Dynamic Controllers Simulation x i Newtonian laws gravity ground - - PowerPoint PPT Presentation

Dynamic Controllers

xi ∆x

xi+1

Newtonian laws gravity

. . .

degrees of freedom

ground contact forces

Simulation

equations of motion

Simulation + Control

xi ∆x

xi+1

Newtonian laws gravity ground contact forces internal forces

. . .

actuators

An Uncontrolled System

dynamic simulator collision handling contact force next state current state

A Controlled System

dynamic simulator dynamic controller collision handling contact force joint torque next state current state Control Forward Simulation

Feedback vs Feedforward

• Feedback control takes account of the current

state to adjust the output.

• Feedforward control applies a pre-defined signal

without responding to the current state of the system.

General Controllers

PD Servo

• Use proportional-derivative (PD) controllers to

compute joint torques.

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

desired pose or trajectory

τ = −kp(x − ¯ x) − kd ˙ x

Gain and Damping

high gain high damping

τ = −kp(x − ¯ x) − kd ˙ x τ = −kp(x − ¯ x) − kd ˙ x

Problems with PD Servos

• Gain and damping coefficients are difficult to

determine for complex articulated systems.

• High-gain feedback controllers are less stable.
• Precision and stability are often in conflict.

Track a Trajectory

• Can track a sequence of poses by varying the

target pose over time.

• Tuning coefficients is even harder when tracking

a time sequence.

τ n = −kp(xn − ¯ xn) − kd ˙ xn

¯ xn t

Tracking Mocap

Motion Capture-Driven Simulations That Hit and React by Zordan and Hodgins

Gravity Compensation

• PD control never achieves the desired joint value

under gravity.

• Adjust the target position to compensate the

effect of gravity.

τ = −kp(x − ¯ x) − kd ˙ x

gravity compensated target target = desired pose

Stable PD

τ n = −kp(xn − ¯ xn) − kd ˙ xn τ n = −kp(xn+1 − ¯ xn+1) − kd ˙ xn+1 ˙ xn+1 = ˙ xn + ∆t¨ xn xn+1 = xn + ∆t ˙ xn τ n = −kp(xn + ∆t ˙ xn − ¯ xn+1) − kd( ˙ xn + ∆t¨ xn) Instead of tracking current pose track next pose

Stable PD

Stable PD

Feedforward Control

• Feedforward controller executes motor control in

an open-loop manner.

• Inverse dynamics on the input trajectory is

computed during motor learning.

• Explains stereotyped and stylized patterns in

movements; consistent with internal model theory.

Internal Model

Feedforward control, based on an internal model of the motor system and memory of

• bject properties, is used to specify motor commands in advance of the movement.

Low-Impedance Control

• 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 latency in

the dynamic system.

• Explains well-trained motion exhibits low joint

stiffness.

Jacobian Transpose Control

• How much torque does each joint need to exert

in order to result in force f at point p?

• Use Jacobian matrix to map the force in the

Cartesian space to joint torques.

desired force: f point of application: p

τ = J(q, p)T f

Generalized Force

• The virtual displacement of ri can be written in

terms of generalized coordinates

• The virtual work of force Fi acting on ri is

δri = ∂ri ∂q1 δq1 + ∂ri ∂q2 δq2 + . . . + ∂ri ∂qn δqn

Fiδri = Fi

• j

∂ri ∂qj δqj

Functional Controllers

Construct a Complex Controller

• Build on top of basic control mechanisms to

control more complex behaviors.

• Organize basic controllers in a hierarchical

structure.

• Use a finite state machine to switch between

different controllers.

State Machine Controller

• State machine
• Control action
• PD servos

State Machine Controller

• State machine
• Control action
• PD servos

State Machine Controller

• State machine
• Control action
• PD servos

State Machine Controller

Standing Balance

• For static balance, projection of the mass point

must be within the support polygon.

• Use hip strategy or ankle strategy to maintain

balance.

applied at ankle or hip τ = −kp(c − ¯ c) − kd ˙ c

DEMO

• Balance App in DART

Optimal Torques

• Formulate an optimization to solve for motion

and torque (sometimes contact force as well).

• Use equations of motion as constraints to ensure

physical realism.

• Objective function could be defined by task

goals, energy expenditure, or aesthetics preferences.

Challenges in Optimization

• Can’t handle contacts.
• Constraint is highly nonlinear in q.
• Domain is high dimensional.

min

q,τ E(q, τ)

subject to M(q)¨ q + C(q, ˙ q) + G(q) = τ

Contact Constraints

• Maintain balance using sustained frictional

contacts with the environment.

• The character pushes against the environment

and uses the resulting force to control its motion.

fi ∈ Ki min

¨ q,τ,f E(¨

q, τ, f) subject to M(q)¨ q + C(q, ˙ q) + G(q) = τ + JT f

Contact Constraints

Momentum Constraints

• Control changes in linear and angular momenta

to regulate the center of mass and center of pressure simultaneously.

Momentum Constraints

• Use a simplified model to explain

complex dynamic balance.

• An inverted pendulum is a mass point

attached to a massless rod.

• Dynamic equation of the simple model:
• Compute the torque required at the pivot

point to maintain a balanced position.

Balanced Locomotion

¨ θ = g l sin θ + τ ml2

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 of

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

• n the center of mass
• This allows the character to

change the future point of support

DEMO

• SIMBICON demos

Low-dimensional Control

• 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-of-motion