Kinematic Modeling, Analysis, and Real-Time Implementation of a Planar Five-Bar Pantograph Robot

This paper presents the kinematic analysis and real-time implementation of a planar five-bar robot, also known as a pantograph mechanism. Although the mechanism is made from only a few links and joints, its motion is not simple: the same end-effector position can often be reached in different ways, and some configurations can cause loss of control or poor motion quality. For this reason, a usable control system must do more than compute whether a point is reachable. It must also choose the correct motion branch, avoid singular configurations, and keep the robot inside a safe operating region. The paper first develops the position and velocity models of the mechanism in a step-by-step way. It then uses these models to identify the different motion branches, classify the singular configurations, and evaluate the quality of the reachable workspace. Based on this analysis, a verified operating region is defined for safe and reliable motion. The paper also shows how the continuous kinematic model can be implemented in a sampled control loop using only motor encoder measurements. The implementation includes homing, branch selection, reference limiting with virtual walls, comparison of several inverse-kinematics methods, and an estimator for the end-effector position. Numerical results confirm that the proposed implementation keeps the mechanism on the desired branch, avoids unsafe regions, and estimates the end-effector position with accuracy limited mainly by encoder resolution. The paper also extends the main results to the more general asymmetric five-bar mechanism.