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Kinematics on nub

Sebastian Chaparro

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Index

1. Introduction
2. Forward Kinematics
3. Inverse Kinematics
4. IK Heuristic Methods
5. Using constraints
6. Demos

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Introduction

• How to bring to life animated objects?

• One approach is skeletal animation where the object to animate is represented by a skeleton and a skin.

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Skeleton

• Set of rigid multibody system called bones (or links) attached by joints. A skeleton usually is represented as a hierarchical structure.

• Joints are parametrized by Degrees of Fredom (DOF).

• An end effector $\mathbf{s}$ is a point of interest that depends on joint configurations: $ \mathbf{s} = f(\mathbf{ \theta }) $

• Skeleton is used to define or modify movements (e.g. [Keyframe animation](https://www.utdallas.edu/atec/midori/Handouts/keyframing.htm), [Procedural animation](https://www.alanzucconi.com/2017/04/17/procedural-animations/), [MoCap data](https://en.wikipedia.org/wiki/Motion_capture)).

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Skin

• Once motion is set, it is required to bind the skeleton with a surface (2D or 3D mesh).

• Skeleton motion must deform the mesh smoothly (e.g when we bend the elbow, the skin around the bones stretches and shrinks).

• To do so, given a mesh vertex it is defined an influence weight per skeleton joint. The vertex is deformed according to joint transformations.

• Check this short example video.

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Skinning

<iframe width="100%" height="500px" data-src="videos/Skinning.webm"></iframe>

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Forward Kinematics (FK)

• Given the joint configurations $ \mathbf{ \theta }$ find the End effector Position $ \mathbf{s} = f(\mathbf{ \theta })$


• Direct joint manipulation


• Exhaustive


• Not Redundant

<iframe class="fragment" data-fragment-index="5" width="100%" height="500px" data-src="videos/FK.webm"></iframe>

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Inverse Kinematics (IK)

• Given the state of the Final effector $ \mathbf{s} $ find joint configurations: $\mathbf{ \theta } = f^{-1}( \mathbf{ s}) $


• Indirect joint manipulation based on Goal Reaching


• Root joint is Fixed


• Not Exhaustive


• Redundant

<iframe class="fragment" data-fragment-index="6" width="100%" height="500px" data-src="videos/IK.webm"></iframe>

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Requirements

IK on interactive applications must be:

• R1 Efficient: Take as little time as possible.

• R2 Accurate: Reach the goal position / orientation.

• R3 Scalable: Work with Big amounts of DOF.

• R4 Robust: Reach the goal when managing constraints.

• R5 Able to Generate natural poses.

• R6 Generic: Deal with arbitrary Figures.

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IK Methods

Kind	R1	R2	R3	R4	R5	R6
Analitycal	X	X	-	X	X	-
Numerical	-	X	X	X	-	X
** Numerical Heuristic (FABRIK) **	X	X	X	-	X	X

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IK Heuristic Methods

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Cyclic Coordinate Descent (CCD)

Proposed by Wang and Chen on 1991

• Works only on Kinematic chains.
• Let $ \mathbf{v\_{ie}} $ the vector formed by the $ith$ joint and the end effector position (Yellow one).
• Let $ \mathbf{v\_{it}} $ the vector formed by the $ith$ joint and the target position (Green one).
• Modify each Joint configuration per iteration to reduce the error:

$$ cos(\theta \_{i}) = \frac{ \mathbf{v\_{ it }} } { \left| \mathbf{v \_{ it }} \right| } \frac{ \mathbf{v \_{ie}} }{ \left| \mathbf{v\_ {ie}} \right| } , \mathbf{r} = \mathbf{v\_{ it }} \times \mathbf{v\_{ ie }}$$

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Cyclic Coordinate Descent (CCD)

<iframe width="100%" height="500px" data-src="videos/CCD_Solver_1.webm"></iframe>

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Forward and Backward Reaching Inverse Kinematics (FABRIK)

Proposed by Andreas Aristidou on 2009

• "Minimize error by adjusting each joint angle one at a time".
• Let $ \mathbf{p}\_i$ the position of the $ ith $ joint in a chain, with $ i \in \\{ 1,2,...,n \\}$, $\mathbf{p}\_1$ the root of the chain, $\mathbf{p}\_n$ the end effector and $\mathbf{t}$ the target position.
• Move the structure while keeping distances $ d\_i = \left| \mathbf{p}\_i - \mathbf{p}\_{i+1} \right| $ between Joints (bones are rigid) via finding a point on a line.
• A full iteration is composed of two stages:
  • Foward stage: Assume that the target $\mathbf{t}$ is reached by end effector $\mathbf{p}\_n$ and adjust the distances of the remaining Joints.
  • On Backward stage: move the root $\mathbf{p}\_1$ to its initial position and adjust the distances of the remaining Joints.

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FABRIK

<iframe width="100%" height="500px" data-src="videos/FABRIK_Solver_1.webm"></iframe>

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Using constraints

• When end effectors are manipulated we expect to obtain intuitive poses.


• There could exist many solutions (i.e many different poses) that satisfy the IK problem.


• Limiting the movement of the skeleton could enhance IK performance.

<iframe class="fragment" data-fragment-index="4" width="100%" height="500px" data-src="videos/multiple_solutions.webm"></iframe>

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Using constraints

Limiting the joint movement locally by enclosing its related segment on a volume.

• [Fast and Easy Reach-Cone Joint Limits](https://pdfs.semanticscholar.org/d535/e562effd08694821ea6a8a5769fe10ffb5b6.pdf)
• [A joint-constraint model using signed distance fields](https://link.springer.com/article/10.1007/s11044-011-9296-1)

Using physical attributes.

• [An Efficient Energy Transfer Inverse Kinematics Solution](https://pdfs.semanticscholar.org/aac6/cbd168f0e01911edbe564f59d7c1a00b7535.pdf)

Locking a joint position or orientation.

• [Nailing and pinning: Adding constraints to inverse kinematics](https://otik.uk.zcu.cz/bitstream/11025/11239/1/Greeff.pdf)

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Using local constraints - Example

• Assume that Node 0 rotate only around a fixed axis (1 DOF).


• Assume that Node 0 rotation is enclosed by a minimum and a maximum angle.


• With local constraints there's a unique solution when target is reachable.

<iframe class="fragment" data-fragment-index="5" width="100%" height="500px" data-src="videos/constraint_1.webm"></iframe>

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Hinge constraint

1-DOF rotational constraint. i.e the node will rotate only around a single direction. Furthermore, the rotation made by the constrained node is enclosed on a minimum and maximum angle.

<iframe width="100%" height="400px" data-src="videos/hinge_interactive.webm"></iframe>

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Ball and socket constraint

3-DOF rotational constraint (the node could rotate around any direction) that decomposes a rotation into two components called Swing (2-DOF) and Twist (1-DOF) rotations and limits each of them (see FABRIK paper).

<iframe width="100%" height="400px" data-src="videos/BallAndSocket.webm"></iframe>

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DEMOS

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Getting skeleton from Data

There are different kind of files as Collada or BVH to "transport 3D assets between applications". Here we are interested on skeletal structure.

BVH Demo <iframe width="100%" height="400px" data-src="videos/bvh_example.webm"></iframe>

Collada Demo <iframe width="100%" height="400px" data-src="videos/dae_example.webm"></iframe>

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Building & Interacting

Allow the user to define and interact with the skeletal structure easyly.

Saying HI! <iframe width="100%" height="500px" data-src="videos/demo_2.webm"></iframe>

Multiple end Effectors <iframe width="100%" height="500px" data-src="videos/demo_skeleton.webm"></iframe>

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Skinning

Allow the user to define a skeleton and bind it to a mesh.

<iframe width="100%" height="500px" data-src="videos/builder_demo_high_speed.webm"></iframe>

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Procedural Animation

Fish Demo <iframe width="100%" height="500px" data-src="videos/fish_demo.webm"></iframe>

Flock Demo <iframe width="100%" height="500px" data-src="videos/flock_demo.webm"></iframe>

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Procedural Animation

Eagle Demo

<iframe width="100%" height="500px" data-src="videos/eagle_demo.webm"></iframe>

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Procedural Animation

Multilegged gait simulation

<iframe width="100%" height="500px" data-src="videos/procedural_demo.webm"></iframe>

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References

• Introduction to IK

• Inverse Kinematics Techniques in Computer Graphics: A Survey

• Forward and Backward Reaching Inverse Kinematics (FABRIK)

• Cyclic Coordinate Descent (CCD)