## 2 Maximum Performance Evaluation  
### Optimization Formulation  
The dexterity motion of a human finger is not achieved by lumped actuation, but by spatially distributed actuation of opponents pollicis (Figure 1a). Based on the understanding of such spatially distributed actuation, the DAM was designed to actuate one joint by thrusting two sliders along a link [22] (Figure 1b)). Because its joint angular velocity and torque vary depending on the position of the sliders (i.e., the application point of force), the DAM is a redundantly actuated system. Figure 1: Actuation mechanism inspired by finger movement: (**a**) Spatially distributed actuation of a finger [24]; (**b**) A distributed actuation mechanism [22]. In principle, the optimization formulation can be expressed to precisely evaluate the maximum performance of a manipulator [25], as follows:  
\[ \begin{array}{ll}\text{Find}&x_{1},x_{2},\cdots,\textit{ and }x_{n}\\ \text{To maxmize}&f(\mathbf{x})=\mid\mathbf{v}_{e}(\mathbf{x})\mid\\ \text{Subject to}&h=\mathbf{d}_{k}\cdot\mathbf{v}_{e}=\mid\mathbf{d}_{k}|| \mathbf{v}_{e}|&k=1,\cdots,K\\ &g_{v}(\mathbf{x})\leq 0&\
u=1,\cdots,N\\ &x_{i}^{(l)}\leq x_{i}\leq x_{i}^{(u)}&i=1,\cdots,n\end{array}  \tag{1}\]  
where the superscripts (\(l\)) and (\(u\)) denote the lower and upper bounds of the design variable \(x_{i}\); the objective function \(f\) is the norm of the end-effector velocity; \(h\) is the equality constraint that matches the direction of the end-effector velocity \(\mathbf{v}_{e}\) with the base direction \(\mathbf{d}_{k}\); and \(g_{v}\) is the inequality constraint that represents motor performance. Figure 2 shows a DAM-based three-revolute-joint manipulator (DAM-3R), which was investigated in this study. The position and velocity of the sliders can be treated as control variables that determine the end-effector velocity, \(\mathbf{v}_{e}\)= (\(v_{x}\), \(v_{y}\)). For the DAM-3R, the design variable vector becomes \(\mathbf{x}=\) [\(\theta_{1}\), \(s_{1}\), \(s_{2}\), \(s_{3}\), \(\dot{s}_{1}\), \(\dot{s}_{2}\), \(\dot{s}_{3}\), \(\dot{s}_{1}\)\({}^{b}\), \(\dot{s}_{2}\)\({}^{b}\), \(\dot{s}_{3}\)\({}^{b}\)], where \(\theta_{1}\) is the first joint angle; \(s_{1}\), \(s_{2}\), and \(s_{3}\) are the positions of the front sliders; \(s_{1}\), \(\dot{s}_{2}\), and \(\dot{s}_{3}\) are the thrusting speeds of the front sliders; and \(\dot{s}_{1}\)\({}^{b}\), \(\dot{s}_{2}\)\({}^{b}\), and \(\dot{s}_{3}\)\({}^{b}\) are the thrusting speeds of the back sliders. In this study, the base direction \(\mathbf{d}_{k}\) was defined as follows:  
\[\mathbf{d}_{k}=\left[\cos\alpha,\ \sin\alpha\right]^{\text{T}} \tag{2}\]  
where \(\alpha\) is at intervals of 22.5 degrees from 0 to 360 degrees. This generated a total of 16 base directions (i.e., \(K\) = 16 in (1)). Because the speed and force of the sliders must operate within motor characteristics curves in four quadrants, \(g_{v}\) was formulated as follows:  
\[ \begin{array}{ll}g_{1}=F_{j}\pm\frac{F_{\max}}{\dot{s}_{\max}}\dot{s}_{j}\pm F _{\max}\leq 0&j=1,2,3\\ g_{2}=F_{j}^{b}\pm\frac{F_{\max}}{\dot{s}_{\max}}\dot{s}_{j}\pm F_{\max}\leq 0&j=1,2,3 \end{array}  \tag{3}\]  
where the subscript \(j\) indicates the joint number; \(F_{j}\) and \(F_{j}^{b}\) are the thrusting force of the front slider and back slider, respectively; \(F_{\max}\) is the maximum thrusting force of the front and back sliders; and \(s_{\max}\) is the maximum thrusting speed of the front and back sliders. In this study, gravitational force was applied as an external force to reflect inertia loads. For comparison, a joint actuation mechanism (JAM)-based three-revolute-joint manipulator (JAM-3R) was also considered with power-equivalent motors and the same geometric specifications (Figure 3 and Table 1). When the end-effector is free to rotate, the first joint angle \(\theta_{1}\) becomes a control variable. Accordingly, the JAM-3R is also a redundant manipulator, but it has less redundancy than the DAM-3R.