Notes_10_01
1 of 7
State Space Model For Spring-Mass-Damper
x
k
m
c
mx  cx  kx  0
q  x
x  
k
c
x  x
m
m
q x 
{y}      
q  x 
q   x
use states y to compute derivative of states {y }  f {y}
q  x   0
{y }         k
q x  m
1  x   0
 
 mc  x   mk
1   y1 
 
 mc  y 2 
{y }  f {y}
{y }  [A]{y}
a e t 
assumed solution {y}   1 t 
a 2 e 
{y }  [A]{y}
 a e t 
{y }   1 t    {y}
 a 2 e 
 {y}  [A]{y}
([A]   [I 2 ]){y}  {0}
det([ A]   [I 2 ])  0
1
 

det  k
0
c
 m (  m  ) 
for c = 0, λ  
[A]{y}   {y}  {0}
2 
c
k
 0
m
m
k
k
i
 i n
m
m
n 
 is an eigenvalue, NOT  from EOM
2
λ
c
k
 c 
 
 
2m
m
 2m 
k
m
2
critical damping
underdamped
k
 c CR 

  0
m
 2m 
 1
c CR  2 m  n
λ    n  i  n 1   2

c
c CR

c
2 m n
d   n 1   2
Notes_10_01
2 of 7
State Space Model For Double Spring-Mass-Damper
x1
k1
x2
k2
m1
k3
m2
c1
c2
c3
m1x1  k1x1  c1x 1  k 2 (x1  x 2 )  c 2 (x 1  x 2 )
m 2 x 2  k 3 x 2  c 3 x 2  k 2 (x 2  x 1 )  c 2 (x 2  x 1 )
x1 

x 2 
q  
 x1 
 
q x 
{y}      2 
q   x 1 
x 
 2
 x 1 

x 2 
q   
use states y to compute derivative of states {y }  f {y}
0

 x 1  
0
x    k  k
q    
1
2
{y }      2    


m
1
q  x1   
k
x  
 2
 2

m2

{y }  [A]{y}
0



1
0
0
 c1  c 2
k2

 
m1
 m1
 k  k3 
c

  2
 2
m2
 m2 

 x 
1
 1 
c2
 x 2 

  x   [A]{y}
m1
 1 
 c 2  c 3  x 
  2 
 
 m 2 
0



{y }  f {y}
assumed solution {y}  {a i e t }
{y }   {y}
for c1 = c2 = c3 = 0,
a = (m1k2+ m1k3+ m2k1+ m2k2) / (m1m2), b = (k1k2+ k1k3+ k2k3) / (m1m2)
λi
a
4b 
1  1  2 
2 
a 
Notes_10_01
3 of 7
State Space Model for Cylindrical Coordinate Manipulator
Y
Y
Y
r
r3
T
F
a


X
Both links

X
Link 2
X
Link 3
Main body link 2 - Shaft and end-effector link 3
Mass centers at a and r3 from waist rotation axis, a=constant, r3 = variable
Masses m2 and m3 - centroidal mass moments of inertia J2 and J3
 CCW from positive x axis – a and r3 radial from rotation axis
T is rotary actuator torque of ground on body 2 about waist measured CCW positive
F is radial actuator force of body 2 on body 3 measured positive outward
Gravity g acts along negative y axis
m 2 a 2  m 3 r32  J 2  J 3

0



r3 
q  
  
q    
r3 
T  2m 3 r3 r3   gm 2 a  m 3 r3  cos 
0    



  
2
m 3  r3  

F  m 3 r3   m 3 g sin 


 
q r 
{y}      3 
q    
r3 
use states y to compute derivative of states {y }  f {y}
Notes_10_01







0 0 1 0 r3 


0 0 0 1   



 
   

r3 
 

q  r3  
{y }        

1
q    
m 2 a 2  m 3 r32  J 2  J 3 0  T  2m 3 r3 r3   gm 2 a  m 3 r3  cos 
r3  

 
  
0
m 3  F  m 3 r3  2  m 3 g sin 








4 of 7
Notes_10_01
5 of 7
State Space Model for Two Link Anthropomorphic Manipulator (Double
Pendulum)
Y
C
3
d2
m 3 , J3
a2
d3
a3
T3
m 2 , J2
2
T2
B
A
X
Two solid rigid bars with revolute joints A and B
Lengths d2 and d3 - mass centers at a2 and a3 from proximal ends
Masses m2 and m3 - centroidal mass moments of inertia J2 and J3
2 CCW from positive x axis
T2 is torque of ground on bar 2 about pin A, CCW positive
3 CCW from centerline of bar 2
T3 is torque of bar 2 on bar 3 about pin B, CCW positive
Gravity g acts along negative y axis
J B  m 3 a 32  J 3
J A  J B  m 2 a 22  m 3 d 22  J 2  2m 3 d 2 a 3 cos  3
C  J B  m 3 d 2 a 3 cos 3
D  m 3 d 2 a 3 sin 3
G 2  m 2 a 2  m 3 d 2 g cos 2
G 3  m 3 a 3 g cos ( 2  3 )
1
2

 2 
 J A C  T2  
 D 3  2D 2  3 
 G 2  G 3  

   





 
2


 3 
 C J B   T3  
 D 2
  G 3 
 2 

 3 
q  
 
q     2 
 3 
 2 
 
q  3 
{y}       
q   2 
 3 
Notes_10_01
use states y to compute derivative of states {y }  f {y}


 2 




0 0 1 0    3 


0 0 0 1  



 2 
 2  

 3 
 

q   3  
{y }        

1
q  2   J
2









D


2
D


C
T
G

G

  2  
 2
3
2 3
3 
 3   A

 
   C J   T   
 D 22
  G 3  
B   3 







6 of 7
Notes_10_01
7 of 7
Numerically Evaluate Linear State Matrix
assume y   A LINEAR y using ns number of states and n number of time samples
 y 1   y 1 
   
 y 2   y 2  
     
   
   
 y ns  t1  y ns  t 2
Y   [A
ns x n
 y1   y1 
 y 1  

   
 y 
 2 
 y 2   y 2 
    [A LINEAR ]    


   
   
 y ns  
   
tn 
 y ns  t1  y ns  t 2
 y1  
y  
 2 
  
   
 y ns  
tn 
] Y
LINEAR
ns x n
ns x ns
Note that samples need not be computed using fixed h, and do not need to be continuous.
for n = ns
 
 Y 1
[A LINEAR ]  Y
 y 1   y 1   y 1   y 1  
        
 y   y   y   y 
  2   2   2   2  
 




        
       
 y ns  t1  y ns  t 2  y ns  t 3  y ns  t 4 
 y1   y1   y1   y1  
        
 y 2   y 2   y 2   y 2  
            
        
       
 y ns  t1  y ns  t 2  y ns  t 3  y ns  t 4 
1
for n  ns
A LINEAR   Y  YT YYT 
1
extract  from A LINEAR 
f n   abs
2
h
1
2 max f n 
prefer
h
1
10 max f n 