Constraints Dynamics

Constrained Dynamics ƒ The penalty method: ƒ use springs to enforce constraints ƒ does not satisfy constraints exactly ƒ leads to stiff equations

ƒ Constraint forces: directly calculate the forces required to maintain constraints: ƒ Cancel out any forces acting against the constraint.

Example: A bead on a wire Legal positions x must satisfy:

1 C ( x ) = ( x ⋅ x − 1) = 0 2 Legal velocities must satisfy:

C ( x ) = x ⋅ x = 0 Legal accelerations must satisfy:

C( x ) = x ⋅  x + x ⋅ x = 0 Start with a legal position and velocity, and make sure that the last equation is always satisfied.

Example: A bead on a wire ƒ A 2D particle constrained to move on the unit circle: ƒ Express constraint as a scalar behavior function: 1 C ( x ) = ( x ⋅ x − 1) = 0 2

ƒ Goal: compute a force to maintain the constraint.

Example: A bead on a wire ƒ Add a constraint force to ensure legal acceleration:  x=

f + fˆ m

// fˆ is the (unkown) constraint force

ƒ Substitute: C(x ) = x ⋅ ƒ Therefore:

f + fˆ + x ⋅ x = 0 m

fˆ ⋅ x = −f ⋅ x − m x ⋅ x One equation, two unknowns!

Principle of Virtual Work ƒ In order to uniquely solve the previous equation another equation is needed ƒ Principle of virtual work: the constraint does not add (or remove) any energy to the total energy of the system. m ƒ The kinetic energy is T = x ⋅ x 2 ƒ Its time derivative is T = m x ⋅ x = m f ⋅ x + m fˆ ⋅ x This term should be zero

ƒ We got our extra equation !

General Case ƒ General framework:

C ( x ) = 0, C ( x ) = 0 ƒ Assume (x ) = 0 ƒ Find force that will satisfy C ƒ Use the principle of virtual work

ƒ How to deal with: ƒ Many particles ? ƒ Many constraints ?

Principle of Virtual Work ƒ For every legal velocity fˆ ⋅ x should vanish

fˆ ⋅ x = 0, ∀x | x ⋅ x = 0 ƒ Conclusion: fˆ must be in the direction of x: fˆ = λ x ƒ Substituting this term back, we get:

λ=

−f ⋅ x − m x ⋅ x x⋅x

(

)

ƒ Now we can compute fˆ = λ x and  x = f + fˆ / m

Constrained Dynamics System ƒ Represent all particles as a 3n vector q ƒ Define a diagonal mass matrix M and its inverse W ƒ f is the vector of known forces ƒ fˆ is the vector of unknown constraint forces ƒ The global equation governing the system:

 = Wf q

Constrained Dynamics System ∂C ƒ Legal velocity C = q = Jq = 0 ∂q

ƒ J is called the Jacobian of the system  = Jq   + Jq  ƒ Legal acceleration C ƒ Substitute the forces:  = Jq   + JW(f + fˆ ) C

 = 0 and rearranging: ƒ Requiring C

  − JWf JWfˆ = − Jq

Principle of Virtual Work fˆ ⋅ q = 0, ∀q | Jq = 0

ƒ All and only vectors satisfying this can be expressed as: fˆ = J T λ   − JWf ƒ Substituting fˆ we get: JWJ T λ = − Jq ƒ λ are known as Lagrange multipliers ƒ Using λ we can find fˆ and continue