tower3D.tens

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[CONSTANTS]

  % Parametrization of a two-prism tower taken from 
  %    C. Sultan, M. Corles and R. E. Skelton
  %    The pre-estressability problem of tensegrity structures: some
  %    analytical results, International journal of solids and structures
  %    vol 38, 2011
  %
  % We model a SVDB struture (section 4) which essentially has the same
  % solutions as a SVD structure (section 3)
  % We compute the constants that give a stable configuration and then
  % leave some ranges around the equilibrium point to allow form motion.
  % In this way we are sure to have valid ranges for the variables.

  % The two driving parameters
  alpha := 60*pi/180
  delta := 30*pi/180
 
  % Structural parameters (taken from the matlab file)
  b := 0.27                   % Size of base/platform
  l := 0.4                    % Length of struts
  P := PARAMETER(PRETENSION)  % Pretension (taken from parameter file)
  rs := 0.0025                % Radius to represent struts
  rc := 0.00075               % Radius to represent cables

  % small value (used to compare alpha with pi/3 = 60 deg)
  zero:=1e-10

  % h is a zero-centered parametrization of the structure height 
  u := sin(delta)*cos(alpha+pi/6)   % auxiliary
  h := (abs(alpha-pi/3)<zero ? l*cos(delta)/2 :
                               cos(delta)*(l*u+sqrt(b^2/3-3*l^2*u^2)-b/sqrt(3))/(2*u))

  % Length of the different strings
  D := sqrt(h^2+b^2/3+l^2-2*l*b*sin(delta)*sin(alpha)/sqrt(3)-2*l*h*cos(delta))
  V := sqrt(b^2+l^2-2*l*b*sin(delta)*sin(alpha+pi/6))
  S := sqrt(h^2+b^2/3+l^2*sin(delta)^2-2*l*b*sin(delta)*cos(alpha-pi/6)/sqrt(3))

  % print D

  % Tension of the different strings
  T_d_0 := 1
 
  T_v_0 := (abs(alpha-pi/3)<zero ? T_d_0*V*(3*l*sin(delta)/(2*b)-1)/D :
                                   T_d_0*V*((l*cos(delta)/h-1)*sin(alpha-pi/6)-cos(alpha))/(D*sqrt(3)*cos(alpha+pi/6)))
  T_s_0 := (abs(alpha-pi/3)<zero ? T_d_0 :
                                   T_d_0*S*(l*cos(delta)/h-1)/D)

  T_b_0 := (abs(alpha-pi/3)<zero ? T_d_0*(2*b^2-9*l*b*sin(delta)+9*l^2*sin(delta)^2)/(6*D*b) :
                                   T_d_0*(3*l^2*sin(delta)*cos(delta)
                                          +6*b*h*cos(alpha-pi/3)
                                          -6*l*h*sin(delta)
                                          -2*sqrt(3)*b*l*cos(delta)*sin(alpha))/(6*D*sqrt(3)*h*cos(alpha+pi/6)))

  % Compression of the struts (fixed an error from the paper!)
  C_0 := (abs(alpha-pi/3)<zero ? T_d_0*(3*l^2*sin(delta)/(2*b)-l/2)/D :
                                 T_d_0*(2*sqrt(3)*h*b*sin(delta)
                                        -sqrt(3)*l*b*sin(2*delta)/2
                                        +6*h^2*cos(delta)*sin(alpha-pi/3)
                                        -6*l*h*(cos(delta)^2*sin(alpha-pi/3)+sin(alpha+pi/6)/sqrt(3))
                                        +2*sqrt(3)*l^2*cos(delta)*cos(alpha)
                                        +6*l^2*cos(delta)^3*sin(alpha-pi/3))/(6*D*h*sin(pi/3-alpha)))
                                    
  % Normalize the tension/compression (to remove the pre-tension)
  n := sqrt(6*(T_d_0^2+T_v_0^2+T_s_0^2++T_b_0^2+C_0^2))
  T_d := P*T_d_0/n
  T_v := P*T_v_0/n
  T_s := P*T_s_0/n
  T_b := P*T_b_0/n
  C := P*C_0/n
  
  % print T_d
  % print T_v
  % print T_s
  % print T_b
  % print C

  % Physical quantities used to define the stiffness of the strings
  E := 14e10  % N/m^2 Young's modulus
  s := 2.5e9  % N/m^2 maximum tendon stress 
  g := 4      % safety margin

  % The stiffness as defined int the paper
  K_d_0 := (E*P*g/s)*T_d
  K_v_0 := (E*P*g/s)*T_v
  K_s_0 := (E*P*g/s)*T_s

  % The rest lengths for the different cables
  R_d := K_d_0*D/(T_d+K_d_0)
  R_v := K_v_0*V/(T_v+K_v_0)
  R_s := K_s_0*S/(T_s+K_s_0)

  % print R_d
  % print R_v
  % print R_s
  
  % The stiffness as in appears in our models
  K_d := K_d_0 / R_d
  K_v := K_v_0 / R_v
  K_s := K_s_0 / R_s

  % print K_d
  % print K_v
  % print K_s

  % Margin. Used to increase the force, length, rest-length
  % from those in the initial configuration.
  % By using a large margins we ensure that the motion of
  % the structure is not hindered by theses limits, i.e.,
  % these limits will neve be active but the minimum
  % limit might be (cables must have at least 0 tension, etc).
  M := 20

  % Factor used to determine the minimum cable length. This
  % must be 0 or larger (we used 0.1 sometimes)
  ml := 0

  % range of compression forces supported by the struts
  % For struts the rest lengt is set to 0 and the stiffnes
  % is set to vary so that with a constant length, we can
  % have different forces.
  % If both the rest length and the stiffness are free we
  % have an extra (useless) degree of freedom.
  min_fs := -M*C  % compression is negative!
  max_fs :=  0

  % data for the cables on the base/platform: range of supported tensions
  % These cables have constant length. We additionally fix the stiffness
  % so that varying the rest length we can have different forces.
  % If both the rest length and the stiffness are free we
  % have an extra (useless) degree of freedom.
  min_f_b := 0
  max_f_b := M*T_b
  
  % data for diagonal cables
  min_lc_d := ml*D
  max_lc_d := M*D
  min_r_d := 0
  max_r_d := M*R_d
  
  % data for vertical cables 
  min_lc_v := ml*V
  max_lc_v := M*V
  min_r_v := 0
  max_r_v := M*R_v

  % data for saddle cables 
  min_lc_s := ml*S
  max_lc_s := M*S
  min_r_s := 0
  max_r_s := M*R_s

[STRUCTURE]

  strut s1 : n1 n2
             length l
             force [min_fs,max_fs]
             radius rs
          
  strut s2 : n3 n4
             length l
             force [min_fs,max_fs]
             radius rs
          
  strut s3 : n5 n6
             length l
             force [min_fs,max_fs]
             radius rs

  strut s4 : n7 n8
             length l
             force [min_fs,max_fs]
             radius rs
          
  strut s5 : n9 n10
             length l
             force [min_fs,max_fs]
             radius rs
          
  strut s6 : n11 n12
             length l
             force [min_fs,max_fs]
             radius rs


  % Cables in the same order as in the matlab file

  cable c1  : n1 n11                         % diagonal
              length [min_lc_d,max_lc_d]
              stiffness K_d
              rest [min_r_d,max_r_d]
              radius rc
               
  cable c2  : n1 n6                          % vertical
              length [min_lc_v,max_lc_v]
              stiffness K_v
              rest [min_r_v,max_r_v]
              radius rc
              
  cable c3  : n3 n7                          % diagonal
              length [min_lc_d,max_lc_d]
              stiffness K_d
              rest [min_r_d,max_r_d]
              radius rc
              
  cable c4  : n3 n2                          % vertical
              length [min_lc_v,max_lc_v]
              stiffness K_v
              rest [min_r_v,max_r_v]
              radius rc
              
  cable c5  : n5 n9                          % diagonal
              length [min_lc_d,max_lc_d]
              stiffness K_d
              rest [min_r_d,max_r_d]
              radius rc
              
  cable c6  : n5 n4                          % vertical
              length [min_lc_v,max_lc_v]
              stiffness K_v
              rest [min_r_v,max_r_v]
              radius rc
              
  cable c7  : n7 n4                          % saddle
              length [min_lc_s,max_lc_s]
              stiffness K_s
              rest [min_r_s,max_r_s]
              radius rc
              
  cable c8  : n7 n2                          % saddle
              length [min_lc_s,max_lc_s]
              stiffness K_s
              rest [min_r_s,max_r_s]
              radius rc
              
  cable c9  : n9 n6                          % saddle
              length [min_lc_s,max_lc_s]
              stiffness K_s
              rest [min_r_s,max_r_s]
              radius rc
              
  cable c10 : n9 n4                          % saddle
              length [min_lc_s,max_lc_s]
              stiffness K_s
              rest [min_r_s,max_r_s]
              radius rc
              
  cable c11 : n11 n2                         % saddle
              length [min_lc_s,max_lc_s]
              stiffness K_s
              rest [min_r_s,max_r_s]
              radius rc
              
  cable c12 : n11 n6                         % saddle
              length [min_lc_s,max_lc_s]
              stiffness K_s
              rest [min_r_s,max_r_s]
              radius rc

  cable c13 : n11 n8                         % vertical
              length [min_lc_v,max_lc_v]
              stiffness K_v
              rest [min_r_v,max_r_v]
              radius rc
              
  cable c14 : n2 n8                          % diagonal
              length [min_lc_d,max_lc_d]
              stiffness K_d
              rest [min_r_d,max_r_d]
              radius rc
              
  cable c15 : n7 n10                         % vertical
              length [min_lc_v,max_lc_v]
              stiffness K_v
              rest [min_r_v,max_r_v]
              radius rc
              
  cable c16 : n4 n10                         % diagonal
              length [min_lc_d,max_lc_d]
              stiffness K_d
              rest [min_r_d,max_r_d]
              radius rc
              
  cable c17 : n9 n12                         % vertical
              length [min_lc_v,max_lc_v]
              stiffness K_v
              rest [min_r_v,max_r_v]
              radius rc
              
  cable c18 : n6 n12                         % diagonal
              length [min_lc_d,max_lc_d]
              stiffness K_d
              rest [min_r_d,max_r_d]
              radius rc
              
  cable c19 : n1 n3                          % base/platform
              length b
              force [min_f_b,max_f_b]
              radius rc
              hidden
                    
  cable c20 : n1 n5                          % base/platform
              length b
              force [min_f_b,max_f_b]
              radius rc
              hidden
      
  cable c21 : n3 n5                          % base/platform
              length b
              force [min_f_b,max_f_b]
              radius rc
              hidden

  cable c22 : n8 n10                         % base/platform
              length b
              force [min_f_b,max_f_b]
              radius rc
              hidden
                    
  cable c23 : n8 n12                         % base/platform
              length b
              force [min_f_b,max_f_b]
              radius rc
              hidden
      
  cable c24 : n10 n12                        % base/platform
              length b
              force [min_f_b,max_f_b]
              radius rc
              hidden
              
[ADDONS]
  platform : body "bodies/triangle.off" red
             sphere 2*rs (0,0,0.005) black decoration
             sphere 2*rs (b,0,0.005) black decoration
             sphere 2*rs (b*cos(pi/3),-b*sin(pi/3),0.005) black decoration
             (0,0,0) n8
             (b,0,0) n10
             (b*cos(pi/3),-b*sin(pi/3),0) n12

[COUPLE]
  % couple stuts
  orientation: s1,[Rz(2*pi/3)]s2,[Rz(4*pi/3)]s3,s5
  orientation: s3,s4
  orientation: s2,s6
  force      : s1,s2,s3,s4,s5,s6
  
  % couple diagonal
  length : c1,c3,c5,c14,c16,c18
  rest   : c1,c3,c5,c14,c16,c18
  force  : c1,c3,c5,c14,c16,c18
  
  % couple vertical
  length : c2,c4,c6,c13,c15,c17
  rest   : c2,c4,c6,c13,c15,c17
  force  : c2,c4,c6,c13,c15,c17
  
  % couple saddle
  length : c7,c8,c9,c10,c11,c12
  rest   : c7,c8,c9,c10,c11,c12
  force  : c7,c8,c9,c10,c11,c12

  % couple base/platform
  force  : c19,c20,c21,c22,c23,c24
  
[FIXED POINTS]

  % n1 is implicitly fixed

  n3(z)=0

  n5(y)=0
  n5(z)=0

[OBSTACLES]

  plane: box (-0.2,-0.2,-0.005)+(b+0.4,b*sin(pi/3)+0.4,0.001) green
  anchor1: sphere 2*rs (0,0,-0.004) red decoration
  anchor2: sphere 2*rs (b,0,-0.004) red decoration
  anchor3: sphere 2*rs (b*cos(pi/3),b*sin(pi/3),-0.004) red decoration
  
[COLLISIONS]

  check: all
  do not check: connected