2. Cable-Driven Parallel Manipulators Kinematics
Let’s consider the general structure of CDPM illustrated in
Figure 2 that contains a moving platform connected with a fixed platform through several flexible cables. The reference frame
{W} is attached to the fixed platform and the moving frame
{R} is attached to the center of mass of moving platform. For convenience in notation, the right-superscript denotes the frame where the mentioned object reference to, and any vector without a right-superscript is implied to reference to
{W}. The general coordinate of
{R} with respect to
{W} is defined by the vector
x = [
P θ]
T where
P = [
xp yp zp]
T and
θ = [
αp βp γp]
T are the positioning vector of center of mass and rotational vector that represents Euler angle (
z-y-x) of the moving platform, respectively. Let
Ai and
Bi be the positioning vectors that represent the end of the ith cable at the pole and at the moving platform, respectively. According to
Figure 2, we have
ai = AiBi and
bi = PBi. If the general coordinate of the moving platform is given, then the direction of tension exerted on any cable can be obtained through Equation (1).
where
is the rotational matrix that represents the orientational of {
R} respect to {
W} and it can be defined by (2).
where
Cx and
Sx represent for cos(
x) and sin(
x), respectively.
In order to balance the external wrench
ψe that exerted on
P, the actuators mounted in the fixed frame are controlled to create cable tension forces. Moreover, the cable tension forces also affect cable lengths, while cable lengths are desired results in the kinematic problem of CDPM. Hence, the derivation of the relationship between
ψe and cable tension forces is an interesting issue. Let
m and
n be the degree of freedom of moving platform and the number of cables used in system, respectively. Considering the static equilibrium system of the moving platform, the relationship between the cable tension force and external wrench is established in Equation (3).
where
W ∈
Rmxn is the wrench matrix that has ith column vector is defined in (4), and
t = [
t1 …
t2] is the vector that represents the set of magnitude of cable tension forces with
ti ≥ 0, ∀
i = {0, …,
n}.
3. The Properties of Wrench-Closure Workspace
In the following section, for any vector v ∈ Rn and scalar a, we denote that v = a means that vi = a, ∀i = {0…, n} and the other binary conditional operators are analogous to that.
According to Equation (3), let
ω = {
w1,…,
wn} be the set of column vectors of given wrench matrix; the external wrench
ψe can be considered as a non-negative combination of
ω since
t ≥
0. Let
Sψ be the set of all nonnegative combinations of
ω and can be expressed as (5).
where
pos(
ω) denotes the nonnegative linear span of
ω.
Again, on the problem of kinematics, the wrench matrix
W depends on the pose
x of the moving platform. Hence,
Sψ also depends on
x. On the other hand, in this paper, we are interested in the set of poses, in which there always exists at least one set of non-negative tension forces balance arbitrary, given external wrench
ψe. The set of these poses is also known as the wrench-closure workspace, and this can be expressed as (6).
We can express (6) as the condition for pose
x to be in WCW by (7).
In convexity analysis, if ω non-negatively spans ℝm, then the spanning set ω must hold at least m + 1 vectors, and if ω just has m + 1 vectors, then they should be affinely independent.
According to the results from Theorem 3.6 of Davis [
19] and Theorem 2.3 of Conn, Scheinberg and Vicente [
20], we can extract the condition (7) into two simpler conditions, which is stated in Theorem 1.
Theorem 1. A general coordinate x of moving platform belongs to the wrench-closure workspace of a certain CDPM that must satisfy these following conditions: The Equation (8) is the necessary condition and it is equivalent to the wrench matrix
W has full-rank. If
dim(
W) <
m then the system (3) has no solution, even if
t is arbitrarily chosen in
Rn. On the contrary, if the condition (8) is satisfied, then
ω(
x) non-negatively linear span the set which always contains a simplex cone. The condition (9) is sufficient for the set
ω(
x) to nonnegatively linear span
Rm. The Equation (9) is equivalent to Equation (10), in which the geometrical interpretation is easily archived.
where m-first column vectors in
W are linearly independent.
For convenience, let ωA = {w1,…, wm} be the set of m-first column vectors in W and ωB = {wm+1, …, wn} be the set of remaining vectors. Geometrically, the condition (10) can be interpreted that the intersection between interior of cones spanned by ωB and -ωA must exist.
For instance, let us consider a 3 × 5 wrench matrix
W = [
w1 w2 …
w5] whose column vectors are divided into two sets,
ωA = {
w1,
w2,
w3} and
ωB = {
w4,
w5}. These sets and their positive spanning sets are illustrated in
Figure 3. Since
ωA is a linearly independent set (
w1,
w2 and
w3 do not lie in a common plane), the Equation (8) is satisfied. Secondly, the intersection between
pos(−
ωA) and
pos(
ωB) is not empty, hence, the Equation (9) is also satisfied. Therefore, the given wrench matrix that has column vectors illustrated in
Figure 3 has a general coordinate presented in the wrench-closure workspace.
The special case of this problem is CDPM contains only one redundant actuator, i.e.,
n − m = 1. In that case, the condition (10) is simplified to (11).
where
λ = [
λ1 λ
2 …
λm]
T and
λi =
ti/tn.
Indeed, vector wn non-negatively spans a ray and this must belong to a cone spanned by −ωA to satisfy the Equation (10).
For instance, let us consider a 3 × 4 wrench matrix
W = [
w1 w2 w3 w4] which has column vectors illustrated in
Figure 4. Similar to the last example, since
w1,
w2 and
w3 are linearly independent, we have that
span({
w1,
w2,
w3,
w4}) =
R3. Moreover,
w4 belongs to the negative spanning set of
w1,
w2 and
w3 then
pos({
w1,
w2,
w3,
w4}) =
R3.
4. Simplify Problem by Using Linear Equivalence
Let M ∈ Rmxm be the square matrix established by taking m-column vectors in W and H ∈ Rmx(n−m) consist of the remaining column vectors in W. According to Theorem 1, if x ∈ WCW ⊂ Rm then ω must contain m linearly independent vectors, i.e., there exist a certain set of m-column vectors in W that form the nonsingular matrix M.
Theorem 2. Given that N ∈ Rmxm is the non-singular matrix, if ω non-negatively spans Rm then N.ω = {Nw1, Nw2, …, Nwn} also non-negatively spans Rm. If ω does not nonnegatively span Rm, then N.ω also does not non-negatively span Rm.
Proof of Theorem 2. The proof of the first part of Theorem 2 can be derived from the proof of Theorem 4.6 by Regis in [
21]. The minimum condition in order for
ω nonnegatively span
Rm is that
ω must contain at least
m + 1 affinely independent vectors. Hence, let consider {
w1,
w2, …,
wm+1} be the set of
m + 1 affinely independent vectors in
ω. Suppose {
Nw1,
Nw2, …,
Nwm+1} is affinely dependent then
∃i ∈ {1, …,
m+1} such that
Nwi = Σ
j = {1, …, i−1, i+1, …,m} Nwj. Since
N is invertible, we have
N−1Nwi =
N−1Σ
j = {1, …, i−1, i+1, …, m} Nwj. Then, by simplifying both sides, we have
wi = Σ
j = {1, …, i−1, i+1, …, m}wj. However, it violates the assumption in Theorem 2. Following the procedure analogous to that employed for the proof of the second part. □
According to Theorem 2, we can state that if
ω does or does not non-negatively span
Rm than
M−1ω also does or does not nonnegatively span
Rm. Thus, Equation (3) can be replaced by any of the Equations (12)–(14), but the condition represented by (9) is not influent.
where
tA = [t
1 t
2 … t
m]
T and
tB = [t
m+1 t
m+2 … t
n]
T.
Based on Equation (12), we can write
M−1ω = {
e1,
e2, …,
em,
M−1wm+1, …,
M−1wn}, where {
e1,
e2, …,
em} is the standard basis of
Rm. Following the Equation (10), the set of m-first column vectors in
W is now represented as
ωA = {
e1,
e2, …,
em} and the set of remaining vectors is
ωB = {
M−1wm+1, …,
M−1wn}. Geometrically,
ωA non-negatively spans closed nonnegative hyperoctant of
Rm,
Rm+. Hence, if
x belongs to WCW, then the intersection between the interior of cone
ωB and open negative hyperoctant
Rm− should not be empty. For instance,
Figure 5 illustrated the specific case of 2 redundant actuators-planar CDPM when a general coordinate is presented in wrench-closure workspace and linear equivalence was applied to the wrench matrix generated by the considered general coordinate.
In the special case, when
n −
m = 1, the sufficient condition for the general coordinate belonging to WCW is simplified as (15).
In the general case, when
n −
m > 1, we cannot determine whether
x does or does not belong to WCW only based on the signs of elements in
ωB. However, the linear equivalence nonnegative span set of
ω simplifies the condition (10). Since
ωA already non-negatively spans
Rm+, in Equation (14), it is equivalent to
ItA > 0, ∀
tA > 0, the sufficient condition for the general coordinate
x belongs to WCW is defined by (16).
For a more compact system of inequalities, we can rewrite (16) as (17).
Theorem 3. A general coordinate x belongs to the wrench-closure workspace of a certain CDPM if (8) and (17) are satisfied.
In the next section, we will introduce the procedure in order to determine whether (17) does or does not have solution.
5. Checking the Feasibility of a System of Inequalities
Based on the matrix theory for solving the system of linear equations, L. Dines developed his own theory on determining the general solution of system of linear inequations. He also proposed the method used to check the feasibility of them. In this section, we will introduce L. Dines’ theory, make our conclusions and establish our algorithm to solve the system (17).
Let’s consider the system of inequations in form of
Ax > 0 where
A is a coefficient matrix. L. Dines proposed the concepts I-positive, I-negative and I-definite to establish his own work on checking the feasibility of
Ax > 0. These concepts can be summarized by the Definition 1.
Definition 1. If matrix A (18) contains a column in which all elements are positive (or negative) then A is said to be I-positive (or I-negative) with respect to that column. In both cases, generally, matrix A is said to be I-definite with respect to that column. Additionally, if matrix A is I-definite (I-positive or I-negative), then it must contain at least one column in which all elements have same sign (positive or negative).
Conclusion 1. According to definition 1, we can conclude that if matrix A is I-definite then Ax > 0 is feasible.
Proof of Conclusion 1. Let {a1, a2, …, an} be the set of column vectors of A, if A is I-definite then there exist an column vector ai, i ∈ {1, 2, …, n}, in which all elements has same sign. Hence, we always have a set of solution x = [x1 x2 … xi … xn]T = [0 0 … a … 0]T, where a > 0, in a feasible region. □
However, conclusion 1 is not sufficient to check the feasibility of system Ax > 0, i.e., If the matrix A is not I-definite, then we cannot conclude that Ax > 0 is feasible or not. In this case, L. Dines introduced the concepts of I-complement and I-minors. The I-complement of the rth column of matrix A, A1r, is the matrix that can be established as follows.
Separate the elements of rth column of A, [a1r a2r … amr]T, into three sets based on their sign: positive set P = {ai1r, ai2r, …, aiPr}, negative set N = {aj1r, aj2r, …, ajNr} and zero set Z = {ak1r, ak2r, …, akZr}.
Then, we orderly top-down establish each row of
A1r by orderly combining
air and
ajr, and used them to compute each element of the corresponding row by the second-order determinant (19), where value of
k corresponding to the order of elements in the considered row.
The last rows of A1r are derived from the rows of A that contain akr and formed by taking these rows without elements akr.
By that procedure, the matrix A1r which has P = N + Z rows and n − 1 columns is established. As we can see, if the matrix A has n columns, then there were n I-complements {A11, A12, …, A1n}, and the set of these matrices is called I-minors of n − 1 columns of the matrix A. By employing the above procedure to each I-minor of n − 1 columns of the matrix A, I-minors of n − 2 columns of the matrix A can be archived. Recursively, we can obtain I-minors of n − h columns of the matrix A where h < n.
Based on Theorem 1 stated by Dines in [
18], we can derive the sufficient condition in order to check the feasibility of system of inequalities in form of
Ax > 0.
Theorem 4. The system of inequalities in form of Ax > 0 is feasible if there exists at least one I-minor of n-h column which is I-definite where h < n.
Since the system (17) in form Ax > 0, so we can apply Theorem 4 to it and make our conclusion.
Conclusion 2. The system of inequalities (17) is feasible if there exists at least one I-minor of h columns which is I-positive.
After the whole process of checking the feasibility of given system (17), we have a sufficient condition to determine whether the pose
x which generates the wrench matrix
W belongs to WCW or does not. The overview of the whole process to check the feasibility of certain pose is illustrated in
Figure 6.
6. Case Studies and Simulation
In this section, the proposed wrench closure condition is applied to search for the wrench-closure workspace of two configurations CDPM defined in
Figure 1. For convenience, we define three types of WCW:
Total orientation wrench closure workspace (TOWCW): Other than the definition in [
22], in this article, TOWCW presents the whole WCW when position and orientation of moving platform are considered as variables. Notice that, because of the limitation in visualization, we can plot the TOWCW of 3-dof CDPM.
Constant orientation wrench closure workspace (COWCW): For 6-dof CDPM, we can visualize the projection of its TOWCW on up to 3-dimensional workspace. If the projected space was only related to the position of moving platform, this projection is called COWCW. For 3-dof CDPM, COWCW is the section obtained at specific γp.
Constant position wrench closure workspace (CPWCW): Similar to COWCW, CPWCW is the projection of TOWCW when the projected space was only related to the orientation of moving platform, i.e., position of moving platform is constant.
As shown in
Figure 7, WCW of the first configuration CDPM shown in
Figure 1 is generated by point-wisely employing proposed WWC in the grid map, which has a dimension of 1 m × 1 m × 2π and resolution of 0.02 m × 0.02 m × 0.02 rad. Total orientation WCW of planar CDPM is a subset of 3-dimensional space (
xP-yP-γP); it is illustrated in
Figure 7a. Moreover, 2-dimensional sections of total orientation WCW of planar CDPM obtained at
γP = 0°, 0.052 rad and −0.052 rad are also plotted in
Figure 7b–d. The orange, yellow and purple closed curves in
Figure 7a represent the boundaries of slices that present in
Figure 7b–d, respectively. However, this configuration is simple, since it has 3-dof and only one redundant actuator. Hence, it cannot verify the feature “applicable for a completely or redundantly restrained CDPM” of proposed WWC. The second configuration CDPM shown in
Figure 1 is a 2-redundantly actuated spatial CDPM, and it can be used to reinforce the flexibility of proposed WWC.
As shown in
Figure 8, WCW of the second configuration CDPM shown in
Figure 1 is generated by applying proposed WWC. Constant orientation WCW with
αP = 0,
βP = 0 and
γP = 0.052 rad, and constant position WCW with
xP = 0,
yP = 0 and
zP = 0 are shown in
Figure 8a,b, respectively. As shown in
Figure 7c and
Figure 8a, the cross-section perpendicular to the
z-axis of COWCW of the spatial CDPM is trickly equivalent to the COWCW of the planar CDPM due to their similar dimensions. This comment is also verified by Pham et al. in [
16]. In
Figure 8b, the CPWCW is not continuous and in practice, this type of CDPM just can work on a small set of orientational spaces.
In simulations, the time-consumption of proposed WCC used to determine whether a pose presents in WCW is also considered. Time-consumption is not the same in each performance due to two factors: size of wrench matrix and the interruption of algorithm. The first factor depends on configuration of CDPM, it is related to the number of used cables and DoF of moving platform. In the worst case, when a pose belongs to WCW and the algorithm for checking the feasibility of system of inequations must evaluate all I-minor of given wrench matrix. The second factor is the interruption of the algorithm. Our algorithm will orderly evaluate I-minor of the given wrench matrix until one of them is I-definite. In the best case, the wrench matrix is not full-rank, the pose absolutely does not belong to WCW and the algorithm does not take place.