The state of Rhode Island did not send any delegates to the Constitutional Convention in Philadelphia in 1787 because they were afraid of exactly what ended up happening. That is, they were afraid that the convention would create a new set of laws that would give too much power to the national government.
Many state governments liked the status quo under the Articles of Confederation. They did not want the national government to become more powerful. They suspected that the elites who were gathering in Philadelphia were likely to create such a government. They thought that by refusing to participate they could prevent this from happening since the Articles could only be amended with unanimous approval of the states. As it was, the Convention circumvented that law and did indeed create the new Constitution that gave much more power to the national government.
Not only was Rhode Island the only colony/state to decline to send a representative to the Constitutional Convention of 1787, it was also the last of the newly-established states to sign and ratify the document that emerged out of those proceedings. As the document linked below illustrates, Rhode Island stood out among the already-fractious political entities for its staunch adherence to the Articles of Confederation, which strictly limited the powers of a central government, one of the most contentious issues debated at the convention and in political gatherings preceding it.
The direction in which the independent states were heading was strongly suggested by the known positions of many of the most prominent members of the pro-independence movement. Rhode Island's leading officials understood this, and they remained opposed to that direction--in effect, the movement towards a coalition of states bound by a strong central government. Any movement towards greater centralized control of the emerging country was bound to elicit demonstrations of annoyance from the citizens of that particular state. Rhode Islanders relished their reputation for independent thought and governance--a sentiment that had earned it the moniker "Rogues Island" more than a century before. The decision to boycott the Constitutional Convention, therefore, was entirely consistent with the colony/state's history, including its early opposition to the British Crown. Rhode Islanders simply rejected the notion of governance from outside their very limited borders, and did not want to give their imprimatur to the constitution that would emerge.