArIshvaraM is by far the most popular. I think the raga is adopted from Hindustani music.
arddhanArIshvaraM ArAdhayAmi satatam
Raga: kumudakriya / Tala: rUpakam
P: arddhanArIshvaraM ArAdhayAmi satatam atri bhRgu vasiSThAdi munibRnda vanditam shrI
A: arddhayAma alaHNkAra vishESaprabhAvam arddhanArIshvarI priya karam abhayakaram shivam
C: nAgEndra maNi bhUSitam nandituragArOhitam shrI guruguhapUjitam kumudakriyA rAganutam AgamAdi sannutam anantavEda ghOSitam amarEshAdisEvitam AraktavarNashObhitam
I offer my prayers to Lord Arddhanarishvara incessantly. He is surrounded by a host of eminent sages, such as Atri, Bhrigu, and Vasishta.
He has the glory of being exceptionally decorated during the time of Arddha Jama worship. He delights Arddhanarishvari. He is Shiva, offering refuge to his devotees.
He is adorned with the precious gem Nagendramani, and is mounted upon the sacred bull. He is worshipped by the prosperous Guruguha, and is extolled by the melody of Kumudakriya. He is praised by all the Agamas, and is proclaimed by all the Vedas. He is venerated by Indra and others, and his form shines with the red hue.