Abstract
Forward kinematics analysis of parallel manipulators requires solving highly complicated nonlinear equations, which deriving a closedform 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 threedegreeoffreedom spatial parallel manipulator from a new perspective. The manipulator is a 3CRRR 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 projectionbased interpretation, to reduce the complexity of kinematic equations. Afterward, the position of the endeffector 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.
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Abbreviations
 \(O_{{\rm inertia}}\) :

origin of the reference Cartesian coordinate frame XYZ
 \(O_i\) :

origin of the coordinate frame corresponding to the ith limb
 \(O_p\) :

origin of the coordinate frame attached to the endeffector
 Pr :

unit vector denoting the screw of a prismatic joint
 Rev\(_i\) :

unit vector denoting the screw of the ith 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
 \(S_i\) :

ith screw
 \(S_{{\mathrm{lmss}}_i}\) :

ith screw associated with the limb motion screw system
 \(S_{{\mathrm{lcss}}_i}\) :

ith screw associated with the limb constraint screw system
 \(S_{{\mathrm{pcss}}_i}\) :

ith screw associated with the platform constraint screw system
 \(S_{{\mathrm{pmss}}_i}\) :

ith screw associated with the platform motion screw system
 \(\alpha _i, \beta _i, \gamma _i\) :

direction cosines of the ith screw associated with the platform constraint screw system in the reference coordinate frame
 \(A_i\) :

maximum reach point of the ith limb on the ground plane
 \(B_i\) :

instantaneous contact point of the ith limb to the ground plane
 \(C_i\) :

upper end of the ith limb
 \(D_i\) :

projection of \(C_i\) on the ground plane
 \(L_i\) :

length of the ith limb
 \(x_p, y_p, z_p\) :

components of the position of the endeffector with respect to the reference coordinate frame
 \(proj_i\) :

projection of the ith 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 xaxis of the reference coordinate frame
 dw :

distance between \(D_1\) and \(D_2\) or \(D_3\) along the yaxis of the reference coordinate frame
 \(\theta _i\) :

angle between the motion direction of the ith limb on the ground plane and the positive direction of the xaxis of the reference coordinate frame
 e :

vertical distance between \(C_i\) and the moving platform
 proj\(_p\) :

projection of the endeffector position on the ground plane
 \(R_i\) :

joint variable corresponding to the ith limb
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Nazari, A.A., Hasani, A. & Beedel, M. Screw theorybased mobility analysis and projectionbased kinematic modeling of a 3CRRR parallel manipulator. J Braz. Soc. Mech. Sci. Eng. 40, 357 (2018). https://doi.org/10.1007/s4043001812773
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DOI: https://doi.org/10.1007/s4043001812773