Animation Principles in Motion Planning and Realistic Motion
Animation
Szirmay-Kalos László
Animation = time dependence
T
1
(
t
)
T
2
(
t
)
T
V
(
t
)
Transformations
shape
color
Optical attributes, etc.
Real-time animation and
discrete time simulation
void IdleFunc( )
{//
idle call back
static float tend = 0;
const float dt = 0.01; // dt is
”
infinitesimal
”
float
tstart
= t
end
;
t
end
= glutGet(GLUT_ELAPSED_TIME)
/1000.0f
;
for(float t
= tstart; t < tend; t += dt) {float Dt = min(dt, tend – t);
for (Object * obj : objects) obj->Control(Dt);
for (Object * obj : objects) obj->Animate(Dt);
}
glutPostRedisplay();
}
tstart
t
end
dt
Realistic motion
•
Physical laws:
–
Newton law
–
Collision detection and response: conservation of linear
momentum and angular momentum
•
No deformation of rigid bodies
•
Physiological laws
–
Skeleton: bones meet in
joints of given degrees
of freedom
–
Skin is deformed
with
bone motion
•
Principle of lazyness
Newton law
r
(
t
) =
r
L
T
M
(
t
)
F
/
m
=
r
=
r
L
d
2
d
t
2
d
2
d
t
2
T
M
(
t
)
Force acts via some
elastic phenomenon, i.e.
changes continously
T
M
(
t
) C
2
continuous
m
T
M
(t): Motion planning
•
Requirement: generally C
2
, sometimes (C
1
,C
0
)
•
Matrix elements are not independent
–
Plan in the space of independent parameters
p
(t)
axis+angle
1
T
M
=
sx
sy
sz
1
1
1
1
px py pz
1
R
r
’=
r
cos(
)+
w
0
(
r
w
0
)(1-cos(
))+
w
0
r
sin(
)
1. scaling:
sx, sy, sz
2. orientation:
wx, wy, wz,
3. position:
px, py, pz
Motion definition in parameter space
•
p
(t) elements are of C
2
, sometimes (C
1
,C
0
)
•
p
(t) definition:
–
splines
–
With formula:
script animation
–
keyframe animation
–
With curves:
path animation
–
With forces:
physical animation
–
From measurements:
motion capture animation
Keyframe
animáció
Resulting animation curves
Additional spline animation
Path animation
Transformation in path animation
Explicit up vector
Frenet frame:
zm
=
r
’(
t
)
zm
=
r
’(
t
)
xm
=
zm
up
xm
=
zm
r
’’(
t
)
ym
=
zm
xm
ym
=
zm
xm
zm
xm
ym
T
M
=
xm
0
(
t
) 0
ym
0
(
t
) 0
zm
0
(
t
) 0
r
(
t
) 1
Vertical direction
is the direction
of the force
r
(
t
) curve:
r
(
t
)
nose
:
z
Physical animation
•
Forces (e.g. gravity,
turbulence, etc.)
•
Mass, inertia
•
Linear momentum and
angular mumentum
•
Collision detection
•
Collision response
–
Conservation laws
wrong
Dynamics of point like objects
F
(
r
,
v
,t) force
m
for( t
= 0; t < T; t += dt) {F
=
force
(
r
,
v
)
a
=
F
/m
v
+=
a
·dt
if ( CollisionDetection )
CollisionResponse
r
+=
v
·dt
}
v
n
= CollisionNormal
v’
v’ =
[
v - n(v
·
n)]-[n(v
·
n)
·
bounce]
v
ray:
r
+
v
·
t
intersection: t
*
If
t* < dt Collision
CollisionDetection
CollisionResponse
d
r
/dt =
v
d
v
/dt =
F
(r
,
v
,
t)/m
-n(v
·
n)
r
v
Continuous-Discrete collision detection
for a half-space
r
(
t
i
)
r
(
t
i+
1
)
n
·
(
r
-
r
0) = 0
n
·
(
r
-
r
0) > 0
n
·
(
r
-
r
0) < 0
v
sugár:
r
+
v
·t
metszés:
t
*
Ha
t
* < d
t
Collision
Character animation
r
w
=
r
L
·
R
hand
·
T
lowerarm
·
R
elbow
·
T
upperarm
·
R
shoulder
·
T
spline
·
T
man
r
w
r
L
Homogeneous coordinates
Skeleton
Walking cycle
Transformation hierarchy
2
l
1
l
1
1
l
1
(
x,y
,z
)
T
ranslate
(
l
1
, 0, 0
)
;
R
otate
(theta
2
, 0, 0, 1)
*
T
ranslate
(
l
1
, 0, 0
)
Translate(
l
1
, 0, 0
)
*
Rotate(theta
1
, 0, 0, 1)
Transformation hierarchy
l
1
2
1
l
1
2
1
Rotate(theta
2
, 0, 0, 1) *
T
ranslate
(
l
1
, 0, 0
)
*
Rotate(theta
1
, 0, 0, 1);
Rotate(theta
2
, 0, 0, 1) *
T
ranslate
(
l
1
, 0, 0
)
*
Rotate(theta
1
, 0, 0, 1) *
Translate(
x0, y0, z0
);
(
x
0
,y
0,z0
)
PMan
T0 =
Pman
moves ahead
Translate
(
forward, up, 0
)
T1= shoulder position
Translate(leftShoulderPos)
T2
= arm swinging
Rotate(armAngle, armRotAxis)
…
T3 =
hand position
Translate(armLength, 0, 0)
T0
T1
T2
Pman
Head
Torso
Leg1
Leg2
Arm
Arm2
T0
T1,
T2
T1
Pman
drawing and animation
class Pman {float
armAngle, dArmAngle, forward, up
;
const …
armLength, armRotAxis, rightArmJoint
, …;
public:
void Draw
Arm
(float dt
, mat4 M
) {M = Rotate(
armAngle
,
armRotAxis
)
* M
;
// T
2
DrawRefArm(M);
}
void Draw
Pman
(
) {// set matrices from animation parameters
mat4 M = Translate
(
forward, up
, 0
);
// T0
DrawRefBody(M);
DrawArm(dt, Translate(
rightArmJoint
)
* M);
// T1
DrawLeg(dt, Translate(
rightLegJoint
)
* M);
…
}
void Animate(float dt) { // calculate animation parametersarmAngle += dArmAngle * dt;
if (armAngle > 0.5 || armAngle < -0.5) dArmAngle *= -1;
forward += 5 * dt;
}
};
Pman
Arm1
Arm2
Body
Push/Pop
Inverse kinematics
T0 = moves ahead (forward, up) ???
T2 = leg rotation (ang)
Leg is not sliding
forward += leg * fabs(sin(angNew) - sin(angOld));
up = leg * cos(angNew);
leg
forward
up
ang
Inverse kinematics
l
1
2
1
l
2
up(
1
,
2
) =
l
1
sin
1
+ l
2
sin(
1
+
2
)
forward(
1
,
2
) =
l
1
cos
1
+ l
2
cos(
1
+
2
)
Skinning
3. hw: Spiderman in Torus
The game is inside a torus that is procedurally textured and diffuse-
specular. A textured spherical ball is rolling inside the torus (path
animaton), and there are two bouncing light sources. When the ball
comes, spiderman should jump into the direction given by a mouse
click to avoid collision and death. We see the scene from the point of
view of spiderman, who always looks at the direction of the ball. The
scene is Phong shaded (per-pixel shading).
T
orus
x
y
z
R
r
Ball Rolling inside the Torus
n
v
v +
(-
n
) = 0
Torus parameter space
r
(
u,v
)
u
v
u
(
t
)
,v
(
t
)
Exploring the intricate world of animation through topics like time dependence, real-time simulation, Newton's laws, and motion planning. Dive into the details of keyframe animation, resulting animation curves, and spline animation to grasp the essence of creating dynamic visual experiences.
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Presentation Transcript
Animation Szirmay-Kalos L szl
Animation = time dependence T1(t) TV(t) T2(t) Transformations shape color Optical attributes, etc.
Real-time animation and discrete time simulation tstart tend dt // idle call back void IdleFunc( ) { static float tend = 0; const float dt = 0.01; // dt is infinitesimal float tstart = tend; tend = glutGet(GLUT_ELAPSED_TIME)/1000.0f; for(float t = tstart; t < tend; t += dt) { float Dt = min(dt, tend t); for (Object * obj : objects) obj->Control(Dt); for (Object * obj : objects) obj->Animate(Dt); } glutPostRedisplay(); }
Realistic motion Physical laws: Newton law Collision detection and response: conservation of linear momentum and angular momentum No deformation of rigid bodies Physiological laws Skeleton: bones meet in joints of given degrees of freedom Skin is deformed with bone motion Principle of lazyness
Newton law m d2 dt2 r F/m = = rL d2 dt2TM(t) r(t) = rL TM(t) TM(t) C2 continuous Force acts via some elastic phenomenon, i.e. changes continously
TM(t): Motion planning Requirement: generally C2 , sometimes (C1,C0) Matrix elements are not independent Plan in the space of independent parameters axis+angle 1. scaling: 2. orientation: wx, wy, wz, 3. position: sx, sy, sz p(t) px, py, pz 1 1 1 px py pz 1 sx sy sz R TM= 1 1 r = rcos( )+w0(r w0)(1-cos( ))+w0 rsin( )
Motion definition in parameter space p(t) elements are of C2 , sometimes (C1,C0) p(t) definition: splines With formula: script animation keyframe animation With curves: path animation With forces: physical animation From measurements: motion capture animation
Keyframe anim ci 1. key 2.key 3.key 4.key 5.key
Transformation in path animation ym r(t) curve: nose: z xm xm0(t) 0 ym0(t) 0 zm0(t) 0 r(t) 1 TM= r(t) zm Explicit up vector zm = r (t) xm = zm up ym = zm xm Frenet frame: zm = r (t) xm = zm r (t) ym = zm xm Vertical direction is the direction of the force
dynamics Physical animation Forces (e.g. gravity, turbulence, etc.) Mass, inertia Linear momentum and angular mumentum Collision detection Collision response Conservation laws wrong
Dynamics of point like objects F(r,v,t) force m CollisionDetection r ray: r+v t v v dr/dt = v dv/dt = F(r,v,t)/m intersection: t* If t* < dt Collision for( t = 0; t < T; t += dt) { F = force (r, v) a = F/m v += a dt if ( CollisionDetection ) CollisionResponse r += v dt } CollisionResponse n = CollisionNormal v -n(v n) v v = [v - n(v n)]-[n(v n) bounce]
Continuous-Discrete collision detection for a half-space n (r - r0) > 0 r(ti) sug r: r+v t v n (r - r0) = 0 metsz s: t* Ha t* < dt Collision r(ti+1) n (r - r0) < 0
Character animation rL rw rw = rL Rhand Tlowerarm Relbow Tupperarm Rshoulder Tspline Tman Homogeneous coordinates
Transformation hierarchy l1 (x,y,z) Translate(l1, 0, 0); Rotate(theta2, 0, 0, 1) * Translate(l1, 0, 0) l1 2 l1 Translate(l1, 0, 0) * Rotate(theta1, 0, 0, 1) 1
Transformation hierarchy Rotate(theta2, 0, 0, 1) * Translate(l1, 0, 0) * Rotate(theta1, 0, 0, 1); 2 l1 1 2 Rotate(theta2, 0, 0, 1) * Translate(l1, 0, 0) * Rotate(theta1, 0, 0, 1) * Translate(x0, y0, z0); l1 1 (x0,y0,z0)
PMan Pman T0 T1, T2 Head Torso Leg1 Leg2 Arm Arm2 T0 = Pman moves ahead Translate(forward, up, 0) T2 T1 T1= shoulder position Translate(leftShoulderPos) T0 T2 = arm swinging Rotate(armAngle, armRotAxis) T1 T3 = hand position Translate(armLength, 0, 0)
Pman Pman drawing and animation class Pman { float armAngle, dArmAngle, forward, up; const armLength, armRotAxis, rightArmJoint, ; public: void DrawArm(float dt, mat4 M) { M = Rotate(armAngle, armRotAxis) * M; // T2 DrawRefArm(M); } void DrawPman( ) { // set matrices from animation parameters mat4 M = Translate(forward, up, 0); // T0 DrawRefBody(M); DrawArm(dt, Translate(rightArmJoint) * M); // T1 DrawLeg(dt, Translate(rightLegJoint) * M); } void Animate(float dt) { // calculate animation parameters armAngle += dArmAngle * dt; if (armAngle > 0.5 || armAngle < -0.5) dArmAngle *= -1; forward += 5 * dt; } }; Arm1 Arm2 Body Push/Pop
Inverse kinematics T0 = moves ahead (forward, up) ??? T2 = leg rotation (ang) leg ang Leg is not sliding up forward forward += leg * fabs(sin(angNew) - sin(angOld)); up = leg * cos(angNew);
Inverse kinematics 1 l1 up( 1, 2) = l1 sin 1 + l2 sin( 1 + 2) 2 l2 forward( 1, 2) = l1 cos 1 + l2 cos( 1 + 2)
3. hw: Spiderman in Torus The game is inside a torus that is procedurally textured and diffuse- specular. A textured spherical ball is rolling inside the torus (path animaton), and there are two bouncing light sources. When the ball comes, spiderman should jump into the direction given by a mouse click to avoid collision and death. We see the scene from the point of view of spiderman, who always looks at the direction of the ball. The scene is Phong shaded (per-pixel shading).
Torus ) ( y 2 + + = 2 2 2 2 0 R x z y r r x R z = + ( , ) ( r cos( u 2 )) cos( 2 ) x u v R r u v = ( , ) sin( R + 2 ) y u v = ( , ) ( cos( 2 )) sin( 2 ) z u v r u v
Ball Rolling inside the Torus Torus parameter space v r r r d ( ( ), ( )) d d u t d v t u v = = + v d d t u t v t u(t),v(t) u n r(u,v) v v + (-n ) = 0
