Ever since the very creation of the word robot, people think that robots should look and act like humans. However, until recently, this has only been a fantasy. Making a true robot that can actually walk like a human, or remotely look like a human, has been trapped in the realm of science fiction movies and books. The recent amazing humanoid robotic development efforts have been conducted by large corporations and research universities with multi-million dollar budgets, though humanoid robots can actually be built at home by a common man. While there is no single correct definition of "robot" a typical robot will have several or possibly all of the following properties

It is artificially created.

It can sense its environment, and manipulate or interact with things in it.

It has some ability to make choices based on the environment, often using automatic control or a pre-programmed sequence.

It is programmable.

It moves with one or more axes of rotation or translation.

It makes dexterous coordinated movements.

It moves without direct human intervention.

It appears to have intent or agency.

The last property, the appearance of agency, is important when people are considering whether to call it a machine robot, or just a machine.

The hexapod is an insect inspired-robot which has six legs that permits navigation flexibility on various terrains. The principle benefit of this form of a walker is its balance. The character inspired the researchers and new progressive thoughts are available in thoughts but occasionally they are easy and effective, occasionally bulky and vital. One of the first walks machines turned into reality was created in 1870 by Russian Mathematician Chebyshev

Legged gadgets were used for at least a hundred years and are advanced to wheels in a few aspects: Legged locomotion has to be robotically superior to wheel or to tracked locomotion over a variety of soil situations and truly superior for crossing obstacles. Wheeled walkers are the simplest and most inexpensive, and also tracked walker are superb for moving, but they cannot cover almost all sorts of terrain. There are distinct sorts of legged taking walks robots. They are roughly divided into groups based on the range of legs they possess. Bipeds have two legs, quadrupeds four, hexapods six, and octopods have eight legs. Bipeds’ walker is dynamically strong but statically volatile, therefore, such robots are more difficult to balance, and dynamic balance can only be achieved through walking ^{3, 4}.^{ }^{}

Mechanical engineering is not more involved in robotics since mechatronics and robotics found a vast application in implementing the concept of running a robot model using servo motors and drives since it needs more amount of energy to run the robot model. The two six-legged walkers are linked by using link mechanism and by coupling two kinematic walker with separate motor for each walker. By using separate motor we can run each walker in the desired position like front and back, thereby we can able to control the walker to turn left and right motion. In this project, a 6-legs robotic walker was designed and fabricated ^{2, 4}.^{}

A large number of research papers have been studied on four-bar link mechanism. A review of related literature has been described. Hamid and Saeed

Kumar and Mohanty

Abdulkadar and Deshmukh

It would be difficult to compete with the efficiency of a wheel on smooth hard surfaces but as condition increases rolling friction, this linkage walking machines become more viable and wheels of similar size cannot handle obstacles. Based on the literature review, the researcher has designed and fabricated four-bar walking machine. This linkage mechanism is capable and enough to carry a battery and a small motor

In toy industries, robotic toys have got many applications and demand. It can also be used for military purposes. By placing bomb detectors in the machines we can easily detect the bomb without a risk to human life. It can be used as heavy tanker machines for carrying bombs as well as carrying other military goods. It can be used for space research where the wheeled machines have difficulties to navigate. It is also applicable in the goods industries for the short distance transportation of goods within the factory. The mountain roads or other difficulties where ordinary vehicles cannot be moved easily can be replaced by our four-leg mechanical spider. Heavy loads can be easily transported if we made this as a giant one. It has got the further applications for the study of linkage mechanism and kinematic motions. The geometry and conditions can be changed according to the application needs. This walking machine can travel on rough surfaces very easily, therefore, this machine can be used efficiently with rough surfaces where ordinary moving machine cannot travel

Nelson

The Japanese craftsman Hisashige Tanaka

Arinjay and Kho

Roy et al. ^{ 14}.

Legged mechanisms have a long history. In 1850, Chebyshev designed a four-legged machine with mechanically synchronized legs, based on straight-line linkages, ^{ 15}.

A purely mechanical solution is nice, and reliable, but is obviously limited to flat ground. The pioneers in terms of control were Franck and McGhee (1968),

Another important contribution comes from the MIT Humanoid Robotics group. R. Brooks

The Ambulatory Robotics Lab (1991-2003) of McGill University developed several robots with dynamic walking, under the direction of M. Buehler. The most famous is the Scout’s series (

Mobile robots are employed increasingly for various applications such as search and rescue missions, disaster relief operations, and surveillance. Where wheeled vehicles have difficulty in moving on uneven terrain such as rocky or sandy areas, legged animals are able to move smoothly and energy-efficient in various environments. In fact, wheeled transport devices require building and maintaining expensive level roads, which may not be possible in many environments. Therefore, researchers are developing legged robots mimicking the mechanical structure, moving method, and control system of the animals. As in the natural creatures, legged robots are developed with various numbers of legs, employing different software packages for simulations and analysis. Singh and Bera

Simulation, design, and control of six-legged robots are still a current research field, attracting many researchers in recent years ^{12, 19, 20, 21, 22}. While many unmanned robots are already used practically in the real world environment, further research is required for bringing six-legged robots in the real-world applications

A mechanical mechanism or linkage made of rigid links can be used to provide the required foot motion using rotary motors as the derivers. A single degree-of-freedom walking leg mechanism using planar Peaucellier-Lipkin type eight-link crank driven linkage has been proposed in

In most of the previous studies on the multi-legged robots, the legs orientation with respect to the system body is frontal, where legs are positioned perpendicular to the advancement direction, or circular, in which the legs are situated radially respect to the robot body. In this work, the proposed walking robot presents sagittal disposal, where the robot moves parallel to the legs plane. A minimally actuated six-legged walker with a single motor and four-link mechanism design is proposed for this purpose. To the best of authors’ knowledge, kinematics of such a system has not been analyzed before. In this context, first, the system mechanism and design are described. The kinematic equations are derived using a vector analysis method and some hypotheses on the interaction with the ground. The tripod walking system is simulated based on the analysis and the design parameters. Finally, the working prototype of the system is introduced. The experiments verified the walking ability and stability of the system.

Various mechanical linkages have been used as the power transport system in legged robots

Arrangement of six legs, as depicted in

Both of the driver shafts are connected to a single rotary motor using a pulley. From

Feasibility of walking can be shown experimentally or using simulation. Forward kinematics of the system represents the system motion based on a given input that is the motor speed in this case. The interaction of the robot with the ground needs to be considered and the body frame of the robot is not fixed. Suppose the points O, A, J, and B are attributed to the legs according to

u→1L1 | Vector from point FL1 to AL1 | u→1R2 | Vector from point FR2 to AR2 |

u→2L1 | Vector from point O to JL1 | u→2R2 | Vector from point O to JR2 |

u→3L1 | Vector from point BL to AL1 | u→1R2 | Vector from point BR to AR2 |

u→4L1 | Vector from point AL1 to JL1 | u→4R2 | Vector from point AR2 to JR2 |

u→5L1 | Vector from point O to BL | u→5R2 | Vector from point O to BR |

u→0 | Vector from point FL1 to O | u→s | Vector from point FL1 to FR2 |

The loop closure equation for the mechanism can be obtained using vector analysis as follows. Let point O be the origin of the O_xy frame of coordinates attached to the body. The configuration equations of the L1 and R2 legs as four link mechanisms is given as:

As the points, BR, BL, and O are connected to the frame we have

Note that each vector

where

When the driver link BA is rotated by

Equations (5) and (6) are solved using MATLAB to obtain new vectors with the given rotation of the driver or motor.

An additional equation is related to the constraint that foot R2 should be on the ground. The equation can be expressed as

Knowing the link vectors, the vectors

Then, knowing

As the solution gives the position of each point for any rotation of the shaft, the velocity and acceleration of each point can now be measured by differentiation. The robot overall velocity can be given by the following equation:

A vector analysis of the velocity and acceleration vectors reveals the unknown values with respect to the system input, which is the rotation of driver or motor shaft. As the legs move symmetrically, the results of one leg are simply repeated for other legs.

When the driver link rotates with an angular velocity,

Then, considering Figure 7, the normal and tangent components will be

The tangential component

Thus, knowing the

As represented in

Sketch line 1 normal to the driver link

Draw line 2 from the tip of

Assign vector

Depict line 4 at J and normal to OJ

Draw line 5 parallel to JA from the intersection of lines 4 and 3. Then

Draw line 6 connecting the tip of

Set line 7 from F normal to AF. Draw vector

Considering O_XY (attached to the robot body as mentioned before), draw line 8 parallel to link JA and draw vector

Assign

Note that the BC implies that the foot velocity, for the supportive leg, is zero. It is supposed that the swinging and supporting legs are switching in steps A and B.

STEP A: In step A, three feet including L1, L3, and R2 are on the ground and other feet move forward. The time to pass the trajectory depends on the speed of the motor, which is set to 15 RPM. Therefore, at time t=2 s step A is finished and the moving feet reach the ground. At the moment a mechanical impact will happen. In

STEP B: The start of step B is considered at the end of step A.

The torque T applies a force on the legs which is shown as

From equation:

We have:

The normal component of the force (normal to the link) is approximately

To calculate the strength of the material a constant force

Supposing quasi-static conditions the sum of the moments about the pins at point J is zero

Similarly, the sum of the moments about point F is zero, so

The reaction of pin support at the point F is

The reaction of the pin support at the point J

To check, the sum of the forces about the y axis is zero:

Considering the first span of the beam 0 ≤ x_{1} < 140, the bending moment M is M(x_{1}) = R_{J}*(x_{1})

The values of M at the edges is

M_{1}(0) = 5.40*(0) = 0 (N*mm)

M_{1}(140) = 5.40*(140) = 756.49 (N*mm)

For the second span of the beam 140 ≤ x_{2} < 228 we have M(x_{2}) = R_{J}*(x_{2}) - P_{1}*(x_{2} - 140)

The values of M at the edges of the span:

M_{2}(140) = 5.40*(140) - 14*(140 - 140) = 756.49(N*mm)

M_{2}(228) = 5.40*(228) - 14*(228 - 140) = 0 (N*mm)

The maximum bending stress is given by

In which:

M=the internal bending moment about the section’s neutral axis, calculated above as 756.49 (N.mm).

c=the perpendicular distance from the neutral axis to the farthest point on the section, approximately 5mm.

I=the moment of inertia of the section area about the neutral axis. The second moment of the area at the maximum M is I=85 mm^{4}

So

Note that yield strengths for Ductile Iron typically is 275 MPa. Therefore the safety factor is obtained as

The values measured based on the worst-case shows the legs stand the exerted forces.

In order to obtain numerical values, the driver link is supposed to be rotating with a constant angular velocity of 15 RPM provided by a velocity controlled motor.

The dynamic equation of the system during walking is not presented in this paper, as a model-based controller has not been planned. The equations may be described by the motion equations as well as impact mechanics for the interaction of the feet with the ground. The dynamic equations are achieved by deriving the governing equations of motion for the mechanism. Well-known methodologies are Newton-Euler and Euler-Lagrange method. The Newton-Euler method is based on the vector approach and consists of three translation equations and three rotation equations for each link. The equations are combined with constraint equations of the joints and loop closure. The Euler-Lagrange equation is generally written in the matrix form of:

As an approximation, the overall mass of the stationary parts can be supposed at point O, and half of the mass of each link can be considered as lumped masses at the ends of the links. The following masses can be defined

MJL1 | Lumped mass at point JL1 | 0.5*Mass of upper arm +0.5*Mass of forearm |

MAL1 | Lumped mass at point AL1 | 0.5*Mass of driver link |

MFL1 | Lumped mass at point FL1 | 0.5*Mass of forearm |

MO | Lumped mass at point O | Mass of fixed parts+0.5*Mass of upper arms and drivers |

As L1 and L3 legs move together, one can combine the mass of L3 in L1 (likewise R3 in R1). A lumped inertia I_{o} is considered to include inertial effects. The numerical values are as follows:

Dynamic effects are generally ignored at low speeds. With the solar panel, the robot can be used in remote lands, where short and occasional repositioning is required. Further investigations are in progress to use the robot in the real environment, various surfaces, etc.

The main focus of this paper was on the kinematics of a single actuator hexapod robot. A six-legged robot was designed and fabricated based on crank rocker mechanisms. The overall dimensions of the mechanism were selected so that the crank rocker function is ensured. Then, the six-legged robot was designed with connecting six of the mechanisms. The kinematic equations were derived using vector analysis and hypothesis on the boundary conditions representing the ground interaction with the feet. The robot gait analysis was performed with the velocity analysis and simulations, and the overall motion of the robot was presented. The simulation results showed that the robot was able to walk based on the described gait. The robot was fabricated and tested successfully. The robot is proposed for moving in remote environments where low-speed repositioning is required. As future work, it is suggested that the gait should be investigated experimentally for various surfaces with different friction properties.