The design of dual-axis electromagnetic actuators (DEA) are relied on the structure of electromagnetic oscillatory actuators (EOA) which vacillate between desired bound. Following the Fig. 1, a DEA comprises double perpendicular frames (one fixed frame and one rotated frame) which are aligned together via another free-moving frame⁽⁵⁾. This free-moving frame is appended on two axes so that it is able to revolve around these axes. Besides, each coil which are mounted to fixed frame or rotated frame has 820 turns with a resistance of 58 Ω. The max power to supply the coil can reach 6 V maximum. These dual energized coils will move a permanent magnet which is inserted inside the free-moving frame. Moreover, each side of the DEA comprises a built-in permanent magnet that place in contrary direction to the magnet of free-moving frame to converge and retain the magnetic field in the linear tendency as shown in Fig. 1. The Neodymium (ND35) is utilized for all permanent magnets due to the most powerful commercialized product. In addition, all frames are designed and conducted with ABS material to reduce the heat tolerances during printing process. The working principle of the proposed DEA is based on engendering Lorentz force among the permanent magnets and energized coils. This force is altered the direction by varying the current direction flown inside the coils and defined following left-hand rule. Consequently, free-moving frame is able to revolve around x-axis when supplying the current to the upper coil. In the same way, the energized coil of the rotated frame creates a torque in order to rotate its frame around the y-axis.

Fig. 1 
Structure of dual-axis electromagnetic actuator

In this study, we design the hexapod with 2-DOF leg is based on the structure of the DEA. The robot consisted of six DEA modules aligned to the main body of the robot that illustrated in the Fig. 2.

Fig. 2 
The overview of the hexapod robot using six DEAs

Each three parallel actuated modules are arranged symmetrically at both side of the body. The robot has 22 mm height, 52 mm width, 71 mm length and 22 g weight. The gaps between two narrow legs are 63 mm and 111 mm of each side and two symmetric sides, respectively. At the end of each rotated frame, a double rods are mounted to the end of rotated frame to lift the body from the ground. A rubber pad is fixed at the bottom of the double rods to maintain the stability and friction force between the leg and ground.

The forward kinematic model of the hexapod robot is determined using Denavit-Hartenberg (DH) rule. Four reference systems maintain path from the body frame to the foot, with last three of them relative the leg, as shown in Fig. 3.

Fig. 3 
Coordinate reference system of the body and legs of the hexapod robot

We call PjG is the j-th leg foot-tip position in the global reference system and j2P is the j-th foot-tip position in the foot coordinate system (second following reference system). The relation between the foot coordinate frame and the body frame can be obtained as follows:

Where, TGB refers to the transformation matrix between the body coordinate frame and the global coordinate frame, TjB0 refers to the transformation matrix between the j-th leg coordinate frame and body coordinate frame, Tj01 and Tj12 are the transformation matrix between the neighboring coordinates frames of the j-th leg and are given as:

Where, Cθ_ijand Sθ_ijdenote cosθ_ij and sinθ_ij, respectively; α_ij, a_ij, d_ij and θ_ij are the Denavit-hartenberg parameters with the specific values provided in the Table 1.

Following the path planning shown in Fig. 4, the motion of the robot is operated by six legs which conduct in a sequence to lift and move the body forward that called supporting stage and transferring stage, respectively.

Fig. 4 
Path planning of transferring stage and supporting stage

In our study, the tripod gait is implemented to the robot so leg groups are defined like as 1-4-5 and 2-3-6. Firstly, group 1-4-5 supports to hold and move body forward while group 2-3-6 steps up to transfer the next transition phase. Then, the subsequent phase is conducted by converting the stage of each group from the preceding phase. The motion of both phases are presented in Fig. 5. Finally, the robot maintains reverting stages of each group in a sequence to move the robot forward continuously.

Fig. 5 
Two stages of the tripod gait for the proposed hexapod robot

In this study, the tripod gait of micro hexapod robot is simulated to verify the walking ability of the proposed prototype. Firstly, a 3D CAD model of the robot is designed with actual parameters by utilizing CAD software. Then, the model is converted to xml-type model for importing to the Simscape Multibody physics engine from Matlab/Simulink. In detail, the process for converting a 3D CAD model to Simscape Multibody model is described in Fig. 6.

Fig. 6 
Flowchart of the process for converting a 3D CAD model to Simscape Multibody model

The result of the modeling for MHR consisted of three subsystems is described in Fig. 7. The Robot subsystem is the modeling of the MHR includes of the masses, inertial moments, physical dimension of body and legs. We also design the Floor subsystem and Contact-force subsystem to model the interaction among the legs and ground. Besides, a 3D mechanics explorer shown in Fig. 8 is enable to observe the motion of the robot intuitively.

Fig. 7 
Modeling of the micro hexapod robot using Simscape Multibody physics engine

Fig. 8 
3D mechanics explorer of the Simscape Multibody simulation

The gait simulation is implemented in this study to verify the motion planning of two stages: transferring and supporting for all legs the MHR presented in the previous chapter. The result of the tripod gait implemented by reverting the stages of each phase is shown in Fig. 9 It indicates that the MHR successfully generates the straight walking when applying the desired paths.

Fig. 9 
Simulation result of the tripod gait

In this paper, we also implement the tripod gait test for the physical prototype of the MHR. Following the experimental result presented in Fig. 10, the MHR completed 21 steps for 20 seconds with 15 mm displacement. Thus, the robot reaches 0.75 mm/s in speed and the conduct a distance of 0.71 mm each step. The results validate the performance of the proposed MHR using DEAs when conducting the tripod gait on the flat ground.

Fig. 10 
Experimental result of the tripod gait when micro hexapod robot after conducts 21 steps