To Design a Small Pneumatic Actuator Driven Parallel Link Mechanism for Shoulder Prostheses for Daily Living Use

3.1. Arm structure

The details of the arm structure (in the following explanation, denoted as the Arm) are shown in Fig. 2. The Arm is composed of three segments. Segment 1 links two disks called the Base 1 and the Platform 1 with the Backbone 1 and three pneumatic actuators, placed equiangularly with respect to the center of the Base 1. The Backbone 1 is fixed to the center of the Base 1, and connected to the center of the Platform 1 with two passive revolute joints. To simplify the analysis, the Backbone 1 was assumed as a compression spring that can only move along longitudinal direction, but not as a rubber rod as described in the design concept section. By assembling Base 1 and Platform 1 with a compressed Backbone 1, actuators and wires that connect the pneumatic actuators with two disks are constantly loaded. This allows the Platform 1 to move along the longitudinal direction of the Backbone 1, turn around the joint of Platform 1 and Backbone 1, as a result of length changes of the three actuators.

The Platform 1 disk of Segment 1 is also used as the Base 2 disk of the Segment 2, which has a similar structure with the Segment 1, but with a different length. Segment 3 contains only a rigid rod (Rod) fixed to the center of outside the Platform 2, i.e. the Base 3 of the Segment 3. For the convenience of description, let h ₁, h ₂ and l _R be the initial length of the Backbone 1, 2 and Rod respectively.

3.2. Kinematics

In order to analyze the behavior of end-effector and working space of the Arm, forward and inverse kinematics model of the parallel link mechanism were derived.

The coordinate system of the Arm is shown in Fig. 3. Without loss of the generality, the thicknesses of all disks, and the shaft diameters of Backbone 1, 2, Rod were set to 0. The global coordinate system O _B1-XYZ is located at the center of the Base 1, with the Z-axis directed along the Backbone 1. The contact points of three pneumatic actuators to Base 1 (B ₁₁ , B ₁₂ , B ₁₃) were aligned equiangularly along the peripheral of a circle with a radius r _B , and B ₁₁ is on the X-axis.

The local coordinate system O _P1-x _P1 y _P1 z _P1 locates at the center of disk Platform 1, h ₁ away from O _B1along the Z-axis. The contact points of pneumatic actuators to Platform 1 (P ₁₁ , P ₁₂ , P ₁₃) are on radius r _P . In turn, the contact points of three pneumatic actuators to Base 2 (B ₂₁ , B ₂₂ , B ₂₃) are equiangularly set on circumference of a circle, radius r _B , and B ₂₁is on the x _P1-axis. The local coordinate systemO _P2 -x _P2 y _P2 z _P2 is set at the center of Platform 2, and the distance from O _P1is h ₂. The contact points (P ₂₁ , P ₂₂ , P ₂₃) are aligned equiangularly along the peripheral of a circle with a radius r _P , and P ₂₁ is on the x _P2-axis.

Finally, the Rod of length l _R is fixed up at O _P2 along the z _P2-axis.

Suppose the actuators are activated, and their lengths change (expressed discretely: l _i gets to l _i ’, i =1,..., 6). Therefore, h ₁ and h ₂ are converted to h ₁ ’ and h ₂ ’, two passive joints (Joint 1) of Platform 1 and the one (Joint 2) of Platform 2 rotate by α,β, γ, σ (see Fig. 2, 3), respectively.

At first, the position of O _P1, P _1i ,P _1i ’(i=1, 2, 3)andthe rotation matrix R ₁ of Joint 1 in O _P1-x _P1 y _P1 z _P1, i.e. ^P1 O _P1, ^P1 P _1i , ^P1 P _1i ’ and ^P1 R ₁ can be presented as follows:

And,

Therefore, the coordinate of each element of ^P1 P _1i ’ is:

Next, P _1i ’, B _1i ’in O _B1-XYZ, i.e. ^B1 P _1i ’ and ^B1 B _1i ’ can be defined as:

So, each element of ^B1 P _1i ’ can be expressed as:

Accordingly, the length of each actuator can be defined:

By Equation (5), (7) and (8), the equations describing the relation between the lengths of pneumatic actuators, wires attached in Segment 1, Backbone 1, and the angles of two revolute joints can be defined as follows:

Equation (9) is a simultaneous equation with three unknown variables, α,β,h ₁’. In the case that the lengths l ₁’, l ₂’, l ₃’ are given, it is possible to calculate α,β,h ₁’ by using the Newton method (Ku, 1999; Merlet, 1993; Press et al., 1992). Similarly, for the Segment 2 (i.e. in O _P1-x _P1 y _P1 z _P1), the following equation can be derived. In the case that the length l ₄’, l ₅’, l ₆’, are given, it is possible to calculate γ, σ,h ₂’.

The position of the Rod end P _RE ’ in O _P2 -x _P2 y _P2 z _P2, i.e. ^P2 P _RE ’ can be defined as:

Let x, y, z be the coordinate of ^B1 P _RE ’, then end position of the Rod, x, y, z in O _B1-XYZ can be described as:

3.3. Jacobian matrix

In order to evaluate the motion characteristics of the Arm, it is necessary to develop theJacobian matrix of the Arm structure.

As l _i ’(i=1, ,6), α,β,h ₁’, γ, σand h ₂’ can be taken as functions of time t, noticing l _i ’,x, y, zare functions of α,β,h ₁’, γ, σ, h ₂’, the Equation (9), (10), (12) can be differentiated with respect to t, to get the following equations. The A, B, Care the matrixes with element a _ij , b _ij (i,j=1,2,3), c _ij (i=1,2,3, j=1, , 6), respectively.

Next, the orientation of ^B1 P _RE ’ in O _B1-XYZ can be expressed by Equation (15), where the element of the matrix ^P1 R ₁ ^P2 R ₂ is presented by r _ij (i,j=1,2,3).

By using Equation (15), the Euler angles (ϕ,θ, ψ) can be acquired (Yoshikawa, 1988).

Equation (17) can be derived by differentiating Equation (16) with regard to time t (Yoshikawa, 1988):

ϕ,θ, ψ are functions of α,β, γ, σ. Therefore, by using a matrix D with element d _ij (i=1, 2, 3, j=1, 2, 4, 5), (ϕ,θ, ψ) ^T can be expressed as:

Equation (14) and (18) can be integrated to the following equation.

Let inverse matrix of A and B be A ^-1 and B ^-1, which comprise elements a ^-1 _ij , b ^-1 _ij (i,j=1,2,3), then α,β,h ₁’, γ, σ, h ₂’can be derived from Equation (13).

Therefore, by using Equation (19) and (20), the vector representing the posture of end-effector can be acquired.

From the above equation, the Jacobian matrixJ ₁ can be calculated.

3.4. Static mechanics

Let force generated and virtual displacement of each actuator be τ andΔl, and the force generated and virtual displacement of Rod end P _RE ’ be F and Δx. From virtual work principle, the relationship between these two pairs is:

By using Equation (21)

Therefore, from Equation (22) and (23), the relation between F and τis acquired.

4.1. An estimative index of manipulability: condition number

The condition number (Arai, 1992) was employed as evaluation indicator for the motion characteristics of the Arm mechanism. The condition number is based on the singular value of the Jacobian matrix. The Equation (24) can be described as the expression for how τis convertedinto F. Furthermore, a singular value decomposition, expressed by Equation (25), can make the property of Jeven clearer.

Here, Uand V are 6x6 orthogonal matrixes, which can be described by Equation (26).

Substituting Equation (25) into (24), the relation between τand F can be rewritten as Equation (27).

Equation (26) and (27) can be rewritten using the elements of Uand V.

Considering the function of the manipulator, it is preferable that the forces that could be generated at the end of the Rod in all direction are as uniform as possible. That is, it is the ratio of the maximum singular value to the minimum one, i.e., the condition number, should be close to 1 as much as possible.

Since the condition number of J ^T reflects the both the forceand torque working at the end of the Rod, in order to conduct proper evaluations, it is necessary to separate the influence of forceand torque. Therefore, in the Equation (24), J ^T is separated into the part contributing to the force and the one contributing to the torque, and singular value decomposition was conducted at two parts separately. Therefore, J ^T is separated as follows.

Here,J _f ^T andJ _m ^T are 3x6 matrixes, so, three singular values σ _fi , σ _mi (i=1,2,3) exist in each of J _f ^T and J _m ^T . Thus, we use the following three condition numbers as estimative index:

4.2. An outline of the evaluation process

The following is an outline of the evaluation process.

1. Setting up a coordinate space Σ _CS1;

2. Setting up an initial configuration (physical dimension of the Arm mechanism), and modelling the Arm and human body in Σ _CS1;

3. Defining EFAA (Expected Frequently Accessed Area) and RA (Reachable Area) of the Arm in Σ _CS1;

4. For different length of pneumatic actuators, reflecting translational motion of the actuators, numerically calculating andplotting the Rod end positionP _RE ’;

5. Calculating the estimative indexes for all the P _RE ’ in EFAA and RA;

6. Changing the parameters of the Arm mechanism, and going back to recalculating Step 4;

7. After a certain number of loops of execution (Step 4 to 6), evaluating all the configurations to decide configurations optimal for the spatial accessibility (plot number in EFAA) and manipulability.

4.4. Important areas in the working space of the Arm

Two important areas in the working space of the Arm are defined for the analysis and evaluation of the Arm. One is the area close to the chest and the median sagittal plane, which is expected to be accessed very frequently during most daily living tasks. This area is the EFAA defined before, and expressed as Σ _EFAA . Another is the area that represents the reachable area [RA] of the end effector, defined as Σ _RA . Geometries of Σ _EFAA and Σ _RA are illustrated as in Fig. 6. The volumes of Σ _EFAA and Σ _RA are set to 12000cm³ and 75000cm³, respectively.

4.5. Calculation and analysis

All the calculation was calculated numerically by using Matlab (MathWorks, Inc.). A simplified Arm, as shown in Fig. 7, is drawn for visualization in Matlab. P _RE ’ is calculated by substituting the initial values to Equation (12), with variable length of actuators, which stands for the translational motion of the pneumatic actuators. Three different values were set for each l _i ^’ (i=1, ,6) in Fig. 3(b). Supposing that the resting length of the actuators are LW _i (i=1, ,6), and the maximum,minimum, middle increment of the actuators are L _max , L _min , L _mid , then the three different values are LW _i + L _max , LW _i + L _min , LW _i +L _mid 　(Fig. 8). Thus, for each Arm configuration, a total number of 3⁶=729 sets of calculation were calculated for P _RE ’, then the configuration could be evaluated.

The estimative index described in section 4.1 was adapted as follows, for evaluating different aspects of the system.

• N _EFAA : Number of points plotted in Σ _EFAA (Spatial accessibility)

• N _RA : Number of points plotted in Σ _RA (Spatial accessibility)

• M _t (C): 15percent trimmed mean condition number C (Eq. 30) of points plotted in Σ _EFAA (Manipulability)

• M _t (C _f ): 15percent trimmed mean condition number C _f (Eq. 31) of points plotted in Σ _EFAA (Manipulability)

• M _t (C _m ): 15percent trimmed mean condition number C _m (Eq. 32) of points plotted in Σ _EFAA (Manipulability)

After the calculation and evaluation, the physical dimensions of the Arm structure were changed, and the calculation and evaluation for the new configuration were repeated. Note, in this chapter, only the results of changing h ₂ and l _R are to be reported.

From this process, optimal configurations of the Arm structure could be determined.

5.1. Results of the initial configuration

A plot of P _RE ’ for the Arm structure with the initial configuration is shown in Fig. 9, where points in red, blue and gray stand for the P _RE ’ located in Σ _EFAA , in Σ _RA and outside of the both areas, respectively. Basically, the group of points is longitude-axis-symmetric.

Table 1 shows the values of estimative indexes defined in section 4.5

5.2. Results of other configurations generated by changing parameters

Suppose that the parameters h ₂ and l _R are changed as in Equation (34), where, beginning with 170mm, 250mm (parameters of the initial configuration) h ₂ and l _R increase or decrease incrementally by 25mm and 50mm, respectively, and i is an integer, taking value 0, 1, 2.

Therefore, a total number of 25 combinations could be made, and identified with No.1-1, , No.1-25, as shown in Table 2. The estimative indexes were calculated for all the combinations. The N _EFAA and N _RA are shown in Fig. 10. The horizontal axis stands for the ID of combination, and vertical axis represents N _EFAA (Fig. 10(a)) or N _RA (Fig. 10(b)).

Fig. 10(b) shows a tendency that as the Arm length l _R gets longer, N _RA decreases almost monotonically, but there is a steep descent after No.1-21.This can be attributed to the fact that a certain Arm length l _R would make the Rod end more likely to go over the Σ _RA .

Whereas, as shown in Fig. 10(a), there are roughly two stages. In the stage after No. 1-13, N _EFAA decreases while vibrating irregularly and strongly. This can be attributed to the reason same as before: a certain Arm length l _R would make the Rod end more likely to go over the Σ _EFAA . A different point is that the value of No. 1-16, with a smallest h ₂ (h ₂ = 120mm), shows a large local maximum. It seems h ₂ affected N _EFAA more than N _RA . In the stage before No.1-12, N _EFAA gradually increases as the Arm length gets longer, which means that it is necessary to precisely investigate the possibility of the Arm with shorter length, i.e., smaller l _R and h ₂. Thus, the Arm with the parameters shown in Equation (33) was investigated.

A total number of 100 combinations could be made, and identified with No.2-1, , No.2-100, as shown in Table 3. The estimative index was calculated for all the combinations. The results are shown in Table 3 and Fig. 11, 12.

As shown in Fig. 11, within the range, as the Arml _R gets longer, N _EFAA and N _RA changes in the opposite direction: N _EFAA increases and N _RA decreases. Thus, it is reasonable that, prospective solution for the Arm should be specified within this range.

In Fig. 12, the horizontal axis stands for the ID of combination, and vertical axis represents M _t (C) (Fig. 12(a)), M _t (C _f ) (Fig. 12(b))and M _t (C _m ) (Fig. 12(c)). There is a steep increase between No. 2-87 and No. 2-88. Since, the smaller value of M _t (C), M _t (C _f ) and M _t (C _m ) means the better manipulability of the shoulder prosthesis, prospective solutions should be chosen from the combinations before the No. 88.

Apparently, better accessibility requires a bigger value of N _EFAA and N _RA, however, from the aforementioned results, it is clear that within the range investigated (as shown in Fig. 11), they can not be satisfied simultaneously. That is, N _EFAA and N _RA should be traded-off depending on which area (EFAA or RA) is more important.

For this purpose, thresholds were determined as follows to reflect different weighting policies and the constraint from manipulability, i.e., M _t(C), M _t(C _f), M _t(C _m). The average μ ₀and standard deviation σ ₀ of N _EFAA, N _RA, M _t(C), M _t(C _f), M _t(C _m) were calculated. For M _t(C), M _t(C _f), M _t(C _m), threshold value was set as μ ₀+0.5σ ₀, which stands for the largest mean value that could be allowed. IN the EFAA-favoured policy, N _EFAA should be larger than μ ₀+0.5σ ₀(upper bound), but N _RA should be at least larger than μ ₀-0.5σ ₀ (lower bound). Similarly, in the RA-favoured policy, N _RA should be larger than μ ₀+0.5σ ₀, but, N _EFAA should be at least larger than μ ₀-0.5σ ₀. Equation (34)-(a) and (b) show the threshold values reflecting the EFAA-favoured and RA-favoured policies, respectively.

By using the Equation (34), 8 configuration candidates were selected. In Table 4, the configurations painted red and blue are selected by Equation (34)-(a) and Equation (34)-(b), respectively.

Furthermore, since M _t (C), M _t (C _f ), M _t (C _m ) are the mean values for C, C _f , C _m , for the configurations selected by the threshold values shown in the Equation (34), there is a possibility that, at some postures, the manipulability might be extremely bad. Therefore, it is necessary to specify the lower bound for the worst manipulability that could be allowed. The following three additional estimative indexes were defined, which means the configuration with a smaller 3^rd quartile (the value of a point that divides a data set into 3/4 and 1/4 of points) would be tolerated. They are only defined for EFAA, because this area requires precise manipulation more than RA.

• Q ₃(C): the 3rd quartile of C (Eq. 30) of points in Σ _EFAA

• Q ₃(C _f ): the 3rd quartile of C _f (Eq. 31) of points in Σ _EFAA

• Q ₃(C _m ): the 3rd quartile of C _m (Eq. 32)of points in Σ _EFAA