| Comparing with the traditional 6-DOF (Degree of Freedom) parallel robot, a limited-DOF parallel robot has many advantages such as having less driving limbs, being simple in control, easy in manufacture, low in cost and so on. Recently, limited-DOF parallel manipulator has become a focus among the international mechanics and robotics fields. The main study of the limited-DOF parallel manipulator character includes the kinematics, singularity configurations, workspaces and so on.Kinematics is an important part of mechanism analysis, and Jacobian matrix plays an important role in the kinematics analysis of parallel manipulators. The traditional Jacobian matrix of the limited-DOF parallel manipulator is not a square matrix, that is, the corresponding relationship between manipulator's input and output is not satisfied. The essential reason for this phenomenon is that the elements of the moving platform output velocity are not independent. In this paper, the translational Jacobian matrix was used to denote the relationship between the linear velocity of the moving platform and the independent output velocity vector of the manipulator, while the rotational Jacobian matrix was used to denote the relationship between the angular velocity of the moving platform and the independent output velocity vector of the manipulator. Through the translational and rotational Jacobian matrices, an n×n square Jacobian matrix of n DOF (n<6) parallel manipulator was obtained, and the manipulator velocity was analyzed. On the basis of the translational and rotational Jacobian matrices, the translational and rotational Hessian matrices were obtained, which denoted the relationship between the linear and angular acceleration of the moving platform and the independent output acceleration vector of the manipulator, respectively. The acceleration of the limited-DOF parallel manipulator was further analyzed.As a whole, a limited-DOF parallel robot has more complex kinematic characters than a 6-DOF parallel robot. Many special characters may show up because of the moving platform constraint, which leads to the lack of the limited-DOF parallel prototype and dissatisfaction of the practical wide requirement. Some 3-DOF and 4-DOF parallel manipulators were synthesized in this paper, including the parallel manipulator with a constrained limb, the limited-DOF parallel manipulator with less limbs and the parallel manipulator with a non-linear limb and so on. On the basis of the translational and rotational Jacobian matrices, the kinematics of these parallel manipulators was analyzed in detail.Singularity configuration of the parallel manipulator is an important issue in the design, application and analysis of parallel robots. Jacobian algebra method is the most general and common method to study the parallel manipulator singularity configuration. According to the translational and rotational Jacobian matrices and the n×n square Jacobian matrix received from these two matrices, the limited-DOF parallel manipulators, especially the 3-DOF and 4-DOF parallel manipulators with different movement types were analyzed in this paper. The involved manipulators were the three-dimension translation parallel manipulator, the three-dimension rotation parallel manipulator, the 3-DOF parallel manipulator with mixed movement, the three-dimension translation and one-dimension rotation 4-DOF parallel manipulator, the three-rotation and one-dimension translation 4-DOF parallel manipulator, and the 4-DOF parallel manipulator with coupling translation and rotation.Accurate and convenient workspace analysis is greatly helpful for the evaluation of a new manipulator. The general limited-DOF parallel manipulator workspace was solved; 3-DOF and 4-DOF parallel manipulators were used as examples to explain the detailed process. Parameters were defined to describe the workspace, the influence of joint distribution to limited-DOF parallel manipulators, especially to the 3-DOF parallel manipulator workspace was discussed.
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