**2. Simple mechanical design**

The following questions arise: What would be the simplest walking robot design that can effectively overcome obstacles? What would be the minimal number of degrees of freedom for such a robot? Can a simple control system work when overcoming different types of obstacles? How can 3D printing technology bring additional advantages in the development of robots, based on minimalist

internal movements and algorithms are needed to sustain balance.

mental two-legged robots which can sustain static balance.

legs. The robot has 8 degrees of freedom, four for each leg.

primates and cattle (grazing animals).

two-layer micro actuators driven by hydrogels.

overcome obstacles including climbing stairs.

also important issues that have been studied in recent years.

reconfigurable robots.

**94**

The stability of a walking robot is a major issue, because it defines the conditions under which it will not lose balance. There are two types of stability - static and dynamic. Static walking means that the robot can be stopped at any moment during the gait cycle without losing balance. Dynamic walking means that additional

Two-legged robots usually have dynamic stability and a relatively large number of degrees of freedom [5–7]. They can go around or climb obstacles, but need a complex control system and consume a lot of energy. Their reliability is lower, due to the large number of electrical and mechanical components. There are experi-

Alternative design solutions with a minimum number of mechanical elements [8] and nature-inspired robots are being sought [9]. In [10] is presented an ultralight, inexpensive two-legged robot "SLIDER" with a design of the leg without a knee. This non-anthropomorphic design with straight legs reduces the weight of the legs significantly, while maintaining the same functionality as anthropomorphic

The four-legged 3D printed robot presented in [9] is 3D printed with PLA (polylactic acid). It has a simple design and can walk without any form of software or controller. The robot consists of a rectangular body and four legs, each with a degree of freedom that rotates and raises the leg. At the end of each of the legs is mounted a rubber foot to improve traction. Although there are only 4 degrees of freedom, the robot realizes a gait which is similar to the gaits used by walking

In [8] is presented a robot with one motor and several clutches. By sequential action of the clutches, the proposed robot can rotate in different directions and can walk. It can be combined with other identical modules to build more complex

Walking mechanisms that do not need motors are studied [11, 12]. However,

3D printing technology is used to create and test the qualities of prototypes of walking robots [9, 10, 13, 14]. Conventional materials such as PLA [9] and ABS (Acrylonitrile Butadiene Styrene) [13] are most commonly used. In [14] a methodology for 3D printing of hermetic soft drives with built-in air couplings is proposed. Two materials are used, hard and flexible, and printing is done with a printer with two extruders. Additive manufacturing is evolving and finds more and more applications in robotics. In [15] the main focus is on developing a methodology for creating a 3D printed, low-budget robotic arm with six degrees of freedom that can be used with an external artificial intelligence system. In [16] is used a custom 3D printer and CAD model of a structure for a specialized device, which consists of

Maintaining stability when moving [17, 18] and overcoming obstacles [2, 19] are

For these reasons, here it will be discussed the design of a new 3D printed model of a walking robot, based on a minimalist approach [20, 21]. Mies van der Rohe's motto "Less is more" reflects the approach to the robot's design. Using only two motors, the robot can walk forward and backward, rotate 360 deg. around itself and

their passive movement is realized only on slopes and is difficult to control.

approach?

*Collaborative and Humanoid Robots*

It is well known that a robot needs at least 6 degrees of freedom to reach any point in its workspace with any orientation. 3 for changing the position and 3 for realizing the random orientation. Since the walking robot moves on a surface, it can be concluded that 3 degrees of freedom are enough - X, Y axis and orientation. After all there are examples of mobile robots with two motors that achieve satisfactory results 1. A new simple design of a two-motored robot, called "Big Foot", is suggested.

The robot's body is made up of a round base {1} and a platform {2} in which all the main elements are located. The platform is mounted in the center of the circular base, and the two bodies can rotate relative to each other around the vertical axis R1 (see **Figure 1**). The movement around R1 is realized by means of a controlled motor {6}. The stator of this motor is fixed on the platform {2}, and the rotor is connected by means of a reducer to the base {1}. The motor {7} is located in the platform {2} and drives the shaft {8} by means of a gear mechanism. This shaft performs the second important rotation R2, which is perpendicular to R1. Two arms {3} are fixed to the shaft {8}, and two feet {4} are mounted at the ends of the arms. For proper walking, the feet {4} and the round base {1} need to move with a constant orientation with respect to each other. To achieve this, a gear mechanism {5} is used, which has a gear ratio of 1. It consists of 3 gears with the same module and number of teeth which are mounted in the arm {3} (Figur 1) The 3D printed model is powered by a rechargeable battery, and the control is carried out remotely via Bluetooth communication with a PC or a smartphone. Different variants of the control software are developed using sensors of different types. Video with the robots movements is available from: Video 1.

The key elements for walking are the body {2}, arms {3} and feet {4}. This is the basic structure of the robot. While walking, the body and feet remain parallel **Figure 2**.

Initially the body {2} is fixed (**Figure 2a**). The arm {3} is rotating and thanks to the gears z1, z2, z3 the feet are moving parallel to the fixed body before reaching the ground. Afterwards, the feet {4} are fixed (**Figure 2b**), the arm {3} is rotating and this time the body {2} is in motion, remaining parallel to the feet. The trajectories can be seen in the following videos: Video 2 and Video 3.

The trajectory of any edge point of gear z2 is interesting. The trajectory resembles a heart and is called the cardioid - a type of cycloid. It can be followed in the animation: Video 4.

The rotation mechanism is presented in **Figure 3**. Here the gear motor {6} works in a mode where the rotor is fixed and the body rotates as the stator operates. Thus the robot can rotate to any angle without any of the wires tangling up.

**Figure 1.** *Structure of the big foot robot and a picture of the 3D printed prototype.*

#### **Figure 2.**

*The mechanism for maintaining a parallel movement between the body and the feet. a) Fixed body {2}. b) Fixed feet {4}.*

Walking on flat terrain is accomplished by repeating between two phases:

Phase 1 – the two feet {4} are acting as supports. The motor {6} by means of the shaft {8} drives the arms {3}, which rotate around point A. The body of the robot is moved, where all its points move along the trajectories of arcs of a circle with radius *RAB* ¼ *AB* ¼ *L*<sup>3</sup> and angle *φ<sup>B</sup>* ¼ *αmax* � *αmin*. The body of the robot travels forward with one step S (**Figure 4**).

To simplify the theoretical model, it is assumed that the mass of all moving parts during this phase is concentrated at point C1. The coordinates of this point (mass center) are given in **Figure 4**. When designing the robot, it is aimed to keep the center of gravity C1 as low as possible. This increases the stability of the robot.

The horizontal movement of the robot's body is evaluated by:

$$X\_{c1} = X\_B = X\_A + L\_3 \cos\left(a\right),\tag{1}$$

Where YB is the vertical coordinate of point B, hc1 is the vertical distance to the center of mass at point C1. The height hc1 does not change during movement. The robot moves at low speed and therefore the inertial forces are not taken into

*G*<sup>1</sup> = *m*1*g* is the robot's body weight, and g is the Earth's gravitational

After differentiating (1) and (2) is obtained the velocity of the robot:

*Vx* <sup>¼</sup> *<sup>X</sup>*\_ *<sup>C</sup>*<sup>1</sup> <sup>¼</sup> *<sup>α</sup>*\_*L*<sup>3</sup> sin ð Þ¼ *<sup>α</sup> <sup>ω</sup>L*<sup>3</sup> sin ð Þ *<sup>α</sup> Vy* <sup>¼</sup> *<sup>Y</sup>*\_ *<sup>C</sup>*<sup>1</sup> ¼ �*α*\_*L*<sup>3</sup> cosð Þ¼� *<sup>α</sup> <sup>ω</sup>L*<sup>3</sup> cosð Þ *<sup>α</sup>*

*MA*<sup>1</sup> ¼ ½ � *L*<sup>3</sup> cosð Þþ *α d G*1, (3)

*MB*<sup>1</sup> ¼ *dG*1, (4)

(5)

account. The torques at points A and B are determined by:

*3D Printed Walking Robot Based on a Minimalist Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.97335*

 

acceleration.

**97**

**Figure 3.**

*Rotating mechanism.*

XA is the horizontal coordinate of point A with respect to a fixed coordinate system, and α = α(*t*) is the current angle of rotation of the unit {3} with length L3 with respect to the horizon. The vertical displacement of point C1 is determined by:

$$Y\_{c1} = Y\_B - h\_{c1} = Y\_A + L\_{\eth} \sin\left(a\right) - h\_{c1} \tag{2}$$

*3D Printed Walking Robot Based on a Minimalist Approach DOI: http://dx.doi.org/10.5772/intechopen.97335*

Walking on flat terrain is accomplished by repeating between two phases: Phase 1 – the two feet {4} are acting as supports. The motor {6} by means of the shaft {8} drives the arms {3}, which rotate around point A. The body of the robot is moved, where all its points move along the trajectories of arcs of a circle with radius *RAB* ¼ *AB* ¼ *L*<sup>3</sup> and angle *φ<sup>B</sup>* ¼ *αmax* � *αmin*. The body of the robot travels forward

*The mechanism for maintaining a parallel movement between the body and the feet. a) Fixed body {2}. b)*

To simplify the theoretical model, it is assumed that the mass of all moving parts during this phase is concentrated at point C1. The coordinates of this point (mass center) are given in **Figure 4**. When designing the robot, it is aimed to keep the center of gravity C1 as low as possible. This increases the stability of the robot.

XA is the horizontal coordinate of point A with respect to a fixed coordinate system, and α = α(*t*) is the current angle of rotation of the unit {3} with length L3 with respect to the horizon. The vertical displacement of point C1 is determined by:

*Xc*<sup>1</sup> ¼ *XB* ¼ *XA* þ *L*<sup>3</sup> cosð Þ *α* , (1)

*Yc*<sup>1</sup> ¼ *YB* � *hc*<sup>1</sup> ¼ *YA* þ *L*<sup>3</sup> sin ð Þ� *α hc*1*:* (2)

The horizontal movement of the robot's body is evaluated by:

with one step S (**Figure 4**).

*Collaborative and Humanoid Robots*

**Figure 2.**

**96**

*Fixed feet {4}.*

Where YB is the vertical coordinate of point B, hc1 is the vertical distance to the center of mass at point C1. The height hc1 does not change during movement. The robot moves at low speed and therefore the inertial forces are not taken into account. The torques at points A and B are determined by:

$$M\_{A1} = [L\_3 \cos \left(a\right) + d] \mathbf{G}\_1,\tag{3}$$

$$M\_{B1} = dG\_1,\tag{4}$$

*G*<sup>1</sup> = *m*1*g* is the robot's body weight, and g is the Earth's gravitational acceleration.

After differentiating (1) and (2) is obtained the velocity of the robot:

$$\begin{cases} V\_x = \dot{X}\_{C1} = \dot{a}L\_3 \sin \left( a \right) = aL\_3 \sin \left( a \right) \\ V\_y = \dot{Y}\_{C1} = -\dot{a}L\_3 \cos \left( a \right) = -aL\_3 \cos \left( a \right) \end{cases} \tag{5}$$

**Figure 4.** *Phase of support steps.*

Here ω is the angular velocity of the arms {3}. From (5) the magnitude and velocity of a point from the robot's body can be determined. This phase ends when the round base {1} reaches the ground. Then the body stops moving and a second phase beggins.

Phase 2 - The robot body is stationary, the arms {3} rotate about an axis at point B, the feet {4} are moving. The mass of the moving parts is less than the mass of the robot's body. It is assumed that it is concentrated at point B. During this phase, the robot does not move and therefore its speed is zero. The feet move progressively along a trajectory, which is an arc of a circle. The torques at points A and B are determined by:

$$M\_{A2} = \mathbf{0},\tag{6}$$

HR and BR are dimensions according to **Figure 5**. The coefficient usually assumes

In order for the robot to move, it is necessary for the body and its feet to reach the ground. This is only possible under certain conditions for the sizes L2, L3 and L4.

> *L*<sup>2</sup> ≤ *L*<sup>3</sup> þ *L*<sup>4</sup> *L*<sup>4</sup> ≤*L*<sup>2</sup> þ *L*<sup>3</sup>

Dimensions *L1* and *L5* are important for increasing the robot's stability, but their excessive increase reduces the maneuverability of the robot and increases its overall

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>3</sup> � ð Þ *L*<sup>2</sup> � *L*<sup>4</sup>

2

*hBmax* ¼ *L*<sup>3</sup> � *L*<sup>2</sup> þ *L*<sup>4</sup> (11)

*hSmax* ¼ *L*<sup>2</sup> þ *L*<sup>3</sup> � *L*<sup>4</sup> (12)

� � (13)

From **Figure 6** can be determined the step *S*, at which the robot moves

*L*2

During the phase of support feet (phase 1) the arm is rotated at an angle *φB*:

*S* 2ð Þ *L*<sup>2</sup> � *L*<sup>4</sup>

*φ<sup>B</sup>* ¼ 2*artan*

Accordingly, in phase 2, the arm {3} rotates at an angle φ<sup>s</sup> ¼ 2π � φB.

q

(9)

(10)

positive values less than one. Kro > 1 only for climbing and jumping robots.

� � � �

*S* ¼ 2

The conditions are set by the inequalities:

*3D Printed Walking Robot Based on a Minimalist Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.97335*

*Basic dimensions of the 3D printed prototype.*

Maximum lift height of the body *hBmax*

Maximum lift height of the feet *hSmax*

dimensions.

**99**

**Figure 5.**

$$M\_{B2} = L\_3 G\_2 \cos\left(a\right),\tag{7}$$

G2 = *m*2*g* is the mass of the moving elements in this phase. The loading in the shafts is cyclic, with shaft A being more loaded (see formulas (3), (4), (6) and (7). During the transition from phase 1 to phase 2 and vice versa, shock loads occur in the construction of the robot, which are not taken into account.

#### **3. Determining the basic dimensions**

When walking on a relatively flat ground the robot switches between two phases where the contact area with the ground is large. Movement is balanced and reliable. In **Figure 5** are presented the basic dimensions of the 3D printed prototype. Five lengths (*L*1-*L*5; **Figure 3**) and their proportions determine the qualities of the robot and its capability to walk and overcome obstacles.

Obviously, the larger the model, the higher the obstacles that it can overcome. Therefore, the height of the obstacle ho should be compared with the height—*HR* and the length—*BR* of the robot. Thus, different designs of one robot and even a variety of different robots can be objectively compared.

A dimensionless coefficient is suggested with the help of which the scale of a robot and an obstacle can be compared:

$$K\_m = \frac{h\_o}{\sqrt{H\_R B\_R}}.\tag{8}$$

**Figure 5.** *Basic dimensions of the 3D printed prototype.*

Here ω is the angular velocity of the arms {3}. From (5) the magnitude and velocity of a point from the robot's body can be determined. This phase ends when the round base {1} reaches the ground. Then the body stops moving and a second

Phase 2 - The robot body is stationary, the arms {3} rotate about an axis at point B, the feet {4} are moving. The mass of the moving parts is less than the mass of the robot's body. It is assumed that it is concentrated at point B. During this phase, the robot does not move and therefore its speed is zero. The feet move progressively along a trajectory, which is an arc of a circle. The torques at points A and B are

G2 = *m*2*g* is the mass of the moving elements in this phase. The loading in the shafts is cyclic, with shaft A being more loaded (see formulas (3), (4), (6) and (7). During the transition from phase 1 to phase 2 and vice versa, shock loads occur in

When walking on a relatively flat ground the robot switches between two phases where the contact area with the ground is large. Movement is balanced and reliable. In **Figure 5** are presented the basic dimensions of the 3D printed prototype. Five lengths (*L*1-*L*5; **Figure 3**) and their proportions determine the qualities of the robot

Obviously, the larger the model, the higher the obstacles that it can overcome. Therefore, the height of the obstacle ho should be compared with the height—*HR* and the length—*BR* of the robot. Thus, different designs of one robot and even a

A dimensionless coefficient is suggested with the help of which the scale of a

*Kro* <sup>¼</sup> *ho* ffiffiffiffiffiffiffiffiffiffiffiffi *HRBR*

the construction of the robot, which are not taken into account.

**3. Determining the basic dimensions**

robot and an obstacle can be compared:

**98**

and its capability to walk and overcome obstacles.

variety of different robots can be objectively compared.

*MA*<sup>2</sup> ¼ 0, (6)

p *:* (8)

*MB*<sup>2</sup> ¼ *L*3*G*<sup>2</sup> cosð Þ *α* , (7)

phase beggins.

*Phase of support steps.*

*Collaborative and Humanoid Robots*

**Figure 4.**

determined by:

HR and BR are dimensions according to **Figure 5**. The coefficient usually assumes positive values less than one. Kro > 1 only for climbing and jumping robots.

In order for the robot to move, it is necessary for the body and its feet to reach the ground. This is only possible under certain conditions for the sizes L2, L3 and L4. The conditions are set by the inequalities:

$$\begin{cases} L\_2 \le L\_3 + L\_4\\ L\_4 \le L\_2 + L\_3 \end{cases} \tag{9}$$

Dimensions *L1* and *L5* are important for increasing the robot's stability, but their excessive increase reduces the maneuverability of the robot and increases its overall dimensions.

From **Figure 6** can be determined the step *S*, at which the robot moves

$$S = 2\sqrt{L\_3^2 - \left(L\_2 - L\_4\right)^2} \tag{10}$$

Maximum lift height of the body *hBmax*

$$h\_{B\text{max}} = L\_3 - L\_2 + L\_4 \tag{11}$$

Maximum lift height of the feet *hSmax*

$$h\_{\text{Smax}} = L\_2 + L\_3 - L\_4 \tag{12}$$

During the phase of support feet (phase 1) the arm is rotated at an angle *φB*:

$$\varphi\_B = 2 \arctan\left(\frac{\mathcal{S}}{2(L\_2 - L\_4)}\right) \tag{13}$$

Accordingly, in phase 2, the arm {3} rotates at an angle φ<sup>s</sup> ¼ 2π � φB.

**Figure 6.** *Scheme for determining the geometric parameters for walking on flat terrain.*

When attacking an obstacle with the robot's body, the height of the obstacle h0 must be less than the maximum possible lifting of the robot body (h0 < hBmax). When attacking with the feet {4} the maximum height of the obstacle h0 is determined in a similar way ℎ*o*<ℎSmax.

After differentiating formula (10) with respect to L2 and knowing the values of lengths *L3* and *L4*, is obtained

$$\frac{d\mathcal{S}}{dL\_2} = -\frac{2(L\_2 - L\_4)}{\sqrt{L\_3^2 - L\_2^2 + 2L\_4L\_2 + L\_4^2}}, \quad L\_3^2 - L\_2^2 + 2L\_4L\_2 + L\_4^2 > 0\tag{14}$$

which shows that when *L*<sup>2</sup> = *L*4, the function has an extreme, in this case it is a maximum (**Figure 7**).

In the specific example when the length of the link L2 = 17 [mm] the robot will move with maximum step S and as fast as possible under equal other conditions. In this case, the body and feet of the robot are raised to the same height hBmax = hSmax. The driving mechanisms and the battery are located in the body of the robot, therefore the displaced masses in the two phases differ significantly. From the point of view of energy saving, it is more profitable to lift the body less, but this in turn leads to a reduction in velocity of the robot. An approach is applied in which the potential energy in the two phases of movement on flat terrain is equated.

The energy needed to lift the body during phase 1 is:

$$\mathbf{E\_{p1}} = \mathbf{m\_1}\mathbf{g}\mathbf{h\_{Bmax}}\tag{15}$$

*m*1*hBmax* ¼ *m*2*hSmax hBmax* þ *hSmax* ¼ 2*L*<sup>3</sup>

From (18), the height at which the body is lifted when the maximum potential

ð Þ *m*<sup>1</sup> þ *m*<sup>2</sup>

*hBmax* <sup>¼</sup> <sup>2</sup>*L*3*m*<sup>2</sup>

The weights of the 3D printed prototype are distributed in the two masses, respectively m1 = 245 [g] and m2 = 30 [g]. The maximum lifting height of the body hBmax = 12[mm] is obtained. At L3 = 55 [mm] all parameters of the prototype are determined. These proportions of the lengths of the links not only improve the loading of the links and improve the distribution of energy in the two phases of movement on flat terrain, but also have a positive effect on overcoming high

When overcoming obstacles with height *hb < h0 < hS*, there are two ways to attack the obstacle: with the body (the round base {1} **Figure 1**) or with the feet. If the height of the obstacle h0 is greater than the maximum lift of the body hBmax, the robot cannot climb on it during phase 1. In practice, it turns out that the

(18)

(19)

 

energies, for the two phases are equalized, is as follows

*3D Printed Walking Robot Based on a Minimalist Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.97335*

**4. Passive adaptation to obstacles**

*Graph of the function for changing the step S(L2).*

obstacles.

**101**

**Figure 7.**

The energy that the motor delivers in order to move the feet during phase 2 is:

$$E\_{p2} = m\_2 g h\_{\text{Smax}} \tag{16}$$

The equalized energies are as follows

$$E\_{p1} = E\_{p2} \to m\_1 h\_{Bmax} = m\_2 h\_{Smax} \tag{17}$$

From (17) and geometrical considerations from **Figure 6**, the following system is obtained

*3D Printed Walking Robot Based on a Minimalist Approach DOI: http://dx.doi.org/10.5772/intechopen.97335*

**Figure 7.** *Graph of the function for changing the step S(L2).*

$$\begin{cases} m\_1 h\_{B\text{max}} = m\_2 h\_{S\text{max}} \\ h\_{B\text{max}} + h\_{S\text{max}} = 2L\_3 \end{cases} \tag{18}$$

From (18), the height at which the body is lifted when the maximum potential energies, for the two phases are equalized, is as follows

$$h\_{B\text{max}} = \frac{2L\_3 m\_2}{(m\_1 + m\_2)}\tag{19}$$

The weights of the 3D printed prototype are distributed in the two masses, respectively m1 = 245 [g] and m2 = 30 [g]. The maximum lifting height of the body hBmax = 12[mm] is obtained. At L3 = 55 [mm] all parameters of the prototype are determined. These proportions of the lengths of the links not only improve the loading of the links and improve the distribution of energy in the two phases of movement on flat terrain, but also have a positive effect on overcoming high obstacles.

#### **4. Passive adaptation to obstacles**

When overcoming obstacles with height *hb < h0 < hS*, there are two ways to attack the obstacle: with the body (the round base {1} **Figure 1**) or with the feet.

If the height of the obstacle h0 is greater than the maximum lift of the body hBmax, the robot cannot climb on it during phase 1. In practice, it turns out that the

When attacking an obstacle with the robot's body, the height of the obstacle h0 must be less than the maximum possible lifting of the robot body (h0 < hBmax). When attacking with the feet {4} the maximum height of the obstacle h0 is

After differentiating formula (10) with respect to L2 and knowing the values of

<sup>3</sup> � *<sup>L</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*L*4*L*<sup>2</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup>

Ep1 ¼ m1ghBmax (15)

*Ep*<sup>2</sup> ¼ *m*2*ghSmax* (16)

*Ep*<sup>1</sup> ¼ *Ep*<sup>2</sup> ! *m*1*hBmax* ¼ *m*2*hSmax* (17)

<sup>4</sup> > 0 (14)

4

which shows that when *L*<sup>2</sup> = *L*4, the function has an extreme, in this case it is a

In the specific example when the length of the link L2 = 17 [mm] the robot will move with maximum step S and as fast as possible under equal other conditions. In this case, the body and feet of the robot are raised to the same height hBmax = hSmax. The driving mechanisms and the battery are located in the body of the robot, therefore the displaced masses in the two phases differ significantly. From the point of view of energy saving, it is more profitable to lift the body less, but this in turn leads to a reduction in velocity of the robot. An approach is applied in which the potential energy in the two phases of movement on flat terrain is equated.

The energy that the motor delivers in order to move the feet during phase 2 is:

From (17) and geometrical considerations from **Figure 6**, the following system is

determined in a similar way ℎ*o*<ℎSmax.

*L*2 <sup>3</sup> � *<sup>L</sup>*<sup>2</sup>

¼ � <sup>2</sup>ð Þ *<sup>L</sup>*<sup>2</sup> � *<sup>L</sup>*<sup>4</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*Scheme for determining the geometric parameters for walking on flat terrain.*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*L*4*L*<sup>2</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup>

<sup>q</sup> , *<sup>L</sup>*<sup>2</sup>

The energy needed to lift the body during phase 1 is:

The equalized energies are as follows

lengths *L3* and *L4*, is obtained

*Collaborative and Humanoid Robots*

*dS dL*<sup>2</sup>

**Figure 6.**

maximum (**Figure 7**).

obtained

**100**

robot can adapt to the obstacle and climb it by attacking it with the feet. It does not need special sensors and control algorithms. This process is illustrated in **Figure 8**. The robot body (round base) collides with the vertical section of the obstacle. Then there is a sliding of the feet on the horizontal terrain and the body is sliding on the vertical obstacle, the arm {3} performs a planer movement. It can be determined the instantaneous center of velocities of the arm {3} by taking into account the motion of points A and B from it. With respect to the absolute coordinate system, the instantaneous center of velocities of the link AB has coordinates:

$$\begin{cases} X\_Q = X\_B - L\_3 \sin\left(a\right) \\ Y\_Q = L\_3 \cos\left(a\right) \end{cases} \tag{20}$$

*X*0 *<sup>Q</sup>* � *<sup>L</sup>*<sup>3</sup> 2 <sup>2</sup>

*3D Printed Walking Robot Based on a Minimalist Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.97335*

described passive adaptation is available from Video 5.

jumps to point *B*<sup>0</sup>

**5. Overcoming obstacles**

• Overcoming the obstacle

• Rolling over of the robot

cannot climb.

**Figure 9.**

**103**

*Five consecutive stages when climbing an obstacle.*

þ *Y*<sup>0</sup> *Q* <sup>2</sup>

Trajectory (*TMCa*) is an arc from the red circle, which is described according to the absolute coordinate system *OXY*. Trajectory (*TMCr*) is an arc from the blue circle, which is described with respect to the relative coordinate system, connected with the moving arm AB. In this situation the arms perform a planar movement and the body and feet of the robot are moving along X and Y axes respectively. When the body touches the ground, the instantaneous velocity center switches again and

From the reasoning made so far, it can be seen that depending on the height of the obstacle, it is possible to overcome it when attacking with the body or to adapt to it and attack it with the feet. If the height of the obstacle h0 is less than the maximum height reached by the feet hSmax, several scenarios are possible:

• Repeated sliding of the robot's feet and body on the obstacle, during which it

1.The feet are in contact with the obstacle and the round base with the ground. Because of the rotation of the arm {3} sliding starts between the base and the

**Figure 9** illustrates 5 stages when climbing an obstacle which differ in the

elements of contact between the robot, the obstacle and the terrain.

<sup>¼</sup> *<sup>L</sup>*<sup>3</sup> 2 <sup>2</sup>

1. The feet begin to rotate and attack the obstacle. Video of the

(23)

In this situation, the instantaneous velocity center of the arm jumps from point *A0* to point *Q1* and starts to move along an arc of a circle (**Figure 8**). The circle has radius *L3* and center [�*XB*,0] and its equation excluding the angle α, is derived from (20):

$$\left(\mathbf{X}\_{Q} + \mathbf{X}\_{B}\right)^{2} + \mathbf{Y}\_{Q}^{2} = L\_{3}^{2} \tag{21}$$

The relative instantaneous velocity center with respect to the coordinate system [*A*0,X<sup>0</sup> ,*Y*<sup>0</sup> ], and connected to the arm AB, is defined by the system of equations

$$\begin{array}{c|c} X\_Q' = L\_3 \sin^2(a) & \rightarrow \begin{vmatrix} X\_Q' = \frac{L\_3}{2} (\cos(2a) + 1) \\ \end{vmatrix} \\ Y\_Q' = L\_3 \sin(a) \cos(a) & \rightarrow \begin{vmatrix} X\_Q' = \frac{L\_3}{2} \sin(2a) \end{vmatrix} \end{array} \tag{22}$$

The relative trajectory of the instantaneous velocity center is also an arc of a circle, and its equation is derived from (22) after excluding α:

#### **Figure 8.**

*Instantaneous velocity center and adaptive movements in case of collision between the robot's body and the obstacle.*

*3D Printed Walking Robot Based on a Minimalist Approach DOI: http://dx.doi.org/10.5772/intechopen.97335*

$$\left(X\_Q' - \frac{L\_3}{2}\right)^2 + \left(Y\_Q'\right)^2 = \left(\frac{L\_3}{2}\right)^2\tag{23}$$

Trajectory (*TMCa*) is an arc from the red circle, which is described according to the absolute coordinate system *OXY*. Trajectory (*TMCr*) is an arc from the blue circle, which is described with respect to the relative coordinate system, connected with the moving arm AB. In this situation the arms perform a planar movement and the body and feet of the robot are moving along X and Y axes respectively. When the body touches the ground, the instantaneous velocity center switches again and jumps to point *B*<sup>0</sup> 1. The feet begin to rotate and attack the obstacle. Video of the described passive adaptation is available from Video 5.
