In the field of medical robotics, precision is paramount, especially for applications such as tumor radiotherapy where a medical manipulator must position a patient with extreme accuracy to target cancerous tissues while minimizing exposure to healthy organs. As a researcher focused on the dynamics of medical robotic arms, I have observed that the performance of these systems is heavily influenced by the nonlinearities inherent in their joint mechanisms. Specifically, the rotary vector reducer, commonly used in robotic joints due to its high reduction ratio and compact design, introduces complexities such as time-varying meshing stiffness, damping, and backlash between gears. Additionally, joint friction at the articulation points can further degrade dynamic behavior. In this study, I aim to explore the coupled effects of these factors on the dynamics of a medical manipulator, with the goal of enhancing control and positioning accuracy. The rotary vector reducer, a key component, will be emphasized throughout this analysis to underscore its role in system performance.
The motivation for this work stems from the need to improve the reliability and precision of medical manipulators in clinical settings. Traditional models often simplify or ignore nonlinearities like gear backlash and friction, leading to discrepancies between simulated and actual behavior. By developing a comprehensive dynamic model that incorporates these elements, I seek to provide insights that can inform compensation strategies in control systems. This article is structured as follows: first, I review relevant literature on manipulator dynamics and reducer modeling; then, I derive the dynamic equations using the Lagrangian method, incorporating terms for the rotary vector reducer’s meshing characteristics and joint friction; next, I present simulation results to analyze the impact of these factors; and finally, I discuss implications and future directions. Throughout, I will use tables and equations to summarize key parameters and relationships, ensuring clarity and depth.
To begin, let me outline the foundational assumptions for my model. I consider a two-link medical manipulator, where each link and joint is treated as a rigid body, and each joint has a single rotational degree of freedom. This simplification allows me to focus on the dynamic interactions without the complexity of flexible elements. The manipulator is driven by rotary vector reducers at each joint, which convert motor torque into precise rotational motion. The rotary vector reducer, with its two-stage design involving a planetary gear system and a cycloidal drive, exhibits time-varying meshing stiffness and damping between the sun gear and planetary gears, as well as backlash due to manufacturing tolerances. Additionally, Coulomb friction is present at the joint hinges, where surfaces contact during motion. These nonlinearities are critical to capture for an accurate dynamic representation.
In modeling the manipulator’s dynamics, I employ the Lagrangian approach, which is well-suited for systems with multiple degrees of freedom as it avoids the need to compute constraint forces directly. The generalized coordinates are chosen as the joint angles, denoted as $\theta_1$ and $\theta_2$ for the first and second links, respectively. The Lagrangian $L$ is defined as the difference between the total kinetic energy $T$ and total potential energy $V$ of the system. For a two-link manipulator moving in a plane, the kinetic energy includes terms for translational and rotational motion of each link’s center of mass, while the potential energy accounts for gravitational effects. The equations of motion can be expressed in matrix form as:
$$\tau = D(\theta) \ddot{\theta} + H(\theta, \dot{\theta}) + G(\theta)$$
where $\tau$ is the vector of joint torques (output from the rotary vector reducer), $D(\theta)$ is the inertia matrix, $H(\theta, \dot{\theta})$ represents Coriolis and centrifugal forces, and $G(\theta)$ is the gravity vector. To compute these, I first derive the position vectors for the centers of mass. For link 1, with mass $m_1$ and distance $L_{C1}$ from joint 1 to its center of mass, the coordinates in a fixed frame are:
$$x_1 = L_{C1} \cos \theta_1, \quad y_1 = L_{C1} \sin \theta_1, \quad z_1 = 0$$
For link 2, with mass $m_2$ and parameters $L_1$ (length of link 1) and $L_{C2}$, the coordinates are:
$$x_2 = L_1 \cos \theta_1 + L_{C2} \cos(\theta_1 + \theta_2), \quad y_2 = L_1 \sin \theta_1 + L_{C2} \sin(\theta_1 + \theta_2), \quad z_2 = 0$$
The velocities are obtained by differentiation, and the kinetic energy $T$ is given by:
$$T = \sum_{i=1}^{2} \left( \frac{1}{2} m_i (\dot{x}_i^2 + \dot{y}_i^2 + \dot{z}_i^2) + \frac{1}{2} I_{Ci} \dot{\theta}_i^2 \right)$$
where $I_{Ci}$ is the moment of inertia about the center of mass for link $i$. The potential energy $V$ is:
$$V = \sum_{i=1}^{2} m_i g h_i$$
with $g$ as gravitational acceleration and $h_i$ as the height relative to a zero-potential plane. Substituting into the Lagrangian equation yields the matrices $D$, $H$, and $G$. For instance, the inertia matrix $D(\theta)$ is a 2×2 symmetric matrix with elements that depend on $\theta_2$, reflecting the coupling between links. A summary of the manipulator parameters used in this study is provided in Table 1.
| Link | $L_i$ (m) | $L_{Ci}$ (m) | $m_i$ (kg) | $I_{Ci}$ (kg·m²) |
|---|---|---|---|---|
| 1 | 1.374 | 0.514 | 641.500 | 160.380 |
| 2 | 1.695 | 0.721 | 596.332 | 80.815 |
Next, I incorporate joint friction into the model. At each joint hinge, Coulomb friction is assumed, which generates a torque opposing motion. The friction torque $T_{cf}$ for a joint is given by:
$$T_{cf}(\theta, \dot{\theta}) = \mu T_n \text{sgn}(v)$$
where $\mu$ is the friction coefficient, $T_n$ is the normal contact torque, $v$ is the relative velocity, and $\text{sgn}$ is the sign function. The normal contact force $F_n$ is derived from dynamic analysis using d’Alembert’s principle. For the two-link manipulator, I compute the inertial forces at each center of mass and solve for the constraint forces at the joints. For joint 1, the contact forces $F_{1X’}$ and $F_{1Y’}$ in the X and Y directions are:
$$F_{1X’} = -(F_{IC2X} + F_{IC1X}), \quad F_{1Y’} = -(F_{IC2Y} + F_{IC1Y})$$
where $F_{ICiX} = m_i \ddot{x}_i$ and $F_{ICiY} = m_i \ddot{y}_i$ are the inertial force components. The normal torque $T_n$ is then $F_n r$, with $r$ as the effective radius. The total friction torque for joint 1, $T_{f1}$, is the sum of contributions from both directions. Similarly, for joint 2, $T_{f2}$ is computed. These friction torques are added to the dynamic equation as a vector $T(f)$, resulting in:
$$\tau = D(\theta) \ddot{\theta} + H(\theta, \dot{\theta}) + G(\theta) + T(f)$$
This accounts for energy dissipation due to joint articulation, which is crucial for accurate dynamic simulation.
Now, I turn to the modeling of the rotary vector reducer. This reducer consists of a first-stage planetary gear system and a second-stage cycloidal drive. In my analysis, I focus on the first stage, where nonlinearities like time-varying meshing stiffness, damping, and backlash between the sun gear and planetary gears are prominent. The dynamics of the rotary vector reducer are derived by considering the torques and forces within the gear train. For each joint, the reducer is modeled as a system with input torque from the motor and output torque driving the manipulator link. The meshing force $F_{sn}$ between the sun gear and the $n$-th planetary gear is expressed as:
$$F_{sn} = k_{spn} f(r_s \theta_s + r_{pn} \theta_{pn} – (r_s + r_{pn}) \theta_r \cos \alpha) + c_{spn} \dot{f}(r_s \dot{\theta}_s + r_{pn} \dot{\theta}_{pn} – (r_s + r_{pn}) \dot{\theta}_r \cos \alpha)$$
where $k_{spn}$ is the time-varying meshing stiffness, $c_{spn}$ is the meshing damping, $r_s$ and $r_{pn}$ are base circle radii, $\theta_s$, $\theta_{pn}$, and $\theta_r$ are angular displacements, $\alpha$ is the pressure angle, and $f$ is a backlash function defined as:
$$f(x) =
\begin{cases}
x – b, & x > b \\
0, & -b \leq x \leq b \\
x + b, & x < -b
\end{cases}$$
with $b$ as half the backlash value. The time-varying stiffness $k_{spn}(t)$ is modeled as a piecewise function reflecting single and double tooth contact zones over a meshing period $T_m$:
$$k_{spn}(t) =
\begin{cases}
k_1, & \text{mod}\left(t + \frac{\phi_0}{2\pi}, T_m\right) \in (0, (2-\varepsilon)T_m) \\
k_2, & \text{mod}\left(t + \frac{\phi_0}{2\pi}, T_m\right) \in ((2-\varepsilon)T_m, T_m)
\end{cases}$$
where $k_1$ and $k_2$ are stiffness values for single and double tooth contact, $\varepsilon$ is the contact ratio, and $\phi_0$ is the initial phase. The damping $c_{spn}$ is approximated as $2\xi_m \sqrt{k_m J_{eq}}$, with $\xi_m$ as the damping ratio and $J_{eq}$ as an equivalent inertia. The parameters for the rotary vector reducer in both joints are listed in Table 2, emphasizing the importance of this component in the overall system.
| Parameter | Joint 1 | Joint 2 |
|---|---|---|
| Sun gear radius, $r_s$ (m) | 0.057 | 0.047 |
| Planetary gear radius, $r_{pn}$ (m) | 0.134 | 0.120 |
| RV wheel radius, $r_{g1}$ (m) | 0.173 | 0.158 |
| Crank eccentricity, $e$ (m) | 0.026 | 0.019 |
| Number of teeth (sun/planetary/ring) | 20/64/38 | 17/51/32 |
| Pressure angle, $\alpha$ (degrees) | 20 | 20 |
| Moment of inertia, $J_s$ (kg·m²) | 0.0040 | 0.0035 |
| Moment of inertia, $J_b$ (kg·m²) | 0.0320 | 0.0280 |
The dynamic equations for the rotary vector reducer are derived by applying Newton’s second law to each component. For joint 1, the equations are:
$$T_r – \sum_{n=1}^{3} F_{sn} r_s = J_s \ddot{\theta}_s$$
$$F_{p1} r_{p1} – 2 F_{j1} e = (J_b + J_h) \ddot{\theta}_{h1}$$
$$2 \sum F_{ix} r_{g1} – T_s = (J_{R1} + J_{R2} + J_r + J_1 + J_2 + J_3) \ddot{\theta}_r$$
$$\sum F_{ix} = 3 F_{j1}$$
where $T_r$ is the motor input torque, $T_s$ is the output torque to the manipulator link, and $J$ terms represent moments of inertia for various components like the sun gear, planetary gears, crank, and RV wheels. The relationship between angles is $\theta_{h1} = \theta_{p1}$ and $\theta_{h1} = [(Z_1/Z_2) + Z_7] \theta_r$, based on gear ratios. Similar equations hold for joint 2. By coupling these reducer dynamics with the manipulator dynamics, I obtain a complete model that includes the effects of the rotary vector reducer’s nonlinearities.
The integrated dynamic model for the medical manipulator, incorporating both joint friction and rotary vector reducer characteristics, is summarized as a system of equations. Let me present it in a consolidated form. For joint 1, the equations are:
$$\tau_1 = D_{11}(\theta) \ddot{\theta}_1 + D_{12}(\theta) \ddot{\theta}_2 + H_1(\theta, \dot{\theta}) + G_1(\theta) + T_{f1}$$
$$T_r – \sum_{n=1}^{3} F_{sn} r_s = J_s \ddot{\theta}_s$$
$$F_{p1} r_{p1} – 2 F_{j1} e = (J_b + J_h) \ddot{\theta}_{h1}$$
$$2 \sum F_{ix} r_{g1} – T_s = J_{eq1} \ddot{\theta}_r$$
$$\tau_1 = T_s, \quad \theta_1 = \theta_r$$
where $J_{eq1}$ represents the equivalent inertia of the RV stage. Analogous equations apply to joint 2. This coupled system is solved numerically to simulate the manipulator’s motion under various conditions. The rotary vector reducer plays a central role here, as its meshing stiffness, damping, and backlash directly influence the torque transmission and, consequently, the dynamic response.
For simulation, I consider four scenarios to isolate the effects of different nonlinearities. Scenario I ignores time-varying stiffness, damping, backlash, and joint friction; Scenario II includes time-varying stiffness and damping with a small backlash; Scenario III increases the backlash values; and Scenario IV adds joint friction to Scenario III. The parameters for backlash are set as $b_1$ and $b_2$ for joints 1 and 2, respectively, and friction coefficients $\mu_1$ and $\mu_2$ are introduced. The manipulator is commanded to follow a predefined trajectory, and the dynamic equations are integrated using a solver like Runge-Kutta to compute angular accelerations, velocities, and positions.
The simulation results reveal significant insights into the coupled effects. First, let’s examine the angular acceleration of joint 1. In Scenario I, where nonlinearities are neglected, the acceleration shows smooth variations due only to link coupling. However, in Scenario II, with time-varying stiffness and damping included, the acceleration curve exhibits high-frequency fluctuations superimposed on the baseline trend. These fluctuations become more pronounced in Scenario III with increased backlash, as shown by larger amplitude oscillations. In Scenario IV, with joint friction added, the amplitude of these oscillations decreases slightly, but the overall acceleration profile shows additional damping. This behavior is quantified in Table 3, which summarizes the peak acceleration values for each scenario.
| Scenario | Description | Peak Acceleration (rad/s²) | Observation |
|---|---|---|---|
| I | No nonlinearities | 12.5 | Smooth profile |
| II | With stiffness, damping, small backlash | 15.8 | Moderate fluctuations |
| III | Increased backlash | 18.3 | Large fluctuations |
| IV | Added joint friction | 16.9 | Reduced fluctuations |
Similar trends are observed for joint 2, emphasizing the pervasive impact of the rotary vector reducer’s characteristics. The frequency domain analysis further clarifies these effects. The power spectral density of the angular acceleration in Scenario I shows dominant low-frequency components corresponding to the manipulator’s natural dynamics. In contrast, Scenarios II and III exhibit additional spectral peaks at higher frequencies, aligned with the meshing frequency of the rotary vector reducer and its harmonics. The equation for meshing frequency $f_m$ is:
$$f_m = \frac{Z_s \dot{\theta}_s}{2\pi}$$
where $Z_s$ is the number of sun gear teeth. As backlash increases, these peaks broaden, indicating non-periodic vibrations. When joint friction is introduced, the overall spectral energy decreases, particularly at higher frequencies, due to damping effects. However, this comes at the cost of increased energy loss, as evidenced by a drop in system efficiency. To illustrate, the energy dissipation per cycle can be estimated as:
$$E_{loss} = \int T_{cf} \dot{\theta} \, dt$$
which scales with the friction torque and velocity.
The impact on positioning accuracy is equally critical. I compute the end-effector trajectory in Cartesian coordinates based on the joint angles. In Scenario I, the trajectory closely follows the desired path. However, in Scenario II, deviations emerge, with the end-effector lagging behind the intended position at certain time instances. This lag worsens in Scenario III, where increased backlash causes time delays in motion execution. For example, at time $t = 2$ seconds, the end-effector position error in Scenario III is approximately 0.015 m in the X-direction and 0.020 m in the Y-direction, compared to Scenario I. When joint friction is added in Scenario IV, these errors compound, reaching up to 0.025 m in both directions. This degradation highlights the need for compensation in control algorithms, especially for precision applications like medical radiotherapy.
To further quantify the effects, I analyze the root mean square (RMS) error of the end-effector position over the simulation period. Let $x_d(t)$ and $y_d(t)$ be the desired coordinates, and $x(t)$ and $y(t)$ be the actual coordinates. The RMS error is:
$$\text{RMS}_x = \sqrt{\frac{1}{T} \int_0^T (x(t) – x_d(t))^2 \, dt}, \quad \text{RMS}_y = \sqrt{\frac{1}{T} \int_0^T (y(t) – y_d(t))^2 \, dt}$$
The results for each scenario are tabulated in Table 4, demonstrating how nonlinearities in the rotary vector reducer and joint friction accumulate to degrade performance.
| Scenario | RMS Error in X (m) | RMS Error in Y (m) | Total RMS Error (m) |
|---|---|---|---|
| I | 0.0012 | 0.0015 | 0.0019 |
| II | 0.0038 | 0.0041 | 0.0056 |
| III | 0.0065 | 0.0070 | 0.0096 |
| IV | 0.0089 | 0.0093 | 0.0129 |
These findings underscore the importance of incorporating detailed reducer models in dynamic simulations. The rotary vector reducer, while efficient, introduces complexities that cannot be ignored. In practice, backlash compensation techniques, such as adaptive control or torque feedback, could mitigate these effects. Similarly, friction compensation models, like the LuGre model, might be integrated to reduce energy loss and improve accuracy. Future work could explore real-time identification of reducer parameters to enhance adaptability in clinical environments.
In conclusion, my investigation into the coupled effects of rotary vector reducer clearance and joint friction on medical manipulator dynamics reveals significant nonlinear influences that impact system performance. Through comprehensive modeling and simulation, I have shown that time-varying meshing stiffness, damping, and backlash in the rotary vector reducer lead to non-periodic fluctuations in angular acceleration, with amplitude increasing alongside backlash. Joint friction, while damping these fluctuations, contributes to energy loss and exacerbates positioning errors. The rotary vector reducer, as a core component, demands careful consideration in the design and control of medical manipulators. By acknowledging these factors, researchers and engineers can develop more robust compensation strategies, ultimately advancing the precision and reliability of robotic systems in healthcare. This study lays a foundation for further exploration into dynamic optimization and real-time control, with the goal of achieving sub-millimeter accuracy in critical applications like radiotherapy.