**6. The extraction of Denavit-Hartenberg parameters for T-S Robot**

**Figure 4.** (a) Introduction a new geometric model as Reuleaux Triangle-Star Robot with telescopic arms. (b) Introduc‐

In the boundary of kinematics science, place, speed, acceleration, and all the derivations higher than local variable (in proportion to time) are examined. As defined [2, 8], the robotic kine‐ matics is the study of the robotic movement without the consideration of the forces and torques applied to it. As a matter of fact, the examination of robotic motion geometry is regarded as

**1.** Direct kinematics: local calculation and the central manipulator orientation in relation to

**2.** Reverse kinematics: if the location and central manipulator orientation are given, the calculation of all the possible joint angles involved in directing the robot toward a desired

A systematic, practical approach has been represented by Denavit-Hartenberg to determine

In this paper, direct kinematics and reverse kinematics are examined through this approach and then position analysis, speed, input, and output acceleration are closely studied. The interesting point is that the direct kinematics of Parallel arms is as complicated as the reverse kinematics of Serial arms and the simplicity of the reverse kinematics of parallel mechanisms

**a.** Reuleaux Triangle-Star Robot with kinematics structure {RT-S (3-PRP)}

an invariable coordinate frame work proportional to the intended time.

The kinematics problem includes the following sections [2, 8]:

location and orientation is called reversed kinematics.

rotation and transformation between the two adjacent links in a robot [2].

is as much as the simplicity of the direct kinematics of Serial arms [2, 5].

tion of a new geometric model as Circle-Star robot with telescopic arms.

**5. Kinematic analysis of the T-S Robot**

**b.** Circle-Star Robot with kinematics structure {C-S (3-PRP)}

These robots are as follows:

72 Recent Advances in Robotic Systems

basic frame work.

To achieve *ai k* , *α<sup>i</sup> k* , *Si k* , *θ<sup>i</sup> k* parameters, first, the position of coordinate axis according to Denavit-Hartenberg model as shown in **Figure 6** (**a**, **b**) is determined. Then by placing these parameters in Denavit-Hartenbary matrix, which is obtained by relation (2) according to **Figure 5**. Consequently, the achieved transmitted matrixes and robotic kinematics analysis are carried out.

$$q = \left[q\_1, q\_2, q\_3, \right]^r, \ r = \left[r\_1, r\_2, r\_3, \right]^r, \ r = f\left(q\right) \tag{2}$$

**Figure 5.** Kinematic chain and parameters representation of the Hartenbary–Denavit model for two adjacent links [1].

**Figure 6.** (a) Kinematic chain and parameters of the proposed T-S (3-PRP) robot with telescopic arms (b) Kinematic chain and parameters of the T-S robot with rigid arms.

$$\begin{aligned} \begin{bmatrix} H\_{i,i+1}^{k} \end{bmatrix} = \begin{bmatrix} R\_{i,i+1} & P\_{i,i+1} \\ \begin{bmatrix} 0 \end{bmatrix} & 1 \end{bmatrix}, \quad i = 0, 1, 2, \text{( joint)}\\ k &= 1, 2, 3, 4, \dots \text{(N.chain)} \end{aligned} \tag{3}$$

$$H\_{i,i+1} = \begin{bmatrix} \cos\theta\_i & -\sin\theta\_i\cos\alpha\_i & \sin\theta\_i\sin\alpha\_i & a\_i\cos\theta\_i\\ \sin\theta\_i & \cos\theta\_i\cos\alpha\_i & -\cos\theta\_i\sin\alpha\_i & a\_i\sin\theta\_i\\ 0 & \sin\alpha\_i & \cos\alpha\_i & S\_i\\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{4}$$

*ai* : offset distance between two adjacent joint axes, where *ai* =| *pi* − *oj* |.

*αi* : twist angle between two adjacent joint axes. It is the angle required to rotate the *zi* axis into alignment with the *zi* + 1 –axis about the positive *xi* + 1–axis according to the right-hand rule.

*θi* : joint angle between two incident normals of a joint axis. It is the angle required to rotate the *xi* –axis into alignment with the *xi* + 1 –axis about the positive *zi* –axis according to the right-hand rule.

*Si* : translational distance between two incident normals of a joint axis. *Si* = | *pi* − *oi* | is positive if the vector *pi* − *oi* points in the positive *z <sup>i</sup>* –direction; otherwise, it is negative (**Figure 6**).

To analyze the robot kinematically, first, the central manipulator point is transferred to the corner of the triangle to which the reference coordinate device is joined and through the extraction of *ai k* , *α<sup>i</sup> k* , *Si k* , *θ<sup>i</sup> k* parameters, **Table 1**, related to Denavit–Hartenberg parameters, is achieved. In this case, *k* = 1, 2, 3 is the number of central manipulator point transmission, **Figure 7**. These transmissions are as follows: Path 1: *C*, *P*1, *A*1, Path 2: *C*, *P*3, *A*3, *A*1, Path 3 : *C*, *P*2,, *A*<sup>1</sup>

**Figure 7.** Transmission paths from point C to *A*1.

**Figure 6.** (a) Kinematic chain and parameters of the proposed T-S (3-PRP) robot with telescopic arms (b) Kinematic

[ ] ( ) ,1 ,1

=

cos sin cos sin sin cos sin cos cos cos sin sin

é ù ê ú - <sup>=</sup>

*i i i i ii i i i i i ii i*

 qa

> qa

00 0 1

ë û

*k*

1,2,3,4,...

*i ii*

 a (N.chain)

*a*

 =| *pi* − *oj* |.

*S*

 q

 q

–direction; otherwise, it is negative (**Figure 6**).

parameters, **Table 1**, related to Denavit–Hartenberg parameters, is

(4)

axis into


–axis according to the right-hand

 = | *pi* − *oi*

<sup>=</sup> (3)

, 1 0,1,2,.,

*i i i jo*

*intH*

0 sin cos

: twist angle between two adjacent joint axes. It is the angle required to rotate the *zi*

alignment with the *zi* + 1 –axis about the positive *xi* + 1–axis according to the right-hand rule.

: joint angle between two incident normals of a joint axis. It is the angle required to rotate the

To analyze the robot kinematically, first, the central manipulator point is transferred to the corner of the triangle to which the reference coordinate device is joined and through the

achieved. In this case, *k* = 1, 2, 3 is the number of central manipulator point transmission,

a

*<sup>a</sup> <sup>H</sup>*

 q a

 q a

<sup>+</sup>

0 1

+ + <sup>+</sup> <sup>=</sup> é ù ê ú ë û

*R P*

*k ii ii*

q

q

: offset distance between two adjacent joint axes, where *ai*

–axis into alignment with the *xi* + 1 –axis about the positive *zi*

points in the positive *z <sup>i</sup>*

: translational distance between two incident normals of a joint axis. *Si*

chain and parameters of the T-S robot with rigid arms.

74 Recent Advances in Robotic Systems

, 1

*i i*

*ai*

*αi*

*θi*

*xi*

*Si*

rule.

if the vector *pi*

extraction of *ai*

− *oi*

*k* , *α<sup>i</sup> k* , *Si k* , *θ<sup>i</sup> k*


**Table 2.** Kinematics parameters of T-S robot.

In the above indexes, the first shows the number of the direction, and the second shows the number of the parameter. To avoid making mistakes, the number of direction indexes (first index) is shown to the power of *k*.

By placing **Table 2** parameters, in matrix 2, the amount of *Hi*,*i*+1 *<sup>k</sup>* matrix is achieved in which, *i* =1, 2, 3, 4 is the number of coordinate systems joined to the links and *k* = 1, 2, 3 is the number of the central manipulator reference point directions joined to C point. The speed-acceleration points are obtained through placing the position analysis matrixes two by two in an equal way.
