Cosserat model and neural-network control for a parallel continuum robot
A parallel continuum robot moves a platform by pushing and retracting flexible rods. It has no joints: its shape is whatever the rods adopt as they bend, and computing that shape requires solving nonlinear differential equations. This project did two things. It implemented the physical model with Cosserat theory, and it trained a neural network that learns directly from the real robot which rod lengths take the end effector to a given position. The network reaches a mean error of 0.5 cm.
Context
Cosserat rod theory describes a slender body through the position and orientation of each of its cross-sections, and can include bending, torsion, shear and extension. It is the most complete model available for continuum robots, but solving it online is costly for controlling a robot in real time.
The alternative is to learn the input-output relation from data. If the data come from the physical robot, the network also absorbs what the model does not describe, such as backlash, friction and assembly errors.
The robot is the lab's rod-driven platform, in a three-filament version.
Cosserat model
The model was implemented in MATLAB in three variants: with fixed rod lengths and external forces and moments as input, with variable lengths, and with an iterative scheme that finds end-effector position and orientation for given lengths.
Robot shape for different combinations of external force and moment
Changes to the prototype
The microcontroller was replaced with an Arduino, the module that steered the rods was removed, legs were added to the base, and a structure was built for two webcams, one overhead and one lateral.
Rod actuators and new electronics
Structure holding the cameras around the robot
Data and training
The two cameras detect a colored marker on the end effector and return its position in centimeters after a manual calibration.
Frames from the two cameras with the end effector detected
The robot sweeps a grid of rod-length combinations and the measured position is recorded at each one. Those pairs train a multilayer perceptron with three hidden layers of 256, 128 and 64 neurons, which takes the desired position and returns the three lengths.
The whole flow, from camera calibration to robot control, was integrated into a Streamlit application.
Control application, showing the desired position and the one detected by the cameras
Results
Mean absolute error on the test set was 0.5 cm. In validation on the robot, over much of the workspace the point reached matches the desired one within that margin. In some regions the error reaches 2 cm per axis.
Three causes were identified. The training grid is coarse and leaves some zones poorly represented. Camera calibration is manual and depends on lighting. And the robot is more sensitive in certain configurations, where a small change in length produces a large displacement of the end effector.
What is missing
Control is open loop. The network predicts the lengths and the robot executes them, but the error measured by the cameras is not used to correct. Closing that loop is the most direct next step.
The Cosserat model and the neural network remained two parallel developments. The network was trained only on real data, so the physical model was not used to generate data or as the starting point of a hybrid model, and its prediction was not compared against the measurements either. The system controls only the position of the end effector, not its orientation.
How it fits in Robiolab
Continuum and flexible robots are the lab's most active line, and this project addresses its central problem: how to control a body whose shape cannot be written in a closed-form equation. It is the third stage of a sequence on the same robot. The platform solved actuation, the high-speed redesign changed the motors and electronics, and this work adds the model, camera sensing and learned control.