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Matrix modeling of inverse dynamics of spatial and planar parallel robots

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Abstract

Recursive matrix relations for kinematics and dynamics analysis of two known parallel mechanisms: the spatial 3-PRS and the planar 3-RRR are established in this paper. Knowing the 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 can be verified in the framework of the Lagrange equations with their multipliers. Finally, compact matrix equations and graphs of simulation for power requirement comparison of each of three actuators in two different actuation schemes are obtained. For the same evolution of the moving platform, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems. The study of the dynamics of the parallel mechanisms is done mainly to solve successfully the control of the motion of such robotic systems.

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Abbreviations

a k,k−1,b k,k−1,c k,k−1 :

orthogonal rotation matrices

R :

general rotation matrix of moving platform

θ 1,θ 2,θ 3 :

three constant orthogonal matrices

\(\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\) :

three right-handed orthogonal unit vectors

l 1,l 2 :

length of two links of each leg

θ :

angle of inclination of three sliders

φ k,k−1 :

relative rotation angle of T k rigid body

\(\vec{\omega}_{k, k - 1}\) :

relative angular velocity of T k

\(\vec{\omega}_{k0}\) :

absolute angular velocity of T k

\(\tilde{\omega}_{k, k - 1}\) :

skew-symmetric matrix associated to the angular velocity \(\vec{\omega }_{k, k - 1}\)

\(\vec{\varepsilon}_{k, k - 1}\) :

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 center A k of joint

\(\vec{v}_{k, k - 1}^{A}\) :

relative velocity of the center A k

\(\vec{\gamma}_{k, k - 1}^{A}\) :

relative acceleration of the center A k

\(\vec{r}_{k}^{C}\) :

position vector of the mass center 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

\(m_{p}, \hat{J}_{p}\) :

mass and central tensor of inertia of moving platform

\(f_{10}^{A}, f_{10}^{B}, f_{10}^{C}, m_{10}^{A}, m_{10}^{B},m_{10}^{C}\) :

forces and torques of fixed actuators

\(m_{21}^{A}, m_{21}^{B}, m_{21}^{C}, m_{32}^{A}, m_{32}^{B},m_{32}^{C}\) :

torques of mobile actuators

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  1. Department of Mechanics, University “Politehnica” of Bucharest, Bucharest, Romania

    Stefan Staicu

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Staicu, S. Matrix modeling of inverse dynamics of spatial and planar parallel robots. Multibody Syst Dyn 27, 239–265 (2012). https://doi.org/10.1007/s11044-011-9281-8

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