3-Axis 7-DOF Stability and PID Design

Spring-damper system extended to 28-state space in MATLAB; instability found without control and fixed with PID.

Period 2021.02 – 2021.08 · Affiliation Keimyung University (undergraduate researcher, Robot Dynamics and Intelligent Control Lab) · Role Mathematical modeling, state-space design, MATLAB stability simulation, PID controller design

Background and Goals

The ultimate goal was a self-stabilizing (disturbance rejection) system in which the robot body returns to equilibrium after an external force and is ready for the next one. Starting from a simple 1-axis model, I extended it progressively to a 3-axis system and quantified how the added complexity affects stability.

Approach

Equations of motion and modeling of the 1-axis 3-DOF system (left) and the MATLAB step response (right).
  1. 1-axis 3-DOF modeling: Derived the equations of motion for masses m₁, m₂, and m₃ connected in series by springs (k) and dampers (c).
  2. First stability check: Tracked the position change of the central mass m₂ when a 455 N external force was applied to m₁. Confirmed that even without a controller the step response amplitude decayed and stability was maintained.
  3. Extension to 3 axes and 7 DOF: Extended the structure so that six masses (m₁–m₆) are arranged radially around a central mass (m₇).
  4. State-space integration: Converted the equations of motion of each axis to \(\dot{x} = Ax + Bu,\; y = Cx + Du\) and combined the two 14×14 systems block-diagonally into a 28×28 combined matrix, analyzing them simultaneously in a single environment.
  5. PID design: Confirmed that after the 3-axis extension stability was no longer guaranteed without control, then designed a PID controller for each axis and derived the gains through simulation.
Concept diagram of the 3-axis 7-DOF system (left), modeling (center), and the state variable matrix of the state equation (right).

Results

Discovered through simulation that the stability present in the 1-axis case is no longer guaranteed without control once the system is extended to 3 axes and 7 DOF, and worked through the full engineering cycle of “problem identification → analysis → PID solution.” Established a systematic methodology for determining gains via simulation.

Tech Stack

MATLAB (Control System Toolbox) · Equations of motion / state-space representation · Spring-damper vibration systems · PID control · Linear algebra