Physically Based Motion Transformation
Zoran Popovi´c Andrew Witkin
∗
Computer Science Department
Carnegie Mellon University
Abstract
We introduce a novel algorithm for transforming character anima-
tion sequences that preserves essential physical properties of the
motion. By using the spacetime constraints dynamics formulation
our algorithm maintains realism of the original motion sequence
without sacrificing full user control of the editing process.
In contrast to most physically based animation techniques that
synthesize motion from scratch, we take the approach of motion
transformation as the underlying paradigm for generating computer
animations. In doing so, we combine the expressive richness of an
input animation sequence with the controllability of spacetime op-
timization to create a wide range of realistic character animations.
The spacetime dynamics formulation also allows editing of intu-
itive, high-level motion concepts such as the time and placement of
footprints, length and mass of various extremities, number of body
joints and gravity.
Our algorithm is well suited for the reuse of highly-detailed cap-
tured motion animations. In addition, we describe a new methodol-
ogy for mapping a motion between characters with drastically dif-
ferent numbers of degrees of freedom. We use this method to re-
duce the complexity of the spacetime optimization problems. Fur-
thermore, our approach provides a paradigm for controlling com-
plex dynamic and kinematic systems with simpler ones.
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional
Graphics and Realism—Animation; G.1.6 [Numerical Analysis]:
Optimization.
Keywords: Human Body Simulation, Physically Based Anima-
tion, Animation with Constraints
1 Introduction
Controllable automatic synthesis of realistic character motion is a
difficult problem. The motion of a character with many degrees
of freedom (DOFs) needs to be consistent with the laws of physics.
More importantly, in order for the motion to look realistic,the entire
musculoskeletal structure must be taken into account. Controlling
this complex motion generation process adds further difficulties. In
this paper, we present a solution to the problem of generating both
controllable and realistic character animations. Instead of motion
∗
now at Pixar Animation Studios.
synthesis, we take the approach of motion transformation. For ex-
ample, we transform a human running sequence by restricting the
range of motion for a knee joint to obtain a realistic run with a limp.
Any dynamically sound motion, such as captured motion or the
result of a physical simulation, can be used as an input to our trans-
formation algorithm. The first step of our algorithm constructs a
simplified character model and fits the motion of the simplified
model to the captured motion data. From this fitted motion we
obtain a physical spacetime optimization solution that includes the
body’s mass properties, pose, and footprint constraints, muscles and
the objective function. To edit the animation we modify the con-
straints and physical parameters of the model and other spacetime
optimization parameters (e.g. limb geometry, footprint positions,
objective function, gravity.) From this altered spacetime parame-
terization we compute a transformed motion sequence. Finally, we
map the motion change of the simplified model back onto the orig-
inal motion to produce a final animation sequence.
Once the spacetime model has been constructed from the input
data, our algorithm can be turned into a motion library, since each
change in the physical formulation of the model produces a new
motion sequence. Thus, acaptured motion sequence of a human run
can be turned into a running motion library capable of generating
all possible runs that fit the needs of the animator.
Our algorithm presents the first solution to the problem of edit-
ing captured motion taking dynamics into consideration. We also
describe a novel methodology for mapping motion between charac-
ters with drastically different kinematic structure. In addition, we
introduce a method for simplification of complex dynamic systems
without losing the fundamental dynamic properties of motion.
The next section describes how this work relates to other re-
search efforts. We follow with the algorithm outline, and proceed
to describe each stage of the algorithm in detail. In Section 8 we
report the results of the algorithm’s application on two human cap-
tured motion sequences. We conclude the paper with the main con-
tributions and future research directions.
2 Related Work
Forward dynamics methods compute motions of objects that obey
the laws of physics. For rigid objects (e.g. [5, 4, 21]), or secondary
motion of cloth (e.g. [12, 6]), forward dynamics techniques are
ideal because obeying physical laws is synonymous with realism.
However, active characters create motion with their own muscles.
The specific motion of real creatures depends on their intricate mus-
culoskeletal structure. Determining exact muscle forces that would
make the animation look realistic is extremely difficult. In addition,
with dynamics methods each animation frame depends on the pre-
vious frame (and consequently on all other preceding frames). The
smallest change of dynamic properties of any single frame drasti-
cally affects all consecutive frames, resulting in lack of controlla-
bility.
Spacetime constraints approach effectively addresses the need
for both realism and controllability of character motion [33, 9, 20,
28]. In the spacetime framework the user first specifies pose con-
straints that must be satisfied by the resulting motion sequence (e.g.
the character pose at the beginning and end of the animation). In ad-
dition to these constraints, the user also specifies an objective func-
tion that is a metric of performance or style such as total power con-
sumption of all of the character’s muscles. The algorithm takes this
spacetime specification and finds the motion trajectories that min-
imize the objective function while satisfying the constraints. High
realism and intuitive control give this method great appeal. The
downside, however, is that the current algorithms do not scale up to
the complexity of characters one would like to animate. The time
complexity of the spacetime formulation and convergence difficul-
ties remain a huge impediment. Another problem of these methods
is that they are extremely sensitive to the starting position of the
optimization process — if optimization begins far away from the
solution, the optimization methods often cannot converge to the op-
timal motion. As a result, spacetime optimization methods have
not been successfully applied to automatic generation of human
motion. Our work draws from the ideas of spacetime constraints
and tries to address the issues that prevent this method from being
applied to complex character models.
Robot controller design has also been applied to the domain of
realistic computer animation (e.g. [26, 32, 31, 18]). These methods
use controllers that drive the actuator forces based on the current
state of the environment. These forces, in turn, produce desired
motion. Intuitively, controllers can be thought of as a set of in-
stinctual reflexes that control muscles and collectively produce a
character’s continuous motion. Once the controllers have been fine-
tuned and synchronized to each other, this method can produce a
wide range of expressive animations [26, 19]. Furthermore, a num-
ber of different animations can be created without any additional
work, because the controllers adjust to the changes in the environ-
ment. Recently, van de Panne introduced an interesting method for
generating motion from footsteps that includes some rudimentary
physical properties[30]. Although a controller transformation algo-
rithm has been reported [17], determining controllers that produce
realistic character motion is extremely difficult, and has not been
formalized.
Another way to generate realistic motion is to attain it from the
real world. Recent availability of real-time 3-D motion capture sys-
tems provides such an alternative. Motion capture systems use sen-
sors to record absolute positions of key points on the character’s
body over a period of time. Not surprisingly, the resulting clip mo-
tion is convincing and rich with expressive detail that is almost im-
possible to generate by any computer methods. Although this type
of animation is quite realistic, it yields highly unstructured and un-
correlated motion, even when converted to joint angles within a
hierarchical character model.
Recently, a number of motion capture editing methods have been
proposed [34, 8, 15, 14, 16, 27]. These methods don’t generate mo-
tion “from scratch” like earlier described methods, but transform
existing motion sequences. Even though some of the methods solve
for the entire transformed motion sequence (and thus use the term
spacetime), they do not include any notion of dynamics. Recently
Gleicher [16] introduced a method for remapping captured motion
onto drastically different characters. While his method is capable
of producing many interesting motions, it has no means of mak-
ing the motion physically realistic. For example, a tall and lanky
character would utilize his muscles (and therefore move) in very
different ways than a more compact character. A property of all
motion editing methods that ignore inherent dynamics is that while
they can effectively transform motion by small amounts, larger de-
formations reveal undesirable, unrealistic artifacts.
An alternative approach to editing realistic motion sequences is
to extract the physical model from captured data, and perform all
editing on the computer model instead. Spacetime optimization is
a good candidate for this task. Of course, the primary problem
of spacetime methods is that they do not guarantee a solution for
3
Spacetime edit
Spacetime Motion
Reconstruction
Spacetime
2
Simplification
1
Final Motion
Original Motion
Motion Model
Fitting
Transformed
Model
Complex
Simplified
Model
4
1
Figure 1: Algorithm outline.
motion problems of complex characters such as humans. We cir-
cumvent this issue by developing smaller, abstracted models. This
model simplification is motivated by biomechanics research [7].
Blickhan and Full demonstrate the similarity in the multi-legged
locomotion of kinematically different animals. They show striking
similarities between a human run, a horse run and the monopode
bounce (i.e. pogostick). This similarity motivates our approach to
reducing the DOF count of complex kinematic structures such as
humans.
Biomechanics also studies the postulated optimality of motion
in nature [1, 2, 24]. There has been a considerable amount of work
in the area of human performance in sports [22]. This work also
reaffirms that spacetime optimization is a good choice for realistic
motion synthesis.
3 Algorithm Outline
Much like the motion capture editing methods, our algorithm does
not synthesize motion from ground zero. Instead, it transforms the
input motion sequence to satisfy the needs of the animation. Al-
though our algorithm was motivated by the desire to enable real-
istic high-level control of high quality captured motion sequences,
the same methods can be applied to motion of arbitrary source.
At its core our algorithm uses spacetime optimization because
the spacetime formulation maintains the dynamic integrity of mo-
tion and provides intuitive motion control. Because such methods
have not been shown to be feasible for human motion models, we
must also find a way to simplify the character model.
The entire transformation process breaks down to four main
stages (Figure 1):
Character Simplification. Create an abstract character model
containing the minimal number of degrees of freedom nec-
essary to capture the essence of the input motion. Map the
input motion onto the simplified model.
Spacetime Motion Fitting. Find the spacetime optimization prob-
lem whose solution closely matches the simplified character
motion.
Spacetime Edit. Change spacetime motion parameters, introduce
new pose constraints, change the character kinematics, objec-
tive function, etc.
Motion Reconstruction. Remap the change in motion introduced
by the spacetime edit onto the original motion to produce the
final animation.
The character simplification and spacetime motion fitting stages
require a significant amount of human intervention. However, once
c)b)
a)
Figure 2: Kinematic character simplification: a) elbows and spine
are abstracted away, b) upper body reduced to the center of mass,
c) symmetric movement abstraction.
the spacetime model is computed it can be reused to generate a
wide range of different animations. The spacetime edit and motion
reconstruction stages are fully automated. They also take much less
time to compute than the first two stages, which enables computa-
tion of transformed motion sequences at near-interactive speeds.
4 Character Simplification
Instead of solving spacetime constraint optimizations on the full
character, we first construct a simpler character model which we
then use for all spacetime optimizations. There are two important
reasons for character simplification:
• DOF reduction improves performance and facilitates conver-
gence of the spacetime optimization.
• creation of an abstract model that contains only DOFs essen-
tial for the given motion captures the more fundamental prop-
erties of the body movement. As a result, detailed motion in
the input sequence will be preserved during the transforma-
tion process.
Simplified models capture the minimum amount of structure neces-
sary for the input motion task, and therefore capture the “essence”
of the input motion. Subsequent motion transformations modify
this abstract representation while preserving the specific feel and
uniqueness of the original motion. Our simplificationprocess draws
from certain ideas in the biomechanics research [7]. We take the
view that, abstractly speaking, highly dynamic natural motion is
created by “throwing the mass around,” or changing the relative
position of body mass. With this in mind a human arm with more
than 10 DOFs can be represented by a rigid object with only three
shoulder DOFs without losing much of the mass displacement abil-
ity. Simplification of certain body parts also depends on the type
of the input motion. For example, while the above mentioned arm
simplification may work well for the human run motion, it would
not appropriately represent the ball-throwing motion.
Simplification reduces the number of kinematic DOFs, as well
as muscle DOFs by a factor of two to five. Since each DOF is rep-
resented by hundreds of unknown coefficients during the optimiza-
tion, simplification can reduce the size of the optimization by as
many as 1000 unknowns. More importantly, a character with fewer
DOFs also creates constraints with significantly smaller nonlinear-
ities. In practice, the optimization has no convergence problems
with the simplified character models.
Character simplification is performed manually. We apply three
basic principles during this process:
DOF removal. Some body parts are fused together removing
DOFs that link them together. Elbow and wrist DOFs are
usually removed for running and walking motion sequences
where they have little impact on the motion.
Node subtree removal. In some cases of high-energy motion the
entire subtree of the character hierarchy can be replaced with
a single object, usually a mass point with three translational
DOFs. For example, the upper body of a human character
can be reduced to a mass point for various jumping motion
sequences where the upper body catapults in the direction of
the jump.
Exploit symmetric movement. Broad jump motions contain in-
herent symmetry since both legs move in unison. Thus, we
can abstract both legs with one, turning the character into a
monopode.
The simplification process ensures that the overall mass distribution
is preserved, so if a number of nodes are represented with a single
object we match the mass, center of mass and moments of inertia
of the new structure to be as close to the original as possible.
Once the character model has been simplified the original mo-
tion can be mapped onto it. Since the simplified character has
significantly fewer DOFs, this mapping is over-determined. We
define handles to aid us in the motion transfer process by corre-
lating essential properties between complex and simplified motion
sequences.
4.1 Handles
Handles are multi-valued time-varying functions that can be evalu-
ated on both complex and simplified character models. All handles
depend on the character pose defined by the vector of values for
each DOF q(t
i
). They represent intuitive measurements of various
body properties such as 3D point positions, 3D directions, distance
between two assigned body points.
Some 3D position handles are simply points on the character’s
body (e.g. the foot-ground contact position). Others, like the center
of mass position handle, depend on a much larger set of DOFs.
Direction handles are often used to represent the orientation of the
character. When kinematic topology has been drastically changed
during the simplification, it is often useful to use distance handles
to correlate various body points.
In order to match two animations, we ensure equality between
the corresponding handles. For example, if we reduced the human
upper body down to a mass point, we would use the center of mass
handle to correlate two animations. When two legs are reduced to
one, we would use the foot-floor contact point handle, defined as
the midpoint between two foot contact points and equate it with the
foot point handle of the monopode.
Let us define the collection of all handles of the original (com-
plex) motion as h
o
(q
o
(t)),andleth
s
(q
s
(t)) be the corresponding
simplified motion handles. We find the motion of the simplified
character by solving
E
d
= [h
o
(q
o
(t
i
)) − h
s
(q
s
(t
i
))]
2
(1)
min
q
s
(t
i
)
E
d
(2)
for each frame t
i
. This process is equivalent to solving an inverse
kinematics problem for each time frame of the animation. Natu-
rally, there should be at least as many handles as there are DOFs
in the simplified character. That way the simplified motion is fully
determined by h
s
(q
s
(t)).
5 Spacetime Motion Fitting
Handles help us map the original motion onto the simplified char-
acter. However, the resulting motion is no longer dynamically cor-
rect. Before we can edit the motion with spacetime constraints we
need to create not only dynamically correct but also realistic mo-
tion of the simplified model. In other words, we need to find the
spacetime optimization problem whose solution comes very close
to the simplified model motion we computed in section 4.1. Sec-
tion 5.1 describes the spacetime constraints formulation of motion.
Subsequent sections describe our approach to finding the appropri-
ate muscles, spacetime constraints and the objective function which
would yield the motion closely matching the input sequence.
5.1 Spacetime Constraints Formulation
We obtain the body dimensions and mass distributions from biome-
chanics sources [11, 23]. All other concepts of spacetime optimiza-
tion have their intuitive counterparts in real-life.
A character is an object performing motion of its own accord. It
has a finite number of kinematic DOFs and a number of muscles.
DOFs usually represent joint angles of the character’s extremities,
while muscles exert forces or torques on different parts of the body,
thus actuating locomotion. Given that both body and muscle DOFs
change through time we refer to them collectively as q(t), or sepa-
rately as kinematic q
k
(t), and muscle DOFs q
m
(t).
The task of motion synthesis is to find the desired motion of a
character. This “goal motion” is rarely uniquely specified; rather,
one looks for a motion that satisfies some set of requirements. Gen-
erally these requirements are represented either through constraints,
external forces or through the objective function.
For instance, the requirements of a sequence that animates a per-
son getting up from a chair would include the fact that the person is
sitting in the chair at time t
0
and standing up at final time t
1
.Were-
fer to such requirements as pose constraints (C
p
), and we insist that
the character must use its own muscles to satisfy these constraints.
In addition to pose constraints, the environment imposes a num-
ber of mechanical constraints (C
m
) onto the body. For example,
in order to enforce the upright position of a human, we need to
constrain both of her feet to the floor. The floor exerts forces onto
the feet ensuring that the feet never penetrate the floor surface. All
mechanical constraints provide external forces necessary to satisfy
the constraints. There may also be other external forces within the
environment such as gravity and wind.
Finally, we also need to ensure dynamic correctness of the mo-
tion. We do this by constraining the acceleration of each DOF. Intu-
itively, we make sure that F = ma holds for all degrees of freedom
at all times. In this document we call such constraints dynamics
constraints (C
d
). As long as these constraints are satisfied, we know
that the resulting motion is physically possible, given the muscles’
ability to generate forces.
When motion is defined in this way, it straightforwardly maps
onto a non-linearly constrained optimization problem: we optimize
the objective function E(q(t), t) parameterized in space and time,
subject to the pose, mechanical and dynamics constraints:
min
q(t)
E(q(t), t) subject to
C
p
(q(t), t) = 0
C
m
(q(t), t) = 0
C
d
(q(t), t) = 0
(3)
This optimization is a variational calculus problem, as we solve for
functions, not values. Such problems are solved by continuous op-
timization methods, which require computation of first derivatives
of all constraints and the objective function. We use a sparse SQP
[13] method to solve spacetime optimization problems.
The following sections describe specifics of determining mus-
cles, constraints and the objective function that produce realistic
motion and closely match the input animation.
5.2 Muscles
Muscles are the primary source of character locomotion. The
biomechanics community has developed a number of complex mus-
cle models, which closely match empirical data [29, 10, 3]. While
these models tend to be very accurate, their complexity makes them
difficult to differentiate and use in full body optimizations. Since
our character model is drastically simplified, it would not make
much sense to apply realistic muscles on simplifiedkinematic struc-
ture. Instead, we use simple structures that account for entire mus-
cle groups, yet still induce forces onto DOFs similar to those of real
muscles.
We use generalized muscle forces Q to represent the abstract
muscle. These muscles apply accelerations directly onto DOFs,
much like robotic servo-motors positioned at joints apply forces on
robotic limbs. Having a generalized muscle at each character DOF
presents the minimum set of muscles that ensures the full range of
character motion. Unfortunately, the ability to apply arbitrary gen-
eralized force onto each joint is a poor model of natural muscles.
For example, sudden non-smooth muscle forces generate extremely
jerky, unnatural motion much like motion generated by bang-bang
controllers [25].
In addition, arbitrary impulse muscle forces tend to produce
highly unstable spacetime optimization problems with poor con-
vergence properties, because the problem becomes badly scaled.
This becomes apparent when we compare the relative change in
motion resulting from changing a single coefficient of a kinematic
DOF q
k
(t) and a generalized muscle DOF q
m
(t) by the same fixed
amount δq. Naturally, motion changes are orders of magnitude
more drastic when we displace q
m
(t) coefficients, since muscles
directly affect accelerations of many kinematic DOFs. This imbal-
ance in sensitivity between the coefficients of kinematic and gener-
alized muscle DOFs makes it difficult for any optimization methods
to converge to a solution.
To circumvent the problems of simple generalized force mus-
cles described above, yet still maintain a simple and differentiable
muscle model, we use a damped servo model often used in robotic
simulations [26, 19]. Each kinematic DOF q
k
i
has a corresponding
damped generalized muscle force
Q
k
i
= k
s
(q
k
i
− q
m
i
) − k
d
( ˙q
k
i
−˙q
m
i
) (4)
where q
m
i
is the additional muscle DOF that is often interpreted as
the desired value of q
k
i
. Differentiation with respect to DOFs and
their velocities is straightforward. This formulation does not ex-
hibit scaling problems since q
m
i
is of the same scale as q
k
i
,andthe
velocity dependent damping encourages smoothness. To further en-
sure smooth muscle forces akin to those found in nature, we always
include a muscle smoothness metric within the objective function
(see Section 5.4.)
5.3 Constraints
Most of the pose and mechanical constraints fall out of the nature
of the input motion. For example, in a run or walk sequence we
specify mechanical point constraints during each period the foot is
in contact with the floor. Similarly, a leg kick animation defines
a pose constraint at the time the leg strikes the target. We avoid
specifying extraneous constraints that are not essential for the input
motion, since they reduce the flexibility of the subsequent space-
time editing process.
The model simplification process may also introduce additional
constraints. For example, if the upper body was reduced to a mass
point, the mass point DOFs need to be restricted to stay within the
bounds of the upper body center of mass. This ensures that move-
ment of mass points can never cause an improper human configu-
ration.
Additional pose constraints can be introduced for further control
during motion editing. For example, we can introduce a hurdle
obstacle into the human jump motion environment, which forces
the character to clear a certain height during flight.
5.4 Objective Function
There has been much research into the optimality of motion in na-
ture [1, 2, 24]. We, however, avoid guessing the right objective
function altogether. Instead, we rely on the fact that the starting
motion is very close to the optimum. At first, our objective mea-
sured the deviation from the original motion (E
d
) as described by
Equation 1. We also include the muscle smoothness objective com-
ponent E
m
=
¨
q
2
m
at all times. The spacetime objective is a weighted
sum of the two objective components.
E = w
d
Z
E
d
+
Z
E
m
(5)
Once the Newtonian constraint residuals become small, we gradu-
ally decrease w
d
all the way to zero. The existence of the E
d
com-
ponent early in the optimization process prevents the optimization
method from diverging from the initial motion until the spacetime
constraints are satisfied (i.e. until the dynamics integrity of the mo-
tion has been established). This approach ensures that the space-
time minimization process stays near the input motion, while at the
same time keeps the muscle forces smooth.
Upon convergence, we end up with a spacetime problem defi-
nition whose solution is very close to the original motion. With
the spacetime optimization problem successfully constructed, the
intuitive “control knobs” of the spacetime constraints formulation
can be edited to produce a nearly inexhaustible number of different
realistic motion sequences.
6 Spacetime Edit
A spacetime constraints parameterization provides powerful and in-
tuitive control of many aspects of the dynamic animation: pose and
environment constraints, explicit kinematic and dynamic properties
of the character, and the objective function.
By changing existing constraints the user can rearrange foot
placements both in space and time. For example, a human run se-
quence can be changed into a zig-zag run on an uphill slope by
moving the floor contact constraints wider apart and progressively
elevating them. The constraint timing can also be changed: extend-
ing the floor contact time duration of one leg creates an animation
that gives the appearance of favoring one leg. We can also introduce
new obstacles along the running path, producing new constraints
that, for example, require legs to clear a specified height during the
flight phase of the run. We can also affect the environment of the
run by changing the gravity constant, producing a human running
sequence on the moon surface, for example.
Changes can also be made on the character model itself. We can
change the limb dimensions or their mass distribution characteris-
tics, and observe the resulting dynamic change of the motion. We
can remove body parts, restrict various DOFs to specific ranges, or
remove DOFs altogether, effectively placing certain body parts in a
cast. For example, we can create different gimpy run sequences by
shortening the leg, making one leg heavier, reducing the range of
motion for the knee DOF, removing the knee DOF. Various muscle
properties of the character can also affect the look of transformed
motion. We can limit the force output of the muscles, forcing the
character to compensate by using other muscles.
Finally, the overall “feel” of the motion can be changed by
adding additional appropriately weighted objective components.
For example, we can produce a softer looking run by adding an
objective component that minimizes floor impact forces. Or we can
make the run look more stable by including a measure of static bal-
ance in the objective.
After each edit we re-solve the spacetime optimization problem
and produce a new transformed animation. Since the optimization
starting point is near the desired solution, and all dynamic con-
straints are satisfied at the outset, optimization converges rapidly.
In practice, while the initial spacetime optimization may take more
than 15 minutes to converge, the spacetime optimizations during
the editing process take less than two minutes.
7 Motion Reconstruction
In order to create the transformed animation of the full character
model, we reconstruct the final motion from the original motion and
two simplified spacetime motions. We apply the transformation to
the original sequence so that we modify the fundamental dynamic
properties of motion, while preserving the specific intricate details
in the original.
The reconstruction relies on both spacetime constraints and mo-
tion handles as described in Section 4.1. All spacetime constraints
are mapped to their full character equivalents. For example, foot
placement constraints are mapped onto foot constraints of the full
character. Having completed the spacetime editing stage, we have
three distinct sets of handles
1
• original motion handles h
o
(q
o
)
• spacetime fit handles h
s
(q
s
)
• transformed spacetime handles h
t
(h
t
)
We define the final motion handles as
h
f
(q
f
) = h
o
(q
o
) + (h
t
(q
t
) − h
s
(q
s
)) (6)
essentially displacing the original handles by the difference be-
tween the two spacetime solutions. Since the right side of the
equation is known, it would seem that solving for the inverse-
kinematics-like problem of finding q
f
that satisfies equation 6
would complete the reconstruction. Unfortunately, the number of
handles is considerably smaller than the number of DOFs in the
full character, so this problem is highly under-determined, and we
cannot directly solve for q
f
without accounting for the extra DOFs.
We formulate the reconstruction process as a sequence of per-
frame subproblems:
min
q
f
E
dm
(q
o
, q
f
)
subject to
C(q) = 0
h
f
(q
f
) = h
o
(q
o
) + (h
t
(q
t
) − h
s
(q
s
))
(7)
Simply stated, we follow the transformed handles and satisfy all
constraints (C(q)) while we try to be as close as possible to the
original motion.
We first formulate a measure of closeness to the original motion.
A simple objective function that measures the deviation of each
DOF E
dd
= (q
f
− q
o
)
2
produces undesirable results. Each DOF
needs to be carefully scaled both with respect to what it measures
(joint angles measure radians, translational DOFs measure meters),
and with respect to its importance within the character hierarchy.
For example, the change of the hip joint DOF affects the overall
motion significantly more than the same amount of change applied
to the ankle joint. In order to avoid these problems, we designed
a completely new objective E
dm
that measures the amount of dis-
placed mass between the two poses.
1
For clarity, we omit the explicit time dependency of handles and DOFs.
