The objective of this part of my PhD was to develop a four-link inverted pendulum model that behaved similarly to a human when exposed to shipboard motions.


Based on Kaneís method and the principle of virtual work, Eulerís equations of motion of a system can be written as:

for translational motion, and:

for rotational motion of rigid bodies. The index k is cycled over the number of bodies in the system, and the index i is cycled over the total number of generalized coordinate derivatives, . The terms wk and vkG are the angular velocity and the velocity of the centre of mass, respectively.

These equations can be reformulated as:


A contains the mass and inertia-related terms, and B and C contain the linear and nonlinear velocity-related terms. The right-side term f contains the external forces and moments acting on each body. Wk and Vk are partial velocity and partial angular velocity matrices.

In the formulation above, the matrices are all algorithmically generated based on the mass, inertia, geometry, and previous state of the system. Generation the equations of motion this way has the advantage that it is fairly simple to code and debug. The major downside is that it does not provide an algebraic representation of the equations for use in deriving a control system.


The model was validated by constraining its joints and comparing it to simpler published models. In all validation cases the results were very close.


The first model was called GRM3D, which consisted of a 3D rigid body, with no articulation, exposed to 6 DOF ship motion:

The results matched very well:

The second model it was compared to was called PSM3D, a single-segment, single-joint spatial inverted pendulum model with 6 DOF ship motion.

Again the results were very good.

To validate the motion of all four links the model was compared to a cart/inverted pendulum model with published equations for 4 links.

Here are the results: