Mechanism Simulation and Optimal Design of Five-Link Bionic Joint Driven by Two Antagonistic Pneumatic Muscles

Abstract

The five-link parallel mechanism is proposed to improve joint bionic performance, and the kinematics is established for the closed chain joint actuated by two antagonistic artificial pneumatic muscles (PMs). Interference and singularity constraints are analyzed, and the joint torque model is given based on the spring-damp dynamics. Through extracting the spring force term from torque equations, the compliance of bionic joint is derived and expressed as the ratio of angle to spring torque. Energy consumption is analyzed using the PM length varying. Based on MATLAB/SimMechanics, the relationships between the axil installation parameters and swing performances are illustrated through the simulations, including the effect of the installation height and width varying on the angle scope, swing response, compliance, and energy consumption. The bionic shoulder and elbow joints are optimally designed. Compared to the conventional joints, the swing angular range of the proposed joints is enhanced, and the contraction amount of PMs is reduced. The optimal mechanism is more humanoid.

1. Introduction

The bionic joint is one of the keys of the manipulator developed to help labor working in the unstructured environment. Lightweight, anthropopathy swing trajectory, and compliance are the main performance indices for bionic design. High flexibility of the artificial pneumatic muscle (PM) actuators can make manipulator joints more anthropopathy [1 –4].

A pair of antagonistic parallel installation PMs is necessary for actuating joint due to the PM only outputting unidirectional force like cable-driven system [5]. Sekine designed a PM driven parallel link mechanism to get the 2-DOF (degree of freedom) shoulder, in which stability is bad and computing and control are complicated [6]. One DOF PM driven joint is practical currently.

By far, the PM joint mechanisms swing in a plane can be mainly classified into three models, PMs connected with the rotating axis (a) through a pulley; (b) by public hinge; (c) by multilink. About the model (a), Sui [7] designed a bionic elbow joint and Chang [8] designed a 2-DOF rehabilitation robot arm and the self-organizing fuzzy sliding mode controller for this arm. Shin et al. [9] designed the variable radius pulley to improve the output torque. Obviously, the elbow designed using the model (a) is not very bionic, due to the human joint swing around the noncircle condyle. About the model (b), Hosoda et al. [10] designed a bipedal robot that can walk, run, and jump. About the model (c), Choi et al. [11] designed a 2-DOF bionic arm. Mechanism parameters are very important to the maximum swing angle, flexibility, and controlling accuracy of joint [12]. The output torque of model (a) joint is the most [13]. Both the output torque and the swing angle of the model (b) joint are the smallest. The swing angle of the model (c) joint is the biggest [14]. The model (c) joint is the most close to the skeletal muscle structure of the human joint, so it is the research hotspot. However, there are some drawbacks of the model (c) joint as below. The model (c) mechanism is symmetrical, while the human body bone is asymmetrical. The model (c) can be equivalent to the four-link mechanism; namely, connecting rod shaft is located in the connecting line between PMs' spindles.

Most of the literatures about stiffness or compliance of the bionic joints focuses on the control scheme [14, 15], and few is about how mechanism parameters affect the compliance or stiffness. Meanwhile, the stiffness of cable-driven parallel mechanism is not totally credible by traditional theoretical analysis, so the experimental analysis method is adopted [16].

The literature shows that some of bionic indices are conflicting and required to be simultaneously optimized. General analytical solution is nonexistent. The multiobjective optimization schemes under constrains are proposed to obtain an optimal bionic joint, such as evolutionary algorithms, neural network, and self-adaptive greedy scheduling [17]. However, these exiting numeric optimal schemes just give the general results, and cannot illustrate the effect of processing given parameter on bionic performance.

This paper firstly makes the rotating axis nonlocated in the line between PMs' spindles, so the bionic joint can be equivalent to the five-link parallel mechanism with more optimization space. And a simulation platform is developed in order to avoid the complicated analytic solutions of multiparameter mechanism optimum. The main contribution of this paper is proposing a novel five-link joint mechanism and using the axis installation height and width to improve the bionic characteristics including swing trajectory, compliance, and energy consumption one by one.

The rest of this paper is organized as follows. Firstly, the mechanism model and kinematics of the five-link bionic joint actuated by two PMs are established. The velocity kinematics between the swing angle and contraction of PMs is deduced. And then four swing constrains about mechanism are given. Secondly, based on the single PM dynamics model, the compliance of the bionic joint is derived. Thirdly, the simulation platform is set up using SimMechanics. The joint bionic swing performance, such as angle scope, joint compliance, energy costing, and rotation velocity, is analyzed under different mechanism parameters in details. The relationships between parameters and bionic swing are discussed in detail. Fourthly, according to the human arm joints parameters, the optimal design for shoulder and elbow joint are accomplished. The full paper is summarized and the conclusions are drawn in the last section.

2. Kinematics and Constrains

2.1. Mechanism Configuration and Kinematics

The mechanisms with one DOF and coordinate system are shown in Figure 1 swing within the x-y plane. Figures 1 (a) and 1 (b) show the initialization and swing states under h > 0 and h < 0 configuration, respectively. The proposed mechanism can be equivalent to the closed chain five-link (OA-AD-DC-CB-BO, common four-link is AD-DC-CB-BA) with two parallel PM actuators. The joint rotates towards the contraction PM side, meanwhile the antagonistic PM extents.

Figure 1:

Bionic joint mechanism and coordinate system: (a) h > 0; (b) h < 0.

In Figure 1, M denotes the load, OE the arm bone (rigid and fixed with DC), AD the left PM (such as triceps), and BC the right side PM (such as biceps). r_l and r_r are the distance between the PMs installed point A, B and the joint rotation point O. L is the initial installed length of PMs. θ is the swing angle of the joint and the positive direction is in the counter-clockwise. h is the rotation shaft installed height and positive above line AB. Noteworthy, for the common four-link joint, h is 0 and r_l equal to r_r.

We use ∑ _xoy to denote the fixed base Cartesian coordinate system and ∑ _{x
_m
o
_m
y
_m} the floating coordinate system. According to the robotics kinematics, the rotation matrix between ∑ _xoy and ∑ _{x
_m
o
_m
y
_m} is

R = [\begin{bmatrix} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{bmatrix}] .

(1)

Without loss of generality, set the forward swinging as an anticlockwise direction. Using (1), the PM length varying value (relative to the initial installed length) can be derived as

\begin{matrix} Δ L_{r} = ({[r_{r} (1 - \cos θ) + (h + L) \sin θ]}^{2} {+ {[- r_{r} \sin θ + h (1 - \cos θ) - L \cos θ]}^{2})}^{1 / 2} - L, \end{matrix}

(2)

\begin{matrix} Δ L_{l} = ({[r_{l} (\cos θ - 1) + (h + L) \sin θ]}^{2} {+ {[r_{l} \sin θ + h (1 - \cos θ) - L \cos θ]}^{2})}^{1 / 2} - L, \end{matrix}

(2′)

where ΔL_r, ΔL_l are the contraction of the right and left PM, respectively.

In the literature, regardless of mechanism models, for reducing computation complexity, the relationship of the swing angle and the PM length varying is always simply defined as

\begin{matrix} Δ L_{r} = r (\frac{π}{2} - θ), \end{matrix}

(3)

\begin{matrix} (3^{}) & Δ L_{l} = r (\frac{π}{2} + θ), \end{matrix}

(3′)

where r = r_l = r_r is the simplified rotating radius of the joint and $r π / 2$ is the precontraction of the PM.

Equation (2) is more precise than (3) but more complicated. Differentiating (2), we can get the angular velocity as

\dot{θ} = \frac{Δ L_{r} + L}{[r_{r}^{2} + h (h + L)] \sin θ + r_{r} L \cos θ} Δ {\dot{L}}_{r} .

(4)

Equations (2) and (4) show that the mechanism parameters, r_l, r_r, and h, have the distinct effect on the joint swing.

2.2. Constraints Analysis

The first constraint condition (denoted by C1) is about the antagonistic installation. In the process of swing, one PM should be shortened while the other elongated relative to the PM initial installed length. Due to the antagonistic installation, two PMs length varying must meet

Δ L_{l} Δ L_{r} < 0 .

(5)

The second constraint condition (denoted by C2) is about the mechanism interference. Link OE cannot swing over point B or A, so

- \arctan (\frac{r_{l}}{h}) \leq θ \leq \arctan (\frac{r_{r}}{h}) .

(6)

Similarly, AD or BC cannot swing over point O, so

\arctan (\frac{r_{r} L}{(h^{2} + h L + r_{r}^{2})}) \leq θ \leq - \arctan (\frac{r_{l} L}{(h^{2} + h L + r_{l}^{2})}) .

(7)

The third constraint condition (denoted by C3) is about the maximum shrinkage and stretch constraint of PMs

\begin{matrix} (s_{i} - s_{\max}) L_{r}^{0} \leq Δ L_{r} < 0, \\ 0 < Δ L_{l} \leq (s_{i} + s_{a}) L_{l}^{0}, \end{matrix}

(8)

where s_i is the precontraction rate in initial installation state and s_max the realizable maximum shrinkage rate. L_r⁰ and L_l⁰ are the length of right and left PM under s_i = 0, respectively, and s_a the maximum prestretch rate.

The fourth constraint condition (denoted by C4) is about kinematics singularity point. When point C moves to line OB prolongation or point D to line OA prolongation, respectively, ΔL_r and ΔL_l will reach the extreme value, and then

\frac{\partial Δ L_{r}}{\partial θ} = 0, \frac{\partial Δ L_{l}}{\partial θ} = 0 .

(9)

The singular point will bring the stick lock or velocity change suddenly, so it should be avoided.

Define θ_ci^a, θ_ci^c as the maximum reachable angle under the constraint Ci in the anticlockwise and clockwise swing, respectively. If θ_ci^a < θ_cj^a, the constraint Ci is stronger than Cj (i, j = 1,2, 3,4). The effective swing angle range of the bionic joint should meet all four constraints; thus

[θ_{\min} = \max {θ_{c i}^{c}}, θ_{\max} = \min {θ_{c i}^{a}}] .

(10)

Above four constrains make the joint mechanism design more complex.

3. Compliance and Energy Efficiency Analysis

3.1. Dynamics of Pneumatic Muscle

Based on the work principle of the PM, the outputting force depends upon the inside pressure P. According to the spring-damp system dynamics [19], dynamics model of a single PM hanging a mass vertically is

m Δ \ddot{L} + B (P) Δ \dot{L} + K (P) Δ L + m g = F (P)

(11)

in which

\begin{matrix} (11^{'}) & K (P) = k_{0} + k_{1} P, \\ B (P) = {\begin{cases} b_{0 i} + b_{1 i} P & (Inflation processing) \\ b_{0 d} + b_{1 d} P & (Deflation processing), \end{cases} \end{matrix}

(11′)

where m denotes the load mass (the mass of adjacent bionic joint and arm, ignoring the mass of current bionic joint and arm), ΔL the shrinkage of the PM, and K(P), B(P) are the elasticity and damping coefficients varying with P. The coefficients k and b should be got by experiments and linear fitting.

3.2. Compliance of the Bionic Joint

According to (11) and the initial installation state, the PMs output torque are

\begin{matrix} τ_{r} = [F_{r} (P_{r}) - K_{r} (P_{r}) (s_{i}^{r} L_{0}^{r} - Δ L_{r}) - B_{r} (P_{r}) Δ {\dot{L}}_{r}] r_{r}, \\ τ_{l} = [F_{l} (P_{l}) - K_{l} (P_{l}) (s_{i}^{l} L_{0}^{l} - Δ L_{l}) - B_{l} (P_{l}) Δ {\dot{L}}_{l}] r_{l}, \end{matrix}

(12)

where P_r and P_l are the pressure of the right and left PM, respectively, which can be computed by

\begin{matrix} P_{r} = P_{i}^{r} + Δ P, \\ P_{l} = P_{i}^{l} - Δ P, \end{matrix}

(13)

where P_i is the initial pressure of PMs.

Therefore, the composition torque of the bionic joint can be determined by

τ_{t} = F_{r} (P_{r}) r_{r} - F_{l} (P_{l}) r_{l} - K_{r} (P_{r}) (s_{i}^{r} L_{0}^{r} - Δ L_{r}) r_{r} + K_{l} (P_{l}) (s_{i}^{l} L_{0}^{l} - Δ L_{l}) r_{l} - B_{r} (P_{r}) Δ {\dot{L}}_{r} r_{r} + B_{l} (P_{l}) Δ {\dot{L}}_{l} r_{l} .

(14)

Extracting the spring force, the 3rd and 4th items, from (14), and combining (11′) and (13), yield the spring torque τ_sj

τ_{s j} = S_{j} θ = (k_{0}^{l} + k_{1}^{l} P_{i}^{l}) r_{l} (s_{i}^{l} L_{0}^{l} - Δ L_{l}) - (k_{0}^{r} + k_{1}^{r} P_{i}^{r}) r_{r} (s_{i}^{r} L_{0}^{r} - Δ L_{r}) - (k_{1}^{l} Δ L_{l} r_{l} + k_{1}^{r} Δ L_{r} r_{r}) Δ P,

(15)

where S_j is the joint stiffness.

At the initial state, the shrinkage of left PM is the same as right PM for isomorphic mechanism, namely, s_i^lL₀^l = s_i^rL₀^r and ΔL_l = ΔL_r = 0. So in the initial position, output torque of the joint is 0, τ_sj = 0, and we can get

(k_{0} + k_{1} P_{i}^{r}) r_{r} = (k_{0} + k_{1} P_{i}^{l}) r_{l} .

(16)

The compliance is the characteristics describing the relationship between the joint rotations deform to the passive torque, always defined as the inverse of the joint stiffness. Combining (15) and (16), we can yield the compliance of bionic joint as

C_{j} = \frac{1}{S_{j}} = θ \times ((k_{0}^{l} + k_{1}^{l} P_{i}^{l}) r_{l} (s_{i}^{l} L_{0}^{l} - Δ L_{l}) - (k_{0}^{r} + k_{1}^{r} P_{i}^{r}) r_{r} (s_{i}^{r} L_{0}^{r} - Δ L_{r}) {- [k_{1}^{l} (s_{i}^{l} L_{0}^{l} - Δ L_{l}) r_{l} + k_{1}^{r} (s_{i}^{r} L_{0}^{r} - Δ L_{r}) r_{r}] Δ P)}^{- 1} .

(17)

Equation (17) shows that the compliance is not relationship with the mass. So ignoring the mass of current bionic joint and arm is bad for getting the more precious dynamics but does not affect the compliance analysis.

3.3. Energy Efficiency Analysis

The joint's energy cost is the key characteristics to bionic performance and meanwhile is difficult to count. Under the same load, the energy consumption is approximately proportional to the PM length varying ΔL and the load gravitational potential energy varying ΔE_p. Therefore, we adopt ΔL and ΔE_p to scale the energy consumption. Pressure adjustments in both the inflation and deflation processes need supplying energy to the solenoid valve. For the agonist-antagonist pneumatic muscles joint, ΔL can be simplified as the sum of the contraction and elongation, such as ΔL_r – ΔL_l. The zero potential level is defined as when θ = 0, then the gravitational potential energy of the load can be obtained by

Δ E_{p} = m g (h + L) (1 - \cos θ) .

(18)

Equation (18) shows ΔE_p will increase with the bigger h under given θ and L.

4. Analysis of Mechanism Parameters

4.1. Simulation Platform

Avoiding the difficulty of analytic solutions, we use the simulation to figure out the relationships between the mechanism parameters and the bionic swing. We establish the simulation platform using Simulink/SimMechanics shown in Figure 2.

Figure 2:

The simulation platform based on SimMechanics.

There is no PM module given in MATLAB/Simulink. Regardless of the radial varying, the PM is equivalent to the prismatic actuator with the driver module of (11). In Figure 2, “R” denotes the rotation joint, “Weld” fixed connection, and “Body” the link. “Joint Sensor” is used to get the joint angle and velocity. This simulation platform input is the slope increasing pressure that is transformed to the force supplied on the prismatic according to (11), and output includes each joint angle, velocity, and length varying of the PM. During simulation, the joint swing can stop when any constrain is not satisfied. This platform can be used to analyze the influence of the mechanism parameters to the bionic joint rotation quantitatively. The sample frequency is set to 100 Hz, and data float precision is 0.001.

4.2. Comparing the Nonlinear Contraction Model to the Circle Model

In order to illustrate the differentia between (2) and (3), we conduct the joint swing simulation with parameters shown in Table 1, and results are shown in Figure 3.

Table 1:

Simulation parameters.

r_l = r_r = r	h	L	θ	$s_{\max}^{l} = s_{\max}^{r}$	s_a^l = s_a^r	Angle step

0.035 m	0	0.250 m	${- π / 2, π / 2}$	20%	2%	0.001 rad

Figure 3:

The relationships of ΔL – θ.

Figure 3 shows that the relationship reflected by (3) is close to (2) only when the angle is very small. As the joint angle increases, errors produced by (3) increase obviously. This is one of the reasons for large errors generated in trajectory tracking process in the literature.

4.3. The Effect of h on the Rotation Angle Scope

For anticlockwise swing, the right PM is the master diver. For symmetric mechanism, the joint's effective swing angle can reach the maximum when the antagonistic PM's maximum contraction equals the other PM's maximum elongation. So for getting the max swing angle, the below mechanism parameters are adopted

\begin{matrix} s_{i}^{l} = s_{i}^{r} = 9 %, L_{0}^{l} = L_{0}^{r} = \frac{L}{1 - s_{i}} = 0.275 m, \\ Δ L_{\max}^{l} = (1 + s_{a}) L_{0}^{l} - L = 0.030 m, \\ Δ L_{\max}^{r} = (1 - s_{\max}) L_{0}^{r} - L = - 0.030 m . \end{matrix}

(19)

Other parameters are the same as Table 1. During the simulation, the master diver is inflated. Considering all the constraints, the relationships between bionic joint effective swing angles with varying h are shown in Figure 4.

Figure 4:

The relationships of ΔL – θ under different h.

Figure 4 shows the following rules:

The joint effective swing angle increases when decreasing h under h > 0.

Under h ≠ 0, the contraction and velocity of the right and left PMs are different.

When h decreases from positive to negative, $| d Δ L / d θ |$ of the master PM increases; that means the angle error caused by the PM contraction error will decrease. As a result, this is beneficial to get more precise angular position control. Therefore, the value of h should be increased for the low position precision and high response speed demand; otherwise, h should be decreased.

In Figure 4, the black solid points connected by the dashed line denote the max effective swing angle under different h. The dashed line shows that the relationship of h to max θ is unsymmetrical, and the effective swing angle will reach the max when h = 0.

4.4. The Effect of r on the Rotation Angle Scope

4.4.1. Symmetrical Mechanism Anticlockwise Rotation

The joint width is set to r_l = r_r = r and varies between 0.025~0.04 m (step 0.005 m). Other parameters are the same as Table 1. The simulation results under all constraints are shown in Figure 5.

Figure 5:

The relationships of ΔL – θ under different r with symmetrical mechanism.

As shown in Figure 5, the effective swing angle increases with decreasing r. Meanwhile, curve slope becomes smaller at the bigger θ, which means the rotation is more sensitive to the contraction of the PM. So it is difficult to achieve the precise position control at the bigger θ. Without considering the control accuracy, the rotation scope can be increased by reducing r.

4.4.2. Unsymmetrical Mechanism Anticlockwise Rotation

The joint total width, r_l + r_r, is kept as 0.070 m constant, and other parameters are the same as Table 1. The simulation results are shown in Figure 6.

Figure 6:

The relationships of ΔL – θ under different r with unsymmetrical mechanism.

For the isomorphic PMs and the same installation prestretch rate, θ_max increases with increasing r_r under r_r < r_l. θ_max decreases with increasing r_r under r_r > r_l. For the unsymmetrical mechanism, the reason restricting the maximum angle of the bionic joint is the maximum elongation ((s_i + s_a)L₀) of the left PM.

4.4.3. Unsymmetrical Mechanism Bidirectional Rotation

This section studies the unsymmetrical joint swing from left to right under r_r ≥ r_l. Set r_r = 0.065~0.035 m (step 0.01 m), other parameters same as Section 4.4.2. The simulation results are shown in Figure 7.

Figure 7:

The relationships of ΔL – θ under different r with unsymmetrical mechanism during bidirectional rotation.

Figure 7 shows the following rules:

$| d Δ L / d θ |$ of the master PM increases when increasing r_r.

Both $| θ_{\min} |$ and θ_max decrease when increasing r_r.

For the isomorphic PMs, increasing r_r can make $θ_{\max} > | θ_{\min} |$ .

For the isomeric PMs and r_l = r_r mechanism, we can make the initial contraction of the right PM less than left PM to increase the maximum swing angle θ_max.

4.5. The Effect of r on the Energy Consumption

Equation (18) indicates the effect of h on energy consumption. In addition, the effect of ΔL_r – ΔL_l on energy consumption should be studied. Given h = 0, ΔL_r – ΔL_l will vary with r. For the unsymmetrical mechanism, r_l + r_r is kept as 0.070 m constant and other parameters are the same as Table 1. Set r_l = 0.055~0.015 m (step is 0.01 m), and the simulation results are shown in Figure 8.

Figure 8:

The relationships of ΔL_r – ΔL_l with θ under different r.

Figure 8 shows that ΔL_r – ΔL_l decreases with increasing r_l. Attentively, the lesser ΔL means the energy saving. The energy consumption can be reduced properly through setting r_l > r_r.

4.6. The Effect of h and r on the Compliance

When the humanoid robot works in the human being living environment, the collision and contact between robot and human are inevitable. Therefore, for ensuring the human safe, the compliance is very important to bionic joint.

The simulation of the bionic joint compliance during swinging in anticlockwise is shown in Figure 9, and dynamics parameters are listed in Table 2.

Table 2:

PM's dynamics parameters.

L (m)	r_l + r_r (m)	m (kg)	k ₀	k ₁	a ₀	b _0i	b _1i	b _0d	b _1d

0.250	0.070	5.000	4.820	1.320	14.890	1.220	1.350	1.310	−2.300

Figure 9:

The relationships between the compliance with r and h. (a) Unsymmetrical mechanism with h = 0; (b) symmetrical mechanism with r_l = r_r = 0.035 m.

Figure 9 shows the following rules:

the joint initial compliance is relevant to r, not to h.

For the symmetrical mechanism, the joint compliance increases when increasing h from the negative to the positive.

For a pair of the isomorphic PMs, the initial compliance increases when increasing r_l.

The larger compliance is present with the larger θ.

Under collision, joint angle will be increased by impact torque suddenly. The joint compliance should quickly increase with increasing angle to give buffer for human. For safe coexitings of humanoid robot and human being, we should increase h or r_l to get the abrupt curve between compliance with swing angle.

5. Optimal Design of Shoulder and Elbow Joints

5.1. Optimal Object

The primary objective of this paper is to design the more humanoid arm just considering the case swinging in the vertical plane. Mechanism design object is reference to the males aged 18–60 in southeast region of China. The optimal object parameters are shown in Table 3.

Table 3:

Human arm and swing parameters [18].

	Length	Mean diameter	The max forward angle	The max backward angle

Upper arm	0.313 m	0.0828 m	3.140 rad	−0.785 rad
Lower arm	0.237 m	0.0792 m	2.090 rad	0

The angle scopes of both the human shoulder and elbow joints are asymmetry. This paper's optimal goal is to make the joint sizes and rotation angle scope of the bionic joint arm close to human arm. At the same time, try to reduce energy consumption and improve the joint's compliance as possible.

Noteworthy, the effects of mechanism parameters on different bionic performance are conflicting, so it is difficult to get the optimal effect for all bionic characteristics. We just can get the general optimal design by trials and errors.

5.2. Design of the Shoulder Joint

According to Table 3, design r_l + r_r = 0.082 m. In view of the maximum angle of shoulder swing forward (anticlockwise) is bigger than backward (clockwise); set r_r > r_l. Because θ_max has an increasing tendency when decreasing r_r as shown in Figure 7, design r_r = 0.030 m, r_l = 0.052 m. FESTO Company's MAS-10-xx type PMs were employed as actuator with L₀^l = L₀^r = 0.273 m, L_l = L_r = 0.250 m, and s_a = 9%, s_max = 20%. The maximum pretension rate of PMs produced by FESTO is about 2%.

As discussed in Section 4.3, set h < 0 to avoid the singular point appearing before C1. Without considering the contraction constraint of left PM, θ_max under different h simulation is shown in Figure 10.

Figure 10:

$θ_{\max} - h$ curves only under the right PM contraction constrain.

As shown in Figure 10, θ_max of C2 is not monotonically increasing with decreasing $| h |$ . The swing angle reaches the comprehension maximum at the point of intersection, under both the shrinkage and the mechanism interference constraints. Therefore, we set h = – 0.002 m.

If the left PM installation and work performances are the same as the right, namely, using isomorphic PMs, Figure 11 shows that the left PM oversteps constraints when the right PM reaches the extension or shrinkage values. To solve this problem, we propose an optimal design procedure as follows,

Figure 11:

The relationships of ΔL_r, ΔL_l with θ under same PMs.

Step 1. With the given parameters of the right PM, we can obtain the maximum contraction |ΔL_max^r| = 0.030 m and the maximum anticlockwise swing angle θ_max = 1.392 rad according to Figure 10.

Step 2. Calculating (2) with θ_max, we can obtain the maximum elongation of the left PM ΔL_max^l = 0.053 m. Thus, the length of the left PM without inflated is L₀^l = L + ΔL_max^l = 0.303 m.

Step 3. According to Table 3, we set $θ_{\min} = - θ_{\max} / 4 = - 0.348$ rad.

Step 4. Substituting θ_min = – 0.348 rad into (2′), we can obtain the maximum contraction of the slave PM ΔL_min^l = – 0.018 m.

Step 5. Set ΔL_{s_max + s_a}^l = ΔL_max^l – ΔL_min^l = 0.0707 m, and $s_{\max}^{l} + s_{a}^{l} = Δ L_{s_{\max} + s_{a}}^{l} / L_{0}^{l} = 23.349 %$ .

Step 6. Calculating the precontraction ratio of the left PM, $s_{i}^{l} = ((L_{0}^{l} - Δ L_{l i}) / L_{0}^{l}) = 17.437$ %, the maximum contraction ratio $(s_{\max}^{l} - s_{i}^{l}) = (Δ L_{\min}^{l} / L_{0}^{l}) = 5.91$ % in anticlockwise swing, we can obtain the maximum contraction ratio of the left PM s_max^l = s_i^l + 5.91% = 23.34%.

According to ΔL_max^l = (s_i^l + s_a^l)L₀^l, the maximum elongation ratio of the left PM $s_{a}^{l} = (Δ L_{\max}^{l} / L_{0}^{l}) - s_{i}^{l} \approx 0$ , which meets the constraint that the PM original physical length may not be elongated.

In general, the left PM chosen should meet the parameters s_max^l = 23.34%, L₀^l = 0.303 m, s_i^l = 17.44%. This optimal design with asymmetry PMs is easier to get the maximum forward swing angles.

The response rates $d θ / d Δ L_{r}$ under different r in anticlockwise swing are shown in Figure 12.

Figure 12:

The shoulder joint $d θ / d Δ L_{r} - Δ L_{r}$ under different r.

As shown in Figure 12, the accelerating performance of the common bionic joint (r_l = r_r = 0.041 m, h = 0) is poor, and appropriately decreasing h is beneficial to the better accelerating performance. However, the singular point is more likely to appear if h is increased in the symmetrical mechanism. Curve $d θ / d Δ L_{r} - Δ L_{r}$ becomes very steep after the contraction ΔL_r reaching −0.020 m round, which is disadvantaged to the position control. Therefore, we designed r_l = 0.03 m < r_r = 0.052 m with h = 0.020 m, and both θ_max and the accelerating performance are improved.

5.3. Design of the Elbow Joint

According to Table 3, the width of the elbow is set as r_l + r_r = 0.075 m. The elbow can only flex forward unilaterally. So the single PM driver method is applied. The spring can be installed on the left side replacing left PM, which can eliminate the constraint of the maximum elongation and also can reduce the energy consumption. Without the elongation constraint of left side, r_r should be reduced as small as possible to improve θ_max. Considering the installation size, we set r_r = 0.026 m, L = L₀^r = 0.250 m, s_i^r = 0, s_max^r = 20%.

According to Figure 1 and (18), the maximum θ_max only can be $π / 2$ (when h = 0) and ΔE_p is larger under h ≥ 0. However, θ_max can be increased more than $π / 2$ under h < 0. We conduct simulation of the elbow bionic joint in anticlockwise swinging under all the constraints and different h shown in Figure 13.

Figure 13:

The relationships of shrink – h.

As shown in Figure 13, θ_max gets the maximum when h = – 0.019 m and the absolute value of slope is bigger than the others. This indicates that h = – 0.019 m is beneficial to getting the accurate position control and effective using the PM contraction. With the proposed elbow joint parameters, θ_max = 2.08 rad close to the expectation value 2.09 rad. In addition, the fixed point of the left spring under the x axis (y < 0) can avoid the mechanism interference.

5.4. Comparative Analysis

Conventional 4-link joint is symmetric and h is 0. We can get the r_l and r_r of conventional joint by averaging the mean diameter of arm in Table 3. For fair comparing, the conventional and proposed joint use the same PM under same arm length.

The results comparing the human and conventional bionic joint to the optimal design are shown in Table 4.

Table 4:

The comparison of three joints mechanism and swing parameters.

	r_l (m)	r_r (m)	h (m)	$θ_{\max}$ (rad)	$θ_{\min}$ (rad)	$θ_{\max} - θ_{\min}$	$θ_{\max} / \| θ_{\min} \|$

Shoulder joint
Human	∖	∖	∖	3.14	−0.785	3.93	4
Conventional	0.041	0.041	0	0.81	−0.81	1.62	1
Proposed	0.052	0.030	−0.002	1.39	−0.35	1.74	3.97
Elbow joint
Human	∖	∖	∖	2.09	0	2.09	∖
Conventional	0.038	0.038	0	1.46	0	1.46	∖
Proposed	0.049	0.026	−0.019	2.08	0	2.08	∖

Compared to the conventional mechanism, the angle range of the proposed shoulder is enhanced about 7% and the elbow about 42%. The contraction of two antagonistic PMs can be reduced about 7% (utmost) during the same swing scope. The optimization mechanism enlarges the swing angle range. For the bionic shoulder joint, the compliance can also be improved under r_r < r_l. For the bionic elbow joint, only a single PM is employed to drive the mechanism, which saves cost and makes θ_max reaching the maximum angle of the human elbow flexion.

The mechanism and swing process are shown in Figure 14.

Figure 14:

The sketch of proposed and conventional bionic arm. (a) Proposed virtual prototype; (b) proposed joint swing processing; (c) Conventional joint swing processing.

Both proposed shoulder and elbow joints have nonzero h, and different r_l and r_r as shown in Table 4. In Figure 14 (b), the 5-link mechanism of shoulder joint is not obvious due to h = – 0.002 m only. However, the optimal r_l and r_r make shoulder joint swing more like human. Comparing Figures 14 (b) to 14 (c), we are confident that the optimal h, r_l and r_r parameters make 5-link joint more advanced than conventional 4-link joint.

6. Conclusions

Taking anticlockwise swinging for example, this paper has analyzed the impact of three mechanism parameters on the bionic joint swing characteristics, and the shoulder and elbow joint are optimally designed. The analytic and simulation results have shown the following:

Swing space of the bionic joint can be improved by appropriately reducing h.

The anticlockwise swing maximum angle can be increased by reducing r_r, and the energy consumption can be reduced synchronously.

Compliance of the bionic joint will be better through increasing h or r_l properly.

It is conducive to use contraction of the PM effectively by employing the isomerism installation for the bionic joint driven by two antagonistic PMs.

Above-mentioned work is beneficial to develop the more anthropopathy artificial muscle joint. It is a great encouragement to our future work, intending to make the precision control of the PM joint angle easy and human-like.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 50905170), Natural Science Foundation of Zhejiang Province (no. LQ13E050004), and Research Project of General Administration of Quality Supervision, Inspection and Quarantine of China (no. 201210076-2).

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