1. Jason Clark (jason@essentialmath.com)
Inverse Kinematics
Jason Clark

2. Essential Math for Games
Inverse Kinematics
Actors are animated in game.
Animations are independent of the world
Want to have more realistic response to what the Actors are interacting with.
Need a mechanism for reacting to goals specified during the game.
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Kinematics
Have a hierarchical skeleton structure
Each joint is defined local to its parent
Rotation
Constant Translation
defines set for entire structure. Is the global position of end joint
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Kinematics
Consider the kinematic chain below.
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Inverse Kinematics
Animations can be thought of as driving the kinematics in a game
Want reaction to world. I.e. Position a limb based on a goal defined by game situation.
Need method given goal position to update to achieve it.
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Inverse Kinematics
Kinematics.
Therefore Inverse Kinematics is As a general problem this is hard. G D1 Q2 D3 E Q1 Q3 D2
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Types of solutions
Analytical
Equation that can be directly solved
Preferred, practically impossible as a general solution
Numerical
Expensive
In-accurate
Unfortunately only practical option for general solver.
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Three Common Methods
Jacobian (Numerical)
Cyclic Coordinate Descent. (Numerical)
Anthropomorphic. (Analytical? Depends)
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The Jacobian
What is it? A linear approximation to
Matrix of partial derivatives of entire system.
In this case, defined relative to
Defines how changes relative to instantaneous changes in the system.
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The Jacobian
Linear approximation
Actual E
Linear Approximation
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The Jacobian
Limiting discussion to positional goals only.
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The Jacobian
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Not quite right
Jacobian defines changes in relative to changes in
Want to know how a desired change in maps to changes in
Recall that the equation of inverse kinematics is defined as
Therefore we need
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14. Inverse Jacobian Problems
Typically system is over specified (under defined)
Ameliorate problem by limiting joints. E.g. Specify elbow as a 1DOF joint.
No guarantee it is invertible.
Typically not a square matrix.
Singularities.
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15. Essential Math for Games
Cheat. Pseudo Inverse
Square so can be inverted.
But why is it okay to use this?
Principle of Virtual Work
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16. Principle of Virtual Work
Tends towards the right solution. (I.e. It’s the right idea, if not completely accurate)
Take small steps to minimize error.
Calculate error, and if too large adjust step size, or size of
Error =
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17. Calculating the Jacobian
Given angle and axis of rotation
is relative to as defined by
Velocity of relative to change in
Gives one column of Jacobian V1 V2 N2 E N1
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Algorithm
Only considering positional goals.
Define the following variables – end effector – ith Joint position – ith Joint axis of rotation
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Algorithm
Calculate the Jacobian
Calculate:
Determine error =
If error > tolerance then repeat 4 till within tolerance.
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Algorithm
Calculate:
Apply to entire system
Repeat until Or max steps is reached.
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Addendum
Minimizes joint angle rates. Configuration produced may not be natural looking
Add gain values for each joint. Add “stiffness” I.e. Bias which joints are moved
Add constraints, Clamp joint angles.
Both these ideas slow down convergence of the algorithm. Need more steps
as force transforms to internal velocities
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One last thing
As stated previously, the Jacobian is a linear approximation to the problem.
Using the pseudo inverse is an approximation to solve for intractability of the problem.
So already cheating, why not cheat more? Can simply use as approximation. Slower convergence, but much cheaper to compute than
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23. Cyclic Coordinate Descent
The Jacobian is hard. It deals with entire system as a whole. Good results, more complicated to code, and expensive.
That’s too much work, lets deal with each joint individually.
Heuristic only. Not as mathematically grounded as the Jacobian.
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CCD
Reduce problem to local consideration at each joint. Start at leaf and move up.
Calculate change in joint needed to minimize “error”. Error =
Calculate
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CCD
Calculate rotation necessary to align into
Modify current joint by and adjust entire chain below. This is important.
Move up chain and repeat.
Repeat over entire chain till.
Done!
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Addendum
Pretty easy. Much simpler than the Jacobian.
Problems. Takes more updates of chain and steps than Jacobian does. Simpler individual steps, need more of them.
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Addendum
Moves end joints first. Heuristic inherently favors them. Local minima problem
Results in unnatural configurations. E.g Wrist bent oddly to reach goal. Worse than Jacobian about this.
Basically Gauss-Seidel. SOR can help.
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Demo
Snake
From Jeff Lander
darwin3d.com/gdm/1998.htm
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Anthropomorphic
Other 2 solutions are general purpose solutions
Most games have people. We know something about them.
Standard principle of computer science. Use knowledge of problem domain to simplify problem
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Rainbow 6: Lockdown
Leave wrist alone. Inaccurate, but good enough and difficult to get right.
Treat elbow as a 1DOF joint. Not precisely accurate. But good enough.
Treat the shoulder as 3DOF joint with no constraints. Again, not entirely true but good enough.
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Rainbow 6: Lockdown
Leg can be run through same algorithm as Arm. (mostly)
Knee can be simulated as a 1DOF joint. (Elbow)
Hip can be simulated as a 3DOF joint (Shoulder)
Have second pass to fix Ankle.
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Algorithm
Elbow defines the distance between the shoulder and the end effector.
Simple Trig problem
Law of Cosines B A Q C
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Algorithm
After that, simply run one step of the CCD algorithm on the shoulder.
Done! Simple, accurate within a tolerable margin, and cheap.
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References
Welman, Chris, “Inverse Kinematics and Geometric Constraints for Articulated Figure Manipulation”, Simon Fraser University
Meredith, Michael and Maddock, Steve, “Real-Time Inverse Kinematics: The Return of the Jacobian” University of Sheffield
Tolani, Goswami, and Badler, “Real-Time Inverse Kinematics Techniques for Anthropomorphic Limbs”, University of Pennsylvania
Blow, Jonathan, “Oh My God! I Inverted Kine”. Game Developer. September 1998
Parent, Rick, “Computer Animation: Algorithms and Techniques”, Morgan-Kaufmann, San Francisco, 2001.
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35. Questions?