From cat feeders and cat flaps to robot toys, humans are deploying increasingly autonomous systems to look after their pets. In parallel, industry is developing the next generation of autonomous systems to look after humans in the home -most notably robot arms that might assist with all manner of domestic tasks. How might the animals and humans in these spaces engage with these and with each other? This research focuses upon the role that 'Trust' plays in autonomous animal-centric robotic systems and the ways in which we as researchers can further understand how to design, develop, and evaluate such systems by taking a Responsible Research and Innovation (RRI) approach. Understanding, designing for, and negotiating 'Trust' is complex, particularly in contexts where animals, humans, and intelligent systems (including Robotics, AI, & IoT) come together in a social context, in which bonds are created, friendships develop, and mutual care plays a part in the interaction and developing relationship between people and animals.