**1. Introduction**

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

Yi, B.-J., Freeman, R. A. & Tesar, D. (1989). Open-loop stiffness control of overconstrained

Yoshikawa, T. (1984). Analysis and control of robot manipulators with redundancy, Int.

Zhao, J., Feng, Z., Zhou, K. & Dong, J. (2005). Analysis of the singularity of spatial parallel manipulator with terminal constraints, Mech. Mach. Theory 40(3): 275–284. Zhao, Y. & Gao, F. (2010). The joint velocity, torque, and power capability evaluation of a

Zhao, Y., Liu, J. F. & Huang, Z. (2011). A force analysis of a 3-rps parallel mechanism by

redundant parallel manipulator. accepted in Robotica.

using screw theory. accepted in Robotica.

1345.

Symp. Rob. Res. pp. 735–747.

mechanisms/robotic linkage systems, Proc. IEEE Int. Conf. Robot. Autom. 3: 1340 –

The kinematic and dynamic modelling of robotic manipulators has, as a specific field of robotics, represented a complex problem. To deal with this, the researchers have based their works on a great variety of mathematical theories (Seiling, 1999). One of these tools is the Dual Algebra, which is a concept originally introduced in 1893 by William Kingdon Clifford (Fisher, 1998; Funda, 1988). A dual number is a compact form that can be used to represent the rigid body motion in the space (Keller, 2000; Pennestrí & Stefanelli, 2007), it has therefore, a natural application in the analysis of spatial mechanisms specifically mechanical manipulators (Bandyopadhyay, 2004, 2006; Bayro-Corrochano & Kähler, 2000, 2001).

Several works related to dual-number in kinematics, dynamics and synthesis of mechanisms have been developed (Cheng, 1994; Fisher, 1998, 1995, 2003) in (Moon & Kota, 2001) is presented a methodology for combining basic building blocks to generate alternative mechanism concepts. The methodology is based on a mathematical framework for carrying out systematic conceptual design of mechanisms using dual vector algebra. The dual vector representation enables separation of kinematic function from mechanism topology, allowing a decomposition of a desired task into subtask, in order to meet either kinematic function or spatial constraints. (Ying et al, 2004) use dual angles as an alternative approach to quantify general spatial human joint motion. Ying proposes that dual Euler angles method provides a way to combine rotational and translational joint motions in Cartesian Coordinate systems, which can avoid the problems caused by the use of the joint coordinate system due to non-orthogonality. Hence the dual angles method is suitable for analyzing the motion characteristics of the ankle joint. The motion at the ankle joint complex involves rotations about and translations along three axes. In the same field of biomechanics (KiatTeu et al, 2006) present a method that provides a convenient assessment of golf-swing effectiveness. The method can also be applied to other sports to examine segmental rotations. In general, this method facilitates the study of human motion with relative ease. The use of a biomechanical model, in conjunction with dual-number coordinate transformation for motion analysis, was shown to provide accurate and reliable results. In particular, the advantage of using the dual Euler angles based on the dual-

Kinematic and Dynamic Modelling of

**2. Mathematical preliminaries** 

The tangent vectors to u and v are:

ˆ *u v*

*e e*

From the obvious

as a column vectors:

Cartesian Frame:

and translation.

Serial Robotic Manipulators Using Dual Number Algebra 69

(Bandyopadhyay & Ghosal, 2004) performs a study in order to determinate principal twist of the end-effector of a multi-degree of freedom manipulator, which plays a central role in

(Yang & Wang, 2010) solve the direct and inverse kinematics of a SCARA robot. They proved that the dual number allows compact formulations considerably facilitating the analysis of robot kinematics. To deal with coordinate transformation in three dimensional Cartesian space, the homogeneous transformation is usually applied. It is defined in the four-dimensional space and its matrix multiplication performs the simultaneous rotation

Let us consider a transformation of coordinates between the Cartesian Coordinate system

It is well known that the point (,) *u v* is localized by the vector *rx y* <sup>î</sup> from the origin of

; *r r ai bi aj*

*u v* 

The unit vectors of the Cartesian frame can be written in the form of column vectors:

; <sup>0</sup>

Cartesian Frame:

1 0 ; 0 1 *i j*

We can describe the oblique frame (,) *u v* in terms of the tangent vectors ai and bi+aj written

*a b ai bi aj <sup>a</sup>* 

The column vectors 1 0 *<sup>T</sup>* , 0 1 *<sup>T</sup>* can be combined into a single matrix describing the

1 0 0 1 

*x au bv y u av* ; 0 (1)

*r au bv avj* ( ) <sup>î</sup> (2)

it is clear that the coordinates (u, v) are not orthogonal.

(3)

(4)

(5)

the analysis, design, motion planning and determination of singularities.

(x,y) and the oblique coordinate system (u,v) given by the equations:

With a, b real numbers. The geometry is depicted in Figure 1.

2 2

*b*

*b a*

the Cartesian coordinate. From the transformation (1):

number coordinate transformation approach, is that it allows, for a complete 3D motion, to use only six parameters for each anatomical joint. KiatTeu infers that the method has proved to be an effective means to examine high-speed movement in 3D space. It also provides an option in assessing the contributions of the individual segmental rotations in production of the relevant velocity of the end-effector.

(Page et al, 2007) present the location of the instantaneous screw axis (ISA) in order to obtain useful kinematic models of the human body for applications such as prosthesis and orthoses design or even to help in disease diagnosis techniques. Dual vectors are used to represent and operate with kinematic screws with the purpose of locating the instantaneous screw axes which characterize this instantaneous motion. A photogrammetry system based on markers is used to obtain the experimental data from which the kinematic magnitudes are obtained. A comprehensive analysis of the errors in the measurement of kinematic parameters was developed, obtaining explicit expressions for them based on the number of markers and their distribution.

### **1.1 Dual-number representation in robotics**

The dual-number representation has been extended to other fields of mechanics; rigid body mechanics is an area where the dual number formulation has been applied, especially in the kinematics and dynamics of mechanisms.

The homogeneous transformation is a point transformation; in contrast, a line transformation can also naturally be defined in 3D Cartesian space, in which the transformed element is a line in 3D space instead of a point. In robotic kinematics and dynamics, the velocity and acceleration vectors are often the direct targets of analysis. The line transformation will have advantages over the ordinary point transformation, since the combination of the linear and angular quantities can be represented by lines in 3D space. Since a line in 3D space is determined by four independent parameters. (Gu, 1988) presents a procedure that, offers an algorithm which deals with the symbolic analysis for both rotation and translation. In particular, the aforementioned is effective for the direct determination of Jacobian matrices and their derivatives. The dual-number transformation procedure, based on these properties and the principle of transference, can be used for finding Jacobian matrices in robotic kinematics and their derivatives in robotic dynamics and control modeling. A related work was performed in (Pennock & Mattson, 1996) where the forward position problem of two PUMA robots manipulating a planar four bar linkage payload is solved using closed-form solutions for the remaining unknown angular displacements based in orthogonal transformation matrices with dual-number. (Brosky & Shoham, 1998; Sai-Kai, 2000) introduce the generalized Jacobian matrix which consists of the complete dual transformation matrices. The generalized Jacobian matrix relates force and moment at the end-effector to force and moment at the joints for each axe. Furthermore, the generalized Jacobian matrix also relates motion in all directions at the joints to the motion of the end-effector, an essential relation required at the design stage of robot manipulators. An extension of these kinematics and statics schemes into dynamics is possible by applying the dual inertia operator. (Brodsky, 1999) formulated the representation of rigid body dynamic equations introducing the dual inertia operator. Brodsky gives a general expression for the three-dimensional dynamic equation of a rigid body with respect to an arbitrary point. Then the dual Lagrange equation is established by developing derivative rules of a real function with respect to dual variables.

(Bandyopadhyay & Ghosal, 2004) performs a study in order to determinate principal twist of the end-effector of a multi-degree of freedom manipulator, which plays a central role in the analysis, design, motion planning and determination of singularities.

(Yang & Wang, 2010) solve the direct and inverse kinematics of a SCARA robot. They proved that the dual number allows compact formulations considerably facilitating the analysis of robot kinematics. To deal with coordinate transformation in three dimensional Cartesian space, the homogeneous transformation is usually applied. It is defined in the four-dimensional space and its matrix multiplication performs the simultaneous rotation and translation.
