Character Animation: Dynamic Approaches Simulate articulated rigid - - PowerPoint PPT Presentation

Character Animation:

Dynamic Approaches

• Simulate articulated rigid body system
• Feedback and feedforward control
• Simple models
• Optimal control

Simulation

xi ∆x

xi+1

Newtonian laws gravity

. . .

degrees of freedom actuators

ground contact forces

Simulation + Control

xi ∆x

xi+1

Newtonian laws gravity ground contact forces internal forces

. . .

actuators

General framework

ODE integration collision handling contact force next state current state

General framework

ODE integration Dynamic controller collision handling contact force joint torque next state current state Control Forward simulation

Articulated bodies

• It is more intuitive to control an articulated

body system is the reduced coordinates

Maximal coordinates

(x0, R0) (x1, R1) (x2, R2)

Reduced coordinates

θ1, φ1 θ2 state variables: 18 state variables: 9

x, y, z, θ0, φ0, ψ0

Forward dynamics

• Given current state and joint torques, compute

joint acceleration via equations of motion

• Compute next state from the joint acceleration

via any integration method

• What if there’s additional force applied to the

system?

General framework

ODE integration Dynamic controller collision handling contact force joint torque next state current state Control Forward simulation

Inverse dynamics

• Given current state and joint acceleration,

compute the joint torques

• What if there’s external force applied to the

system?

• Simulate articulated rigid body system
• Feedback and feedforward control
• Simple models
• Optimal control

PD servos

• A proportional-derivative servo (PD servo)

generates torques when current state deviates from desired state

• Each joint is actuated by a PD servo to track a

desired pose or trajectory

• Determine the gains and damping coefficients

can be tedious

τ = kp(θd − θ) − kv ˙ θ

Tracking motion capture

Feedback control

• Problem with feedback control
• Motion precision depends on the gain
• High-gain feedback controllers are less stable,

especially when there is delay

• Difficult to tune the gains for complex

articulated system

Feedforward control

• Feedforward controller executes motor control in

a open-loop manner

• Explains stereotyped and stylized patterns in

movements; consistent with internal model theory

• Inverse dynamics on the input trajectory is

computed during motor learning

Low-impedance controller

• Combine feedforward torques and low-gain

feedback controllers

• Feedback control is still necessary to deal with

small deviations from desired trajectory

• Feedforward torques deal with the slowness of

the human nervous system

• Explains well-trained motion exhibits low joint

stiffness

Controlling human motion

• Based on feedback or feedforward controls, we

can manually construct controllers for human behaviors

• Construct a finite state machine based on trial

and error, intuition and heuristics, and side- by-side comparisons with video footage

Human model

• 17 rigid bodies
• 30 controlled dofs
• body segment and

density from biomechanical data

• mass and inertia

calculated from polygonal model

Hierarchy of control laws

• State machine
• Control action
• Low level control

Hierarchy of control laws

• State machine
• Control action
• Low level control

Hierarchy of control laws

• State machine
• Control action
• Low level control

Controller framework

• A framework for composing controllers
• A supervisor controller determines which

individual controller to activate

• A controller is described by its pre-condition,

post-condition, and expected performance

• Simulate articulated rigid body system
• Feedback and feedforward control
• Simple models
• Optimal control

Inverted pendulum

• A mass point is attached to a massless rod
• Dynamic equation of the simple model
• Compute the torque required at the pivot

point to maintain a balanced position

• For static balance, projection of the mass

point must be within the support polygon

¨ θ = g l sin θ + m τ l2

Biped locomotion control

• SIMBICON
• Simple finite state machine
• Torso and swing hip control
• Balance feedback

Simple state machine

• Each state consists of a pose

representing target angles

• All independent joints are

controlled by PD servos

• Transitions between states
• ccur after fixed duration
• f time or foot contact

events

Torso and swing-hip control

• Both torso and swing-hip

have target angles expressed in world coordinates

• The stance-hip torque is left

as a free variable

τA = −τtorso − τB

Balance feedback

• Swing-hip target angle is

continuously modified based on the center of mass

• This allows the character to

change the future point of support

• Simulate articulated rigid body system
• Feedback and feedforward control
• Simple models
• Optimal control

General framework

ODE integration torques optimization collision handling joint torque next state current state

solve joint torques that

• ptimizes task goals and
• beys physics

Optimal contacts

• Maintain balance using sustained frictional

contacts with the environment

• The character pushes against the environment

and uses the resulting force to control its motion

• Contact forces are restricted by friction cones

General framework

ODE integration torques optimization collision handling contact force joint torque next state current state

solve joint torque that

• ptimizes task goals and
• beys physics based on

contact information

Optimal momentum control

• Control changes in linear and angular momenta

to control the center of mass and center of pressure simultaneously

Optimal plan

• Plan joint torques by
• ptimizing a low-dimensional

physical model

• To generate the plan, full-body

dynamics are approximated by a set of closed-form equations-

• f-motion

Stochastic optimal control

• Automatically learn a robust control strategies

under various sources of uncertainty

• Use a probabilistic formulation in which all prior

beliefs over unknown quantities are modeled by probability distributions

• Optimize a controller that maximizes expected

return, which is computed by Monte Carlo methods