**3. Kinematic analysis**

One of the most important issues in the study of parallel mechanisms is the kinematic analysis, where the generated results form the base for the application of the mechanism.

*i a* represents the vector *OAi* (Fig. 1). [cos sin 0]*<sup>T</sup> i ii a g* , in which,

Re . Re . Re . Re . 1 23 4 45 , 45 , 135 , <sup>135</sup> *d dd d and* ,

and

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

Cartesian coordinates *A* (*O, x ,y, z*) and *B* (*P, u, v, w*) represented by {A} and {B} are

*<sup>i</sup> d* . Assuming that each limb is connected to the fixed base by a universal joint, the orientation of ith limb with respect to the fixed base can be described by two Euler angles,

*<sup>i</sup>* is inactive.

is the vector along *A Bi i*

*<sup>i</sup>* around *ni*

*<sup>i</sup>* and *<sup>i</sup> d* are active joints actuated by the rotary

represents the unit

, which is perpendicular

with the length of

Fig. 3. Schematic of the universal joint, and the joints variables.

vector along the axes of ith rotary actuator and *<sup>i</sup> d*

(Fig. 3). It is to be noted that

Fig. 4. Schematics of well-known Stewart platforms.

*<sup>i</sup>* around the axis *is*

and linear actuators, respectively. However,

rotation

to *<sup>i</sup> d*  and *is*

connected to the base and moving platforms, respectively. In Fig.3, *is*

, followed by rotation

By replacing the passive universal joints in the Stewart mechanism with active joints in the above mentioned mechanisms, the number of legs could be reduced from 6 to 3 or 4. This

$$\mathcal{Y}\_1^{\text{Non}-\text{Red.}} = 0^\circ \,, \ \mathcal{Y}\_2^{\text{Non}-\text{Red.}} = \mathbf{120^\circ} \,, \ \text{and} \ \ \mathcal{Y}\_3^{\text{Non}-\text{Red.}} = -\mathbf{120^\circ} \,,$$

where *g* and *r* are the radius of the fixed and moving platforms, respectively. *Bbi* represents the position of the ith joint on the platform in the moving frame {B}, *<sup>B</sup> <sup>i</sup> <sup>i</sup> <sup>B</sup> b PB* . *Bbi* is constant and is equal to [cos sin 0] *B T <sup>i</sup> i i b h* . We can represent *<sup>A</sup> B ij R r* , the rotation matrix from *B* to*A* , using Euler angles as

$${}^{A}\_{B}\mathbf{R} = \begin{bmatrix} \mathbf{c}a\_{2}\mathbf{c}a\_{3} & \mathbf{c}a\_{3}\mathbf{s}a\_{2}\mathbf{s}a\_{1} - \mathbf{s}a\_{3}\mathbf{c}a\_{1} & \mathbf{c}a\_{3}\mathbf{s}a\_{2}\mathbf{c}a\_{1} + \mathbf{s}a\_{3}\mathbf{s}a\_{1} \\ \mathbf{c}a\_{2}\mathbf{s}a\_{3} & \mathbf{s}a\_{3}\mathbf{s}a\_{2}\mathbf{s}a\_{1} + \mathbf{c}a\_{3}\mathbf{c}a\_{1} & \mathbf{s}a\_{3}\mathbf{s}a\_{2}\mathbf{c}a\_{1} - \mathbf{c}a\_{3}\mathbf{s}a\_{1} \\ -\mathbf{s}a\_{2} & \mathbf{c}a\_{2}\mathbf{s}a\_{1} & \mathbf{c}a\_{2}\mathbf{c}a\_{1} \end{bmatrix} \tag{1}$$

where 1 1 s sin and 1 1 c cos , and so on. 1 , <sup>2</sup> , and 3 are three Euler angles defined according to the *zyx* convention. Thus, the vector *<sup>B</sup> i b* would be expressed in the fixed frame {A} as

$$
\overrightarrow{db}\_{\hat{i}} = \,^A\_B R \,\overrightarrow{PB\_{\hat{i}}}\,\Big|\_B . \tag{2}
$$

Let *p* and *ir* denote the position vectors for *P* and *Bi* in the reference frame*A* , respectively. From the geometry, it is obvious that

$$
\vec{r}\_i = \vec{p} + \vec{b}\_i \tag{3}
$$

Subtracting vector *<sup>i</sup> a* from both sides of (3) one obtains

$$
\vec{r}\_i - \vec{a}\_i = \vec{p} + \vec{b}\_i - \vec{a}\_i \,. \tag{4}
$$

Left hand side of (4) is the definition of *<sup>i</sup> d* , therefore

$$d\_i^2 = \left(\vec{p} + \vec{b}\_i - \vec{a}\_i\right) \cdot \left(\vec{p} + \vec{b}\_i - \vec{a}\_i\right) \,. \tag{5}$$

Exploiting Higher Kinematic Performance –

singularities are clearly shown.

physical meaning of reciprocal screws.

matrix by means of the screw theory.

DOF PM by means of the screw theory.

where

is a 6 1 vector:

 *<sup>i</sup>* and

Joint velocity vector in the redundant mechanism, Re . *<sup>d</sup> q*

Re .

Re .

**4. Jacobian analysis using screw theory** 

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 49

Singularities of a PM pose substantially more complicated problems, compared to a serial manipulator. One of the first attempts to provide a general framework and classification may be traced back to Gosselin and Angeles (1990) ,who derived the input–output velocity map for a generic mechanism by differentiating the implicit equation relating the input and output configuration variables. In this way, distinct jacobian matrices are obtained for the inverse and the direct kinematics, and different roles played by the corresponding

For singularity analysis other methods rather than dealing with jacobian matrix are also available. Pendar et al. (2011) introduced a geometrical method, namely *constraint plane method*, where one can obtain the singular configurations in many parallel manipulators with their mathematical technique. Lu et al. (2010) proposed a novel analytic approach for determining the singularities of some 4-DOF parallel manipulators by using *translational/rotational jacobian matrices*. Piipponen (2009) studied kinematic singularities of planar mechanisms by means of *computational algebraic geometry* method. Zhao et al. (2005) have proposed *terminal constraint* method for analyzing the singularities based on the

Gosselin & Angeles (1990) have based their works on deriving the jacobian matrix. They performed this by defining three possible conditions. In these conditions the determinant of forward jacobian matrix or inverse jacobian matrix is investigated. They have shown that having dependent Plücker vectors in a parallel manipulator is equivalent to zero

In this section the jacobian analysis of the proposed PMs are approached by using the theory of screws. Zhao et al. (2011) proposed a new approach using the screw theory for force analysis, and implemented it on a 3-DOF 3-RPS parallel mechanism. Gallardo-Alvarado et al. (2010) presented a new 5-DOF redundant parallel manipulator with two moving platforms, where the active limbs are attached to the fixed platform. They find the jacobian

A class of series-parallel manipulators known as 2(3-RPS) manipulators was studied by Gallardo-Alvarado et al. (2008) by means of the screw theory and the principle of virtual work. Gan et al. (2010) used the screw theory to obtain the kinematic solution of a new 6- DOF 3CCC parallel mechanism. Gallardo-Alvarado et al. (2006) analyzed singularity of a 4-

Hereafter we derive the jacobian matrices of the proposed mechanisms using screw theory.

12 3 41 2 3 4 [ ] *d T q d*

 *<sup>d</sup> <sup>d</sup> <sup>d</sup>*

respectively. However, joint velocity vector in the non-redundant mechanism,

12312 3 [ ]

  , is an 8 1 vector:

*Non d* Re . *q* ,

(13)

*Non d <sup>T</sup> q d d d* . (14)

*<sup>i</sup> d* are the angular and linear velocities of the rotary and linear actuators,

determinant of the forward jacobian matrix and then a platform singularity arises.

Using Euclidean norm, *<sup>i</sup> d* can be expressed as:

$$d\_i = \sqrt{\left(\left(x - x\_i\right)^2 + \left(y - y\_i\right)^2 + \left(z - z\_i\right)^2\right)}\_{\text{'}}\tag{6}$$

where,

$$\begin{cases} x\_i = -h \left( \cos \gamma\_i r\_{11} + \sin \gamma\_i r\_{21} \right) + g \cos \gamma\_i \\ y\_i = -h \left( \cos \gamma\_i r\_{12} + \sin \gamma\_i r\_{22} \right) + g \sin \gamma\_i \ . \end{cases} \tag{7}$$
 
$$z\_i = -h \left( \cos \gamma\_i r\_{13} + \sin \gamma\_i r\_{23} \right)$$

Coordinates ( ,,,) *CAxyz i iiii* are connected to the base platform with their *<sup>i</sup> x* axes aligned with the rotary actuators in the *is* directions, with their *<sup>i</sup> <sup>z</sup>* axes perpendicular to the fixed platform. Thus, one can express vector *<sup>i</sup> d* in *Ci* as

$$\begin{aligned} ^{C\_i}\overrightarrow{d\_i} &= d\_i \begin{bmatrix} \sin\psi\_i \\ -\sin\theta\_i \cos\psi\_i \\ \cos\theta\_i \cos\psi\_i \end{bmatrix} \end{aligned} \tag{8}$$

and from the geometry is clear that

$$
\overrightarrow{r\_i} = \overrightarrow{a\_i} + \prescript{A}{C\_i}{R}^{C\_i} \overrightarrow{d\_i} \ \tag{9}
$$

where *<sup>i</sup> A <sup>C</sup> R* is the rotation matrix from *Ci* to *A* ,

$$\begin{aligned} \, \_C^A R = \begin{bmatrix} \cos \gamma\_i & -\sin \gamma\_i & 0\\ \sin \gamma\_i & \cos \gamma\_i & 0\\ 0 & 0 & 1 \end{bmatrix}. \end{aligned} \tag{10}$$

By equating the right sides of (3) and (4), and solving the obtained equation, *<sup>i</sup>* and *<sup>i</sup>* can be calculated as follows:

$$\varphi\_{i} = \sin^{-1}\left(\frac{\cos\gamma\_{i}(\mathbf{x} - \mathbf{x}\_{i}) + \sin\gamma\_{i}(y - y\_{i})}{d\_{i}}\right),\tag{11}$$

$$\theta\_i = \sin^{-1}\left(\frac{\sin\gamma\_i(\mathbf{x} - \mathbf{x}\_i) - \cos\gamma\_i(y - y\_i)}{d\_i\cos\nu\_i}\right). \tag{12}$$
