2.1. Kinematic relations

Figure 1 shows a schematic of a general $n$ -DOF manipulator driven by $m$ inextensible and non-slack tendons. Each tendon can apply only tensile force to pull the target link through a set of consecutive pulleys. Due to the unilateral nature of the force applied by the tendon, the total number of tendons, $m$ , must be greater than the number of joints, $n$ , in order to control the robot manipulator [Reference Lee and Tsai24]. It is assumed that the maximum number of tendons is no larger than $2n$ because, if $m \geq 2n$ , each joint can be controlled independently by two or more antagonistic tendons and thus the design problem becomes trivial. Therefore, the number of tendons satisfies $n+1 \leq m \leq 2n$ . If $\alpha := m-n$ represents the tendon redundancy, it must be within the range $1 \leq \alpha \leq n$ .

Suppose tendon $i$ , driven by input motor $i$ , runs to pass over the pulleys consecutively, starting from the $\underline {i}$ -th joint and terminating at the $\overline {i}$ -th joint of the manipulator. Then the tendon displacement, $\Delta s_i$ , can be written as a combination of the angular displacement of the joints en route as follows:

(1) \begin{equation} \Delta s_{i}=r_{i}\Delta q_{i}=\sum _{j=\underline {i}}^{\overline {i}}{(-1)^{\epsilon _{ij}} {r}_{ij} \Delta \theta _{j}}, \end{equation}

where $r_{i}$ and $\Delta q_i$ denote the radius of pulley attached at motor $i$ and the angular displacement of motor $i$ , respectively; $r_{ij}$ denotes the radius of the pulley placed coaxially at joint $j$ along tendon $i$ and $\Delta \theta _{j}$ represents the angular displacement of joint $j$ ; and $\epsilon _{ij}$ is 0 (resp. 1) if tendon $i$ rotates the pulley of $r_{ij}$ positively (resp. negatively). Refer to [Reference Tsai and Lee27] for more details regarding the kinematic relations above. Augmenting (1) for all the tendons yields

(2) \begin{equation} \Delta {\mathbf {s}}={\mathbf {R}} \Delta \mathbf {q}={\mathbf {A}} \Delta {\boldsymbol \theta }, \end{equation}

where $\Delta {\mathbf {s}} =[ \Delta s_1\, \Delta s_2\,\cdots \Delta s_m]^T$ denotes the vector of tendon displacements; ${\mathbf {R}}\in \mathbb {R}^{m\times m}$ is the diagonal matrix whose $(i,i)$ component is $r_{i}$ and $\Delta \mathbf {q} =[\Delta q_1\,\Delta q_2\,\cdots \Delta q_m]$ denotes the vector of motor displacements; ${\mathbf {A}} \in \mathbb {R}^{m \times n}$ denotes the relational matrix whose $(i,j)$ component is $(-1)^{\eta _{ij}} {r}_{ij}$ from (1) if tendon $i$ and joint $j$ are associated, otherwise, zero; and $\Delta {\boldsymbol \theta } =[ \Delta \theta _1\, \Delta \theta _2\,\cdots \Delta \theta _n]^T$ is the vector of joint displacements. The entries in the $i$ -th row of $\mathbf {A}$ carry information about radii, associativity, and directionality of the pulleys that tendon $i$ passes over, while the entries in the $j$ -th column of $\mathbf {A}$ provide information regarding the pulleys that are coaxially located at joint $j$ . Because each tendon must pass over every pulley within the starting and terminating tendon points, consecutive nonzero elements must appear in each row of $\mathbf {A}$ .

The above displacement relation in (2), if rewritten with velocity quantities, becomes

\begin{equation*} \dot {\mathbf {s}} = {\mathbf {R}}\dot {\mathbf {q}}={\mathbf {A}}\dot {{\boldsymbol \theta }}, \end{equation*}

which gives

(3) \begin{equation} \dot {\mathbf {q}}={\mathbf {R}}^{-1} {\mathbf {A}}\dot {{\boldsymbol \theta }}. \end{equation}

By invoking the virtual work principle, a relation dual to (3) is obtained as follows:

(4) \begin{equation} {\boldsymbol \tau } ={\mathbf {B}} {\mathbf {t}}, \end{equation}

where ${\mathbf {B}} := {\mathbf {A}}^T{\mathbf {R}}^{-1} \in \mathbb {R}^{n \times m}$ , which is unitless and rectangular, is called as the control structure matrix (or simply the structure matrix), and ${\boldsymbol \tau } \in \mathbb {R}^n$ and ${\mathbf {t}} \in \mathbb {R}^m$ denote the vectors of joint torque and motor torque, respectively. The general solution for $\mathbf {t}$ that reproduces the given joint torque $\boldsymbol \tau$ is found as

(5) \begin{equation} {\mathbf {t}}={\mathbf {B}}^{\dagger }{\boldsymbol \tau }+ \mathbf {N} {\boldsymbol \eta } \,\,\,\, \longleftrightarrow \,\,\, {\mathbf {t}} = {\mathbf {t}}_{p} + {\mathbf {t}}_{n}, \end{equation}

where ${\mathbf {B}}^{\dagger }:= {\mathbf {B}}^T({\mathbf {B}}{\mathbf {B}}^T)^{-1}$ and $\mathbf {N} \in \mathbb {R}^{m \times \alpha }$ , respectively, denote the right pseudo inverse of $\mathbf {B}$ and the null space matrix whose columns consisting of the basis vectors of the null space of $\mathbf {B}$ , and ${\boldsymbol \eta }\in \mathbb {R}^{\alpha }$ is a vector of free parameters [Reference Strang36]. Hence, every row vector of $\mathbf {B}$ is orthogonal to the columns of $\mathbf {N}$ . The particular solution, ${\mathbf {t}}_{p}$ , corresponds to the minimum norm solution of $\mathbf {t}$ , but it does not guarantee the positivity of the input actuation by itself. And ${\mathbf {t}}_{n}$ , which is composed of the columns of $\mathbf {N}$ , serves as a complementary solution, ensuring that the entire solution remains positive. Because ${\mathbf {t}}_{n}$ does not alter the joint torque, it works as the internal force. Details regarding the feasibility and controllability issues regarding the kinematic structure shall be addressed in the following subsection.

2.2. Controllability

To reproduce an arbitrary joint torque $\boldsymbol \tau$ using motor torque $\mathbf {t}$ , a positive solution for $\mathbf {t}$ in (5) must exist after suitably adjusting the free parameter $\boldsymbol \eta$ . A tendon-driven manipulator with this capability is said to be controllable. This requirement can be met if a linear combination of columns of $\mathbf {N}$ can produce a vector whose entries are of a same sign [Reference Murray, Li and Sastry37]. For the case of single redundancy (i.e., $\alpha =1$ ), a controllable tendon-driven manipulator must have a null space basis vector having entries of the same sign.

The controllability is closely related to the appearance of structure matrix $\mathbf {B}$ . In particular, if the tendons having a single redundancy ( $\alpha =1$ ) are driven by the motors placed at the base, the tendon-driven structure that is constructed by a minimum number of pulleys shows the following $n$ -th order pseudo-triangular form of structure matrix [Reference Lee and Tsai24, Reference Morecki, Busko, Gasztold and Jaworek26]:

(6) \begin{equation} \widehat {{\mathbf {B}}}(n) :=\begin{bmatrix} b_{1,1}&b_{1,2}&\cdots &b_{1, n-1}&b_{1,n}&b_{1,n+1}\\ 0&b_{2,2}&\cdots &b_{2,n-1}&b_{2,n}&b_{2,n+1}\\ \vdots &\vdots &\ddots &\vdots &\vdots &\vdots \\ 0&0&\cdots &b_{n-1,n-1}&b_{n-1,n}&b_{n-1,n+1}\\ 0&0&\cdots &0&b_{n,n}&b_{n,n+1}\\ \end{bmatrix}, \end{equation}

where $b_{ij}$ is the $(i,j)$ th nonzero element. Investigating the form of $\widehat {{\mathbf {B}}}(n)$ above, we notice that the last joint relevant to the last row of $\widehat {{\mathbf {B}}}(n)$ is driven by two antagonistic tendons, the second to last joint is driven by these two tendons together with an additional new tendon, and so on. This monotonic staircase pattern allows the manipulator to be readily controlled by the actuators at the base. Ou and Tsai [Reference Ou and Tsai25] proposed the following closed form solution, $\widehat {{\mathbf {B}}}^*(n)$ , for $\widehat {{\mathbf {B}}}(n)$ that not only satisfies the isotropic condition $\widehat {{\mathbf {B}}}(n) \widehat {{\mathbf {B}}}^T(n) = {\mathbf {I}}_{n}$ but also admits $\mathbb {I}= [1\,1\,\cdots \,1]^T$ as a null space vector:

(7) \begin{equation} \widehat {{\mathbf {B}}}^*(n)= \left [\begin{array}{cccc} \sqrt {\frac {2n^2}{n(n+1)}}&-\sqrt {\frac {2}{n(n+1)}}&-\sqrt {\frac {2}{n(n+1)}}&\cdots \\ 0&\sqrt {\frac {2(n-1)^2}{(n-1)n}}&-\sqrt {\frac {2}{(n-1)n}}& \cdots \\ \vdots &\vdots &\vdots &\ddots \\ 0&0&0&\cdots \\ 0&0&0&\cdots \end{array}\right . \left . \begin{array}{cccc} -\sqrt {\frac {2}{n(n+1)}}&-\sqrt {\frac {2}{n(n+1)}}&-\sqrt {\frac {2}{n(n+1)}}&-\sqrt {\frac {2}{n(n+1)}}\\ -\sqrt {\frac {2}{(n-1)n}}&-\sqrt {\frac {2}{(n-1)n}}&-\sqrt {\frac {2}{(n-1)n}}&-\sqrt {\frac {2}{(n-1)n}}\\ \vdots &\vdots &\vdots & \vdots \\ 0&\frac {\sqrt {2}}{\sqrt {3}}&-\frac {1}{\sqrt {3}}&-\frac {1}{\sqrt {3}}\\ 0&0&1&-1 \end{array} \right ]. \end{equation}

Once the structure matrix is found, it means the tendon routes and relative size of pulleys of the tendon-driven manipulator has been designed.

However, in more general cases of redundancy ( $\alpha \gt 1$ ), unlike the case of $\alpha =1$ , it is hard to imagine how the shape of $\mathbf {B}$ and its associative null space basis vectors are related. Motivated by this fact, we define the simplest possible forms for admissible null space matrix $\mathbf {N}$ when the tendons are driven at the base and connected with minimal number of pulleys for the case of $\alpha \gt 1$ , which greatly helps explore the compatible shapes of $\mathbf {B}$ .

The diagonal block, $\mathbf {N}_{ii} \in \mathbb {R}^{e_i}$ , has $e_i$ elements, which we call the index of $\mathbf {N}_{ii}$ . Thus, the sum of the indices of all diagonal blocks equals the total number of tendons as

(8) \begin{equation} \sum _{i=1}^{\alpha } e_i = m. \end{equation}

It is obvious that any tendon-driven manipulator that has a CBTF of $\mathbf {N}$ is controllable because the positive tension requirement can always be satisfied by the combination of columns of $\mathbf {N}$ . Besides, for any null space matrix $\mathbf {N}$ of a controllable tendon-driven manipulator, there is a linear transform that converts $\mathbf {N}$ into its CBTF equivalent form, as described in the following lemma.

Figure 2.

The controllable block triangular form (CBTF) of the null space matrix $\mathbf {N}$ . Each diagonal block consists of elements of the same sign. The index $e_i$ denotes the size of the $i$ -th diagonal block.

If we examine the shape of the CBTF equivalent $\mathbf {N}$ for any given structure of a tendon-driven manipulator, we can easily observe how the joint torque and the motor torque are coupled. Let us apply the transformation in Lemma 1 to (5), which gives the following equation after the premultiplication of $\mathbf {P}$ :

(9) \begin{equation} {\mathbf {t}}'=\mathbf {P}{\mathbf {B}}^{\dagger }{\boldsymbol \tau }+ {\mathbf {N}}_{CBTF}\, {\boldsymbol \eta }', \end{equation}

where ${\mathbf {t}}' :=\mathbf {P}{\mathbf {t}}$ and ${\boldsymbol \eta }' := \mathbf {Q}^{-1} {\boldsymbol \eta }$ . The above equation shows the maximally decoupled relation between the free parameter ${\boldsymbol \eta }'$ and ${\mathbf {t}}'$ , which is good for knowing better the underlying physical structure. However, from the viewpoint of control implementation, simply transforming $\mathbf {N}$ into $\mathbf {N}_{CBTF}$ and having the relation (9) do not help solve (5) for the positive $\mathbf {t}$ . To utilize the advantages of the CBTF of $\mathbf {N}$ for the control, a physical tendon-driven manipulator must be developed via considerate and purposeful design so that its null space matrix $\mathbf {N}$ is of a CBTF.

In the sections to follow, a general design method for tendon-driven manipulators with $1 \leq \alpha \leq n$ is proposed to obtain the desired structure matrix $\mathbf {B}$ , satisfying physical and functional requirements, for a given CBTF of $\mathbf {N}$ .

3.1. Requisites for structure matrix $\mathbf {B}$

In Design Problem I, $\mathbf {B}$ will be found by using the properties of the orthogonality between $\mathbf {B}$ and $\mathbf {N}$ and some physical constraints. The followings are the fundamental physical constraints:

(Constraint 1) Each tendon route starts from the base.

(Constraint 2) Tendon connections are minimal for a given $\mathbf {N}$ .

(Constraint 3) Each tendon must pass over all of the pulleys of the incident joints continuously without skipping any joint in the middle.

With regard to the structural form of $\mathbf {B}$ , Constraint 1 means that every column of $\mathbf {B}$ must start with a nonzero element; Constraint 2 means that the total number of nonzero elements of $\mathbf {B}$ is minimized for a given $\mathbf {N}$ ; and the last constraint dictates that nonzero elements must appear continuously in each column of $\mathbf {B}$ . Due to the annihilation condition between the row space of $\mathbf {B}$ and the column space of $\mathbf {N}$ as

(10) \begin{equation} {\mathbf {B}}\mathbf {N} =\mathbf {0}, \end{equation}

the minimally connected admissible form of $\mathbf {B}$ must be an upper block triangular matrix, defined below.

Figure 3.

The complementary controllable block triangular form (CBTF) of structure matrix $\mathbf {B}$ . Each diagonal block exhibits itself a pseudo triangular form.

The size of block ${\mathbf {B}}_{i,j} \in \mathbb {R}^{(e_{i}-1) \times e_{j}}$ in Fig. 3 must comply with the corresponding blocks of $\mathbf {N}$ so that the annihilation (10) is met. Each pair of diagonal blocks $({\mathbf {B}}_{i,i}, \mathbf {N}_{i,i})$ satisfying ${\mathbf {B}}_{i,i} \mathbf {N}_{i,i} = \mathbf {0}$ can be viewed as a sub-structure that itself becomes a minimal tendon connection with single redundancy, where ${\mathbf {B}}_{i,i}$ is to be of a pseudo triangular form. Based on this observation, we need to seek or design $\mathbf {B}$ confined to a class of complementary CBTFs for a given $\mathbf {N}$ in a CBTF.

In general, all solutions for $\mathbf {B}$ in (10) may be written as

(11) \begin{equation} {\mathbf {B}} = {\mathbf {H}} ({\mathbf {I}}_{m} - \mathbf {N} \mathbf {N}^\ddagger ), \end{equation}

where ${\mathbf {I}}_{m}$ denotes the $m$ dimensional identity matrix; $\mathbf {N}^\ddagger := (\mathbf {N}^T\mathbf {N})^{-1}\mathbf {N}^T$ represents the left pseudo inverse of $\mathbf {N}$ [Reference Zhang38]; and ${\mathbf {H}} \in \mathbb {R}^{n \times m}$ is an arbitrary matrix that yields a complementary CBTF of $\mathbf {B}$ . However, it is not easy to restrict $\mathbf {H}$ such that $\mathbf {B}$ in (11) is to be of a complementary CBTF. Rather than using (11), an algorithmic approach that gives all admissible solutions for $\mathbf {B}$ is more desirable. So, consider a decomposition of $\mathbf {B}$ :

(12) \begin{equation} {\mathbf {B}} = {\mathbf {C}} {\mathbf {B}}^0, \end{equation}

where ${\mathbf {B}}^0 \in \mathbb {R}^{n \times m}$ , to be determined later, denotes a known particular solution to (10), which is also a complementary CBTF matrix, and ${\mathbf {C}} \in \mathbb {R}^{n \times n}$ is an arbitrary nonsingular upper triangular matrix:

(13) \begin{equation} {{\mathbf {C}}}:=\begin{bmatrix} c_{1,1}&c_{1,2}&\cdots &c_{1,n}\\ 0 & c_{2,2}&\cdots &c_{2,n}\\ \vdots &\vdots &\ddots &\vdots \\ 0&0&\cdots &c_{n,n}\\ \end{bmatrix}. \end{equation}

Because any complementary CBTF matrix still maintains its form even after premultiplication by an upper triangular matrix $\mathbf {C}$ , the general complementary CBTF solution $\mathbf {B}$ of (10) can be expressed as (12).

The immediate question is how to obtain the particular solution ${\mathbf {B}}^0$ for given $\mathbf {N}$ . To answer the question, we multiply the $i$ -th row blocks of $\mathbf {B}$ and $\mathbf {N}$ , which yields

(14) \begin{equation} {\mathbf {B}}_{i,i}\mathbf {N}_{i,i}=\mathbf {0} \end{equation}

and

(15) \begin{equation} \begin{split} & \begin{bmatrix}{\mathbf {B}}_{i,i+1}&{\mathbf {B}}_{i,i+2}&\cdots &{\mathbf {B}}_{i,\alpha }\end{bmatrix} \begin{bmatrix}\mathbf {N}_{i+1,i+1}&\mathbf {N}_{i+1,i+2}&\cdots &\mathbf {N}_{i+1,\alpha }\\ \mathbf {0}&\mathbf {N}_{i+2,i+2}&\cdots &\mathbf {N}_{i+2,\alpha }\\ \vdots &\vdots &\ddots &\vdots \\ \mathbf {0}&\mathbf {0}&\cdots &\mathbf {N}_{\alpha, \alpha } \end{bmatrix} \\ & = -{\mathbf {B}}_{i,i} \left [ \mathbf {N}_{i,i+1} \,\mathbf {N}_{i,i+2} \,\cdots \,\mathbf {N}_{i,\alpha } \right ]. \end{split} \end{equation}

For the simplicity of notation, (15) is re-expressed as

(16) \begin{equation} {\mathbf {B}}_{[i, \, i+1:\alpha ]}\, {\mathbf {N}}_{[i+1:\alpha, \, i+1:\alpha ]} = -{\mathbf {B}}_{i,i} \,{\mathbf {N}}_{[i,\,i+1:\alpha ]}, \end{equation}

where the subscript $[a:b, c:d]$ denotes the submatrix of $\mathbf {N}$ composed of block elements from the $a$ -th to the $b$ -th rows and from the $c$ -th to the $d$ -th columns.

The fact that ${\mathbf {B}}_{i,i}$ must be of a pseudo triangular form in $\mathbb {R}^{(e_1-1) \times e_i}$ independent of overall tendon redundancy and the result in (7) suggest that ${\mathbf {B}}_{i,i}^0$ , a particular solution corresponding to ${\mathbf {B}}_{ii}$ in (14), can be chosen as

(17) \begin{equation} {\mathbf {B}}_{i,i}^0 = \widehat {{\mathbf {B}}}^*({e_i}-1) {\mathbf {D}}_i^{-1} \end{equation}

where ${\mathbf {D}}_i \in \mathbb {R}^{e_i \times e_i}$ is the diagonal matrix whose diagonal entries consist of the elements in $\mathbf {N}_{i,i}$ . It is straightforward to prove from (7) that ${\mathbf {B}}_{i,i}^0\mathbf {N}_{i,i} = \mathbf {0}$ . By combining the obtained ${\mathbf {B}}_{i,i}^0$ , a particular solution ${\mathbf {B}}_{[i, \, i+1:\alpha ]}^0$ of (16) is determined as

(18) \begin{equation} {\mathbf {B}}_{[i, \, i+1:\alpha ]}^0 = -{\mathbf {B}}_{i,i}^0 \,{\mathbf {N}}_{[i,\,i+1:\alpha ]} \left ({\mathbf {N}}_{[i+1:\alpha, \, i+1:\alpha ]}\right )^\ddagger, \end{equation}

where $(\!\cdot \!)^\ddagger$ means the left pseudo inverse of the argument. Ultimately, (17) and (18) for $i=1,2,\cdots, \alpha$ become a complete particular solution ${\mathbf {B}}^0$ of (10).

Since a particular solution has been found, the next task is to determine $\mathbf {C}$ such that the ultimate design ${\mathbf {B}}={\mathbf {C}}{\mathbf {B}}^0$ generates the desired joint torque capacity. For this purpose, define the unbiased joint torque vector ${\mathbf {u}}_\tau \in \mathbb {R}^{n}$ such that

(19) \begin{equation} {\boldsymbol \tau }={\mathbf {M}}{\mathbf {u}}_{\tau }=\begin{bmatrix} \mu _1 & 0&\cdots & 0 &0 \\ 0 & \mu _2 &\cdots &0& 0 \\ \vdots & \vdots & \ddots & \vdots &\vdots \\ 0 & 0 &\cdots & 0 & \mu _n \\ \end{bmatrix} {\mathbf {u}}_{\tau } \end{equation}

where $\mathbf {M}$ is the designer’s weighting matrix that assigns the torque capacity of the joints. High weights must be given to the manipulator’s joints that are likely to take high loads under nominal operation. By combining (5) and (19), the transmission from ${\mathbf {u}}_\tau$ to ${\mathbf {t}}_p$ becomes

(20) \begin{equation} \begin{split} T_{u_{\tau } \rightarrow {t_p}} &= \frac {|| {\mathbf {t}}_p ||^2}{||{\mathbf {u}}_\tau ||^2} = \frac { {\mathbf {u}}_\tau ^T {\mathbf {M}}^T {\mathbf {B}}^{\dagger T} {\mathbf {B}}^\dagger {\mathbf {M}}{\mathbf {u}}_\tau }{{\mathbf {u}}_\tau ^T{\mathbf {u}}_\tau }\\ &= \frac { {\mathbf {u}}_\tau ^T {\mathbf {M}}^T ({\mathbf {C}}{\mathbf {B}}^0)^{\dagger T} ({\mathbf {C}}{\mathbf {B}}^0)^\dagger {\mathbf {M}}{\mathbf {u}}_\tau }{{\mathbf {u}}_\tau ^T{\mathbf {u}}_\tau }. \end{split} \end{equation}

Figure 4.

Mappings between ${\mathbf {t}}$ , ${\mathbf {u}}_{\tau }$ , and $\boldsymbol \tau$ .

The transmission $T_{u_{\tau } \rightarrow {t_p}}$ means the required amount of motor torque for a given unbiased joint torque. It should be $T_{u_{\tau } \rightarrow {t_p}}=1$ to ensure the uniform distribution of motor torque under any choice of ${\mathbf {u}}_\tau$ being $||{\mathbf {u}}_\tau || = 1$ . This happens when a unit sphere in ${\mathbf {u}}_\tau$ maps onto a unit sphere in ${\mathbf {t}}_p$ , so

(21) \begin{equation} {\mathbf {M}}^T {\mathbf {B}}^{\dagger T} {\mathbf {B}}^\dagger {\mathbf {M}}= {\mathbf {M}}^T ({\mathbf {C}}{\mathbf {B}}^0)^{\dagger T} ({\mathbf {C}}{\mathbf {B}}^0)^\dagger {\mathbf {M}} = {\mathbf {I}}_m. \end{equation}

The above isotropic condition leads to an ellipsoidal mapping between $\boldsymbol \tau$ and $\mathbf {t}$ due to (19). The ellipsoidal mapping indicates that the relative capacity of joint torques between the joints can be set as the way the designer wants under the motor torques of uniform capacity. Refer to Fig. 4 that depicts the related mapping diagram. Following the definition of the right pseudo inverse, (21) can be modified as follows:

(22) \begin{equation} {\mathbf {B}}{\mathbf {B}}^T=({\mathbf {C}}{\mathbf {B}}^0 )({\mathbf {C}}{\mathbf {B}}^0)^T = {\mathbf {M}}{\mathbf {M}}^T ={\mathbf {M}}^2. \end{equation}

The fact that the right-hand side of this equation is a positive definite diagonal matrix leads to: (i) the rows of ${\mathbf {C}}{\mathbf {B}}^0$ must be orthogonal to each other and (ii) the norm of row $i$ equals $\mu _i$ . These observations are used to determine $\mathbf {C}$ (and eventually $\mathbf {B}$ ) in conjunction with the structural conditions for realizability, which are used in the next subsection.

3.2. Determining structure matrix $\mathbf {B}$

To determine the $\mathbf {C}$ of an upper triangular form, Eq. (22) is solved row by row from the last row to the first. Let ${\mathbf {b}}_{i}^{0}$ and ${\mathbf {b}}_i$ be the $i$ -th row vectors of ${\mathbf {B}}^{0}$ and $\mathbf {B}$ , respectively. The last row, ${\mathbf {b}}_n^0$ , of ${\mathbf {B}}^0$ contains only two nonzero elements as

(23) \begin{equation} {\mathbf {b}}_{n}^{0} = \begin{bmatrix} 0&\,0&\cdots &0&\,b_{n,n+\alpha -1}^0&\,b_{n,n+\alpha }^0 \end{bmatrix}. \end{equation}

Because ${\mathbf {b}}_n = c_{n,n}{\mathbf {b}}_n^0$ from (12), the last row of (22) yields the following:

(24) \begin{equation} c_{n,n} =\pm \frac {\mu _{n}}{||{\mathbf {b}}_{n}^{0}||}. \end{equation}

The sign of $c_{n,n}$ in (24) decides the positive direction of the torque generated by the tension and can be selected according to the designer’s preference. Hence, ${\mathbf {b}}_n$ can be completely determined.

Next, the $(n-1)$ -th row, ${\mathbf {b}}_{n-1}$ , of $\mathbf {B}$ is written as

\begin{equation*} {\mathbf {b}}_{n-1}=c_{n-1,n-1}{\mathbf {b}}_{n-1}^{0}+c_{n-1,n}{\mathbf {b}}_{n}^{0}, \end{equation*}

which satisfies the orthogonality condition: ${\mathbf {b}}_{n}{\mathbf {b}}_{n-1}^{T} = 0$ as discussed at the end of the previous subsection. Rewriting the above equation yields

(25) \begin{equation} \begin{bmatrix} {{\mathbf {b}}}_{n}({\mathbf {b}}_{n-1}^{0})^{T}&{{\mathbf {b}}}_{n}({\mathbf {b}}_{n}^{0})^{T} \end{bmatrix} \begin{bmatrix} c_{n-1,n-1}\\ c_{n-1,n} \end{bmatrix}=0, \end{equation}

which admits the solution for $c_{n-1,n-1}$ and $c_{n-1,n}$ as

(26) \begin{equation} \begin{bmatrix} c_{n-1,n-1}\\ c_{n-1,n} \end{bmatrix}=\lambda _{n-1} \boldsymbol {\xi }_{n-1}, \end{equation}

where $\boldsymbol {\xi }_{n-1}=[{\xi }_{n-1,1} {\xi }_{n-1,2} ]^T \in \mathbb {R}^2$ denotes the null space basis of $\left [{{\mathbf {b}}}_{n}({\mathbf {b}}_{n-1}^{0})^{T},\,{{\mathbf {b}}}_{n}({\mathbf {b}}_{n-1}^{0})^{T}\right ] \in \mathbb {R}^{1 \times 2}$ ; and $\lambda _{n-1}$ is some number to be determined by the magnitude condition. Since the norm of ${\mathbf {b}}_{n-1}$ must be $\mu _{n-1}$ from (22), $\lambda _{n-1}$ is determined as

(27) \begin{equation} \lambda _{n-1}= \pm \frac {\mu _{n-1}}{||{\mathbf {b}}_{n-1}^0\xi _{n-1,1}+{\mathbf {b}}_{n}^0 \xi _{n-1,2}||}. \end{equation}

Thus, ${\mathbf {b}}_{n-1}$ is completely determined except for its sign, but again the sign can be chosen by the designer.

Following a similar procedure as above, the $i$ -th row vector, ${\mathbf {b}}_i$ , of $\mathbf {B}$ can be written as

\begin{equation*} {\mathbf {b}}_{i}=\sum _{j=i}^{n} c_{i,j }{\mathbf {b}}_{j}^{0}. \end{equation*}

Since ${\mathbf {b}}_{i}$ is orthogonal to rows ${\mathbf {b}}_{i+1}, {\mathbf {b}}_{i+2}, \ldots, {\mathbf {b}}_{n}$ , which are known already, we can obtain

(28) \begin{equation} \begin{bmatrix} {\mathbf {b}}_{i+1}({\mathbf {b}}_{i}^{0})^{T}&{\mathbf {b}}_{i+1}({\mathbf {b}}_{i+1}^{0})^{T}&\cdots &{\mathbf {b}}_{i+1}({\mathbf {b}}_{n}^{0})^{T}\\ {\mathbf {b}}_{i+2}({\mathbf {b}}_{i}^{0})^{T}&{\mathbf {b}}_{i+2}({\mathbf {b}}_{i+1}^{0})^{T}&\cdots &{\mathbf {b}}_{i+2}({\mathbf {b}}_{n}^{0})^{T}\\ \vdots &\vdots &\ddots &\vdots \\ {\mathbf {b}}_{n}({\mathbf {b}}_{i}^{0})^{T}&{\mathbf {b}}_{n}({\mathbf {b}}_{i+1}^{0})^{T}&\cdots &{\mathbf {b}}_{n}({\mathbf {b}}_{n}^{0})^{T}\\ \end{bmatrix} \begin{bmatrix} c_{i,i}\\ c_{i,i+1}\\ \vdots \\ c_{i,n} \end{bmatrix}=\mathbf {0}. \end{equation}

Its nontrivial solution is determined as

(29) \begin{equation} \begin{bmatrix} c_{i,i}\,& c_{i,i+1}\,& \cdots \,& c_{i,n} \end{bmatrix}^T = \lambda _{i} \boldsymbol {\xi }_{i}, \end{equation}

where $\boldsymbol {\xi }_{i}=[{\xi }_{i,1}\,{\xi }_{i,2}\,\cdots {\xi }_{i,n-i+1}]^T \in \mathbb {R}^{n-i+1}$ denotes the null space basis of the matrix in the left hand side of (28). Because $||{\mathbf {b}}_i||=\mu _i$ , the constant $\lambda _{i}$ is determined as

\begin{equation*} \lambda _{i}= \pm \frac {\mu _{i}}{|| \sum _{j=i}^n \boldsymbol {\xi } _{i,j-i+1}{\mathbf {b}}_{j}^0||}. \end{equation*}

After completing this process, $\mathbf {C}$ is obtained and consequently is $\mathbf {B}$ by ${\mathbf {B}}={\mathbf {C}}{\mathbf {B}}^0$ . There are a total of $2^n$ possible $\mathbf {B}$ ’s depending upon the choice of sign for $\lambda _i$ . However, physically they differ only by the positive sign of joint angles without influencing the fundamental kinematic structure, which is said to be isomorphic. This implies that Design Problem I yields practically a unique solution for $\mathbf {B}$ in a complementary CBTF, which shows a minimal tendon connection, for a given CBTF matrix of $\mathbf {N}$ and a joint torque capacity $\mathbf {M}$ .

3.3. Post-design adjustment

In realizing the designed $\mathbf {B}$ into hardware, the physical size of pulleys along any tendon route can be scaled up or down by inversely scaling the size of motor pulley, so that $\mathbf {B}$ remains unchanged. Such relative scaling between pulleys is what we call the post-design adjustment.

Suppose ${\mathbf {R}}=\mbox {diag}\{r_1, r_2, \ldots, r_m\}$ is the matrix consisting of radii of motor pulleys. Then the physical structure matrix ${\mathbf {A}}^{T}$ becomes

\begin{equation*} {\mathbf {A}}^{T} = {\mathbf {B}}{\mathbf {R}}, \end{equation*}

where ${\mathbf {A}}^T$ contains the physical size of joint pulleys. And, varying the size of $\mathbf {R}$ turns in other possible size of joint pulleys. This kind of post-design adjustment is a necessary technical step when determining the realizable size of pulleys, a proper set of motors, and even their gear ratios.

Although the post-design adjustment, however, can help tune the proportion of pulley size between tendon routes in general, it becomes of no use if the resulting structure matrix $\mathbf {B}$ is inherently unrealizable. The unrealizable designs of $\mathbf {B}$ include: (i) severe variation in the entries of any column of $\mathbf {B}$ and (ii) presence of a zero entry in between nonzero entries in the columns of $\mathbf {B}$ . The former case corresponds to a large difference in the pulley size within a tendon route, while the latter indicates a disconnected tendon route or a tendon route having a zero-radius pulley which is not allowed. In Design Problem II, to be discussed in the next section, an optimization scheme is integrated by treating both $\mathbf {B}$ and $\mathbf {N}$ as design variables to produce inherently better realizable designs.

4.1. Determining weight of joint torque

We begin to solve the problem by selecting the weight of joint torque $\mathbf {M}$ in the optimal sense, if not specified or given in advance.

Assuming that the link lengths and joint structure of the tendon-driven manipulator are given, the forward kinematic map from the joint variables to the task variables can be obtained. Let ${\mathbf {J}}({\boldsymbol \theta })$ be the kinematic Jacobian matrix of a tendon-driven manipulator at configuration $\boldsymbol \theta$ such that

\begin{equation*} \dot {{\mathbf {x}}} = {\mathbf {J}}({\boldsymbol \theta }) \dot {{\boldsymbol \theta }}, \end{equation*}

where $\dot {{\mathbf {x}}} \in \mathbb {R}^l$ denotes the Cartesian velocity and $l$ is the dimension of the Cartesian space. Following the virtual work principle [Reference Goldstein, Safko and Poole40],

(30) \begin{equation} {\boldsymbol \tau } = {\mathbf {J}}^T({\boldsymbol \theta }) {\mathbf {F}}, \end{equation}

where ${\mathbf {F}}\in \mathbb {R}^l$ is the force exerted by the end-effector. Refer to Fig. 5 for a signal flow diagram of the overall tendon-driven manipulator system By combining (19) and (30), the transmission from $\mathbf {F}$ to the unbiased joint torque ${\mathbf {u}}_{\tau }$ becomes

(31) \begin{equation} \begin{split} T_{F \rightarrow u_{\tau } } &= \frac {||{\mathbf {u}}_\tau ||^2}{|| {\mathbf {F}} ||^2} = \frac {{\mathbf {u}}_\tau ^T{\mathbf {u}}_\tau }{{\mathbf {F}}^T{\mathbf {F}}} \\ &= \frac { {\mathbf {F}}^T {\mathbf {J}}({\boldsymbol \theta }){\mathbf {M}}^{-T}{\mathbf {M}}^{-1} {\mathbf {J}}^{T}({\boldsymbol \theta }) {\mathbf {F}}}{{\mathbf {F}}^T{\mathbf {F}}}. \end{split} \end{equation}

The isotropic transmission (i.e., $T_{F \rightarrow u_{\tau } } = 1$ ) means that the mapping from $\mathbf {F}$ to ${\mathbf {u}}_\tau$ becomes uniform and no directional tendency exists. If so, the mapping from $\mathbf {F}$ to $\boldsymbol \tau$ ultimately becomes ellipsoidal, inflated, and/or deflated in particular directions from the unit sphere as designated by the weight of joint torque $\mathbf {M}$ , as shown in Fig. 6. This effectively means that the end-effector load is distributed to every joint in a manner as designated by the weight $\mathbf {M}$ .

This ideal situation occurs only when the following holds:

(32) \begin{equation} {\mathbf {J}}({\boldsymbol \theta }){\mathbf {M}}^{-T}{\mathbf {M}}^{-1}{\mathbf {J}}^{T}({\boldsymbol \theta })= {\mathbf {I}}_l, \end{equation}

However, there is no such a solution for $\mathbf {M}$ that satisfies (32) for all joint configurations simultaneously. Thus, by relaxing the ideal isotropic condition, we search for the diagonal matrix $\mathbf {M}$ that yields the smallest condition number of the matrix ${\mathbf {M}}^{-1}{\mathbf {J}}^{T}({\boldsymbol \theta })$ on average for all configurations by the following optimization problem:

(33) \begin{equation} \begin{split} &\min _{{\mathbf {M}}}\,{\sum _{i}^{N_c}\frac {1}{N_c}{\left (1-\frac {1}{\mbox {cond}({\mathbf {M}}^{-1}{\mathbf {J}}^{T}({\boldsymbol \theta }_{i}))}\right )^{2}}}, \\ &\,\mbox {subject to}\,\,\,\,\,{\mathbf {M}}\gt \mathbf {0}, \end{split} \end{equation}

where $\mbox {cond}(\!\cdot \!)$ denotes the condition number or the ratio of the maximum to the minimum singular values of the argument matrix [Reference Strang36], and ${\boldsymbol \theta }_{i}, i=1,\ldots, N_c$ , are the $N_c$ number of joint configurations that roughly cover the entire workspace. Because the condition number can increase too abruptly as the configuration moves toward singularity, the above cost function is constructed to use the inverse of the condition number for numerical stability. Once $\mathbf {M}$ is found via a conventional nonlinear optimization tool, it is supplied as input data for the optimal design of the tendon-driven manipulator.

Figure 6.

Mappings between $\mathbf {F}$ , ${\mathbf {u}}_{\tau }$ , and $\boldsymbol \tau$ .

4.2. Design via optimization

As the central part of Design Problem II, a constrained optimization problem is constructed to find $\mathbf {B}$ and $\mathbf {N}$ as follows:

(34) \begin{equation} \min _{{\mathbf {B}},\,\mathbf {N}} \,\,L= {\Phi ({\mathbf {B}}) + \rho \Psi (\mathbf {N})}, \end{equation}

subject to

(35a) \begin{align} \mathbf {N}^{T}\mathbf {N}&={\mathbf {I}}_{\alpha },\end{align}

(35b) \begin{align} {\mathbf {B}}{\mathbf {B}}^{T}&={\mathbf {M}}^{2},\end{align}

(35c) \begin{align} {\mathbf {B}}\mathbf {N}&=\mathbf {0},\end{align}

(35d) \begin{align} \mathbf {N}_{i,i}&\gt \mathbf {0},\end{align}

where $\mathbf {N}$ and $\mathbf {B}$ , respectively, are assumed to be the CBTF and its complementary CBTF matrices with prescribed indices. The above optimization problem can be solved numerically by using any nonlinear constrained optimization solver supplied with initial guesses of $\mathbf {N}$ and $\mathbf {B}$ .

The minimizing cost function $L$ consists of two scalar functions, $\Phi ({\mathbf {B}})$ and $\Psi (\mathbf {N})$ . The first scalar function $\Phi ({\mathbf {B}})$ is defined as

\begin{align*} \Phi ({\mathbf {B}}) \triangleq &\sum _{i=1}^{m} \sum _{j=1,k=1}^n {(1-(\frac {b_{j,i}}{b_{k,i}})^{2})^{2} + (1-(\frac {b_{k,i}}{b_{j,i}})^{2})^{2}}, \end{align*}

where $b_{j,i}$ denotes the $(j,i)$ -th nonzero element of $\mathbf {B}$ . (Ignore $b_{j,i}$ if the $(j,i)$ -th element is supposed to be a zero entity.) The cost effect of $\Phi ({\mathbf {B}})$ is to obtain regularized elements in each column of $\mathbf {B}$ by penalizing the relative magnitude ratios. The other scalar function $\Psi (\mathbf {N})$ in (34) is defined by the sum of the magnitudes of the off-diagonal blocks of $\mathbf {N}$ as

(36) \begin{equation} \Psi (\mathbf {N}) \triangleq \sum _{i=2}^{\alpha }\sum _{j=1}^{i-1}{\mathbf {N}_{j,i}^T\mathbf {N}_{j,i}}, \end{equation}

which is intended to decrease the cross coupling between groups of joints, thereby enhancing the transparency of tendon actuation in physical tendon-driven manipulators. The weight, $\rho$ , is a selectable number that sets the relative importance of $\Phi ({\mathbf {B}})$ and $\Psi (\mathbf {N})$ ; at an extreme, $\rho$ can be set to 0 when tendon cross coupling is not a critical issue. The designers can modify the cost function $L$ or select any other reasonable cost function as wished, as long as it fits the specific design purpose.

The accompanying constraints (35) in the optimization reflect the physical requirements for the design of tendon-driven manipulators; (35a) enforces each column of $\mathbf {N}$ to be orthonormal; (35b) imposes the specification on the joint torque capacity; (35c) means the annihilation between $\mathbf {B}$ and $\mathbf {N}$ ; and (35d) keeps the entries of $\mathbf {N}_{i,i}$ to be of a same (positive) sign. The constraints (35b) and (35c) are practically imposed in the form of Design Problem I at each iterative step of solving (34), as they represent the same conditions for Design Problem I.

These constraints restrict the dimension of the feasible space of the solution variables [Reference Lay, Lay and McDonald41]. To determine the dimension of the feasible space, let us first count the solution variables that are composed of nonzero elements in $\mathbf {N}$ and $\mathbf {B}$ . Then,

(37) \begin{equation} \mbox {Num}(\mathbf {N}) = m(\alpha +1)-\sum _{i=1}^{\alpha }{ie_{i}} \end{equation}

and

(38) \begin{equation} \mbox {Num}({\mathbf {B}}) = \frac {n(n+1)}{2}+m(\alpha +1)-\alpha ^{2}-\sum _{i=1}^{\alpha }{ie_{i}}. \end{equation}

Remember that $e_i$ , denoting the $i$ -th index of $\mathbf {N}$ , satisfies $\sum _{i=1}^\alpha e_i = m$ . Next, let us count the number of constraints. Constraint (35a) permits the following number of algebraic relations:

(39) \begin{equation} \frac {\alpha (\alpha +1)}{2}. \end{equation}

Constraints (35b) and (35c), respectively, allow the following number of relations:

\begin{equation*} \frac {n(n+1)}{2}\,\,\,\,\mbox {and}\,\,\,\,m(\alpha +1)-\alpha ^{2}-\sum _{i=1}^{\alpha }{ie_{i}}. \end{equation*}

Because $\mbox {Num}({\mathbf {B}})$ is exactly equal to the sum of the number of relations in (35b) and (35c), the number of independent solution variables, $n_{d}^{*}$ , is

(40) \begin{equation} n_{d}^{*}=\frac {(2m-\alpha )(\alpha +1)}{2}-\sum _{i=1}^{\alpha }ie_{i}. \end{equation}

For instance, for an $n$ jointed tendon-driven manipulator with a single redundancy (i.e., $\alpha =1$ and thus $m=n+1$ ), $n_{d}^{*}$ becomes $m-1$ . This is equal to the number of independent nonzero elements in $\mathbf {N}$ , so $\mathbf {B}$ becomes entirely dependent on the choice of $\mathbf {N}$ .

If $n_{d}^{*}$ is large, additional conditions could be assigned to improve the design. It is possible to include, for example, a constraint that the one vector, $\mathbb {I}$ , must belong to the null space; with this, the concentration of tension in any particular tendon could be mitigated when producing joint torque [Reference Sheu, Huang and Lee30]. Since this constraint generates an additional $m-1$ relations, a tendon-driven manipulator with $\alpha =1$ would not have any independent design variables left over, meaning that the design problem becomes an algebraic problem, not an optimization one.

5.1. Examples: design problem I

Assume that we want to design a tendon-driven structure with $\alpha =2$ for a 4-DOF manipulator whose kinematic parameters regarding link lengths and joint alignments are already known. The weight of the joint torque is given as ${\mathbf {M}}=\mbox {diag}\{4,\,3,\,2,\,1\}$ , so that the inner joints can take a greater load than the outer ones. The following three possible CBTFs of null space matrices – whose columns are not yet normalized – are selected to find the corresponding structure matrices by applying the procedure in Design Problem I.

Unlike the others, $\mathbf {N}_2$ shows a decoupled form with no off-diagonal block. As shall be clear soon, a decoupled form of $\mathbf {N}$ is destined to produce a decoupled structure matrix $\mathbf {B}$ , which would not be an acceptable design if the actuators have to be placed at the base. Note that all of the given $\mathbf {N}$ ’s need to be normalized to make their column lengths equal to unity before moving on to the design procedure.

With the null space matrix $\mathbf {N}_1$ and the weight of the joint torque $\mathbf {M}$ , a particular solution for the structure matrix is obtained using (17) and (18):

The combination matrix is then found from (24) to (29) as

By multiplying the particular solution and the combination matrix, the structure matrix is finally determined as

This structure matrix satisfies the designated weight of the joint torque because

\begin{equation*} {\mathbf {B}}{\mathbf {B}}^T=\begin{bmatrix} 16 &\,0 &\,0 &\,0 \\ 0 &\,9 &\,0 &\,0 \\ 0 &\,0 &\,4 &\,0\\ 0 &\,0 &\,0 &\,1 \end{bmatrix}, \end{equation*}

which is equal to ${\mathbf {M}}^2$ .

Now, if the decoupled form of the null space matrix, $\mathbf {N}_2$ , is used for the design, the structure matrix is obtained as follows:

As expected, the designed structure matrix has a decoupled form. Since the last three tendons do not affect the first two joints, the actuators corresponding to these three tendons must be placed in the middle of the manipulator. This result violates our first design constraint, so the design should not be accepted. However, it does meet the assigned weight of the joint torque. It is worth noting that the result obtained with $\mathbf {N}_2$ is equivalent to that obtained by designing two separate 2-DOF manipulators, each driven by three tendons.

If $\mathbf {N}_3$ is used, the following structure matrix is obtained:

Note that the second row of the upper off-diagonal block matrix is zero. The appearance of these zero elements in the middle of a column implies the disconnected tendon, which is not physically acceptable. This situation can be predicted by investigating the orthogonality condition between $\mathbf {N}$ and $\mathbf {B}$ .

In conclusion, there are situations where the desired forms of $\mathbf {N}$ are given or known a priori. If so, the tendon-driven manipulators can be designed simply by solving Design Problem I. The only care needed is to check whether the resulting design is physically acceptable or not.

5.2. Examples: design problem II

The numerical example in this subsection involves designing a tendon-driven structure for a common 3-DOF robot manipulator shown in Fig. 7 using the procedure for Design Problem II. The Denavit–Hartenberg kinematic parameters are known and presented in Table I. The example is intended to illustrate the procedure to solve Design Problem II and then to verify the important characteristics of the proposed design method. Assume that five tendons drive the robot manipulator, that is, $n=3$ , $m=5$ , and $\alpha =2$ .

Table I.

Denavit-Hartenberg parameters of the 3-DOF manipulator.

Figure 7.

A 3-DOF manipulator used for designing its tendon-driven structure.

To begin the design by following the procedure for Design Problem II, we first need to determine the weight of joint torque $\mathbf {M}$ by solving (33). For this purpose, we divide each joint angle from $-\pi$ [rad] to $\pi$ [rad] at equal intervals of $\pi /10$ [rad], to prepare the joint configurations ( ${\boldsymbol \theta }_i$ ) to be used for computing ${\mathbf {J}}({\boldsymbol \theta }_i)$ . Here, the singular configurations which yield $\det ({\mathbf {J}}{\mathbf {J}}^{T})\lt 10^{-10}$ are neglected. After inputting a total of 378 configurations into (33) and solving it via the fmincon function [Reference Byrd, Gilbert and Nocedal42], a numerical optimization routine on Matlab, the following weight of the joint torque is obtained: ${\mathbf {M}}= \mbox {diag}\{12.74,\, 9.80,\, 7.54\}$ .

Another information to be provided is the indices ( $e_i$ ) of $\mathbf {N}$ that define the specific shape of $\mathbf {N}$ and thus its complementary $\mathbf {B}$ . Among the possible sets of indices, that is, $\left \{ (e_1, e_2) | \,(2, 3), \,(3,2), \,(4,1) \right \}$ , the chosen set is $(e_1, e_2)=(2, 3)$ . With this set, joint 1 and the other two joints are likely to be actuated in a partially decoupled fashion, which is suitable for the kinematic structure of the spatial 3-DOF robot manipulator.

Now, let us determine the optimal CBTF of $\mathbf {N}$ and its complementary CBTF of $\mathbf {B}$ by numerically solving (34) with constraints (35). The following initial null space matrix $\mathbf {N}$ , not normalized yet, is provided to initiate the optimization:

(44)

This initial null space matrix is inadequate for a practical reason: increasing the force in the last three tendons causes much greater force in the first two tendons, which is undesirable, due to large coupling through the off-diagonal components of $\mathbf {N}_{init}$ . The complementary structure matrix of $\mathbf {N}_{init}$ is determined by the procedure of Design Problem I:

The initial structure matrix ${\mathbf {B}}_{init}$ shows that the maximum variation in pulley size appears in the fifth tendon route and is less than 2 (that is, $7.303/3.999$ ). And there would be no disconnected tendons by ${\mathbf {B}}_{init}$ . Thus, the physical realization of the design with ${\mathbf {B}}_{init}$ may be possible, though not the optimal.

Using $\mathbf {N}_{init}$ and ${\mathbf {B}}_{init}$ , the function values of $\Phi$ and $\Psi$ contributing to the overall cost function $L$ are calculated to be $5.216$ and $0.987$ , respectively. The weight $\rho$ in the optimization (34) is set as $5.287$ in order to equalize the effect of $\Phi$ and $\Psi$ . Once performing the optimization via the fmincon in Matlab, we obtain the following result:

The function values of $\Phi$ and $\Psi$ after the optimization become $0.375$ and $0.484$ , respectively, which are much smaller than those using the initial conditions. Compared with $\mathbf {N}_{init}$ , the relative magnitudes of the off-diagonal elements in $\mathbf {N}_{opt}$ become much smaller, which is desirable. The structure matrix ${\mathbf {B}}_{opt}$ also shows an improvement from ${\mathbf {B}}_{init}$ , with the pulley sizes along each tendon route becoming further regularized as each column is composed of elements with similar magnitudes. Note that the number of independent variables is $n_d^* = 4$ from (40). Thus, if wished, additional constraints can be imposed to gain extra benefits until some independent design variables remain.

Figure 8 presents the tendon connection diagrams for optimal design. To clearly illustrate the designed tendon-driven structure, planar tendon connection diagrams using ${\mathbf {B}}_{init}$ and ${\mathbf {B}}_{opt}$ compared in Fig. 8(a), where the scaling factors ${\mathbf {R}}_{init}=\mbox {diag}\{0.620$ , $0.620$ , $0.081$ , $0.089$ , $0.089\}$ [m] and ${\mathbf {R}}_{opt}=\mbox {diag}\{0.1$ , $0.1$ , $0.93$ , $0.121$ , $0.121\}$ [m] were, respectively, used as the post-design adjustment by considering the realizability of the design results. It can be seen that the pulleys for ${\mathbf {B}}_{opt}$ are more regular than those for ${\mathbf {B}}_{init}$ . The 3-D shape of the designed tendon-driven structure is presented in Fig. 8(b).

Figure 8.

Tendon connection diagrams of the designs ${\mathbf {B}}_{init}$ and ${\mathbf {B}}_{opt}$ for a 3-DOF manipulator using post-design adjustments ${\mathbf {R}}_{init}=\mbox {diag}\{0.620$ , $0.620$ , $0.081$ , $0.089$ , $0.089\}$ [m] and ${\mathbf {R}}_{opt}=\mbox {diag}\{0.1$ , $0.1$ , $0.93$ , $0.121$ , $0.121\}$ [m], respectively.

5.3. Example: comparison with other method

The previous design for the 3-DOF manipulator using the proposed method is now compared with the SVD-based approach proposed in [Reference Sheu, Huang and Lee30]. The fundamental design principle of the SVD-based approach is to find $\mathbf {B}$ so that the mapping between the end-effector force and the tension, or ${\boldsymbol \tau }_{m} = {\mathbf {B}}^\dagger {\mathbf {J}}^T({\boldsymbol \theta }^*) {\mathbf {F}}$ , becomes isotropic at a chosen reference configuration ${\boldsymbol \theta }^*$ , while not much attention on the null space matrix $\mathbf {N}$ is paid as long as the span of $\mathbf {N}$ possesses the vector $\mathbb {I}= [1 \,1 \, \cdots \, 1]^T$ .

The requirement of isotropy in the SVD-based approach can be translated into the following equation:

\begin{equation*} \mbox {cond}\left ( {\mathbf {B}}^\dagger {\mathbf {J}}^T({\boldsymbol \theta }^*) \right ) = \mbox {cond}\left ({\mathbf {V}}_B {\mathbf {D}}_B^{\dagger }{\mathbf {U}}_B^{T}{\mathbf {V}}_J {\mathbf {D}}_J^{T}{\mathbf {U}}_{J}^T \right )=1, \end{equation*}

where the SVD of $\mathbf {B}$ and ${\mathbf {J}}({\boldsymbol \theta }^*)$ , defined as ${\mathbf {B}} := {\mathbf {U}}_B {\mathbf {D}}_B {\mathbf {V}}_B^T$ and ${\mathbf {J}}({\boldsymbol \theta }^*) := {\mathbf {U}}_J {\mathbf {D}}_J {\mathbf {V}}_J^T$ , is imposed [Reference Strang36]. By enforcing ${\mathbf {U}}_B ={\mathbf {V}}_J$ , the following is obtained:

(42) \begin{equation} {\mathbf {D}}_B^{\dagger }{\mathbf {D}}_J^{T}=\frac {1}{a_c}{\mathbf {I}}_{n}, \end{equation}

where $a_c$ is a positive real number. If $a_c$ is assigned, ${\mathbf {D}}_B$ can be determined. Therefore, the remaining part of the design is to find the $m \times m$ elements of ${\mathbf {V}}_B$ , the span of which contains the vector $\mathbb {I}$ . Combining the constraints, a total of $(m-1)(m-2)/2$ independent variables in ${\mathbf {V}}_B$ remain. These remaining variables are further reduced and the design becomes completed if a preferred form of $\mathbf {B}$ is selected by the designer from an admissible set. See [Reference Sheu, Huang and Lee30] for more information.

To numerically obtain the tendon-driven structure of the 3-DOF robot manipulator using the SVD-based approach, ${\boldsymbol \theta }^*=(\theta _{1}, \theta _{2}, \theta _{3})=(\frac {\pi }{6}, 0,-\frac {\pi }{6})$ is selected as the reference configuration. The manipulator Jacobian is computed as

\begin{align*} \begin{split} {\mathbf {J}}({\boldsymbol \theta }^*)= \left [\begin{array}{ccc} -1.260& 0.693& 0.693\\ \phantom {-}3.182& 0.400& 0.400\\ \phantom {-}0 &3.386 &1.386 \end{array}\right ] \end{split} \end{align*}

whose ${\mathbf {U}}_J$ , ${\mathbf {D}}_J$ , and ${\mathbf {V}}_J$ , respectively, are

\begin{align*} \begin{split} & {\mathbf {D}}_J = \begin{bmatrix} 3.819& 0& 0\\ 0 &3.409& 0\\ 0 & 0 &0.416 \end{bmatrix}; \,{\mathbf {V}}_J = \left [ \begin{array}{ccc} -0.181& -0.983& \phantom {-}0.018\\ -0.897& \phantom {-}0.173& \phantom {-}0.408\\ -0.404& \phantom {-}0.058& -0.913 \end{array} \right ]; \\ & {\mathbf {U}}_J = \left [\begin{array}{ccc} -0.176& \phantom {-}0.410& -0.895\\ -0.287& -0.891& -0.352\\ -0.942& \phantom {-}0.195& \phantom {-}0.275 \end{array}\right ]. \end{split} \end{align*}

From the constructive equation ( ${\mathbf {U}}_B = {\mathbf {V}}_J$ ) and the relation (42), ${\mathbf {U}}_B$ and ${\mathbf {D}}_B$ become

\begin{equation*} {\mathbf {U}}_B = \left [ \begin{array}{ccc} -0.181& -0.983& \phantom {-}0.018\\ -0.897& \phantom {-}0.173& \phantom {-}0.408\\ -0.404& \phantom {-}0.058& -0.913 \end{array} \right ];\, {\mathbf {D}}_B=\begin{bmatrix} 3.819& 0& 0&0&0\\ 0 &3.409& 0&0&0\\ 0 & 0 &0.416&0&0 \end{bmatrix}. \end{equation*}

To determine ${\mathbf {V}}_B$ , the following form of $\mathbf {B}$ is chosen from the admissible forms for $m=5$ and $n=3$ , as defined in [Reference Sheu, Huang and Lee30].

\begin{equation*} {\mathbf {B}}=\begin{bmatrix} b_{1,1}& b_{1,2}& b_{1,3}& b_{1,4}& b_{1,5}\\ 0& 0& b_{2,3}& b_{2,4}& b_{2,5}\\ 0& 0& 0& b_{3,4}& b_{3,5} \end{bmatrix}, \end{equation*}

where $b_{i,j}$ denotes a nonzero number. Note that this form of $\mathbf {B}$ happens to be exactly the same as that for our design.

The relation ${\mathbf {B}} = {\mathbf {U}}_B {\mathbf {D}}_B {\mathbf {V}}_B^T$ yields five constraints due to the five zeros in the chosen $\mathbf {B}$ . In addition, by assigning an arbitrary number to any one place of $b_{i,j}$ , we can determine ${\mathbf {V}}_B$ completely. Thus, by setting $b_{1,1} = 1$ , the final design result is

\begin{equation*} \mathbf {N}_{svd}= \left [ \begin{array}{cc} 0.447& \phantom {-}0.844\\ 0.447& \phantom {-}0.075\\ 0.447& -0.306\\ 0.447& -0.306\\ 0.447& -0.306 \end{array} \right ] \,\mbox {and}\, {\mathbf {B}}_{svd}=\left [\begin{array}{ccccc} 1& -3.018& 0.319& 1.026& \phantom {-}0.673\\ 0& \phantom {-}0& 0.817& 1.948& -2.764\\ 0& \phantom {-}0& 0& 1.131& -1.131 \end{array} \right ], \end{equation*}

where the subscript “svd” denotes that the terms are related to the SVD-based approach. The designed ${\mathbf {B}}_{svd}$ shows that the size of pulleys in the fifth tendon route varies considerably because the maximum to minimum ratio of elements in the last column rises up to nearly 4. Refer to Fig. 9 for the tendon connection diagram using ${\mathbf {B}}_{svd}$ , where the pulley size was scaled with ${\mathbf {R}}_{svd}=\mbox {diag}\{0.3,\,0.3,\,0.3,\,0.3,\,0.3\}$ . It shows a much larger variation in pulley size than with ${\mathbf {B}}_{opt}$ in Fig. 8. It should also be noted that assigning 1 to an arbitrary element $b_{i,j}$ does not always produce a solution; for instance, if $b_{1,3}=1$ , no solution can be found. The uncertainty regarding how to impose additional constraints becomes more severe as $n$ and $m$ increase.

Figure 9.

Planar tendon connection diagram of the design ${\mathbf {B}}_{svd}$ for a 3-DOF manipulator using post-design adjustment ${\mathbf {R}}_{svd}=\mbox {diag}\{0.3,\,0.3,\,0.3,\,0.3,\,0.3\}$ [m].

Figure 10.

Joint torque ellipsoids from motor torque ( $||{\mathbf {t}}||=1$ ) through ${\mathbf {B}}_{svd}$ and ${\mathbf {B}}_{opt}$ . Axes are normalized by the maximum singular value of their structure matrices.

Next, the torque transmission capability can be assessed by examining the shape of the ellipsoidal mapping from the motor torque of the unit sphere ( $||{\mathbf {t}}||=1$ ) through the structure matrix $\mathbf {B}$ . Figure 10 shows the two normalized ellipsoids obtained by using ${\mathbf {B}}_{svd}$ and ${\mathbf {B}}_{opt}$ . The ellipsoid for ${\mathbf {B}}_{svd}$ is narrow with a small volume, while the principal directions are not aligned with the physical joint axes. The three lengths of its principal axes are $(1, \frac {\sigma _{2}}{\sigma _1},\frac {\sigma _{3}}{\sigma _1}) = (1, 0.9, 0.109)$ , where $\sigma _i$ denotes the singular value of the structure matrix in descending order. The fact that the shortest length is only 0.109 indicates that it is difficult to generate a certain joint torque using the tension. This can cause a serious problem in bearing loads and controlling the robot manipulator in the joint space. In contrast, the ellipsoid for ${\mathbf {B}}_{opt}$ is broader with a larger volume, and the principal directions align exactly with the physical joint axes. The three lengths of its principal axes are $(1, \frac {\sigma _{2}}{\sigma _1},\frac {\sigma _{3}}{\sigma _1})= (1, 0.769, 0.592)$ . The ratios are exactly the same as the weight of joint torque $\mathbf {M}$ , which was assigned at the beginning of the design process. Thus, joint torques can be generated more easily by the proposed design.

5.4. Complicated example: design of a 6-DOF tendon-driven manipulator

Now, we illustrate the design of rather a complicated high DOF tendon-driven manipulator using the proposed design method. The target system is a 6-DOF articulated tendon-driven manipulator consisting of three sets of yaw-pitch joints, as shown in Fig. 11. We allow tendon redundancy as $\alpha =3$ . Assume that the kinematic dimensions of links and joint alignments are given a priori, as summarized in Table II.

Figure 11.

A six DOF manipulator used for designing its tendon-driven structure.

First, we need to determine the best weight of the joint torque using (33). Considering the distribution of load bearing between the joints, an additional inequality condition for the relative weight factor of the torque capacity is added such that $\frac {\mu _{i}}{\mu _{i+1}} \geq \beta \gt 1, i=1,\cdots, 5$ , where the relative weight factor $\beta$ is set as 1.3 in the example. It is intended that the torque capacity of a joint be at least 30 percent greater than that of the adjacent outer joint. By solving (33) with the inequalities, the weight of the joint torque can be determined as ${\mathbf {M}}= \mbox {diag}\{ 7.506,\, 5.774,\, 4.442,\, 3.417,\, 2.628,\, 2.022 \}$ .

Next, the indices of $\mathbf {N}$ are set as $(e_{1},\,e_{2},\,e_{3})=(3,\,3,\,3)$ . This is for a physical reason; with these indices, every three tendons take control of each group of yaw-pitch joints that are collocated at a place. Since the desired CBTF of $\mathbf {N}$ has been determined by the indices, we can move on to the main optimal design procedure.

By setting the initial $\mathbf {N}$ , not normalized yet, as

the structure matrix is determined via Design Problem I as follows:

The pair, $\mathbf {N}_{init}$ and ${\mathbf {B}}_{init}$ , is not suitable because some off-diagonal elements of $\mathbf {N}_{init}$ are large, and the variation in pulley size along the tendon routes is rather high. The values of $\Phi$ and $\Psi$ with $\mathbf {N}_{init}$ and ${\mathbf {B}}_{init}$ are calculated to be $2.725\times 10^{3}$ and $1.242$ , respectively. When we set $\rho$ as $21.935$ , the optimal solution is found as

and

with which the values of $\Phi$ and $\Psi$ become $15.670$ and $0.980$ , respectively. On average, the off-diagonal components of $\mathbf {N}_{opt}$ are lower than those of $\mathbf {N}_{init}$ . In addition, the pulley sizes along each tendon route are much more regularized for ${\mathbf {B}}_{opt}$ than for ${\mathbf {B}}_{init}$ .

Table II.

Denavit-Hartenberg parameters of the 6-DOF manipulator.

Figure 12.

Planar tendon connection diagram of the designed structure for the 6-DOF manipulator with ${\mathbf {B}}_{init}$ and ${\mathbf {B}}_{opt}$ .

Figure 12 displays planar tendon connection diagrams using ${\mathbf {B}}_{init}$ and ${\mathbf {B}}_{opt}$ , where the scales of pulley size were adjusted, respectively, with ${\mathbf {R}}_{init}=\mbox {diag}\{0.2,$ $0.2,$ $0.2,$ $0.2,$ $0.2,$ $0.2,$ $0.266,$ $0.4,$ $0.4\}$ [m] and ${\mathbf {R}}_{opt}=\mbox {diag}\{0.2,$ $0.369,$ $0.2,$ $0.2,$ $0.235,$ $0.287,$ $0.294,$ $0.4,$ $0.4\}$ [m] by considering the visual proportion of the manipulator. As shown in the figure, the size of the pulleys for ${\mathbf {B}}_{init}$ is irregular. In particular, the size of pulleys along tendons for $t_2$ , $t_7$ , $t_8$ , and $t_9$ for ${\mathbf {B}}_{init}$ varies significantly. In contrast, the the size of pulleys designed optimally using ${\mathbf {B}}_{opt}$ looks natural, thus the proposed method produces an acceptable design.