A passive end effector of a surgical system includes a base, a first mechanism, and a second mechanism. The base attaches to an end effector coupler of a robot arm positioned by a surgical robot. The first mechanism extends between a rotatable connection to the base and a rotatable connection to a tool attachment mechanism. The second mechanism extends between a rotatable connection to the base and a rotatable connection to the tool attachment mechanism. The first and second mechanisms pivot about the rotatable connections to constrain movement of the tool attachment mechanism to a range of movement within a working plane. The tool attachment mechanism is configured to connect to a surgical saw including a saw blade for cutting.