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ROBOTARM.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME ROBOTARM
* Problem :
* *********
* Minimize the time taken for a robot arm to travel between two points.
* This is problem 8 in the COPS (Version 2) collection of
* E. Dolan and J. More'
* see "Benchmarking Optimization Software with COPS"
* Argonne National Labs Technical Report ANL/MCS-246 (2000)
* SIF input: Nick Gould, December 2000
* classification OOR2-AN-V-V
* The number of subintervals
*IE NH 50 $-PARAMETER
*IE NH 100 $-PARAMETER
*IE NH 200 $-PARAMETER
IE NH 400 $-PARAMETER
* approximation of pi
RF PI/4 ARCTAN 1.0
RM PI PI/4 4.0
* Total length of arm
RE L 5.0
* Upper bounds on the controls
RE MAXURHO 1.0
RE MAXUTHE 1.0
RE MAXUPHI 1.0
* Other useful values
IE 0 0
IE 1 1
RE ONE 1.0
RE THREE 3.0
RI NH NH
RD 1/NH NH 1.0
RM -MAXURHO MAXURHO -1.0
RM -MAXUTHE MAXUTHE -1.0
RM -MAXUPHI MAXUPHI -1.0
RM -PI PI -1.0
RM 2PI PI 2.0
R/ 2PI/3 2PI THREE
RM 4PI/3 2PI/3 2.0
RD 0.5/L L 0.5
VARIABLES
* Step size
X STEP
DO I 0 NH
* The length and the angles theta and phi for the robot arm
X RHO(I)
X THE(I)
X PHI(I)
* The derivatives of the length and the angles
X RHO.(I)
X THE.(I)
X PHI.(I)
* The controls
X URHO(I)
X UTHE(I)
X UPHI(I)
* The moments of inertia
X ITHE(I)
X IPHI(I)
ND
GROUPS
* Objective function: minimize final time
ZN TF STEP NH
* The linear parts of the x, y, vx, vy constraints {j in 1..nh}:
DO J 1 NH
IA J-1 J -1
* linear part of rho constraint {j in 1..nh}:
* - rho[j] + rho[j-1] + 0.5*step*(rho_dot[j] + rho_dot[j-1]) = 0
XE RHO(J) RHO(J-1) 1.0 RHO(J) -1.0
* linear part of the constraint {j in 1..nh}:
* - the[j] + the[j-1] + 0.5*step*(the_dot[j] + the_dot[j-1]) = 0
XE THE(J) THE(J-1) 1.0 THE(J) -1.0
* linear part of phi constraint {j in 1..nh}:
* - phi[j] + phi[j-1] + 0.5*step*(phi_dot[j] + phi_dot[j-1]) = 0
XE PHI(J) PHI(J-1) 1.0 PHI(J) -1.0
* linear part of u_rho constraint {j in 1..nh}:
* - rho_dot[j] + rho_dot[j-1] + (0.5/L)*step*(u_rho[j] + u_rho[j-1]) = 0
XE RHO.(J) RHO.(J-1) 1.0 RHO.(J) -1.0
* linear part of u_the constraint {j in 1..nh}:
* - the_dot[j] + the_dot[j-1] +
* 0.5*step*(u_the[j]/I_the[j] + u_the[j-1]/I_the[j-1]) = 0
XE THE.(J) THE.(J-1) 1.0 THE.(J) -1.0
* linear part of u_phi constraint {j in 1..nh}:
* - phi_dot[j] + phi_dot[j-1] +
* 0.5*step*(u_phi[j]/I_phi[j] + u_phi[j-1]/I_phi[j-1]) = 0
XE PHI.(J) PHI.(J-1) 1.0 PHI.(J) -1.0
ND
* linear part of inertia constraints {i in 0..nh}:
* -I_the[i] + 1/3((L-rho[i])^3+rho[i]^3)*(sin(phi[i]))^2 = 0
* -I_phi[i] + 1/3((L-rho[i])^3+rho[i]^3)/ = 0
DO I 0 NH
XE ITHE(I) ITHE(I) -1.0
XE IPHI(I) IPHI(I) -1.0
ND
CONSTANTS
BOUNDS
XR ROBOTARM 'DEFAULT'
XL ROBOTARM STEP 0.0
DO I 0 NH
XL ROBOTARM RHO(I) 0.0
ZU ROBOTARM RHO(I) L
ZL ROBOTARM THE(I) -PI
ZU ROBOTARM THE(I) PI
XL ROBOTARM PHI(I) 0.0
ZU ROBOTARM PHI(I) PI
ZL ROBOTARM URHO(I) -MAXURHO
ZU ROBOTARM URHO(I) MAXURHO
ZL ROBOTARM UTHE(I) -MAXUTHE
ZU ROBOTARM UTHE(I) MAXUTHE
ZL ROBOTARM UPHI(I) -MAXUPHI
ZU ROBOTARM UPHI(I) MAXUPHI
ND
XX ROBOTARM RHO(0) 4.5
XX ROBOTARM THE(0) 0.0
ZX ROBOTARM PHI(0) PI/4
XX ROBOTARM RHO(NH) 4.5
ZX ROBOTARM THE(NH) 2PI/3
ZX ROBOTARM PHI(NH) PI/4
XX ROBOTARM RHO.(0) 0.0
XX ROBOTARM THE.(0) 0.0
XX ROBOTARM PHI.(0) 0.0
XX ROBOTARM RHO.(NH) 0.0
XX ROBOTARM THE.(NH) 0.0
XX ROBOTARM PHI.(NH) 0.0
START POINT
Z ROBOTARM STEP 1/NH
RE RHOI 4.5
DO I 0 NH
RI I I
R/ I/NH I NH
R* (I/NH)^2 I/NH I/NH
R* THEI 2PI/3 (I/NH)^2
R= PHII PI/4
R* THE.I 4PI/3 I/NH
R- T1 L RHOI
R* T1^2 T1 T1
R* T1^3 T1^2 T1
R* T2^2 RHOI RHOI
R* T2^3 T2^2 RHOI
R+ IPHII T1^2 T2^2
R/ IPHII IPHII THREE
R( SPHII SIN PHII
R* ITHEI IPHII SPHII
R* ITHEI ITHEI SPHII
* The length and the angles theta and phi for the robot arm
Z ROBOTARM RHO(I) RHOI
Z ROBOTARM THE(I) THEI
Z ROBOTARM PHI(I) PHII
* The derivatives of the length and the angles
X ROBOTARM RHO.(I) 0.0
Z ROBOTARM THE.(I) THE.I
X ROBOTARM PHI.(I) 0.0
* The controls
X ROBOTARM URHO(I) 0.0
X ROBOTARM UTHE(I) 0.0
X ROBOTARM UPHI(I) 0.0
* The moments of inertia
Z ROBOTARM ITHE(I) ITHEI
Z ROBOTARM IPHI(I) IPHII
ND
ELEMENT TYPE
EV PROD X Y
EV RATIO X Y
EV RATIO Z
EV PHI RHO
EP PHI L
EV THE RHO PHI
EP THE L
ELEMENT USES
DO I 0 NH
XT SRHO.(I) PROD
ZV SRHO.(I) X STEP
ZV SRHO.(I) Y RHO.(I)
XT STHE.(I) PROD
ZV STHE.(I) X STEP
ZV STHE.(I) Y THE.(I)
XT SPHI.(I) PROD
ZV SPHI.(I) X STEP
ZV SPHI.(I) Y PHI.(I)
XT SURHO(I) PROD
ZV SURHO(I) X STEP
ZV SURHO(I) Y URHO(I)
XT SUTHE(I) RATIO
ZV SUTHE(I) X STEP
ZV SUTHE(I) Y UTHE(I)
ZV SUTHE(I) Z ITHE(I)
XT SUPHI(I) RATIO
ZV SUPHI(I) X STEP
ZV SUPHI(I) Y UPHI(I)
ZV SUPHI(I) Z IPHI(I)
XT ITHE(I) THE
ZV ITHE(I) RHO RHO(I)
ZV ITHE(I) PHI PHI(I)
ZP ITHE(I) L L
XT IPHI(I) PHI
ZV IPHI(I) RHO RHO(I)
ZP IPHI(I) L L
ND
GROUP USES
DO J 1 NH
IA J1 J -1
* nonlinear part of rho constraint {j in 1..nh}:
* - rho[j] + rho[j-1] + 0.5*step*(rho_dot[j] + rho_dot[j-1]) = 0
XE RHO(J) SRHO.(J1) 0.5 SRHO.(J) 0.5
* nonlinear part of the constraint {j in 1..nh}:
* - the[j] + the[j-1] + 0.5*step*(the_dot[j] + the_dot[j-1]) = 0
XE THE(J) STHE.(J1) 0.5 STHE.(J) 0.5
* nonlinear part of phi constraint {j in 1..nh}:
* - phi[j] + phi[j-1] + 0.5*step*(phi_dot[j] + phi_dot[j-1]) = 0
XE PHI(J) SPHI.(J1) 0.5 SPHI.(J) 0.5
* nonlinear part of u_rho constraint {j in 1..nh}:
* - rho_dot[j] + rho_dot[j-1] + (0.5/L)*step*(u_rho[j] + u_rho[j-1]) = 0
ZE RHO.(J) SURHO(J1) 0.5/L
ZE RHO.(J) SURHO(J) 0.5/L
* nonlinear part of u_the constraint {j in 1..nh}:
* - the_dot[j] + the_dot[j-1] +
* 0.5*step*(u_the[j]/I_the[j] + u_the[j-1]/I_the[j-1]) = 0
XE THE.(J) SUTHE(J1) 0.5 SUTHE(J) 0.5
* nonlinear part of u_phi constraint {j in 1..nh}:
* - phi_dot[j] + phi_dot[j-1] +
* 0.5*step*(u_phi[j]/I_phi[j] + u_phi[j-1]/I_phi[j-1]) = 0
XE PHI.(J) SUPHI(J1) 0.5 SUPHI(J) 0.5
ND
DO I 0 NH
* nonlinear part of inertia constraints {i in 0..nh}:
* -I_the[i] + 1/3((L-rho[i])^3+rho[i]^3)*(sin(phi[i]))^2 = 0
* -I_phi[i] + 1/3((L-rho[i])^3+rho[i]^3) = 0
XE ITHE(I) ITHE(I) 1.0
XE IPHI(I) IPHI(I) 1.0
ND
OBJECT BOUND
* Solution
*LO SOLUTION 9.14687D+00 $ (NH=50)
*LO SOLUTION 9.14267D+00 $ (NH=100)
*LO SOLUTION 9.14138D+00 $ (NH=200)
*LO SOLUTION 9.14108D+00 $ (NH=400)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS ROBOTARM
TEMPORARIES
R SINP
R COSP
M COS
M SIN
INDIVIDUALS
T PROD
F X * Y
G X Y
G Y X
H X Y 1.0
T RATIO
F X * Y / Z
G X Y / Z
G Y X / Z
G Z - X * Y / Z ** 2
H X Y 1.0 / Z
H X Z - Y / Z ** 2
H Y Z - X / Z ** 2
H Z Z 2.0 * X * Y / Z ** 3
T PHI
F ( ( L - RHO ) ** 3 + RHO ** 3 ) / 3.0
G RHO RHO ** 2 - ( L - RHO ) ** 2
H RHO RHO 2.0 * L
T THE
A SINP SIN( PHI )
A COSP COS( PHI )
F SINP * SINP *
F+ ( ( L - RHO ) ** 3 + RHO ** 3 ) / 3.0
G RHO SINP * SINP *
G+ ( RHO ** 2 - ( L - RHO ) ** 2 )
G PHI 2.0 * SINP * COSP *
G+ ( ( L - RHO ) ** 3 + RHO ** 3 ) / 3.0
H RHO RHO SINP * SINP * 2.0 * L
H PHI RHO 2.0 * SINP * COSP *
H+ RHO ** 2 - ( L - RHO ) ** 2
H PHI PHI 2.0 * ( COSP * COSP - SINP * SINP ) *
H+ ( ( L - RHO ) ** 3 + RHO ** 3 ) / 3.0
ENDATA