In the field of radiation therapy for cancer treatment, precision and accuracy are paramount. The medical manipulator, which positions the patient’s target area within the beam irradiation zone, is a critical component. Its performance directly influences treatment efficacy and safety. A key element in the joints of such manipulators is the RV reducer, a high-precision gear transmission system known for its compact size, high reduction ratio, and torque capacity. However, inherent nonlinearities such as time-varying meshing stiffness, damping, tooth side clearance within the RV reducer, and friction at the joint articulations can significantly degrade the dynamic performance and positioning accuracy of the manipulator. In this comprehensive study, we develop a detailed nonlinear dynamic model for a two-link medical manipulator that explicitly incorporates these factors. Our aim is to thoroughly investigate their individual and coupled effects on the system’s dynamic characteristics, including angular acceleration, vibration spectra, energy distribution, and end-effector trajectory errors. The insights gained are crucial for informing high-precision control strategies and compensation algorithms in clinical settings.
The modeling approach is multi-faceted. First, we derive the required joint torques for the manipulator’s rigid-body motion using the Lagrangian formulation. Second, we model the friction torque at the revolute joints using a Coulomb friction model based on contact forces obtained from a dynamic force analysis. Third, we construct a detailed dynamic model for the RV reducer, focusing on the nonlinear interactions in the first-stage planetary gear set between the sun gear and the planet gears. Finally, we integrate these sub-models into a coupled system of equations that describes the complete manipulator dynamics. Extensive simulation studies are then conducted to analyze the system’s response under various conditions. To structure the vast amount of parameters and data involved, we present them in tabular form.
| Link Number (i) | Length Li (m) | Distance to CoM LCi (m) | Mass mi (kg) | Moment of Inertia ICi (kg·m²) |
|---|---|---|---|---|
| 1 | 1.374 | 0.514 | 641.500 | 160.380 |
| 2 | 1.695 | 0.721 | 596.332 | 80.815 |
The Lagrangian method is employed to derive the equations of motion for the two-link manipulator, considering only its large-scale rigid-body motion. The generalized coordinates are chosen as the joint angles $\theta_1$ and $\theta_2$. The kinetic energy $T$ and potential energy $V$ of the system are calculated based on the position vectors of the links’ centers of mass. The Lagrangian $L = T – V$ is then formed, and the standard Euler-Lagrange equation is applied:
$$ \tau_i = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) – \frac{\partial L}{\partial q_i}, \quad i=1,2 $$
where $\tau_i$ represents the generalized torque at joint $i$, and $q_i$ are the generalized coordinates ($\theta_1, \theta_2$). After considerable algebraic manipulation, the equations can be compactly written in the standard matrix form for robotic manipulators:
$$ \boldsymbol{\tau} = \mathbf{D}(\boldsymbol{\theta}) \ddot{\boldsymbol{\theta}} + \mathbf{H}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \mathbf{G}(\boldsymbol{\theta}) $$
Here, $\boldsymbol{\tau} = [\tau_1, \tau_2]^T$ is the vector of joint torques (output torques from the RV reducers), $\boldsymbol{\theta}$, $\dot{\boldsymbol{\theta}}$, $\ddot{\boldsymbol{\theta}}$ are the position, velocity, and acceleration vectors, respectively. $\mathbf{D}(\boldsymbol{\theta})$ is the $2 \times 2$ inertia matrix, $\mathbf{H}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}})$ is the $2 \times 1$ vector of Coriolis and centrifugal forces, and $\mathbf{G}(\boldsymbol{\theta})$ is the $2 \times 1$ gravity vector. The elements of these matrices are functions of the link parameters listed in Table 1.
Joint friction is a significant source of nonlinearity and energy loss. We model the revolute joints as having surface contact, and the friction torque is calculated using a Coulomb model. The first step is to determine the normal contact forces at the joint interfaces. This is achieved by applying D’Alembert’s principle to perform a dynamic force analysis on each link, considering inertial forces. The contact forces $F_{jX}’$ and $F_{jY}’$ at joint $j$ in the X and Y directions are solved from equilibrium equations. The resultant normal contact force $F_{nj}$ is then:
$$ F_{nj} = \sqrt{(F_{jX}’)^2 + (F_{jY}’)^2} $$
The Coulomb friction torque $T_{cfj}$ for joint $j$ is given by:
$$ T_{cfj}(\dot{\theta}_j) = \mu_j \, F_{nj} \, r_j \, \text{sgn}(\dot{\theta}_j) $$
where $\mu_j$ is the coefficient of friction, $r_j$ is the effective radius of the contact surface, and $\text{sgn}(\cdot)$ is the signum function. The total friction torque vector for the two joints is denoted as $\mathbf{T}_f = [T_{f1}, T_{f2}]^T$.
The core of the dynamic complexity arises from the RV reducer. The RV reducer is a two-stage reduction system. For our model, we focus on the nonlinearities in the first-stage planetary gear train (sun gear and three planet gears), while treating subsequent stages (cycloidal pin wheel and output mechanism) as rigid bodies for load transformation. The kinematic relationship between the input (sun gear) and output (carrier) is crucial. The sun gear is driven by the motor, and the output carrier is connected to the manipulator link. The meshing force between the sun gear and the $n$-th planet gear ($n=1,2,3$) is modeled as a nonlinear spring-damper element with backlash. The force on the sun gear from planet $n$, $F_{sn}$, is:
$$ F_{sn} = k_{spn}(t) \, f(\delta_{spn}) + c_{spn} \, \dot{f}(\delta_{spn}) $$
where $\delta_{spn} = r_s \theta_s + r_{pn} \theta_{pn} – (r_s + r_{pn}) \theta_r \cos \alpha$ is the relative displacement along the line of action. $r_s$, $r_{pn}$, $\theta_s$, $\theta_{pn}$, $\theta_r$ are base radii and angular displacements of the sun, planet, and carrier, respectively. $\alpha$ is the pressure angle. The function $f(x)$ represents the backlash nonlinearity:
$$ f(x) = \begin{cases}
x – b, & x > b \\
0, & -b \le x \le b \\
x + b, & x < -b
\end{cases} $$
and its derivative $\dot{f}(x)$ is used for damping. The parameter $b$ represents half the total tooth side clearance. The time-varying meshing stiffness $k_{spn}(t)$ is approximated as a piecewise constant function switching between single-tooth and double-tooth meshing stiffness ($k_1$ and $k_2$) according to the gear mesh cycle $T_m$ and contact ratio $\varepsilon$:
$$ k_{spn}(t) = \begin{cases}
k_1, & \text{mod}\left(t + \frac{\phi_0}{2\pi}, T_m\right) \in (0, (2-\varepsilon)T_m) \\
k_2, & \text{mod}\left(t + \frac{\phi_0}{2\pi}, T_m\right) \in ((2-\varepsilon)T_m, T_m)
\end{cases} $$
The meshing damping $c_{spn}$ is related to the average stiffness $k_m$ and damping ratio $\xi_m$:
$$ c_{spn} = 2 \xi_m \sqrt{ \frac{k_m \, r_s^2 \, r_{pn}^2 \, J_s \, J_{bn}}{r_s^2 J_s + r_{pn}^2 J_{bn}} } $$
where $J_s$ and $J_{bn}$ are the moments of inertia of the sun gear and planet gears. The dynamic equations for the RV reducer subsystem are derived by applying Newton’s second law to the sun gear, each planet gear (considering both revolution and rotation), and the output carrier. The final relationships connect the motor input torque $T_r$, the internal meshing forces, and the output torque $T_s$ (which is $\tau_1$ for joint 1). A similar, independent set of equations is derived for the RV reducer at the second joint. The parameters for both RV reducers are summarized below.
| Parameter | Symbol (Joint 1) | Value (Joint 1) | Symbol (Joint 2) | Value (Joint 2) |
|---|---|---|---|---|
| Sun Gear Base Radius | $r_s$ | 0.057 m | $r’_s$ | 0.047 m |
| Planet Gear Base Radius | $r_{pn}$ | 0.134 m | $r’_{pn}$ | 0.120 m |
| Pressure Angle | $\alpha$ | 20° | $\alpha’$ | 20° |
| Sun Gear Teeth | $Z_1$ | 20 | $Z’_1$ | 17 |
| Planet Gear Teeth | $Z_2$ | 64 | $Z’_2$ | 51 |
| Pin Wheel Teeth | $Z_7$ | 38 | $Z’_7$ | 32 |
| Average Meshing Stiffness | $k_m$ | Calculated per ISO | $k’_m$ | Calculated per ISO |
| Damping Ratio | $\xi_m$ | 0.05 | $\xi’_m$ | 0.05 |
| Component | Moment of Inertia – Joint 1 (kg·m²) | Moment of Inertia – Joint 2 (kg·m²) |
|---|---|---|
| Sun Gear ($J_s$, $J’_s$) | 0.0040 | 0.0035 |
| Planet Gear ($J_{b}$, $J’_{b}$) | 0.0320 | 0.0280 |
| Planet Revolution ($J_3$, $J’_3$) | 0.1300 | 0.1100 |
| Other Components (Aggregated) | See model equations | See model equations |
The complete coupled dynamic model of the medical manipulator, integrating the rigid-body dynamics, joint friction, and the detailed RV reducer dynamics, is represented by the following composite system of equations:
$$
\begin{aligned}
&\boldsymbol{\tau} = \mathbf{D}(\boldsymbol{\theta}) \ddot{\boldsymbol{\theta}} + \mathbf{H}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \mathbf{G}(\boldsymbol{\theta}) + \mathbf{T}_f, \\
&T_{r} – \sum_{n=1}^{3} F_{sn} r_s = J_s \ddot{\theta}_s, \\
&F_{p1} r_{p1} – 2F_{j1} e = (J_b + J_h) \ddot{\theta}_{h1}, \\
&2 \sum F_{ix} r_{g1} – T_s = (J_{R1}+J_{R2}+J_r+J_1+J_2+J_3) \ddot{\theta}_r, \\
&\sum F_{ix} = 3F_{j1}, \\
&\text{(Analogous equations for the joint 2 RV reducer…)}, \\
&\tau_1 = T_s, \quad \tau_2 = T’_s, \\
&\theta_1 = \theta_r, \quad \theta_2 = \theta’_r.
\end{aligned}
$$
This system is highly nonlinear due to the piecewise stiffness $k_{spn}(t)$, the backlash function $f(x)$, and the signum function in friction. We solve it numerically using a variable-step solver suitable for stiff systems. For simulation, we define four distinct operational scenarios to isolate and analyze the effects of different nonlinearities:
- Case I: Baseline model. Excludes RV reducer time-varying stiffness, damping, clearance, and joint friction. Only includes rigid-body dynamics and coupling.
- Case II: Includes RV reducer time-varying meshing stiffness and damping. Includes a small tooth side clearance ($b_1=0.02\,\text{mm}$, $b_2=0.02\,\text{mm}$). Excludes joint friction.
- Case III: Includes RV reducer stiffness and damping. Includes larger tooth side clearance ($b_1=0.05\,\text{mm}$, $b_2=0.04\,\text{mm}$). Excludes joint friction.
- Case IV: Includes all nonlinearities: RV reducer stiffness, damping, larger clearance ($b_1=0.05\,\text{mm}$, $b_2=0.04\,\text{mm}$), and joint friction ($\mu_1=\mu_2=0.02$).
The manipulator is commanded to follow a specific smooth trajectory for both joints. The dynamic responses are analyzed in time and frequency domains, and the end-effector path is computed.
The angular acceleration responses of joint 1 and joint 2 are profoundly affected by the RV reducer nonlinearities. In Case I, the acceleration profiles show relatively smooth variations dictated solely by the inertial coupling between the links. The introduction of the time-varying meshing stiffness, damping, and clearance in Case II causes high-frequency, aperiodic fluctuations superimposed on the base trajectory. These fluctuations are impact-induced vibrations resulting from the teeth entering and exiting contact due to clearance. The time-domain plots reveal a marked increase in the number of peaks and the amplitude of oscillations compared to Case I. When the clearance is increased in Case III, these fluctuations become more severe, with larger amplitude spikes, indicating stronger impacts within the RV reducer transmission. The inclusion of joint friction in Case IV alters the response: the amplitude of the high-frequency fluctuations is noticeably attenuated. This is because friction acts as an additional damping mechanism, dissipating some of the impact energy. However, this damping comes at the cost of a constant torque bias and energy loss.
Frequency-domain analysis via Fast Fourier Transform (FFT) provides deeper insight. The spectrum for Case I shows dominant low-frequency components related to the commanded trajectory and link coupling. For Cases II, III, and IV, the spectra exhibit a broad distribution of energy across a wide range of higher frequencies. These are not discrete harmonics but continuous bands, confirming the aperiodic, chaotic-like nature of the response induced by backlash. The power spectral density (PSD) of the angular acceleration quantifies the energy distribution across frequencies. As the clearance increases from Case II to Case III, the overall energy level in the medium to high-frequency range rises significantly, indicating that more kinetic energy is being channeled into destructive vibratory modes due to heavier impacts in the RV reducer. The PSD plot for Case IV shows a reduction in this high-frequency energy compared to Case III, again highlighting the damping effect of friction, but the low-frequency energy is also affected, signifying overall system energy loss.
| Case | Key Features Included | Angular Acceleration Fluctuation | High-Freq. Energy (PSD) | End-Effector Path Error | Remarks |
|---|---|---|---|---|---|
| I | Rigid-body coupling only | Low, smooth, periodic | Very Low | Reference trajectory | Idealized baseline |
| II | RV reducer stiffness, damping, small clearance | Moderate, aperiodic fluctuations | Moderate | Visible deviation & time lag | Clearance-induced impacts begin |
| III | RV reducer stiffness, damping, large clearance | High, severe aperiodic fluctuations | High | Increased deviation & lag | Stronger impacts degrade performance |
| IV | All above + joint friction | Fluctuations attenuated but biased | Reduced from Case III | Largest cumulative error | Friction damps vibration but adds steady-state error |
The most critical performance metric for a medical manipulator is the accuracy of its end-effector (patient support platform) path. We compute the Cartesian coordinates of the end-effector from the joint angles. In Case I, the path follows the intended trajectory. In Case II, a clear deviation is observed; at any given time instant, the end-effector is not at its intended position. Furthermore, the motion exhibits a phase lag, meaning it arrives at points later than scheduled. This is a direct consequence of the compliance and impacts in the RV reducer, which introduce effective backlash and torque transmission errors. The problem exacerbates in Case III with larger clearance, where both the positional deviation and the time lag increase. Finally, in Case IV, joint friction introduces an additional, persistent bias error. While friction may slightly reduce the oscillatory jitter around the path, it causes the manipulator to consistently undershoot or overshoot due to stiction and Coulomb effects, leading to the largest overall root-mean-square error in positioning. This coupling effect between RV reducer clearance and joint friction is particularly detrimental for precision tasks.
The mathematical representation of the end-effector position for our two-link planar manipulator is:
$$ x_e = L_1 \cos(\theta_1) + L_2 \cos(\theta_1 + \theta_2) $$
$$ y_e = L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2) $$
The errors $\Delta x_e = x_e^{\text{actual}} – x_e^{\text{desired}}$ and $\Delta y_e$ are computed for each case. The RV reducer nonlinearities directly corrupt the joint angles $\theta_1$ and $\theta_2$, which propagate through these kinematic equations to the endpoint.
In conclusion, our detailed nonlinear dynamic modeling and simulation study unequivocally demonstrates the significant coupling effects of RV reducer tooth side clearance and joint friction on the dynamics of a medical manipulator. The time-varying meshing stiffness and damping inherent in the RV reducer, when combined with even small clearances, generate aperiodic fluctuations in joint accelerations and inject high-frequency vibrational energy into the system. Increasing the clearance amplifies these detrimental effects. Joint friction, while providing some damping to these vibrations, introduces its own set of problems: steady-state torque offset, increased energy consumption, and most importantly, additional bias error in the end-effector positioning. For the demanding application of radiation therapy patient positioning, where sub-millimeter accuracy is often required, these effects are unacceptable. Therefore, advanced control strategies must be developed to actively compensate for these nonlinearities. Potential directions include adaptive control that estimates and cancels friction, and robust or backlash-inverse control techniques that mitigate the effects of the RV reducer clearance. Future work will involve experimental validation on a physical manipulator prototype and the implementation of such advanced control algorithms based on the insights from this model. The comprehensive understanding of how the RV reducer’s internal dynamics couple with joint friction to affect overall system performance is a vital step towards achieving the ultra-high precision demanded by modern medical robotics.