Modal-Space Control for Articulated Characters

晃晃

Modal-Space Control for Articulated Characters

SUMIT JAIN and C. KAREN LIU

Georgia Institute of Technology

We present a novel control algorithm for simulating an articulated character

performing a given reference motion and its variations. The unique feature

of our controller is its ability to make a long-horizon plan at every time

step. Our algorithm overcomes the computational hurdle by applying modal

analysis on a time-varying linear dynamic system.We exploit the properties

of modal coordinates in two ways. First, we design separate control strate-

gies for dynamically decoupled modes. Second, our controller only applies

long-horizon planning on a subset of modes, largely reducing the size of the

control problem.With this decoupled and reduced control system, the char-

acter is able to execute the reference motion while reacting to unexpected

perturbations and anticipating changes in the environment.We demonstrate

our results by simulating a variety of reference motions, such as walking,

squatting, jumping, and swinging.

Categories and Subject Descriptors: I.3.7 [Computer Graphics]: Three-

Dimensional Graphics and Realism—Animation

General Terms: Algorithms, Design, Experimentation

Additional Key Words and Phrases: Character animation, motion capture,

modal analysis

ACM Reference Format:

Jain, S. and Liu, C. K. 2011. Modal-Space control for articulated

characters. ACM Trans. Graph. 30, 5, Article 118 (October 2011),

12 pages.

DOI = 10.1145/2019627.2019637

http://doi.acm.org/10.1145/2019627.2019637

1. INTRODUCTION

The ability to respond to the changes in the environment and predict

the consequences of our own action is fundamental for everydaymo-

tor tasks. Physically simulating a virtual characterwho exhibits both

reactive and anticipatory behaviors presents immense challenges in

many facets. First, the human motor system is both under-actuated

and redundant. The former leads to complex issues with balance

while the latter results in a high-dimensional and underconstrained

problem. Second, the interaction with the environment via contacts

is discrete in nature. The discontinuity introduced by the change

of contact states further complicates both simulation and control

problems.

One possible approach to achieving both a reactive and anticipa-

tory virtual character is to formulate a long-term planning problem,

such as spacetime optimization, and update the plan at every time

step according to the current state of the character and of the envi-

ronment. Motion produced by long-term planning usually appears

more compliant because the character is not always in the urgency

of matching the immediate goal. In addition, the frequent replan-

ning allows the character to respond to unexpected perturbations in

a timelymanner. Though straightforward, this problemis extremely

difficult and prohibitively expensive to solve in practice. Long-term

planning on a full human dynamic system requires us to resolve all

the aforementioned challenges. To date, offline solutions to opti-

mal trajectory problems are very sensitive to parameters and initial

conditions of the problem. We certainly cannot apply such brittle

solutions at every time step in an online fashion.

This article tackles a more feasible problem: designing a control

system capable of long-term planning and frequent replanning for

simulating a specific motion sequence. We introduce a new control

systemthat tracks the referencemotionwhile reacting to unexpected

perturbations and adapting to anticipated changes in the environ-

ment. Our key insight is that the long-term planning can be largely

simplified by approximating the dynamic system using modal anal-

ysis. In our formulation, we do not solve one long-term planning

problem in the generalize coordinates, rather, we formulate a set of

control strategies in a reduced and dynamically decoupled modal

coordinates. Modal analysis offers two advantages to our problem.

—Independent control. In the modal space, each mode is gov-

erned by an independent equation of motion. This reduces a

N-dimensional optimal control problem to N independent one-

dimensional problems.

—Model reduction. Modal analysis organizes modes by the natu-

ral frequencies of the dynamic system. Typically a few modes

are sufficient to capture the dynamic behaviors of the system.

This property potentially reduces the dimension of the control

variables.

In spite of these great advantages, modal analysis is only suited

for linear dynamic systems.We circumvent the issue by linearizing

the nonlinear dynamic equations around the current state at each

time step, resulting in a time-varying linear dynamic model.

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