Abstract
To avoid the unit inconsistency problem in the conventional Jacobian matrix, new formulation of a dimensionally homogeneous inverse Jacobian matrix for parallel manipulators with a planar mobile platform by using three end-effector points was presented (Kim and Ryu, 2003). This paper presents force relationships between joint forces and Cartesian forces at the three End-Effector points. The derived force relationships can then be used for analyses of the input/output force transmission. These analyses, forward and inverse force transmission analyses, depend on the singular values of the derived unit consistent Jacobian matrix. Using the proposed force relationship, a numerical example is presented for actuator size design of a 3-RRR planar parallel manipulator.
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Abbreviations
- T j :
-
Three distinct and noncollinear End-Effector points (j=1, 2, 3) Center of the base reference frame Center of the mobile frame
- x, y, ζ :
-
Axes of the base reference frame
- x’y’,z’:
-
Axes of the mobile reference frame attached to the mobile platform
- A i :
-
Center of the base universal joint of legi
- B i :
-
Center of the platform universal (or spherical) joint of legi
- q :
-
The vector defined by the coordinates of three points describing the motion of the mobile platform (e.g.
- q=[x1, y1, z1, x2, y2, z2, x3, y3, z3]T)k i, j :
-
The coefficients that are functions of the geometry of the mobile platform jointsB i and the pre-selected three pointsT j
- λ i :
-
The magnitude of the actuating length
- n i :
-
The unit vector of articular coordinatei
- t :
-
Time differentiation of three with respect to the fixed world coordinate system
- Λ :
-
Vector of articular coordinates,
- Λ = [λ1, λ2, λ3, λ4, λ5, λ6]T J :
-
Jacobian matrix of the manipulator
- l i,1,l i,2 :
-
The link lengths
- σ min,σ max :
-
The minimum and maximum singular values of the Jacobian matrix
- F 1,F 2,F 3 :
-
The external forces acting on three points
- τ:
-
The joint forces
- Φ q :
-
The constraint Jacobian matrix
- c :
-
The twist of end effector (C)
- V c ,ω :
-
The linear velocity and angular velocity of the center of the mobile
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Kim, SG., Ryu, J. Force transmission analyses with dimensionally homogeneous Jacobian matrices for parallel manipulators. KSME International Journal 18, 780–788 (2004). https://doi.org/10.1007/BF02990296
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DOI: https://doi.org/10.1007/BF02990296