Hi, welcome to AditNow.
There is a three page history of the Tivydale Coal Company in the book "The Last of the Day-Hole Miners" by John Senior ISBN 1-874754-78-0. Got my copy from Moore Books as advertised on this site.
If it's the same Fred Robinson then the section on the Tivydale Coal Company opens with:
"In 1928, Mr Fred Robinson of Cawthorne, commenced mining as the Tivydale Coal Company. Previously employed by the Cannon Hall Estate, he had managed 3 small drifts which supplied coal for domestic use on the estate. mining the Silkstone Seam in an area of Cawthorne village known as Tivydale."
There are still quite a few remains at the various sites the company worked. Most notably the canteen at Norcroft Bridge, and a magazine at Tivydale.
Eventually the company employed over 200 men, so quite a large enterprise. A Mr Charles Robinson is credited with supplying the info for the book.