# Screw theory-based mobility analysis and projection-based kinematic modeling of a 3-CRRR parallel manipulator

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## Abstract

Forward kinematics analysis of parallel manipulators requires solving highly complicated nonlinear equations, which deriving a closed-form solution is often a real challenge. Being used in closed loop position control of mechanisms, the forward kinematics solution of parallel manipulators is of great importance. Here, we investigate the mobility, forward kinematics, and inverse kinematics of a previously introduced three-degree-of-freedom spatial parallel manipulator from a new perspective. The manipulator is a 3-CRRR parallel mechanism proposed for object manipulation tasks. The mobility of the mechanism is, first, discussed using screw theory, showing that the robot has only three translational degrees of freedom. Next, the forward kinematics of the robot is analyzed based on a geometric approach. Using this method, which is the main novelty of our article, the spatial representation of the manipulator is transformed to a simpler planar representation by a projection-based interpretation, to reduce the complexity of kinematic equations. Afterward, the position of the end-effector is extracted by some algebraic expressions written based on geometrical properties of the robot. Then, the inverse kinematics of the mechanism is analyzed through the same approach. Finally, the kinematic modeling is verified using numerical and analytical methods. The results show that the obtained kinematic model has high accuracy.

## Keywords

Parallel manipulator Forward kinematics Inverse kinematics Mobility analysis Screw theory## List of symbols

- \(O_{{\rm inertia}}\)
origin of the reference Cartesian coordinate frame

*XYZ*- \(O_i\)
origin of the coordinate frame corresponding to the

*i*-th limb- \(O_p\)
origin of the coordinate frame attached to the end-effector

*Pr*unit vector denoting the screw of a prismatic joint

- Rev\(_i\)
unit vector denoting the screw of the

*i*-th revolute joint*a*distance between two adjacent revolute joints in a limb

*b*horizontal distance between the upper and the lower revolute joints of a limb

*h*vertical distance between the upper and the lower revolute joints of a limb

- \(S_i\)
*i*-th screw- \(S_{{\mathrm{lmss}}_i}\)
*i*-th screw associated with the limb motion screw system- \(S_{{\mathrm{lcss}}_i}\)
*i*-th screw associated with the limb constraint screw system- \(S_{{\mathrm{pcss}}_i}\)
*i*-th screw associated with the platform constraint screw system- \(S_{{\mathrm{pmss}}_i}\)
*i*-th screw associated with the platform motion screw system- \(\alpha _i, \beta _i, \gamma _i\)
direction cosines of the i-th screw associated with the platform constraint screw system in the reference coordinate frame

- \(A_i\)
maximum reach point of the

*i*-th limb on the ground plane- \(B_i\)
instantaneous contact point of the

*i*-th limb to the ground plane- \(C_i\)
upper end of the

*i*-th limb- \(D_i\)
projection of \(C_i\) on the ground plane

- \(L_i\)
length of the

*i*-th limb- \(x_p, y_p, z_p\)
components of the position of the end-effector with respect to the reference coordinate frame

- \(proj_i\)
projection of the

*i*-th limb on the ground plane- \(x_{A_i}, y_{A_i}, z_{A_i}\)
components of the position of \(A_i\) with respect to the reference coordinate frame

- \(x_{B_i}, y_{B_i}, z_{B_i}\)
components of the position of \(B_i\) with respect to the reference coordinate frame

- \(x_{C_i}, y_{C_i}, z_{C_i}\)
components of the position of \(C_i\) with respect to the reference coordinate frame

- \(x_{D_i}, y_{D_i}, z_{D_i}\)
components of the position of \(D_i\) with respect to the reference coordinate frame

*dl*distance between \(D_1\) and \(D_2\) or \(D_3\) along the

*x*-axis of the reference coordinate frame*dw*distance between \(D_1\) and \(D_2\) or \(D_3\) along the

*y*-axis of the reference coordinate frame- \(\theta _i\)
angle between the motion direction of the

*i*-th limb on the ground plane and the positive direction of the*x*-axis of the reference coordinate frame*e*vertical distance between \(C_i\) and the moving platform

- proj\(_p\)
projection of the end-effector position on the ground plane

- \(R_i\)
joint variable corresponding to the

*i*-th limb

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