Mechanisms do not satisfy vector closure condition
The mechanism does not satisfy the 1st point of vector closure condition
A planar RDWM does not satisfy the 1st point of vector closure condition is shown in Fig. 7. Here, the unit of the values of parameters related to length are assumed to be [cm]. The processes for deriving the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) and checking the vector closure condition are shown below:
Defining \(^ T{\varvec{ p}}_{ {T}{Ci}}\), \({R}_{{i}}\), \(^ T{\varvec{ p}}_{ {T}{Di}}\) and \(^ T{\varvec{ p}}_{ {Bi}\text {j}}\)
Select the parameters for the mechanism in Fig. 7 as follows:
$$\begin{aligned} \left. \begin{array}{lll} ^ T{\varvec{ p}}_{ {TC}\text {1}} = \left[ -6 \quad -10 \right] ^\text {T}, \quad ^ T{\varvec{ p}}_{ {TC}\text {2}} = \left[ 6 \quad -10 \right] ^\text {T},\quad \quad \qquad \\ ^ T{\varvec{ p}}_{ {TC}\text {3}} = \left[ 6 \quad10 \right] ^\text {T}, \quad ^T{\varvec{ p}}_{ {TC}\text {4}} = \left[ -6 \quad10 \right] ^\text {T}.\qquad \qquad \end{array} \right. \end{aligned}$$
Radii of the pulleys: \({R}_\text {1}= {R}_\text {2} = {R}_\text {3} = {R}_\text {4}= 2.\)
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {TD}\text {1}}=\left[ -2 \quad 0 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {TD}\text {2}}=\left[ -2 \quad 0 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TD}\text {3}}=\left[ 2 \quad 0 \right] ^\text {T},& \quad ^T{\varvec{ p}}_{ {TD}\text {4}}=\left[ 2 \quad 0 \right] ^\text {T}.\end{array} \right. \end{aligned}$$
Position of wire end points on the top plate w.r.t the top plate coordinate:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {B}\text {11}} = \left[ -8 \quad -10 \right] ^\text {T}, &\quad ^T{\varvec{ p}}_{ {B}\text {12}} = \left[ -4 \quad -10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {21}} = \left[ 4 \quad -10 \right] ^\text {T}, & \quad ^T{\varvec{ p}}_{ {B}\text {22}} = \left[ 8 \quad -10 \right] ^\text {T}, \\ ^T{\varvec{ p}}_{ {B}\text {31}} = \left[ 8 \quad 10 \right] ^\text {T}, &\quad ^T{\varvec{ p}}_{ {B}\text {32}} = \left[ 4 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {41}} = \left[ -4 \quad 10 \right] ^\text {T}, &\quad ^T{\varvec{ p}}_{ {B}\text {42}} = \left[ -8 \quad 10 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Set of position \(^ B{\varvec{ p}}_{ {T}}\) and orientation \(^ B{\varvec{ R}}_{ {T}}\) of the top plate
$$\begin{aligned} ^ B{\varvec{ p}}_{ {T}}= \left[ 50 \quad 50 \right] ^\text {T}, \quad ^{B}{\varvec{ R}}_{ {T}}=\left[ \begin{array}{ll} 1 &{} \quad 0\\ 0&{} \quad 1 \end{array} \right] . \end{aligned}$$
(26)
Position of wire end points on the top plate
From Eq. (10), we have:
$$\begin{aligned} \left. \begin{array}{llll} ^ B{\varvec{ p}}_{ {B}\text {11}} = \left[ 42 \quad 40 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {12}} = \left[ 46 \quad 40 \right] ^\text {T}, \qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {21}} = \left[ 54 \quad 40 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {22}} = \left[ 58 \quad 40 \right] ^\text {T}, \qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {31}} = \left[ 58 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {32}} = \left[ 54 \quad 60 \right] ^\text {T}, \qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {41}} = \left[ 46 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {42}} = \left[ 42 \quad 60 \right] ^\text {T}. \qquad \qquad \end{array} \right. \end{aligned}$$
Position of wire end points on the frame
The DAMs are arranged on the frame such that the positions of the wire end points on the frame are as below:
$$\begin{aligned} \left. \begin{array}{ll} ^ B{\varvec{ p}}_{ {A}\text {11}} = \left[ 42 \quad 20 \right] ^\text {T}, &\quad ^ B{\varvec{ p}}_{ {A}\text {12}} = \left[ 46 \quad 20 \right] ^\text {T}, \\ ^ B{\varvec{ p}}_{ {A}\text {21}} = \left[ 54 \quad 20 \right] ^\text {T}, &\quad ^B{\varvec{ p}}_{ {A}\text {22}} = \left[ 58 \quad 20 \right] ^\text {T}, \\ ^ B{\varvec{ p}}_{ {A}\text {31}} = \left[ 58 \quad 90 \right] ^\text {T}, &\quad ^ B{\varvec{ p}}_{ {A}\text {32}} = \left[ 54 \quad 90 \right] ^\text {T}, \\ ^ B{\varvec{ p}}_{ {A}\text {41}} = \left[ 46 \quad 90 \right] ^\text {T}, &\quad ^ B{\varvec{ p}}_{ {A}\text {42}} = \left[ 42 \quad 90 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Calculating the wire vectors \(^ B{\varvec{ e}}_{{i}\text {j}}\)
The wire vectors \(^ B{\varvec{ e}}_{{i}{j}}\) can be calculated using Eq. (11), and the results are shown below:
$$\begin{aligned} \left. \begin{array}{lll} ^ B{\varvec{ e}}_{\text {1}\text {1}} = \;^ B{\varvec{ e}}_{\text {1}\text {2}} = \;^ B{\varvec{ e}}_{\text {2}\text {1}} = \;^ B{\varvec{ e}}_{\text {2}\text {2}} = \left[ 0 \;\; -1 \right] ^\text {T},\quad \;\qquad \quad \qquad \quad \qquad \\ ^ B{\varvec{ e}}_{\text {3}\text {1}} = \;^ B{\varvec{ e}}_{\text {3}\text {2}} = \;^ B{\varvec{ e}}_{\text {4}\text {1}} = \;^ B{\varvec{ e}}_{\text {4}\text {2}} = \left[ 0 \;\; 1 \right] ^\text {T}.\;\quad \quad \quad \quad \quad \quad \qquad \qquad \end{array} \right. \end{aligned}$$
Checking if the two wires in the DAMs are in parallel
With this configuration, the two wires in the DAMs are in parallel.
Derivation of the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\)
From Eqs. (6) and (16) with n = 3, N = 4, it is easy to see that the matrices \({\varvec{{W}}}_2\) and \({\varvec{{W}}}_{2}^{'}\) have size 7 × 8. As mentioned in the previous section, both of them will not be used in the proposed judgment method. Therefore they are not necessary to be derived, only the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) will be used for the judgment and it is derived as follows:
From Eq. (12), the vectors \({^ B}{\varvec{{e}}}_{{i}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{ll} ^ B{\varvec{ e}}_{\text {1}} = \;^ B{\varvec{ e}}_{\text {2}} = \left[ 0 \quad -1 \right] ^\text {T},\qquad \qquad \qquad \quad \qquad \\ ^ B{\varvec{ e}}_{\text {3}} = \;^ B{\varvec{ e}}_{\text {4}} = \left[ 0 \quad 1 \right] ^\text {T}, \qquad \quad \qquad \qquad \qquad \qquad \end{array} \right. \end{aligned}$$
From Eq. (13), the vectors \({^ B}{\varvec{ p}}_{ {TCi}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{lll} ^ B{\varvec{ p}}_{ {TC}\text {1}} = \left[ 44 \quad 40 \right] ^\text {T},\quad ^ B{\varvec{ p}}_{ {TC}\text {2}} = \left[ 56 \quad 40 \right] ^\text {T}, \qquad \quad \qquad \\ ^ B{\varvec{ p}}_{ {TC}\text {3}} = \left[ 56 \quad 60 \right] ^\text {T},\quad ^ B{\varvec{ p}}_{ {TC}\text {4}} = \left[ 44 \quad 60 \right] ^\text {T}.\qquad \qquad \qquad \end{array} \right. \end{aligned}$$
Then the wire matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) from Eq. (17) becomes:
$$\begin{aligned} \begin{array}{llll} {\varvec{{W}}}_{{A}\text {2}}^{'}=\left[ \begin{array}{lllllllllll} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ &{} &{} &{} \\ -1 &{} -1 &{} \quad 1&{} \quad 1\\ &{} &{} &{} \\ 6 &{} -6 &{} \quad 6 &{} -6 \end{array} \right] _{3 \times 4}, \end{array} \end{aligned}$$
(27)
The matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) has size \(3 \times 4\), with the row is equal to the number of DOFs in the planar global motion space and the column given by the four DOFs in the local motion space. It is about a quarter of the size of the wire matrix \({\varvec{{W}}}_2\).
Vector closure condition check
Easily to see in this case that rank (\({\varvec{{W}}}_{{A}\text {2}}^{'}\)) = 2, it is smaller than the number of dimension space of this wire mechanism (=3). Therefore, this mechanism can only realize the motion in two dimensions and it cannot realize motion in the remaining one. The first row of \({\varvec{{W}}}_{{A}\text {2}}^{'}\) corresponds to the resultant force in X direction, and all the elements of the row are zero, which causes the rank (\({\varvec{{W}}}_{{A}\text {2}}^{'}\)) to reduce one. The resultant force in X direction is zero so the motion in X direction can not be realized. This mechanism does not satisfy the 1st point which also means it does not satisfy vector closure condition.
The mechanism does not satisfy the 2nd point of vector closure condition
A planar RDWM does not satisfy the 2nd point of vector closure condition is shown in Fig. 8. Here, the unit of the values of parameters related to length are also assumed to be [cm]. The processes for deriving the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) and checking the vector closure condition are shown below:
Defining \(^ T{\varvec{ p}}_{ {T}{Ci}}\), \({R}_{{i}}\), \(^ T{\varvec{ p}}_{ {T}{Di}}\) and \(^ T{\varvec{ p}}_{ {Bi}\text {j}}\)
Select the parameters for the mechanism in Fig. 8 as follows:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {TC}\text {1}} = \left[ 16 \quad -20 \right] ^\text {T},&\quad ^ T{\varvec{ p}}_{ {TC}\text {2}} = \left[ 20 \quad -10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TC}\text {3}} = \left[ -4 \quad 20 \right] ^\text {T},&\quad ^ T{\varvec{ p}}_{ {TC}\text {4}} = \left[ -16 \quad 20 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Radii of the pulleys: \({R}_\text {1}= {R}_\text {2} = {R}_\text {3} = {R}_\text {4}= 2.\)
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {TD}\text {1}}=\left[ -2 \quad 0 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {TD}\text {2}}=\left[ 0 \quad -2 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TD}\text {3}}=\left[ 2 \quad 0 \right] ^\text {T}, &\quad^ T{\varvec{ p}}_{ {TD}\text {4}}=\left[ 2 \quad 0 \right] ^\text {T}.\end{array} \right. \end{aligned}$$
Position of wire end points on the top plate w.r.t the top plate coordinate:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{{B}{11}} = [ 14 \quad -20 ]^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B}{12}} = [ 18 \quad -20 ]^{\text{T}}, \\ ^ T{\varvec{ p}}_{ {B}{21}} = [ 20 \quad -12 ] ^{\text{T}}, &\quad ^ T{\varvec{ p}}_{ {B} {22}} = [ 20 \quad -8 ] ^{\rm T}, \\ ^ T{\varvec{ p}}_{ {B}{31}} = [ -2 \quad 20] ^{\rm T}, &\quad ^ T{\varvec{ p}}_{ {B}{32}} = \left[ -6 \quad 20 \right] ^{\rm T}, \\ ^ T{\varvec{ p}}_{ {B} {41}} = \left[ -14 \quad 20 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B}\text {42}} = \left[ -18 \quad 20 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Set of position \(^ B{\varvec{ p}}_{ {T}}\) and orientation \(^ B{\varvec{ R}}_{ {T}}\) of the top plate
$$\begin{aligned} ^ B{\varvec{ p}}_{ {T}}= \left[ 50 \quad 50 \right] ^\text {T},\quad ^{B}{\varvec{ R}}_{ {T}}=\left[ \begin{array}{ll} 1 &{} \quad 0\\ 0&{} \quad 1 \end{array} \right] . \end{aligned}$$
(28)
Position of wire end points on the top plate
From Eq. (10), we have:
$$\begin{aligned} \left. \begin{array}{llll} ^ B{\varvec{ p}}_{ {B}\text {11}} = \left[ 64 \quad 30 \right] ^\text {T}, \quad ^ B{\varvec{ p}}_{ {B}\text {12}} = \left[ 68 \quad 30 \right] ^\text {T}, \qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {21}} = \left[ 70 \quad 38 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {22}} = \left[ 70 \quad 42 \right] ^\text {T},\qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {31}} = \left[ 48 \quad 70 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {32}} = \left[ 44 \quad 70 \right] ^\text {T},\qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {41}} = \left[ 36 \quad 70 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {42}} = \left[ 32 \quad 70 \right] ^\text {T}.\quad \qquad \qquad \end{array} \right. \end{aligned}$$
Position of wire end points on the frame
The DAMs are arranged on the frame such that the positions of the wire end points on the frame are as below:
$$\begin{aligned} \left. \begin{array}{llll} ^ B{\varvec{ p}}_{ {A}\text {11}} = \left[ 64 \quad 0 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {A}\text {12}} = \left[ 68 \quad 0 \right] ^\text {T},\quad \qquad \qquad \\ ^ B{\varvec{ p}}_{ {A}\text {21}} = \left[ 118 \quad 2 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {A}\text {22}} = \left[ 118 \quad 6 \right] ^\text {T},\quad \qquad \qquad \\ ^ B{\varvec{ p}}_{ {A}\text {31}} = \left[ 48 \quad 100 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {A}\text {32}} = \left[ 44 \quad 100 \right] ^\text {T},\quad \qquad \qquad \\ ^ B{\varvec{ p}}_{ {A}\text {41}} = \left[ 36 \quad 100 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {A}\text {42}} = \left[ 32 \quad100 \right] ^\text {T}.\qquad \qquad \end{array} \right. \end{aligned}$$
Calculating the wire vectors \(^ B{\varvec{ e}}_{{i}\text {j}}\)
The wire vectors \(^ B{\varvec{ e}}_{{i}{j}}\) can be calculated using Eq. (11), and the results are shown below:
$$\begin{aligned} \left. \begin{array}{ll} ^ B{\varvec{ e}}_{\text {1}\text {1}} = ^ B{\varvec{ e}}_{\text {1}\text {2}} = \left[ 0 \;\; -1 \right] ^\text {T},&\quad^ B{\varvec{ e}}_{\text {2}\text {1}} =^ B{\varvec{ e}}_{\text {2}\text {2}} = \left[ 4/5 \;\; -3/5 \right] ^\text {T}, \\ ^ B{\varvec{ e}}_{\text {3}\text {1}} = ^ B{\varvec{ e}}_{\text {3}\text {2}} =^ B{\varvec{ e}}_{\text {4}\text {1}} =^ B{\varvec{ e}}_{\text {4}\text {2}} = \left[ 0 \;\; 1 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Checking if the two wires in the DAMs are in parallel
With this configuration, the two wires in the DAMs are in parallel.
Derivation of the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\)
From Eqs. (6) and (16) with n = 3, N = 4, it is easy to see that the matrices \({\varvec{{W}}}_2\) and \({\varvec{{W}}}_{2}^{'}\) have size \({7 \times 8}\). Similarly to the previous example, these matrices are not necessary to be derived, only the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) will be used for the judgment and it is derived as follows:
From Eq. (12), the vectors \({^ B}{\varvec{{e}}}_{{i}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{lll} ^ B{\varvec{ e}}_{\text {1}} = \left[ 0 \quad -1 \right] ^\text {T},\;^ B{\varvec{ e}}_{\text {2}} = \left[ 4/5 \quad -3/5 \right] ^\text {T}, \;^ B{\varvec{ e}}_{\text {3}} = ^ B{\varvec{ e}}_{\text {4}} = \left[ 0 \quad 1 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
From Eq. (13), the vectors \({^ B}{\varvec{ p}}_{ {TCi}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{lll} ^ B{\varvec{ p}}_{ {TC}\text {1}} = \left[ 66 \quad 30 \right] ^\text {T}, \quad ^ B{\varvec{ p}}_{ {TC}\text {2}} = \left[ 70 \quad 40 \right] ^\text {T},\quad \quad \qquad \\ ^ B{\varvec{ p}}_{ {TC}\text {3}} = \left[ 46 \quad 70 \right] ^\text {T}, \quad^ B{\varvec{ p}}_{ {TC}\text {4}} = \left[ 34 \quad 70 \right] ^\text {T}.\qquad \qquad \end{array} \right. \end{aligned}$$
Then the wire matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) from Eq. (17) becomes:
$$\begin{aligned} \begin{array}{lll} {\varvec{{W}}}_{{A}\text {2}}^{'}=\left[ \begin{array}{lllllllllll} 0 & \quad {} 4/5 &\quad {} 0 & \quad {} 0\\ &{} &{} &{} \\ -1 &{} -3/5 &\quad {} 1 &\quad {} 1\\ &{} &{} &{} \\ -16 &{} -4 &{} -4 &{} -16 \end{array}\right] _{3 \times 4}, \end{array} \end{aligned}$$
(29)
Similarly to the previous example, the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) related to the global motion in this case also equals just a quarter of the normal wire matrix \({\varvec{{W}}}_{\text {2}}\).
Vector closure condition check
Easily to see in this case that rank (\({\varvec{{W}}}_{{A}\text {2}}^{'}\)) = 3. It equals to the number of dimension space of the wire mechanism (=3) so it satisfies the 1st point of vector closure condition. However considering the 3rd row of \({\varvec{{W}}}_{{A}\text {2}}^{'}\), this row corresponds to the resultant moment. It has all elements with negative values so with any positive wire tension \({\varvec{{T}}}_{\text {0}}\), the resultant moment will be produced only negative value. Therefore no wire tension \({\varvec{{T}}}_{\text {0}}\) satisfies \({\varvec{{W}}}_{{A}\text {2}}^{'}{\varvec{{T}}}_{\text {0}}=0\). This mechanism does not satisfy the 2nd point which also means it does not satisfy vector closure condition.
Mechanisms satisfy vector closure condition
The planar RDWM with DAMs
A planar RDWM with DAMs is shown in Fig. 9. Here, the unit of the values of parameters related to length are assumed to be [cm]. The processes for deriving the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) and checking the vector closure condition are shown below:
Defining \(^ T{\varvec{ p}}_{ {T}{Ci}}\), \({R}_{{i}}\), \(^ T{\varvec{ p}}_{ {T}{Di}}\) and \(^ T{\varvec{ p}}_{ {Bi}\text {j}}\)
Select the parameters for the mechanism in Fig. 9 as follows:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {TC}\text {1}} = \left[ -6 \quad 10 \right] ^\text {T}, &\quad^ T{\varvec{ p}}_{ {TC}\text {2}} = \left[ -10 \quad -6 \right] ^\text {T},\qquad \qquad \\ ^ T{\varvec{ p}}_{ {TC}\text {3}} = \left[ 10 \quad -6 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {TC}\text {4}} = \left[ 6 \quad 10 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Radii of the pulleys: \({R}_\text {1}= {R}_\text {2} = {R}_\text {3} = {R}_\text {4}= 2.\)
$$\begin{aligned} \left. \begin{array}{lll} ^ T{\varvec{ p}}_{ {TD}\text {1}}=\left[ 2 \quad 0 \right] ^\text {T}, \quad ^ T{\varvec{ p}}_{ {TD}\text {2}}=\left[ 0 \quad 2 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TD}\text {3}}=\left[ 0 \quad -2 \right] ^\text {T}, \qquad ^ T{\varvec{ p}}_{ {TD}\text {4}}=\left[ 2 \quad 0 \right] ^\text {T}.\end{array} \right. \end{aligned}$$
Position of wire end points on the top plate w.r.t the top plate coordinate:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {B}\text {11}} = \left[ -4 \quad 10 \right] ^\text {T},\quad ^ T{\varvec{ p}}_{ {B}\text {12}} = \left[ -8 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {21}} = \left[ -10 \quad -4 \right] ^\text {T}, \quad ^ T{\varvec{ p}}_{ {B}\text {22}} = \left[ -10 \quad -8 \right] ^\text {T},\\ ^ T{\varvec{ p}}_{ {B}\text {31}} = \left[ 10 \quad -8 \right] ^\text {T}, \quad ^ T{\varvec{ p}}_{ {B}\text {32}} = \left[ 10 \quad -4 \right] ^\text {T},\\ ^ T{\varvec{ p}}_{ {B}\text {41}} = \left[ 8 \quad 10 \right] ^\text {T}, \quad ^ T{\varvec{ p}}_{ {B}\text {42}} = \left[ 4 \quad 10 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Set of position \(^ B{\varvec{ p}}_{ {T}}\) and orientation \(^ B{\varvec{ R}}_{ {T}}\) of the top plate
$$\begin{aligned} ^ B{\varvec{ p}}_{ {T}}= \left[ 50 \quad 50 \right] ^\text {T},\quad ^{B}{\varvec{ R}}_{ {T}}=\left[ \begin{array}{cc} 1 &{} 0\\ 0 \quad &{} 1 \end{array} \right] . \end{aligned}$$
(30)
Position of wire end points on the top plate
From Eq. (10), we have:
$$\begin{aligned} \left. \begin{array}{c} ^ B{\varvec{ p}}_{ {B}\text {11}} = \left[ 46 \quad 60 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {12}} = \left[ 42 \quad 60 \right] ^\text {T},\;\qquad \;\qquad \\ ^ B{\varvec{ p}}_{ {B}\text {21}} = \left[ 40 \quad 46 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {22}} = \left[ 40 \quad 42 \right] ^\text {T},\;\qquad \;\qquad \\ ^ B{\varvec{ p}}_{ {B}\text {31}} = \left[ 60 \quad 42 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {32}} = \left[ 60 \quad 46 \right] ^\text {T},\;\;\qquad \qquad \\ ^ B{\varvec{ p}}_{ {B}\text {41}} = \left[ 58 \quad 60 \right] ^\text {T},\,\quad ^ B{\varvec{ p}}_{ {B}\text {42}} = \left[ 54 \quad 60 \right] ^\text {T}.\;\;\qquad \qquad \end{array} \right. \end{aligned}$$
Position of wire end points on the frame
The DAMs are arranged on the frame such that the positions of the wire end points on the frame are as below:
$$\begin{aligned} \left. \begin{array}{ll} ^ B{\varvec{ p}}_{ {A}\text {11}} = \left[ 46 \quad 90 \right] ^\text {T},&\quad ^ B{\varvec{ p}}_{ {A}\text {12}} = \left[ 42 \quad 90 \right] ^\text {T}, \\ ^ B{\varvec{ p}}_{ {A}\text {21}} = \left[ 10 \quad 23.5 \right] ^\text {T}, &\quad ^ B{\varvec{ p}}_{ {A}\text {22}} = \left[ 10 \quad 19.5 \right] ^\text {T}, \\ ^ B{\varvec{ p}}_{ {A}\text {31}} = \left[ 90 \quad 19.5 \right] ^\text {T}, &\quad ^ B{\varvec{ p}}_{ {A}\text {32}} = \left[ 90 \quad 23.5 \right] ^\text {T}, \\ ^ B{\varvec{ p}}_{ {A}\text {41}} = \left[ 58 \quad 90 \right] ^\text {T}, &\quad ^ B{\varvec{ p}}_{ {A}\text {42}} = \left[ 54 \quad 90 \right] ^\text {T}.\end{array} \right. \end{aligned}$$
Calculating the wire vectors \(^ B{\varvec{ e}}_{{i}\text {j}}\)
The wire vectors \(^ B{\varvec{ e}}_{{i}{j}}\) can be calculated using Eq. (11), and the results are shown below:
$$\begin{aligned} \left. \begin{array}{c} ^ B{\varvec{ e}}_{\text {1}\text {1}} = \;^ B{\varvec{ e}}_{\text {1}\text {2}} = \left[ 0 \quad 1 \right] ^\text {T},\quad \;\qquad \quad \qquad \quad \qquad \\ ^ B{\varvec{ e}}_{\text {2}\text {1}} = \;^ B{\varvec{ e}}_{\text {2}\text {2}} = \left[ -4/5 \quad -3/5 \right] ^\text {T},\quad \;\qquad \qquad \\ ^ B{\varvec{ e}}_{\text {3}\text {1}} = \;^ B{\varvec{ e}}_{\text {3}\text {2}} = \left[ 4/5 \quad -3/5 \right] ^\text {T},\quad \quad \qquad \qquad \\ ^ B{\varvec{ e}}_{\text {4}\text {1}} = \;^ B{\varvec{ e}}_{\text {4}\text {2}} = \left[ 0 \quad 1 \right] ^\text {T}.\quad \;\qquad \quad \qquad \quad \qquad \end{array} \right. \end{aligned}$$
Checking if the two wires in the DAMs are in parallel
With this configuration, the two wires in the DAMs are in parallel.
Derivation of the matrix \({\varvec{{W}}}_{{A}}^{'}\)
From Eqs. (6) and (16) with n = 3, N = 4, it is easy to see that the matrices \({\varvec{{W}}}_2\) and \({\varvec{{W}}}_{2}^{'}\) have size 7 × 8. Similarly to the previous examples, these matrices are not necessary to be derived, only the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) will be used for the judgment and it is derived as follows:
From Eq. (12), the vectors \({^ B}{\varvec{{e}}}_{{i}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{ll} ^ B{\varvec{ e}}_{\text {1}} = \left[ 0 \quad 1 \right] ^\text {T}, \quad ^ B{\varvec{ e}}_{\text {2}} = \left[ -4/5 \quad -3/5 \right] ^\text {T}, \\ ^ B{\varvec{ e}}_{\text {3}} = \left[ 4/5 \quad -3/5 \right] ^\text {T}, \quad ^ B{\varvec{ e}}_{\text {4}} = \left[ 0 \quad 1 \right] ^\text {T} \end{array} \right. \end{aligned}$$
From Eq. (13), the vectors \({^ B}{\varvec{ p}}_{ {TCi}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{lll} ^ B{\varvec{ p}}_{ {TC}\text {1}} = \left[ 44 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {TC}\text {2}} = \left[ 40 \quad 44 \right] ^\text {T},\qquad \quad \qquad \\ ^ B{\varvec{ p}}_{ {TC}\text {3}} = \left[ 60 \quad 44 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {TC}\text {4}} = \left[ 56 \quad 60 \right] ^\text {T}.\qquad \qquad \end{array} \right. \end{aligned}$$
Then the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) from Eq. (17) becomes:
$$\begin{aligned} \begin{array}{lll} {\varvec{{W}}}_{{A}\text {2}}^{'}=\left[ \begin{array}{lllllllllll} 0 &{} -\dfrac{4}{5} &{} \quad \dfrac{4}{5} &\quad{} 0\\ &{} &{} &{} \\ 1 &{} -\dfrac{3}{5} &{} -\dfrac{3}{5} &\quad {} 1\\ &{} &{} &{} \\ -6 &{} \quad \dfrac{6}{5} &{} -\dfrac{6}{5} &\quad {} 6 \end{array}\right] _{3 \times 4}, \end{array} \end{aligned}$$
(31)
Similarly to the previous examples, the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) related to the global motion in this case also equals just a quarter of the normal wire matrix \({\varvec{{W}}}_{\text {2}}\).
Vector closure condition check
From the expression in the judgment of a candidate section, only the matrix that contributes to global motion \({\varvec{{W}}}_{{A}\text {2}}^{'}\) is needed to check the vector closure condition. Using the matrix of global motion from Eq. (31), we have:
-
(i)
rank (\({\varvec{{W}}}_{{A}\text {2}}^{'}\)) = 3.
-
(ii)
With \({\varvec{{T}}}_{{S}\text {2}}=\left[ \begin{array}{cccccccc} 3&5&5&3 \end{array}\right] ^\text {T}>{\varvec{{0}}}\), easy to get \({\varvec{{W}}}_{{A}\text {2}}^{'}{\varvec{ T}}_{{S}\text {2}}={\varvec{{0}}}\).
The above analysis shows that rank (\({\varvec{{W}}}_{{A}\text {2}}^{'}\)) equals to the number of dimension spaces of the wire mechanism. As there is a vector \({\varvec{ T}}_{{S}\text {2}}\) that satisfies \({\varvec{{W}}}_{{A}\text {2}}^{'}{\varvec{ T}}_{{S}\text {2}}={\varvec{{0}}}\), the matrix \({\varvec{{W}}}_{{A}\text {2}}^{'}\) satisfies the vector closure condition and the mechanism can produce the resultant force in any direction within its motion space. The results mean that the planar RDWM with four sets of DAMs in this example has the same structure as that of a conventional wire mechanism with four sets of single actuator modules, where each wire is equivalent to a set of two wires for each DAM in the planar RDWM, when judging the vector closure condition.
The 3D RDWM with DAMs
A 3D RDWM with DAMs is shown in Fig. 10. Here, the unit of the values of parameters related to length are assumed to be [cm]. Because of space limitations, the DAMs are not shown; only the end points of the wires on those modules are shown, with the important representative vectors. The processes for deriving the matrix \({\varvec{{W}}}_{{A}\text {3}}^{'}\) and checking the vector closure condition are shown below:
Defining \(^ T{\varvec{ p}}_{ {T}{Ci}}\), \({R}_{{i}}\), \(^ T{\varvec{ p}}_{ {T}{Di}}\), and \(^ T{\varvec{ p}}_{ {Bi}\text {j}}\)
Select the parameters for the mechanism in Fig. 10 as follows:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {TC}\text {1}} = \left[ -6 \quad 10 \quad 10 \right] ^\text {T}, &\quad ^T{\varvec{ p}}_{ {TC}\text {2}} = \left[ -10 \quad -6 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TC}\text {3}} = \left[ 10 \quad -6 \quad 10 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {TC}\text {4}} = \left[ 6 \quad 10 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TC}\text {5}} = \left[ -10 \quad 6 \quad -10 \right] ^\text {T}, &\quad ^T{\varvec{ p}}_{ {TC}\text {6}} = \left[ 0 \quad -10 \quad -10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TC}\text {7}} = \left[ 10 \quad 6 \quad -10 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Radii of the pulleys: \({R}_\text {1}= {R}_\text {2} = {R}_\text {3} = {R}_\text {4}= {R}_\text {5} = {R}_\text {6} = {R}_\text {7}= 2.\)
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {TD}\text {1}} = \left[ 2\quad 0 \quad 0 \right] ^\text {T}, &\quad ^T{\varvec{ p}}_{ {TD}\text {2}} = \left[ 0 \quad 2 \quad 0 \right] ^\text {T},\\ ^ T{\varvec{ p}}_{ {TD}\text {3}} = \left[ 0 \quad -2 \quad 0 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {TD}\text {4}} = \left[ 2 \quad 0 \quad 0 \right] ^\text {T},\\ ^ T{\varvec{ p}}_{ {TD}\text {5}} = \left[ 0 \quad 2 \quad 0 \right] ^\text {T}, & \quad ^ T{\varvec{ p}}_{ {TD}\text {6}} = \left[ -2 \quad 0 \quad 0 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {TD}\text {7}} = \left[ 0 \quad -2 \quad 0 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
The positions of wire end points on the top plate w.r.t the top plate coordinate are given by:
$$\begin{aligned} \left. \begin{array}{ll} ^ T{\varvec{ p}}_{ {B}\text {11}} = \left[ -4 \quad 10 \quad 10 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B}\text {12}} = \left[ -8 \quad 10 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {21}} = \left[ -10 \quad -4 \quad 10 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B}\text {22}} = \left[ -10 \quad -8\quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {31}} = \left[ 10 \quad -8 \quad 10 \right] ^\text {T}, &\quad^ T{\varvec{ p}}_{ {B}\text {32}} = \left[ 10 \quad -4 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {41}} = \left[ 8 \quad 10 \quad 10 \right] ^\text {T}, &\quad^ T{\varvec{ p}}_{ {B}\text {42}} = \left[ 4 \quad 10 \quad 10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B}\text {51}} = \left[ -10 \quad 8 \quad -10 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B}\text {52}} = \left[ -10 \quad 4 \quad -10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B6}\text {1}} = \left[ -2 \quad -10 \quad -10 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B6}\text {2}} = \left[ 2 \quad -10 \quad -10 \right] ^\text {T}, \\ ^ T{\varvec{ p}}_{ {B7}\text {1}} = \left[ 10 \quad 4 \quad -10 \right] ^\text {T}, &\quad ^ T{\varvec{ p}}_{ {B7}\text {2}} = \left[ 10 \quad 8 \quad -10 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Set of position \(^ B{\varvec{ p}}_{ {T}}\) and orientation \(^ B{\varvec{ R}}_{ {T}}\) of the top plate
$$\begin{aligned} ^ B{\varvec{ p}}_{ {T}}= \left[ 50 \quad 50 \quad 50 \right] ^\text {T},\quad ^ B{\varvec{ R}}_{ {T}}=\left[ \begin{array}{ccc} 1 &\quad {} 0 & \quad {} 0\\ 0&\quad {} 1 &\quad {} 0\\ 0&\quad {} 0 &\quad {} 1 \end{array} \right] . \end{aligned}$$
(32)
Position of wire end points on the top plate
From Eq. (10), we have:
$$\begin{aligned} \left. \begin{array}{ccc} ^ B{\varvec{ p}}_{ {B}\text {11}} = \left[ 46 \quad 60 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {12}} = \left[ 42 \quad 60 \quad 60 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {B}\text {21}} = \left[ 40 \quad 46 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {22}} = \left[ 40 \quad 42 \quad 60 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {B}\text {31}} = \left[ 60 \quad 42 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {32}} = \left[ 60 \quad 46 \quad 60 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {B}\text {41}} = \left[ 58 \quad 60 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {42}} = \left[ 54 \quad 60 \quad 60 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {B}\text {51}} = \left[ 40 \quad 58 \quad 40 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {52}} = \left[ 40 \quad 54 \quad 40 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {B}\text {61}} = \left[ 48 \quad 40 \quad 40 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {62}} = \left[ 52 \quad 10 \quad 40 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {B}\text {71}} = \left[ 60 \quad 54 \quad 40 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {B}\text {72}} = \left[ 60 \quad 58 \quad 40 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Position of wire end points on the frame
The DAMs are arranged on the frame such that the positions of the wire end points on the frame are as shown below:
$$\begin{aligned}\left. \begin{array}{ll} ^ B{\varvec{ p}}_{ {A}\text {11}} = \left[
46 \quad 90 \quad 100 \right] ^\text {T},&\quad ^ B{\varvec{ p}}_{
{A}\text {12}} = \left[ 42 \quad 90 \quad 100 \right] ^\text {T},\\
^ B{\varvec{ p}}_{ {A}\text {21}} = \left[ 10 \quad 46 \quad 100
\right] ^\text {T},&\quad ^ B{\varvec{ p}}_{ {A}\text {22}} = \left[
10 \quad 42 \quad 100 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{
{A}\text {31}} = \left[ 90 \quad 42 \quad 100 \right] ^\text
{T},&\quad ^ B{\varvec{ p}}_{ {A}\text {32}} = \left[ 90 \quad 46
\quad 100 \right] ^\text {T},\\ ^ B{\varvec{ p}}_{ {A}\text {41}} =
\left[ 58 \quad 90 \quad 100 \right] ^\text {T},&\quad ^ B{\varvec{
p}}_{ {A}\text {42}} = \left[ 54 \quad 90 \quad 100 \right] ^\text
{T},\\^ B{\varvec{ p}}_{ {A}\text {51}} = \left[ 10 \quad 58 \quad 0
\right] ^\text {T},&\quad ^ B{\varvec{ p}}_{ {A}\text {52}} = \left[
10 \quad 54 \quad 0 \right] ^\text {T},\\ ^B{\varvec{ p}}_{ {A}\text
{61}} = \left[ 48 \quad10 \quad 0 \right] ^\text {T},&\quad ^
B{\varvec{ p}}_{ {A}\text {62}} = \left[ 52 \quad 10 \quad 0 \right]
^\text {T},\\ ^ B{\varvec{ p}}_{ {A}\text {71}} = \left[ 90 \quad 54
\quad 0 \right] ^\text {T},&\quad ^ B{\varvec{ p}}_{ {A}\text {72}}
= \left[ 90 \quad 58 \quad 0 \right] ^\text {T}.\quad \end{array}
\right. \end{aligned}$$
Calculating the wire vectors \(^ B{\varvec{ e}}_{{i}\text {j}}\)
The wire vectors \(^ B{\varvec{ e}}_{{i}{j}}\) can be calculated using Eq. (11), and the results are shown below:
$$\begin{aligned} \left. \begin{array}{llllll} ^ B{\varvec{ e}}_{\text {1}\text {1}} =\,^ B{\varvec{ e}}_{\text {1}\text {2}} = \left[ 0 \quad 3/5 \quad 4/5 \right] ^\text {T},\quad \qquad \quad \\ ^ B{\varvec{ e}}_{\text {2}\text {1}} = \,^ B{\varvec{ e}}_{\text {2}\text {2}} = \left[ -3/5 \quad 0 \quad 4/5 \right] ^\text {T},\quad \qquad \quad \\ ^ B{\varvec{ e}}_{\text {3}\text {1}} = \,^ B{\varvec{ e}}_{\text {3}\text {2}} = \left[ 3/5 \quad 0 \quad 4/5 \right] ^\text {T},\quad \qquad \quad \\ ^ B{\varvec{ e}}_{\text {4}\text {1}} = \,^ B{\varvec{ e}}_{\text {4}\text {2}} = \left[ 0 \quad 3/5 \quad 4/5 \right] ^\text {T},\quad \qquad \quad \\ ^ B{\varvec{ e}}_{\text {5}\text {1}} = \,^ B{\varvec{ e}}_{\text {5}\text {2}} = \left[ -3/5 \quad 0 \quad -4/5 \right] ^\text {T},\quad \qquad \quad \\ ^ B{\varvec{ e}}_{\text {6}\text {1}} = \,^ B{\varvec{ e}}_{\text {6}\text {2}} = \left[ 0 \quad -3/5 \quad -4/5 \right] ^\text {T},\;\;\qquad \quad \\ ^ B{\varvec{ e}}_{\text {7}\text {1}} = \,^ B{\varvec{ e}}_{\text {7}\text {2}} = \left[ 3/5 \quad 0 \quad -4/5 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Checking if the two wires in the DAMs are in parallel
With this configuration, the two wires in the DAMs are in parallel.
Derivation of the matrix \({\varvec{{W}}}_{{A}}^{'}\)
From Eqs. (6) and (16) with n = 6, N = 7, it is easy to see that the matrices \({\varvec{{W}}}_3\) and \({\varvec{{W}}}_{3}^{'}\) have size 13 × 14. Similarly to the previous examples, these matrices are not necessary to be derived, only the matrix \({\varvec{{W}}}_{{A}\text {3}}^{'}\) will be used for the judgment and it is derived as follows:
From Eq. (12), the vectors \({^ B}{\varvec{{e}}}_{{i}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{ll} ^ B{\varvec{ e}}_{\text {1}} = \left[ 0 \quad 3/5 \quad 4/5 \right] ^\text {T},&\quad ^ B{\varvec{ e}}_{\text {2}} = \left[ -3/5 \quad 0 \quad 4/5 \right] ^\text {T}, \\ ^ B{\varvec{ e}}_{\text {3}} = \left[ 3/5 \quad 0 \quad 4/5 \right] ^\text {T},&\quad ^ B{\varvec{ e}}_{\text {4}} = \left[ 0 \quad 3/5 \quad 4/5 \right] ^\text {T},\\ ^ B{\varvec{ e}}_{\text {5}} = \left[ -3/5 \quad 0 \quad -4/5 \right] ^\text {T},&\quad^ B{\varvec{ e}}_{\text {6}} = \left[ 0 \quad -3/5 \quad -4/5 \right] ^\text {T},\\ ^ B{\varvec{ e}}_{\text {7}} = \left[ 3/5 \quad 0 \quad -4/5 \right] ^\text {T}. \end{array} \right. \end{aligned}
$$
From Eq. (13), the vectors \({^ B}{\varvec{ p}}_{ {TCi}}\) can be derived as:
$$\begin{aligned} \left. \begin{array}{lll} ^ B{\varvec{ p}}_{ {TC}\text {1}} = \left[ 44 \quad 60 \quad 60 \right] ^\text {T},\quad ^ B{\varvec{ p}}_{ {TC}\text {2}} = \left[ 40 \quad 44 \quad 60 \right] ^\text {T},\qquad \quad \qquad \\ ^ B{\varvec{ p}}_{ {TC}\text {3}} = \left[ 60 \quad 44 \quad 60 \right] ^\text {T},\quad^ B{\varvec{ p}}_{ {TC}\text {4}} = \left[ 56 \quad 60 \quad 60 \right] ^\text {T}.\quad \qquad \qquad \\ ^ B{\varvec{ p}}_{ {TC}\text {5}} = \left[ 40 \quad 56 \quad 40 \right] ^\text {T},\quad ^ B{\varvec{ p}}_{ {TC}\text {6}} = \left[ 50 \quad 25 \quad 40 \right] ^\text {T},\qquad \quad \qquad \\ ^ B{\varvec{ p}}_{ {TC}\text {7}} = \left[ 60 \quad 56 \quad 40 \right] ^\text {T}. \end{array} \right. \end{aligned}$$
Then the matrix \({\varvec{{W}}}_{{A}\text {3}}^{'}\) from Eq. (17) becomes:
$$\begin{aligned}
{\varvec{W}}_{{A}{3}}^{\prime}=\left[\begin{array}{ccccccc} 0 &\quad
-\dfrac{3}{5} & \quad \dfrac{3}{5} & \quad 0 & \quad-\dfrac{3}{5}
&\quad 0 & \quad\dfrac{3}{5}\\
\dfrac{3}{5} & \quad 0 & \quad 0 & \quad\dfrac{3}{5} & \quad 0
&\quad -\dfrac{3}{5} & \quad 0\\
\dfrac{4}{5} &\quad\dfrac{4}{5} &
\quad\dfrac{4}{5} &\quad\dfrac{4}{5} &\quad -\dfrac{4}{5} &\quad
-\dfrac{4}{5} &\quad-\dfrac{4}{5}\\
2 &\quad- \dfrac{24}{5} &\quad-\dfrac{24}{5} &\quad 2 &\quad
-\dfrac{24}{5} & \quad 2 &\quad-\dfrac{24}{5}\\
\dfrac{24}{5} &
\quad 2 &\quad -2 &\quad-\dfrac{24}{5} &\quad -2 &\quad 0 & \quad 2\\
-\dfrac{18}{5} & \quad-\dfrac{18}{5} & \quad \dfrac{18}{5} & \quad
\dfrac{18}{5} &\quad \dfrac{18}{5} & \quad 0 &\quad -\dfrac{18}{5}\\
\end{array} \right] _{6\times 7}, \end{aligned}$$
(33)
The matrix \({\varvec{{W}}}_{{A}\text {3}}^{'}\) has size 6 × 7, where the number of rows is equal to the six DOFs in the 3D global motion space and the number of columns is equal to the seven DOFs in the local motion space. It is about a quarter of the size of the wire matrix \({\varvec{{W}}}_3\).
Vector closure condition check
From the expression in the judgment of a candidate section, only the matrix that contributes to global motion \({\varvec{{W}}}_{{A}\text {3}}^{'}\) is needed to check the vector closure condition. Using the matrix of global motion from Eq. (33), we have:
-
(i)
rank (\({\varvec{{W}}}_{{A}\text {3}}^{'}\)) = 6.
-
(ii)
With \({\varvec{ T}}_{{S}\text {3}}=\left[ \begin{array}{cccccccc} 12&5&5&12&5&24&5\end{array}\right] ^\text {T}>{\varvec{{0}}}\), easy to get \({\varvec{{W}}}_{{A}\text {3}}^{'}{\varvec{ T}}_{{S}\text {3}}={\varvec{{0}}}\).
The above analysis shows that rank (\({\varvec{{W}}}_{{A}\text {3}}^{'}\)) equals to the number of dimension spaces of the wire mechanism. As there is a vector \({\varvec{ T}}_{{S}\text {3}}\) that satisfies \({\varvec{{W}}}_{{A}\text {3}}^{'}{\varvec{ T}}_{{S}\text {3}}={\varvec{{0}}}\), the matrix \({\varvec{{W}}}_{{A}\text {3}}^{'}\) satisfies the vector closure condition and the mechanism can produce the resultant force in any direction within its motion space. The results mean that the 3D RDWM with seven sets of DAMs in this example has the same structure as a conventional wire mechanism with seven sets of single actuator modules, wherein each wire in the conventional wire mechanism is equivalent to a set of two wires of each DAM in the 3D RDWM, in judgment with the vector closure condition.
Remark
The above examples show the advantages of the proposed judgment method for RDWM’s configurations. It specially shows the effects in the case of dealing with the multi-dimension mechanisms as the 3D case in the last example. In that case, we only have to deal with a 6 × 7 matrix instead of a 13 × 14 one. This will let the designers to focus on thinking about arranging the configurations and then they can check those configurations with faster, simpler and easier way.