## 3 Kinematic Models of the Human and Robot Thumbs  
Human thumb anatomy and motion are more complex than other hand digits. The four thumb bones, starting at the wrist, are the trapezium, first metacarpal, proximal phalange, and distal phalange, and the three thumb joints are carpometacarpal (CMC), metacarpophalangeal (MCP), and interphalangeal (IP), as shown in Figure 1a. The saddle\(-\)shaped CMC joint, also known as the trapeziometacarpal joint (TM joint), has three DOFs and allows the thumb to perform complex motions, thus enabling the hand to engage with a variety of objects through various grasps such as power, pinch, or lateral. The majority of clinical\(-\)based thumb models identify the CMC joint movements as flexion/extension (FE), abduction/adduction (AA), and pronation/supination (PS) [41; 42; 43; 44; 45]. Even though the metacarpal bone has three motion modes (AA, FE, and PS), it only has two degrees of actuation, since the PS motion is coupled with the FE and AA motion according to [PERSON] et al. [46]. [PERSON] et al. [41], using cadaver thumbs, found that the thumb FE and AA axes are non\(-\)orthogonal and non\(-\)intersecting, making accurate modeling of the thumb more challenging than the other digits. The thumb MCP and IP joints have two DOFs (FE and AA) and one DOF (FE), respectively [42; 47]. Developing an exact kinematic model of the CMC joint is difficult due to its complex motion. As such, various approximate kinematic joint definitions have been used to describe its motion [20; 21; 48; 49]. For example, the CyberGlove\(\circledR\)[21; 45] uses an approximate thumb model, where the CMC joint FE and AA axes are considered orthogonal and intersecting, and the FE and PS motions are coupled and defined as a roll motion instead. The model is based on the location of the flex sensors used to track the CMC joint motion. [PERSON] et al. [49] developed a vision\(-\)based hand\(-\)tracking system to teleoperate an underactuated ARH. They modeled the human thumb as a kinematic chain with two DOFs (FE and AA) for the CMC joint and one DOF (FE) for each of the MCP and IP joints. They assumed the CMC joint PS motion and the MCP joint AA motion to be fixed, and the FE motion of the MCP joint was coupled with the IP joint. In this research, the developed human thumb kinematic model is based on clinical anatomical models (see Figure 1a) with assumptions. The FE and AA axes are assumed to be intersecting and orthogonal. The PS motion is considered fixed due to its relatively small motion of (\(\sim\)17 deg). The FE axes for each of the CMC, MCP, and IP joints are considered to be parallel to each other, as shown in Figure 1b,c. The thumb kinematic base coordinate system is established at the intersection of the thumb and index metacarpal bones, as shown in Figure 1b. The human thumb is kinematically modeled as a serial linkage with four rotational joints and is shown schematically in Figure 1c. The CMC joint is modeled as two serially intersecting orthogonal rotational joints, \(J_{FE}\) and \(J_{AA}\), connected by the metacarpal phamleg. The kinematic parameters of the human thumb are defined following the modified Denavit-Hartenberg (mDH) kinematic convention [50] and presented in Table 1. \(L_{mp}\), \(L_{pp}\), and \(L_{dp}\) are the lengths of the metacarpal, proximal phalange, and distal phalange, respectively, and \(L_{t}\) is the offset distance to the contact surface point of the thumb. The MCP joint of each of the remaining fingers (index, middle, ring, and pinky), similar to that of the thumb, is a two\(-\)degree\(-\)of\(-\)freedom joint. However, the abduction/adduction motion is small, and it is reasonable to model the MCP as a single degree of freedom for the flexion/extension motion, as shown in Figure 1b for the index finger. The FE axis of the distal IP, proximal IP, and MCP joints of the fingers are modeled as  
Figure 1: Human thumb (**a**) Anatomy, (**b**) Visual of assumed kinematic model, and (**c**) Kinematic model schematic.