Abstract
Recursive matrix relations for kinematics and dynamics analysis of a three-prismatic-revolute-cylindrical (3-PRC) parallel kinematic machine (PKM) are performed in this paper. Knowing the translational motion of the platform, we develop first the inverse kinematical problem and determine the positions, velocities and accelerations of the robot’s elements. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work and the results in the framework of the Lagrange equations with their multipliers can be verified. Finally, compact matrix equations and graphs of simulation for input force and power of each of three actuators are obtained. The investigation of the dynamics of this parallel mechanism is made mainly to solve successfully the control of the motion of such robotic system.
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Abbreviations
- a k,k−1,b k,k−1,c k,k−1 :
-
orthogonal transformation matrices
- θ 1,θ 2 :
-
two constant orthogonal matrices
- \(\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}\) :
-
three right-handed orthogonal unit vectors
- \(l = 2r\sqrt{3}\) :
-
length of the side of moving platform
- l 2 :
-
length of the limb of each leg
- θ :
-
angle of inclination of three sliders
- \(\lambda^{A}_{10},\lambda^{B}_{10},\lambda^{C}_{10}\) :
-
displacements of three prismatic actuators
- ϕ k,k−1 :
-
relative rotation angle of T k rigid body
- \(\vec{w}_{k,k-1}\) :
-
relative angular velocity of T k
- \(\vec{w}_{k0}\) :
-
absolute angular velocity of T k
- \(\tilde{w}_{k,k-1}\) :
-
skew-symmetric matrix associated to the angular velocity \(\vec{w}_{k,k-1}\)
- \(\vec{\varepsilon}_{k,k-1}\) :
-
relative angular acceleration of T k
- \(\vec{\varepsilon}_{k0}\) :
-
absolute angular acceleration of T k
- \(\tilde{\varepsilon} _{k,k - 1}\) :
-
skew-symmetric matrix associated to the angular acceleration \(\vec{\varepsilon} _{k,k - 1}\)
- \(\vec{r}_{k,k - 1}^{A}\) :
-
relative position vector of the centre A k of joint
- \(\vec{v}_{k,k - 1}^{A}\) :
-
relative velocity of the centre A k
- \(\vec{\gamma} _{k,k - 1}^{A}\) :
-
relative acceleration of the centre A k
- \(\vec{r}_{k}^{C}\) :
-
position vector of the mass centre of T k rigid body
- \(m_{k},\hat{J}_{k}\) :
-
mass and symmetric matrix of tensor of inertia of T k about the link-frame x k y k z k
- \(f_{10}^{A},f_{10}^{B},f_{10}^{C}\) :
-
input forces of prismatic actuators
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Li, Y., Staicu, S. Inverse dynamics of a 3-PRC parallel kinematic machine. Nonlinear Dyn 67, 1031–1041 (2012). https://doi.org/10.1007/s11071-011-0045-z
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DOI: https://doi.org/10.1007/s11071-011-0045-z