In recent years, the embodied robot has emerged as a transformative platform for artificial intelligence, drawing significant attention from both academia and industry. Unlike conventional robotic arms, an embodied robot requires highly integrated and backdrivable rotational joints that deliver both high torque density and precise motion control. However, the design of controllers for such joints remains a challenging task due to strong nonlinearities, parameter uncertainties, and time-varying external loads. This thesis focuses on the design, optimization, and experimental validation of a minimum-loss PID passivity-based controller (PID-PBC) for rotational joints in an embodied robot. The research begins with the establishment of a comprehensive Port-Hamiltonian (PH) model for the joint, covering the permanent magnet synchronous motor (PMSM) and a novel precision cycloidal reducer. Subsequently, an interconnection and damping assignment passivity-based control (IDA-PBC) strategy is developed, which is then extended to a PID-PBC scheme augmented by a novel feedback load observer. To further enhance the adaptability of the controller, a multi-objective optimization problem is formulated for the equilibrium point assignment, and an improved multi-objective walrus optimization algorithm (IMOWaOA) is proposed to solve it. In addition, a fuzzy optimization method is designed to tune the controller parameters in real time. Finally, a custom driver board is designed and a test bench is built. Experimental results confirm that the proposed methodology significantly improves the response speed, stability, and disturbance rejection capability of the joint control system, demonstrating great potential for applications in the field of embodied robot.
1. Introduction
1.1 Research Background and Significance
The embodied robot is regarded as a future disruptive product, following the trajectories of computers, smart phones, and new energy vehicles. The rotational joint is the core functional unit of an embodied robot, as it integrates an electric motor, a reducer, and encoders into a monolithic mechatronic system. The quality of the rotational joint directly determines the dynamic performance and safety of the entire embodied robot. Therefore, the design of high-performance controllers for robotic joints has become a key research topic.
Traditional signal-based control methods, including proportional-integral-derivative (PID) control, sliding mode control (SMC), and model predictive control (MPC), are widely adopted in motor drive systems. However, these methods often rely heavily on accurate system parameters and thus may suffer from performance degradation when parameter variations occur. For applications requiring high reliability and smooth interaction, such as medical assistance and human-robot collaboration, the demand for stronger stability guarantees motivates the exploration of energy-based control approaches.
Passivity-based control (PBC) is an energy-shaping approach that exploits the physical structure of the system to ensure closed-loop stability. By treating the system as a network of energy-processing components, PBC methods provide a constructive framework for controller design. This thesis investigates the application of passivity-based control, specifically the PID-PBC methodology, to the rotational joints of an embodied robot, with the goal of improving both stability and dynamic responsiveness.
1.2 Literature Review
1.2.1 Joint Control Methods
In the field of PMSM control, PI controllers are the most commonly used method due to their simple structure and ease of implementation. However, their performance may degrade in the presence of nonlinearities and uncertainties. Advanced control strategies such as sliding mode control, model predictive control, and active disturbance rejection control have been extensively studied to address these issues. These methods often offer improved performance but require substantial computational resources and can be sensitive to modeling errors.
1.2.2 Passivity-Based Control
The concept of passivity has its roots in network theory. In 1989, R. Ortega formally introduced passivity-based control as a systematic method for stabilizing nonlinear systems. The interconnection and damping assignment passivity-based control (IDA-PBC) method, proposed by Ortega et al. in 2001, has become a canonical approach for port-Hamiltonian systems. This method assigns a desired energy function to the closed-loop system, thereby reshaping the system’s energy landscape to guarantee asymptotic stability.
1.2.3 Meta-heuristic Optimization Algorithms
Meta-heuristic algorithms, such as particle swarm optimization (PSO), grey wolf optimizer (GWO), and chimp optimization algorithm (ChOA), have been successfully applied to complex optimization problems. The walrus optimization algorithm (WaOA), proposed by Trojovský and Dehghani in 2023, simulates the behavior of walruses in foraging, migrating, and evading predators. Despite its efficiency, the original WaOA has limitations in handling multi-objective problems and may suffer from premature convergence. To overcome these issues for the controller optimization tasks in an embodied robot, this thesis proposes an improved multi-objective version of WaOA.
1.3 Main Contributions
– Establishment of the overall PH model for the rotational joint of an embodied robot.
– Design and validation of a minimum-loss PID-PBC controller with a novel feedback load observer.
– Development of a multi-objective optimization framework for the equilibrium point determination using an improved walrus optimization algorithm.
– Design of a fuzzy auto-tuning method for the PID-PBC controller parameters and its experimental demonstration.
2. PH-Based Mathematical Modeling of the Rotational Joint
2.1 Motivations and Model Selection
The physical system of an embodied robot’s rotational joint is a typical multi-domain, multi-level electromechanical system. Modeling such a system using conventional Euler-Lagrange or Hamiltonian equations is insufficient for capturing the energy exchange behavior between different physical domains. The Port-Hamiltonian (PH) framework, which introduces port variables to model energy interaction with the environment, is chosen as the mathematical foundation in this thesis.
2.2 PH Model of the Permanent Magnet Synchronous Motor
For a surface-mounted PMSM, the dynamics can be expressed in the d-q reference frame as follows:
\[
\begin{aligned}
U_d &= R_s i_d + L_d \frac{di_d}{dt} – p_n \omega_m L_q i_q \\
U_q &= R_s i_q + L_q \frac{di_q}{dt} + p_n \omega_m L_d i_d + p_n \omega_m \psi_f \\
T_{out} &= \frac{3}{2} p_n [\psi_f i_q + (L_d – L_q) i_d i_q] – B \omega_m
\end{aligned}
\tag{1}
\]
The Hamiltonian function for the PMSM subsystem is defined as:
\[
E_{Ha}(x) = \frac{1}{2} x^T C^{-1} x = \frac{x_1^2}{2L_d} + \frac{x_2^2}{2L_q} + \frac{x_3^2}{2J}
\tag{2}
\]
Choosing the state vector as \( x = (L_d i_d, L_q i_q, J\omega_m)^T \), the PMSM can be represented in PH form:
\[
\dot{x} = [J(x) – R(x)]\nabla E_{Ha}(x) + O(x)u
\tag{3}
\]
with the interconnection matrix \( J(x) \), damping matrix \( R(x) \), and input matrix \( O(x) \) appropriately defined based on the motor parameters.
2.3 PH Model of the Novel Precision Cycloidal Reducer
The cycloidal reducer is modeled based on its kinematic relationships. Let \( \theta_{cs} \), \( \theta_{cg} \), and \( \theta_z \) represent the spin angle of the cycloidal disc, the revolution angle of the cycloidal disc, and the angle of the output shaft, respectively. The model can be expressed as:
\[
\begin{aligned}
\dot{x} &= A x + B u = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} x + \begin{bmatrix} 1 \\ (C_z – C_c)/C_c \\ -(C_z – C_c)/C_c \end{bmatrix} \omega_{in} \\
y &= O \nabla E_{Hc}(x)
\end{aligned}
\tag{4}
\]
where \( A \) is a zero matrix due to no internal energy storage, and the Hamiltonian energy is defined as \( E_{Hc}(x) = \frac{1}{2}(\theta_{cs}^2 + \theta_{cg}^2 + \theta_z^2) \).
2.4 Overall Model of the Rotational Joint
Based on the cascade property of PH systems, the overall model is obtained by interconnecting the PMSM model and the cycloidal reducer model. The state vector is extended to \( x = (L_d i_d, L_q i_q, J\omega_m, \theta_{cs}, \theta_{cg}, \theta_z)^T \). The complete PH model takes the form:
\[
\begin{aligned}
\dot{x} &= [J_{all}(x) – R_{all}(x)] \nabla E_{Hall}(x) + B_{all} u \\
y &= B_{all}^T \nabla E_{Hall}(x)
\end{aligned}
\tag{5}
\]
The detailed matrices are constructed by combining the individual models of the two subsystems, taking into account the velocity coupling between the motor shaft and the reducer input.
2.5 PH Difference Model
To facilitate the design of a controller for a non-zero equilibrium point, a PH difference model is established. The error dynamics are expressed as:
\[
\dot{\tilde{x}} = [J(x) – R(x)](\nabla H – \nabla H_{ref}) + O(x)\tilde{u}
\tag{6}
\]
It can be proven that the PH difference model preserves passivity under certain convexity conditions on the Hamiltonian function.
3. Minimum-Loss PID-PBC Controller Design
3.1 Theoretical Preliminaries
A system is passive if there exists a non-negative storage function \( E(t) \) such that:
\[
E[x(T)] – E[x(0)] \leq \int_0^T u^T(t) y(t) dt
\tag{7}
\]
The feedback stabilization principle for PH systems asserts that if the closed-loop system can be written in the form:
\[
\dot{x} = [J_{dir}(x) – R_{dir}(x)] \frac{\partial H_{dir}}{\partial x}
\tag{8}
\]
with \( J_{dir} = -J_{dir}^T \) and \( R_{dir} = R_{dir}^T \geq 0 \), then the system is stable in the sense of Lyapunov.
3.2 Minimum-Loss IDA-PBC Design
3.2.1 Equilibrium Point Design
To improve the energy efficiency of the embodied robot, the minimum-loss control strategy is adopted. The equilibrium point for the current values is derived by minimizing the total power loss \( P_m = P_{Cu} + P_{Fe} \):
\[
\begin{aligned}
i_{d0} &= 0, \\
i_{q0} &= \frac{T_e}{p_n \psi_f}, \\
\omega_{m0} &= \omega_0
\end{aligned}
\tag{9}
\]
3.2.2 IDA-PBC Controller Equation
The IDA-PBC controller is derived based on the PH model and the desired closed-loop energy function. The resulting controller is:
\[
\begin{aligned}
u_d &= L_d i_{d0} – J_{12} i_q – r_1 (i_d – i_{d0}) – p_n \omega_m L_q i_q \\
u_q &= r_1 i_{d0} + r_2 i_q + J_{12} i_{d0} – p_n \omega_m (L_d i_d + \psi_f)
\end{aligned}
\tag{10}
\]
3.3 PID-PBC Design with Novel Load Observer
3.3.1 PID-PBC Method
The PID-PBC methodology extends the PH framework by adding a PID controller on the passive output. The controller structure is defined as:
\[
\begin{aligned}
\dot{\rho} &= -K_I \tilde{y} – K_P \tilde{y} \\
u &= -K_D \dot{\tilde{y}} + \rho
\end{aligned}
\tag{11}
\]
where \( K_P, K_I, K_D \) are symmetric positive definite matrices.
3.3.2 Novel Feedback Load Observer
A novel feedback load observer is developed to estimate the load torque in real-time. The observer dynamics are given by:
\[
\begin{aligned}
\dot{\hat{x}} &= A\hat{x} + Bu + K_1(y-\hat{y}) + K_2(\dot{y}-\dot{\hat{y}}) + K_3 \int (y-\hat{y}) dt \\
\hat{y} &= C\hat{x}
\end{aligned}
\tag{12}
\]
The observer gains are selected to place the error dynamics in the left half-plane. Compared with the conventional Luenberger observer, the proposed observer has a faster response and better disturbance estimation capability. The improvement in transient response is around 80% in simulation.
3.4 Stability Analysis
For the closed-loop system, the Lyapunov function is chosen as:
\[
H_d(x,\rho) = H(x) + \frac{1}{2}\rho^T K_I^{-1}\rho
\tag{13}
\]
One can show that \( \dot{H}_d \leq 0 \), which implies asymptotic stability by the LaSalle invariance principle. This confirms that the proposed PID-PBC controller preserves the stability guarantees of the passivity-based framework.
3.5 Simulation and Numerical Analysis
3.5.1 Simulation Setup
The simulations are conducted in Matlab/Simulink using the joint parameters listed in the following table.
| Parameter | Symbol | Value |
|---|---|---|
| Stator resistance | \( R_s \) | 2.875 Ω |
| d-axis inductance | \( L_d \) | 8.5 mH |
| q-axis inductance | \( L_q \) | 8.5 mH |
| Number of pole pairs | \( p_n \) | 4 |
| Flux linkage | \( \psi_f \) | 0.175 Wb |
| Damping coefficient | \( B \) | 0.008 N·m·s |
| Moment of inertia | \( J \) | 0.003 kg·m² |
| Number of cycloidal teeth | \( C_z \) | 40 |
| Number of pin teeth | \( C_c \) | 39 |
3.5.2 Response Speed Test
The closed-loop response of the PID-PBC controller is compared with that of the IDA-PBC controller and a conventional PID controller. The desired speed is set to 50 rpm. The simulation results show that the PID-PBC controller settles in approximately 0.055 s without overshoot, whereas the IDA-PBC controller takes about 0.06 s and the PID controller exhibits 4% overshoot. This demonstrates the superior dynamic response of the PID-PBC method.
3.5.3 Load Disturbance Rejection Test
Under a load step change from 3 N·m to 5 N·m, the PID-PBC controller exhibits a speed deviation of less than 2 rpm, which is approximately 30% lower than that of the PID controller. The recovery time is also reduced by 35%. These results confirm the strong disturbance rejection capability of the passivity-based approach.
3.5.4 Parameter Sensitivity Analysis
The sensitivity of the PID-PBC controller parameters is analyzed. Key findings include:
– \( K_P \): If \( K_P < -R_s \), the system becomes unstable. For \( K_P \geq -R_s \), smaller \( K_P \) improves the response speed but degrades disturbance rejection.
– \( K_I \): Larger \( K_I \) improves both the response speed and disturbance rejection. However, excessive values provide diminishing returns.
– \( K_D \): Larger \( K_D \) improves the response speed but introduces overshoot.
These observations form the basis for the fuzzy tuning strategy presented in the next section.
4. Optimization of the PID-PBC Controller
4.1 Multi-Objective Equilibrium Point Optimization
To broaden the applicability of the controller for different operational requirements of the embodied robot, a multi-objective optimization problem is formulated for determining the equilibrium point of the PID-PBC controller. The objective functions incorporate six common control strategies:
\mathbf{F} =
\begin{bmatrix}
F_1 & = & \frac{T}{I} & \text{(MTPA)} \\
F_2 & = & \frac{T}{P_S} & \text{(MTPW)} \\
F_3 & = & \frac{P}{V_{rms}I_{rms}} & \text{(MPF)} \\
F_4 & = & -\frac{1}{P_{Cu}+P_{Fe}} & \text{(MLC)} \\
F_5 & = & \frac{P_{out}}{P_{in}} & \text{(MOE)} \\
F_6 & = & P_{out} & \text{(MOP)}
\end{bmatrix}
\tag{14}
\]
The optimization problem is subject to voltage, current, and power constraints:
\[
\begin{aligned}
& \frac{3}{2}p_n[\psi_f i_q + (L_d – L_q)i_d i_q] = T \\
& (R_s i_d – \omega_m L_q i_q)^2 + (R_s i_q + \omega_m L_d i_d + \omega_m \psi_f)^2 \leq U_{max}^2 \\
& \frac{3}{2}(R_s i_d^2 + R_s i_q^2 + \omega_m \psi_f i_q) \leq P_{max} \\
& i_d^2 + i_q^2 \leq I_{max}^2
\end{aligned}
\tag{15}
\]
4.2 Improved Multi-Objective Walrus Optimization Algorithm
4.2.1 Basic Walrus Optimization Algorithm
The WaOA is a population-based meta-heuristic algorithm inspired by the foraging, migration, and predator-avoidance behaviors of walruses. The position of each walrus represents a candidate solution, and the walrus with the longest tusks acts as the leader guiding the population.
4.2.2 Enhancements
The algorithm is improved in three aspects:
1. Multi-objective capability: The concept of an archive is introduced to store non-dominated solutions, extending the WaOA to multi-objective problems.
2. Niederreiter sequence initialization: A low-discrepancy sequence is used to initialize the population, ensuring a more uniform distribution and faster convergence.
3. Refraction-based oppositional learning: This mechanism generates oppositional solutions, effectively broadening the search space and avoiding premature convergence.
4.2.3 Benchmark Test Results
The improved algorithm, denoted as IMOWaOA, is tested on a set of benchmark functions. The convergence curves demonstrate that the IMOWaOA converges significantly faster than the original WaOA, PSO, and ChOA, while maintaining good solution accuracy. For the simplified equilibrium point optimization problem, the IMOWaOA finds the optimal solution within 4 iterations in most cases.
4.2.4 Multi-Objective Optimization Results
The IMOWaOA is applied to the standard multi-objective equilibrium point optimization problem. The resulting archive contains 50 Pareto-optimal solutions. A subset of these solutions is summarized in the following table.
| No. | \( i_d \) (A) | \( i_q \) (A) | \( F_1 \) | \( F_2 \) | \( F_3 \) | \( F_4 \) | \( F_5 \) | \( F_6 \) |
|---|---|---|---|---|---|---|---|---|
| 1 | -0.376 | 3.699 | -0.15574 | -0.01664 | -0.81513 | 6.34372 | -0.30276 | -31.67 |
| 2 | -0.113 | 3.642 | -0.15650 | -0.01680 | -0.80886 | 6.28197 | -0.30438 | -31.67 |
| 3 | -0.174 | 3.655 | -0.15640 | -0.01678 | -0.81067 | 6.29033 | -0.30416 | -31.67 |
| 4 | -0.079 | 3.636 | -0.15654 | -0.01681 | -0.80778 | 6.27891 | -0.30446 | -31.67 |
| 5 | -0.002 | 3.619 | -0.15658 | -0.01682 | -0.80500 | 6.27589 | -0.30454 | -31.67 |
| 6 | -0.134 | 3.647 | -0.15647 | -0.01680 | -0.80951 | 6.28448 | -0.30431 | -31.67 |
| 7 | -0.038 | 3.627 | -0.15657 | -0.01682 | -0.80634 | 6.27658 | -0.30452 | -31.67 |
| 8 | -0.199 | 3.661 | -0.15634 | -0.01677 | -0.81136 | 6.29487 | -0.30404 | -31.67 |
| 9 | -0.295 | 3.681 | -0.15606 | -0.01671 | -0.81363 | 6.31755 | -0.30344 | -31.67 |
| 10 | -0.250 | 3.672 | -0.15621 | -0.01674 | -0.81264 | 6.30592 | -0.30375 | -31.67 |
4.3 Fuzzy Optimization of PID-PBC Parameters
4.3.1 Design of Fuzzy Sets and Membership Functions
A two-input, three-output fuzzy system is designed. The inputs are the speed error \( e \) and its derivative \( de/dt \). The outputs are the adjustments for \( k_p \), \( k_i \), and \( k_d \). Each variable is divided into nine fuzzy subsets: {BF, MF, SF, S, Z, B, SZ, MZ, BZ}. Gaussian membership functions are adopted for the speed error, while triangular membership functions are used for the error derivative.
4.3.2 Fuzzy Rule Base
The rule base is designed based on the parameter sensitivity analysis results from Section 3. A 9×9 rule table is constructed for each controller parameter. An illustrative example of the rule base for \( \Delta K_P \) is given as follows.
| \( de/dt \) \ \( e \) | BF | MF | SF | S | Z | B | SZ | MZ | BZ |
|---|---|---|---|---|---|---|---|---|---|
| BF | BZ | BZ | MZ | MZ | MZ | SZ | B | Z | Z |
| MF | MZ | MZ | SZ | SZ | SZ | B | B | Z | S |
| SF | MZ | MZ | SZ | SZ | SZ | B | Z | S | SF |
| S | SZ | SZ | B | B | B | Z | Z | S | SF |
| Z | S | SZ | B | B | SF | S | S | SF | MF |
| B | BF | B | Z | Z | SF | S | S | SF | MF |
| SZ | MF | Z | S | S | MF | SF | SF | MF | BF |
| MZ | SF | Z | S | S | MF | SF | MF | MF | BF |
| BZ | Z | SF | SF | SF | BF | MF | BF | BF | BF |
4.3.3 Fuzzy Inference and Defuzzification
The Mamdani-type fuzzy inference is adopted, and the center-of-gravity method is used for defuzzification:
\[
u = \frac{\sum_{i=1}^{n} x_i \sigma(x_i)}{\sum_{i=1}^{n} \sigma(x_i)}
\tag{16}
\]
4.3.4 Simulation Results
The performance of the fuzzy-optimized PID-PBC controller is compared with the “fastest” tuned parameter set and the “smoothest” tuned parameter set. The simulation results at a target speed of 50 rpm under a load step disturbance are summarized below.
The fuzzy-optimized curve reaches the setpoint in approximately 0.04 s without overshoot. This is 20% faster than the fastest curve and 35% faster than the smoothest curve. Under the load disturbance, the speed deviation is comparable to that of the fastest curve, but without overshoot during the recovery process. These results validate the effectiveness of the fuzzy optimization approach.
5. Driver Circuit Design and Experimental Validation
5.1 Driver Circuit Design
A custom driver board is designed to meet the requirements of the rotational joint of the embodied robot. The board is based on the STM32F405RGT6 microcontroller and the DRV8301 gate driver chip, with a 4-layer PCB layout. The driver provides:
– Three-phase inverter for PMSM control with a DC bus voltage of 24 V.
– On-board voltage regulation to supply 5 V and 3.3 V for the control electronics.
– Interfaces for an encoder, brake, and host communication.
5.2 Test Bench Setup
An experimental platform is built to validate the controller. The test bench includes the fabricated driver board, the developed rotational joint prototype, a HBM T40B digital torque sensor for output torque and speed measurement, and a magnetic powder brake as the programmable load.
5.3 Optimization Effect Verification
The fuzzy optimization method is experimentally verified at a target speed of 10 rpm for the joint output. The optimized curve shows a 50% reduction in settling time compared with the non-optimized case, reduced speed fluctuation, and no overshoot. The results confirm the practical effectiveness of the proposed optimization framework.
5.4 Speed Control Performance Comparison
In the speed regulation test, the PID-PBC controller and the conventional PID controller are compared. The joint output speed is set to 15 rpm, corresponding to a motor speed of approximately -1700 rpm. The PID-PBC controller reaches steady-state with a transient response comparable to that of the PID controller but without any overshoot. The steady-state errors are small for both controllers, and the speed curves are stable. The experimental results demonstrate the practical usability of the proposed PID-PBC controller.
6. Conclusion and Outlook
In this thesis, the design and optimization of a passive controller for the rotational joints of an embodied robot are thoroughly investigated. The main conclusions are summarized below:
① A comprehensive PH model for the rotational joint is established by integrating the PMSM model and the cycloidal reducer model. The model effectively captures the energy exchange behavior of the system and serves as a solid foundation for the subsequent controller design.
② A minimum-loss PID-PBC controller combined with a novel feedback load observer is developed. Simulation results demonstrate that the PID-PBC controller not only preserves the stability advantages of the passivity-based control but also achieves a 10% improvement in response speed over IDA-PBC while eliminating overshoot.
③ An improved multi-objective walrus optimization algorithm (IMOWaOA) is proposed for the equilibrium point optimization. The algorithm converges faster than the original WaOA, PSO, and ChOA, and is capable of obtaining a well-distributed Pareto front.
④ A fuzzy optimization method is designed for online tuning of the PID-PBC parameters. The method integrates the parameter influence laws derived from sensitivity analysis and effectively improves the dynamic response and disturbance rejection of the system.
⑤ The driver board and test platform are developed, and the experimental results confirm the correctness and effectiveness of the proposed controller and the optimization methods in a realistic environment for the embodied robot.
For future work, the following aspects are worth further investigation:
– The current model assumes ideal kinematic behavior of the reducer. A more detailed model that accounts for friction, backlash, and compliance would improve the model accuracy.
– The present controller is focused on speed regulation. Position control and torque control modes should be explored to extend the applicability of the proposed method in the embodied robot.
– The integration of learning-based techniques with passivity-based control is a promising direction, enabling more adaptive and intelligent behavior for the embodied robot.