Partial Covering of a Circle by 6 and 7 Congruent Circles

Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for n = 2 to 5. In the present paper, numerical solutions are given to this problem for n = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of n = 6 is very complicated, whilst the case n = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.

Self-deployable tensegrity structures for adaptive morphing of helium-filled aerostats

In this paper, the authors propose, investigate, and discuss a concept of novel type of deployable helium-filled aerostat as a low-cost mean of transport. Internal construction of the aerostat is based on ultra-light tensegrity structure equipped with prestressed tensioned elements of controllable lengths. Such tensegrity structure allows for adaptive morphing of the aerostat understood as simultaneous controllable modifications of aerostat volume and shape during the flight. The controlled volume changes enable influencing buoyancy force and obtaining desired vertical motion during the ascending and descending process. In turn, external shape changes allow for lowering the aerodynamic drag and energy usage needed to uphold stable horizontal position or maintain the desired flight path. Moreover, such internal structure allows for convenient storage, transportation and deployment of the aerostat construction on the ground or in required point at the atmosphere. The article presents an analysis of the exemplary operational mission of the aerostat. The authors introduce the mechanical model capturing interaction of the internal tensegrity structure and aerostat envelope based on the finite-element method, as well as dynamic model allowing for simulation of the aerostat’s vertical and horizontal motion influenced by buoyancy and drag forces. Both these models are used to positively verify the feasibility of the proposed concept of deployable tensegrity-based aerostat with adaptive morphing and its efficiency in realization of the assumed flight mission.

Orientation Control of Self-Righting Tensegrity Landers

Squishy Robotics, Inc. has developed a spherical sensor robot that can be rapidly deployed by air drops of up to 1,000 ft for emergency response situations to improve situational awareness for first responders. Although the tensegrity structure has successfully been shown to survive the drop, some payloads require orientation when they land. For example, a payload that contains sensors and communication equipment to relay the data may need the robot to be oriented such that the antennas are pointing upward, or some sensors are positioned in a specific plane for operation. This requirement presents a challenge for a tensegrity-based delivery system because the structure absorbs energy using passive compliance and bounces several times upon landing. Although active systems using motors and actuators could be used to control orientation after landing, they increase the overall weight and complexity of the system. This paper describes the research on a passive control solution that achieves the correct orientation by placing weights on selected rods forming an asymmetrically weighted tensegrity structure that preferentially rolls and orients itself during the impact process. The design approach is applied to three robot sizes and the self-righting behavior is validated through experimental results.

Robotic Tensegrity Structure with a Mechanism Mimicking Human Shoulder Motion

This paper proposes a three degrees-of-freedom tensegrity structure with a mechanism inspired by the ligamentous structure of the shoulder. The proposed mechanism simulates the wide motion ranges of the human shoulder joint and is composed of three rigid bodies and sixteen steel wires with three mutually perpendicular rotating axes. Since it belongs to the class 1 tensegrity structure that the rigid bodies do not make any contact with each other, the joint has a certain amount of flexibility, which not only can help protect its mechanism from external impacts but also can prevent human injury that might happen when the mechanism and humans interact each other. Moreover, the proposed mechanism can be manufactured by using fewer materials than a fully rigid mechanism, and thus, it can be made in a lightweight fashion and reduce the inertial effects as well. Finally, to actuate the robotic shoulder, the cables connected to each motor are able to drive the rotating shafts of the joint mechanism.