Original: Acceleration Skinning: Kinematics-Driven Cartoon Effects for Articulated Characters
Revised: Acceleration Skinning: Kinematics-Driven Cartoon Effects for Articulated Characters

Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: A BSTRACT
Revised: A BSTRACT

Original: Cartoon effects described in animation principles are key to adding2 fluidity and style to animated characters.
Revised: Cartoon effects described in animation principles are key to adding fluidity and style to animated characters.

Original: This paper extends theexisting framework of Velocity Skinning to use skeletal acceleration,4 in addition to velocity, for cartoon-style effects on rigged characters.
Revised: This paper extends the existing framework of Velocity Skinning to use skeletal acceleration, in addition to velocity, for cartoon-style effects on rigged characters.

Original: This Acceleration Skinning is able to produce a variety of cartoon6 effects from highly efficient closed-form deformers while remainingcompatible with standard production pipelines for rigged characters.
Revised: This Acceleration Skinning is able to produce a variety of cartoon effects from highly efficient closed-form deformers while remaining compatible with standard production pipelines for rigged characters.

Original: This paper showcases the introduction of the framework along withproviding applications through three new deformers.
Revised: The paper showcases the introduction of the framework along with providing applications through three new deformers.

Original: Specifically, a10 followthrough effect is obtained from the combination of skeletal acceleration and velocity.
Revised: Specifically, a followthrough effect is obtained from the combination of skeletal acceleration and velocity.

Original: Also, centrifugal stretch and centrifugal lift12 effects are introduced using rotational acceleration to model radialstretching and lifting effects.
Revised: Also, centrifugal stretch and centrifugal lift effects are introduced using rotational acceleration to model ra- dial stretching and lifting.

Original: The paper also explores the applica- 14 tion of effect-specific time filtering when combining deformationstogether allow for more stylization and artist control over the results.
Revised: The paper also explores the application of effect-specific time filtering when combining deformations together allowing for more stylization and artist control over the results.

Original: 1 I NTRODUCTION
Revised: 1 I NTRODUCTION

Original: The fundamental principles of animation laid out by Walt Disney19 Studios remain an important driving factor for animators today [26].
Revised: The fundamental principles of animation laid out by Walt Disney Studios remain an important driving factor for animators today [26].

Original: These principles, created by the pioneers of traditional hand-drawn21 animation, have been guiding artists on how to create effects such assquash and stretch and followthrough to add vitality to animation.
Revised: These principles, created by the pioneers of traditional hand-drawn animation, have been guiding artists on how to create effects such as squash and stretch and followthrough to add vitality to animation.

Original: Many techniques have been proposed to adapt such cartoon effectsto suit the world of 3D animation production [14], however their25 application still remains a non-trivial task.
Revised: Many techniques have been proposed to adapt such cartoon effects to suit the world of 3D animation production [14], however their application still remains a non-trivial task.

Original: Currently, professionalanimators generally create such effects either through rigged anima-27 tion or using custom, manually controlled deformers [17].
Revised: Currently, professional animators generally create such effects either through rigged anima- tion or using custom, manually controlled deformers [17].

Original: Whilethese approaches allow fine artistic control, they become tedious29 in adapting the work to multiple characters and effects.
Revised: While these approaches allow fine artistic control, they become tedious in adapting the work to new characters and animated sequences.

Original: As an alternative, physically-based simulation can produce the effect in a31 fully automatic fashion, but often requires expert tuning and costlycomputation which does not adapt easily to the usual production33 pipeline, including the iterative animation workflow.
Revised: As an alternative, physically-based simulation can produce the effect in a fully automatic fashion, but often requires expert tuning and costly computation which does not adapt easily to the usual production pipeline, including the iterative animation workflow.

Original: Recently, a technique called Velocity Skinning [23] (VS) intro-35 duced a method able to automatically represent a subset of cartooneffects, including squash and stretch and drag / floppy effect, that can37 be applied on top of a character deformed by skinning solely using a38 standard rig.
Revised: Recently, a technique called Velocity Skinning [23] (VS) intro- duced a method able to automatically represent a subset of cartoon effects, including squash and stretch and drag / floppy effect, that can be applied on top of a character deformed by skinning solely using a standard rig.

Original: As its name suggests, it relies on the use of velocity at39 the joint level to drive the deformations, providing real-time effects40 that automatically adapt to the animation and hierarchy of the rigged41 character.
Revised: As its name suggests, it relies on the use of velocity at the joint level to drive the deformations, providing real-time effects that automatically adapt to the animation and hierarchy of the rigged character.

Original: The approach provides simple and direct artist control42 over the deformation that works with animator workflow, and does43 not require costly physical simulation.
Revised: The approach provides simple and direct artist control over the deformation that works with animator workflow, and does not require costly physical simulation.

Original: In this paper, we propose a natural extension we call [-Acceleration 
-] Skinning (AS) which builds off of this previous work.
Revised: In this paper, we propose a natural extension we call [-Acceleration 
-]Accelera- tion Skinning (AS) which builds off of this previous work.

Original: Namely, we ex46 plore and showcase additional effects for skinning animation based47 on the use of acceleration in the pipeline, in conjunction with the VS48 approach.
Revised: Namely, we explore and showcase additional effects for skinning animation based on the use of acceleration in the pipeline, in conjunction with the VS approach.

Original: To this end, we propose a more general framework that49 combines both, as described in Sec. 3.
Revised: To this end, we propose a more general framework that combines both, as described in Sec. 3.

Original: This generalization allows50 a wider set of deformers while preserving the original advantages51 of the VS approach, such as its high efficiency and art directabil52 ity.
Revised: This generalization allows a wider set of deformers while preserving the original advantages of the VS approach, such as its high efficiency and art directability.

Original: Foremost, we showcase the additional use of acceleration, in53 combination with velocity, to model overshooting deformation, a54 common effect which is often called followthrough in animation 55 guides (Sec. 4).
Revised: Foremost, we showcase the additional use of acceleration, in combination with velocity, to model overshooting deformation, a common effect which is often called followthrough in animation guidebooks (Sec. 4).

Original: Further, as acceleration provides a natural separation56 of coordinate frames, e.g. associated to centrifugal effects when57 rotational motions are involved, we also propose additional parame58 terized deformation effects.
Revised: Further, as acceleration provides a natural separation of coordinate frames, e.g. associated to centrifu- gal effects when rotational motions are involved, we also propose additional parameterized deformation effects.

Original: Specifically, we introduce centrifugal59 stretch , for the elongation of an object that rotates, and centrifugal60 lift , for the elevation of a rotating object, such as a twirling skirt61 (Sec. 5).
Revised: Specifically, we intro- duce centrifugal stretch , for the elongation of an object that rotates, and centrifugal lift , for the elevation of a rotating object, such as a twirling skirt (Sec. 5).

Original: Finally, we extend the general (VS/AS) domain further by62 exploring the use of time filters to highlight and tune effects based63 on artist control (Sec. 6).
Revised: Finally, we extend the general (VS/AS) domain further by exploring the use of time filters to highlight and tune effects based on artist control (Sec. 6).

Original: 64 In total, the contributions of this paper are to extend the capa65 bilities of VS and skinning overall, and add a set of interesting,66 AS-specific effects with little overhead in implementation and cost67 while still supporting a convenient workflow for artist-driven cartoon 68 effects.
Revised: In total, the contributions of this paper are to extend the capa- bilities of VS and skinning overall, and add a set of interesting, AS-specific effects with little overhead in implementation and cost while still supporting a convenient workflow for artist-driven cartoon animation.

Original: 2 R ELATED W ORK
Revised: 2 R ELATED W ORK

Original: Much of the production animation created today for characters de71 pends on skinning, such as Linear Blend Skinning (LBS) [16], where the surface geometry (skin) of the character is moved in relation to73 an articulated skeleton.
Revised: Much of the production animation created today for characters de- pends on skinning, such as Linear Blend Skinning (LBS) [16], where the surface geometry (skin) of the character is moved in relation to an articulated skeleton.

Original: The dependence between the joints of theskeleton and the vertices of the skin is defined by scalar skinning75 weights and in combination with the skeleton and joints constitutethe rig for character skinning.
Revised: The dependence between the joints of the skeleton and the vertices of the skin is defined by scalar skinning weights and in combination with the skeleton and joints constitute the rig for character skinning.

Original: LBS is a straightforward implemen-77 tation of skinning where the deformation is computed as a linearcombination of the joint transforms.
Revised: LBS is a straightforward implemen- tation of skinning where the deformation is computed as a linear combination of the joint transforms.

Original: A large set of work in computer79 animation research has improved upon LBS skinning techniquesbut LBS remains a common tool in practice.
Revised: A large set of work in computer animation research has improved upon LBS skinning techniques but LBS remains a common tool in practice.

Original: As such, our gen-81 eral formulation for Acceleration Skinning is derived for the basic
Revised: As such, our gen- eral formulation for Acceleration Skinning is derived for the basic

Original: LBS formulation, but the acceleration-skinning deformers proposed83 may also be applied to other methods, such as the Dual Quaternionapproach [8], another common skinning technique.
Revised: LBS formulation, but the acceleration-skinning deformers proposed may also be applied to other methods, such as the Dual Quaternion approach [8], another common skinning technique.

Original: Traditional skinning defines (static) skin poses of the character 86 that can be derived from interpolating keyframes from an animation.
Revised: Skinning techniques define (static) skin deformation of the char- acter. This deformation is guided from a skeletal pose traditionally derived from interpolating keyframes.

Original: However, this approach only interpolates in-between poses and88 does not model dynamic effects, such as drag at high speeds, or followthrough due to inertia.
Revised: However, this approach only interpolates in-between poses and does not model dynamic effects, such as drag at high speeds, or followthrough due to inertia.

Original: To this end, physically-based simulation90 has been used to represent so-called secondary actions on top ofkeyframe motion.
Revised: To this end, physically-based simulation has been used to represent so-called secondary actions on top of keyframe motion.

Original: We distinguish three main approaches to compute92 secondary motion.
Revised: We distin- guish three main approaches to compute secondary motion.

Original: First, projection-based dynamics [18] and its extensions [3] have been proposed to modeling muscle and soft tissue94 giggling effects [10,19,25] and can be tuned to exaggerate specificcartoon-like effects [2,4].
Revised: First, projection-based dynamics [18] and its extensions [3] have been pro- posed to modeling muscle and soft tissue giggling effects [10,19,25] and can be tuned to exaggerate specific cartoon-like effects [2,4].

Original: Second, layering techniques have been96 proposed where distinct effects are handled as simple physical models, using jiggling implicit surfaces [22], spring-driven bones [11],98 or custom volume effects such as squash and stretch [1].
Revised: Second, layering techniques have been proposed where distinct effects are handled as simple physical models, using jiggling implicit surfaces [22], spring-driven bones [11], or custom volume effects such as squash and stretch [1].

Original: Third,reduced deformable models have been introduced [6,28,30] where100 precomputed shape analysis allows for model reduction, and therefore sped-up computation.
Revised: Third, reduced deformable models have been introduced [6,28,30] where precomputed shape analysis allows for model reduction, and therefore sped-up computation.

Original: Work is on-going and new techniques[-
-] for producing simulation effects are still being proposed, such asefficient simulated cartoon deformations applied on top of skinning104 animation [31], which defines the space of secondary effects as theorthogonal subspace of the rig.
Revised: Related techniques[-
-] for producing simulation effects are still being proposed as well, such as efficient simulated cartoon deformations applied on top of skinning animation [31], which defines the space of secondary effects as the orthogonal subspace of the rig.

Original: In total, these physically-based ap- 106 proaches are able to model rich and detailed dynamic exaggerateddeformations, possibly with collision handling, but they share two108 main drawbacks.
Revised: In total, these physically-based approaches are able to model rich and detailed dynamic exaggerated deformations, possibly with collision handling, but they share two main drawbacks.

Original: First, physically based simulations are limitedwith respect to efficiency.
Revised: First, physically based simulations are limited with respect to efficiency.

Original: Even fast approaches, that can tolerate110 models with thousands of vertices, cannot scale easily to a largenumber of meshes at the same time –as may be required for video112 games applications– or very detailed mesh of several millions ofvertices, while geometric methods such as skinning are still orders114 of magnitude faster.
Revised: Even fast approaches, that can tolerate models with thousands of vertices, cannot scale easily to a large number of meshes at the same time –as may be required for video games applications– or very detailed mesh of several millions of vertices, while geometric methods (such as VS and AS as well as traditional skinning) are still orders of magnitude faster.

Original: Second, simulation requires numerical timeintegration computed from the initial conditions.
Revised: Second, simulation requires numerical time in- tegration computed from the initial conditions.

Original: This integration116 imposes the animation to be baked before being visualized, whichhamper the efficiency of the animator by limiting the possibility118 of navigating and editing details at arbitrary instances along theanimation timeline.
Revised: This integration imposes the animation to be baked before being visualized, which hampers the efficiency of the animator by limiting the possibility of navigating and editing details at arbitrary instances along the animation timeline.

Original: As such, standard production pipelines [17,21]120 often favor procedural deformers that are independent of past statewhen possible.
Revised: As such, standard production pipelines [17,21] often favor procedural deformers that are independent of past state when possible.

Original: Aside from simulation, geometric and kinematics-based defor-123 mations have also been proposed in the literature.
Revised: Aside from simulation, geometric and kinematics-based defor- mations have also been proposed in the literature.

Original: For instance,sketch- [9,15] and example-based [5,24] techniques aim to precisely125 define shape deformations for expressive effects, but require manualinput for each shape.
Revised: For instance, sketch- [9,15] and example-based [5,24] techniques aim to precisely define shape deformations for expressive effects, but require manual input for each shape.

Original: In contrast, procedural approaches allow more127 automatic adaptation while remaining compute efficient, as theytake advantage of the skeletal structure position and kinematics.
Revised: In contrast, procedural approaches allow more automatic adaptation while remaining compute efficient, as they take advantage of the skeletal structure position and kinematics.

Original: For129 example, Nobel et al.
Revised: For example, Nobel et al.

Original: [20] introduce a local bending behavior parameterized by the direction and velocity between consecutive bones.
Revised: [20] introduce a local bending behavior parameterized by the direction and velocity between consecutive bones.

Original: 131 Yong et al.
Revised: Yong et al.

Original: [12] propose a squash and stretch deformer parameterized by root joint velocity, and a drag effect on end-effector joints of133 a skinned character.
Revised: [12] propose a squash and stretch deformer parameterized by root joint velocity, and a drag effect on end-effector joints of a skinned character.

Original: Procedural bone stretching was also proposedby Kwon and Lee [13] for local squash and stretch effects.
Revised: Procedural bone stretching was also proposed by Kwon and Lee [13] for local squash and stretch effects.

Original: Recently,135 Rohmer et al.
Revised: Recently, Rohmer et al.

Original: [23] introduced Velocity Skinning (VS) that can beseen as a generic framework to define local deformers such as squash137 and stretch and drag based on joint velocity.
Revised: [23] introduced Velocity Skinning (VS) that can be seen as a generic framework to define local deformers such as squash and stretch and drag based on joint velocity.

Original: Like the proposed workhere, the VS approach fits well to standard animation pipeline in139 reusing the existing rig (i.e. original skeleton and skinning weights)140 for automatic parameterization of deformation.
Revised: Like the proposed work here, the VS approach fits well to standard animation pipeline in reusing the existing rig (i.e. original skeleton and skinning weights) for automatic parameterization of deformation.

Original: VS also provides a141 closed form procedural deformation that can be computed extremely142 efficiently for each vertex in parallel, similar to raw skinning.
Revised: VS also provides a closed form procedural deformation that can be computed extremely efficiently for each vertex in parallel, similar to basic skinning.

Original: Our143 approach follows VS, but extends their approach to  acceleration144 terms.
Revised: Our approach follows VS, but extends the approach to add acceleration terms.

Original: This allows additional deformation behaviors that cannot be145 achieved in using velocity alone, such as the proposed followthrough 146 effect that requires deformationjust as the velocity approaches zero.
Revised: This allows additional deformation behaviors that cannot be achieved in using velocity alone such as the proposed followthrough effect, that requires deformation as the velocity approaches zero.

Original: 147 We also note some cartoon effects have been generated using time148 filtering [27] or oscillating splines [7] applied to vertex trajectories.
Revised: We also note some cartoon effects have been generated using time filtering [27] or oscillating splines [7] applied to vertex trajectories.

Original: 149 Our approach also parallels these works in applying low-pass time150 filters in order to control the relative timing and magnitude of effects151 relative to the instantaneous joint values of the skeleton.
Revised: Our approach also parallels these works in applying low-pass time filters in order to control the effect-specific timing and magnitude of effects relative to the instantaneous joint values of the skeleton.

Original: 3 A CCELERATION S KINNING
Revised: 3 A CCELERATION S KINNING

Original: In LBS, the deformed position p of an arbitrary vertex at an initial154 position p 0 can be expressed as the linear blending
Revised: In LBS, the deformed position p of an arbitrary vertex at an initial position p 0 can be expressed as the linear blending

Original: [Equation]
Revised: [Equation]

Original: where j is the index of a joint, b j ∈ [ 0 , 1 ] are the skinning weights156 that should satisfy ∑ j b j = 1.
Revised: where j is the index of a joint, b j ∈ [ 0 , 1 ] are the skinning weights that should satisfy ∑ j b j = 1.

Original: p j are the rigid skinning deformation of the position p 0 relative to the joint j .
Revised: p j are the rigid skinning deformation of the position p 0 relative to the joint j .

Original: As shown in Rohmer et158 al [23], differentiating this expression in time, and reorganizing the159 terms along the skeleton hierarchy leads to the following “skinning160 like” relation in velocity
Revised: As shown in Rohmer et al [23], differentiating this expression in time, and reorganizing the terms along the skeleton hierarchy leads to the following “skinninglike” relation in velocity

Original: [Equation]
Revised: [Equation]

Original: where v is the net velocity of the vertex at position p , v / j is the162 contribution coming from the rigid transformation of the joint j163 only to the net velocity.
Revised: where v is the net velocity of the vertex at position p , v / j is the contribution coming from the rigid transformation of the joint j only to the net velocity.

Original: And ˜ b j are the kinematics-weights that are164 obtained from a combination of skinning weights
Revised: And ˜ b j are the kinematics-weights that are obtained from a combination of skinning weights

Original: [Equation]
Revised: [Equation]

Original: with Desc ( j ) being the descendants of the joint j in the skeleton166 hierarchy.
Revised: with Desc ( j ) being the descendants of the joint j in the skeleton hierarchy.

Original: 167 We note that this derivation is valid for any arbitrary degree of168 differentiation in time.
Revised: Wenote that this derivation is valid for any arbitrary degree of differentiation in time.

Original: Therefore, relation in Eq.
Revised: Therefore, relation in Eq.

Original: (2) can also be169 differentiated, which leads to an acceleration-like skinning
Revised: (2) can also be differentiated, which leads to an acceleration-like skinning

Original: [Equation]
Revised: [Equation]

Original: With this time a being the net acceleration of the vertex at position171 p , and a / j is the contribution coming from the joint j to the net172 acceleration.
Revised: with a being the net acceleration of the vertex at position p , and a / j is the contribution coming from the joint j to the net acceleration.

Original: 173 Both velocity v / j and acceleration a / j contributions are related to174 rigid motions, and we can therefore breakdown their expressions in175 relation to linear/angular velocity and acceleration of their respective176 joint as
Revised: Both velocity v / j and acceleration a / j contributions are related to rigid motions, and we can therefore breakdown their expressions in relation to linear/angular velocity and acceleration of their respective joint as

Original: [Equation]
Revised: [Equation]

Original: v Lj and a Lj are respectively the linear velocity and acceleration of178 joint j . ω
Revised: v Lj and a Lj are respectively the linear velocity and acceleration of joint j . ω

Original: j is the angular velocity, and α j = ˙ ω
Revised: j is the angular velocity, and α j = ˙ ω

Original: j is the angular accel179 eration vector.
Revised: j is the angular acceleration vector.

Original: r j = p − proj j ( p ) is the relative vector between p and its orthogonal projection proj j ( p ) onto the rotation axis passing by181 the joint j and oriented along ω j .
Revised: r j = p − proj j ( p ) is the relative vector between p and its orthogonal projection proj j ( p ) onto the rotation axis passing by the joint j and oriented along ω j .

Original: Note that while velocity depends182 only on two components – linear and angular –, the acceleration183 depends on three, respectively, the linear component ( a Lj ), angu184 lar component ( α j × r j ), and centripetal component( ω j ×  ω j × r j ).
Revised: Note that while velocity depends only on two components (linear and angular), the acceleration depends on three, respectively, the linear component ( a Lj ), angular component ( α j × r j ), and centripetal component ω j × ( ω j × r j ).

Original: These components are illustrated in Figure 2 for velocity and accel- 186 eration.
Revised: These components are illustrated in Figure 2 for velocity and acceleration.

Original: Aligned with Rohmer et al.
Revised: Aligned with Rohmer et al.

Original: [23] the general idea in accelera-188 tion skinning is to define “deformers” ( ψ s) as procedural functionsrepresenting a specific type of parameterized deformation at the190 individual joint level.
Revised: [23] the general idea in accelera- tion skinning is to define “deformers” ( ψ s) as procedural functions representing a specific type of parameterized deformation at the individual joint level.

Original: Subsequently, the deformers are combinedover the skeleton hierarchy as Eq.
Revised: Subsequently, the deformers are combined over the skeleton hierarchy as Eq.

Original: (4) in order to distribute globally192 the total deformation over the skinning surface as
Revised: (4) in order to distribute globally the total deformation over the skinning surface as

Original: [Equation]
Revised: [Equation]

Original: d ( p ) is the net deformation to p such that the final deformed po- 194 sition is p final=
Revised: d ( p ) is the net deformation to p such that the final deformed position is p final

Original:  p + d ( p ) .
Revised: = p + d ( p ) .

Original: Further, the deformer ψ can itself bedecomposed into a linear sum of individual sub-effects ψ effect suchthat
Revised: Further, the deformer ψ can itself be decomposed into a linear sum of individual sub-effects ψ effect such that

Original: [Equation]
Revised: [Equation]

Original: In this paper, we adopt the deformers from the Velocity Skin- 198 ning work (e.g. drag) and propose three unique deformers that areconstructed using the components of the derived acceleration terms200 from Eq.
Revised: In this paper, we adapt the deformers from the Velocity Skinning work (e.g. drag) and propose three unique deformers that are constructed using the components of the derived acceleration terms from Eq.

Original: Namely, we propose formulations for new deformersthat we call respectively ψ ft for followthrough, ψ cs for centrifugalstretch, and ψ cl for centrifugal lift.
Revised: Namely, we propose formulations for new deformers that we call respectively ψ ft for followthrough, ψ cs for centrifugal stretch, and ψ cl for centrifugal lift.

Original: These are described in the following sections in detail.
Revised: These are described in the following sections in detail.

Original: As each deformer ψ effect is defined in ageneric way for any joint j we will omit to explicitly mention the205 dependence to the index j in its velocity and acceleration parametersfor notation clarity.
Revised: As each deformer ψ effect is defined in a generic way for any joint j we will omit to explicitly mention the dependence to the index j in its velocity and acceleration parameters for notation clarity.

Original: 4 F OLLOWTHROUGH DEFORMER
Revised: 4 F OLLOWTHROUGH DEFORMER

Original: Followthrough is a fundamental principle of animation and has been209 used extensively in animation throughout history as a way to conveythe inertia of the animated shape (see Figure 3.
Revised: Followthrough is a fundamental principle of animation and has been used extensively in animation throughout history as a way to convey the inertia of the animated shape (see Figure 3).

Original: Drag and follow- 211 through are two effects that often go hand in hand.
Revised: Drag and fol- lowthrough are two effects that often go hand in hand.

Original: We describeour approach to represent followthrough effect next, relying on the213 combination of velocity and acceleration information.
Revised: We describe our approach to represent followthrough effect next, relying on the combination of velocity and acceleration information to extract the salient details needed to construct the effect.

Original: Recall the velocity-based floppy deformer ψ floppy was proposedin VS to model a drag effect.
Revised: A velocity-based floppy deformer ψ floppy was proposed in VS to model a drag effect.

Original: This deformer was split into two216 parts either coming from the linear velocity or the angular velocity.
Revised: This deformer was split into two parts either coming from the linear velocity or the angular velocity.

Original: The linear velocity related effect was associated with the deformerψ floppy , lin representing a translation, while the angular velocityrelated effect was associated with the deformer ψ floppy , ang representing bending.
Revised: The linear velocity related effect was associated with the deformer ψ floppy , lin representing a translation, while the angular velocity related effect was associated with the deformer ψ floppy , ang representing bending.

Original: [Equation]
Revised: [Equation]

Original:  where R ( θ , u ) is the rotation matrix parameterized by an angle θ222 and an axis u , and Id is the identity matrix.
Revised: [Equation] where R ( θ , u ) is the rotation matrix parameterized by an angle θ and an axis u , and I is the identity matrix.

Original: K lin and K ang areuser223 defined coefficients allowing the scaleof the magnitude of each 224 deformation.
Revised: K lin and K ang are coefficients that an animator may use to scale the magnitude of each deformation separately.

Original: A straightforward extension of suchmodel to  followthrough con226 sists in substituting the linear and angular acceleration terms instead227 of their velocity counterpart in these floppy deformers.
Revised: A straightforward extension of such to model followthrough con- sists of substituting the linear and angular acceleration terms instead of their velocity counterpart in these floppy deformers.

Original: However,228 this substitution leads to undesirable artifacts.
Revised: However, this substitution leads to undesirable artifacts.

Original: Consider a  motion 229 –either linear or angular– with three phases: an acceleration phase230 at its start; a constant velocity phase; and a deceleration phase at231 the end before the motion stops.
Revised: Consider a generic motion (linear or angular) with three phases: an acceleration phase at its start; a constant velocity phase; and a deceleration phase at the end before the motion stops.

Original: These three phases are illustrated 232 in Figure 4 (and Figure 1,left) in the case of an angular motion, 233 comparing the velocity-based deformer to the acceleration-based234 one.
Revised: These three phases are illustrated for a simple angular motion in Figure 4 (and Figure 1, left) comparing the velocity-based deformer to the acceleration-based one.

Original: For such  case, the proposed acceleration-basedfloppy deformer235 will first start by applying a draglike deformation,then continue236 with no deformation, and end during the deformation phase with an237 opposite effect of drag, i.e. an overshot of the deformation beyond the final pose.
Revised: For such a case, the proposed acceleration-based deformer will first start by applying a draglike deformation, continue with no deformation, and end during the deformation phase with an opposite effect of drag, i.e. an overshot of the deformation beyond the final pose.

Original: This last phase corresponds to the expected behavior for fol-240 lowthrough representation.
Revised: This last phase corresponds to the expected behavior for fol- lowthrough representation.

Original: However, the current deformer alsocontains a draglike behavior  which is not expected in the first phase.
Revised: However, the proposed deformer also contains a draglike behavior followed by no deformation which is not expected.

Original: To avoid this unexpected behavior, we modified the proposed deformer by filtering based on the relative direction between the current244 velocity and acceleration parameters.
Revised: To avoid this unexpected behavior, we modified the proposed deformer by filtering based on the relative direction between the current velocity and acceleration parameters.

Original: More precisely, we introducea smooth indicator function D that we characterize by the conditionthat the velocity component is aligned and opposite with the acceler- 247 ation component.
Revised: More precisely, we introduce a smooth indicator function D that we characterize by the condition that the velocity component is aligned and opposite with the acceleration component.

Original: Specifically, we propose the followingindicatorfunction
Revised: Specifically, we propose the following

Original: [Equation]
Revised: [Equation]

Original: where the parameter D ∈ [ 0 , 1 ] allows to adapt how fast the tran- 250 sition to the deceleration phase is computed.
Revised: where the parameter D ∈ [ 0 , 1 ] allows to adapt how fast the transition to the deceleration phase is computed.

Original: We then define thefollowthrough (ft) deformer, ψ ft , as
Revised: We then define the followthrough (ft) deformer, ψ ft , as

Original: [Equation]
Revised: [Equation]

Original:  Results obtained using this deformer are illustrated on a circularmotion in Figure 5.
Revised: (10) Results obtained using this deformer are illustrated on a circular motion in Figure 5.

Original: In addition, we can further take advantage of the smooth indicator255 to separate the contribution of drag from the followthrough.
Revised: In addition, we can further take advantage of the smooth indicator to separate the contribution of drag from the followthrough.

Original: Doingso, we propose additional pure acceleration drag (ad) deformer ψ adas
Revised: Doing so, we propose additional pure acceleration drag (ad) deformer ψ ad as ψ ad , lin ( v L , a L ) [Equation] [Equation] ψ ad , ang ( ω , α ,

Original: [Equation]
Revised: [Equation]

Original: (11) which will apply a drag effect during the acceleration phase of themotion (see Figure 6,top).
Revised: ) = [Equation] [Equation] [Equation]

Original:  Note that both deformers ψ ft and ψ adcan be used together on an animation and parameterized separately261 in adapting their respective coefficients K lin and K ang for additionaluser control (see example in Figure 6,bottom).
Revised: ang [Equation] (11) which will apply a drag effect during the acceleration phase of the motion (see Figure 6, top). Note that both deformers ψ ft and ψ ad can be used together on an animation and parameterized separately in adapting their respective coefficients K lin and K ang for additional user control (see example in Figure 6,bottom).

Original: Note that this no-263 tion of combining drag with followthrough remains consistent withtraditional animation concepts [29] (Figure 3, top).
Revised: Note that this no- tion of combining drag with followthrough remains consistent with traditional animation concepts [29] (Figure 3, top).

Original: 5 C ENTRIFUGAL - BASED DEFORMERS
Revised: 5 C ENTRIFUGAL - BASED DEFORMERS

Original: Centrifugal motion during rotation of a physical system moves away267 radially from the axis.
Revised: Centrifugal motion during rotation of a physical system moves away radially from the axis.

Original: This movement is countered by centripetalacceleration in general.
Revised: This movement can be countered by centripetal acceleration if conditions are favorable.

Original: We exploit this relationship to proposetwo 269 new deformers.
Revised: However, we can also exploit this relationship to propose new deformation effects.

Original: Namely, as noted in Eq.
Revised: Namely, as noted in Eq.

Original: (5), the net acceleration of avertex following a rotating motion contains a centripetal component271 acting radially around the rotation axis, and expressed as ω ×  ω × r  .
Revised: (5), the net acceleration of a vertex fol- lowing a rotating motion contains a centripetal component acting radially around the rotation axis, and expressed as ω × ( ω × r ) .

Original: In contrast to the angular component, this component exists even when the magnitude of the velocity remains constant, as in the[Figure] case of a circular motion with constant angular velocity.
Revised: This component exists even when the magnitude of the velocity remains constant, as in the case of a circular motion with constant angular velocity.

Original: Wetake 275 employ this accelerationterm in two distinct effects, first, with a[Figure] generic centrifugal stretch deformer ψ cs , and second, in a centrifugal [Figure]
Revised: We employ this acceleration in two distinct effects, first, with a generic centrifugal stretch deformer ψ cs , and second, in a centrifugal “lift” deformer ψ cl .

Original: “lift” deformer, ψ cl .

Original: By their nature, both are applicable for rotating[Figure] motion only.
Revised: By their nature, both are applicable for rotating motion only.

Original: 5.1 Centrifugal stretch
Revised: 5.1 Centrifugal stretch

Original: [Figure] When observing how chefs make pizza, we see that after kneading281 and pressing the dough, they proceed to toss it in the air with a spin282 of their wrists.
Revised: When observing how chefs make pizza, we see that after kneading and pressing the dough, they proceed to toss it in the air with a spin of their wrists.

Original: This spin is associated with centripetal acceleration,283 which in return leads to a centrifugal effectexperienced by any posi284 tion placed in this rotating frame that stretches the dough further out.
Revised: This spin is associated with centripetal acceleration, which in return leads to a centrifugal effect in the rotating frame that stretches the dough radially out.

Original: 285 Inspired from this notion, we propose the creation of the centrifugal286 stretch deformer that reproduce such  effect with
Revised: Inspired from this notion, we propose the creation of the centrifugal stretch deformer that reproduces such an effect with

Original: [Equation]
Revised: [Equation]

Original: Note, as the centrifugal effect acts outward from the axis of rotation 288 as visualized in Figure2, weneed to negate the centripetal acceler289 ation that is directed inward.
Revised: Note, as the centrifugal effect acts outward from the axis of rotation (as visualized in Figure 2), we negate the centripetal acceleration which is directed inward.

Original: ψ cs creates an effect that translates290 vertices outward away from the point of rotation.
Revised: ψ cs creates an effect that translates vertices outward away from the point of rotation.

Original: As the magnitude291 of acceleration increases linearly with distance, the vertices farther292 away from the center of rotation experience larger deformation than293 those closer to the center.
Revised: As the magnitude of acceleration increases linearly with distance, the vertices farther away from the center of rotation experience larger deformation than those closer to the center.

Original: A simple example appears in Figure 7.
Revised: A simple example appears in Figure 7.

Original: 5.2 Centrifugal lift
Revised: 5.2 Centrifugal lift

Original: Consider the motion of a twirling cloth, such as a dancer’s skirt. [Figure]
Revised: Inspired by the motion of a twirling cloth, such as a dancer’s skirt, we propose a centrifugal lift deformer.

Original: As the dancer twirls, their skirt gently lifts.
Revised: As the dancer twirls, the skirt gently lifts.

Original: The same centrifugal[Figure] force that caused the pizza dough to expand also applies to this [Figure]
Revised: The same centrifugal force that caused the pizza dough to expand also applies to this shape.

Original: shape.

Original: However, in the case of a cloth, the garment surface is more[Figure] inextensible, and internal stress will prevent elongation in its local [Figure]
Revised: However, in the case of a cloth, the garment (surface) is more inextensible, and internal stress will prevent elongation in its local tangent plane.

Original: tangent plane. As a result, the rotating dress may first unwrap from[Figure] its wrinkled configuration onto its extended shape before its only
Revised: As a result, the rotating dress may first unwrap from its wrinkled configuration onto its extended shape before its remaining degree of freedom are “lifted” in the relative normal direction.

Original: [Figure]

Original: remaining degree of freedom are “lifted” in the relative normal

Original: [Figure] direction. While a general solution to account for centripetal lift is difficult305 without a dynamic model, we can offer a simple solution that canbe used to approximate this centrifugal lifting effect on a conical- 307 like shapes.
Revised: While a general solution to account for such effect is difficult without a dynamic model, we can offer a simple solution that can be used to approximate this centrifugal lifting effect on conicallike shapes.

Original: Cones with rotation about their center have consistentnormals that allow us to filter the previously defined deformers to309 act only along the normal direction to create the effect.
Revised: Cones with rotation about their center have consistent normals that allow us to filter the previous stretch deformer to act only along the normal direction and create a lift effect.

Original: Our basicformulation of pure centrifugal lift ψ cl is thus
Revised: Our basic formulation of this centrifugal lift ψ cl is thus

Original: [Equation]
Revised: [Equation]

Original: where n is the local normal of an ideal cone at the current vertex312 position p ,and K cl is a user-defined parameter enabling to allowtuned exaggeration of the lift. 
Revised: where n is the local normal of an ideal cone at the current vertex position p , K cl is a user-defined parameter enabling to allow tuned exaggeration of the lift. χ ( r ) = ∥

Revised: r ∥ − r 0 represents the axial co- ordinates along the cone of base radius r 0 .

Revised: This ensures that no deformation is applied at the base of the cone.

Original: Further, any model that can be reasonably approximated by the315 cone can enjoy this deformer by using the cone as a proxy geometry and mapping the projected geometry to the latter.
Revised: Further, any model that can be reasonably approximated by the cone can enjoy this deformer by using the cone as a proxy geometry and mapping the projected geometry to the latter.

Original: For example, the317 stylized dress in Figure 8 maps to the nearest point in the conicalapproximation to yield a set of normals that are used for the skirtdeformation with ψ cl .
Revised: For example, the stylized dress in Figure 8 maps to the nearest point in the conical approximation to yield a set of normals that are used for the skirt deformation with ψ cl .

Revised: Note that the top part of the dress is stitched to the character thanks to the component χ ( r ) in Eq.

Original: This effect can further be seen as a moregeneric deformation filter that can be applied to any other deformerψ in constraining it to act in the normal direction, and can be defined322 as the functional 
Revised: This effect can further be seen as a more generic deformation filter that can be applied to any other deformer ψ in constraining it to act in the normal direction, and can be defined as the functional ψ clfilter : ψ (cid:55)→ [Equation] n

Original: [Equation]
Revised: [Equation]

Revised: The untuned filter has frames distributed nearly equally throughout the clip.

Revised: The tuned filter has more frames, with more easing at the end

Revised: It also has a arc motion.

Original: where ψ can be an arbitrary combination of the previously defined324 deformers.
Revised: where ψ can be an arbitrary combination of the previously defined deformers.

Original: 6 T IME FILTERING
Revised: 6 T IME FILTERING

Original: As a final extension to the AS approach, which can also be applied327 in the VS framework but did not appear in the previous work, weexplore the use of time filtering to offer more fine grain and use- 329 ful control over the mixing of various deformers.
Revised: As a final extension to the AS approach, which can also be applied in the VS framework but did not appear in the previous work, we explore the use of time filtering to offer more fine grain and useful control over the mixing of various deformers.

Original: To this end, wepoint out that time filtering is a useful tool in general.
Revised: To this end, we point out that time filtering is a useful tool in general.

Original: In all our 331 examples, we receive input motion provided by smooth interpolatedcurves from existing keyposes.
Revised: Time filtering brings two benefits.

Original: But more generally, we may also 333 allow non-smooth inputs, for example, coming from direct interaction where the user articulates the skeleton using a mouse. In the 335 first case, velocity and acceleration can be analytically describedfrom the keyframes, while in the latter case we consider numerical 337 differentiation from the joint frames which is smoothed by a lowpass filter to limit spurious high frequency noise.
Revised: First, filtering inputs using a low-pass filter allows our method to handle non-smooth inputs robustly, for example, coming from direct interaction where the user articulates the skeleton using a mouse. Second, when combining effects, applying different time filters allows a controlled delay on the timing of when a specific effect will occur relative to another.

Original: To this end, we 339 consider in our case a simple auto-regressive first order  filter
Revised: We considered in our case a simple auto-regressive first order low-pass filter

Original: [Equation]
Revised: [Equation]

Original: where x represents the input value (linear/angular veloc-

Original: [Figure]

Original: ity/acceleration), y its filtered value, ∆ t being the frame duration,[Figure] and γ ∈ [ 0 , 1 ] being a user-defined parameter that controls the cutoff 
Revised: where y is the value to be filtered, x represents the input (lin- ear/angular velocity/acceleration), ∆ t is the time duration, and γ ∈ [ 0 , 1 ] is a user-defined parameter that controls the cutoff fre- quency.

Original: [Figure]

Original: frequency. Qualitatively,  small value of γ will lead to very smooth, [Figure] butalso delayed signal, while  values near one will cause snappier 
Revised: Qualitatively, a small value of γ will lead to a smooth but delayed signal, while γ values near one will lead to snappier response times with higher frequencies.

Original: [Figure]

Original: response time with higher frequencies.

Original: [Figure]

Original: Assuming different effects can take as inputs a signal filtered with

Original: [Figure]

Original: different γ values, this allows to apply a controlled delay on the [Figure] time where a specific effect will take place relative to another one. [Figure]
Revised: To apply a controlled delay to different effects, the animator sets different γ values to each.

Original: Empirically, we find useful a long and smooth centrifugal stretch [Figure] deformation, while followthrough works best as a faster and more [Figure]
Revised: Empirically, we found useful a long and smooth centrifugal stretch deformation ( γ = 0 . 6), while followthrough works best as a faster and more temporary effect ( γ = 0 .

Original: temporary effect.

Original: This authoring of the timing of the effect is sup-

Original: [Figure] ported trivially through the described time filtering, by considering[Figure] a small γ -value for ω used in ψ ft ( ω , r ) , and a larger γ -value to354 compute α used in ψ cs ( α , r ) .
Revised: This authoring of the timing of the effect is supported trivially through the described time filtering, by considering a small γ -value for ω used in ψ ft ( ω , r ) , and a larger γ -value to compute α used in ψ cs ( α , r ) .

Original: As shown in Figure 9, adjusting these355 values allows artistic control over the resulting animation timing on356 each individual effect, while preserving a simple combination of the357 effects as a linear summation, ψ
Revised: As shown in Figure 9, adjusting these values allows artistic control over the resulting animation timing on each individual effect, while preserving a simple combination of the effects as a linear summation.

Original: = ψ ft + ψ cs in this case.

Original: 7 R ESULTS
Revised: 7 R ESULTS

Original: We illustrate and compare some AS results obtained on more com360 plex animations that can also be seen in the associated video.
Revised: We illustrate and compare some AS results obtained on more com- plex animations that can also be seen in the associated video.

Original: All361 the illustrations are computed in real time (approx.
Revised: All the illustrations are computed in real time (approx.

Original: 60 fps) on a 362 common laptop using a non-optimized CPU implementation.
Revised: 60 fps) on a machine with an Intel 2.70 GHz multi-core processor and 32GB RAM, using a non-optimized CPU implementation.

Original: The363 cow and flower examples have a rig of  about 10 joints and up to a364 thousandof triangles.
Revised: The cow and flower examples have a rig of with about ten joints and up to a thousand triangles.

Original: Although we did not propose a GPU imple365 mentation in this paper, the computational time of AS is similar to366 VS and shares the same properties: it is fully compatible with highly efficient computation in a single-pass vertex shader that has been 368 shown to be applicable up to millions of vertices [23]. 
Revised: Although we did not propose a GPU implementation in this paper, the computational time of AS is similar to VS and shares the same properties: it is fully compatible with highly effi- cient computation in a single-pass vertex shader that was shown to be applicable up to millions of vertices [23]. In our implementation, the computational cost associated to the application of AS-deformers is roughly similar to the one spent in computing LBS. As thus, a full AS computation (LBS + AS-deformers) is twice slower than a raw

Original: However,the non-sparsity of the weights ˜ b limits the possibility of extremeoptimization that can be applied on raw LBS.
Revised: LBS. However, please note that the non-sparsity of the weights ˜ b limits the possibility of extreme optimization that can be applied on basic LBS.

Original: Figure 11 shows the comparison of a motion made by an artist372 of a flower pot “jumping” from left to right.
Revised: Figure 11 shows the comparison of a motion made by an artist of a flower pot “jumping” from left to right.

Original: The top row shows theraw animation set by the artist using LBS, while the bottom row374 illustrates the result obtained after addingthe AS deformers.
Revised: The top row shows the basic animation set by the artist using LBS, while the bottom row illustrates the result obtained after adding AS deformers.

Original: Thebending of the flower obtained during the acceleration (drag effect)376 and deceleration phases (followthrough effect) are highlighted withthe red rectangles.
Revised: The bending of the flower obtained during the acceleration (drag effect) and deceleration phases (followthrough effect) are highlighted with the red rectangles.

Original: Figure 12 shows a comparison between VS and378 AS on an animated cow illustrating the conceptual example shownin Figure 3.
Revised: Figure 12 shows a comparison between VS and AS on an animated cow illustrating the conceptual example shown in Figure 3.

Original: The main difference here between AS and VS can be380 noticed at the end of the motion during the deceleration phase wherethe followthrough deformer is acting in the AS animation.
Revised: The main difference here between AS and VS can be noticed at the end of the motion during the deceleration phase where the followthrough deformer is acting in the AS animation.

Revised: We note that an intuitive analogy can be made between the related deformers and material properties.

Revised: While velocity-based deformers represent the effect of friction (or fluid drag) during a motion, acceleration-based deformers relate to momentum and inertial effects.

Revised: In addition, the time-filtering models delay these effects and can therefore be linked to modifying the strain rates of visco-elastic materials.

Revised: While not explicit in their connection to these physical properties, we find these analogies useful for aiding in animators use of the described deformers.

Original: While both VS and AS propose a notion of “stretch” deformers, 383 they differ sharply in their effect and are applicable in separate scenarios (see Figure 13).
Revised: Note, while both VS and AS propose a notion of “stretch” de- formers, they differ sharply in their effect and are applicable in separate scenarios (see Figure 13).

Original: VS stretch relies on a scaling transformation385 to produce a squash and stretch effect.
Revised: VS stretch relies on a scaling transformation to produce a squash and stretch effect.

Original: It requires an axis built froman ad-hoc frame using the skeletal structure and a notion of relative387 centroid associated to a limb and its descendant.
Revised: It requires an axis built from an ad-hoc frame using the skeletal structure and a notion of relative centroid associated to a limb and its descendant.

Original: VS stretch is relevant to illustrate cartoon effects that are elongated along the line389 of action and allow stretching as well as squeezing in that direction.
Revised: VS stretch is relevant to illustrate cartoon effects that are elongated along the line of action and allow stretching as well as squeezing in that direction.

Original: On the contrary, AS centrifugal stretch relies on local translation391 only that does not require the precomputation of such centroid , andits direction is fully defined from the joint transformation.
Revised: On the contrary, AS centrifugal stretch relies on local translation only that does not require the precomputation of such centroid , and its direction is fully defined from the joint trans- formation.

Original: However,393 the lack of notion of centroid does not allow to represent directly anotion of squeezing centered around the moving limb.
Revised: However, the lack of notion of centroid does not allow to represent directly a notion of squeezing centered around the moving limb.

Original: 8 C ONCLUSION
Revised: 8 C ONCLUSION

Original: We introduce Acceleration Skinning as a real-time skinning tech-397 nique that employs the key components of articulated skeletal ac- [Table] celeration for cartoon effects.
Revised: We introduce Acceleration Skinning as a real-time skinning tech- nique that employs the key components of articulated skeletal acceleration for cartoon effects.

Original: The general idea is to utilize specific399 acceleration terms to expand the gamut of deformation effects avail- [Table] able through the skinning pipeline.
Revised: The general idea is to utilize specific acceleration terms to expand the gamut of deformation effects available through the skinning pipeline.

Original: The approach is meant to be401 employed in conjunction with the recent work in Velocity Skinning [23].
Revised: The approach is meant to be employed in conjunction with the recent work in Velocity Skinning [23].

Original: In this paper, we showcase its efficacy by creating three403 new deformation effects, including automatic followthrough, a common design tool called out by classic 2D animators.
Revised: In this paper, we showcase its efficacy by creating three new deformation effects, including automatic followthrough, a common design tool called out by classic 2D animators.

Original: Beyond, AS is405 a general approach with an expandable collection of effects.
Revised: Beyond, AS is a general approach with an expandable collection of effects.

Original: This method is not intended as a replacement to physical simu-407 lation of deformation, but rather as an artist-driven tool to createstylized secondary motion.
Revised: This method is not intended as a replacement to physical simu- lation of deformation, but rather as an artist-driven tool to create stylized secondary motion.

Original: In support of such, it is easy to control,409 and works “out of the box” on standard skeletal rigs.
Revised: In support of such, it is easy to control, and works “out of the box” on standard skeletal rigs.

Original: As the effects[Table] are tied to the skinning rig and not the animation, AS (and VS) are411 useful for real-time animation effects, even for unseen motion inputs.[Table]
Revised: As the effects are tied to the skinning rig and not the animation, AS (and VS) are useful for real-time animation effects, even for unseen motion inputs.

Original: AS extends the capabilities of LBS and VS to empower animators[Table] through a simple tool that bootstraps existing pipelines.Further, in
Revised: AS extends the capabilities of LBS and VS to empower animators through a simple tool that bootstraps existing pipelines.

Original: [Table] this paper, we explore the power of effect-specific time filtering and[Table] its relevance to enhancing artist control.
Revised: Further, in this paper, we explore the power of effect-specific time filtering and its relevance to enhancing artist control.

Original: However, unlike physical[Table] modeling, the technique is not able to handle collisions, interaction 
Revised: However, unlike physical modeling, the technique is not able to handle collisions, interaction forces, and other nuanced dynamic effects that are generated by direct simulation.

Original: [Table]
Revised: In addition, our implementation of the lift de- former is not fully generic as the conical approximation is currently pre-computed manually.

Original: forces, and other nuanced dynamic effects that are generated by [Table] direct simulation. [Table]
Revised: Automatic conical fit could be explored in the future to apply this deformer in a seamlessly.

Original: While we describe effects that hearken back to traditional anima-[Table] tion, the technique is not limited to this restriction.
Revised: While we describe effects that hearken back to traditional anima- tion, the technique is not limited to this restriction.

Original: For example,421 one possible future extension may be to model oscillatory motion[Table] using a skinning system.
Revised: For example, one possible future extension may be to model oscillatory motion using a skinning system.

Original: Hypothetically, if we were to expand the[Table] system to include higher order derivatives, more oscillations could [Table]
Revised: Hypothetically, if we were to expand the system to include higher order derivatives, more oscillations could be modeled.

Original: be modeled. Therefore, a possible direction for future work lies in[Table] employing this technique for higher derivatives.
Revised: Therefore, a possible direction for future work lies in employing this technique for higher derivatives.

Original: Because the work

Original: [Table]

Original:  can be applied in real-time, we are excited to investigate interactive[Table] applications, for example the use of such in gaming.As an AS
Revised: Because the work can be applied in real-time, we are excited to investigate interactive applications, for example the use of such in gaming.

Original: [Table] rig works independently of the animation, it may be applied as an[Table] add-on for interactive character animation, such as in games or in a430 virtual reality (VR) setting.
Revised: As an AS rig works independently of the animation, it may be applied as an add-on for interactive character animation, such as in games or in a virtual reality (VR) setting.

Original: In summary we introduce the AS method that is a natural exten432 sion to VS, pushing further into the potential for deformable effects433 layered over LBS, and providing a wider collection of stylized de434 formations for artists to employ. 
Revised: In summary we introduce the AS method that is a natural exten- sion to VS pushing further into the potential for deformable effects layered over LBS and providing a wider collection of stylized de- formations for artists to employ. A CKNOWLEDGMENTS

Revised: The authors acknowledge the foundational efforts of the original Velocity Skinning team and Bellairs Workshop on Animation 2020.

Revised: We also thank Eric Patterson, Jerry Tessendorf