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Vector Units and Quaternions

Jim Van VerthRed Storm Entertainment

jimvv@redstorm.com

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About This Talk• Will discuss how to do quaternion

math on PS2

• Assume that you already know and want to use quaternions

• Assume that you already know something about how the VU works

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About Me• Lead engineer at Red Storm

Entertainment

• Not a quaternion god

• Not a vector unit god

• Not really familiar with VCL

• Just a 3D guy trying to get by…

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About the code• Most examples written in macro mode

(VU0)

• Easy to translate to micro mode

• Examples that would be faster in micro mode are discussed separately

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Matrices on PS2• PS2 is really well set up to do

matrices

• Multiplies are highly parallel

• Not so good for quaternions

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Matrix Multiply• This is what we’re

up against• Takes 4/7 cycles to

transform a point• Takes 16/19 cycles

to concat matrices (9/12 cycles for 3x3 matrix)

vmulax ACC, vf2, vf1x vmadday ACC, vf3, vf1y vmaddaz ACC, vf4, vf1zvmaddw vf6, vf5, vf1w

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Why Quaternions?• Quaternions take up less space: 4

floats vs. 9 (best case)

• Quaternions interpolate well

• Avoid floating point drift (normalize vs. Gram-Schmidt orthogonalization)

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Quaternions on VU• Fit very well

• Four floats, aligned to 16-bit boundary

• Work just like homogeneous point

• Make sure stored (x,y,z,w) not (w,x,y,z)

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Quaternion Multiplication• If quaternion is (x, y, z,w) or (v, w) then

) , ( 212121122121 vvvvvv wwwwqq

• All standard vector operations> Add, scale, dot product, cross product

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Quaternion Mult on PS2• Interleaves dot

product and rest via accumulator

• Takes advantage of linearity of cross product

• Cycle count: 8/11• Less than matrix!

vmul vf3, vf1, vf2vopmula.xyz acc, vf1, vf2vmaddaw.xyz acc, vf2, vf1wvmaddaw.xyz acc, vf1, vf2wvopmsub.xyz vf3, vf2, vf1 vsubaz.w acc, vf3, vf3zvmsubax.w acc, vf0, vf3xvmsuby.w vf3, vf0, vf3y

w= w1·w2 v1 • v2

v = w1·v2 + w2·v1 + v1 v2

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Vector Rotation• Formula for vector rotation:

0

1

wp

qpqp

• Two mults takes 16 cycles, plus the inverse

• Can do better

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Vector Rotation, Take Two• If q is normalized, then can do:

)( )(2 )( 2 pvvpvpvpvp ww

• This is faster than two straight multiplies on serial processor

• Faster on vector processor, too!

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Vector Rotation on VU• p in vf1, q in vf2vmul vf11, vf1, vf2

vopmula.xyz acc, vf2, vf1vopmsub.xyz vf5, vf1, vf2vmul.w vf6w, vf2w, vf2wvadd.w vf7w, vf2w, vf2w vmulax.w accw, vf0w, vf11xvmadday.w accw, vf0w, vf11yvmaddz.w vf11w, vf0w, vf11zvopmula.xyz acc, vf2, vf5vmaddaw.xyz acc, vf5, vf7wvmaddaw.xyz acc, vf1, vf6wvmaddaw.xyz acc, vf2, vf11wvopmsub.xyz vf3, vf5, vf2

p = (vp)·v + w2·p + 2w·(v p) + v (v p)

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Vector Rotation on VU• First part builds all

the pieces• Second part adds

‘em all together• Cycles: 13/16• Better than straight

multiply• Worse than matrix

vmul vf11, vf1, vf2vopmula.xyz acc, vf2, vf1vopmsub.xyz vf5, vf1, vf2vmul.w vf6w, vf2w, vf2wvadd.w vf7w, vf2w, vf2w vmulax.w accw, vf0w, vf11xvmadday.w accw, vf0w, vf11yvmaddz.w vf11w, vf0w, vf11zvopmula.xyz acc, vf2, vf5vmaddaw.xyz acc, vf5, vf7wvmaddaw.xyz acc, vf1, vf6wvmaddaw.xyz acc, vf2, vf11wvopmsub.xyz vf3, vf5, vf2

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Full Transforms• Combination of translation vector t,

quat r, 3 scale factors s

• Once again, want to transform point

• Basic formula:

1rprtp )(s

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Point Transformation• p in vf1, q in vf2• scale in vf3• translation in vf4• Takes four extra

cycles for scale (including stalls), one extra for xlate

• Cycle count: 18/21

vmul vf1, vf1, vf3vmul vf11, vf1, vf2vopmula.xyz acc, vf2, vf1vopmsub.xyz vf5, vf1, vf2vmul.w vf6w, vf2w, vf2wvadd.w vf7w, vf2w, vf2w vmulax.w accw, vf0w, vf11xvmadday.w accw, vf0w, vf11yvmaddz.w vf11w, vf0w, vf11zvopmula.xyz acc, vf2, vf5vmaddaw.xyz acc, vf5, vf7wvmaddaw.xyz acc, vf1, vf6wvmaddaw.xyz acc, vf2, vf11wvmaddaw.xyz acc, vf4, vf0wvopmsub.xyz vf3, vf5, vf2

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Transform Concatenation• Look at formula:

12122 rtrtt

rrr

)( 2

12

21

s

sss

• Have to transform point and multiply two quaternions and multiply scales

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Transform Concatenation• Takes 8 cycles for quat multiply, 18

for transform, 1 for scale

• Have three stall cycles available

• Bottom line: 24/27 cycles

• Much slower than matrix multiplication

• Not recommended

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Matrix Conversion• Quat-vector transformation not as

efficient as matrix-vector transformation (13 cycles vs. 4)

• To do multiple points, want to convert quaternion to a 4x4 matrix

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Matrix Conversion• Corresponding 4x4 matrix to

normalized quat q = (x,y,z,w) is:

1000

02212222

02222122

02222221

22

22

22

yxwxyzwyxz

wxyzzxwzxy

wyxzwzxyzy

qM

• Not obvious how to do this efficiently

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Matrix Conversion• Two approaches• One works well in macro mode• One in micro mode

> uses Lower instructions to achieve better parallelism

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Matrix Conversion (macro)• Idea: matrix is built from two other

matrices

wzyx

zwxy

yxwz

xyzw

wzyx

zwxy

yxwz

xyzw

wzyx

qM

q ),,,(

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Matrix Conversion (macro)• Simplification: matrix multiply is series

of row vector multiplies

• Create right matrix, generate left matrix via accumulator tricks

wzyx

zwxy

yxwz

xyzw

qR

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Matrix Conversion (macro)• Look at one row in matrix multiply:

vmulax ACC, vf5, vf1x

vmadday ACC, vf6, vf1y

vmaddaz ACC, vf7, vf1z

vmaddw vf9, vf8, vf1w

• Or could just do:vmulaw ACC, vf8, vf1w vmadday ACC, vf6, vf1y vmaddaz ACC, vf7, vf1zvmaddx vf9, vf5, vf1x

• Is linear, so order doesn’t matter

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Matrix Conversion (macro)• Idea: all values we need for left matrix are

in quaternion• Load accumulator with mula by w value

(always positive)• vmadd or vmsub to multiply by positive or

negative value and accumulate

vmulaw.xyz acc, vf2, vf5wvmaddax.xyz acc, vf3, vf5xvmadday.xyz acc, vf4, vf5yvmsubz.xyz vf13, vf1, vf5z

4vf

3vf

2vf

1vf

13vf

5vf

yxwz

wzyx

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Matrix Conversion (macro)• More simplification:

> Last row of Mq always (0,0,0,1), don’t compute!

> Last column always 0 too, don’t compute!

> Last row of Rq just the quat in VU format

• Just build:

~

~

~

~

zyx

wxy

xwz

yzw

qR

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Matrix Conversion (macro)vaddw.x vf1, vf0, vf4

vaddz.y vf1, vf0, vf4

vsuby.z vf1, vf0, vf4

vsubz.x vf2, vf0, vf4

vaddw.y vf2, vf0, vf4

vaddx.z vf2, vf0, vf4

vaddy.x vf3, vf0, vf4

vsubx.y vf3, vf0, vf4

vaddw.z vf3, vf0, vf4

vmr32.w vf12, vf0

vmr32.w vf13, vf0

vmr32.w vf14, vf0

• Stage one:> Load quat in vf4> Build right matrix> Clear right column of

result

vf1=(w,z,-y,~)

vf2=(-z,w,x,~)

vf3=(y,-x,w,~)

vf4=(x,y,z,w)

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Matrix Conversion (macro)vmulaw.xyz acc, vf1, vf4w vmaddaz.xyz acc, vf2, vf4zvmsubay.xyz acc, vf3, vf4yvmaddx.xyz vf12, vf4, vf4x vmulaw.xyz acc, vf2, vf4wvmaddax.xyz acc, vf3, vf4xvmadday.xyz acc, vf4, vf4yvmsubz.xyz vf13, vf1, vf4zvmulaw.xyz acc, vf3, vf4wvmaddaz.xyz acc, vf4, vf4zvmadday.xyz acc, vf1, vf4yvmsubx.xyz vf14, vf2, vf4xvmove.xyzw vf15, vf0

• Stage two:> Matrix multiply to get first

three rows

> Clear bottom row

• Note: accumulate only on xyz (w already cleared)

• Cycles: 25/28

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Matrix Conversion (micro)• Lots of duplicate calculations in matrix

1000

02212222

02222122

02222221

22

22

22

yxwxyzwyxz

wxyzzxwzxy

wyxzwzxyzy

qM

• Idea: calculate only what we need, use shifting and accumulator tricks to parallelize efficiently

• Devised by Colin Hughes of SCEE

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Matrix Conversion (micro)

mula acc, vf1, vf1 loi SQRT_2muli vf3, vf1, I mr32.w vf24, vf0madd vf2, vf1, vf1 nopaddw vf4, vf0, vf0w nopopmula acc, vf3, vf3 move vf27, vf0msubw vf5, vf3, vf3w mr32.w vf26, vf0maddw vf6, vf3, vf3w mr32.w vf25, vf0addaw.xyz acc, vf0, vf0w nopmsubax.yz acc, vf4, vf2x nopmsuby.z vf26, vf4, vf2y mr32 vf3, vf5msubay.xz acc, vf4, vf2y mr32 vf7, vf6msubz.y vf25, vf4, vf2z mr32.y vf24, vf5msubz.x vf24, vf4, vf2z mr32.x vf26, vf5addy.z vf24, vf0, vf6y mr32.z vf25, vf3addx.y vf26, vf0, vf6x mr32.x vf25, vf7

• Three parts• Calculate

elements• Clear matrix• Shift, add

and copy into place

• 16/19 cycles

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Matrix Conversion• If you’re converting a quaternion and

going to use it immediately, can make some assumptions

• Don’t create bottom row (just use vf0)

• Don’t clear right column (just use xyz)

• Saves four cycles in macro mode case

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Transform to Matrix• Use one of the quaternion matrix

techniques

• Scale first three rows by each scale factor

• Replace last row with translation

• Results:> 29/32 for macro mode> 20/23 for micro mode

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Normalization• Need to normalize quaternion to keep

it useful for rotation> (Also avoids floating point drift)

• Fortunately PS2 has reciprocal square root instruction

• Unfortunately it takes a while

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Normalizationvmul vf2, vf1,

vf1

vaddaz.w acc, vf2, vf2

vmaddax.w acc, vf0, vf2

vmaddy.w vf2, vf0, vf2

vrsqrt Q, vf0w, vf2w

vwaitq

vmulq vf1, vf1, Q

• Compute dot product

• Compute 1/length• Scale quaternion• With stalls, takes

24/27 cycles

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Normalization• Another approach

>From “The Inner Product”, March 2002 Game Developer by Jonathan Blow

>Approximate 1/x via Newton-Raphson iteration

>First iteration takes (looks like) 4/7 cycles on VU0

>Second iteration takes as long as RSQRT >Recommend: if x > 0.91521198, use

approx>Otherwise use RSQRT

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Interpolation• This is where it’s at

• It would be great if it was fast

• Um, well…

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Interpolation• First look at spherical linear interp

θsin

)θsin())1(θsin(),,slerp(

rqrq

ttt

• That’s a lot of sines• Could precompute , 1/sin • But at least 28 cycles for one of the

other sines• We (RSE) don’t use slerp anyway

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Interpolation• Lerp, then

rqq

rqrq

tt

ttt )1(),,lerp(

is simply (q in vf1, r in vf2, t in vf3w)vaddax acc, vf1, vf0x

vmsubaw acc, vf1, vf3w

vmaddw vf1, vf2,vf3w

• Need to normalize afterwards

• Makes 30/33 cycles

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Interpolation• Not quite that simple

• Problem: if q•r < 0, interpolation will take long way around sphere

• Need to negate one quat

• Gives the same orientation, but the interpolation takes the short route

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Linear Interpolationvmul vf4, vf1, vf2 vaddaz.w acc, vf04, vf4 vmaddax.w acc, vf00, vf4 vmaddy.w vf4, vf00, vf4 vnop vnop vnop cfc2 t0,$16 and t0,t0,0x0002 vaddax acc, vf1, vf0x beq t0,zero,Add vmsubaw acc, vf2,vf3w b Finish Add: vmaddaw acc, vf2,vf3w Finish: vmsubw vf1, vf1, vf3w

• Compute dot product

• Check for negative • Interpolate• Follow up with

normalization• Takes 43/46 cycles

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Linear Interpolation• There’s more we can do• Jonathan Blow’s article, again• Use spline to correct error in lerp• More investigation needed• Initial results: takes about 24-26 more

cycles• Looks faster than slerp, more

accurate than lerp

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How We’re Using All This• A bit research-y at the moment

• VU0-based math library

• Optimization in specific routines

• In particular, concatenation and interpolation for bones animation

• More memory savings: store quat as 4.12 fixed-point shorts

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Conclusions• Quaternions useful on PS2

• Cheaper to concatenate (alone)

• Convert to matrix to transform

• Use linear interpolation

• Check out Jonathan Blow’s article

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References• Shoemake, Ken, “Animating Rotation with

Quaternion Curves,” Computer Graphics, Vol. 19, No. 3 (July 1985).

• EE Core Instruction Set Manual• VU User’s Manual• Sony newsgroups• Blow, Jonathan, “Hacking Quaternions,” Game

Developer, Vol. 9, No. 3 (March 2002). [get updated source from www.gdmag.com/code.htm]

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Questions?

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• Please hand in comment sheets

• Slides available at:

http://obiwan.redstorm.com/~jimvv

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Vector Units and Quaternions

Jim Van VerthRed Storm Entertainment

jimvv@redstorm.com