1 Introduction
1.1 Previous work
1.2 New contribution
1.3 Outline
2 Synthesis of minimum energy adaptive structures
2.1 Total energy minimization (TEO)
2.2 Loading conditions
2.3 Structural adaptation phases
2.4 Model assumptions
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Following (Skelton and Oliveira 2009), a general definition of tensegrity structures is adopted in this study. That is, a tensegrity structure is a pin-jointed system consisting of cables and struts which require appropriate prestress to maintain stable equilibrium under loading, i.e., cable elements do not slack. Only kinematically determinate tensegrity systems are considered in this work, i.e., which do not contain mechanisms.
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Elements are pin-jointed and all loads are transferred to nodes as point loads.
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The formulation is implemented with the assumption of small strains and small displacements. To prevent potential stability issues caused by finite mechanisms that could develop through cable slackness, cable elements are kept in tension under all loading events through appropriate prestress and control actions.
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The structure dynamic response is assumed to be controlled by other means; hence, seismic design criteria are not considered. In addition, fatigue is not considered as a limit state because the structure is designed to be controlled under strong loading events which occur rarely.
3 Prestress applied through actuation
4 Embodied energy minimization
4.1 Objective function
4.2 Equilibrium and compatibility constraints
4.3 Ultimate limit state (ULS) constraints
4.3.1 Element stress and buckling constraints
4.3.2 Fail-safe constraints
4.4 Serviceability limit state (SLS) constraints
4.5 Actuator control command constraints
4.6 Auxiliary constraints
4.6.1 Actuator embodied energy auxiliary constraints
4.6.2 Load actuation threshold (LAT) auxiliary constraints
4.7 Embodied energy minimization, full model formulation (MINLP)
\( {\displaystyle \begin{array}{c}\min \\ {}\mathrm{x}\end{array}} \) | \( {E}_{embd}=\sum \limits_{i=1}^{n^e}{\alpha}_i{L}_i{\rho}_i{e}_i^e+c\sum \limits_{i=1}^{n^e}{F}_i{n}_i{e}_i^a \) | Objective function | |
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s.t. | |||
\( {\displaystyle \begin{array}{c}\mathbf{A}\Delta {\mathbf{F}}^0=\mathbf{0}\\ {}\Delta {\mathbf{F}}^0=\overline{\mathbf{K}}{\mathbf{L}}^{-1}\left({\mathbf{A}}^{\mathrm{T}}\Delta {\mathbf{u}}^0-\Delta {\mathbf{L}}^0\right)\end{array}} \) | Prestress | ||
\( {\mathbf{Ku}}_{jL}^{P\_C}={\mathbf{P}}_{jl}^P \) | ∀j, ∀ l | Equilibrium constraints | |
\( {\mathbf{Ku}}_{jL}^{L\_C}={\mathbf{P}}_{jl}^L \) | ∀j, ∀ l | ||
\( {\mathbf{Ku}}_{jL}^P={\mathbf{P}}_{jl}^{permanent} \) | ∀j, ∀ l | ||
\( {\mathbf{Ku}}_{jL}^L={\mathbf{P}}_{jl}^{live} \) | ∀j, ∀ l | ||
\( {\mathbf{Ku}}_{LAT}^L={\mathbf{P}}_{LAT}^{live} \) | |||
\( {\mathbf{F}}_{jl}^{P\_C}=\overline{\mathbf{K}}{\mathbf{L}}^{-1}\left({\mathbf{A}}^{\mathrm{T}}{\mathbf{u}}_{jl}^{P\_C}\mathbf{\Delta }{\mathbf{L}}_{jl}^P\right) \) | ∀j, ∀ l | ||
\( {\mathbf{F}}_{jl}^{L\_C}=\overline{\mathbf{K}}{\mathbf{L}}^{-1}\left({\mathbf{A}}^{\mathrm{T}}{\mathbf{u}}_{jl}^{L\_C}\mathbf{\Delta }{\mathbf{L}}_{jl}^L\right) \) | ∀j, ∀ l | ||
\( {\mathbf{F}}_{jl}^P=\overline{\mathbf{K}}{\mathbf{L}}^{-1}{\mathbf{A}}^{\mathrm{T}}{\mathbf{u}}_{jl}^P \) | ∀j, ∀ l | ||
\( {\mathbf{F}}_{jl}^L=\overline{\mathbf{K}}{\mathbf{L}}^{-1}{\mathbf{A}}^{\mathrm{T}}{\mathbf{u}}_{jl}^L \) | ∀j, ∀ l | ||
\( {\underset{\_}{\sigma}}^c{\alpha}_i\le {F}_{ijl}^{\varPsi }{\le}_{\sigma}^{-c}{\alpha}_i \) | ∀i ∈ Scable, ∀ j, ∀ l, ∀Ψ∈{a, c, e} | Element stress and buckling constraints | |
\( {\underset{\_}{\sigma}}^s{\alpha}_i\le {F}_{ijl}^{\varPsi }{\le}_{\sigma}^{-s}{\alpha}_i \) | ∀i ∈ Sstrut, ∀ j, ∀ l, ∀Ψ∈{a, c, e} | ||
\( -{F}_i^b\le {F}_{ijl}^{\varPsi } \) | ∀i ∈ Sstrut, ∀ j, ∀ l, ∀Ψ∈{a, c, e} | ||
\( {\underset{\_}{\sigma}}^c{\alpha}_i\le {F}_{ijl}^{\varPsi }{\le}_{\sigma}^{-c}{\alpha}_i \) | ∀i ∈ Scable, ∀ j, ∀ l, ∀Ψ∈{b, d, f} | Fail-safe constraints | |
\( {\underset{\_}{\sigma}}^s{\alpha}_i\le {F}_{ijl}^{\varPsi }{\le}_{\sigma}^{-s}{\alpha}_i \) | ∀i ∈ Sstrut, ∀ j, ∀ l, ∀Ψ∈{b, d, f} | ||
\( -{F}_i^b\le {F}_{ijl}^{\varPsi } \) | ∀i ∈ Sstrut, ∀ j, ∀ l, ∀Ψ∈{b, d, f} | ||
\( -{F}_i\le {F}_{ijl}^{\varPsi}\le {F}_i \) | ∀i, ∀ j, ∀ l, ∀Ψ | Auxiliary constraints for actuator embodied energy | |
Fmin ≤ Fi ≤ Fmax | ∀i | ||
\( -{u}^{SLSO}\le {u}_{ijl}^{\varPsi}\le {u}^{SLSO} \) | ∀i ∈ Scdof, ∀ j, l = {SLS}, Ψ = c | Displacement constraints | |
\( -{u}^{SLS}\le {u}_{ijl}^{\varPsi}\le {u}^{SLS} \) | ∀i ∈ Scdof, ∀ j, l = {SLS}, Ψ = e | ||
\( -{u}^{SLS}\le {u}_{ijl,L\ AT}^L\le {u}^{SLS} \) | ∀i ∈ Scdof, ∀ j, l = {SLS} | ||
\( -\Delta {L}_{\mathrm{limit}}^0{n}_i\le \Delta {L}_i^{\varPsi}\le \Delta {L}_{\mathrm{limit}}^0{n}_i \) | ∀i, Ψ = a | Actuator layout constraints | |
\( -\Delta {L}_{\mathrm{limit}}^0{n}_i\le \Delta {L}_{ijl}^{\varPsi}\le \Delta {L}_{\mathrm{limit}}{n}_i \) | ∀i, ∀ j, ∀ l, ∀Ψ∈(c, e) | ||
\( \Delta {L}_{il}^P=\Delta {L}_{jl}^P,\mathrm{if}\ {\delta}_i={\delta}_j \) | l = {ULS | ||
\( \sum \limits_i^{n^e}{n}_i\le {n}^a \) | |||
ni ∈ {0, 1} | ∀i | ||
\( {\alpha}_{\mathrm{min}}^c\le {\alpha}_i\le {\alpha}_{\mathrm{max}}^c \) | ∀i ∈ Scable | Bounds for element cross-section areas | |
\( {\alpha}_{\mathrm{min}}^s\le {\alpha}_i\le {\alpha}_{\mathrm{max}}^s \) | ∀i ∈ Sstrut |
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Prestress state (ΔF0, Δu0, ΔL0) under no external load
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Noncontrolled state (FP, uP) under Ppermanent
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Controlled state (FP _ C, uP _ C, ΔLP) under Ppermanent
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Noncontrolled state (FL, uL) under Plive
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Controlled state (FL _ C, uL _ C, ΔLL) under Plive
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Displacement \( {\mathbf{u}}_{LAT}^L \) under load activation threshold \( {\mathbf{P}}_{LAT}^{live} \)
Continuous variable | |||||
V | α | \( \tilde{\mathbf{F}} \) | |||
N | ne | ne | |||
V | ΔF0 | Δu0 | ΔL0 | ||
N | ne | nf | ne | ||
V | FP | uP | FP _ C | uP _ C | ΔLP |
N | ne(np + 1) | nf(np + 1) | ne(np + 1) | nf(np + 1) | ne(np + 1) |
V | FL | uL | FL _ C | uL _ C | ΔLL |
N | 2nenp | 2nfnp | 2nenp | 2nfnp | 2nenp |
V | \( {\mathbf{u}}_{LAT}^L \) | ||||
N | 2nfnp | ||||
Binary variable | |||||
V | n | ||||
N | ne |
5 Operational energy minimization
5.1 Control through force and shape influence matrices
5.2 Objective function
5.3 Optimization constraints
5.4 Operational energy minimization, full model formulation (NLP)
Continuous variable | ||||
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V | ΔLL | WF(I) | WF(II) | WΔF |
N | nanpnk | nanpnk | nanpnk | nanpnk |
6 Total energy optimization (TEO)
7 Numerical examples
Cables | Struts | ||||
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Element type | ① | ② | ③ | ④ | ⑤ |
Self-stress | 1 | cosϕ | -sinϕ | -1 | -cosϕ |
7.1 Parameter settings
7.2 Utilization factors
7.3 Benchmark with passive tensegrity and equivalent adaptive truss solutions
7.4 Tensegrity roof configuration
7.4.1 Dimensions and boundary conditions
Load combination case | Load factor | Permanent load | Load factor | Live load |
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LC1 | 1.35 | Self-weight + dead load | 1.5 | – |
LC2 | 0.9 | Self-weight + dead load | 1.5 | L1 = 0.98 kN/m2 |
LC3 | 1.35 | Self-weight + dead load | 1.5 | L2 = 0.98 kN/m2 |
LC4 | 0.9 | Self-weight + dead load | 1.5 | L3 = L1 + L2 |
7.4.2 Adaptive vs passive tensegrity
LAT | Mass (kg) | Mass saving | Embodied energy (MJ) | Operational energy (MJ) | Energy saving | Actuation time (hours) | Computation time (seconds) | |
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ATS | 88% | 1.70 × 104 | 1.5% | 6.21 × 105 | 2.83 × 104 | −3.0% | 2.12 × 103 | 58 |
PTS | 100% | 1.74 × 104 | – | 6.36 × 105 | – | – | – | 0.21 |
7.4.3 Adaptive tensegrity vs equivalent truss system
LAT | Mass (kg) | Mass saving | Embodied energy (MJ) | Operational energy (MJ) | Energy saving | Actuation time (hours) | Computation time (seconds) | |
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AT | 90% | 1.20 × 104 | 7.1% | 4.29 × 105 | 1.11 × 104 | 4.7% | 1.74 × 103 | 5.53 |
PT | 100% | 1.29 × 104 | – | 4.72 × 105 | – | – | – | 0.20 |
7.5 Tensegrity tower configuration
7.5.1 Dimensions and boundary conditions
Load combination case | Load factor | Permanent load | Load factor | Live load |
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LC1 | 1.35 | self -weight + dead load | 1.5 | – |
LC2 | 1.35 | self -weight + dead load | 1.5 | L1 = 2.94 kN/m2 |
LC3 | 1.35 | self -weight + dead load | 1.5 | L2 = 2.94 kN/m2 |
7.5.2 Adaptive vs passive tensegrity
LAT | Mass (kg) | Mass saving | Embodied energy (MJ) | Operational energy (MJ) | Energy saving | Actuation time (hours) | Computation time (seconds) | |
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ATS | 88% | 8.36 × 104 | −16.2% | 3.05 × 106 | 1.70 × 105 | −22.6% | 3.19 × 103 | 5273.36 |
PTS | 100% | 7.20 × 104 | – | 2.63 × 106 | – | – | – | 0.22 |
7.5.3 Adaptive tensegrity vs equivalent truss system
LAT | Mass (kg) | Mass saving | Embodied energy (MJ) | Operational energy (MJ) | Energy saving | Actuation time (hours) | Computation time (seconds) | |
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AT | 70% | 4.01 × 104 | 29.2% | 1.46 × 106 | 2.39 × 105 | 17.6% | 9.85 × 103 | 2.73 |
PT | 100% | 5.66 × 104 | – | 2.07 × 106 | – | – | – | 0.20 |
7.6 On energy requirements of adaptive tensegrity structures
LAT | Mass (kg) | Mass saving | Embodied energy (MJ) | Operational energy (MJ) | Energy saving | Actuation time (h) | Computation time (s) | |
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ATS | 0% | 4.130 × 104 | 42.59% | 1.507 × 106 | 3.81 × 107 | −1408% | 7.22 × 104 | 978.70 |
20% | 4.133 × 104 | 42.55% | 1.508 × 106 | 3.80 × 107 | −1404% | 7.22 × 104 | 12,599.25 | |
50% | 5.661 × 104 | 21.24% | 2.066 × 106 | 4.90 × 106 | −186% | 2.21 × 104 | 1183.93 | |
88% | 8.357 × 104 | −16.17% | 3.050 × 106 | 1.70 × 105 | −22.6% | 3.19 × 103 | 5273.36 | |
PTS | 100% | 7.194 × 104 | – | 2.626 × 106 | – | – | – | 0.21 |
LAT | Mass (kg) | Mass saving | Embodied energy (MJ) | Operational energy (MJ) | Energy saving | Actuation time (h) | Computation time (s) | |
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AS | 0% | 1.492 × 104 | 73.65% | 0.544 × 106 | 1.02 × 107 | −419.13% | 7.22 × 104 | 702.12 |
20% | 1.494 × 104 | 73.60% | 0.545 × 106 | 9.38 × 106 | −380.60% | 6.58 × 104 | 879.96 | |
50% | 2.855 × 104 | 49.57% | 1.042 × 106 | 1.24 × 106 | −10.22% | 2.21 × 104 | 1.57 | |
70% | 4.007 × 104 | 29.22% | 1.463 × 106 | 2.39 × 105 | 17.64% | 9.85 × 103 | 2.73 | |
PS | 100% | 5.661 × 104 | – | 2.066 × 106 | – | – | – | 0.20 |
7.7 On solution quality
Embodied energy minimization (MINLP) | Solver | Objective function Eemb(MJ) | Computation time (s) |
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Tensegrity roof | BARON | – | – |
Knitro | 6.21 × 105 | 48.55 | |
Bonmin | 6.21 × 105 | 2947.49 | |
FilMINT | 7.49 × 105 | 63.61 | |
Tensegrity tower | BARON | – | – |
Knitro | 3.05 × 106 | 4953.72 | |
Bonmin | – | – | |
FilMINT | – | – |
Embodied energy minimization (MINLP) | Algorithm | Objective function Eemb(MJ) | Computation time (s) |
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Tensegrity roof | Knitro-BNB | 6.21 × 105 | 48.55 |
Knitro-HQG | 6.26 × 105 | 113.27 | |
Knitro-MISQP | – | – | |
Tensegrity tower | Knitro-BNB | 3.05 × 106 | 4953.72 |
Knitro-HQG | – | – | |
Knitro-MISQP | – | – |
Operational energy minimization (NLP) | Solver | Objective function Eop(MJ) | Computation time (s) |
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Tensegrity roof | Knitro-IPM | 2.83 × 104 | 0.038 |
Knitro-IPM (multi-start) | 2.83 × 104 | 8.25 | |
BARON | 2.83 × 104 | 0.37 | |
Tensegrity tower | Knitro-IPM | 1.70 × 105 | 0.095 |
Knitro-IPM (multi-start) | 1.70 × 105 | 29.13 | |
BARON | 1.70 × 105 | 35.50 |