Articulated Body Dynamics The Basics

Articulated Body DynamicsThe Basics

Comp 768

October 23, 2007

Will Moss

October, 23 2007 2

Overview

•Motivation•Background / Notation•Articulate Dynamics Algorithms

– Newton-Euler Algorithm– Composite-Rigid Body Algorithm– Articulated-Body Algorithm (Featherstone)– Lagrange Multiplier approach (Baraff)

• Originally a problem from robotics– Given a robotic arm

with a series of joints that can apply forces to themselves (called motors), find the forces to get the robot arm into the desired configuration

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History

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Applications

•Computer Graphics– Humans, animals, birds,

robots, etc.– Wires, chains, ropes, etc.– Trees, grass, etc.– Many more

• http://vrlab.epfl.ch/~alegarcia/VHOntology/long.html

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Basics

•An articulated body is a group of rigid bodies (called links) connected by joints

•Multiple types of joints– Revolute (1 degree of freedom)– Ball joint (3 degrees of freedom) – Prismatic, screw, etc.

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Notation

• In rigid body dynamics we had two equations

– Fs is the vector of spatial forces– Is is the spatial inertia matrix (6 x 6)– as is the spatial acceleration

• This is called spatial algebra– Combines the linear and angular components of the

physical quantities into one 6 dimensional vector

)()()( tatItF sss I(t)ω(t)τ(t)

Ma(t)F(t)

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Notation

•Transitioning this to articulated bodies

– Qi is the force on link i

– H is the joint-space inertia matrix (n x n)– are the coordinates, velocities and

accelerations of the joints– C term produces the vector of forces that produce zero

acceleration

ik

n

j

n

j

n

kjijkjiji gqqCqHQ

1 1 1

qqq and ,

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Forward vs. Inverse Dynamics

• Inverse Dynamics– The calculation of forces given a set of

accelerations

•Forward Dynamics– The calculation of accelerations given a

set of forces

ik

n

j

n

j

n

kjijkjiji gqqCqHQ

1 1 1

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Algorithms

• Inverse Dynamics–Newton-Euler Algorithm

•Forward Dynamics–Composite-Rigid-Body Algorithm–Articulated-Body Algorithm– Lagrange Multiplier Algorithm

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Newton-Euler Algorithm

•Goal– Given the accelerations and velocities at the

joints, find the forces required at the joints to generate those accelerations

•Recursive approach– Finds the accelerations and velocities of link i in

terms of link i - 1

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•Method1.Calculate the velocities and accelerations at each

link2.Calculate the required net force acting on each

link to generate those accelerations3.Calculate the joint forces required to generate the

net forces on each link

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1. Find the velocities and accelerations of the links

link the

ofon accelerati and velocity theare and

andjoint

for the variablesphysical theare and ,

,joint ofmotion allowed

thedescribingr unit vecto a is where

av

qqq

i

si

)0( 01 vqsvv iii - i

)0( 01 aqsqsvaa iiiiii-i

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2. Find the forces on each link

iiiiiiil

i vIvaIvIdt

df

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3. Find the forces on the joints

•This can be reformulated in link coordinates to speed up the calculation

•Runs in O(n)

ljji1ii fff

ln

jn

ljj ff wherefff i1ii

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Forward vs. Inverse Dynamics

• Inverse Dynamics– The calculation of forces given a set of

accelerations

•Forward Dynamics– The calculation of accelerations given a

set of forces

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Composite-Rigid-Body Algorithm

– Q is the vector of the forces on the links– H is the joint-space inertia matrix (n x n)– C vector of forces that produce zero acceleration–

•Algorithm– Calculate the elements of C– Calculate the elements of H– Solve the set of simultaneous equations

ik

n

j

n

j

n

kjijkjiji gqqCqHQ

1 1 1

)qC(q,qH(q)Q

onsaccelerati and s velocities,coordinate theare and , qqq

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•Solve for C– Setting the acceleration to zero, we get– We can, therefore, interpret C as the forces which

produce no acceleration– We can use a forward-dynamics solver (like Newton-

Euler) to solve for the forces given the position, velocity and an acceleration of zero

)qC(q,Q

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•Solve for H– If we set C to 0, we observe that is the vector of

joint forces that will impart an acceleration of onto a stationary robot• Therefore, the ith column of H is the vector of forces required to

produce a unit of acceleration about joint i and no other acceleration.

– Treat the links i…n as a rigid-body with inertia defined by

– Treat the links from 1…i-1 are therefore unmoving

iC

i

n

ijj

Ci IIII

1

qqH )(q

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•Solve for H (cont.)– Given that

– Since none of the links from 1 … i-1 are moving, every joint transmits onto the subsequent link, so we can solve for H by solving

– Which is a complete solution for H since it is symmetric– Runs in O(n2)

iC

ii sIf

ijfsH iS

jji for

if

sS of transposespacial theiss Where

onacceleratiunit a is

and joint at forceunit a is where

i

i

s

if

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•Once you have H and C, solve the system of equations using any solver– O(n3), but the constant is small enough that for n less

than ~12 the O(n2) term dominates

•Like Newton-Euler, this can be reformulated in link coordinates– Faster for n ≤ 16

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Articulated-Body Algorithm

• (Re)consider the equation of motion of an articulated body

•This is true for any link in the articulated body

paIf A

ngaccelerati fromlink thekeep torequired force or the force, bias theis

and intertiabody articulate theis

link, a toapplied force a is where

p

I

fA

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•Consider an articulate robot as a single joint attached to an articulated body– The problem simplifies to the forward dynamics of a

one-joint robot (much simpler than the general case)– The first joint is simply a one-joint robot– The second joint is a one-joint robot with a moving base

(slightly more complicated, but still much simpler that the general case)

– Solving this requires two tasks• Solving the one-joint robot forward dynamics problem• Finding the articulated-body inertias (I) and bias forces (p)

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• Solving the one-joint robot problem

sIs

pqsvaIsQq AS

bbAS

joint at theon acceleratiq

joint at thevelocity q

joint at the forceQ

base theofon acceleratia

base theofvelocity v

s of transposespatial s

joint theofmotion allowed

thedescribingr unit vecto theis s

force biasp

robot theofrest

theof inertiabody darticulateI

b

b

S

A

q ,q Q, ,p

October, 23 2007 24


• Finding the articulated-body inertia (IA) and bias force (p)

sIs

pvvIsQsvvIppp S

vSvv

2

221221221

sIsIssI

III S

SA

1

2221

iiiv

i vIvp p Q,

Where is called the velocity-product force and is defined to be

vip

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• These formulas can be reformulated recursively, so allow us to find and in terms of only and

• Our algorithm is then– Calculate the series of articulated body inertias and bias forces

– Using these inertias and bias forces, calculate the joint accelerations

• Since these are both defined recursively, they each take O(n), making the entire algorithm O(n)

AiI

ip A

iI 1

1ip

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Lagrange-Multiplier Method

•The preceding methods are reduced-coordinate formulations– These methods remove some of the dof’s by enforcing a

set of constraints (a joint can only rotate in a certain direction constraining the motion of the joint and the link)

– Finding a parameterization for the generalized coordinates in terms of the reduced coordinates is not always easy

•The Lagrange Multiplier Method considers all the d.o.f.’s of the system

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• Consider the equation of motion of i bodies

– M describes the mass properties of the system is an d x d matrix where d is the number of dof’s of body i when not constrained

• Also consider a constraint i that removes m dof’s from the system, we can write it as

– Where each jik is a m x d matrix that represents the constraint on link k where• d is again the number of dof’s of body k and• m is the number of dof’s removed by the constraint

iii FvM

0cvjvjvj ininkik1i1

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• To simplify the notation, we replace the q individual constraint equations

• With – Where J is a q x n matrix of the individual jik matrices

• Where q is the total number of constraints on the system and

• n is the number of bodies

– c is a q dimensional vector

0cvjvjvj

0cvjvjvj

0cvjvjvj

nnkk11

nnkk11

nnkk11

qqqq

2222

1111

0cvJ

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• Just as we did when solving the constrained particle dynamics problems, we require that the constraint does no work. This results is a constraint force of the form:

– Where λi is an m (dof’s removed by constraint i) dimensional column vector and is referred to as the Lagrange multiplier

• The problem is now just to find a λ so the constraint forces and any external forces satisfy the constraints

λJλ

j

j

F Ti

inT

i1T

ci

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• If we introduce an external force acting on the system and combine the equations, we get

• Solving for and plugging into our constraint equation, we get

• For constraints that act on two bodies, the matrix system is tightly banded and can be solved in O(n)– Using banded Cholesky decomposition, for example

cFJMλJJM ext1T1

extT FλJvM

v

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• For more complicated constraints, is no longer sparse and we reformulate the equation as

• If we required acyclic constraints, then sparse-matrix theory tells us that has perfect elimination order– This means that if we factor into three matrices LDLT, L will be as sparse as H

and can be computed in O(n)

– We can then solve for λ, by solving each piece of LDLT

separately, each also in O(n) time and combining the solutions

cFJM

0

λ

yLDL ext1

T

0J

JM T

T1JJM

cFJM

0

λ

y

0J

JMext1

T

0J

JM T

October, 23 2007 32

Summary

• Inverse Dynamics– Newton-Euler is the standard implementation in O(n)

•Forward dynamics– Composite-rigid-body algorithm is simpler and faster for

n < 9, runs in O(n3)– Articulated-body algorithm is faster for n > 9, runs in

O(n)– Lagrange multiplier method is somewhat simpler than

ABA and speed is comparable, runs in O(n)

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References / Thanks• R. Featherstone, Robot Dynamics Algorithms,

Boston/Dordrecht/Lancaster: Kluwer Academic Publishers, 1987.

• D. Baraff, "Linear-Time Dynamics using Lagrange Multipliers," Proc. SIGGRAPH '96, pp. 137-146, New Orleans, August 1996.

• Thanks to Nico for his slides from last year

Articulated Body Dynamics The Basics

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