**9. References**

82 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

*i i j j k k i kj jk j ik ki k ji ij*

*i k j k j j k j k*

 

 

 

> 

*H VP H V P*

 

 

*H H VP H VP*

 22 2 22 2 22 2 22 2 1 2,0 1 2,0 1 2,0 1 2,0 1 2 22 2 22 2 22 2 22 2 1 2,0 1 2,0 1 2,0 1 2,0 1 *j i k ik k i ki k j i j i i j i j*

*H H VP H VP*

0 *B A B B B A BA B B MF F f MT DF T t AA B A AA AA B A*

<sup>0</sup> *BA B B MF F f AA B A*

<sup>0</sup> *B A BA B B MT DF T t AA AA B A*

 <sup>1</sup> 22 2 *F M* 2 11*<sup>f</sup>* 

 <sup>1</sup> 1 1 21 *FM F* 1 0 20*<sup>f</sup>* 

 <sup>1</sup> 2 2 2 22 *T M t DF* 2 1 1 12 

 <sup>1</sup> 1 1 2 1 11 *T M T t DF* 1 0 2 0 01 

The presented method, based on dual-number representation, has demonstrated be a powerful tool for solving a great variety of problems, that imply motions simultaneity off rotation and translation of rigid bodies in the space; the aforementioned, allows establishing dual rotation matrices. Robotics is a field wherein dual numbers have been employed to describe the motion of a rigid body, in particular of serial robotic arms. The methodology proposed is useful for robotic arms with cylindrical, prismatic and rotational joints. Once

ˆ , if the dual part of ˆ

developed methodology can be generalized to different topologies, which is a great

advantage that allows that only one program solves a great variety of topologies.

is zero, the mechanism has only

is zero, only exist prismatic joints. So the

2 2 1 1 2 2 1 1 2 2 1 1 2 22 2 22 2 1 2,0 1 2,0 1 2 22 2 22 2 1 2,0 1 2,0 1 2 22 2 22 2 1 2,0 1 2,0 1 2 22 2 22 2 22 2 22 2 1 2,0 1 2,0 1 2,0 1 2,0 1

 

*f t f t f t PPP PP P P PP*

 

2

From dynamic equilibrium:

**7. Conclusions** 

established the dual angles ˆ

 and 

revolute joints, otherwise if the primary part of ˆ

*H*


**0**

**5**

*Italy*

**On the Stiffness Analysis and Elastodynamics of**

Accurate models to describe the elasticity of robots are becoming essential for those applications involving high accelerations or high precision to improve quality in positioning and tracking of trajectories. Stiffness analysis not only involves the mechanical structure of a robot but even the control system necessary to drive actuators. Strategies aimed to reduce noise and dangerous bouncing effects could be implemented to make control systems more robust to flexibility disturbances, foreseing mechanical interaction with the control system because of regenerative and modal chattering (1). The most used approaches to study elasticity in the literature encompass: the *Finite Element Analysis* (FEA), the *Matrix Structural Analysis* (MSA), the *Virtual Joint Method* (VJM), the *Floating Frame of Reference Formulation*

FEA is largely used to analyze the structural behavior of a mechanical system. The reliability and precision of the method allow to describe each part of a mechanical system with great detail (2). Applying FEA to a robotic system implies a time-consuming process of re-meshing in the pre-processing phase every time that the robot posture has changed. Optimization all over the workspace of a robot would require very long computational time, thus FEA models

The MSA includes some simplifications to FEA using complex elements like beams, arcs, cables or superelements (3–5). This choice reduces the computational time and makes this method more efficient for optimization tasks. Some authors recurred to the superposition principle along with the virtual work principle to achieve the global stiffness model (6–8). Others considered the minimization of the potential energy of a PKM to find the global stiffness matrix (9), while some approaches used the total potential energy augmented adding

The first papers on VJM are based on pseudo-rigid body models with "virtual joints" (12–14). More recent papers include link flexibility and linear/torsional springs to take into account bending contributes (15–19). These approaches recur to the Jacobian matrix to map the stiffness of the actuators of a PKM inside its workspace; especially for PKMs with reduced mobility, it implies that the stiffness is limited to a subspace defined by the dofs of its end-effector. Pashkevich *et al*. tried to overcome this issue by introducing a multidimensional

Finally, the FFRF and ANCF are powerful and accurate formulations, based on FEM and continuum mechanics, to study any flexible mechanical system (21). The FFRF is suitable

are often employed to verify components or subparts of a complex robotic system.

the kinematic constraints by means of the Lagrange multipliers (10; 11).

lumped-parameter model with localized 6-dof virtual springs (20).

(FFRF) and the *Absolute Nodal Coordinates Formulation* (ANCF).

**1. Introduction**

*University of Catania, Department of Mechanical and Industrial Engineering*

**Parallel Kinematic Machines**

Alessandro Cammarata

