feasibility study for a colossus tensegrity...

Feasibility Study for a Colossus

Tensegrity Design

[email protected]

mailto:[email protected]

Outline

• What?– What assumptions?

– What results?

• How?– How to compute tensegrity equilibria?

– How to compute minimal mass?

– How to compute optimal complexity?

• Why?– Why are tensegrity models reliable?

– Why do these structures yield minimal mass?

– Why are tensegrity structures more controllable with less power?

• Next?– Optimal tensegrity telescope (a new topology)

– Information architecture (What to measure? What cables to control?)

– Feedback control to maintain shape

Assumptions:

• Design of Parabolic Dish only (other structures later)

• Existing 60 mirrors/hardware to be mounted on new dish

• Gravity forces only

– in two positions (0, 50 deg azimuths)

– F(gravity)=f(new dish)+f(secondary)+f(60 mirrors/instruments)+5%margin

• Weight of all mirrors/instruments same as given,

– but attached to a new dish structure

– total weight distributed evenly among all nodes of new dish

• Design criteria: minimal mass (at both 0, 50) subject to constraints:

– yield and buckling constraints

– All nodes on the front side form a parabolic surface in both positions

– Mass of our design = max{mass at 0, mass at 50}

• Axially-loaded members:

– Hookean material (pipes and cables of same steel)

• Tensegrity topology not optimized, results are for an existing topology

Parabolic Structure

(1200 tons)

Instruments

(39 tons)

Primary & Secondary Mirror

(3584=3540+44 tons)

Assumptions: External Force= F(gravity)

Mirrors=60 x 59 tons = 3540 tons

Secondary mirror: 44 tons

Instrumentation: 39 tons

Existing design

F = [ m]g

= [(3584+weight of new dish)tons+5% margin]g

F/pq=(Force at each

of the pq nodes)

New design (of complexity= pq units)

Parabolic Structure

Instruments

(39 tons)

)(4

1 22 yxa

Lz d

[m]2.154

2

a

RLd

[m]2.234

22

L

rRa

[deg]50,0

)tons3584(]N[106.33 6F

]Kg/m[7862 3 bs

Material Properties

Specifications

]/m[1006.2 211 NEE bs

]/m[109.6 28 Nbs (tensegrity systems, pp.104-105, 2009)

]m[752 RD]m[5.72 rd

]m[15L

Assumptions: Dish Dimensions

Assumptions (Use an existing tensegrity unit, DHT)

3 variations below (blue bars, red cables)

Primal DHT unit Dual DHT unit Modified Dual DHT

Unit (best of the 3)

p unitsq units

We have NOT optimized the tensegrity geometry

We chose an existing geometry DHT (3 variations below)

mass of Hollow Pipe m(P), mass of solid rod m(B)

Assumptions: pipes compared to solid bars

0rir)(

4

44

02

3

i

i

bi rr

b

Ef

)( 22

0 iibi rrbm

)1(0 aarri

wall thin:1a

2

2

1

1)()(

a

aBmPm

ibi

440

1

1)()(

aBrPr i

44

4

1)()(

a

aBrPr ii

10/9a

)(32.0)( BmPm ii

)(31.1)(0 BrPr i

)(18.1)( BrPr ii

Outline


– What results?










What Results? Mod. DHT (p,q)=(5,3)

blue bars (pipes),

red cables

What Results?Mod. DHT (p,q)=(5,3)

0 50

Results: Total Mass with Hollow Pipe

( ) (3584 653) tons 5%margin 39.6e6NF gravity

Mod. DHT: (For Comparison)

)10,10(),( qp( ) 2.7090 5 m P e kg ( ) 6.6776 5m P e kg

09.0 rri

Mod. DHT: (Our Selected Design)

)3,5(),( qptons653( ) 2.4180 5 m P e kg ( ) 5.9362 5 m P e kg

5

0 11curre 10nt kgde ( 1200tons)sign: m

(Total units pq=100)

(Total units pq=15)

compare (p,q) = (5,3)

with (10,10)

0 50

Repeat with a different complexity (p,q):

ibL

isL

50,04.0283e+001 4.0283e+001 4.0283e+001 4.0283e+001 4.0283e+001

8.4443e+000 8.4443e+000 8.4443e+000 8.4443e+000 8.4443e+000

3.6084e+001 3.6084e+001 3.6084e+001 3.6084e+001 3.6084e+001

9.1124e+000 9.1124e+000 9.1124e+000 9.1124e+000 9.1124e+000

4.4084e+001 4.4084e+001 4.4084e+001 4.4084e+001 4.4084e+001

-----------------------------------------------------------------------------------------------

3.1327e+001 3.1327e+001 3.1327e+001 3.1327e+001 3.1327e+001

1.0146e+001 1.0146e+001 1.0146e+001 1.0146e+001 1.0146e+001

2.5705e+001 2.5705e+001 2.5705e+001 2.5705e+001 2.5705e+001

1.2048e+001 1.2048e+001 1.2048e+001 1.2048e+001 1.2048e+001

-------------------------------------------------------------------------------------------------

1.8442e+001 1.8442e+001 1.8442e+001 1.8442e+001 1.8442e+001

1.8793e+001 1.8793e+001 1.8793e+001 1.8793e+001 1.8793e+001

4.4084e+000 4.4084e+000 4.4084e+000 4.4084e+000 4.4084e+000

2.2528e+001 2.2528e+001 2.2528e+001 2.2528e+001 2.2528e+001

2.0513e+001 2.0513e+001 2.0513e+001 2.0513e+001 2.0513e+001

2.0513e+001 2.0513e+001 2.0513e+001 2.0513e+001 2.0513e+001

4.0768e+001 4.0768e+001 4.0768e+001 4.0768e+001 4.0768e+001

3.6674e+001 3.6674e+001 3.6674e+001 3.6674e+001 3.6674e+001

2.2528e+001 2.2528e+001 2.2528e+001 2.2528e+001 2.2528e+001

--------------------------------------------------------------------------------------------

1.8305e+001 1.8305e+001 1.8305e+001 1.8305e+001 1.8305e+001

1.5865e+001 1.5865e+001 1.5865e+001 1.5865e+001 1.5865e+001

1.5865e+001 1.5865e+001 1.5865e+001 1.5865e+001 1.5865e+001

1.8305e+001 1.8305e+001 1.8305e+001 1.8305e+001 1.8305e+001

3.2101e+001 3.2101e+001 3.2101e+001 3.2101e+001 3.2101e+001

2.6886e+001 2.6886e+001 2.6886e+001 2.6886e+001 2.6886e+001

--------------------------------------------------------------------------------------------

1.3246e+001 1.3246e+001 1.3246e+001 1.3246e+001 1.3246e+001

1.3085e+001 1.3085e+001 1.3085e+001 1.3085e+001 1.3085e+001

1.3246e+001 1.3246e+001 1.3246e+001 1.3246e+001 1.3246e+001

2.1599e+001 2.1599e+001 2.1599e+001 2.1599e+001 2.1599e+001

1.3085e+001 1.3085e+001 1.3085e+001 1.3085e+001 1.3085e+001

Results: Lengths of pipes Lb and cables Ls (p,q=5,3)

p (longitude)

q (latitude)

0 504.3438e-010 3.9899e-010 4.1854e-010 4.1074e-010 4.0779e-010

6.4291e+006 6.4291e+006 6.4291e+006 6.4291e+006 6.4291e+006

5.4081e-010 5.5609e-010 5.3847e-010 5.4176e-010 5.4748e-010

4.5988e+006 4.5988e+006 4.5988e+006 4.5988e+006 4.5988e+006

3.1063e-010 3.1817e-010 3.1616e-010 3.1833e-010 3.1953e-010

----------------------------------------------------------------------------------------------

1.0411e-008 3.1273e-008 8.1268e-008 8.3371e-008 1.3439e-008

1.2640e+007 1.2640e+007 1.2640e+007 1.2640e+007 1.2640e+007

3.2790e-010 3.2903e-010 3.2991e-010 3.3273e-010 3.2922e-010

2.2909e+007 2.2909e+007 2.2909e+007 2.2909e+007 2.2909e+007

-----------------------------------------------------------------------------------------------

9.4527e+006 9.4527e+006 9.4527e+006 9.4527e+006 9.4527e+006

5.9537e+007 5.9537e+007 5.9537e+007 5.9537e+007 5.9537e+007

0 0 0 0 0

if

2.4197e+006 2.4197e+006 2.4197e+006 2.4197e+006 2.4197e+006

4.1364e+006 4.1364e+006 4.1364e+006 4.1364e+006 4.1364e+006

1.4840e+006 1.4840e+006 1.4840e+006 1.4840e+006 1.4840e+006

2.9771e+006 2.9771e+006 2.9771e+006 2.9771e+006 2.9771e+006

2.4163e-009 3.6038e-009 3.1151e-009 3.4290e-009 4.0895e-009

4.8041e+006 4.8041e+006 4.8041e+006 4.8041e+006 4.8041e+006

--------------------------------------------------------------------------------------------

6.9671e-010 7.3032e-010 7.4477e-010 7.1481e-010 7.1429e-010

1.6317e+007 1.6317e+007 1.6317e+007 1.6317e+007 1.6317e+007

1.3359e+007 1.3359e+007 1.3359e+007 1.3359e+007 1.3359e+007

2.4313e+006 2.4313e+006 2.4313e+006 2.4313e+006 2.4313e+006

3.2116e+006 3.2116e+006 3.2116e+006 3.2116e+006 3.2116e+006

6.5213e-010 6.7560e-010 6.7598e-010 6.6416e-010 6.7021e-010

---------------------------------------------------------------------------------------------

3.2181e+007 3.2181e+007 3.2181e+007 3.2181e+007 3.2181e+007

2.6866e-010 3.7886e-010 4.4696e-010 3.8045e-010 2.6850e-010

3.2181e+007 3.2181e+007 3.2181e+007 3.2181e+007 3.2181e+007

2.6870e-010 5.4422e-010 5.4302e-010 2.6179e-010 9.0692e-011

2.3609e-009 1.6145e-009 1.1908e-009 1.6233e-009 2.3584e-009

it

4.3957e+006 2.2491e+006 3.8286e-007 2.7504e+006 3.0415e+005

5.1826e+006 5.6694e+006 4.1413e+006 3.1217e+006 3.4973e+006

9.1838e-007 1.6197e-006 1.9712e+006 6.4620e+006 5.2565e-006

3.3818e+006 3.5960e+006 2.5031e+006 4.0425e+006 3.3977e+006

2.0282e-007 4.9274e-007 1.1671e-006 9.5092e-007 9.1314e-007

-----------------------------------------------------------------------------------------------

2.2320e+006 1.8760e+006 8.1396e-007 2.4841e-006 6.2040e+006

1.0266e+007 1.4985e+007 1.1899e+007 4.0088e+006 8.4078e+006

5.4698e-007 6.8883e-007 7.0909e+006 1.9242e+007 8.9800e-007

1.2812e+007 1.8966e+007 8.8316e+006 1.0440e+007 7.9529e+006

-----------------------------------------------------------------------------------------------

3.0700e+008 6.8074e-006 9.9643e+006 2.2805e-006 2.5715e+007

3.7888e+008 1.8757e+008 1.9995e+008 1.3517e+007 3.1441e+007

0 0 0 0 0

5.2642e+006 7.8287e+006 2.7708e+006 1.0318e+006 1.1172e-006

7.0021e+006 7.0417e-007 3.8201e-007 4.5678e+006 3.2815e+006

9.1986e-007 4.0189e+006 6.2861e+006 7.1020e+006 1.1032e-006

1.4937e+003 5.6999e-006 1.8496e+006 1.4699e-006 8.0730e+005

2.2616e+006 4.3535e-007 9.6190e-007 1.1881e-006 4.0056e+006

7.0954e+006 7.1297e+006 1.9902e+006 3.3282e+005 2.4766e+006

----------------------------------------------------------------------------------------------

9.6540e-007 1.0367e+007 5.2535e+006 2.1376e-006 4.5823e-007

2.0145e+007 7.0038e-007 3.8510e-007 6.3042e+006 2.0551e+007

5.0993e+005 1.1716e+007 1.2760e+007 7.7055e+006 4.2717e+005

9.1214e+006 6.0023e+006 2.4965e-006 5.8016e+005 5.6774e+006

1.0794e-006 3.7541e+006 1.1319e+007 4.1275e-006 1.5252e-006

1.1984e-006 6.0751e-007 6.6721e-007 1.3026e-006 3.5488e-006

--------------------------------------------------------------------------------------------

2.4267e+008 1.1616e+007 1.9284e+007 7.9716e+006 8.7543e-007

4.9973e-007 6.8642e-007 4.0614e+006 7.2110e-007 1.1741e-006

2.7286e+008 2.2906e+007 2.0905e+007 1.2662e+007 3.0892e+007

5.3281e-007 1.5617e+008 1.7558e+008 7.7951e-007 3.7634e-007

2.1227e+008 2.8231e-006 4.8381e-007 1.2235e+007 2.3284e+008

Results: pipe f and cable t forces

p

q

(50 ) (0 )f f f (3) force (required to hold parabolic shape between azimuth 0, and 50 deg)

if

it

4.3957e+006 2.2491e+006 3.8245e-007 2.7504e+006 3.0415e+005

-1.2465e+006 -7.5972e+005 -2.2878e+006 -3.3074e+006 -2.9318e+006

9.1784e-007 1.6192e-006 1.9712e+006 6.4620e+006 5.2559e-006

-1.2170e+006 -1.0028e+006 -2.0957e+006 -5.5634e+005 -1.2011e+006

2.0251e-007 4.9243e-007 1.1667e-006 9.5061e-007 9.1282e-007

-----------------------------------------------------------------------------------------------

2.2320e+006 1.8760e+006 7.3270e-007 2.4007e-006 6.2040e+006

-2.3741e+006 2.3447e+006 -7.4170e+005 -8.6315e+006 -4.2325e+006

5.4665e-007 6.8850e-007 7.0909e+006 1.9242e+007 8.9767e-007

-1.0097e+007 -3.9435e+006 -1.4078e+007 -1.2470e+007 -1.4956e+007

----------------------------------------------------------------------------------------------

2.9755e+008 -9.4527e+006 5.1162e+005 -9.4527e+006 1.6263e+007

3.1935e+008 1.2803e+008 1.4041e+008 -4.6020e+007 -2.8096e+007

0 0 0 0 0

2.8445e+006 5.4091e+006 3.5112e+005 -1.3879e+006 -2.4197e+006

2.8657e+006 -4.1364e+006 -4.1364e+006 4.3134e+005 -8.5495e+005

-1.4840e+006 2.5349e+006 4.8020e+006 5.6179e+006 -1.4840e+006

-2.9756e+006 -2.9771e+006 -1.1274e+006 -2.9771e+006 -2.1698e+006

2.2616e+006 4.3175e-007 9.5878e-007 1.1847e-006 4.0056e+006

2.2913e+006 2.3256e+006 -2.8139e+006 -4.4713e+006 -2.3275e+006

----------------------------------------------------------------------------------------------

9.6470e-007 1.0367e+007 5.2535e+006 2.1368e-006 4.5752e-007

3.8287e+006 -1.6317e+007 -1.6317e+007 -1.0013e+007 4.2346e+006

-1.2849e+007 -1.6430e+006 -5.9885e+005 -5.6533e+006 -1.2932e+007

6.6901e+006 3.5710e+006 -2.4313e+006 -1.8511e+006 3.2461e+006

-3.2116e+006 5.4249e+005 8.1070e+006 -3.2116e+006 -3.2116e+006

1.1978e-006 6.0684e-007 6.6653e-007 1.3019e-006 3.5481e-006

------------------------------------------------------------------------------------------------

2.1048e+008 -2.0564e+007 -1.2897e+007 -2.4209e+007 -3.2181e+007

4.9946e-007 6.8604e-007 4.0614e+006 7.2072e-007 1.1738e-006

2.4068e+008 -9.2751e+006 -1.1275e+007 -1.9518e+007 -1.2885e+006

5.3254e-007 1.5617e+008 1.7558e+008 7.7925e-007 3.7625e-007

2.1227e+008 2.8215e-006 4.8262e-007 1.2235e+007 2.3284e+008

Results: Force changes in pipes and cables

p

q

Results: pipe and cable radii (p,q = 5,3)

3.3659e-005 3.2952e-005 3.3348e-005 3.3192e-005 3.3132e-005

1.6998e-001 1.6998e-001 1.6998e-001 1.6998e-001 1.6998e-001

3.3651e-005 3.3886e-005 3.3615e-005 3.3666e-005 3.3754e-005

1.6239e-001 1.6239e-001 1.6239e-001 1.6239e-001 1.6239e-001

3.2380e-005 3.2575e-005 3.2523e-005 3.2579e-005 3.2610e-005

-----------------------------------------------------------------------------------------

6.5677e-005 8.6464e-005 1.0978e-004 1.1048e-004 7.0005e-005

2.2064e-001 2.2064e-001 2.2064e-001 2.2064e-001 2.2064e-001

2.5062e-005 2.5084e-005 2.5101e-005 2.5154e-005 2.5088e-005

2.7896e-001 2.7896e-001 2.7896e-001 2.7896e-001 2.7896e-001

---------------------------------------------------------------------------------2.7661e-001 2.7661e-001 2.7661e-001 2.7661e-001 2.7661e-001

4.4236e-001 4.4236e-001 4.4236e-001 4.4236e-001 4.4236e-001

0 0 0 0 0

)(Pri

3.3760e-001 2.8552e-001 1.8340e-004 3.0025e-001 1.7315e-001

1.6106e-001 1.6472e-001 1.5228e-001 1.4189e-001 1.4598e-001

2.1602e-004 2.4894e-004 2.6147e-001 3.5183e-001 3.3413e-004

1.5038e-001 1.5270e-001 1.3948e-001 1.5724e-001 1.5055e-001

1.6368e-004 2.0435e-004 2.5351e-004 2.4085e-004 2.3843e-004

--------------------------------------------------------------------------------------------

2.5131e-001 2.4063e-001 1.9530e-004 2.5813e-004 3.2450e-001

2.0945e-001 2.3022e-001 2.1733e-001 1.6557e-001 1.9925e-001

1.6017e-004 1.6967e-004 3.0392e-001 3.9007e-001 1.8130e-004

2.4124e-001 2.6609e-001 2.1981e-001 2.2919e-001 2.1412e-001

-----------------------------------------------------------------------------------------

6.6033e-001 2.5481e-004 2.8028e-001 1.9386e-004 3.5524e-001

7.0260e-001 5.8935e-001 5.9884e-001 3.0535e-001 3.7710e-001

0 0 0 0 0

0 50cable radii

)(Yris

3.3410e-002 3.3410e-002 3.3410e-002 3.3410e-002 3.3410e-002

4.3683e-002 4.3683e-002 4.3683e-002 4.3683e-002 4.3683e-002

2.6165e-002 2.6165e-002 2.6165e-002 2.6165e-002 2.6165e-002

3.7059e-002 3.7059e-002 3.7059e-002 3.7059e-002 3.7059e-002

1.0558e-009 1.2894e-009 1.1988e-009 1.2577e-009 1.3735e-009

4.7077e-002 4.7077e-002 4.7077e-002 4.7077e-002 4.7077e-002

-------------------------------------------------------------------------------------------

5.6693e-010 5.8044e-010 5.8616e-010 5.7424e-010 5.7403e-010

8.6760e-002 8.6760e-002 8.6760e-002 8.6760e-002 8.6760e-002

7.8502e-002 7.8502e-002 7.8502e-002 7.8502e-002 7.8502e-002

3.3490e-002 3.3490e-002 3.3490e-002 3.3490e-002 3.3490e-002

3.8491e-002 3.8491e-002 3.8491e-002 3.8491e-002 3.8491e-002

5.4849e-010 5.5827e-010 5.5843e-010 5.5352e-010 5.5604e-010

-----------------------------------------------------------------------------------------

1.2184e-001 1.2184e-001 1.2184e-001 1.2184e-001 1.2184e-001

3.5205e-010 4.1806e-010 4.5408e-010 4.1894e-010 3.5194e-010

1.2184e-001 1.2184e-001 1.2184e-001 1.2184e-001 1.2184e-001

3.5207e-010 5.0106e-010 5.0051e-010 3.4752e-010 2.0454e-010

1.0436e-009 8.6301e-010 7.4118e-010 8.6538e-010 1.0431e-009

4.9279e-002 6.0096e-002 3.5752e-002 2.1817e-002 2.2702e-008

5.6835e-002 1.8023e-008 1.3275e-008 4.5904e-002 3.8908e-002

2.0600e-008 4.3058e-002 5.3851e-002 5.7239e-002 2.2559e-008

8.3011e-004 5.1279e-008 2.9211e-002 2.6040e-008 1.9298e-002

3.2301e-002 1.4172e-008 2.1065e-008 2.3411e-008 4.2987e-002

5.7212e-002 5.7351e-002 3.0300e-002 1.2391e-002 3.3801e-002

-------------------------------------------------------------------------------------------

2.1103e-008 6.9157e-002 4.9229e-002 3.1402e-008 1.4539e-008

9.6403e-002 1.7975e-008 1.3329e-008 5.3928e-002 9.7369e-002

1.5337e-002 7.3516e-002 7.6723e-002 5.9621e-002 1.4038e-002

6.4868e-002 5.2621e-002 3.3937e-008 1.6360e-002 5.1177e-002

2.2315e-008 4.1615e-002 7.2260e-002 4.3636e-008 2.6526e-008

2.3513e-008 1.6741e-008 1.7544e-008 2.4513e-008 4.0461e-008

------------------------------------------------------------------------------------------

3.3458e-001 7.3204e-002 9.4319e-002 6.0642e-002 2.0096e-008

1.5183e-008 1.7795e-008 4.3285e-002 1.8239e-008 2.3273e-008

3.5479e-001 1.0279e-001 9.8204e-002 7.6428e-002 1.1938e-001

1.5678e-008 2.6841e-001 2.8461e-001 1.8963e-008 1.3176e-008

3.1293e-001 3.6088e-008 1.4940e-008 7.5130e-002 3.2774e-001

pipe radii

0 50

)(Pmi

2.1418e-004 2.0527e-004 2.1024e-004 2.0827e-004 2.0752e-004

1.1450e+003 1.1450e+003 1.1450e+003 1.1450e+003 1.1450e+003

1.9176e-004 1.9445e-004 1.9134e-004 1.9193e-004 1.9294e-004

1.1277e+003 1.1277e+003 1.1277e+003 1.1277e+003 1.1277e+003

2.1691e-004 2.1953e-004 2.1883e-004 2.1958e-004 2.1999e-004

------------------------------------------------------------------------------------------------

6.3415e-004 1.0991e-003 1.7718e-003 1.7945e-003 7.2048e-004

2.3179e+003 2.3179e+003 2.3179e+003 2.3179e+003 2.3179e+003

7.5771e-005 7.5901e-005 7.6003e-005 7.6327e-005 7.5923e-005

4.3997e+003 4.3997e+003 4.3997e+003 4.3997e+003 4.3997e+003

------------------------------------------------------------------------------------------------

6.6217e+003 6.6217e+003 6.6217e+003 6.6217e+003 6.6217e+003

1.7258e+004 1.7258e+004 1.7258e+004 1.7258e+004 1.7258e+004

0 0 0 0 0

2.1545e+004 1.5412e+004 6.3586e-003 1.7043e+004 5.6674e+003

1.0280e+003 1.0752e+003 9.1896e+002 7.9785e+002 8.4448e+002

7.9020e-003 1.0494e-002 1.1577e+004 2.0961e+004 1.8905e-002

9.6701e+002 9.9717e+002 8.3195e+002 1.0573e+003 9.6928e+002

5.5426e-003 8.6390e-003 1.3295e-002 1.2001e-002 1.1760e-002

---------------------------------------------------------------------------------------------

9.2851e+003 8.5125e+003 5.6072e-003 9.7956e-003 1.5480e+004

2.0890e+003 2.5238e+003 2.2489e+003 1.3054e+003 1.8905e+003

3.0947e-003 3.4728e-003 1.1142e+004 1.8355e+004 3.9652e-003

3.2903e+003 4.0031e+003 2.7317e+003 2.9700e+003 2.5923e+003

-----------------------------------------------------------------------------------3.7736e+004 5.6193e-003 6.7985e+003 3.2524e-003 1.0922e+004

4.3537e+004 3.0633e+004 3.1627e+004 8.2233e+003 1.2542e+004

0 0 0 0 0

Results: Pipe mass m(P) and cable mass m(Y)

)(Ymis

6.2111e+002 6.2111e+002 6.2111e+002 6.2111e+002 6.2111e+002

9.6679e+002 9.6679e+002 9.6679e+002 9.6679e+002 9.6679e+002

3.4686e+002 3.4686e+002 3.4686e+002 3.4686e+002 3.4686e+002

1.3829e+003 1.3829e+003 1.3829e+003 1.3829e+003 1.3829e+003

1.0097e-012 1.5060e-012 1.3017e-012 1.4329e-012 1.7089e-012

1.2332e+003 1.2332e+003 1.2332e+003 1.2332e+003 1.2332e+003

--------------------------------------------------------------------------------------------

1.4531e-013 1.5232e-013 1.5534e-013 1.4909e-013 1.4898e-013

2.9496e+003 2.9496e+003 2.9496e+003 2.9496e+003 2.9496e+003

2.4148e+003 2.4148e+003 2.4148e+003 2.4148e+003 2.4148e+003

5.0709e+002 5.0709e+002 5.0709e+002 5.0709e+002 5.0709e+002

1.1747e+003 1.1747e+003 1.1747e+003 1.1747e+003 1.1747e+003

1.9978e-013 2.0697e-013 2.0709e-013 2.0346e-013 2.0532e-013

--------------------------------------------------------------------------------------------

4.8568e+003 4.8568e+003 4.8568e+003 4.8568e+003 4.8568e+003

4.0056e-014 5.6486e-014 6.6640e-014 5.6722e-014 4.0031e-014

4.8568e+003 4.8568e+003 4.8568e+003 4.8568e+003 4.8568e+003

6.6127e-014 1.3393e-013 1.3364e-013 6.4427e-014 2.2320e-014

3.5200e-013 2.4071e-013 1.7755e-013 2.4203e-013 3.5162e-013

1.3513e+003 2.0096e+003 7.1125e+002 2.6486e+002 2.8678e-010

1.6366e+003 1.6458e-010 8.9286e-011 1.0676e+003 7.6696e+002

2.1499e-010 9.3933e+002 1.4692e+003 1.6599e+003 2.5784e-010

6.9386e-001 2.6477e-009 8.5920e+002 6.8279e-010 3.7501e+002

9.4508e+002 1.8192e-010 4.0195e-010 4.9647e-010 1.6738e+003

1.8213e+003 1.8302e+003 5.1087e+002 8.5432e+001 6.3573e+002

-------------------------------------------------------------------------------------------

2.0135e-010 2.1623e+003 1.0957e+003 4.4583e-010 9.5573e-011

3.6417e+003 1.2661e-010 6.9614e-011 1.1396e+003 3.7150e+003

9.2179e+001 2.1178e+003 2.3066e+003 1.3929e+003 7.7219e+001

1.9024e+003 1.2519e+003 5.2070e-010 1.2100e+002 1.1841e+003

3.9482e-010 1.3731e+003 4.1400e+003 1.5097e-009 5.5788e-010

3.6714e-010 1.8611e-010 2.0440e-010 3.9904e-010 1.0872e-009

---------------------------------------------------------------------------------------------

3.6624e+004 1.7532e+003 2.9104e+003 1.2031e+003 1.3212e-010

7.4507e-011 1.0234e-010 6.0554e+002 1.0751e-010 1.7505e-010

4.1182e+004 3.4570e+003 3.1551e+003 1.9110e+003 4.6624e+003

1.3112e-010 3.8434e+004 4.3212e+004 1.9184e-010 9.2619e-011

3.1649e+004 4.2091e-010 7.2134e-011 1.8242e+003 3.4716e+004

0 50

0current design 11e5 kg( 1200tons)m

005+2.9563e )( Yms005+1.0655e )( Yms

005+1.6435e)( Pmb005+3.7213e)( Pmb

005+2.7090e )( Pm 005+6.6776e )( Pm

Results: Mass (p,q = 5, 3)

Pipe mass = 3.72e5kg

Cable mass = 2.96e5kg

Total dish mass = 6.68e5kg

total

cable

pipe

Outline


– What results?










N = [ n1 n2 n3 …………..n2b] : 3x2b

S = [ s1 s2 s3 …………ss] = NCsT : 3xs

B = [b1 b2 b3 ….bb] = NCbT : 3xb

W = [w1 w2 w3 …………w2b] : 3x2b

How to compute tensegrity equilibria?

1 if si terminates on node nj

Csij = -1 if si starts from node nj

0 if si touches not node njnj siwj

w1

w2 w3

bi

Connectivity matrix

C = [Cb, Cs]

N = [ n1 n2 n3 …………..n2b] : 3x2b

S = [ s1 s2 s3 …………ss] = NCsT : 3xs

B = [b1 b2 b3 ….bb] = NCbT : 3xb

W = [w1 w2 w3 …………w2b] : 3x2b

How to compute all equilibria?

nj

siwjw1

w2 w3

bi

b

T

s

T

ˆN

N

K= , ,

= ,

=NC

W

B

ˆ

S

C

T

b b

T

s s C CK C C All equilibria:

Outline


– What results?










iii st iii bf

iiss tscYmi

)(iibb fbcYm

i)(

iibb fbcBmi

2)(

Ec b

b

2

Member force

2 2min

ˆˆsubject to

0, 0

i i i i

S B

i i

J b s

S C B C W

(Tensegrity Systems, 2009, pp.97-101,)

Member Mass(buckling)

b

bb

s

ss cc

,

(yielding)

iisiib scbcm 22

(if yielding is the mode of failure)

(string)(bar)

Bar Radiusb

ii

fYr

)(

Member Lengthisib (string)(bar)

How to minimize mass?

4/1

3

24

)(

b

ii

iE

fbBr

(yielding) (buckling)

-Hookean material and same steel for all members

-Optimization Problem

Failure by yield or buckling?

• failure by yielding

• failure by buckling2 2

2 2

1

modulus of Elasticity, bar material

maximal yield stress, bar material

f force appli

1

ed to bar

L=

/ (4

/

len

(4 )

gth

)

of bar

Ef L

E

Ef L

Outline


– What results?










How to compute optimal complexity?(Optimization over (p,q))

0 50

Mod. DHT with Pipe 09.0 rri

Outline


– What results?










Why are tensegrity models reliable?

• Axially loaded members

– No material bending

• Simplest elements allow most accurate

models

– Cables and rods

• Math can by exploited to greater depth

because the math models are simple

Outline


– What results?










?????? ????

??????

????

simply supported

cantilevered

tension

compression

Why is Tensegrity mass-efficent? (Optimal Material Topology for Primary BC are tensegrity)

qJ qq

qq

q

)(

2/)(tan

tan

:solution

/1/1

/1/1

/1

Optimal Structures in BendingGiven:complexity = q(1+q)

specified length r0

specified aspect ratio ρ

external force u0(θ) (not in yellow)

(β,φ,q)

r0

(ρ, q) = (14, 8)

r0/ρ

b

Find:Minimal volume solution

topology (φ, β)

subject to yield constraints

θ

2β

u0

independent of u0 , θ. Why? (uni-directional member loads)

Minimal Material Volume

W

ln2),,(

hence,

ln2)]([),(

thatnote

]cos)2

1

2

1(sin)

2

1

2

1[(

)(

),,,(),(),,,,,(

/1/1

00

/1/1

00

WWV

qJ

urW

qJ

uWqJuqV

q

qq

qq

stress yieldstring

stress bar yield

(5,5)or )10,10(),(,01.1),(

),(

),(

),(

qif

V

qV

J

qJ

q = 15 q = 30 q = 60

Optimal Cantilever of complexity q=15, 30, 60

ln2)(

02/)(tan

1tan

:qfor Optimal

/1/1

/1/1

/1

qJ qq

qq

q

Optimal Cantilever: optimized over complexity q

Given:complexity = q(1+q)

specified length r0

specified aspect ratio ρ

external force u0(θ) (not in yellow)

(β,φ,q)

r0

(ρ, q) = (14, 8)

r0/ρ

Find:Minimal volume solution

topology (φ, β)

subject to yield constraints

θ

2β

u0

f

Mass of Cantilever

Optimal Complexity q

Worse joints

(cheaper construction)qopt = 4

qopt = 10 qopt = ∞

Joint/compressive element

Mass fraction

no joint mass

complexity

ma

ss

Mass of components Mass of joints

1/ 1/( ) ( 1).q qJ q q q

Outline


– What results?










Why is tensegrity optimal tensile?Sticks and Strings for a Minimal Mass Tensile Material with

a stiffness constraint

Maximal stiffness:

Minimal mass: 0

0

9

Compressive material

Tensile material

Tensile structures: Designing cablesMinimal Mass with a stiffness constraint (=K)

L = q unitsp units

t/p

t/p t/pt/p

t/pt/p

2 21 1

2 2

3

1 1

2

2

tan

ta

3( ) ,

( ( ) )

1(

n

)3

s b

s s b

s s b

s s b

s s b s b

U qtk k U potential energy

L pL

m L

p LL

k k

t

t

K

K

qt

Control stiffness by bar length,

b=(tan)L/q

*

*

5/2 2

5/2

2

0

5/2

2 (1 2 ) tan (1 tan )

(1 tan ) ln 2 / 12 min{ , }

(1 tan ) /

2

[tan ( )]

n n

n

n s s

b b

s

s b

n

n

Efl

g

Why is tensegrity Optimal Compressive

Structure?

Replace this

with this

(repeat n times)

= Optimal complexity:

n* = 3 for e = 0.02(alum bars, spectra strings,

100# force, 3’ length, = 10o)

bar mass (n, a) string mass (n, a , f, E )

yieldbuckle

)(/)( omnm

)(om

α

f

f

f

f

planar case

α

α

Min Mass and Optimal Complexity

T-Bar compressive columns: constant width

Note:

Minimal mass by buckling

≤ minimal mass by yielding

Optimal complexity always

Finite unless:

e = 0 (massless strings)

failure by yielding

buckling force = yield force

Failure by buckling

mass

(n = complexity)

L/D = 10

Min Mass Pipe in Compression

(complexity n = 2 shown)

Outline


– What results?










Structural Systems and Control Laboratory

School of Engineering, UCSD

( d e g re e s ) (

d e g ree s )

Ov

erla

p h

(%

o

f s

ta

ge

h

eig

ht)

t = 0V e r t ic a l s t r in g s

O v e r la p = 1 0 0 %

O v e r lap = 0 %

9 00

Why Tensegrity uses less control power?

A(q)t = 0

AT(q)A(q) = 0

Equilibrium Region

(easily correct for mfg errors)

Outline


– What results?










Next: A mass lower bound for the telescope

(optimal without parabolic shape constraints)

33.6e+6N

Rigid

support/foundation

mass(q=15) = 1.40e+005Kg

mass(q=6) = 1.43e+005Kg

q = 15

F=33.6e+006N

Parabolic shape

constraint

Next: A mass lower bound for the telescope

(optimal without parabolic shape constraints)

Next: Optimal Cantilevered tensegrity as

telescope

Azimuth = 50 deg

Next: Optimal Cantilevered tensegrity as

telescope

Another telescope approachusing rigid Tensegrity Prisms

New (rigid) tensegrity flat plate

topology

Rigid Tensegrity Structures:the first tensegrity plate with no soft modes

New (rigid) tensegrity parabolic plate

Rigid Parabolic surfacefrom Class 1 Tensegrity Prisms

Parabolic Surface

Rigid Telescopefrom Tensegrity Prisms

Outline


– What results?










Choosing Information Precision in Control

wDxMz

uBxCy

wDuBxAx

zp

ypp

ppppp

Plant:

Output:

Measurement:Measurement:

zDxCu

zBxAx

ccc

cccc

Controller G:

)(

00

00

00

)(

)(

)(

)(

)(

)(

,0

t

W

W

W

w

w

w

tw

tw

tw

E

w

w

w

E

p

s

a

T

p

s

a

p

s

a

p

s

a

1

1

0

0:

s

a

W

W$ ppT :

Sensor

Actuator

aw pw sw

Plant

simulation

Controller

zu

YyyE

UuuE

T

T

$$

Find $ and G such that

Example – precision allocation

3 outputs: velocity at r0, r2, r4 :

s/a1s/a2

0rs/a3

s/a4s/a5

s/a6

s/a7s/a8

r

L

),( tr

1r

2r

3r4r

5 torque actuators at r0, r1, r2, r3, r4.

3 force actuators at r1, r2, r3.

5 velocity sensors at r0, r1, r2, r3, r4.

3 displacement sensors at r1, r2, r3.

Consider an undamped simply supported beam with 10 degrees of freedoms.

Mode representation for each mode:

)(2 wubT

iiii 101i

idi icy

uy

Cxy

wMxz

DwwuBAxx

s

pa

2

1

)(

zDxCu

zBxAx

ccc

cccc

Controller G:

$ 1: trPW

IuuE

IyyE

T

T

min $

I30

s. t.r0 r1 r2 r3 r4

ActuatorsF 11.72 13.42 11.72

T 21.37 12.83 13.43 12.83 21.37

Sensorsq 2.195 13.35 2.195

1.544 0.475 0.802 0.475 1.544q

optimal precision distributionWhen = 1, = 0.1,

Example – precision allocation

Application

ESACS for a tensegrity boom

A linearized model is given and the input and output matrices can be determined

according to the sensor/actuator locations.

-500

50100

-500

50

0

50

100

150

200

250

300

350

22222222

16

15

9

5

10

8

55555

9999

21

3

888888

3333

9

5

20

11

5

2

4

4

1

6

3

4444444

18

14

4

1

6

17

13

6666666

12

7

6

11111

777

1

19

23

The bottom nodes are fixed. This structure

has 6 bars, 21 free strings, 6 free nodes.

External disturbance is applied to each node. The

control objective is to keep the top and bottom

surface parallel.

Initially, actuators and sensors are collocated on

each string.

Surface strings: #1 ~ #6

Diagonal strings: #7 ~ #12

Knuckle strings: #13 ~ #18

Reach strings: #19 ~ #21

Outline


– What results?










F=50

mass minfor units 3/2/ DLn

L/D=10 n = 8 n = 8

n = 16

n = 1

n = 2

n = 4

Integrating Structure, Control, IA

F=50

= 8 for min mass (L/D=10), n = 4 for min control3/2/ DLn

n = 8

nc = 4

N(0, E-6)

N(0, 10)

E-2

optimum controller complexity

optimum structure complexity



F=50

= 8 for min mass (L/D=10) and for min control3/2/ DLn

n = 8

N(0, E-6)

N(0, 2xE-7)

N(0, 10)

E-2

nc = 8

Optimize sensor/actuator

precision

optimum structure complexity, and controller complexity

Class 1 Dynamics

b

T

bs

TCCCWNKMN sC K,

T

b

T

s

TT

bs

T NCBNCSBBLCCSWBL , , }M12

1)(

2

1{ 22 .

Desired shape

For some nodes unconstrained

nodesunconstrained

nodes

,Y LNR Y Y

Class 1 Controls

Choose controls to cause the shape error to satisfy a specified

stable linear equation, e.g.

0

b

T

bs

TCCCWNKMN sC K,

T

b

T

s

TT

bs

T NCBNCSBBLCCSWBL , , }m̂12

1)(

2

1{ 22

,Y LNR Y Y

Output Feedback Control

G11

w1

v1 u1

v2

u2+w2v3

u3+w3

Number of prisms = p

String control = ui, i = 1,2,…, 3p

Velocity = vi, i = 1,2,…, 3p

Actuator noise = wi = (0, Wi), i = 1,2,…, 3p

Decentralized Output Feedback

Control

TTTTT

p

SSBWBBWBBSBBSB

WWWdiagWSWG

Gyu

,0

),,,(),(2

1

,

321

qBy

wuBKqqM

T

)(

The control law

yields output covariance = [yyT]= BTB, which

minimizes the control energy [uTu].

Hardware

•3 strings equipped with piezoelectric actuator and sensor

3 ft

Linear actuatorForce sensor

Without controller

Resonance

reduced

30db with

integrator

controller

Vibration Isolation of Tensegrity

Tower

feasibility study for a colossus tensegrity...

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