A design method for tendon-driven serial manipulators using controllable block triangular form of structure null space matrix

Abstract

This paper proposes a generalized method for designing tendon-driven serial-chained manipulators with an arbitrary number of tendon redundancy. First, a special class of tendon-driven structures is defined by introducing the controllable block triangular form (CBTF) of a null space matrix and its complementary CBTF of a structure matrix, satisfying physical constraints related to the minimal connection of tendons and to the placement of actuators. Then it is shown that any general design of tendon-driven serial manipulators can be reduced to the design of such a special class of tendon-driven structures. Two associated design problems are derived and solved. The first design problem is about finding a complementary CBTF structure matrix for a given CBTF null space matrix using algebraic relations, whereas the second one seeks the both matrices that optimize the wanted structural characteristics based on the result of the first design problem. Numerical design examples are provided to show the validity of the proposed method.

1. Introduction

The major advantage of using tendons as a way to transmit force/torque in robotic manipulators includes the energy efficiency and the possibility of innovative mechanical designs due to the fact that the driving actuators can be placed at the base, away from the moving parts [1–8]. However, since any non-closed tendon can deliver only the pulling force, the number of driving tendons must be greater than the number of joints for controllability [9, 10]. The surplus number of tendons or simply the tendon redundancy, in fact, has a significant impact on many aspects from design to control of the tendon-driven manipulators [11, 12]. For example, as the tendon redundancy increases, more efficient and effective tendon-driven structures of serial-chain manipulators in producing input power and torque can be found due to the enlarged tension domain, which benefits the overall performance of the manipulators. However, the trade-off is that the design and control with the tendon-driven structures become extremely complicated with the growth of tendon redundancy [13, 14].

Most existing types of $n$ -jointed tendon-driven serial-chain manipulators belong to $n$ type, $n+1$ type, and $2n$ type, where each type number represents the total number of driving tendons. An $n$ type manipulator has zero tendon redundancy, where each closed tendon drives its corresponding joint bidirectionally. The 3-DOF tendon drive system proposed by Zhu et al. [15], driven by three cables, is an example of the $n$ type tendon-driven manipulator among many others. An $n+1$ type manipulator, which has a single tendon redundancy, is driven by $n+1$ non-closed tendons that can transmit only the pulling force. Due to the inherent coupled nature of tendons in the $n+1$ type manipulator, any robot motion is generated from the cooperative action of multiple tendons. The shoulder/elbow robot [16], having four joints driven by five tendons, falls into this category. The 11-DOF manipulator driven by 12 tendons, developed by Treratanakulwong et al. [17], is also an example of $n+1$ non-closed type. Finally, in a $2n$ type manipulator, each joint is commonly driven by two antagonistic tendons. Because this symmetrical structure is similar to the joint driving mechanism in humans, many human-like or human-mimicking systems tend to adopt the $2n$ type tendon-driven structure. Examples belonging to this type include the coupled tendon-driven manipulator, the Mini 3D CT-Arm [18] and the DLR robotic hand [19]. However, it is noteworthy that the case examples of the tendon-driven manipulator with type number $n+2$ , $n+3$ , or others in between $n+1$ and $2n$ are extremely rare [20–23]. One likely reason for such rarity is the challenge of finding suitable designs where tendons must be optimally coupled, which in turn served as motivation for the current study.

The mathematical design of tendon-driven manipulators has two fundamental issues: (i) the selection of an optimal or comparable topological tendon routing structure and (ii) the selection of the physical dimensions of the pulleys for a given tendon routing topology for efficient and effective tendon actuation. The first issue can be viewed as the enumeration of possible tendon routing structures that are physically realizable and controllable for the given number of joints and the redundancy of actuation in the tendon-driven manipulator [24, 25]. The second issue is related to the actual selection of pulleys to achieve a desired amount of transmitted wrench. However, designs that address both of these issues in general situations have not been well established yet, partly due to the fact that the solution routes for the tendons become too numerous as the degrees of freedom for the manipulator increase, while physically realizable designs that admit suitable pulley sizes are difficult to identify.

Traditionally, mathematical design methods for tendon-driven manipulators have been developed on the basis of a structure matrix, which relates the tension input to the joint torque [26, 27]. Because the structure matrix carries information about the topology of the tendon routes and the physical size of the pulleys, it plays a key role in the design. Lee and Tsai [24] conducted the synthesis of admissible tendon routes by enumeration for the case of single tendon redundancy. They also defined the isomorphism among tendon routing topologies that are equivalent after rearranging the rows and columns of the structure matrix. While the synthesis method in [24] did not account for the pulley sizes of the tendon-driven structures, Ou and Tsai [25] proposed a matrix decomposition method that determines not only the tendon routing topologies but also the pulley sizes along the routes. They sought a structure matrix that yields the isotropic transmission between the tendon input and the end-effector force at a particular manipulator configuration where the manipulator’s Jacobian matrix itself satisfies the unity condition number. Inherently, therefore, the intended isotropy of transmission could not be attained as the manipulator configuration diverged from its design,and their method is ineffective for the design of versatile manipulators with broad operational ranges. Kobayashi et al. [28] studied the adjustability of the joint stiffness in terms of the tendon routing topology. They formulated a static relationship between the elasticity of the tendons and joint stiffness combined with the structure matrix. However, the effect of pulley size was not considered in their analysis. The design method for a tendon-driven manipulator by Chen et al. [29] was based on a two-step decomposition approach. They decomposed the structure matrix into a square transform matrix and a non-square orientation matrix using QR factorization, and then they enumerated admissible orientation matrices producing the isotropic transmission at a chosen reference configuration. However, their method was not general and worked only for manipulators with lower DOFs and single tendon redundancy. A significant period of time later, Sheu et al. [30] proposed a design method using singular value decomposition (SVD) to determine a structure matrix that yields the isotropic transmission. Their design method, while initially intended to address arbitrary tendon redundancy, lacks comprehensive consideration of all aspects of tendon redundancy that could enhance design quality. Ozawa et al. [31] extensively analyzed, classified and designed tendon-driven mechanisms possibly with passive actuation or underactuation. Their design method mainly focused on the synthesis of tendon routing topologies. Dong et al. [32] proposed a design technique that optimizes tendon routing and design variables by using genetic algorithm. However, a key challenge with this approach is its applicability limited to under-actuated robotic systems, along with the issue of high computational complexity. Recently, Cheng et al. [33] introduced a design method utilizing a noncircular pulley, and Zhou et al. [34] explored 12 tendon transmission paths to analyze the impact of tendon routing on tendon tension and joint motion precision. Nonetheless, these approaches face limitations in adapting to more complex multi-joint systems, limiting their broader applicability.

As mentioned above, the design of tendon-driven manipulators becomes tremendously complicated as the number of DOFs and tendon redundancy increase. According to Morecki et al. [26], there are 23,040 possible tendon routes for a 6-DOF tendon-driven manipulator with single tendon redundancy. The number of the potential combinations of cable routing increase exponentially with the addition of each rigid link or joint [35]. Not only the tendon routes but also the pulley sizes must be determined in the design, which requires additional effort. Some mandatory criteria, including the isotropic force/torque transmission characteristics, could be combined to obtain the optimal design of tendon-driven structures, but this still leaves too many remaining choices for high DOF manipulators with arbitrary tendon redundancy.

Therefore, this paper proposes a novel method for designing general tendon-driven manipulators with given tendon redundancy that exhibit the simplest possible topological form of tendon connectivity, the so-called minimal tendon connectivity, which minimizes the effect of friction in the transmission and the complexity of structure. The output of the proposed design method is the physically realizable optimal structure matrix that corresponds to a unique tendon-driven structure of the manipulator. This design method utilizes the strategic choice of the controllable block triangular form of the structure null space matrix, which is to be orthogonally paired with the complementary structure matrix with minimal tendon connectivity. Two standard design problems (i.e., Design Problems I and II) are formulated to find the structure matrix; Design Problem I assumes that the structure null space matrix is fully given, whereas Design Problem II assumes only the shape of the structure null space matrix is given. Detailed solutions for both of these design problems are provided in a step by step manner.

Figure 1. Schematic of a tendon-driven manipulator.

2. Kinematic relations and controllability

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 [24]. 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 [27] 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 [36]. 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 [37]. 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 [24, 26]:

(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 [25] 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 $\mathbf {B}$ .

Definition 1. The null space matrix $\mathbf {N}$ associated with a full-rank structure matrix $\mathbf {B}$ is said to be of a controllable block triangular form (CBTF) if each diagonal block $\mathbf {N}_{ii}$ of $\mathbf {N}$ , as shown in Fig. 2 , is composed of elements with the same sign.

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.

Lemma 2.1. The null space matrix $\mathbf {N}$ of any controllable tendon-driven manipulator can be transformed into a CBTF via a linear transform $\mathbf {N}_{CBTF} = \mathbf {P} \mathbf {N}\mathbf {Q}$ , where $\mathbf {P}\in \mathbb {R}^{m \times m}$ and $\mathbf {Q}\in \mathbb {R}^{\alpha \times \alpha }$ are nonsingular matrices.

Proof. See Appendix I for the details of proof.

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. Design problem I: finding $\mathbf {B}$ for a given $\mathbf {N}$

The characteristics of the design problem under consideration can vary depending on how detailed the information on $\mathbf {N}$ is given. The first class of problem, to be referred to as Design Problem I, is related to finding a suitable or optimal $\mathbf {B}$ for a given CBTF of $\mathbf {N}$ with full numerical data, and the second class, referred to as Design Problem II, is related to simultaneously finding a suitable or optimal $\mathbf {B}$ and $\mathbf {N}$ for a given shape of the CBTF of $\mathbf {N}$ without numerical data. This section is devoted to solving Design Problem I, while the next section addresses Design Problem II based on the result of Design Problem I.

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.

Definition 2. The structure matrix $\mathbf {B}$ is said to be of a complementary CBTF compatible with a given CBTF of $\mathbf {N}$ if not only $\mathbf {B}$ satisfies ( 10 ) and the physical constraints but also each diagonal block ${\mathbf {B}}_{i,i}, i=1,2,\ldots, \alpha, $ of $\mathbf {B}$ is of a pseudo triangular form, as shown in Fig. 3 .

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 $\mathbf {N}$ [38]; 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. Design problem II: finding $\mathbf {B}$ and $\mathbf {N}$ jointly

Design Problem II intends to seek both the structure matrix $\mathbf {B}$ and the null space matrix $\mathbf {N}$ to yield the targeted joint torque capacity $\mathbf {M}$ . It is assumed that a CBTF of $\mathbf {N}$ is given with known indices, but its nonzero elements are not predetermined. Correspondingly, $\mathbf {B}$ must be a matrix with a complementary CBTF for the given form of $\mathbf {N}$ . Thus, the design variables are all the nonzero elements of $\mathbf {B}$ and $\mathbf {N}$ subject to the orthogonality condition, ${\mathbf {B}}\mathbf {N} = \mathbf {0}$ .

We tackle Design Problem II using the constrained optimization approach [39] with a cost function that best reflects the desired characteristics of the tendon-driven manipulator, such as little variation in pulley size along each tendon route and little cross coupling in actuation. Design Problem II recursively invokes Design Problem I as a subroutine.

Figure 5. Signal flow diagram for the tendon-driven manipulator system.

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 [40],

(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 [36], 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 [41]. 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 [30]. 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. Numerical examples

This section provides illustrative examples to show how the two design problems are solved. Through these examples, the characteristics and advantages of the proposed design method are verified in terms of torque transmission, the ability to better use the tendon redundancy, and the flexibility of design. In addition, the proposed design method is quantitatively compared with a previous approach.

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 [42], 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 [30]. 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 [36]. 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 [30] 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 [30].

\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.

6. Concluding remarks

In this paper, a general design method for tendon-driven manipulators with arbitrary redundancy is proposed. By defining the controllable block triangular form (CBTF) of the null space matrix $\mathbf {N}$ and its accompanying complementary CBTF of the structure matrix $\mathbf {B}$ , any design of a tendon-driven manipulator could be reduced to determining these matrices. Two design problems, Design Problems I and II, were formulated based on the prerequisite knowledge associated with the design parameters. Design Problem I seeks the structure matrix $\mathbf {B}$ that is complementary to the given CBTF of null space matrix $\mathbf {N}$ , while all of the physical constraints regarding the properties of the tendon connections and joint torque capacity are satisfied. In contrast, Design Problem II determines not only structure matrix $\mathbf {B}$ but also null space matrix $\mathbf {N}$ , given the indices of $\mathbf {N}$ , via nonlinear optimal programming. The optimal cost for Design Problem II takes into consideration the physical realizability of the design. The details of the solution methods were provided along with the admissible conditions and justifications. And the validity was demonstrated via several numerical examples.

The proposed design method is meaningful in that it enables the general design of tendon-driven manipulators with arbitrary tendon redundancy. With the help of this method, challenging design problems regarding complicated tendon-driven manipulators can become easier and systematic. Moreover, the design by dissecting the structure of $\mathbf {N}$ and $\mathbf {B}$ and the associated mathematical analysis presented here can be adapted or modified to a new class of design and analysis problems related to tendon-driven systems.

Concerning the proposed design method, it is worth making the following remarks for further consideration:

• • The proposed design method assumes all of the tendon driving actuators are located at the base. Even if this is not the case, the proposed design method can still work by dividing the whole design problem into separate sub design problems. That is, each sub design problem can be established by grouping joints and their passing tendons. By solving these sub design problems one by one following the proposed method, the entire design problem can be resolved.

• • The proposed design method assumes the driving actuators are identical and have equal tension generating capabilities. If they differ, the difference must be taken into consideration in the design as a form of another weighting factor.