上海交通大学：《复杂系统动力学计算机辅助分析》课程教学资源_Chapter 3_CHAP3.3-Relative Constraint

D() (RA)》h -v"Bv h=r+As'-r-As Number of constraint equations 5=2 g=(9,于4，=g,)月 a=aa=o.r o x Degrees of freedom 5=4 Parameters of the constraint i,,s=(69,s=k”,y,y

    j i P i P i P i Q j Q j Q j i, j,s  x y , s  x y ,v ,v T T s  2 Degrees of freedom   4 Number of constraint equations   T T i i i q  r Parameters of the constraint   T T j j  j q  r     T T T T T T i j i i j  j q  q q  r r Bj y  x  O j x y j   Cj j r  Q Q j r Q j s P i s i x  i y  Ci Bi ir  P P ir h  i v  j v     0                   i ij j i i t i j t i j t i j v B v RAv h T T ( , ) 2 ( , ) ( , ) 1    P i i i Q j j j h  r  A s  r  A s

Since AA=A RT=-R (A,Rv)A s=vR'Ays'=-v RAus"=-v Bus, (A,RAs”=Rs (A,R）Ay=-B, The constraint equation can be obtained as - 4)-小(R4)A-(R4)As” -vBv h n tev

      ij j T j j i T i i P i T T i P i i T i i P ij j T i P ij j T i P ij j T T i P j j T i i T j ij T i A Rv A v v B v A Rv A s v R s A Rv A s v R A s v RA s v B s A A A R R                         Since  ,   The constraint equation can be obtained as                  0                                                                i ij j P i i Q i i j i i i j j i ij j P i i i i Q i i j i i i j j i ij j P i i i Q i i j j j i ij j t i j i i v B v v B r r v B s v R s v B v RAv r r RAv A s RAv A s v B v RAv r A s r A s v B v RAv h T T T T T T T T T T T T T T ( , ) 

The constraint equation is Differentiation of the equation leads to --语66-] @,-)Ay The Jacobians of with respect to 4,and 4 are -vAv Velocity equation can be written by =v p小=》]，q=ag p(i.i)=0

   0                 ij j T i P i T T i P ij j T j i i T i T t i j i v B v v B r r v B s v R s ( , )           0                   j i ij j T i P j i ij j T j i i T i T j i i T i i T t i j i v A v v A r r v B r r v A s              ( , )                              ij j T i P ij j T i T i T t i j i ij j T i P ij j T j i i T i T i T i T t i j i i j v A v v B v A s v A v v B v A r r v A s q q 0 0 ( , ) ( , )  ,  0 t(i, j)     v T i j t i j t i j t i j i j q  q q , q  q q ( , ) ( , ) ( , )    The constraint equation is Differentiation of the equation leads to The Jacobians of with respect to and are t(i, j)  i q j q Velocity equation can be written by q v q   t(i, j) 

Acceleration constraint equation r4小小4t- v(8,-0.)Av; B吃小4近小6以矿+6扩a -项B, yB(-）+2g4(传-)-（向-Bs -(色-)B,y 功= 6小灯6-]

Acceleration constraint equation                    0                                         j i ij j T i P j i ij j T i P j i ij j T j i i T i T j i i i T i T i j i ij j T i j i T i T j i i i T i T j i i i T i T t i j i i v B v v B r r v A r r v A s v B s v A v v A r r v B r r v A r r 2 2 2 ( , )                                        2 2 2 2 T T T T T P i i i j i i i i j i i j i ij j T i j i ij j                           v B r r v A r r v B s v B v           j  i       2 2 0 T T T T i i i j i i i i j i             v B r r v A r r     

Relative distance constraint ·Definition: The distance between two points of different bodies is constant

2022年3月11日 15 Relative distance constraint • Definition： – The distance between two points of different bodies is constant