Parents can effortlessly assist their child to walk, but the mechanism behind such physical coordination is still unknown. Studies have suggested that physical coordination is achieved by interacting humans who update their movement or motion plan in response to the partner's behaviour. Here, we tested rigidly coupled pairs in a joint reaching task to observe such changes in the partners' motion plans. However, the joint reaching movements were surprisingly consistent across different trials. A computational model that we developed demonstrated that the two partners had a distinct motion plan, which did not change with time. These results suggest that rigidly coupled pairs accomplish joint reaching movements by relying on a pre-programmed motion plan that is independent of the partner's behaviour.