Spacetime constraints A cartoonists view, by Laura Green - - PowerPoint PPT Presentation

Spacetime constraints

Luxo Jr.

Witkin and Kass, Spacetime constraints, SIGGRAPH 88 A cartoonist’s view, by Laura Green

Motivations

Forward simulation generates highly realistic motion, but very hard to control Robotics controllers can generate motion efficiently but the controllers themselves are hard to design

Definition

Cast motion synthesis into an optimization problem The resulting motion is the optimal motion given a set of tasks Provide both realism and control

A B C

Spacetime particle

A particle with a jet engine

Obey equations of motion Reach specific points at specific times Be fuel efficient

mg

Motion

A motion sequence includes

X = {q(t), f(t)} Time varying jet force

f(t) f(t) q(t)

Time varying particle position

q(t)

Constraints

Newtonian constraints

f mg

m¨ q − f − mg = 0

Position constraints

q(0) = qa q(n − 1) = qb

Objective function

t1

t0

|f(t)|2dt

Minimize the fuel consumption

DOF representation

Discretize and q(t) f(t) Spline representations

X = {q0, q1, . . . , qn−1, f0, f1, . . . , fn−1}

¨ qk = qk+1 − 2qk + qk−1 h2

X = {c0, c1, . . . , cm−1, f0, f1. . . . , fp−1}

c0 c1 cm−1 ¨ q(t) = ∂q ∂c ¨ c + ∂ ˙ q ∂c ˙ c

Constraint derivatives

Derivatives of Newtonian constraints

Ci : mqi+1 − 2qi + qi−1 h2 − fi − mg = 0

= 0 i = j

• therwise

i = j ± 1 if ∂Ci ∂qj = −2m h2 = m h2 = 0 i = j

• therwise

if ∂Ci ∂fj = −1

Constraint derivatives

Derivatives of position constraints How does the Jacobian look like?

q0 q1 qn−1 . . . fn−1 f0 f1 . . . Cp0 Cp1 C1 Cn−2 . . .

q0 − qa = 0 qn−1 − qb = 0

Cp1 : Cp0 :

Derivatives of objective function

Objective function derivatives

∂G ∂fj = 2fj ∂G ∂qj = 0 ∂2G ∂fi∂fj = 2 = 0 i = j

• therwise

G(X) =

n−1

• t=0

||f(t)||2

Put it all together

Solve for that min

X G(X)

Ci(X), i = 0 . . . n − 1 q0 = qa qn−1 = qb subject to X

human characters

Complex Newtonian constraints Need to project forces into generalized coordinates Apply Lagrangian dynamics

Generalized forces

ri = ri(q1, q2, . . . , qn) δri =

• j

∂ri ∂qj δqj Fiδri = Fi

• j

∂ri ∂qj δqj =

• j

Qjδqj

Generalized coordinates Virtual displacement represented in generalized coordinates Virtual work of acting on particle Fi ri is the generalized force acting on Qj qj

Lagrangian dynamics

d dt ∂Ti ∂ ˙ qj − ∂Ti ∂qj − Qj = 0 Apply Lagrangian dynamic equation on each joint DOF indicates external forces and muscle force acting on qj Qj

External forces

Gravitational force

• i∈n(j)

mig∂ci ∂qj qj ci ci qj ∂ci ∂qj = 0 ∂ci ∂qj = 0

j n(j)

External forces

F

Contact force

F ∂p ∂qj p

Muscle forces

−ks(qj − ¯ qj)

Muscle force is proportional to the difference between the current joint angle and the desired joint angle

Newtonian constraints

• i∈n(j)

tr ∂Wi ∂qj Mi ¨ WT

i

• +
• i∈n(j)

mig∂ci ∂qj +

• k

Fk ∂pk ∂qj + ks(qj − ¯ qj) = 0

net force gravity contact force muscle force

m¨ q − f − mg = 0 Instead of Newtonian constraints for an articulated body system is a lot more complicated

Advantages of spacetime

Intuitive constraint specification Change the feel of motion by modifying the

• bjective function

walking on hot coals walking on eggs carrying a bowl of hot soup pursued by a bear

Disadvantages of spacetime

Size of the problem is proportional to both time and character’s DOFs If the starting point is too far from the solution, it might not converge Constraints are highly nonlinear in high dimensional space Constraint count is proportional to time

Summary

definition method application forward kinematics inverse kinematics forward dynamics inverse dynamics

• ptimal control