### Physiological structure

The bony complex of the elbow is an intricate mechanical system comprising the forearm bones (radius and ulna) and the upper arm bone (humerus).

From the perspective of anatomical medicine, the muscles associated with elbow flexion/extension include the biceps, triceps, brachioradialis, brachialis, and anconeus. During movement, the brachioradialis and anconeus play coordinating roles, whereas the biceps and triceps are the main muscles responsible for elbow joint flexion and extension, respectively [14, 15], which is one of the reasons we examined the biceps and triceps in the current study. In addition, the brachialis, which is a deep muscle, was initially taken into consideration. However, according to the previous study [16], during elbow joint extension, the brachialis is only a small muscle antagonist to the triceps, and the sEMG signals produced by the brachialis are difficult to measure. Thus, we did not include the brachialis in the analysis. Taking these factors into consideration, we focused on the biceps and triceps as the main research targets in the current study.

To establish a physiological model of the elbow, it is necessary to understand the distribution of the major muscle groups: the long head of the biceps originates from the supraglenoid tubercle while the coracoradialis originates from the coracoid process of the scapula. Both muscles converge on the muscle belly and integrate to become the biceps tendon that inserts on the radial tuberosity. The triceps has three heads; the long head originates on the scapula and the two short heads originate from the humerus. They integrate to become the triceps tendon that inserts on the olecranon, as shown in Fig. 2.

The elbow joint comprises the trochlea on the humeral side and the trochlear notch and olecranon process on the ulnar side. The trochlea is a hyperboloid-shaped cylinder that fits well into the trochlear notch [17]. Because of these structures, the elbow joint can be modeled as a hinge joint [14, 15, 18].

The physiological structure of the forearm comprises the ulna and radius. The tendinous insertion of the triceps averages 20–24 mm by 8–12 mm and extends to the medial margin of the olecranon [19] (see the right side of Fig. 2). We labeled the insertion of the triceps as point *B*. The angle between the straight lines connected by the elbow joint’s center of rotation (*O*), point B, and the axis of the forearm is denoted as *ξ* (see Fig. 3).

The physiological characteristics described above are important for understanding elbow movement and form the basis of our mathematical model. We established a physiological structural model of the elbow joint to solve the dynamics model (see Fig. 4). This model comprises an equivalent line of muscle fibers and bones.

In Fig. 4, *J* and *C* represent the equivalent origination points of the biceps and triceps on the shoulder, respectively; *AO* and *BOE* represent the humerus and the bones of the forearm, respectively; and *F*_{bi} and *F*_{tr} represent the force of the biceps and triceps, respectively.

To simplify the complexity of the structural model, we assumed that the axis of the humerus and forearm create an angle of 0° when the elbow is fully extended [20, 21]. Thus, *ξ* = *ξ*_{2}. *l*_{OH} and *l*_{OI} are the arms of the biceps and triceps muscle forces, respectively, which can be obtained using a geometric model.

### Musculotendinous force

On the basis of the HMM, each musculotendon unit is modeled as a muscular unit with two parallel elements: an active contractile element (CE) producing the active muscle force *F*_{c} and a nonlinear passive elastic element (PE) producing the passive force *F*_{p}, as shown in Fig. 5 [10].

From the HMM, we realize that the muscle force *F* is produced by the combined effects of the muscle contractile force and muscle passive force [12], as expressed by:

$$F = (F_{C} + F_{P} )\cos \varphi$$

(1)

where *F*_{c} is the muscle contractile force, *F*_{p} is the muscle passive force, and \(\varphi\) is the pennation angle. The optimal pennation angles of the biceps and triceps are no more than 10° [22]. When the optimal pennation angle is 10°, the relative change in the rate of the muscle force is approximately 0.047, which shows that the pennation angle has little effect on the muscle force. This can be written as \(\varphi = 0\), and, thus, \(\cos \varphi = 1\) [14, 23].

The model of the muscle contractile force and muscle passive force is expressed as [10,11,12]:

$$\left\{ \begin{array}{l} F_{c} = f_{A} (l) \cdot f_{V} (v) \cdot a(k) \cdot F_{{\rm max} } \\ F_{p} = f_{P} (l) \cdot F_{{\rm max}} \end{array} \right.$$

(2)

where *f*_{A}(*l*), *f*_{V}(*v*), and *f*_{p}(*l*) are, respectively, the normalized muscle contractile force–length relationship, the muscle contractile force–velocity relationship, and the passive elastic force–length relationship. We take *f*_{V}(*v*) = 1 [7, 24, 25], and *a*(*k*) is the muscle activation at time *k*. *F*_{max} is the maximum isometric muscle force while *l* is the normalized muscle fiber length, which is equal to the current muscle fiber length *l*^{m} divided by the optimal muscle fiber length *l*
^{m}_{
0}
:

$$l = \frac{{l^{m} }}{{l_{0}^{m} }}$$

(3)

According to the HMM, the length of the skeletal muscle unit can be calculated as:

$$l^{mt} = l^{t} + l^{m} \cdot \cos \phi$$

(4)

From this equation, we can calculate the current length of muscle fiber *l*^{m}. *l*^{t} is the length of the tendon, which can be regarded as a constant. *l*^{mt} is the length of skeletal muscle, which can be calculated using a geometric model of the human upper limb (see Fig. 4). The skeletal muscle lengths of the biceps and triceps are calculated as:

$$\left\{ \begin{aligned} l_{b}^{mt} &= \sqrt {l_{OK}^{2} + l_{OC}^{2} - 2l_{OK} \cdot l_{OC} \cdot \cos (\pi - \theta - \xi_{1} )} \hfill \\ l_{t}^{mt} &= \sqrt {l_{OB}^{2} + l_{OJ}^{2} - 2l_{OB} \cdot l_{OJ} \cdot \cos \left( {\theta - \xi_{2} + \xi } \right)} \hfill \\ \end{aligned} \right.$$

(5)

Thus, we established the musculotendinous model. Therefore, when determining the constants *F*_{max}, *l*_{t}, and *l*
^{m}_{
0}
, we can calculate *l*^{m} with (4), *l* with (3), and *F* with (1).

### Prediction model

When the upper limb around the elbow joint is involved with rotation, it can be regarded as a fixed axis rotation of the forearm. The total torque can then be calculated as:

$$T = F_{bi} \cdot l_{OH} - F_{tr} \cdot l_{OI} - M_{F} - M_{G}$$

(6)

where *M*_{G} is the gravitational moment of the forearm and hand acting on the elbow joint while *M*_{F} is the external moment acting on the elbow joint.

Assuming that the moment of inertia of the forearm (including the forearm, hand, and load) is *J*, we have the kinetic equation:

$$J \cdot \ddot{\theta }_{k} = T_{k}$$

(7)

where \(\ddot{\theta }_{k}\) is the angular acceleration and *T*_{k} is the total torque at time *k*.

We can then obtain the elbow joint dynamics model in discrete time through joint dynamics analysis:

$$\left\{ \begin{aligned} &\dot{\theta }_{k + 1} = \dot{\theta }_{k} + \ddot{\theta }_{k} \cdot \Delta t \hfill \\ &\theta_{k + 1} = \theta_{k} + \dot{\theta }_{k} \cdot \Delta t + \frac{1}{2} \cdot \ddot{\theta }_{k} \cdot \Delta t^{2} \hfill \\ \end{aligned} \right.$$

(8)

The dynamic model of the elbow joint can be solved using simultaneous Eqs. (6), (7), and (8), which are based on the biceps and triceps muscle force as input and the elbow angle as output.

By combining the three models described above, we obtain the continuous motion estimation model (CMEM) of the elbow joint, which has sEMG signals as input and the elbow rotation angle as output.