Deployment of a tensegrity footbridge

Deployable structures are structures that transform their shape from a compact state to an extended in-service position. Structures composed of tension elements that surround compression elements in equilibrium are called tensegrity structures. Tensegrities are good candidates for deployable structures since shape transformations occur by changing lengths of elements at low energy costs. Although the tensegrity concept was first introduced in 1948, few full-scale tensegrity-based structures have been built. Previous work has demonstrated that a tensegrity-ring topology is potentially a viable system for a deployable footbridge. This paper describes a study of a near-full-scale deployable tensegrity footbridge. The study has been carried out both numerically and experimentally. The deployment of two modules (one half of the footbridge) is achieved through changing the length of five active cables. Deployment is aided by energy stored in low stiffness spring elements. Self-weight significantly influences deployment, and deployment is not reproducible using the same sequence of cable-length changes. Active control is thus required for accurate positioning of front nodes in order to complete deployment through joining both sides at center span. Additionally, testing and numerical analyses have revealed that the deployment behavior of the structure is non-linear with respect to cable-length changes. Finally, modelling the behavior of the structure cannot be done accurately using friction-free and dimensionless joints. Similar deployable tensegrity structures of class two and higher are expected to require simulation models that include joint dimensions for accurate prediction of nodal positions.  Tensegrity structures are structures composed of tension elements (strings, tendons or cables) surrounding compression elements (bars or struts) in equilibrium (Motro et al. 2003). Various definitions of tensegrity structures exist; a concise definition has been proposed by Skelton et al. (2001) who defined a class K tensegrity structure as a stable equilibrium of axially loaded elements, with a maximum of K compressive members at the nodes. Veuve, N., Dalil Safaei, S., and Smith, I.F.C. (2015). Deployment of a Tensegrity Footbridge. Journal of Structural Engineering, 141(11), 04015021 2 Tensegrity structures are identified by the number of infinitesimal mechanisms and states of selfstress (Pellegrino et al. 1986). Pellegrino et al. (1986) introduced a method for computing the number of infinitesimal mechanisms and self-stress states by singular value decomposition (SVD) of the equilibrium matrix. The design process of tensegrity structures includes simultaneous identification of geometry, topology, element axial stiffness, actuator position, and self-stressed state. Tibert and Pellegrino (2003) reviewed form-finding methods. Several studies have included the derivation of tangent stiffness matrix for a pre-stressed framework (Argyris and Scharpf 1972, Murakami 2001 and Guest 2006). Skelton et al. (2014) proposed a topology optimization method for minimum mass design of tensegrity bridges and showed that the optimized design has a multi-scale character. Tensegrities are attractive for deployment and active control since their shape can be changed through changing lengths of elements. In addition to carrying loading, the elements in tensegrity structures can be actuators and sensors (Skelton et al. 2001). In several studies, e.g. (Furuya. 1992,  2002, Furuya et al. 2006, Dalil Safaei et al. 2013), tensegrities have been proposed as deployable booms for space missions. Zolesi et al. (2012) successfully demonstrated deployment repeatability of a large tensegrity reflector antenna through a 1/4 scale prototype equipped with a gravity compensation system. Tensegrities are good candidate for active structures since they can change their shape under changing environments, such as new loading situations. Fest et al. (2004) employed telescopic struts to investigate the active control behavior of a five-module tensegrity structure. In several studies, biomimetic properties of active tensegrity structures have been studied (Adam and Smith 2008; Domer 2003; Domer and Smith 2005). These structures were not deployable. Although studies have been carried out on deployment of tensegrity structures, very few near full-scale deployable tensegrity structures have been built. Physical models are important for exploration, validation and development of ideas. In addition, a physical model should be large enough to be able to study important full-scale challenges. The influence of joint characteristics on deployment of tensegrity structures has not been studied. This is particularly important for structures of class 2 and higher since a pre-defined sequence of length changes of active cables may not result in a successful deployment due to joint configuration and friction effects. In addition, no study has been performed to investigate the influence of self-weight on deployment. This paper studies the deployment of a clustered-cable-spring tensegrity configuration that was adapted from a design proposal by Motro et.al (2006). Clustered cables are continuous cables that slide continuously over joints located at the ends of struts. Design and analysis of tensegrity footbridges have been studied by several researchers. Tensegrity-ring modules were introduced by Pugh (1976). These modules were studied for deployment and named ring modules by Motro et.al (2006) because of their hollow interior space. Nguyen (2011) and Cevaer et al. (2012) Veuve, N., Dalil Safaei, S., and Smith, I.F.C. (2015). Deployment of a Tensegrity Footbridge. Journal of Structural Engineering, 141(11), 04015021 3 studied the structural behavior of a single pentagonal ring module under the assumption that there was no cable continuity. A near full-scale tensegrity footbridge has been built to study strategies for deployment. More specifically, objectives of this paper are as follows:  Describe an experimental study of a clustered-cable-spring configuration for deployment of a near-full-scale tensegrity footbridge.  Demonstrate the need for active control for reproducible deployment behavior.  Evaluate the assumption of dimensionless joints for prediction of deployment behavior.  Study the influence of self-weight on deployment. The next sections provide details, results and evaluations related to these goals. The tensegrity ring footbridge The footbridge is composed of four identical pentagonal ring modules connected together in a “hollow-rope” system (Rhode-Barbarigos et al. 2010, Bel Hadj Ali et al 2010, and Motro et.al 2006). Each pentagonal ring module in Figure 1 has 15 struts and 30 cables. Tensegrity structures are characterized by the number of self-stress states and infinitesimal mechanisms. Analysis of the equilibrium matrix reveals no infinitesimal mechanisms and six independent states of selfstress. Each tensegrity ring module is a class two tensegrity structure (Skelton et al. 2001). The empty space in the tensegrity ring module is employed for footbridge pedestrian traffic.  Deployment illustration of four-module tensegrity footbridge. This paper studies the deployment of one half of a 1⁄4 scale model of the footbridge (adapted from Rhode-Barbarigos et al. (2010) and Rhode-Barbarigos et al. (2012b)) Rhode-Barbarigos et al. (2010) and Bel Hadj Ali et al. (2010) studied the serviceability of a 16 m span and 6.2 m diameter tensegrity footbridge without considering clustered cables and Veuve, N., Dalil Safaei, S., and Smith, I.F.C. (2015). Deployment of a Tensegrity Footbridge. Journal of Structural Engineering, 141(11), 04015021 4 deployability. Rhode-Barbarigos et al. (2012b) then studied several deployment methods and showed that utilizing springs, inspired from (Schenk et al. 2007), and clustered-cables reduces the number of actuators required for controlled deployment. Bel Hadj Ali et al. (2011) proposed an analysis method for clustered tensegrity structures through a modified dynamic relaxation algorithm. Clustered cable elements run continuously through nodes. They influence the mechanics of tensegrity structures through reducing number of kinematic constraints and changing the internal-force distribution of the elements (Moored and Bart-Smith 2009). A 1/4 scale model has been designed, manufactured and assembled in order to study deployment behavior . Each half is composed of 15 springs, 5 continuous cables, 30 struts and 20 cables. Also, each half is controlled by five continuous active cables. Each active continuous cable starts from a node connected to the support and ends at the front nodes. Figure 2 shows continuous actuated cables (active cables) and spring elements. Active cables are connected to the front nodes of the half bridge . The length of each active cable is changed through winding or unwinding the cable on a drum that is fixed on moving support . An actuation step or a control command is the length changes of the five active cables applied during one step of deployment.  (a) Side view of the near-full scale tensegrity footbridge Veuve, N., Dalil Safaei, S., and Smith, I.F.C. (2015). Deployment of a Tensegrity Footbridge. Journal of Structural Engineering, 141(11), 04015021 5  (b) front view with node numbering of the near-full scale tensegrity footbridge The structural weight of each half is approximately 100 kg. Both ends move in rail-support systems . Struts are made of steel hollow tube section profiles with a length of 1.35 m, a diameter of 28 mm and a thickness of 1.5 mm. The steel grade of the struts is S355 with a modulus of elasticity of 210 GPa. Cables have a diameter of 4 mm and are made of stainless steel with a modulus of elasticity of 120 GPa. The value of spring stiffnesses at the support is 2 kN/m and is 2.9 kN/m for other springs. The foot