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My Adventure with Inverse KinematicsPowerPoint Presentation

My Adventure with Inverse Kinematics

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“ways in which system can change” find DOF of system to understand it control & constrain DOF to control it examples fundamental task of IK is to control degrees of freedom to attain goal

What is this lecture about?

- the path I took while solving a problem with inverse kinematics
- how I use math to model and [sometimes] solve problems
- decisions I made, both good and bad
- learn from my successes and failures
- work in progress!

What isn’t this lecture about?

- not an IK tutorial or introduction
- not going to give you the One True Way to do IK
- has nothing to do with graphics
- not about my rock climbing game
- Experimental Gameplay WorkshopFriday, 4pm - 7pm

Takeaway

- You will get insight into how IK on a whole human body is different from IK on a single arm.
- You will see how IK is a building block, but not a complete solution to character animation.

Prerequisites

- must be very comfortable with math
- doesn’t mean you know tons of math
- does mean you follow quickly and aren’t afraid of it

- probably best if you’ve implemented a simple IK system
- foot placement, arm grabbing door, etc.

Talk Structure(linear with occasional branches)

- ...
- attempted solution
- problem statement and discussion
- related theory
- attempted solution
- problem statement and discussion
- related theory
- ...

Problem: How to move a guy with the mouse?

- move like a [rock climbing] human
- interactive
- direct body/limb control
- did not want to use physics

Theory: How to move a guy with the mouse?

- inverse kinematics vs. forward kinematics
- FK: p=f(q)
- IK: q = f-1(p)
- p is Euclidean, angles
- q is configuration space

p

p = (x, y)

q = (q1, q2)

q2

q1

q2

q1

Theory: How to move a guy with the mouse?(cont.)- degrees of freedom (DOF)
- incredibly important concept

- configuration space

- physical system, polynomial, etc.

p = (x, y)

q = (q1, q2)

Solution: How to move a guy with the mouse?

- analytical IK
- invert p = f(q) to get q from p
- relatively easy for the 2D arm
- inverse trig, some vector algebra

- gets complicated as n increases

Problem: Analytical IK can’t handle human.

p

- human has lots of DOF
- even 2D arm with shoulderhas more DOF than neededto reach goal

- solving p = f(q) requirescertain characteristics

q3

q2

q1

p = (x, y)

q = (q1 , q2 , q3)

Theory: Analytical IK can’t handle human.

p

- p = f(q) is a set of nonlinear equations
- generally no closed forms
- must iterate numerically

- must be square, nDOF = ngoal
- over-, under-constrained system
- “redundant manipulator”
- ex. human reaching for point and/or orientation

- vast literature on solving f(q) = p

q3

q2

q1

p = (x, y)

q = (q1 , q2 , q3)

Solution: Analytical IK can’t handle human.

p

- use Cyclic Coordinate Descent(CCD) to solve arms
- iterative, recursive position-spacealgorithm
- fast, easy, robust
- start from last position
- not path independent, but faster

- I draw every outside iteration
- looks like animation, but it’s not

q3

q2

q1

p = (x, y)

q = (q1 , q2 , q3)

g1

g2

Problem: CCD only handles serial chains.p2

p1

- multiple goals are necessaryfor human
- stack up vectors inp=f(q)
- coupled at branches

q3

q5

q4

q2

q1

p = (x1, y1, x2, y2)

q = (q1 , q2 , q3 , q4 , q5)

g1

g2

Theory: CCD only handles serial chains.p2

p1

- coupling between goals
- either solvable alone,neither solvable together
- how to distribute desired-angleerror?

q3

q5

q4

q2

q1

p = (x1, y1, x2, y2)

q = (q1 , q2 , q3 , q4 , q5)

Solution: CCD only handles serial chains.

p2

p1

- simply average desired angles atbranch parents to attainmultiple goals and distributeerror
- q1 in example, imagine p1, p2pulled in opposite directions

q3

q5

q4

q2

q1

p = (x1, y1, x2, y2)

q = (q1 , q2 , q3 , q4 , q5)

Problem: Drift on multiple goals.

- average can’t satisfy all goals
- or even prioritize them
- I specifically want to drag climber hand with mouse and have other end effectors stay on holds

q

Theory: Drift on multiple goals.

- want a rigid constraint at end effectors
- closed loops are hard, CCD doesn’t handle them, nor do common IK algorithms
- must solve simultaneously
- not hierarchical anymore
- closed loops don’t have local behavior

q

Solution: Drift on multiple goals.

- attempt to “Gram-Schmidt” off any movement that would result in error

b

c

a

Problem: Gram-Schmidt removes the wrong direction sometimes.

- could have tried real Jacobian force mapping Gram-Schmidt

Problem: Human is not rooted tree.

- Which end effector (hand, foot) is the root?
- There is no persistent fixed rootin a rock climber
- or walking person

- CCD must start at a fixed root

Theory: Human is not rooted tree.

- switching root is basically a datastructure problem
- relative angles must change
- joint limits (see below)

- behavior will change slightly as well,since CCD isn’t symmetric withrespect to the root

Solution: Human is not rooted tree.

- just type it in
- invert tree to new root
- Caml is pretty good at thissort of thing
- see last year’s GDC talk

- dragging torso is done by inverting to a torso node, then changing the root position
- the hands and feet are all end-effectors
- hack!

Problem: Joint limits

- elbow can’t bend backwards
- should lend believability and intuition

Theory: Joint limits

- inequality constraints: q³qmin
- abandon all (most) hope of analytical solution
- these inequalities are axis aligned planes in configuration space

qmax

q

qmin

Solution: Joint limits

- CCD makes parent-child limits easy
- do min/max clamp in CCD inner iteration
- easy, and rest of system compensates
- issue: CCD doesn’t “know” about the joint limits
- often compensates, but doesn’t know to avoid them

Problem: Child-child limits

- parent-child not enough with moving root
- pelvis example:
- left foot root
- right foot root
- chest root

Problem: Child-child limits

- parent-child not enough with moving root
- pelvis example:
- left foot root
- right foot root
- chest root

Problem: Child-child limits

- parent-child not enough with moving root
- pelvis example:
- left foot root
- right foot root
- chest root

?

Theory: Child-child limits

- both angles are dependent on each other, coupled
- parent is fixed in CCD
- constraints are now non-axisaligned planes in C-space

- simultaneous solution necessaryat branches with child-childlimits

Solution: Child-child limits

- Good solution should:
- satisfy constraints always
- have zero error in goals if possible
- have least squares error if not possible

- Nonlinear Least Squares with Linear Inequalities: minimize (q1 - d1)2 + (q2 - d2)2 w.r.t. qmin12£ q1 - q2 £qmax12
- lots of literature on NLSLI
- I convert to LCP and use Lemke’s algorithm
- works great!

Accomplishments

- We extended CCD to handle:
- multiple end effector goals
- dynamically changing root node
- arbitrary joint limits

Problems

- drift on multi-goals is still a problem
- CCD doesn’t know about joint limits
- sometimes non-physical movement
- CCD isn’t doing physics, not conserving energy, or anything else for that matter

- rooted hierarchies are inconvenient
- how to control redundant DOF?

Body Knowledge System

- system to manage movement and keep it natural
- “situation” & “reaction”
- ex. joint limits handled in body knowledge

- huge rules system
- allocates redundant DOF
- this is the key to natural movement!

Body Knowledge System (cont.)

- IK just a rough initial guess for body knowledge system; should be:
- physically plausible
- no added energy, etc.

- easy to layer conflicting goals
- able to cope with over-constrained goals

- physically plausible
- don’t know enough about this yet, but I think it will be more important than IK for good dynamic animation

What I Didn’t Talk About

- development process stuff
- developing on lower dimensional subproblems
- using visualizations to help debug math
- Mathematica and Matlab for development and debugging
- running multiple techniques on top of each other

- physics-based techniques I tried

What I Didn’t Try

- I should have done a weighted average based on error for multi-goals, duh.
- even weighted average wouldn’t be rigid
- although could make weight very big...
- similar to a stiff system

- even weighted average wouldn’t be rigid
- Wrong: “CCD must have fixed root”
- now that I understand better,this isn’t actualy true
- the root’s position is just more DOF
- CCD probably wouldn’t deal with this very well?

- now that I understand better,this isn’t actualy true

What I Didn’t Try (cont.) data-based methods (model-based) totally ad-hoc methods

- pseudo-inverse methods
- give explicit control over redundant DOF

- functions as lookup tables

References

- two great theses:
- Paolo Baerlocher
- Chris Welman

- google & citeseer
- GDC talks
- David Wu
- Ken Perlin

- Jeff Lander’s articles & code
- Experimental Gameplay Workshop

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