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

Luxo Jr. Witkin and Kass, Spacetime constraints , SIGGRAPH 88 Spacetime constraints A cartoonist’s view, by Laura Green Motivations Definition Cast motion synthesis into an optimization problem The resulting motion is the optimal motion given a Forward simulation generates highly realistic set of tasks motion, but very hard to control Provide both realism and control Robotics controllers can generate motion efficiently but the controllers themselves are hard to design B A C

Spacetime particle Motion A motion sequence includes A particle with a jet Time varying particle position q ( t ) engine f ( t ) q ( t ) Time varying jet force f ( t ) Obey equations of motion Reach specific points at specific times X = { q ( t ) , f ( t ) } Be fuel efficient m g Constraints Objective function Newtonian constraints f Minimize the fuel consumption m ¨ q − f − m g = 0 � t 1 | f ( t ) | 2 dt m g t 0 Position constraints q ( n − 1) = q b q (0) = q a

DOF representation Constraint derivatives Derivatives of Newtonian constraints Discretize and q ( t ) f ( t ) C i : m q i +1 − 2 q i + q i − 1 − f i − m g = 0 X = { q 0 , q 1 , . . . , q n − 1 , f 0 , f 1 , . . . , f n − 1 } h 2 q k = q k +1 − 2 q k + q k − 1 ∂ C i = − 2 m ¨ h 2 i = j h 2 ∂ q j = m if i = j ± 1 Spline representations h 2 X = { c 0 , c 1 , . . . , c m − 1 , f 0 , f 1 . . . . , f p − 1 } otherwise = 0 c + ∂ ˙ q ( t ) = ∂ q c 1 q ¨ ∂ c ¨ ∂ c ˙ ∂ C i c = − 1 i = j ∂ f j if c 0 otherwise c m − 1 = 0 Constraint derivatives Objective function derivatives n − 1 � Derivatives of position constraints Derivatives of objective function || f ( t ) || 2 G ( X ) = t =0 C p 0 : q 0 − q a = 0 ∂ G = 0 C p 1 : q n − 1 − q b = 0 ∂ q j ∂ G How does the Jacobian look like? = 2 f j ∂ f j q 0 q 1 q n − 1 f 0 f 1 f n − 1 . . . . . . C 1 ∂ 2 G = 2 i = j ∂ f i ∂ f j . . . otherwise C n − 2 = 0 C p 0 C p 1

Put it all together human characters Solve for that X min X G ( X ) Complex Newtonian constraints subject to C i ( X ) , i = 0 . . . n − 1 Need to project forces into generalized coordinates Apply Lagrangian dynamics q 0 = q a q n − 1 = q b Generalized forces Lagrangian dynamics Generalized coordinates Apply Lagrangian dynamic equation on each r i = r i ( q 1 , q 2 , . . . , q n ) joint DOF Virtual displacement represented in generalized d ∂ T i − ∂ T i coordinates − Q j = 0 dt ∂ ˙ q j ∂ q j ∂ r i � δ r i = δ q j ∂ q j j indicates external forces and muscle force Q j Virtual work of acting on particle F i r i acting on q j ∂ r i � � F i δ r i = F i δ q j = Q j δ q j ∂ q j j j is the generalized force acting on Q j q j

External forces External forces Gravitational force Contact force F ∂ p m i g ∂ c i q j ∂ q j � q j ∂ q j i ∈ n ( j ) c i F c i j p ∂ c i ∂ c i � = 0 = 0 ∂ q j ∂ q j n ( j ) Muscle forces Newtonian constraints Muscle force is proportional to the difference between the Instead of m ¨ q − f − m g = 0 current joint angle and the desired joint angle Newtonian constraints for an articulated body system is a lot more complicated − k s ( q j − ¯ q j ) � ∂ W i � m i g ∂ c i ∂ p k � M i ¨ � � W T tr + + + k s ( q j − ¯ q j ) = 0 F k i ∂ q j ∂ q j ∂ q j i ∈ n ( j ) i ∈ n ( j ) k net force gravity contact force muscle force

Advantages of spacetime Disadvantages of spacetime Intuitive constraint specification Size of the problem is proportional to both time and character’s DOFs Change the feel of motion by modifying the objective function If the starting point is too far from the solution, it might not converge walking on hot coals Constraints are highly nonlinear in high walking on eggs dimensional space carrying a bowl of hot soup Constraint count is proportional to time pursued by a bear Summary definition method application forward kinematics inverse kinematics forward dynamics inverse dynamics optimal control