Magnetorheological variable stiffness and damping flexible joint with cam working surface for industrial robot

The output torque of variable damping element is mainly affected by the magnetic induced yield stress of MR composite, that is, the magnetic field intensity in the effective damping gap and the rheological properties of MR composite directly affect the performance of variable damping element. If the complex working conditions of MR composite are considered, the difficulty of deriving the mechanical model of variable damping element will be greatly increased. Therefore, the derivation of variable damping principle in this paper is based on the following assumptions:

The MR composite is uniformly distributed in the damping gap, which is not affected by joint rotation and gravity, and the rheological properties of the material are not affected by temperature. Also, the direction and magnitude of the magnetic field in the damping gap are consistent, without considering the non-uniformity of the magnetic field in the gap and the influence of component rotation.

The schematic diagram of the variable damping element is shown in figure 3, and the working area is composed of the gap between the cylinder and the rotary piston. r₁ is the outer diameter of the rotary piston, and r₂ is the inner diameter of the cylinder.

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According to the previous study [29], the output torque M_c of the variable damping element with the working radius of r can be:

$$\begin{equation}{M_c} = 2\pi {r^2}L\tau = \frac{{4\pi {r_1}^2{r_2}^2L\tau \text{ln} \frac{{{r_2}}}{{{r_1}}}}}{{{r_2}^2 - {r_1}^2}} + \frac{{4\pi \eta {r_1}^2{r_2}^2L{\omega _r}}}{{{r_2}^2 - {r_1}^2}}.\end{equation} \tag{ 1 }$$

3.2.1. Design theory.

The original model and equivalent model of the variable stiffness element in the proposed MR flexible joint are shown in figure 4. In figures 4(a), m is the roller mass, x is the input displacement of the variable stiffness element, x₂ is the input displacement of the spring-damper parallel node, x₁ is the output displacement of the variable stiffness element, k₁ and k₂ are the tensile and compressive stiffness of the springs, and c is the damping coefficient of the linear MR damper. In figure 4(b), c_s and k_s are the equivalent damping and equivalent stiffness of the variable stiffness element, respectively. As shown in the figure, the spring (k₂) is connected in parallel with the linear MR damper (c) and then connected in series with the spring (k₁).

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According to Newton's second law of motion, the dynamic model of the original model can be obtained as follows:

$$\begin{equation}\left\{ {\begin{array}{*{20}{c}} {m\ddot x - {k_2}\left(x - {x_2}\right) - c\left(\dot x - {{\dot x}_2}\right) = 0} \\ {{k_1}\left({x_2} - {x_1}\right) - {k_2}\left(x - {x_2}\right) - c\left(\dot x - {{\dot x}_2}\right) = 0} \end{array}} \right..\end{equation} \tag{ 2 }$$

The equivalent dynamic model is:

$$\begin{equation}m\ddot x - {k_{\text{s}}}\left(x - {x_1}\right) - {c_{\text{s}}}\left(\dot x - {\dot x_1}\right) = 0\end{equation} \tag{ 3 }$$

where, $\dot x$ and $\ddot x$ are the input velocity and the input acceleration of the variable stiffness element, respectively. ${\dot x_2}$ is the input velocity of the spring-damper parallel node, and ${\dot x_1}$ is the output velocity of the variable stiffness element.

Applying Laplace transform to equations (2) and (3) to obtain:

$$\begin{equation}\frac{{x\left(s\right)}}{{{x_1}\left(s\right)}} = \frac{{{k_1} - \frac{{{k_1}^2\left({k_1} + {k_2}\right)}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{s^2}}} + \frac{{{k_1}^2c}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{s^2}}}s}}{{m{s^2} + {k_1} - \frac{{{k_1}^2\left({k_1} + {k_2}\right)}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{s^2}}} + \frac{{{k_1}^2c}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{s^2}}}s}}\end{equation} \tag{ 4 }$$

$$\begin{equation}\frac{{x\left(s\right)}}{{{x_1}\left(s\right)}} = \frac{{{k_{\text{s}}} + {c_{\text{s}}}s}}{{m{s^2} + {k_{\text{s}}} + {c_{\text{s}}}s}}.\end{equation} \tag{ 5 }$$

By bringing s= jω into equations (4) and (5), we get:

$$\begin{equation}\frac{{x\left(j\omega \right)}}{{{x_1}\left(j\omega \right)}} = \frac{{{k_1} - \frac{{{k_1}^2\left({k_1} + {k_2}\right)}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{\omega ^2}}} + \frac{{{k_1}^2c}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{\omega ^2}}}j\omega }}{{ - m{\omega ^2} + {k_1} - \frac{{{k_1}^2\left({k_1} + {k_2}\right)}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{\omega ^2}}} + \frac{{{k_1}^2c}}{{{{\left({k_1} + {k_2}\right)}^2} - {c^2}{\omega ^2}}}j\omega }}\end{equation} \tag{ 6 }$$

$$\begin{equation}\frac{{x\left(j\omega \right)}}{{{x_1}\left(j\omega \right)}} = \frac{{{k_{\text{s}}} + {c_{\text{s}}}j\omega }}{{m{s^2} + {k_{\text{s}}} + {c_{\text{s}}}j\omega }}.\end{equation} \tag{ 7 }$$

The original model and the equivalent model describe the same system, so the transfer function is consistent. Therefore, by comparing the terms in equations (6) and (7), it can be concluded that:

$$\begin{equation}{k_s} = {k_1} - \frac{{{k_1}^2\left({k_1} + {k_2}\right)}}{{{{\left({k_1} + {k_2}\right)}^2} + {c^2}{\omega ^2}}}.\end{equation} \tag{ 8 }$$

3.2.2. Variable stiffness model.

The spring pressure of the variable stiffness element is converted to the output torque of the joint through the cam. The joint rotation drives the cam rotation to force the roller to compress the spring, the radial stiffness is converted to the circumferential stiffness, and the output torque M_s of the element is the product of the spring reaction force on the cam surface and the distance from the action point to the joint rotation center. Figure 5 shows the simplified diagram of the output torque of the MR flexible joint variable stiffness element, and the relationship between the output torque and the structure is derived through force analysis.

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Due to the balanced force on the roller, according to Newton's second theorem, it can be obtained that:

$$\begin{equation}\left\{ {\begin{array}{*{20}{c}} {F\text{cos} \beta = {F_{\text{k}}}} \\ {{F_{\text{k}}} = {k_{\text{s}}}\Delta x} \end{array}} \right.\end{equation} \tag{ 9 }$$

where, F is the force exerted by the cam on the roller, β is the angle between the normal of the cam curve at the tangent point and the spring axis, F_k is the spring force, Δx is the compression displacement of spring.

Obtained joint output torque Ms:

$$\begin{equation}{M_{\text{s}}} = \rho \left(\theta \right)F\text{sin} \beta = {k_{\text{s}}}\Delta x\rho \left(\theta \right)\text{tan} \beta \end{equation} \tag{ 10 }$$

$$\begin{equation}\beta = \varphi - \theta \end{equation} \tag{ 11 }$$

where, θ is the rotation angle of the spring retaining rod, ϕ is the angle between the contour normal at the tangent point and the x-axis, ρ(θ) is the polar diameter (the expression of the contour curve in polar coordinates), which is the arm of force.

According to equations (10) and (11), it can be obtained:

$$\begin{align}{M_{\text{s}}} = {k_{\text{s}}}\Delta x\rho \left(\theta \right)\text{tan} \left(\alpha - \theta \right) = \frac{{{k_{\text{s}}}\left[ {\rho \left(\theta \right) - \rho \left(0\right)} \right]\rho \left(\theta \right)\left({k_f} - \text{tan} \theta \right)}}{{1 + {k_f}\text{tan} \theta }}\end{align} \tag{ 12 }$$

where, k_f is the slope of cam contour normal, k_f = tanϕ:

$$\begin{equation}{k_f} = - \frac{{{\text{d}}x}}{{{\text{d}}y}} = - \frac{{\frac{{{\text{d}}x}}{{{\text{d}}\theta }}}}{{\frac{{{\text{d}}y}}{{{\text{d}}\theta }}}}\end{equation} \tag{ 13 }$$

where, x and y are the expression of the cam curve in Cartesian coordinates.

Based on the in the above theoretical analysis, the total output torque M of the MR flexible joint can be obtained as follows:

$$\begin{equation}M = {M_{\text{c}}} + {M_{\text{s}}}.\end{equation} \tag{ 14 }$$

3.2.3. Design of the cam working surface.

From the above analysis, the design of the cam working surface involves the deformation of the spring and the size of the force arm. If the angle of the contour surface changes nonlinearly, the equivalent output stiffness of the MR flexible joint can also change nonlinearly, thus affecting the variable stiffness performance. Within a certain range, only fixed stiffness springs can be used, and various required stiffness changes can be obtained based on different cam working surfaces. Therefore, the design of cam contour surfaces is very important for the performance of MR flexible joint.

The energy conservation method was used to obtain the cam curve. If the torque M_s rotates the variable stiffness element so that its rotation angle is θ, the elastic potential energy E_p(θ) stored by the spring under pressure is:

$$\begin{equation}{E_{\text{p}}}\left(\theta \right) = \frac{1}{2}{k_{\text{s}}}\Delta {\rho ^2} = \frac{1}{2}{k_{\text{s}}}{\left[ {\rho \left(\theta \right) - \rho \left(0\right)} \right]^2}\end{equation} \tag{ 15 }$$

where ρ(0) is the pole diameter of the initial position.

In this case, the work W_p(θ) done by torque M_s is:

$$\begin{equation}{W_{\text{p}}}\left(\theta \right) = \int_0^\theta {{M_{\text{s}}}{\text{d}}\theta } .\end{equation} \tag{ 16 }$$

Without considering the energy loss caused by friction, etc., the work done by the torque is equal to the elastic potential energy stored by the system, and the combined equations (15) and (16) can be obtained:

$$\begin{equation}\frac{1}{2}{k_{\text{s}}}{\left[ {\rho \left(\theta \right) - \rho \left(0\right)} \right]^2} = \int_0^\theta {{M_{\text{s}}}{\text{d}}\theta } \end{equation} \tag{ 17 }$$

$$\begin{equation}\rho \left(\theta \right) = \rho \left(0\right) - \sqrt {\frac{2}{{{k_{\text{s}}}}}\int_0^\theta {{M_{\text{s}}}{\text{d}}\theta } } .\end{equation} \tag{ 18 }$$

From equation (17), the cam working surface can be determined based on the expected output torque function and spring stiffness, and the above derivation results in a theoretical working surface. To obtain the actual working surface, it is necessary to consider the influence of roller radius on the contact point. In the Cartesian coordinate system, the pitch curve is:

$$\begin{equation}\left\{ {\begin{array}{*{20}{c}} {{x_{\text{t}}} = \rho \left(\theta \right)\text{cos} \theta } \\ {{y_{\text{t}}} = \rho \left(\theta \right)\text{sin} \theta } \end{array}} \right.\end{equation} \tag{ 19 }$$

The actual cam curve of the cam shown in figure 6 can be obtained:

$$\begin{equation}\left\{ {\begin{array}{*{20}{c}} {{x_{\text{a}}} = \left(R - A\theta \right)\text{cos} \theta + \frac{r}{{\sqrt {1 + {k_f}^2} }}} \\ {{y_{\text{a}}} = \left(R - A\theta \right)\text{sin} \theta + \frac{{r{k_f}}}{{\sqrt {1 + {k_f}^2} }}} \end{array}} \right..\end{equation} \tag{ 20 }$$

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where, R is the radius of the cam base circle, ${k_f} = \frac{{\text{sin} \theta (R - A\theta ) + A\text{cos} \theta }}{{\text{cos} \theta (R - A\theta ) - A\text{sin} \theta }}$.

Combined with equation (12), can be obtained:

$$\begin{equation}{k_{\text{e}}} = \frac{{{\text{d}}{M_{\text{s}}}}}{{{\text{d}}\theta }} = {A^2}{k_{\text{s}}} = \frac{a}{{{k_{{\text{s}}0}}}}{k_{\text{s}}}\end{equation} \tag{ 21 }$$

where, k_e is the equivalent stiffness of the MR variable stiffness mechanism.

Equation (21) indicates that the equivalent stiffness of the MR flexible joint is related to the parameters of the cam working surface and the stiffness of the MR variable stiffness mechanism, and verifying the feasibility of the proposed variable stiffness structure.

3.2.4. Dynamic simulation of the variable stiffness element.

Adjusting the damping coefficient of the linear MR damper can change the output stiffness of the MR variable stiffness mechanism, and the equivalent output stiffness of the variable stiffness element can be controlled through the motion conversion by the cam mechanism. To verify the correctness of the above theoretical derivation, an equivalent multi body dynamic model of the variable stiffness element was established using ADAMS. Given different damping coefficients, the equivalent stiffness k_s of the MR variable stiffness mechanism, the torque M_s and the equivalent stiffness k_e of the variable stiffness element were simulated and calculated. ADAMS virtual prototype is shown in figure 7, its key parameters are shown in table 1, and simulation results are shown in figures 8–10.

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Table 1. Parameters of virtual prototype of variable stiffness element.

Symbols	Variables	Values
k1	Stiffness of the spring 1	20 (N mm−1)
k2	Stiffness of the spring 2	5 (N mm−1)
c	Damping coefficient of linear MR damper	0–3500 (Ns m−1)

As shown in figure 8, the output torque of the MR flexible joint without considering the variable damping element can be linearly output according to the design while the damping coefficient of the linear MR damper remains unchanged, indicating that the theoretical derivation of the cam working curve is correct. When the damping coefficient of the variable stiffness element is increased at the same rotation angle of the joint, the output torque increases, that is, the joint equivalent stiffness increases.

As shown in figure 9, the joint equivalent stiffness and the damping coefficient have a nonlinear relationship, and the influence of continued increase of damping on the output torque tends to be gentle when the damping coefficient exceeds a certain value. Analyzing figure 10, the variation range of equivalent stiffness k_s of the MR variable stiffness mechanism is between $\frac{{{k_1}{k_2}}}{{{k_1} + {k_2}}} \sim {k_1}$, which is consistent with the theoretical derivation conclusion. The above simulation results verify the structural feasibility of the proposed MR flexible joint with variable stiffness and damping.

Based on the overall structural design and principle analysis, the prototype of the MR flexible joint shown in figure 11 was fabricated.

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Figure 12 shows the equivalent stiffness testing platform of the MR variable stiffness mechanism. The testing system consists of a computer, a DC power supply, a force sensor, upper and lower fixtures, a ZQ tensile machine, and the MR variable stiffness mechanism. The DC power supply provides excitation current to the linear MR damper to adjust the output of the system. And the testing of the variable stiffness and damping performance of the MR flexible joint was completed by the mechanical performance testing platform shown in figure 13. It is mainly composed of MTS tensile machine, DC power supply, PC, and the MR flexible joint prototype. The MTS tensile machine was used for clamping the MR flexible joint, providing excitation at different frequencies and rotation angles, and collecting real-time data such as torque and angular displacement during the testing process. DC power supply was used to regulate the excitation current of the excitation coil and adjust the output performance of the MR flexible joint.

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The equivalent stiffness was tested using the test platform in figure 12. The compression displacement of the tensile machine was set as 11 mm, and the curve of equivalent output stiffness with displacement of the MR variable stiffness mechanism is shown in figure 13. By analyzing figure 14, the output force of the MR variable stiffness mechanism increases with the increase of current under the same compression conditions, indicating that increasing the excitation current can effectively improve the output stiffness of the structure, and its equivalent stiffness has a good controllability. The force versus displacement curve of the MR variable stiffness mechanism changes smoothly, which is basically consistent with the trend shown by Hooke's law. This indicates that the output stiffness characteristics of the MR variable stiffness mechanism are like those of the springs, and proving the good stability of the working performance of the MR variable stiffness mechanism.

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Figure 15 shows the results of equivalent stiffness of the MR variable stiffness mechanism with the current at different compression speeds, and the equivalent stiffness of the mechanism increases with the increase of current. At the initial part of the force-displacement curve, its slope is relatively large, approximately equal to the stiffness value k₁ of the spring, and this part will be extended with the increase of the excitation current. Further analysis that this is because the piston movement driven by the tensile machine needs to overcome the damping force of the linear MR damper, and the damping force increases with the increase of current. Before the piston movement, only the spring k₁ working, therefore the output stiffness at the initial part is relatively high, and the excitation current affects the length of this part. In general, the output force of the mechanism is smooth, and the performance of the MR variable stiffness mechanism is stable. In addition, increasing the compression speed can improve the output force, but to a lesser extent. The equivalent output stiffness is 2.57 N mm⁻¹ at the current of 0 A, the equivalent output stiffness is increased to 4.69 N mm⁻¹ when the current is increased to 2.5 A, and the dynamic range of stiffness is 1.82 (at 100 mm min⁻¹). Through the above analysis, it can be concluded that the mechanism has good controllability of stiffness.

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The MTS test system shown in figure 13 was used to test the variable stiffness performance of the MR flexible joint. After low-pass filtering and visualization processing, the obtained test data showed the performance curve as shown in figure 16. The variable stiffness testing curve has a large DC component, which is due to the parallel connection between the variable stiffness element and the variable damping element, and the torque bias is caused by the damping torque.

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In addition, there is a certain fluctuation in the initial stage of the joint rotation, which is due to the inevitable gap between the piston and the sleeve of the linear MR damper, although filled with MR composite. Therefore, the subsequent analysis was based on a stable stage of 10–20°, and the processing results are shown in figure 17.

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It can be seen from figure that under the same excitation frequency, increasing the excitation current of the variable stiffness element can effectively increase the output torque of the joint. In addition, the torque changes gently with the angle at the stage of 10–20°, indicating that the joint has a good torque ability. When the current is 0 A and 2.5 A, the equivalent output stiffness of the joint is 0.10 Nm rad⁻¹ and 0.22 Nm rad⁻¹ at 0.4 Hz, respectively. The dynamic range of the equivalent stiffness is about 2.2, indicating that the MR flexible joint has good controllability of the equivalent stiffness and variable stiffness ability, which meets the design requirements.

To verify the effectiveness of the variable damping element, the output performance of the MR flexible joint was tested at different frequencies, amplitudes, and currents.

From figure 18, when the rotational amplitude is the same, increasing the excitation current can effectively increase the output torque of the MR flexible joint. The area enclosed by the curve in the figure increases with the increase of current, indicating that its energy dissipation ability increases. That is, by changing the excitation current, the damping output characteristics of the MR flexible joint can be well controlled, reflecting its good controllability. When the MR flexible joint changes direction of rotation, its output torque sinks inward, mainly due to the presence of a cavity in the variable damping element, and the formation of an empty stroke between the piston and sleeve at the reversing position. When the current increases from 0 A to 1.5 A, the torque increases more with the current than when the current increases from 1.5 A to 2.5 A, showing that the magnetic field of the variable damping element is gradually saturated. If the current continues to increase, the output torque will increase, but the increase amplitude will further decrease until the magnetic field is completely saturated.

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Taking figure 18(b) as an example, when the current is 0 A and 2.5 A, the maximum output torque is 3.98 Nm and 9.11 Nm, respectively. According to the damping coefficient calculation formula:

$$\begin{equation}C = \frac{E}{{2{\pi ^2}f{A^2}}}\end{equation} \tag{ 22 }$$

where, E is the area enclosed by the torque versus angle displacement curve in a period, f is the excitation frequency, and A is the excitation amplitude.

It can be obtained that the damping coefficient at 0 A and 2.5 A are 2.60 Nm (rad s⁻¹)^‒1 and 8.56 Nm (rad s⁻¹)^‒1, respectively, and the dynamic range of damping coefficient is 3.29. And according to the data obtained from the experiment, the equivalent damping coefficient of the MR flexible joint changes with the current was fitted by polynomial, as shown in figure 19.

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In order to verify that the proposed MR flexible joint can effectively reduce the collision acceleration to protect workers and the robot body when the robot colliding with the external factors, the collision buffering test system of the robotic arm was setup as shown in figure 20. The test system mainly consists of DC power supply, PC, robotic arm, B&K acceleration sensor, dSPACE, sensor power amplifier and the MR flexible joint. By controlling the excitation current of the variable stiffness and damping elements of the MR flexible joint in the collision buffering system, the output characteristics of the joint prototype were adjusted. The B&K acceleration sensor was used to measure the vibration acceleration value during the collision process, and the sensor power amplifier can provide power to the acceleration sensor while amplifying electrical signals for data collection. PC and dSPACE together constitute a data acquisition system to collect and store the measurement data of the acceleration sensor and perform data post-processing.

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The left end cover of the MR flexible joint was fixedly connected to the end of the robotic arm through a flange, and the acceleration sensor was connected to the right end cover through a mounting rod. At the same time, the mounting rod simulates the collision between the end actuator of the robotic arm and the outside factors. During the collision test, the end of the JH-605 robotic arm was controlled by the teaching box to move in a straight line, and the MR flexible joint moves together. When the operating speed of the robotic arm stabilizes, the mounting rod collides with a specific collision rod. Since the robotic arm has no additional buffering device besides its own structural buffering, the vibration of the robotic arm without the proposed joint is used as the control group in the test.

The MR flexible joint collision buffering test consists of two parts: (1) the collision buffering test of the robotic arm under different equivalent damping: Analyzing the influence of joint damping on the collision buffering performance of the robotic arm. During the test process, the excitation current of the variable damping element is 0 A, 1 A, 2 A, and the excitation current of the variable stiffness element is fixed at 0 A. (2) The collision buffering test of robotic arm under different equivalent stiffness: Analyzing the influence of joint stiffness on the collision buffering performance of the robotic arm. During the test process, the excitation current of the variable stiffness element is 0 A, 1 A, 2 A, and the excitation current of the variable damping element is fixed at 0 A.

Capture the test data of 0.1 s before and after the collision, as shown in figure 21, showing the collision acceleration of the robotic arm under different equivalent damping. From the figure, the peak collision acceleration of rigid collision is 37.87 m s⁻² without the MR flexible joint. After the MR flexible joint prototype is installed, the peak collision acceleration significantly decreases, and the peak collision acceleration increases from 10.26 m s⁻² to 18.25 m s⁻² when the current increases from 0 A to 2 A. The peak optimization rate of vibration acceleration is 51.81% compared with that of rigid collision at the applied current of 2 A. According to the data in figure 21, the peak collision acceleration of the robotic arm increases with the excitation current of the variable damping element, indicating that the output damping of the MR flexible joint will affect the collision buffering performance. The larger the damping coefficient of MR flexible joint, the larger the vibration acceleration during collision. When the damping coefficient is smaller, the frequency of oscillation will be increase and the time of vibration convergence will be extended. Figure 22 shows the power spectral density of the collision acceleration, and the collision energy is mainly concentrated around 75 Hz. Adjusting the damping of the joint can effectively absorb the collision energy, while increasing the damping will reduce the absorbed energy, because increasing the damping will lead to shorter buffering stroke and lower the energy consumption ability of the MR flexible joint.

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Figure 23 shows the collision acceleration of the robotic arm under different equivalent stiffness conditions. As show in the figure, the peak collision acceleration increases with the excitation current of the variable stiffness element. When the excitation current increases from 0 A to 2 A, the peak collision acceleration increases from 10.26 m s⁻² to 22.65 m s⁻². Under the condition of excitation current of 2 A, the optimization rate of the peak collision acceleration is 40.19% compared with rigid collision. Further analysis shows that the greater the equivalent stiffness of the MR flexible joint, the greater the acceleration during collision. At the same time, the greater the equivalent output stiffness of the joint, the longer the collision convergence time after impact. Compared with the peak collision acceleration data under different damping in figure 22, it was found that changing the equivalent output stiffness of the joint has a greater impact on collision buffering performance. Figure 24 shows the collision acceleration power spectral density under different stiffness. It can be seen from the figure that the collision energy of the joint will rapidly increase with the increase of stiffness, because small position deviation will cause large torque under the condition of high stiffness.

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In this paper, the head injury criterion (HIC) [30] was used to evaluate the safety of collision between the end actuator and the external factors. The lower the HIC value, the lower the damage degree of the impact on the actuator. The HIC equation is as follows:

$$\begin{equation}{\text{HIC}} = \left[ \left(\frac{1}{{{t_2} - {t_1}}}\int_{{t_1}}^{{t_2}} a/g{\text{d}}t\right)^{2.5} \left({t_2} - {t_1}\right)\right]\end{equation} \tag{ 23 }$$

where, t₁ and t₂ are the start and end time of evaluation respectively, a is the collision acceleration, and g is the gravitational acceleration.

According to equation (23), reducing the degree of human-machine impact injury (HIC value) can be achieved by reducing the contact stiffness of the collision position, reducing the effective mass (inertia) of the robotic arm, and reducing the operating speed of the robotic arm. In structural design, reducing the effective stiffness in the transmission chain is an effective measure to improve the safety of the robotic arm. The results obtained by calculating the HIC under various conditions in the collision buffering test of the robotic arm are shown in figure 25.

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In the figure, after installing the joint prototype, the buffering effect of the robotic arm on collisions is more obvious. When a certain degree of collision occurs, it can improve the stability and safety of human-machine interaction (considering the reusability and safety of the robotic arm in the test, the collision force is set below 50 N). Compared to rigid collision, the lowest HIC value of vibration at the detecting point is about 7.7%. But for more intense collisions, the vibration effect that joint prototype can reduce in human-machine collisions is no longer significant. At this time, other measures need to be combined, such as adding additional buffer materials to reduce the stiffness of the collision position, algorithmic control of the operating speed, so as to better improve the safety of the robotic arm.

In order to further verify the vibration reduction effect of the proposed intelligent joint on the end actuator of the robotic arm, a simple closed-loop control experiment was conducted. Specifically, using the classic PID algorithm [31] to achieve real-time adjustment of joint damping and stiffness, and providing simple feedback on damping and stiffness through collected acceleration signals. Due to the limitations of experimental conditions and considering the disturbance of stiffness and damping adjustment signals during the experiment, closed-loop control was separately applied to the damping and stiffness element. And considering the time delay of the joint system, the vibration frequency of the collision is appropriately reduced and the duration of vibration is extended, so as to avoid the vibration time being too short to reflect the controllability of the algorithm.

Figure 26 shows the variation of acceleration amplitude and current when the damping element is controlled by PID (the current of variable stiffness element is 0 A). The excitation current is constantly adjusted according to the feedback of the acceleration amplitude, so as to suppress the vibration of the actuator. Due to the protection current set for the joint during the test, the maximum current during the initial vibration period is 2.5 A. As the acceleration amplitude decreases, the excitation current of the damping element also gradually decreases. The application of current is mainly in the early violent vibration, and the changing current is no longer applied to the small residual vibration in the later part. It can also be found from the figure that the change of current is slightly delayed due to the hysteresis of the joint system.

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Figure 27 shows the vibration acceleration curves under different conditions, which are the benchmark test without the joint (Benchmark), no excitation current of the joint (Damping(0 A)-Stiffness(0 A)), the damping element is controlled by PID/no current applied to the stiffness element (Damping(PID)-Stiffness(0 A)) (the stiffness should be small during collision), and apply 2 A current to the damping element/the stiffness element is controlled by PID (Damping(2 A)-Stiffness(PID)), respectively. In order to better demonstrate the effectiveness of the joint and control algorithm, the comparison curve of the benchmark test is added. Obviously, after the installation of flexible joints, the vibration of the end actuator is well suppressed, and the peak acceleration is reduced from about 33.25 m s⁻² of the benchmark test to 18.05 m s⁻². After applying PID closed-loop control to the damping and stiffness elements of the joint, the vibration suppression characteristics of the end actuator have been improved compared to the constant current tests in figures 21 and 23, especially in the later period of vibration, which can effectively shorten the residual vibration time.

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Based on the above analysis, the stiffness and damping of the MR flexible joint have a significant impact on the tracking accuracy of joint position and motor output torque. A large joint stiffness and appropriate joint damping can ensure a small tracking error while also requiring a small torque. After the robot stops moving, the residual vibration at the end of the robot can be effectively suppressed by selecting small joint stiffness and large joint damping, and the robot posture can be maintained with small torque of motor. When a collision occurs, small joint stiffness and damping can effectively reduce the acceleration of collision vibration. And compared with rigid collision, installing the MR flexible joint can significantly reduce human injury during collisions to a certain extent. In summary, the proposed MR flexible joint can adapt to the performance requirements of robot in different motion stages and working conditions by adjusting the stiffness and damping, thereby improving motion ability and interaction stability of the robot.