A non-linear motion base math model refers to the mathematical representation of a motion platform which is typically a Stewart platform or similar 6DOF system—where the relationship between actuator lengths and platform position/orientation is non-linear. This means that small changes in actuator length do not result in proportional changes in platform position or rotation, and the system’s behavior cannot be described using simple linear equations.
Let’s break it down:
🔧 What Makes It Non-Linear?
In a Stewart platform (or any crank/pushrod-based system), the actuators are arranged in a non-parallel, non-uniform geometry. As the platform moves:
- The angles between actuators and the base/top platform change
- The lever arms vary dynamically
- The center of rotation may shift depending on the motion
These factors introduce non-linearities in the kinematic equations, meaning the system’s response is not a straight-line function of the input commands.
🧮 Mathematical Complexity
There are two main mathematical problems involved:
- Forward Kinematics
Given the lengths of the actuators, calculate the platform’s position and orientation (X, Y, Z, roll, pitch, yaw).- This is non-trivial and often solved using iterative numerical methods because there’s no closed-form solution for general configurations.
- Inverse Kinematics
Given a desired platform position and orientation, calculate the required actuator lengths.- This is more commonly used in control systems and can be solved with custom algorithms that account for the platform’s geometry and constraints.
🧠 Why It’s Challenging
- Coupled motion: Moving in one axis (e.g., pitch) may unintentionally affect others (e.g., heave or roll) unless carefully controlled.
- Changing geometry: The effective lever arms and force vectors change throughout the motion envelope.
- Dynamic loads: The system must account for inertia, gravity, and external forces, which vary with position and velocity.
- Real-time control: The math model must run fast enough (often 1,000 Hz or more) to provide smooth, accurate motion.
🧩 How Servos & Simulation Solves It
Servos & Simulation engineers developed a custom software suite that:
- Models the full non-linear geometry of the motion base
- Accepts 6DOF commands (X, Y, Z, roll, pitch, yaw)
- Computes actuator commands in real time
- Allows dynamic repositioning of the coordinate system’s center (e.g., offsetting the center of rotation for pilot realism or antenna testing)
- Includes a Dynamic Evaluator to track actual position using feedback sensors
This approach ensures that even with a non-linear mechanical system, the platform behaves as if it were linear from the user’s perspective—moving cleanly in each axis without cross-axis interference.