Balancing Cube

Six autonomous modules balance a cubic structure

Have you ever seen a troupe of acrobats balancing in formation? How do they keep from falling over? Each member of the group balances their weight against the others, making many tiny adjustments each second.

 

The Balancing Cube can balance on any of its edges or corners. It owes this ability to six rotating mechanisms located on each inner face of the cube. When the mechanisms (simply called modules) are actuated, they exert forces on the cube and shift the center of gravity of the overall system. The modules communicate with each other and coordinate their motion to stabilize the system.

The modules respond by using sensory feedback: when a viewer stands the cube on one of its corners and lets it go – or even pushes it – the cube is able to recover and stabilize.

 

 

How does it work?

The Balancing Cube is an example of a distributed system. Each of the modules is a self-contained unit with onboard power, sensors, a computer, and a motor that can rotate the module relative to the rigid structure. Each module's computer solves an estimation problem: the local sensor data is fused with data communicated from the other units to derive an estimate for the orientation of the cube. Commands are sent to the module's motor based on this estimate, knowledge of other modules' states, and a control law. Even though each module makes its own computations, through the joint action of all modules the cube is balanced.

 

Research

State Estimation for the Cube

For balancing the cube it is necessary to know how the cube is oriented relative to gravity (the tilt) and how fast it is moving. These quantities are estimated from inertial sensors: On each face of the cube, there is an inertial measurement unit with a tri-axis accelerometer and tri-axis rate gyro. Each sensor package is connected to the computer on the rotating module through a slip ring. 

In [1], we present an algorithm for estimating the tilt of the cube based on measurements of multiple accelerometers. The method works for a general rigid body with only rotational degrees of freedom, such as the cube when it is standing on its corner. In particular, the tilt estimate is independent of the dynamics of the rigid body, that is, it is accurate no matter if the cube is static or if it is moving fast.

 

[1] S. Trimpe and R. D'Andrea, Accelerometer-based Tilt Estimation of a Rigid Body with only Rotational Degrees of Freedom, IEEE International Conference on Robotics and Automation, 2010, pp. 2630-2636.

 

Networked Control Systems

The Balancing Cube is an example of a networked control system. The rotating modules - each equipped with local actuation, sensing, and computation - share their data over a network to coordinate themselves and stabilize the cube. We use the Balancing Cube as a testbed to develop and test algorithms for control and estimation over networked systems.

In [2], [3], we present results for the problem of reduced communication state estimation in a networked control system, where the objective is to observe the full state of a dynamic system from multiple network agents while simultaneously seeking to reduced the communication between the agents.

 

[2] S. Trimpe and R. D'Andrea, Reduced Communication State Estimation for Control of an Unstable Networked Control System, 50th IEEE Conference on Decision and Control and European Control Conference, 2011, accepted for publication.

[3] S. Trimpe and R. D'Andrea, An Experimental Demonstration of a Distributed and Event-based State Estimation Algorithm, IFAC World Congress, Milano, Italy, 2011, pp. 8811–8818.

 

Other results: Multi-loop Control, Matrix Exponential

To stabilize the cube we implemented a cascaded control architecture consisting of a fast inner and a slow outer feedback loop. The inner loop controller on each module uses local encoder data to control the module's angular velocity. This controller receives velocity commands from an outer loop controller that has global state information and stabilizes the system. Through this architecture non-linear effects in the drive train like friction or backlash are mitigated from the perspective of the outer controller.

In order to design the stabilizing outer loop controller, one needs a model of the cube including all the inner velocity loops. In [4], we present a systematic approach for obtaining a simplified description of a multi-loop control system at the slow time scale relevant for the outer control loop. The result is based on a limiting property of the matrix exponential, which is proven in the paper.

 

[4] S. Trimpe and R. D'Andrea, A Limiting Property of the Matrix Exponential with Application to Multi-loop Control, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 2009, pp. 6419-6425.

 

 

Student Projects

A list of current student projects can be found on the IDSC Theses and Projects Page and on SiROP.

Many students have been helping to make the Balancing Cube happen.  Below is a list of past projects and participants:

!and yet it moves class 2007/2008

The Balancing Cube project was initiated in the 2007/2008 project class !and yet it moves. The system design was carried out, all components were selected and tested, and a prototype was built during the class.

• Holger Brandsmeier, Kim Chung, Nicklas Hoch, Aurelien Jordan, Johannes Paefgen, Stephan Pleines, Marco Senteler, Fabio Wider, Daniel Wolfertshofer, and Aldo Zgraggen.

Studies on Mechatronics

• Lukas Wunderli, Attitude Estimation for the Balancing Cube, 2009.
• Andre Widmer, Networked and Distributed Control Systems - Physical control systems similar to the Balancing Cube, 2011/2012.
  

Semester Project

• Valentin Baumann, Modeling and Simulation of the Balancing Cube, 2009.
• Lukas Wunderli, Determining the Mode of the Balancing Cube Based on Local Sensor Information, 2009/2010.
• Andreas Köberl, Backlash Modeling for the Balancing Cube, 2010.
• Niklaus Voellmy, User Interface for the Balancing Cube, 2010/2011.
  

Master Thesis

• Dursun Akay, Alternative Modeling Tools and System Identification for the Balancing Cube, 2009.
• Marc Spirig, A Learning Strategy for Swinging Motions of an Inverted Pendulum, 2010/2011.
• Simon Dössegger, Improving the Balancing of the Cube - Adaptation and learning to make the cube balance more steadily, current.

Exchange Student

• Gajamohan Mohanarajah (Tokyo Institute of Technology), Single board computer operating system, 2008.
• Korbinian Nottensteiner (TU München), Moving the Balancing Cube - A gain scheduling approach, 2010/2011.
  

Hilfsassistenz

• Lukas Wunderli, Software Development, 2010.