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make better games Vector Units and Quaternions Jim Van Verth Red Storm Entertainment

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make better games 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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make better games 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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make better games 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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make better games 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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make better games 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, vf1z vmaddw vf6, vf5, vf1w

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make better games 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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make better games 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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make better games Quaternion Multiplication If quaternion is (x, y, z,w) or (v, w) then All standard vector operations >Add, scale, dot product, cross product

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make better games 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, vf2 vopmula.xyz acc, vf1, vf2 vmaddaw.xyz acc, vf2, vf1w vmaddaw.xyz acc, vf1, vf2w vopmsub.xyz vf3, vf2, vf1 vsubaz.w acc, vf3, vf3z vmsubax.w acc, vf0, vf3x vmsuby.w vf3, vf0, vf3y w= w 1 ·w 2 v 1 v 2 v = w 1 ·v 2 + w 2 ·v 1 + v 1 v 2

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make better games Vector Rotation Formula for vector rotation: Two mults takes 16 cycles, plus the inverse Can do better

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make better games Vector Rotation, Take Two If q is normalized, then can do: This is faster than two straight multiplies on serial processor Faster on vector processor, too!

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make better games Vector Rotation on VU p in vf1, q in vf2 vmul vf11, vf1, vf2 vopmula.xyz acc, vf2, vf1 vopmsub.xyz vf5, vf1, vf2 vmul.w vf6w, vf2w, vf2w vadd.w vf7w, vf2w, vf2w vmulax.w accw, vf0w, vf11x vmadday.w accw, vf0w, vf11y vmaddz.w vf11w, vf0w, vf11z vopmula.xyz acc, vf2, vf5 vmaddaw.xyz acc, vf5, vf7w vmaddaw.xyz acc, vf1, vf6w vmaddaw.xyz acc, vf2, vf11w vopmsub.xyz vf3, vf5, vf2 p = (v p)·v + w 2 ·p + 2w·(v p) + v (v p)

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make better games 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, vf2 vopmula.xyz acc, vf2, vf1 vopmsub.xyz vf5, vf1, vf2 vmul.w vf6w, vf2w, vf2w vadd.w vf7w, vf2w, vf2w vmulax.w accw, vf0w, vf11x vmadday.w accw, vf0w, vf11y vmaddz.w vf11w, vf0w, vf11z vopmula.xyz acc, vf2, vf5 vmaddaw.xyz acc, vf5, vf7w vmaddaw.xyz acc, vf1, vf6w vmaddaw.xyz acc, vf2, vf11w vopmsub.xyz vf3, vf5, vf2

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make better games Full Transforms Combination of translation vector t, quat r, 3 scale factors s Once again, want to transform point Basic formula:

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make better games 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, vf3 vmul vf11, vf1, vf2 vopmula.xyz acc, vf2, vf1 vopmsub.xyz vf5, vf1, vf2 vmul.w vf6w, vf2w, vf2w vadd.w vf7w, vf2w, vf2w vmulax.w accw, vf0w, vf11x vmadday.w accw, vf0w, vf11y vmaddz.w vf11w, vf0w, vf11z vopmula.xyz acc, vf2, vf5 vmaddaw.xyz acc, vf5, vf7w vmaddaw.xyz acc, vf1, vf6w vmaddaw.xyz acc, vf2, vf11w vmaddaw.xyz acc, vf4, vf0w vopmsub.xyz vf3, vf5, vf2

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make better games Transform Concatenation Look at formula: Have to transform point and multiply two quaternions and multiply scales

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make better games 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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make better games 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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make better games Matrix Conversion Corresponding 4x4 matrix to normalized quat q = (x,y,z,w) is: Not obvious how to do this efficiently

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make better games 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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make better games Matrix Conversion (macro) Idea: matrix is built from two other matrices

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make better games Matrix Conversion (macro) Simplification: matrix multiply is series of row vector multiplies Create right matrix, generate left matrix via accumulator tricks

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make better games 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, vf1z vmaddx vf9, vf5, vf1x Is linear, so order doesn’t matter

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make better games 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, vf5w vmaddax.xyz acc, vf3, vf5x vmadday.xyz acc, vf4, vf5y vmsubz.xyz vf13, vf1, vf5z

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make better games Matrix Conversion (macro) More simplification: >Last row of M q always (0,0,0,1), don’t compute! >Last column always 0 too, don’t compute! >Last row of R q just the quat in VU format Just build:

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make better games 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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make better games Matrix Conversion (macro) vmulaw.xyz acc, vf1, vf4w vmaddaz.xyz acc, vf2, vf4z vmsubay.xyz acc, vf3, vf4y vmaddx.xyz vf12, vf4, vf4x vmulaw.xyz acc, vf2, vf4w vmaddax.xyz acc, vf3, vf4x vmadday.xyz acc, vf4, vf4y vmsubz.xyz vf13, vf1, vf4z vmulaw.xyz acc, vf3, vf4w vmaddaz.xyz acc, vf4, vf4z vmadday.xyz acc, vf1, vf4y vmsubx.xyz vf14, vf2, vf4x vmove.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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make better games Matrix Conversion (micro) Lots of duplicate calculations in matrix Idea: calculate only what we need, use shifting and accumulator tricks to parallelize efficiently Devised by Colin Hughes of SCEE

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make better games Matrix Conversion (micro) mula acc, vf1, vf1loi SQRT_2 muli vf3, vf1, Imr32.w vf24, vf0 madd vf2, vf1, vf1nop addw vf4, vf0, vf0wnop opmula acc, vf3, vf3move vf27, vf0 msubw vf5, vf3, vf3wmr32.w vf26, vf0 maddw vf6, vf3, vf3wmr32.w vf25, vf0 addaw.xyz acc, vf0, vf0wnop msubax.yz acc, vf4, vf2xnop msuby.z vf26, vf4, vf2ymr32 vf3, vf5 msubay.xz acc, vf4, vf2ymr32 vf7, vf6 msubz.y vf25, vf4, vf2zmr32.y vf24, vf5 msubz.x vf24, vf4, vf2zmr32.x vf26, vf5 addy.z vf24, vf0, vf6ymr32.z vf25, vf3 addx.y vf26, vf0, vf6xmr32.x vf25, vf7 Three parts Calculate elements Clear matrix Shift, add and copy into place 16/19 cycles

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make better games 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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make better games 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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make better games 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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make better games Normalization vmul 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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make better games 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 > , use approx >Otherwise use RSQRT

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make better games Interpolation This is where it’s at It would be great if it was fast Um, well…

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make better games Interpolation First look at spherical linear interp 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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make better games Interpolation Lerp, then 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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make better games Interpolation Not quite that simple Problem: if qr < 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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make better games Linear Interpolation vmul vf4, vf1, vf2 vaddaz.w acc, vf04, vf4 vmaddax.w acc, vf00, vf4 vmaddy.w vf4, vf00, vf4 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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make better games 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 more cycles Looks faster than slerp, more accurate than lerp

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make better games 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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make better games 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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make better games 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

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make better games Questions?

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make better games Please hand in comment sheets Slides available at:

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make better games Vector Units and Quaternions Jim Van Verth Red Storm Entertainment

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