In this section, a mesh force model considering errors is established based on the structural characteristics and kinematic principle of the CORR. In order to analyze the mesh force for simple and convenient, there are three assumptions as shown in the following.
Recently, various models of the human upperlimb anatomy have been derived. The biomechanical models of the arm that stand for precise anatomical models including muscles, tendons and bones are too complex to be utilized in mechanical design of an anthropomorphic robot arm. From the view of the mechanism, we should set up a more practicable model for easy and effective realization.

(1)
The contact distortions between the movable roller and the disk cam, the movable teeth frame, and the inner gear are small and stretchable, and other distortions of the elements of CORR are not considered.

(2)
Because of the small mass of the movable roller, the influence of its inertial force is ignored.

(3)
Ignore the friction forces of movable roller.
The oscillatory roller transmission is equivalent to the parallel camguide linkroller linkages as shown in Figure 3. The entire transmission achieves motion and power transmission through many identical parallel combinations. Therefore, the force analysis of the meshing pair of the movable roller i can be performed, and a model based on this for meshing force can be established [31].
H_{i} is the center point of the movable teeth. F_{ji} is the force between the disk cam and the movable roller. F_{si} and F_{ni} are denoted the forces of the movable teeth frame and the inner gear acting on the movable roller respectively. From Figure 3, the force equilibrium of the movable teeth can be written as
$$F_{si} /\sin \left( {\gamma_{i} + \theta_{i} + \alpha_{i} } \right) = F_{ni} /\sin \left( {{\uppi}/2  \gamma_{i} } \right) = F_{ji} /\sin \left( {{\uppi}/2  \theta_{i}  \alpha_{i} } \right),$$
(4)
where α_{i} is the angle between y axis and the normal vector of the inner gear profile at the mesh point; θ_{i} is denoted the angle between y axis and the connecting line of the movable teeth frame axis and the center of the movable teeth.
$$\theta_{i} = \theta + \left( {i  1} \right)\Delta \theta ,$$
(5)
In Eq. (5), \(\Delta \theta = 2{{\uppi}}/\left( {z + 1} \right)\) denotes the central angle between two adjacent movable teeth, θ is the initial output angle of the movable teeth frame.
The parameter γ_{i} is the angle between two lines, one is the connecting line between the movable teeth frame axis and the center of movable teeth and the other is the connecting line between the centers of the disk cam and movable teeth. According to Figure 3, γ_{i} can be written as follow
$$\gamma_{i} = \text{cos}^{  1} \left( {\left( {s_{i}^{2} + R_{j}^{2}  e^{2} } \right)/2s_{i} R_{j} } \right),$$
(6)
where s_{i} is the distance between the movable teeth frame axis and the center of movable teeth.
$$s_{i} = e\cos \left( {z\theta_{i} } \right) + \sqrt {\left( {R_{j}^{2}  e^{2} \sin^{2} \left( {z\theta_{i} } \right)} \right)} ,$$
(7)
The moments acting on the two disk cams can be calculated. When CORR works stably, the input torque must be equal to the output torque. If one disk cam is selected, considering the force asymmetry of the two disk cams, a coefficient 0.55 is multiplied in the equation. Half of the movable rolling teeth endure meshing force, and the disk cam only receives the forces of movable teeth. The moment equilibrium equation of one disk cam can be calculated as follow
$$0.55M_{0} = \mathop \sum \limits_{i = 1}^{{\left( {z + 1} \right)/2}} F_{ji} e\sin \left( {\gamma_{i}  \theta_{i} + \varphi } \right),$$
(8)
where M_{0} is the input moment by the crankshafts, φ denotes the input angle of the disk cam corresponding to the initial output angle of the movable teeth frame. The expression \(\varphi = \left( {z + 1} \right)\theta\) can be obtained.
When the input moment of the small input gear is given, the input moment M_{0} can be calculated by using the same method as RV reducer, because the first stage of CORR is the same as RV reducer.
The force translation is through the stretch contact distortion between the disk cam, movable teeth, and movable teeth frame according to the structure of CORR. Suppose the distortion is linear and small. It can be described approximately by the small area with width L_{1} and length L_{2} shown in Figure 4, and L_{1} can be determined through Hertz’s formula.
$$L_{1} = \sqrt {\frac{4F}{{{{\uppi}}L_{2} }}\left[ {\lambda \left( {\frac{{1  \mu_{1}^{2} }}{{E_{1} }} + \frac{{1  \mu_{2}^{2} }}{{E_{2} }}} \right)} \right]} ,$$
(9)
where F is denoted the contact force between the two contact elements. μ_{1}, μ_{2} and E_{1}, E_{2} are Poisson’s ratio and Young’s modulus, respectively.
$$\frac{1}{\lambda } = \frac{1}{{\lambda_{1} }} \pm \frac{1}{{\lambda_{2} }},$$
(10)
where λ_{1} is the radius of the movable roller, and λ_{2} denotes the radius of the disk cam and the inner gear, “+” is used for convexity with convexity contact, “−” is used for convexity with concave contact. The contact surface of the movable teeth frame is flat, and the radius of curvature at the mesh point of contact with the movable roller is infinite. So, the equation can be changed to \(1/\lambda = 1/\lambda_{1} .\)
According to Figure 4, the contact distortion of two elements can be written as
$$\delta = \left( {\lambda_{1} \pm \lambda_{2} } \right)  \left( {\sqrt {\lambda_{2}^{2}  L_{2}^{2} } \pm \sqrt {\lambda_{1}^{2}  L_{1}^{2} } } \right),$$
(11)
In the above equation, “+” is used when they are convexity with convexity contact, and “−” is used when they are convexity with concave contact. For plane with convexity contact, the contact distortion can be expressed by Eq. (12).
$$\delta = \lambda_{1}  \sqrt {\lambda_{1}^{2}  L_{1}^{2} } ,$$
(12)
From Figure 5(a), due to the output shaft is fixed, the parameters δ_{ji}, δ_{si} and δ_{ni} are the contact distortions between the movable teeth and the disk cam, movable teeth frame, and inner gear, respectively.
The movable roller will deviate the original position because of those above contact distortions. The excursion will cause a small displacement ε_{i} of the disk cam along the direction of distortion δ_{ji} as shown in Figure 5(b).
$$\varepsilon_{i} = \delta_{ji} + \delta_{si}^{^{\prime}} + \delta_{ni}^{^{\prime}} ,$$
(13)
where \(\delta_{si}^{^{\prime}}\), \(\delta_{ni}^{^{\prime}}\) denote the incidental displacements of δ_{si} and δ_{ni} respectively in the direction of δ_{ji}.
$$\delta_{si}^{^{\prime}} = \delta_{si} \tan \theta_{i} \cos \gamma_{i} ,$$
(14)
$$\delta_{ni}^{^{\prime}} = \delta_{ni} \cos \gamma_{i} /\cos \left( {\theta_{i}  \alpha_{i} } \right).$$
(15)
According to Figures 2 and 5, the displacement ε_{i} can be compensated by a small rotation of the disk cam with a small angle \(\Delta \tau\).
$$\varepsilon_{i} = \Delta \tau \left {OO_{1} } \right = e\Delta \tau \sin \gamma_{i} .$$
(16)
According to Eqs. (4)–(16), the meshing force analysis model of CORR is established. The meshing force is calculated through the following steps as shown in Figure 6.

Step 1. According to the given geometrical and operating state parameters, the relative parameters can be calculated, such as α_{i}, γ_{i} and the curvature radiuses.

Step 2. Obtain the expressions of F_{si} and F_{ni}, in which F_{ji} is the variable, by using Eq. (4).

Step 3. To obtain the expressions of contact distortions δ_{ji}, δ_{si} and δ_{ni}, we can substitute F_{ji}, F_{si} and F_{ni} into Eq. (9), then substitute Eqs. (9) and (10) into Eqs. (11) and (12). In these expressions, F_{ji} is the variable.

Step 4. To obtain the expressions of \(\delta_{si}^{^{\prime}}\) and \(\delta_{ni}^{^{\prime}}\), in which F_{ji} is the variable, substitute δ_{si} and δ_{ni} into Eqs. (14) and (15).

Step 5. Give the initial value of \(\Delta \tau\), and use Eq. (16) to determine ε_{i}.

Step 6. Substitute ε_{i}, δ_{ji}, \(\delta_{si}^{^{\prime}}\) and \(\delta_{ni}^{^{\prime}}\) into Eq. (13) to calculate F_{ji}.

Step 7. Use \(\Delta \sigma = \left {\mathop \sum \limits_{i = 1}^{{\left( {z + 1} \right)/2}} F_{ji} e\sin \left( {\gamma_{i}  \theta_{i} + \varphi } \right)  0.55M_{0} } \right\) as the iterative controlling variable. If \(\Delta \sigma\) satisfy precision, stop the iterative calculation and calculate the meshing forces F_{si} and F_{ni} according to the obtained F_{ji}. If not, make tiny adjustments to \(\Delta \sigma\) and repeat step 5 to step 7.
Geometric errors generally refer to the manufacturing and installation errors of key components in the oscillatory roller transmission. In this paper, the installation error and manufacturing error of the disk cam and the manufacturing error of the movable roller are mainly considered.
3.1 Model of Mesh Force Considering Manufacturing Error of Disk Cam
In actual situations, there are manufacturing errors of the disk cam, which will affect the mesh force. As shown in Figure 7, there are manufacturing errors before the motion of Figure 5. The radius of the disk cam and the contact normal distortions between the movable roller and the disk cam can be shown as Eqs. (17)–(19).
$$R_{{j\_\Delta R_{j} }} = R_{j} \pm \Delta R_{j} ,$$
(17)
$$\delta_{{ji\_\Delta R_{j} }} = \delta_{ji} \pm \Delta \delta_{ji} = \delta_{ji} \pm \Delta R_{j} ,$$
(18)
$$\delta_{{ni\_\Delta R_{j} }} = \delta_{ni} \pm \Delta \delta_{ni} = \delta_{ji} \pm \Delta R_{j} \cos \alpha_{i} ,$$
(19)
where ΔR_{j} is the manufacturing error of the disk cam. The symbol “+” is used for positive errors and “−” is used for negative errors.
Eqs. (17)–(19) are brought back to Eqs. (4)–(16). Then the mesh force is calculated through the above steps. But Step 7 should be added: if δ_{ji_ΔRj} > 0, the process continues calculating. If not, there is no distortion between the movable roller and the disk cam, then the mesh force F_{ji} = 0.
3.2 Model of Mesh Force Considering Installation Error of Disk Cam
As shown in Figure 8, there are installation errors before the motion of Figure 5. The eccentricity of the disk cam and the contact normal distortions between the movable roller and the disk cam become Eqs. (20)–(23).
$$e_{\Delta e} = e \pm \Delta e,$$
(20)
$$\delta_{ji\_\Delta e} = \delta_{ji} \pm \Delta \delta_{ji} = \delta_{ji} \pm \Delta e\cos \theta_{i} ,$$
(21)
$$\delta_{si\_\Delta e} = \delta_{si} \pm \Delta \delta_{si} = \delta_{si} \pm \Delta e\tan \gamma_{i} ,$$
(22)
$$\delta_{ni\_\Delta e} = \delta_{ni} \pm \Delta \delta_{ni} = \delta_{ni} \pm \Delta e\cos \left( {\theta_{i}  \alpha_{i} } \right),$$
(23)
where \(\Delta e\) is the installation errors of the disk cam. “+” is used for positive errors and “−” is used for negative errors.
Eqs. (20)–(23) are brought back to the original formulas, and the other formulas are unchanged. Then calculate the mesh force through the above steps. However, Step 7 should be added: if δ_{ji_Δe} > 0, the process continues calculating. If not, there is no distortion between the movable roller and disk cam, then the mesh force F_{ji }= 0.
3.3 Model of Mesh Force Considering Manufacturing Error of Movable Roller
During the processing of a movable roller, there are inevitable manufacturing errors, which will affect the mesh force. As shown in Figure 9, the manufacturing errors can be expressed as
$$r_{\Delta r} = r \pm \Delta r,$$
(24)
$$\delta_{ji\_\Delta r} = \delta_{ji} \pm \Delta \delta_{ji} = \delta_{ji} \pm \Delta r,$$
(25)
$$\delta_{si\_\Delta r} = \delta_{si} \pm \Delta \delta_{si} = \delta_{si} \pm \Delta r,$$
(26)
$$\delta_{ni\_\Delta r} = \delta_{ni} \pm \Delta \delta_{ni} = \delta_{ni} \pm \Delta r,$$
(27)
where \(\Delta r\) is the manufacturing errors of the movable roller. “+” is used for positive errors, and “−” is used for negative errors.
Eqs. (24)–(27) are brought back to Eqs. (4)–(16). Then calculate the mesh force through the above steps. However, Step 7 should be changed: if δ_{ji_Δr} > 0, the process continues calculating. If not, there is no distortion between the movable teeth and the disk cam, movable teeth frame and inner gear, then the mesh force F_{ji} = 0, F_{si} = 0, F_{ni} = 0.