Prototyping of a novel anthropomorphic mechanical leg

Abstract

This research aims to explore a better kinematic performance and design scheme for a novel mechanical leg using parameter optimization method. Serial structural mechanisms are widely employed in anthropomorphic mechanism legs but with significant disadvantages of the complex structure and large inertia, particularly, for the multi-objective parameter optimization it is hard to select good parameters to achieve excellent performance. In this article, the plane model of the solution space for multiple parameters and a novel statistics parameter optimization method were proposed for a novel mechanical leg. In the position analysis, the structure and workspace for the novel mechanical leg were developed with simple structure, small inertia, and large workspace; and several kinematic performance evaluation indices were also proposed in the kinematics analysis. In the parameter optimization process, the design scheme and prototyping of the mechanical leg have shown a better kinematic performance by considering the assembly technique as compared with the conventional models. The proposed research provides the basis for the applications of the novel mechanical leg, which can be applied in the modern humanoid robot fields to meet the requirements of high stiffness, lower inertia, and good technological efficiency.

Introduction

Humanoid robots are usually designed to model the human bodies, in which mechanical legs are indispensable if the robots are to walk and interact with the environment.^1,2 Currently, most anthropomorphic mechanical legs adopt serial chains. For instance, the anthropomorphic robot developed by Solis et al. ³ has two legs and each leg consists of five serially connected links. Honda in Japan, Sarcos research team in the United States,^4,5 University of Karlsruhe in Germany, and Beijing University of Aeronautics and Astronautics also developed various bionic mechanical legs with serial chains. However, serial chains suffer from several weaknesses in practical application, including small carrying capacity and large motion inertia.^6–8 Compared with serial chains, robotic legs with parallel structure are compact and have large carrying capacity, which are increasingly promising to overcome the issues of serial chains,^9,10 and can be used to conduct the precision polishing work using this kind of robot. ¹¹ Thus, hybrid legs have been introduced in recent years. Liu et al. ¹² designed a hybrid robot system for computed tomography (CT)-guided surgery. Wang et al. ¹³ developed a reconfigurable robot for the search and rescue purpose, including three identical robotic systems for performing distributed activities. Pisla et al. ¹⁴ proposed an active hybrid parallel manipulator (PM) for minimal invasive surgery and studied the structural design and kinematics of a parallel reconfigurable robot. ¹⁵

The optimal design of hybrid legs is one of the essential challenges to robotics community cause it is difficult to select appropriate parameters that ensure a good performance of the mechanical legs only by intuition or experience.^16–18 Currently, there are two main approaches, namely, single-objective parameter optimization and multi-objective parameter optimization. The former study focuses on a single performance index while the latter investigation employs several performance indices as the objective function to optimize the parameters of the manipulator. Yue et al. ¹⁹ adopted the geometric model of the solution space technique to study the distribution of various structural parameters and performance indices. The structural parameters of micro-robot were optimized by Liu et al. ²⁰ using the genetic algorithm. Hao and Merlet ²¹ proposed an optimization method based on the work space theory by considering the workspace and regarding other design objectives. However, these approaches must be initially optimized after the change of the objectives, and these approaches cannot visually and globally show the relationship between the objectives and design parameters as well. Further, the geometric model of the solution space is not suitable for the parameter optimization of the mechanism that has more than four structural parameters. To overcome these problems, the multi-parameter plane model technique was proposed and used for the parameter optimization of the mechanism that has more than four structural parameters, and it can synthesize multiple indexes based on multiple performances using the statistics method. Further, Ceccarelli ²² also conducted the research on designs and prototypes of mobile and humanoid robots, and presented a good performance. ²³ Moreover, this kind of robot could enlarge the application of the precision machining technology^24,25 in industry by combining with different mechanical structures.

Therefore, in this article it proposes a novel parallel anthropomorphic mechanical leg as described in the patent awarded by Li et al. ²⁶ and the related parameter optimization method according to the plane model of the solution space for multiple parameters. The mechanism and structural features of the novel mechanical leg are first presented, and then the position and workspace analysis are conducted using mathematical approach, followed by analyzing the performance of the mechanical leg both in kinematic and static states. Further, the global performance atlases will be presented in detail to explore the distribution role of each structural parameter and the corresponding global performance index. Finally, the parametric design is carried out to obtain the parameter optimization for the mechanical leg.

Structural features of the novel anthropomorphic mechanical legs

A novel anthropomorphic mechanical leg with similar structure to human body was developed, which is driven by three motors to realize the movement of knee joint and ankle joint, respectively. The driving devices are placed on the fixed elements, so that the moving bars of this mechanism are not necessary to carry the weight of the driving motors, thus reducing the number of motors. Compared with conventional transmission system, this anthropomorphic mechanical leg owns the advantages of simple structure, strong carrying capacity, low moving inertia and flexible movement.

To be specific, the mechanism diagram of the anthropomorphic mechanical leg is shown in Figure 1. The mechanism is composed of a fixed platform, an active platform, and two kinematic chains. The two kinematic chains adopt the structural form of 2PUU/PRRRU, that is, the humanoid ankle joint is based on two identical branch chains PUU (Prismatic Pair-Hooke Joint-Hooke Joint) and the humanoid knee joint is based on a branch chain PRRRU (Prismatic Pair-Revolute Pair-Revolute Pair-Revolute Pair-Hooke Joint). It assumes that there are two coordinate systems { R }: O1-1XY₁Z₁ and { S }: O2-2XY₂Z₂, in which { R } is a static coordinate system and { S } is a moving coordinate system with point O₂ as its reference point, as shown in Figure 1. The following structural features can be indicated as:

1. Three linear moving pairs P₁, P₂, and P₃ are parallel with each other which are all fixed on the base.

2. A driving motor controls the input displacement of moving pair P₁ to achieve the rotation of rod FO₂ around point F, thereby realizing the plane rotation of the humanoid knee joint.

3. Two drive motors, respectively, control the input displacement of the moving pairs P₂ and P₃ to achieve the plane JMN motion, respectively, thereby realizing the left and right rollover motion and the up and down pitch motion of the humanoid ankle joint.

4. The initial pose condition of the mechanical leg can be indicated that: (a) the rod FO₂ is perpendicular to plane, MNO₂, and parallel to rod FO₁ and (b) the original lengths of P₁, P₂, and P₃ are L_i0 (i = 1, 2, and 3), respectively, and Δl_i is the corresponding input displacement.

Figure 1.

The structure of humanoid leg.

Position analysis

Position analysis is the fundamental to explore the mechanical movement capability and its applications in industry. ²⁷ As shown in Figure 1, it is to set (0 y z) ^T and (α 0 γ) ^T as the position coordinate and posture coordinate of the reference point O₂, respectively, and then it can determine the position vectors of A, B, C, D, E, F, H, and G in the static coordinate { R } according to the structural features

$$\{\begin{array}{c}{\mathit{A}}_{\mathit{R}}={\left(\begin{array}{ccc}0\hfill & 0\hfill & -a\hfill \end{array}\right)}^T\hfill \\ \multicolumn{1}{c}{{\mathit{B}}_{\mathit{R}}={\left(\begin{array}{ccc}-f/2\hfill & 0\hfill & -b\hfill \end{array}\right)}^T}\\ \multicolumn{1}{c}{{\mathit{C}}_{\mathit{R}}={\left(\begin{array}{ccc}f/2\hfill & 0\hfill & -b\hfill \end{array}\right)}^T}\\ \multicolumn{1}{c}{{\mathit{D}}_{\mathit{R}}={\left(\begin{array}{ccc}0\hfill & L_{10}+\mathrm{\Delta }{l}_1\hfill & -a\hfill \end{array}\right)}^T}\\ \multicolumn{1}{c}{{\mathit{E}}_{\mathit{R}}={\left(\begin{array}{ccc}0\hfill & m-n\sin ({\theta }_1+\theta -\pi /2)\hfill & -n\cos ({\theta }_1+\theta -\pi /2)\hfill \end{array}\right)}^T}\\ \multicolumn{1}{c}{{\mathit{F}}_{\mathit{R}}={\left(\begin{array}{ccc}0\hfill & m\hfill & 0\hfill \end{array}\right)}^T}\\ \multicolumn{1}{c}{{\mathit{H}}_{\mathit{R}}={\left(\begin{array}{ccc}f/2\hfill & L_{30}+\mathrm{\Delta }{l}_3\hfill & -b\hfill \end{array}\right)}^T}\\ \multicolumn{1}{c}{{\mathit{G}}_{\mathit{R}}={\left(\begin{array}{ccc}-f/2\hfill & L_{20}+\mathrm{\Delta }{l}_2\hfill & -b\hfill \end{array}\right)}^T}\end{array}$$

(1)

where n represents the length of rod EF and θ₁ is the deflection angle of rod FO₂ around point F (counter-clockwise rotation is positive and clockwise rotation is negative), and

$${\theta }_1=\arctan \frac{z}{y-m}$$

(2)

Then, the position vectors of M and N in { S } are

$$\{\begin{array}{c}{\mathit{M}}_S={\left(\begin{array}{ccc}-h/2\hfill & 0\hfill & -\sqrt{k^2-h^2/4}\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\mathit{N}}_S={\left(\begin{array}{ccc}h/2\hfill & 0\hfill & -\sqrt{k^2-h^2/4}\hfill \end{array}\right)}^{\mathrm{T}}}\end{array}$$

(3)

and the position vectors of M and N in { R } are

$$\{\begin{array}{c}{\mathit{M}}_R=\mathit{T}{\mathit{M}}_S+{\mathit{O}}_{2R}\hfill \\ \multicolumn{1}{c}{{\mathit{N}}_R=\mathit{T}{\mathit{N}}_S+{\mathit{O}}_{2R}}\end{array}$$

(4)

where O _2R is the position vector of the reference point O₂ in the static coordinate, and T is the conversion matrix. Based on this geometrical structure relationship between the knee joint and ankle joint, the following formula can be taken

$$\{\begin{array}{c}({\mathit{O}}_{2R}-{\mathit{F}}_R){({\mathit{O}}_{2R}-{\mathit{F}}_R)}^{\mathrm{T}}=e^2\hfill \\ \multicolumn{1}{c}{({\mathit{M}}_R-{\mathit{G}}_R){({\mathit{M}}_R-{\mathit{G}}_R)}^{\mathrm{T}}=d^2}\\ \multicolumn{1}{c}{({\mathit{N}}_R-{\mathit{H}}_R){({\mathit{N}}_R-{\mathit{H}}_R)}^{\mathrm{T}}=d^2}\\ \multicolumn{1}{c}{({\mathit{E}}_R-{\mathit{D}}_R){({\mathit{E}}_R-{\mathit{D}}_R)}^{\mathrm{T}}=c^2}\end{array}$$

(5)

The position inverse solution of this mechanism can be derived from equations (1) to (5) as follows

$$\{\begin{array}{c}\mathrm{\Delta }{l}_1=D_1-D_2\hfill \\ \multicolumn{1}{c}{\mathrm{\Delta }{l}_2=D_3-D_4}\\ \multicolumn{1}{c}{\mathrm{\Delta }{l}_3=D_5-D_6}\end{array}$$

(6)

where D₁ = m − nsin(θ − θ₁ − π/2) − L₁₀ and D₁, D₂, D₃, D₄, D₅, and D₆ are the functions of each structural parameter of the mechanical leg. For the anthropomorphic mechanical leg equation (6) is the expression of the position inverse solution of the knee joint and ankle joint, and is also the unique inverse solution.

Workspace analysis

Structural constraint

By setting σ ₁, σ ₂, λ ₁, λ ₂, ζ ₁, and ζ ₂ as the unit vectors along the direction of fixed-length rods, that is, O₁F, BG, CH, FO₂, GM, and HN, respectively, and η₁, η₂, and η₃ as the angles between fixed-length rod O₁F and rod FO₂, fixed-length rod BG and rod GM, and fixed-length rod CH and rod HN, respectively, the following equations can be obtained

$$\{\begin{array}{c}{\eta }_1=\arccos ({\mathit{\sigma }}_1•{\mathit{\lambda }}_2)\hfill \\ \multicolumn{1}{c}{{\eta }_2=\arccos ({\mathit{\sigma }}_2•{\mathit{\xi }}_1)}\\ \multicolumn{1}{c}{{\eta }_3=\arccos ({\mathit{\lambda }}_1•{\mathit{\xi }}_2)}\end{array}$$

(7)

where

$${\mathit{\sigma }}_1=\frac{({\mathit{F}}_R-{\mathit{O}}_{1R})}{m}$$

$${\mathit{\sigma }}_2=\frac{({\mathit{G}}_R-{\mathit{B}}_R)}{|{\mathit{G}}_R-{\mathit{B}}_R|}$$

$${\mathit{\lambda }}_1=\frac{({\mathit{H}}_R-{\mathit{C}}_R)}{|{\mathit{H}}_R-{\mathit{C}}_R|}$$

$${\mathit{\lambda }}_2=\frac{({\mathit{O}}_{2R}-{\mathit{F}}_R)}{e}$$

$${\mathit{\xi }}_1=\frac{({\mathit{M}}_R-{\mathit{G}}_R)}{d}$$

$${\mathit{\xi }}_2=\frac{({\mathit{N}}_R-{\mathit{H}}_R)}{d}$$

The minimum and maximum values of η₁ are expressed as η_1min and η_1max, respectively, and are similar to η₂ and η₃. Then, the angle constraint of the mechanism can be described as

$$\{\begin{array}{c}{\eta }_{1min}\le {\eta }_1\le {\eta }_{1max}\hfill \\ \multicolumn{1}{c}{{\eta }_{2min}\le {\eta }_2\le {\eta }_{2max}}\\ \multicolumn{1}{c}{{\eta }_{3min}\le {\eta }_3\le {\eta }_{3max}}\end{array}$$

(8)

where

$${\mathit{\sigma }}_i=\frac{{\mathit{C}}_i-{\mathit{B}}_i}{c}$$

$${\mathit{\eta }}_i=\frac{{\mathit{B}}_i-{\mathit{A}}_i}{\sqrt{{({\mathit{B}}_i-{\mathit{A}}_i)}^T({\mathit{B}}_i-{\mathit{A}}_i)}}$$

By setting the maximum and minimum input displacements of linear moving pairs, P₁, P₂, and P₃, L_imax and L_imin (i = 1, 2, and 3), respectively, the length constraints of these linear moving pairs can be described as

$$L_{imin}\le L_i\le L_{imax}$$

(9)

Shape of the workspace

Workspace is defined as the set of points that a given reference point on the terminal executor can reach, and it is an essential performance index for structure analysis. It is to set the structural parameters of the knee joint and ankle joint, and the geometric structural constraint conditions according to Table 1. Thus, a workspace skeleton diagram can be drawn according to equations (1)–(9), as shown in Figure 2.

Table 1.

The structural parameters of humanoid leg.

a (mm)	b (mm)	c (mm)	d (mm)	e (mm)	f (mm)	h (mm)	k (mm)	m (mm)	n (mm)	L 10 (mm)	L 20 (mm)	L 30 (mm)	θ (°)	η 1min (°)	η 2min = η3min (°)	η 2max = η3max (°)
150	300	150	440	440	300	300	335	440	175	200	300	300	120	90	0	40

Figure 2.

(a) Shape of position workspace and (b)–(g) shapes of workspace on different constant positions.

As can be seen from Figure 2(a) that the position workspace of the reference point of this anthropomorphic knee joint and ankle joint structure covers a quarter of a circle, and by combining with the findings from Figure 2(b) to (g) it can be indicated that:

1. The posture work space is symmetric about the α axis.

2. With the increase of z, the section of posture workspace decreases gradually.

3. With the increase of y, the range of α increases gradually.

4. The range of position workspace is 440 mm ≤ y ≤ 880 mm and 0 mm ≤ z ≤ 440 mm.

5. When y = 800 mm and z = 0 mm, the section of posture workspace is found to be the largest.

6. The approximate range of flexible posture workspace is −2 rad ≤ α ≤ 2 rad and 0.5 rad ≤ γ ≤ 0.5 rad.

7. The posture workspace is minimized when the mechanical leg is at the edge of the position workspace.

Effect of the parameters on the size of the workspace

To intuitively explore the effect of geometric parameters on the size of the workspace size in this mechanism, the workspace value (WSV), which is defined as the number of points that can be searched out in a certain step length, is employed to evaluate the size of workspace. The standard value of WSV is defined as W₀ by setting the structural parameters according to Table 1, and then the workspace relative value (WSRV) can be expressed as

$$WSRV=\frac{WSV}{W_0}\times 100\%$$

(10)

Thus, according to equations (1)–(10) the effect of structural parameters on size of the workspace size can be determined as illustrated in Figure 3.

Figure 3.

Effects of b, d, f, and h on the size of the workspace, respectively.

It can be found from Figure 3 that the overall effects of structural parameters on the size of the workspace, respectively. To be specific, it can be found that:

1. The WSRV decreases with an increase of b.

2. The WSRV decreases with an increase of f and h, respectively, in which f and h are the same length as shown in Figure 1.

3. When 200 mm ≤ d ≤ 600 mm, the WSRV could increase with an increase of d, however, when 600 mm ≤ d ≤ 800 mm, the WSRV could decrease with an increase of d.

4. When b and f increase, the decrease in the ratio of Δl_i to b or f will cause the range of activity of A_R and B_R to become smaller.

Kinematic performance analysis

It is to set V  = (V_Y V_Z ω_Z ω_X) ^T as the generalized speed vector of the reference point O₂ in the static coordinate { R }, and v  = (v₁ v₂ v₃) ^T as the input speed vector where v_i denotes the input speed of P_i (i = 1, 2, and 3). According to structural features of the knee joint and ankle joint, the following formula can be obtained

$$\{\begin{array}{c}n\sin (\theta -\pi /2-{\theta }_1)+c\cos {\phi }_1=m-L_{10}-\mathrm{\Delta }{l}_1\hfill \\ \multicolumn{1}{c}{n\cos (\theta -\pi /2-{\theta }_1)+c\sin {\phi }_1=a}\end{array}$$

(11)

where φ₁ is the rotation angle of the rod DE around point D. Taking a derivative of equation (10) with respect to time, the rotation angle speed of the coordinate FO₂ around point F can be calculated as follows

$$\begin{array}{cc}\multicolumn{1}{c}{{\omega }_1=}& v_1/(n\cos (\theta -\pi /2-{\theta }_1)\hfill \\ \multicolumn{1}{c}{}& -n\sin (\theta -\pi /2-{\theta }_1)\tan {\phi }_1\hfill \end{array})$$

(12)

where the counter-clockwise direction is positive for ω₁, and vice versa. Then, the linear velocity vector of reference point O₂ is expressed as

$$\mathit{V}\prime =\left(\begin{array}{cc}{V}_Y\hfill & V_Z\hfill \end{array}\right)={\mathit{n}}_{1}e{\omega }_1$$

(13)

where n ₁ is the unit vector along the perpendicular direction to the rod FO₂, and V ′ is only affected by the angle speed, ω₁. Then, the following formula can be derived from equations (11) to (13), as detailed

$$\mathit{V}\prime ={\mathit{n}}_1{K}_1{v}_1$$

(14)

where

$$K_1=e/(n\cos (\theta -\pi /2-{\theta }_1)-n\sin (\theta -\pi /2-{\theta }_1)\tan {\phi }_1)$$

(15)

Based on the structural features of the knee joint and ankle joint, and by assuming the speed vector components of points M and N in the static coordinate as V _M and V _N , respectively, the following formula can be obtained

$$\{\begin{array}{c}{\mathit{V}}_M={\mathit{V}}_{O2}+\mathit{\omega }\times {\mathit{M}}_R\hfill \\ \multicolumn{1}{c}{{\mathit{V}}_N={\mathit{V}}_{O2}+\mathit{\omega }\times {\mathit{N}}_R}\end{array}$$

(16)

where V _O2 and ω are the linear speed vector and angular speed vector of the reference point O₂ in the static coordinate, respectively, and

$$\{\begin{array}{c}{\mathit{V}}_{O2}={\left(\begin{array}{cc}{V}_Y\hfill & V_Z\hfill \end{array}\right)}^T\hfill \\ \multicolumn{1}{c}{\mathit{\omega }={\left(\begin{array}{cc}{\omega }_X\hfill & {\omega }_Z\hfill \end{array}\right)}^T}\end{array}$$

(17)

It is to set e _i as the unit direction vector of moving pairs P_i (i = 1, 2, and 3), and n ₂ and n ₃ as the unit direction vectors of the fixed-length rod GM and HN, respectively. When this mechanism is not in the singular pose, the speed vectors of points M and N along rods GM and HN can be defined as

$$\{\begin{array}{c}\mathit{v}{\prime }_2={\mathit{V}}_M•{\mathit{n}}_2\hfill \\ \multicolumn{1}{c}{\mathit{v}{\prime }_3={\mathit{V}}_N•{\mathit{n}}_3}\end{array}$$

(18)

$$\{\begin{array}{c}\mathit{v}{\prime }_2=v_2{\mathit{e}}_2•{\mathit{n}}_2\hfill \\ \multicolumn{1}{c}{\mathit{v}{\prime }_3=v_3{\mathit{e}}_3•{\mathit{n}}_3}\end{array}$$

(19)

By combining equations (11)–(19), the following formula can be derived

$$\left[\begin{array}{c}{({\mathit{M}}_R\times {\mathit{n}}_2)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{({\mathit{N}}_R\times {\mathit{n}}_3)}^{\mathrm{T}}}\end{array}\right]\mathit{\omega }\prime =\left[\begin{array}{ccc}-K_1{\mathit{n}}_1•{\mathit{n}}_2\hfill & {\mathit{e}}_2•{\mathit{n}}_2\hfill & 0\hfill \\ \multicolumn{1}{c}{-K_1{\mathit{n}}_1•{\mathit{n}}_3}& 0\hfill & {\mathit{e}}_3•{\mathit{n}}_3\hfill \end{array}\right]\mathit{v}$$

(20)

$$\mathrm{where}\mathit{\omega }\prime ={\left(\begin{array}{ccc}{\omega }_Z\hfill & 0\hfill & {\omega }_X\hfill \end{array}\right)}^{\mathrm{T}}$$

(21)

and M _R and N _R are the position vectors of points M and N, respectively, in the static coordinate.

Thus, it can derive the following formula from equation (20)

$$\mathit{\omega }={\mathit{J}}_{\omega }\mathit{v}$$

(22)

where $\mathit{\omega }={\left(\begin{array}{cc}{\omega }_Z\hfill & {\omega }_X\hfill \end{array}\right)}^{\mathrm{T}}$

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{J}}_{\omega }=}& {\left[\begin{array}{cc}{\mathit{T}}_{\omega 1}\left(\begin{array}{cc}1\hfill & 1\hfill \end{array}\right)\hfill & {\mathit{T}}_{\omega 1}\left(\begin{array}{cc}1\hfill & 3\hfill \end{array}\right)\hfill \\ \multicolumn{1}{c}{{\mathit{T}}_{\omega 2}\left(\begin{array}{cc}1\hfill & 1\hfill \end{array}\right)}& {\mathit{T}}_{\omega 2}\left(\begin{array}{cc}1\hfill & 3\hfill \end{array}\right)\hfill \end{array}\right]}^{-1}\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{ccc}-K_1{\mathit{n}}_1•{\mathit{n}}_2\hfill & {\mathit{e}}_2•{\mathit{n}}_2\hfill & 0\hfill \\ \multicolumn{1}{c}{-K_1{\mathit{n}}_1•{\mathit{n}}_3}& 0\hfill & {\mathit{e}}_3•{\mathit{n}}_3\hfill \end{array}\right],\hfill \end{array}$$

$${\mathit{T}}_{\omega 1}={({\mathit{M}}_R\times {\mathit{n}}_2)}^{\mathrm{T}},\mathrm{and}$$

$${\mathit{T}}_{\omega 2}={({\mathit{N}}_R\times {\mathit{n}}_3)}^{\mathrm{T}}$$

By considering that the linear speed vector V ′ is only affected by the angular speed ω₁, it can derive the speed Jacobian matrix of this mechanism with respect to ω₁ from equations (11) and (22) as follows

$$\mathit{V}″={\left[\begin{array}{ccc}{\omega }_1\hfill & {\omega }_{\mathit{X}}\hfill & {\omega }_{\mathit{Z}}\hfill \end{array}\right]}^{\mathrm{T}}=\mathit{Jv}$$

(23)

where J ∈ R ^3×3 is the speed Jacobian matrix of this mechanical arm, and

$$\mathit{J}=\left[\begin{array}{c}\mathit{J}{\prime }_V\hfill \\ \multicolumn{1}{c}{{\mathit{J}}_{\omega }}\end{array}\right]$$

(24)

in which

$$\begin{array}{c}{\mathit{J}\prime }_V=\hfill \\ {\left[\begin{array}{ccc}1/(n\cos (\theta -\pi /2-{\theta }_1)-n\sin (\theta -\pi /2-{\theta }_1)\tan {\phi }_1)\hfill & 0\hfill & 0\hfill \end{array}\right]}^{\mathrm{T}}\hfill \end{array}$$

(25)

When this mechanism is not in the singular pose, to consider that the linear speed and angular speed have different dimensions, equation (23) can be rewritten into the following form

$$\{\begin{array}{c}{\omega }_1={\mathit{J}}_V{v}_1\hfill \\ \multicolumn{1}{c}{\mathit{\omega }={\mathit{J}}_{\omega }\mathit{v}}\end{array}$$

(26)

where J _V and J _ω denote the linear speed and angular speed Jacobian matrices of the knee joint and ankle joint mechanism, respectively. In general, these equations present the relationship between the input speed and output speed.

Evaluation index of kinematic performances

When the mechanism is not in the singular pose, Jacobian matrix J _ω has singular value decomposition. Then, two orthogonal matrices A _ω ∈ R ^3×3 and B _ω ∈ R ^3×3 could satisfy the following equation

$${\mathit{J}}_{\omega }={\mathit{A}}_{\omega }{\mathit{\Lambda }}_{\omega }{\mathit{B}}_{\omega }^{\mathrm{T}}$$

(27)

where ${\mathit{\Lambda }}_{\omega }=\left[\begin{array}{ccc}{\sigma }_{1\omega }\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\sigma }_{2\omega }\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]$ , σ_1ω and σ_2ω are two singular values of J _ω satisfying σ_1ω ≥ σ_2ω. When the mechanism is not in the singular pose, by setting the input vector as unit vector the following equation can be obtained

$${\mathit{v}}^{\mathrm{T}}\mathit{v}=1$$

(28)

By combining equations (24)–(28), it can obtain

$$\frac{\omega {\prime }_Z^2}{{\sigma }_{1\omega }^2}+\frac{\omega {\prime }_X^2}{{\sigma }_{2\omega }^2}=1$$

(29)

where $\omega {\prime }_Z=a_{B11}{\omega }_X+a_{B21}{\omega }_Z$ and $\omega {\prime }_X=a_{B12}{\omega }_X+a_{B22}{\omega }_Z$ .

According to equation (29), when the input speed is a unit vector, the output angular speed distributes in this ellipse, ²⁸ and its axial lengths are σ_1ω and σ_2ω, respectively. To evaluate the speed transmission performance of this mechanical arm, the evaluation index of linear speed transmission performance and angular speed transmission performance, namely, K_V and K_η, are defined as follows

$$\{\begin{array}{c}{K}_V=\Vert {\mathit{J}}_V\Vert \hfill \\ \multicolumn{1}{c}{K_{\eta }={\sigma }_{1\omega }}\end{array}$$

(30)

where J _V denotes the linear speed Jacobian matrix and σ_1ω and σ_2ω are two singular values of J _ω satisfying σ_1ω ≥ σ_2ω. In the workspace of anthropomorphic mechanical leg, the evaluation index of speed transmission performance varies with the posture. Hence, the mean values of K_V and K_η are defined as the evaluation index of global linear-speed-transmission performance R(V) and global angular-speed-transmission performance R(η) which are

$$\{\begin{array}{c}R\left(V\right)=\frac{\sum _V{K}_V}{V}\hfill \\ \multicolumn{1}{c}{R\left(\eta \right)=\frac{\sum _V{K}_{\eta }}{V}}\end{array}$$

(31)

where V denotes the workspace of anthropomorphic mechanical leg and $\sum _L{K}_V$ and $\sum _V{K}_{\eta }$ denote the sums of K_V and K_η for points within the workspace, respectively.

Static performance analysis

It assumes that the base and each moving element are rigid bodies, and the friction force can be ignored. It is to set F  = (F_X F_Y M_Z M_X) ^T as the broad output force vector of reference point O₂, and f ′ = (f₁ f₂ f₃) ^T as driving force, where f _i (i = 1, 2, and 3) is the driving force along P_i(i = 1, 2, and 3). Further, F ′ = (F_X F_Y) ^T is only determined by the moment of the rod FO₂ around the point F(M₁). To simplify the analysis, it is to set F ″= (M₁ M_Z M_X) ^T as the broad output force vector of the reference point O₂. Based on the principle of virtual work, it can generate the relationship between f ′ and F ″. Under the action of external force F ″, the broad virtual displacement of reference point O₂ can be set as δ ι′, and the corresponding virtual displacement input is set as δp ′. Thus, the sum of virtual work done by driving force f of all joints is obtained

$$W{\prime }_1=\mathit{f}\prime •\mathit{\delta }\mathit{p}\prime$$

(32)

where

$$\mathit{\delta }\mathit{p}\prime ={\left(\begin{array}{ccc}\delta {p\prime }_1\hfill & \delta {p\prime }_2\hfill & \delta {p\prime }_3\hfill \end{array}\right)}^T$$

(33)

The sum of virtual work done by external force F ″ is

$$W{\prime }_2=\mathit{F}″•\mathit{\delta }\iota$$

(34)

where

$$\mathit{\delta }\iota \prime ={\left(\begin{array}{ccc}\delta {\iota \prime }_1\hfill & \delta {\iota \prime }_Z\hfill & \delta {\iota \prime }_X\hfill \end{array}\right)}^T$$

(35)

According to the principle of virtual work, when this mechanism is in the equilibrium state, the sum of all elementary work done by each external force equals zero, as follows

$$W{\prime }_1+W{\prime }_2=0$$

(36)

Virtual displacement δ ι′ is not independent on δp ′, and is determined by the speed Jacobian matrix of the parallel mechanism

$$\mathit{\delta }\iota \prime =\mathit{J}•\mathit{\delta }\mathit{p}\prime$$

(37)

Moreover, by combining equations (32)–(37) the following formula can be derived

$$\mathit{F}″=\mathit{G}•\mathit{f}\prime$$

(38)

where

$$\mathit{G}={\left({\mathit{J}}^{-1}\right)}^T$$

(39)

where G is the Jacobian matrix of this mechanism. According to equation (37), it can obtain

$$\mathit{F}\prime ={\left(\begin{array}{cc}{F}_X\hfill & F_Y\hfill \end{array}\right)}^T=\frac{{\mathit{n}}_1{M}_1}{e}$$

(40)

Static performance index

Since the force and moment have different dimensions, equation (39) can be rewritten into the following expression

$$\{\begin{array}{c}{M}_1={\mathit{G}}_F•\mathit{f}\prime \hfill \\ \multicolumn{1}{c}{\mathit{M}={\mathit{G}}_M•\mathit{f}\prime }\end{array}$$

(41)

where G _M is the last two lines of the force Jacobian matrix G . When this mechanism is not in the singular pose, according to the matrix theory the force Jacobian matrix G has singular value decomposition, and then two orthogonal matrices C _M  ∈  R ^3×3 and D _M  ∈  R ^3×3 satisfy the following equation

$${\mathit{G}}_M={\mathit{C}}_M{\mathit{\Lambda }}_M{\mathit{D}}_M^{\mathrm{T}}$$

(42)

where ${\mathit{\Lambda }}_M=\left[\begin{array}{ccc}{\sigma }_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\sigma }_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]$

where σ₁ and σ₂ are two singular values of G _M , satisfying σ₁ ≥ σ₂. When this mechanism is not in the singular pose, it is to set the input vector as unit vector, that is

$${\mathit{f}\prime }^T\mathit{f}\prime =1$$

(43)

By combining equations (41)–(43), the following formula can be obtained

$$\frac{M{\prime }_X^2}{{\sigma }_1^2}+\frac{M{\prime }_Z^2}{{\sigma }_2^2}=1$$

(44)

where $M{\prime }_X=c_{B11}{M}_X+c_{B21}{M}_Z$ and $M{\prime }_Z=c_{B12}{M}_X+c_{B22}{M}_Z$ .

According to equation (44), when the input force is a unit vector the output force distributes in the ellipse^29,30 with its axial length of σ₁ and σ₂, respectively. As the force Jacobian matrix G changes with the moving platform pose, to evaluate the force transmission performance of this mechanism, it is to define the force transmission performance evaluation index and moment transmission performance evaluation index as K_F and K_M, respectively

$$\{\begin{array}{c}{K}_F=\Vert {\mathit{G}}_F\Vert \hfill \\ \multicolumn{1}{c}{K_M={\sigma }_1}\end{array}$$

(45)

In the workspace of anthropomorphic mechanical legs, force and moment transmission performance evaluation indices have different values in different poses. Hence, the average values of K_F and K_M in the workspace of anthropomorphic mechanical leg are defined as the global force transmission performance evaluation index R(F) and torque transmission performance evaluation index R(M), respectively

$$\{\begin{array}{c}R\left(F\right)=\frac{\sum _V{K}_F}{V}\hfill \\ \multicolumn{1}{c}{R\left(M\right)=\frac{\sum _V{K}_M}{V}}\end{array}$$

(46)

where $\sum _V{K}_F$ denotes the sum of all K_F in the workspace and $\sum _V{K}_M$ denotes the sum of all K_M in the workspace.

Global performance atlases

To overcome the issues of current multiple parameter design methods, the plane model of solution space for multi-parameters technique is proposed. In this technique, more than four parameters are dimensionless, and each structural parameter is selected as side length to draw a polygon. Then, it is to draw vertical lines of each side crossing a point in the polygon, and define the intersection of each vertical line and the side as the value of structural parameter. Thus, the entire region within the polygon represents the set of all structural parameters, which can provide the distribution rule of each structural parameter and the corresponding global performance index.

This type of knee joint and ankle joint mainly have nine structural parameters as: a, b, c, f, θ, e, d, k, and h. Considering the structure form of anthropomorphic mechanical leg, structural parameter optimization of thigh and shank will be discussed in detail respectively. As h = f and k = b/cos(θ₁/2), structural parameter, a, has no effect on the kinematic performances of entire anthropomorphic mechanical leg mechanism, so that the structural parameters are actually b, c, d, e, f, and θ. It is to establish the space model for the main structural size parameters of anthropomorphic mechanical leg, and make the structural size parameters dimensionless. First, it is to set

$$P=\frac{b+c+d+e+f+\theta }{6}$$

(47)

Then, k₁ =b/P, k₂ = c/P, k₃ = d/P, k₄ = e/P, k₅ = f/P, and k₆ = θ/P can be obtained. Considering the requirements of structure and assembly technology, it define the value range of k₁, k₂, k₃, k₄, k₅, and k₆ as

$$\{\begin{array}{c}{k}_1\le k_2,k_3,k_4,k_5,k_6\le 6\hfill \\ \multicolumn{1}{c}{0<k_1\le 6}\end{array}$$

(48)

According to equations (47) and (48), it is to draw regular hexagon $\mathrm{MNRTSG}$ with k₁, k₂, k₃, k₄, k₅, and k₆, and then obtained hexagon is the set of all structural parameters. It is to draw vertical lines of each side crossing a random point B in the hexagon, and the value of each structural parameter can be determined, as shown in Figure 4. Each side length in xy coordinate has its vector expression.

Figure 4.

The plane model of multiple parameters for the mechanical leg.

Parametric design

To optimize the operating parameters in industry, different numerical and mathematical models are usually employed.^31–34 In this study, the statistics parameters optimization method based on the plane model of solution space for multiple parameters is proposed for the mechanical leg. The method aims at multiple parameters which have uniformly distributed characteristics, and regarding the values of multiple evaluation indices as objectives. Then it selects proper parameters according to the model calculation and distribution regulation of the values of the statistical sampling, the proposed statistics parameters optimization method can be used to select reasonable structure geometric parameters.

As can be seen from Figure 5 that the maximum values of values of R(V), R(η), R(F), and R(M) are 0.9132 m/s, 1.1937 rad/s, 3.9086 N, and 4.0183 N m, respectively, while the minimum values are 0.3218 m/s, 0.2018 rad/s, 0.5321 N, and 0.3561 N m, respectively. It is to set the middle values of the performance evaluation indices as design objective, that is, R(V)=0.6265 m/s, R(η)=0.6978 rad/s, R(F)=2.2204 N, and R(M)=2.1872 N m. When R(V)≥0.6265 m/s, R(η)≥0.6978 rad/s, R(F)≥2.2204 N, and R(M)≥2.1872 N m, each performance evaluation index is perfect. By uniformly sampling in the range of main design parameters, the distribution rule of each parameter sample is obtained when the design objective is satisfied, and a frequency histogram is drawn as shown in Figure 6, in which horizontal axis denotes the value of structural parameter. By giving a structural parameter, many groups of parameter combination can be obtained, and the performance evaluation index for each structural parameter group can be calculated. Thus, the ratio of the group number of structural parameters that are better than the design objective to the whole group number can be obtained, and b, c, f, θ, e, and d are calculated in order to represent the probability of reaching the objective value.

Figure 5.

Global transmission evaluation index atlases of the upper leg: (a) $R(V)$ (m/s), (b) $R(\eta )$ (rad/s), (c) $R(F)$ (N) and (d) $R(M)$ (N m).

Figure 6.

Histograms of the parameters optimization for the humanoid leg: (a) histograms of length of b, (b) histograms of length of c, (c) histograms of length of f, (d) histograms of length of θ, (e) histograms of length of e, and (f) histograms of length of d.

In Figure 6, f(b), f(c), f(f), f(θ), f(e), and f(d) represent the probability of reaching the objective value. Through above analysis and the consideration of machining and assembly technology, the following structure geometric parameters are selected for this shoulder joint structure: b = 200 mm, c = 150 mm, d = 450 mm, e = 450 mm, f = 310 mm, and θ = 120°. In this way, f(b), f(c), f(d), f(e), f(f), and f(θ) have higher probability, R(V) = 0.8927 m/s, R(η) = 0.8763 rad/s, R(F) = 3.1856 N, and R(M) = 3.9201 N m.

According to the optimizing results for anthropomorphic mechanical leg and the consideration of machining and assembly technology, a design scheme is proposed as shown in Figure 7(a). This anthropomorphic mechanical leg is connected in series with a 3-degree-of-freedom (DOF) spherical parallel mechanism (serving as the prototype of hip joint mechanism), forming a design scheme of anthropomorphic lower limb as shown in Figure 7(b). The knee joint and ankle joint of this anthropomorphic mechanical leg mechanism adopt 4-DOF under-actuated mechanism as the prototype, and achieve the movement of knee joint and ankle joint by three motors. The fixed components such as motors are near waist, which contributes to the reduction of generated moment and moving inertia, and to the increase of the carrying capacity of leg. Thus, the advantages of parallel mechanism can be fully presented. Meanwhile, the lower limb of this anthropomorphic mechanical leg can achieve any gait and actively control various gaits. To sum up, this anthropomorphic lower limb exhibits several advantages, such as simple structure, strong load capacity, low moving inertia, and stable movement.

Figure 7.

The design plan of the humanoid leg: (a) mechanical leg, (b) the biped robot, and (c) prototype of mechanical leg.

Conclusion

1. A novel anthropomorphic mechanical leg was proposed, and the plane model of solution space for multiple parameters and the statistics parameters optimization method based on the plane model of solution space for multiple parameters were proposed. Several kinematic and static transmission performance evaluation indices and global performance evaluation index were defined based upon the kinematic and static analyses.

2. The shapes of the workspace and effects of parameters on the workspace volume were drawn based upon the kinematics and static analyses of the novel mechanical leg, and the global performance evaluation index atlases were drawn based upon the plane model of solution space for multiple parameters.

3. When y = 880 mm and z = 0 mm, the volume of workspace is bigger. The posture work space is symmetric about the α axis and with the increase of z, while the section of posture work space decreases gradually.

4. According to kinematics analyses in the solutions of the plane models for multiple parameters, the maximum and minimum values of R(V), R(η), R(F), and R(M) are 0.9132 m/s, 1.1937 rad/s, 3.9086 N, and 4.0183 N m.

5. Considering the assembly technique, a design scheme and prototyping of mechanical leg was given by using the statistics parameters optimization method, and the present analyses provided a basis for applications of the novel mechanical leg.