Animation

animation of a gyroscope

Working for the European Aeronautic Defence and Space Company EADS Astrium at Immenstaad am Bodensee/Germany and Toulouse/France in 2009, I adapted existing procedures for testing satellite communication and hardware in the Assembly-Integration-Test phase before launch. The projects supported by the team were LISA Pathfinder, Earthcare, Sentinel-2, BepiColombo and Galileo. Each of these satellites introduced new requirements to the telecommand and telemetry database system and new challenges to the real-time software that monitors the hardware.

At the same time, I investigated exact solutions to mechanical problems that arise for articulated structures in space, where gravity and friction are low.

Rigid Body Collision Resolution

Elastischer Stoß

Animation of Skeletons with Hinges and Spherical Joints

Was er für Geld da schreibt, ist gut geschrieben.
Was er verschweigt, die Wahrheit wär's gewesen.
Eugen Berthold Friedrich Brecht

Animation of Skeletons with Hinges and Spherical Joints

skeleton with spherical joint and hinge

During a visit at the Agenţia Spaţială Română in Bucharest/Romania, Alexandru Pandele explained to me the attitude control of the cube satellite that is being built in 2009. The satellite is equipped with two flywheels. The bearing of a wheel is a special hinge.

He motivated me to derive an algorithm to animate a skeleton of rigid bodies that are linked by hinges and spherical joints. Over the course of the simulation, the total linear momentum, and the total angular momentum are invariant. If desired, the algorithm incorporates intrinsic torques of the joints such as friction, and motor control. Otherwise, the total kinetic energy is invariant, too.

The input to the problem are:

• 0th and 1st order initial dynamic states of the objects: position, velocity, orientation, and angular velocity
• mass and inertia tensor of the objects
• location and orientation of the hinges and spherical joints
• friction and motor control at the joints
• connectivity between the objects and joints

The output are:

• linear and angular accelerations of each object

Our derivation shows that these accelerations are determined by linear equations. The article starts by considering skeletons with hinges only. However, the introduction of spherical joints turns out to be simple at a later point: the vector that formerly represented the axis of a hinge is just set to zero.

Timpul învață pe cei fără şcoală.

Animation of Skeleton with Hinges and Spherical Joints	skeleton_anim...pdf	410 kB
Documentation as a Mathematica 5.2 notebook	skeleton_anim...zip	1.1 MB
Demo with source code (C++) *	skeleton_demo.zip	290 kB

* The program checks the math, and produces the animations on this website. Executable compiled with MS C++ Compiler version 14/15. The implementation of the singular value decomposition is from 'Numerical Recipes in C++, 3rd edition' by William Press, Saul Teukolsky, William Vetterling, and Brian Flannery.

There is something about a closet
that makes a skeleton terribly restless.
Wilson Mizner

To verify our derivation, we implement an open source simulation of the dynamics. Our algorithm uses double precision and simple Euler integration with small timesteps. The simulation suffers from numerical error in the integration: After a while, links separate, and the total angular momentum is not invariant.

In the closet

hinge with motor spherical joint with motor oscillating wheel configuration hinge and spherical joint titanic fantasy satellite swing sky and dawn and sun lift coupled pendulum wobble mickey whirl star slide seven sisters la croix occitane ... or load all animations on a separate page

please select an animation

Ştiu că sunt prost.
Dar când mă uit în jur,
prind curaj.
Ion Creangă

Later, Thomas Neukirchner suggested two applications of the simulation software: the modelling of a Foucault pendulum, and a falling cat. Great fun!

We have placed Foucault pendulums at three different lattitudes: At the pole the effect is largest. At the equator, the effect does not exist.

Falling cat

The sequence of the configurations of the extremities is inspired by the cat righting reflex of Wikipedia. Cute cats fall on YouTube - no cats harmed.

The control of the hinges in the 'Falling cat' animation is described in Section 4.4 of my work on the Marsokhod robot.

The configuration of the spherical robot is taken from the design by Masaki Nagai described in his Thesis "Control System for a Spherical Robot". The robot is sometimes referred to 'Sushi' or 'Gimball'.

The mass ratios of sphere, bar, and lever are 1:1/2:3. The inertia tensors assume equal mass distribution throughout the volumes. The floor has a mass close to infinity and the inertia tensor of a sphere. The red and green cones visualize the non-static contact joint between floor and robot. The linear and angular moments are constant = 0.

For shadow rendering, consult the excellent article "Practical and Robust Stenciled Shadow Volumes for Hardware-Accelerated Rendering" by Cass Everitt and Mark J. Kilgard.

Lenea e la om ca rugina la fier.