## 3 The Properties of Wrench-Closure Workspace  
In the following section, for any vector \(v\in R^{n}\) and scalar \(a\), we denote that \(v=a\) means that \(v_{i}=a\), \(\forall i=\{0\,\ldots\,,\,n\}\) and the other binary conditional operators are analogous to that. According to Equation (3), let \(\mathbf{\omega}=\{\mathbf{w}_{1},\,\ldots\,,\,\mathbf{w}_{n}\}\) be the set of column vectors of given wrench matrix; the external wrench \(\mathbf{\psi}_{e}\) can be considered as a non-negative combination of \(\mathbf{\omega}\) since \(\mathbf{t}\geq 0\). Let \(\mathbf{S}_{\mathbf{\psi}}\) be the set of all nonnegative combinations of \(\mathbf{\omega}\) and can be expressed as (5). \[\mathbf{S}_{\mathbf{\psi}}=\left\{\mathbf{\psi}_{e}=\sum_{i=1}^{n}\mathbf{w}_{i}t_{i},\forall t _{i}\geq 0\right\}=pos(\mathbf{\omega}) \tag{5}\]  
where \(pos(\mathbf{\omega})\) denotes the nonnegative linear span of \(\mathbf{\omega}\). Again, on the problem of kinematics, the wrench matrix \(\mathbf{W}\) depends on the pose \(\mathbf{x}\) of the moving platform. Hence, \(\mathbf{S}_{\mathbf{\psi}}\) also depends on \(\mathbf{x}\). 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 \(\mathbf{\psi}_{e}\). The set of these poses is also known as the wrench-closure workspace, and this can be expressed as (6). \[WCW=\{\mathbf{x}:pos(\mathbf{\omega})=\mathbb{R}^{m}\} \tag{6}\]  
We can express (6) as the condition for pose \(\mathbf{x}\) to be in WCW by (7). \[\mathbf{x}\in WCW\Leftrightarrow pos\big{(}\{\mathbf{w}_{i}\}_{i=1}^{n}\big{)}=\mathbb{ R}^{m} \tag{7}\]  
In convexity analysis, if \(\mathbf{\omega}\) non-negatively spans \(\mathbb{R}^{m}\), then the spanning set \(\mathbf{\omega}\) must hold at least \(m+1\) vectors, and if \(\mathbf{\omega}\) just has \(m+1\) vectors, then they should be affinely independent. According to the results from Theorem 3.6 of [PERSON] [19] and Theorem 2.3 of [PERSON], [PERSON] [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:_  
\[span(\mathbf{\omega})=\mathbb{R}^{m} \tag{8}\]  
Figure 2: General kinematic model of CDPM. \[\exists\mathbf{t}>0:\sum_{i=1}^{n}\mathbf{w}_{i}t_{i}=0 \tag{9}\]  
The Equation (8) is the necessary condition and it is equivalent to the wrench matrix \(\mathbf{W}\) has full-rank. If \(dim(\mathbf{W})<m\) then the system (3) has no solution, even if \(\mathbf{t}\) is arbitrarily chosen in \(\mathbf{R}^{n}\). On the contrary, if the condition (8) is satisfied, then \(\mathbf{\omega}(\mathbf{x})\) non-negatively linear span the set which always contains a simplex cone. The condition (9) is sufficient for the set \(\mathbf{\omega}(\mathbf{x})\) to nonnegatively linear span \(\mathbf{R}^{m}\). The Equation (9) is equivalent to Equation (10), in which the geometrical interpretation is easily archived. \[\exists\mathbf{t}>0:-\sum_{i=1}^{m}\mathbf{w}_{i}t_{i}=\sum_{j=m+1}^{n}\mathbf{ w}_{i}t_{i} \tag{10}\]  
where m-first column vectors in \(\mathbf{W}\) are linearly independent. For convenience, let \(\mathbf{\omega}_{A}=\{\mathbf{w}_{1},\,\ldots,\,\mathbf{w}_{m}\}\) be the set of m-first column vectors in \(\mathbf{W}\) and \(\mathbf{\omega}_{B}=\{\mathbf{w}_{m+1},\,\ldots,\,\mathbf{w}_{n}\}\) be the set of remaining vectors. Geometrically, the condition (10) can be interpreted that the intersection between interior of cones spanned by \(\mathbf{\omega}_{B}\) and -\(\mathbf{\omega}_{A}\) must exist. For instance, let us consider a \(3\times 5\) wrench matrix \(\mathbf{W}=[\mathbf{w}_{1}\,\mathbf{w}_{2}\,\ldots\,\mathbf{w}_{5}]\) whose column vectors are divided into two sets, \(\mathbf{\omega}_{A}=\{\mathbf{w}_{1},\,\mathbf{w}_{2},\,\mathbf{w}_{3}\}\) and \(\mathbf{\omega}_{B}=\{\mathbf{w}_{4},\,\mathbf{w}_{5}\}\).