## 5 Framework
### Stage 2: Kinematic Mapping Algorithm of Human to Robot Thumb  
In the second stage of the framework, the estimated human thumb angles are mapped to the robot joint angles. The first two joints (CMC joint) of the human thumb model are orthogonal and intersecting (see Figure 1) whereas those of the robot thumb are non\(-\)orthogonal and non\(-\)intersecting (see Figure 2c,d). Due to these anatomical and kinematic topology differences, the estimated human thumb joint angles cannot be directly mapped to the robot thumb [9]. Hence, a hybrid approach was followed to map the human thumb motion to the robot thumb. Cartesian space mapping was used to map the CMC joint to the first two joints of the robot thumb, and joint space mapping was used to map the MCP and IP joints to the third and fourth joints of the robot thumb. For the Cartesian space mapping, the motion of the CMC joint is decoupled and mapped to the first two joints of the robot thumb using the developed Radial Projection Algorithm (RPA) algorithm. The kinematic model of the human thumb CMC joint (\(H_{cmc}\)) includes the metacarpal phalange, link \(L_{mp}\), as shown in Figure 1d. The kinematic model of the equivalent CMC joint of the robot thumb (\(R_{cmc}\)), which includes the base two joints and links (\(J_{1}\), \(J_{2}\), \(L_{1}\), and \(L_{2}\)), is defined as shown in Figure 2d. The position vectors of the distal end for both \(H_{cmc}\) and \(R_{cmc}\) are defined as \(P_{h}\) and \(P_{r}\), respectively, and shown in Figure 6.  
The \(H_{cmc}\) workspace is the partial surface of a sphere with radius \(r_{0}=L_{mp}\), whereas that of \(R_{cmc}\) is the partial surface of a warped ellipsoid as shown in Figure 7a. Due to the geometric differences between these two workspaces, the distal end of \(R_{cmc}\) (\(P_{r}\)) cannot reach the actual position of the distal end of the metacarpal phalange (\(P_{h}\)). The proposed RPA maps the motion of \(H_{cmc}\) to \(R_{cmc}\) such that \(P_{r}\) follows \(P_{h}\) radially towards the base, as shown in Figure 6. Note that \(P_{r}\) cannot follow \(P_{h}\) throughout its workspace due to the joint range limitations. The workspaces of \(H_{cmc}\) and \(R_{cmc}\) are shown in Figure 7a. The \(H_{cmc}\) workspace radius, \(r_{0}\), is user\(-\)specific and is the length of the user's first metacarpal, \(L_{mp}\). The mapped area using radial projection from the \(H_{cmc}\) to the \(R_{cmc}\) workspace does not change even if the workspace radius \(r_{0}\) changes; however, it depends on the boundary of the \(H_{cmc}\) workspace which is a function of the user\(-\)specific joint limits and offsets. A complete mapping between the workspaces is desirable, since it will allow a user's CMC joint to control the entire \(R_{cmc}\) workspace. As such, to achieve complete mapping, a projected workspace is defined by projecting the \(R_{cmc}\) workspace onto a spherical surface with radius \(r_{0}\), with the bounds of the projected area determined as shown in Figure 7b. The boundary of the projected workspace in the configuration space of the \(H_{cmc}\) is shown in Figure 7c. The rectangle inscribing the projected workspace in the configuration space is considered to be the \(H_{cmc}\) mapped workspace. The \(H_{cmc}\) mapped workspace bounds are determined to be \(\theta_{m,FE,limits}\approx(0^{\circ},70^{\circ})\) and \(\theta_{m,AA,limits}\approx(-10^{\circ},90^{\circ})\).