**3.3 Stage 3**

<9> After *stage 2*, the link height of *Robot A* is set at a low position (**Figure 6**). <10> *Operator A* makes *Robot A* stop, and *Operator B* makes *Robot B* move forward. As a result, the front wheels of *Robot B* are lifted. <11> Both robots move forward. <12> The operators recognize that the front wheels of *Robot B* have passed over the step edge. The operators manipulate each joystick to adjust the difference between

*Cooperative Step Climbing Using Connected Wheeled Robots and Evaluation of Remote… DOI: http://dx.doi.org/10.5772/intechopen.90162*

the speeds of the robots, so that the front wheels of *Robot B* are placed on the upper level of the step. Here, in stages 3 and 4, if *Robot B* is faster than *Robot A*, the tilt of *Robot B* increases. If *Robot A* is faster than *Robot B*, then the tilt of *Robot B* decreases.

### **3.4 Stage 4**

<13> The operators make the robots continue to move forward. The back wheels of *Robot B* come into contact with the step. <14> *Robot A* pulls *Robot B*. *Robot A* supports the climbing of *Robot B* and *Robot A* prevents *Robot B* from tipping over backward. <15> *Robot B* climbs up onto the step. <16> Once the rear wheels of *Robot B* have reached the upper level of the step, the operators stop each robot.

### **4. Theoretical analysis**

When *Robot A* climbs a step, the body of the robot inclines, and its front wheels are lifted due to the difference in velocity between the two connected robots. When the robots are manipulated by the operators, the step climbing ability greatly influences the manipulation time.

In this section, we clarify the relationships among the robot incline, the velocity, and the manipulation time.

### **4.1 Relationships among the manipulation time, the velocity, and the height of the front wheels required to climb the step**

In **Figure 10**, Σ*<sup>B</sup>* is the basic coordinate system for the robots, where *p*<sup>0</sup> is the origin as well as the contact position between the rear wheels of *Robot B* and the ground. In addition, *pi* (*i* = 1–5) are the joints (*p*1, axis of the rear wheels of *Robot B*; *p*2, link position of *Robot B*; *p*3, link position of *Robot A*; *p*4, axis of the rear wheels of *Robot A*; *p*5, axis of the front wheels of *Robot A*; *p*6, tread position of the front wheels of *Robot A*).

The position vectors for the joints in the coordinate system Σ*<sup>B</sup>* are expressed as *<sup>B</sup>pi*<sup>¼</sup> *xi yi* � �*<sup>T</sup>* (*i* = 1–6). In the local coordinate system, for the case in which <sup>Σ</sup>*<sup>i</sup>* is parallel to <sup>Σ</sup>0, <sup>0</sup>*p*<sup>1</sup> <sup>¼</sup> ½ � <sup>0</sup>*RB <sup>T</sup>*, <sup>1</sup>*p*<sup>2</sup> <sup>¼</sup> ½ � *lB* <sup>þ</sup> *lLBhLB <sup>T</sup>*, <sup>2</sup>*p*<sup>3</sup> <sup>¼</sup> ½ � *<sup>d</sup>* <sup>0</sup> *<sup>T</sup>*, <sup>3</sup>*p*4<sup>¼</sup> ½ � *lLA* �*hLA <sup>T</sup>*, <sup>4</sup>*p*<sup>5</sup> <sup>¼</sup> ½ � *lA* �*RA* <sup>þ</sup> *rA <sup>T</sup>*, and <sup>5</sup>*p*<sup>6</sup> <sup>¼</sup> ½ � *rA* <sup>0</sup> *<sup>T</sup>* (**Figure 10**). Then, *ϕ<sup>i</sup>* is the angle between Σ*<sup>i</sup>* and Σ*<sup>i</sup>*�1, and Σ<sup>1</sup> is parallel to Σ<sup>0</sup> in *stage 1*. Thus,

$$
\phi\_1 = 0 \tag{1}
$$

The incline of *Robot A*, P<sup>3</sup> *<sup>k</sup>*¼<sup>1</sup>*ϕi*, is

$$\sum\_{k=1}^{3} \phi\_i = \phi\_2 + \phi\_3 \tag{2}$$

Here, Σ<sup>4</sup> is always parallel to Σ3:

$$
\phi\_4 = 0 \tag{3}
$$

In the basic coordinate system Σ*B*, the homogeneous transformation matrix *<sup>B</sup>T*<sup>4</sup> is as follows:

supports the climbing of *Robot A*, and *Robot B* prevents *Robot A* from tipping over backward. <7> *Robot A* climbs up onto the step. <8> Once the rear wheels of *Robot*

<9> After *stage 2*, the link height of *Robot A* is set at a low position (**Figure 6**). <10> *Operator A* makes *Robot A* stop, and *Operator B* makes *Robot B* move forward. As a result, the front wheels of *Robot B* are lifted. <11> Both robots move forward. <12> The operators recognize that the front wheels of *Robot B* have passed over the step edge. The operators manipulate each joystick to adjust the difference between

*A* have reached the upper level of the step, the operators stop each robot.

**3.3 Stage 3**

**86**

**Figure 9.**

*Entire process of step climbing.*

*Industrial Robotics - New Paradigms*

*Industrial Robotics - New Paradigms*

$$\begin{aligned} \,^B T\_4 : \begin{bmatrix} \cos \phi\_{23} & -\sin \phi\_{23} & \varkappa\_4 \\\\ \sin \phi\_{23} & \cos \phi\_{23} & \varkappa\_4 \\\\ 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{4}$$

Thus, we have:

*DOI: http://dx.doi.org/10.5772/intechopen.90162*

Here

*<sup>B</sup>p*6<sup>¼</sup> *<sup>x</sup>*6*y*<sup>6</sup> � �*<sup>T</sup>*

and

where

**89**

Thus, we have:

sin *<sup>ϕ</sup>*<sup>2</sup> <sup>¼</sup> *hLA* cos *<sup>ϕ</sup>*<sup>23</sup> � *lLA* sin *<sup>ϕ</sup>*<sup>23</sup> � *hLB*

*Cooperative Step Climbing Using Connected Wheeled Robots and Evaluation of Remote…*

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � sin <sup>2</sup>

cos *ϕ*<sup>2356</sup> � sin *ϕ*<sup>2356</sup> *x*<sup>6</sup> sin *ϕ*<sup>2356</sup> cos *ϕ*<sup>2356</sup> *y*<sup>6</sup> 0 01

From (1) and (3), when *p*<sup>6</sup> (the tread position of the front wheels of *Robot A*,

*x*<sup>6</sup> ¼ *lA* cos *ϕ*<sup>23</sup> � �ð Þ *RA* þ *rA* sin *ϕ*<sup>23</sup> þ *lLA* cos *ϕ*<sup>23</sup>

*y*<sup>6</sup> ¼ �*rA* þ *lA* sin *ϕ*<sup>23</sup> þ �ð Þ *RA* þ *rA* cos *ϕ*<sup>23</sup> þ *lLA* sin *ϕ*<sup>23</sup>

sin *ϕ*<sup>23</sup> ¼

� � <sup>þ</sup> *lA*

, **Figure 10**) is at the bottom of the front wheels, P<sup>5</sup>

*ϕ*2

cos *ϕ*<sup>235</sup> ¼ 0 (12)

sin *ϕ*<sup>235</sup> ¼ �1 (13)

þ*hLA* sin *ϕ*<sup>23</sup> þ *lB* þ *lLB* þ *d* cos *ϕ*<sup>2</sup> (14)

�*hLA* cos *ϕ*<sup>23</sup> þ *hLB* þ *RB* þ *d* sin *ϕ*2*:* (15)

*y*<sup>6</sup> ¼ *lA* sin *ϕ*<sup>23</sup> þ �ð Þ *RA* þ *rA* cos *ϕ*<sup>23</sup> þ *RB* � *rA* (16)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � *e*<sup>1</sup> � *y*<sup>6</sup> � �<sup>2</sup> <sup>q</sup>

<sup>2</sup> (18)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � cos <sup>2</sup>*ϕ*<sup>23</sup> <sup>q</sup>

> *lA* <sup>2</sup> <sup>þ</sup> *<sup>e</sup>*<sup>1</sup>

*lA* <sup>2</sup> <sup>þ</sup> *<sup>e</sup>*<sup>1</sup>

cos *ϕ*<sup>2</sup> ¼

The homogeneous transformation matrix, *<sup>B</sup>T*6, is given by

*<sup>B</sup>T*<sup>6</sup> :

Here, *ϕ*<sup>2356</sup> ¼ *ϕ*<sup>2</sup> þ *ϕ*<sup>3</sup> þ *ϕ*<sup>5</sup> þ *ϕ*6.

*ϕ*<sup>2</sup> þ *ϕ*<sup>3</sup> þ *ϕ*<sup>5</sup> ¼ *ϕ*<sup>235</sup> ¼ �90°, where

By substituting (9) for (15), we have:

From (8), (16), and (17), we obtain:

Here, *e*<sup>1</sup> ¼ *RA* � *rA*.

cos *<sup>ϕ</sup>*<sup>23</sup> <sup>¼</sup> *<sup>e</sup>*<sup>1</sup> � *<sup>e</sup>*<sup>1</sup> � *<sup>y</sup>*<sup>6</sup>

*<sup>d</sup>* (9)

*:* (10)

*<sup>k</sup>*¼<sup>1</sup>*ϕ<sup>i</sup>* <sup>¼</sup>

(11)

(17)

Here, *ϕ*<sup>23</sup> ¼ *ϕ*<sup>2</sup> þ *ϕ*3:

$$\propto\_4 = l\_{LA}\cos\phi\_{23} + h\_{LA}\sin\phi\_{23} + l\_B + l\_{LB} + d\cos\phi\_2 \tag{5}$$

$$\{y\_4 = l\_{LA}\sin\phi\_{23} - h\_{LA}\cos\phi\_{23} + h\_{LB} + R\_B + d\sin\phi\_2\tag{6}$$

Then, *y*<sup>4</sup> is equal to *RA* (the radius of the rear wheels of *Robot A*) in *stage 1* (**Figure 10**), and we obtain the following equation from (6):

$$R\_A = l\_{LA} \sin \phi\_{23} - h\_{LA} \cos \phi\_{23} + h\_{LB} + R\_B + d \sin \phi\_2 \tag{7}$$

In this system

$$R\_A = R\_B.\tag{8}$$

**Figure 10.** *Lifting the front wheels of* Robot A *(stage 1).*

*Cooperative Step Climbing Using Connected Wheeled Robots and Evaluation of Remote… DOI: http://dx.doi.org/10.5772/intechopen.90162*

Thus, we have:

$$\sin \phi\_2 = \frac{h\_{LA} \cos \phi\_{23} - l\_{LA} \sin \phi\_{23} - h\_{LB}}{d} \tag{9}$$

Here

*<sup>B</sup>T*<sup>4</sup> :

(**Figure 10**), and we obtain the following equation from (6):

Here, *ϕ*<sup>23</sup> ¼ *ϕ*<sup>2</sup> þ *ϕ*3:

*Industrial Robotics - New Paradigms*

In this system

**Figure 10.**

**88**

*Lifting the front wheels of* Robot A *(stage 1).*

cos *ϕ*<sup>23</sup> � sin *ϕ*<sup>23</sup> *x*<sup>4</sup> sin *ϕ*<sup>23</sup> cos *ϕ*<sup>23</sup> *y*<sup>4</sup> 0 01

*x*<sup>4</sup> ¼ *lLA* cos *ϕ*<sup>23</sup> þ *hLA* sin *ϕ*<sup>23</sup> þ *lB* þ *lLB* þ *d* cos *ϕ*<sup>2</sup> (5)

*y*<sup>4</sup> ¼ *lLA* sin *ϕ*<sup>23</sup> � *hLA* cos *ϕ*<sup>23</sup> þ *hLB* þ *RB* þ *d* sin *ϕ*<sup>2</sup> (6)

*RA* ¼ *lLA* sin *ϕ*<sup>23</sup> � *hLA* cos *ϕ*<sup>23</sup> þ *hLB* þ *RB* þ *d* sin *ϕ*<sup>2</sup> (7)

*RA* ¼ *RB:* (8)

Then, *y*<sup>4</sup> is equal to *RA* (the radius of the rear wheels of *Robot A*) in *stage 1*

(4)

$$
\cos \phi\_2 = \sqrt{\mathbf{1} - \sin^2 \phi\_2}. \tag{10}
$$

The homogeneous transformation matrix, *<sup>B</sup>T*6, is given by

$$\begin{aligned} \,^B T\_6 : \begin{bmatrix} \cos \phi\_{2356} & -\sin \phi\_{2356} & \varkappa\_6 \\ \sin \phi\_{2356} & \cos \phi\_{2356} & \varkappa\_6 \\ 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{11}$$

Here, *ϕ*<sup>2356</sup> ¼ *ϕ*<sup>2</sup> þ *ϕ*<sup>3</sup> þ *ϕ*<sup>5</sup> þ *ϕ*6.

From (1) and (3), when *p*<sup>6</sup> (the tread position of the front wheels of *Robot A*, *<sup>B</sup>p*6<sup>¼</sup> *<sup>x</sup>*6*y*<sup>6</sup> � �*<sup>T</sup>* , **Figure 10**) is at the bottom of the front wheels, P<sup>5</sup> *<sup>k</sup>*¼<sup>1</sup>*ϕ<sup>i</sup>* <sup>¼</sup> *ϕ*<sup>2</sup> þ *ϕ*<sup>3</sup> þ *ϕ*<sup>5</sup> ¼ *ϕ*<sup>235</sup> ¼ �90°, where

$$\cos \phi\_{235} = 0 \tag{12}$$

and

$$
\sin \phi\_{235} = -1 \tag{13}
$$

Thus, we have:

$$\mathbf{x}\_6 = l\_A \cos \phi\_{23} - (-R\_A + r\_A) \sin \phi\_{23} + l\_{LA} \cos \phi\_{23}$$

$$+ h\_{LA} \sin \phi\_{23} + l\_B + l\_{LB} + d \cos \phi\_2 \tag{14}$$

$$y\_{\xi} = -r\_{A} + l\_{A} \sin \phi\_{23} + (-R\_{A} + r\_{A}) \cos \phi\_{23} + l\_{LA} \sin \phi\_{23}$$

$$-h\_{LA} \cos \phi\_{23} + h\_{LB} + R\_{B} + d \sin \phi\_{2}.\tag{15}$$

By substituting (9) for (15), we have:

$$\mathcal{Y}\_6 = l\_A \sin \phi\_{23} + (-R\_A + r\_A) \cos \phi\_{23} + R\_B - r\_A \tag{16}$$

where

$$
\sin \phi\_{23} = \sqrt{\mathbf{1} - \cos^2 \phi\_{23}} \tag{17}
$$

From (8), (16), and (17), we obtain:

$$\cos\phi\_{23} = \frac{e\_1 \cdot \left(e\_1 - y\_6\right) + l\_A\sqrt{l\_A^2 + e\_1^2 - \left(e\_1 - y\_6\right)^2}}{l\_A^2 + e\_1^2} \tag{18}$$

Here, *e*<sup>1</sup> ¼ *RA* � *rA*.

Then, *p*<sup>4</sup> (the position of the axis of the rear wheels of *Robot A*) moves forward in lifting the front wheels of *Robot A* (**Figure 10**). Time *t* ¼ *tm* is the time for the operators to lift the wheels.

*<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> *<sup>y</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> � �*<sup>T</sup>* is the first position of *<sup>p</sup>*4. When the robot manipulation time is *<sup>t</sup>* <sup>¼</sup> 0, the tilt of *Robot A* is zero (*ϕ*<sup>23</sup> <sup>¼</sup> 0). Then, *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> moves to *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* <sup>¼</sup> *<sup>x</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm <sup>y</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* � �*<sup>T</sup>* after *Robot A* moves at a constant velocity of *vA* in *<sup>t</sup>* s, and *<sup>Δ</sup><sup>x</sup>* is the distance between *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> and *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* .

When the operator begins to teleoperate the robots ð Þ *t* ¼ 0 , the position of the axis of the rear wheels of *Robot A*, *x*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> , is obtained from (5), as follows:

$$\mathcal{X}\_{\Phi\_{\left(t=0\right)}} = l\_{LA} + l\_B + l\_{LB} + d \cos \phi\_{2\ \left(t=0\right)} \tag{19}$$

Here,

and

*<sup>t</sup>* <sup>¼</sup> <sup>1</sup> *vA*

**Figure 11.**

**91**

*e*<sup>2</sup> ¼ *hLA* cos *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* � *lLA* sin *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* � *hLB:* (26)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � cos <sup>2</sup>*ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* <sup>q</sup>

> *lA* <sup>2</sup> <sup>þ</sup> *<sup>e</sup>*<sup>1</sup>

> > 2

þ

q

*lA* <sup>2</sup> <sup>þ</sup> *<sup>e</sup>*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>d</sup>*<sup>2</sup> � ð Þ *hLA* � *hLB*

From (26)–(29), the relationships among the velocity of *Robot A* (*vA*), the height of the bottom of the front wheels (*y*6) (**Figure 11(a)**), and the manipulation time

In *stage 1*, the operators lift the bottom of the front wheels of *Robot A* above the

When the horizontal position of the center of gravity is at the contact position between the rear wheels and the road (**Figure 11(b)**), the height of the bottom of the front wheels is *y*<sup>6</sup> ¼ 0.1526 m. On the other hand, *Robot B* does not tip over backward during *stage 3* because the link touches its body when the incline of *Robot B* grows large (**Figure 13**), so stopping further inclination. The operators are able to

*Height range of the front wheels of* Robot A *required to climb a step: (a) situation in which the bottom of the front wheels is equal to the step height and (b) situation in which the horizontal position of the center of gravity*

**4.2 Range of front wheel height within which operators must teleoperate**

step height in order to place the wheels on the step (**Figure 11(a)**). Next, the operators maintain a suitable inclination for *Robot A* in order to prevent it from tipping over backward (**Figure 11(b)**). When *Robot B* stops and *Robot A* continues to move forward, *Robot A* tips over backward (**Figure 12**). Hence, the operators must maintain the height of the front wheels within the range between the step

<sup>þ</sup> *hLA* sin *<sup>ϕ</sup>*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>d</sup>*<sup>2</sup> � *<sup>e</sup>*<sup>2</sup> 2

<sup>2</sup> � *e*<sup>1</sup> � *y*<sup>6</sup> � �<sup>2</sup> <sup>q</sup>

<sup>2</sup> (28)

(27)

*:*

(29)

From (17) and (18), we obtain sin *ϕ*2ð Þ *<sup>t</sup>*¼*tm* and cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼*tm* , as follows:

*Cooperative Step Climbing Using Connected Wheeled Robots and Evaluation of Remote…*

� � <sup>þ</sup> *lA*

sin *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* ¼

q

cos *<sup>ϕ</sup>*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* <sup>¼</sup> *<sup>e</sup>*<sup>1</sup> � *<sup>e</sup>*<sup>1</sup> � *<sup>y</sup>*<sup>6</sup>

Substituting (22) and (23) for (21), we obtain:

for lifting the front wheels (*t* ¼ *tm*) are clarified.

height and the height at which *Robot A* tips over backward.

teleoperate *Robot B* without tipping the robot over backward.

*is at the contact position between the rear wheels and the road.*

*lLA* cos *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* � 1 � � �

*DOI: http://dx.doi.org/10.5772/intechopen.90162*

**for step climbing**

Here, cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> is the value of cos *ϕ*<sup>2</sup> at *t* ¼ 0 s.

After manipulating the robots for *t* ¼ *tm*, the position of the axis of the rear wheels of *Robot A* (*x*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* ) is given by

$$\mathbf{x}\_{4\ (t=tm)} = l\_{LA}\cos\phi\_{23\ (t=tm)} + h\_{LA}\sin\phi\_{23\ (t=tm)} + l\_B + l\_{LB} + d\cos\phi\_{2\ (t=tm)}\tag{20}$$

Here, cos *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* is the value of cos *ϕ*<sup>23</sup> at *t* ¼ *tm*.

Based on (19) and (20), the movement distance of *Robot A* while lifting the front wheels, *Δx* (**Figure 10**), is given as

$$\begin{split} \Delta \mathbf{x} &= \nu\_{A} \mathbf{t} = \mathbf{x}\_{4 \ (t=tm)} - \boldsymbol{\mathcal{X}}\_{4 \ (t=0)} \\ &= l\_{LA} \left\{ \cos \phi\_{23(t=tm)} - \mathbf{1} \right\} + h\_{LA} \sin \phi\_{23(t=tm)} + d \left\{ \cos \phi\_{2 \ (t=tm)} - \cos \phi\_{2(t=0)} \right\} \end{split} \tag{21}$$

Here, *vA* is the constant velocity of *Robot A*. When the front wheels begin to be lifted (*t* ¼ 0, **Figure 10**), the incline of *Robot A* is zero (*ϕ*<sup>2</sup> þ *ϕ*<sup>3</sup> ¼ 0). In this case, from (9) and (10), we obtain:

$$\sin \phi\_{2\ (t=0)} = \frac{h\_{LA} - h\_{LB}}{d} \tag{22}$$

and

$$\cos\phi\_{2\ (t=0)} = \sqrt{\frac{d^2 - (h\_{LA} - h\_{LB})^2}{d}}\tag{23}$$

After manipulating the robots for a time *tm*, the front wheels start to lift. Then, from (9) and (10), we obtain the following:

$$\sin\phi\_{2(t=tm)} = \frac{h\_{LA}\cos\phi\_{23\ (t=tm)} - l\_{LA}\sin\phi\_{23\ (t=tm)} - h\_{LB}}{d} \tag{24}$$

and

$$\cos \phi\_{2\ (t=tm)} = \sqrt{\frac{d^2 - \mathbf{e}\_2^2}{d}}\tag{25}$$

*Cooperative Step Climbing Using Connected Wheeled Robots and Evaluation of Remote… DOI: http://dx.doi.org/10.5772/intechopen.90162*

Here,

Then, *p*<sup>4</sup> (the position of the axis of the rear wheels of *Robot A*) moves forward in lifting the front wheels of *Robot A* (**Figure 10**). Time *t* ¼ *tm* is the time for the

ulation time is *<sup>t</sup>* <sup>¼</sup> 0, the tilt of *Robot A* is zero (*ϕ*<sup>23</sup> <sup>¼</sup> 0). Then, *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> moves to

axis of the rear wheels of *Robot A*, *x*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> , is obtained from (5), as follows:

When the operator begins to teleoperate the robots ð Þ *t* ¼ 0 , the position of the

After manipulating the robots for *t* ¼ *tm*, the position of the axis of the rear

*x*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* ¼ *lLA* cos *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* þ *hLA* sin *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* þ *lB* þ *lLB* þ *d* cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼*tm* (20)

Based on (19) and (20), the movement distance of *Robot A* while lifting the front

Here, *vA* is the constant velocity of *Robot A*. When the front wheels begin to be lifted (*t* ¼ 0, **Figure 10**), the incline of *Robot A* is zero (*ϕ*<sup>2</sup> þ *ϕ*<sup>3</sup> ¼ 0). In this case,

sin *<sup>ϕ</sup>*<sup>2</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *hLA* � *hLB*

s

After manipulating the robots for a time *tm*, the front wheels start to lift. Then,

sin *<sup>ϕ</sup>*<sup>2</sup>ð Þ *<sup>t</sup>*¼*tm* <sup>¼</sup> *hLA* cos *<sup>ϕ</sup>*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* � *lLA* sin *<sup>ϕ</sup>*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* � *hLB*

cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼*tm* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>d</sup>*<sup>2</sup> � ð Þ *hLA* � *hLB*

*d*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>d</sup>*<sup>2</sup> � *<sup>e</sup>*<sup>2</sup> 2

s

*d*

2

cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> ¼

from (9) and (10), we obtain the following:

� �*<sup>T</sup>* is the first position of *<sup>p</sup>*4. When the robot manip-

� �*<sup>T</sup>* after *Robot A* moves at a constant velocity of *vA* in

*x*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> ¼ *lLA* þ *lB* þ *lLB* þ *d* cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> (19)

þ *hLA* sin *ϕ*<sup>23</sup>ð Þ *<sup>t</sup>*¼*tm* þ *d* cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼*tm* � cos *ϕ*<sup>2</sup>ð Þ *<sup>t</sup>*¼<sup>0</sup>

n o

*<sup>d</sup>* (22)

*<sup>d</sup>* (24)

(21)

(23)

(25)

operators to lift the wheels.

*<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* <sup>¼</sup> *<sup>x</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm <sup>y</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm*

*<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> *<sup>y</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup>

*Industrial Robotics - New Paradigms*

*<sup>t</sup>* s, and *<sup>Δ</sup><sup>x</sup>* is the distance between *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> and *<sup>p</sup>*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* .

Here, cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> is the value of cos *ϕ*<sup>2</sup> at *t* ¼ 0 s.

Here, cos *ϕ*<sup>23</sup> ð Þ *<sup>t</sup>*¼*tm* is the value of cos *ϕ*<sup>23</sup> at *t* ¼ *tm*.

wheels of *Robot A* (*x*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* ) is given by

wheels, *Δx* (**Figure 10**), is given as

*Δx* ¼ *vA t* ¼ *x*<sup>4</sup> ð Þ *<sup>t</sup>*¼*tm* � *x*<sup>4</sup> ð Þ *<sup>t</sup>*¼<sup>0</sup> ¼ *lLA* cos *ϕ*<sup>23</sup>ð Þ *<sup>t</sup>*¼*tm* � 1 n o

from (9) and (10), we obtain:

and

and

**90**

$$\mathcal{C}\_2 = h\_{LA} \cos \phi\_{23 \ (t=tm)} - l\_{LA} \sin \phi\_{23 \ (t=tm)} - h\_{LB}. \tag{26}$$

From (17) and (18), we obtain sin *ϕ*2ð Þ *<sup>t</sup>*¼*tm* and cos *ϕ*<sup>2</sup> ð Þ *<sup>t</sup>*¼*tm* , as follows:

$$
\sin \phi\_{23 \ (t=tm)} = \sqrt{1 - \cos^2 \phi\_{23 \ (t=tm)}} \tag{27}
$$

and

$$\cos\phi\_{23\ (t=tm)} = \frac{e\_1 \cdot \left(e\_1 - \chi\_6\right) + l\_A\sqrt{l\_A^2 + e\_1^2 - \left(e\_1 - \chi\_6\right)^2}}{l\_A^2 + e\_1^2} \tag{28}$$

Substituting (22) and (23) for (21), we obtain:

$$t = \frac{1}{v\_A} \left[ l\_{LA} \left( \cos \phi\_{23 \ (t = m)} - 1 \right) - \sqrt{d^2 - \left( h\_{LA} - h\_{LB} \right)^2} + \sqrt{d^2 - e\_2^2} + h\_{LA} \sin \phi\_{23 \ (t = m)} \right]. \tag{29}$$

From (26)–(29), the relationships among the velocity of *Robot A* (*vA*), the height of the bottom of the front wheels (*y*6) (**Figure 11(a)**), and the manipulation time for lifting the front wheels (*t* ¼ *tm*) are clarified.

### **4.2 Range of front wheel height within which operators must teleoperate for step climbing**

In *stage 1*, the operators lift the bottom of the front wheels of *Robot A* above the step height in order to place the wheels on the step (**Figure 11(a)**). Next, the operators maintain a suitable inclination for *Robot A* in order to prevent it from tipping over backward (**Figure 11(b)**). When *Robot B* stops and *Robot A* continues to move forward, *Robot A* tips over backward (**Figure 12**). Hence, the operators must maintain the height of the front wheels within the range between the step height and the height at which *Robot A* tips over backward.

When the horizontal position of the center of gravity is at the contact position between the rear wheels and the road (**Figure 11(b)**), the height of the bottom of the front wheels is *y*<sup>6</sup> ¼ 0.1526 m. On the other hand, *Robot B* does not tip over backward during *stage 3* because the link touches its body when the incline of *Robot B* grows large (**Figure 13**), so stopping further inclination. The operators are able to teleoperate *Robot B* without tipping the robot over backward.

**Figure 11.**

*Height range of the front wheels of* Robot A *required to climb a step: (a) situation in which the bottom of the front wheels is equal to the step height and (b) situation in which the horizontal position of the center of gravity is at the contact position between the rear wheels and the road.*

**Figure 12.** *Tipping over backward of* Robot A*.*

**Figure 13.** *Link touching the front part of* Robot B*, which acts to prevent incline of* Robot A*.*
