Learning-Based Adaptive Robust Control of a 4-DOF Robot Manipulator

Master's Thesis | January - August 2026

Project Overview

This thesis developed and experimentally evaluated a learning-based nonlinear robust controller for a 4-DOF Quanser QArm robotic manipulator. The controller combines the nonlinear dynamics of the robot with online adaptation and disturbance estimation to improve trajectory tracking when the physical system differs from its mathematical model.

The controller was first developed and evaluated in MATLAB/Simulink before being implemented on the physical QArm. Its performance was compared with a conventional PID controller under different trajectory amplitudes, frequencies, and payload conditions.

The experimental results showed a 27.8% overall reduction in RMSE compared with PID, with the nonlinear controller achieving lower RMSE in four of the five experimental tests.

This page highlights the most important aspects of the thesis, including the modeling approach, controller design, implementation, and experimental results, rather than covering the full mathematical development and analysis presented in the complete thesis. The research is also currently in the process of being published with MDPI Robotics.

Objectives

• Develop a nonlinear dynamic model of the 4-DOF QArm.

• Design a nonlinear controller using the robot's dynamic model and measured joint states.

• Incorporate online learning to adjust controller gains during operation.

• Estimate and compensate for unknown disturbances and modeling uncertainties.

• Validate the controller through MATLAB/Simulink simulation.

• Implement the controller on the physical QArm.

• Compare the proposed controller with a conventional PID controller.

• Evaluate tracking performance under different trajectory amplitudes, frequencies, and payload conditions.

Constraints and Challenges 

Robotic manipulators have nonlinear and coupled dynamics, making accurate trajectory tracking more challenging than controlling a simple linear system. The physical QArm can also differ from its mathematical model because of parameter uncertainty, unmodeled dynamics, friction, actuator behavior, and changes in payload.

A conventional PID controller uses fixed gains, so changes in operating conditions can affect its tracking performance. A model-based nonlinear controller can account for the robot's known dynamics, but its performance can still be affected when the mathematical model does not fully represent the physical system.

The main challenge was therefore to develop a controller that could use the known dynamics of the QArm while adapting to effects that were not completely captured by the nominal model. The approach also needed to remain computationally efficient enough for real-time implementation on the physical manipulator.

where T represents the system's kinetic energy and V represents its potential energy. The difference between these two quantities provides the basis for deriving the robot's equations of motion.

The equations of motion were then obtained using the Euler-Lagrange formulation:

Here, q represents the joint positions, τ represents the applied joint torque, and d represents the effects not represented by the model. This formulation allowed the controller to account for the QArm's nonlinear mechanical behavior rather than treating each joint as an independent linear system.

The resulting dynamic model can be represented in the standard form:

where M represents the manipulator's inertia, C represents Coriolis and centrifugal effects, and G represents gravitational effects. The control input is represented by τ, while d represents unknown disturbances and effects that are not captured by the nominal model.

This equation is important to the overall approach because it separates the known dynamics of the robot from the effects the controller needs to estimate and compensate for.

Adaptive Control and Disturbance Estimation

The controller used position and velocity tracking errors to determine how closely the QArm followed its desired trajectory. Rather than relying entirely on fixed feedback gains, the controller continuously adjusted its gains based on the observed tracking behavior.

The adaptive gain laws were defined as:

where k1 and k2 are the adaptive controller gains, alpha1 and alpha2 determine the rate at which the gains adapt, and e1 and e2 represent tracking-error terms.

In practical terms, these equations allow the controller gains to change during operation in response to the tracking behavior of the robot, rather than remaining fixed throughout the trajectory.

The controller also incorporated a disturbance estimator to account for effects that were not represented by the nominal robot model. The resulting control law combined nonlinear model compensation, adaptive feedback, desired trajectory information, and the estimated disturbance:

Here, τ is the resulting control input. The other terms represent the modeled system dynamics, adaptive feedback, desired trajectory information, and estimated disturbance. The term d_hat is particularly important because it allows the controller to compensate for effects that are not fully represented in the nominal dynamic model.

Together, these components allow the controller to use the known behavior of the robot while adapting to deviations from that model during operation.

Simulation and Implementation Approach

Before testing the controller on the physical QArm, the complete control structure was developed and evaluated in MATLAB/Simulink. Simulation provided an initial way to examine the controller's ability to track both fixed setpoints and continuously varying trajectories.

Once the simulation results demonstrated the intended tracking behavior, the controller was implemented on the physical QArm for experimental evaluation under different trajectory amplitudes, frequencies, and payload conditions.

Results and Evaluation 

The controller was evaluated with both simulation and physical hardware implementation results.

Simulation Results

The controller was first implemented in MATLAB/Simulink before being transferred to the physical QArm. The simulation evaluated both fixed setpoint tracking and continuously changing trajectories.

For the setpoint test, each joint was given a commanded position and evaluated based on its ability to converge to and maintain that position. The simulation used different positive and negative joint commands, allowing the controller to be evaluated across different directions of motion. The setpoints were given as:

The four joints successfully reached their commanded positions with minimal steady-state error.

The second simulation evaluated continuous trajectory tracking, where the desired joint positions changed throughout the motion. This required the controller to continuously account for the reference trajectory, nonlinear robot dynamics, tracking errors, adaptive feedback, and disturbance estimation. The trajectories were given as: 

The trajectory tests demonstrated the controller's ability to follow continuously changing reference signals.

The simulation results demonstrated successful convergence to the commanded positions and continuous trajectory tracking across the four joints, providing validation of the controller before hardware implementation.

Experimental Hardware Results

The controller was then evaluated on the QArm's wrist joint under five experimental conditions that varied trajectory amplitude, frequency, and added payload. The experimental testing was limited to the wrist joint because it provided the safest operating configuration for evaluating the controller on the physical QArm, while the same implementation approach can be extended to the remaining joints.

Performance was measured using root mean square error (RMSE), allowing the nonlinear controller to be directly compared with the conventional PID controller.

After running all five tests, the following plots were generated:

Across the five tests, the nonlinear controller achieved a lower RMSE in four of the five conditions. When the results from all five tests were considered together, the nonlinear controller produced a 27.8% overall reduction in RMSE compared with PID.

The largest improvement occurred under the added-payload conditions. With a 32 g payload and a 45° trajectory amplitude, the nonlinear controller reduced RMSE from 0.1745 to 0.0805, corresponding to a 53.9% reduction. With the same payload and a 75° trajectory, RMSE decreased from 0.2552 to 0.1288, a 49.5% reduction.

The experimental responses also showed smoother tracking with less visible stair-stepping in the nonlinear controller's response compared with PID. 

The results demonstrated that incorporating the manipulator's nonlinear dynamics with online adaptation and disturbance estimation improved tracking performance across most of the tested conditions. The hardware experiments also confirmed that the controller could operate on the physical QArm rather than only in simulation.

Conclusions and Future Work 

The project demonstrated an online learning-based nonlinear robust control approach for a 4-DOF robotic manipulator. By combining nonlinear modeling with adaptive control and disturbance estimation, the controller improved trajectory tracking compared with conventional PID control across most of the experimental conditions.

The nonlinear controller achieved a 27.8% overall RMSE reduction, with the largest improvements occurring when an additional payload was applied. The combination of simulation and physical testing also provided a way to evaluate the controller from initial development through real-world implementation.

Future work could expand the experimental evaluation to additional joints, payloads, trajectories, and external disturbances. Further investigation of the QArm's actuator limitations could also help improve performance during more demanding trajectories.