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Dynamics of the spherical 3- \(\mathit{U\underline{P}S/}S\) parallel mechanism with prismatic actuators

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Abstract

Recursive relations in kinematics and dynamics of the symmetric spherical 3- \(\mathit{U\underline{P}S}/S\) parallel mechanism having three prismatic actuators are established in this paper. Controlled by three forces, the parallel manipulator is a 3-DOF mechanical system with three parallel legs connecting to the moving platform. Knowing the position and the rotation motion of the platform, we develop first the inverse kinematics problem and determine the position, velocity, and acceleration of each manipulator’s link. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work, but it has been verified using the results in the framework of the Lagrange equations with their multipliers. Finally, compact matrix relations and graphs of simulation for the input forces and powers are obtained.

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Abbreviations

a k,k−1,b k,k−1,c k,k−1 :

orthogonal transformation matrices

R :

general transformation matrix of the moving platform

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

three orthogonal unit vectors

α i1 ,α i2 ,α i3 (i=A,B,C):

angles giving the position of universal and spherical joints of two platforms

α 1,α 2,α 3 :

Euler angles of successive rotations

β :

initial inclination of three legs

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 of A k 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

l 0 :

radius of the circular moving platform

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

mass and tensor of inertia of moving platform

m k :

mass of T k rigid body

\(\hat{J}_{k}\) :

symmetric matrix of tensor of inertia of T k about the link-frame x k y k z k

f i32 ,p i32 (i=A,B,C):

input forces and powers of three prismatic actuators

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

    Stefan Staicu

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Staicu, S. Dynamics of the spherical 3- \(\mathit{U\underline{P}S/}S\) parallel mechanism with prismatic actuators. Multibody Syst Dyn 22, 115–132 (2009). https://doi.org/10.1007/s11044-009-9150-x

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