Keywords

1 Introduction

Robots are kinds of machine that rely on their own power and control ability to automatically realize various functions. Their mission is to assist or replace human to complete assigned works. Nowadays, robots are widely used in considerable works, such as automobile manufacturing [1], machinery processing [2, 3], palletizing [4, 5], automatic spraying [6], and heavy collaborative lifting [7,8,9]. Robotic joint is a basic component of the robot body. In practice, it should switch its motion frequently between active and passive states, which easily produces vibration during the motion to cause mechanical damage to joints and reduce the motion control accuracy [10, 11]. Consequently, robotic joints are required to have damping ability in the passive motion state. Traditional method is to access elastic elements, like dampers or springs, between the motor and shaft to absorb shock and vibration. However, this method is not adaptive to the damping requirements under various conditions, and also it is unable to realize semi-active vibration control.

The emergence of intelligent materials greatly accelerates the development of novel engineering equipment. As a new smart material, magnetorheological fluids (MRFs) have the ability to rapidly and reversibly change their rheological behaviors between Newtonian fluids and quasi-solid structures with/without a magnetic field [12]. The phenomenon is generally known as the magnetorheological (MR) effect, which is characterized by smooth operation, rapid response and easy control [13, 14]. Thus, it has been extensively integrated in many mechanical devices, such as dampers [15], brakes [16, 17], clutches [18], vibration absorbers [19], robotics applications [20], and haptic devices [21].

Owing to the outstanding features of MRFs, they possess broad application prospects in the robotic advancement. Among the examples are assistive knee braces, exoskeletons and robot grippers. For instance, Chen and Liao [22] developed an MR actuator for assistive knee brace. Li et al. [23] proposed an MR brake for a prosthetic ankle joint to achieve a smooth walk for the user. Pettersson et al. [24] designed a MRFs-based robot gripper which had the ability to handle food products with varying shapes. Kikuchi et al. [25] utilized the MRFs clutch in the design of a leg-shaped robot for brain-injured patients.

The main focus of this study is on the design, modeling and analysis of a MRFs actuator for robotic joints. Figure 1 displays the application of the MRFs actuator in a flexible-joint robot arm. The actuator is installed between a servo motor and a reducer. Once vibration is generated due to the motion state switch of robotic joints, the MRFs actuator can effectively eliminate vibration to enable the joint to move smoothly owing to its soft transmission properties. In this paper, the configuration of the MRFs actuator is described firstly, followed by theoretical modeling of the magnetic circuit and the transmitted torque. Then, the structural model of the actuator is designed and presented. After these, the influences of working induction and speed difference on the total transmitted torque and controllable coefficient are numerically calculated. Finally, an electromagnetic simulation is carried out to verify the designed magnetic circuit of the actuator.

Fig. 1.
figure 1

Configuration of a flexible-joint robot arm utilizing the MRFs actuator

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2 Design and Modeling of a Plate-Shaped MRFs Actuator

Figure 2 shows the configuration of MRFs actuator. In the actuator, a coil is enclosed in the shell cavity to produce an electromagnetic field. The isolation ring made of non-ferromagnetic material is installed to reduce flux leakage. An air gap with a width of 1 mm is formed between shell and plate. The input plate and output plate are respectively engaged with the input shaft and output shaft through eight bolts. The MRFs are confined between input and output plates by two O-rings.

Fig. 2.
figure 2

Configuration of the MRFs actuator

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The magnetic flux forms a closed loop after applying a coil current, as depicted in Fig. 3. Since the rheological effect only occurs to the MRFs within the magnetic circuit region, the actual working area of the actuator refers to the annular region formed between r 1 and r 2.

Fig. 3.
figure 3

Equivalent reluctance of the circuit for the MRFs actuator

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Based on Ohm’s law, magnetic reluctance of each circuit part is calculated by

$$ R_{{{\text{m}}i}} = \frac{{L_{i} }}{{\mu_{i} S_{i} }} $$
(1)

where L is the length and S the cross-sectional area, μ is the magnetic permeability, and the subscript i is the serial number of each part.

Referring to Fig. 3, the magnetic flux forms a closed loop. Hence, the total magnetic reluctance of the circuit is derived as

$$ R_{\text{total}} = R_{\text{m1}} + 2\left( {R_{\text{m2}} + R_{\text{m3}} + R_{\text{m4}} } \right) + R_{\text{m5}} $$
(2)

where \( R_{\text{m1}} \), \( R_{\text{m2}} \), \( R_{\text{m3}} \), \( R_{\text{m4}} \), \( R_{\text{m5}} \) are magnetic reluctances of upper shell, left and right shells, air gap, input and output plates and MRFs, respectively.

The magnetomotive force NI is a product of the coil turns N and the coil current I, which is expressed as

$$ NI = \phi R_{\text{total}} $$
(3)

where \( \phi \) is the magnetic flux of the circuit.

For the MRFs actuator, there are generally two working conditions:

  1. (1)

    Off-field State. The MRFs exhibit Newton fluid state in the absence of a magnetic field. At this time, magnetic particles appear freely distribution state. Thus, a small torque is produced by the MRFs viscosity, which is known as viscous torque T v given below

$$ T_{\text{v}} = \frac{{\pi \eta \Delta \omega r_{2}^{4} }}{2h} $$
(4)

where \( \eta \) is the dynamic viscosity of the MRFs, \( \Delta \omega \) is the angular speed difference between input and output plates, \( r_{2} \) is the outer working radius, \( h \) is the distance between input and output plates (i.e., the working gap size).

  1. (2)

    On-field State. After applying a magnetic field, an obvious change in rheological behavior occurs to the MRFs, forming particle chains along the field direction. Then, a magnetic torque TB is generated which is given below

$$ T_{\text{B}} = \frac{{2\pi \tau_{\text{B}} }}{3}\left( {r_{2}^{3} - r_{1}^{3} } \right) $$
(5)

where \( \tau_{\text{B}} \) is the magnetic yield stress of the MRFs and \( r_{1} \) the inner working radius.

The magnetic yield stress \( \tau_{\text{B}} \) is a function of the magnetic flux intensity B and material parameters of MRFs [26].

$$ \tau_{\text{B}} = kB^{a} $$
(6)

Hence, the total transmitted torque T w is the sum of the magnetic torque T B and the viscous torque T v.

$$ T_{\text{w}} = \frac{{2\pi kB^{a} }}{3}\left( {r_{2}^{3} - r_{1}^{3} } \right) + \frac{{\pi \eta \Delta \omega r_{2}^{4} }}{2h} $$
(7)

Referring to design steps given in [27], structural model of MRFs actuator is designed in Fig. 4. Also, main parameters of the actuator are given in Table 1.

Fig. 4.
figure 4

Structural model of the MRFs actuator: (a) Exploded view; (b) Sectional view.

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Table 1. Main Parameters of the Proposed MRFs Actuator
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3 Performance Analysis of the MRFs Actuator

3.1 Torque Transmission and Controllable Performance

It is apparently in (7) that T w is influenced only by B and Δω once the basic parameters of MRFs and the actuator are given. The variation of total transmitted torque with working induction and speed difference is numerically calculated using Matlab 10.0®, as plotted in Fig. 5.

Fig. 5.
figure 5

Total transmitted torque versus working induction and angular speed difference

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It is shown in Fig. 5 that the total transmitted torque increases nearly exponentially with the working induction. In specific, when the speed difference is 100 rad/s, the torque increment is 10.3 N·m as the induction rises from 0 to 0.5 T. The phenomenon indicates that the torque is largely influenced by the induction. However, the speed difference simply affects the viscous torque which holds a small proportion of the total torque. Thus, it has a very small effect on the total torque, which is exactly the main reason for the MRFs actuator to possess good constant torque characteristics.

The controllable performance of the MRFs actuator can be described by the controllable coefficient λ, which is represented as the ration of T B and T w. Note that the controllability of the transmitted torque becomes better as the controllable coefficient increases.

$$ \lambda = \frac{{T_{\text{B}} }}{{T_{\text{w}} }} = \frac{{4hkB^{a} \left( {r_{2}^{3} - r_{1}^{3} } \right)}}{{4hkB^{a} \left( {r_{2}^{3} - r_{1}^{3} } \right) + 3\eta \Delta \omega r_{2}^{4} }} $$
(8)

Similarly, the variation of controllable coefficient with working induction and angular speed difference is obtained in Fig. 6. As the working induction increases, the controllable coefficient firstly rises rapidly, and then the increasing trend slows down gradually. In addition, there is no slip between the input and output plates once the speed difference is 0. At that time, no viscous torque is produced and the controllable coefficient maintains at a constant value of 1. With the increase of the speed difference, the controllable coefficient reduced slightly and the decreasing trend slows down with increasing working induction. On the whole, the working induction holds a strong impact on the controllable coefficient while the influence of speed difference is relatively slight.

Fig. 6.
figure 6

Controllable coefficient versus working induction and angular speed difference

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3.2 Electromagnetic Simulation of the MRFs Actuator

  1. (1)

    Finite Element Modeling: The structure of the MRFs actuator is two-dimensional axisymmetric, and the magnetic circuit and boundary conditions are all consistent in the tangential direction. Thus, the electromagnetic simulation can be considered as a two-dimension problem. According to the configuration parameters of the actuator, a PLANE13 element is used for finite element modeling with ANSYS 10.0®, as shown in Fig. 7.

    Fig. 7.
    figure 7

    Magnetic loop of the MRFs actuator

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Grid meshing is required after the finite element modeling. The two-dimensional mapped meshing is utilized for regular quadrangles and the boundary meshing for irregular polygons. Additionally, the local refinement is carried out to the working gaps after the gird meshing.

  1. (2)

    Material Property and Boundary Condition: Material properties of each component in the circuit should be endowed prior to the simulation. In the simulation, the relative permeabilities of the non-ferromagnetic materials, including stainless steel and air, are set to 1. The MRFs and the low-carbon steel belong to nonlinear ferromagnetic materials, whose relative permeabilities change with the magnetic field strength. Figure 8 shows the B-H curves of the nonlinear materials in the actuator.

    Fig. 8.
    figure 8

    B-H curves of the nonlinear materials in the MRFs actuator: (a) MRFs; (b) Low carbon steel (ISO C20e).

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Assuming that the magnetic flux lines are completely confined in the internal of the model, the model frame is set as a parallel flux boundary. Also, the coil is applied on the coil domain as a current density load.

Simulation Results and Discussions: Figs. 9 and 10 separately show the magnetic flux lines and magnetic flux intensity of the MRFs actuator at 5 A. The highest magnetic induction intensity appears in the junction of the upper shell and the right shell with its value of 1.81 T, which is lower than the saturation induction of ISO C20e. It proves that a certain margin is presented in the circuit design. Moreover, the magnetic flux lines are strictly limited in the circuit and they are also basically vertical to the plate surface. The effective flux lines account for more than 90% of the total flux lines. Unavoidably, a small amount of flux leakage is found at the junction of non-ferromagnetic and ferromagnetic materials. In general, the simulation results indicate that the circuit meets the design requirements.

Fig. 9.
figure 9

Magnetic flux lines of the MRFs actuator at 5A

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Fig. 10.
figure 10

Magnetic flux intensity of the MRFs actuator at 5A

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The working induction distribution along redial direction at 5A is plotted in Fig. 11. As can be seen, the working induction increases from 0.551T to 0.558T from the inner radius to the outer radius, a variation ratio of nearly 1.3%. The phenomenon shows the magnetic inductions are basically distributed evenly along the radial direction.

Fig. 11.
figure 11

Working induction distribution along radial direction at 5A

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The working induction largely influences the torque transmission ability of the actuator. It is calculated by averaging the induction at each node of the working gap. Figure 12 is the variation of average working induction with coil current. Also, the relationship between transmitted torque and coil current is plotted in Fig. 13. It is obvious that both the working induction and transmitted torque increases with the coil current. However, the variation trend is not a linear relationship due to the effect of material nonlinearity. In specific, the increasing trend slows down as the current increases. At 5A, the working induction reaches about 0.561T that is greater than the saturation induction of MRF-J01® (0.5T). The results demonstrate that the designed circuit fulfills the induction intensity requirement. In addition, the maximum transmitted torque reaches about 11.9 N·m for a coil current of 5A.

Fig. 12.
figure 12

Variation of average working induction with coil current

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Fig. 13.
figure 13

Relationship curve between transmitted torque and coil current

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4 Conclusions

This paper was concerned with the design, modeling and analysis of a MRFs actuator for application in the robotic joints to achieve semi-active vibration control during the motion state transformation. Configuration design, theoretical modeling and numerical simulation were performed to the MRFs actuator successively. The magnetic circuit and the transmitted torque in both off-field and on-field states were theoretically modeled. Numerical calculation results indicated that the working induction had a strong impact on both the total transmitted torque and controllable coefficient while the influences of speed difference were relatively slight. When the speed difference is 100 rad/s, the total torque increment is 408 N·m as the working induction increases from 0 to 0.5 T. In addition, the designed circuit was found to meet the requirement of induction uniformity through an electromagnetic simulation. For a coil current of 5A, the average working induction was about 0.561 T (greater than the saturation induction of the chosen MRFs). It demonstrated that the circuit fulfilled the induction intensity requirement. Also, the maximum transmitted torque reaches about 11.9 N·m.