Most multimode parallel robots can change operation modes by passing through constraint singularities. This paper deals with a comprehensive kinematic study of a three degrees-of-freedom (DOF) multimode three-PRPiR parallel robot developed at Heriot-watt University. This robot is able to reach several operation modes without crossing any constraint singularity by using lockable Pi and R joints. Here, a Pi joint may act as a 1DOF planar parallelogram if its lockable P (prismatic) joint is locked or a 2DOF RR serial chain if its lockable P joint is released. The operation modes of the robot include a 3T operation mode and four 2T1R operation modes with two different directions of the rotation axis of the moving platform. The inverse kinematics and forward kinematics of the robot in each operation mode are dealt with in detail. The joint space and workspace analysis of the robot allow us to know the regions of the workspace that the robot can reach in each operation mode. It is shown that the robot is able to change assembly mode in one operation mode by passing through another operation mode.