Calculations

Calculations

• Ports of an item

  Where an item's in, out and vehicle ports are in its own frame, with their headings.

  Material flow layout geometry addons/flow/lib/catalog.js

  • ModelCraft
  64-bit

• Parameter limits

  Clamps each numeric parameter to its minimum and maximum and rounds whole-number parameters.

  n = min(max, max(min, v)); whole numbers: round(n)

  Material flow layout geometry addons/flow/lib/catalog.js

  • ModelCraft
  64-bit

• Reading a layout

  Checks the positions, rotation, fleet and run settings of a saved layout.

  x, y, z: kept if finite and |v| < 1e6, else 0; rot = ((rot mod 360) + 360) mod 360

  Material flow layout geometry addons/flow/lib/layout.js

  • ModelCraft
  64-bit

• Local point to world

  Turns a point in an item's own frame by the item's rotation and moves it to the item's position.

  x′ = x cos r − y sin r + item x; y′ = x sin r + y cos r + item y; heading = ((h mod 360) + 540) mod 360 − 180

  Material flow layout geometry addons/flow/lib/layout.js

  • ModelCraft
  64-bit

• Placing an item on a port

  The position and rotation that put one of an item's ports on a given point with a given heading.

  rot = ((H − port heading) mod 360 + 360) mod 360; position = P − rotated port offset

  Material flow layout geometry addons/flow/lib/layout.js

  • ModelCraft
  64-bit

• Which ports are joined

  Finds the out and in ports that touch, and merges vehicle path ends into nodes.

  joined when |Δx|, |Δy|, |Δz| ≤ 0.01 m and |Δheading| ≤ 1°; path ends merge when each |Δ| ≤ 0.05 m

  Material flow layout geometry addons/flow/lib/layout.js

  • ModelCraft
  64-bit

• Snap an item to the nearest free port

  Moves an item so one of its free ports sits on the nearest matching free port within a radius.

  d = √(Δx² + Δy² + Δz²); nearest pair with d ≤ radius (0.5 m by default)

  Material flow layout geometry addons/flow/lib/layout.js

  • ModelCraft
  64-bit

• Lengths and times handed to the simulation

  Turns each conveyor into its length, speed and gap, and breakdown times from hours and minutes into seconds.

  zone length = zones × zone length; mtbf (s) = hours × 3600; mttr (s) = minutes × 60

  Material flow layout geometry addons/flow/lib/layout.js

  • ModelCraft
  64-bit

• Where a load is drawn on a belt

  The distance along a belt at which a load is drawn at a given time.

  accumulating: s = min(v × t + c, S); stopping belt: s = D(t) − D at entry; drawn at max(0, s) + L ÷ 2

  Material flow drawing and playback addons/flow/lib/playback.js

  • ModelCraft
  32-bit, display only

• Where a vehicle is drawn

  A vehicle's point on its path segment at a given time.

  f = (t − t₀) ÷ (t₁ − t₀); point = path at f × length

  Material flow drawing and playback addons/flow/lib/playback.js

  • ModelCraft
  32-bit, display only

• Belt strips

  The boxes drawn for a conveyor between points sampled along its path.

  pieces = max(4, ceil(|angle| ÷ 10)) for a curve, else 1; length = √(run² + Δz²); turn = atan2(Δy, Δx)

  Material flow drawing and playback addons/flow/view.js

  • ModelCraft
  32-bit, display only

• Loads and vehicles as instanced boxes

  Sets the position, turn and size of each load and vehicle box for one frame.

  Material flow drawing and playback addons/flow/view.js

  • ModelCraft
  32-bit, display only

• Shortest turn between two directions

  The quaternion that turns one unit vector onto another.

  q = (u × v, 1 + u · v), made unit

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Two-bone reach

  Where the elbow and the end of a two-bone chain go so the end reaches a target.

  cos A = (L₁² + d² − L₂²) ÷ (2 L₁ d); sin A = √(1 − cos² A); elbow = a + dir × L₁ cos A + n × L₁ sin A

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Chain of three or more bones (FABRIK)

  Joint positions of a longer chain whose tip reaches for a target.

  backward sweep from the target, forward sweep from the root, each joint kept its segment length from its neighbor

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Point along a path by length

  The point and direction a given fraction of the way along a polyline.

  s = k × Σ segment lengths; walk the segments subtracting each length until s falls inside one

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Point an axis at a direction

  A matrix whose chosen axis points along a direction, with its up axis leaning to world +Z.

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Blend by influence

  Mixes two matrices by a factor from 0 to 1: place and size linearly, the turn along the shortest arc.

  t = t_a + (t_b − t_a) k; s = s_a + (s_b − s_a) k; q = slerp(q_a, q_b, k)

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• IK on a rig

  Builds the chain for an IK constraint, solves it and swings each bone so it runs to the next joint.

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• World matrix back to a local pose

  Works a bone's local matrix back from a wanted world matrix through its parent.

  local = (parent world × parent rest⁻¹ × rest)⁻¹ × wanted world

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Constraints on placements

  Applies look-at, copy, limit, child-of and follow-path constraints to placements in world coordinates and returns their new local matrices.

  new local = blend(local, parent world⁻¹ × constrained world, influence)

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• World matrix of a placement

  The product of the matrices of a placement and everything it sits in.

  W = M₁ × M₂ × … × M_own

  Inverse kinematics and constraints anim/cns-model.js

  • ModelCraft
  64-bit

• A curve object's line in world coordinates

  The polyline a follow-path constraint runs along.

  Inverse kinematics and constraints anim/cns-model.js

  • ModelCraft
  64-bit

• Canvas position to time and value

  The time and value under a canvas point.

  t = t₀ + (x − pad) ÷ (w − 2 pad) × (t₁ − t₀); v = v₀ + (h − pad − y) ÷ (h − 2 pad) × (v₁ − v₀)

  Keys dragged in the graph editor anim/graph.js

  • ModelCraft
  64-bit

• A key dragged to a new time and value

  Moves a key in time and sets one number of its value.

  time = max(0, round(t × 1000) ÷ 1000); value = v

  Keys dragged in the graph editor anim/graph.js

  • ModelCraft
  64-bit

• A handle dragged

  The handle's new offset from its key in time and value.

  Δt = t − key time (≤ 0 for the left handle, ≥ 0 for the right); Δv = v − key value

  Keys dragged in the graph editor anim/graph.js

  • ModelCraft
  64-bit

• A body from a placement

  The center, half sizes and turn of the box or sphere that stands for a placement.

  scale = lengths of the matrix columns; half = max(1e-4, half box × scale); sphere radius = largest half

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Rotation matrix to quaternion

  The unit quaternion of a rotation given by its three axes.

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Placement matrix from a body state

  The placement's world matrix for a simulated position and turn.

  axes = q applied to X, Y, Z, times the placement's scale; origin = body center − rotated, scaled center offset

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Mass and inertia

  The inverse mass and inverse moments of inertia of a box or a sphere.

  sphere: I = 0.4 m r²; box: I_x = m (b² + c²) ÷ 12, and likewise for y and z

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Depth of a point inside a box

  How far a point is inside a box and the outward normal of the nearest face.

  local = q⁻¹ (p − center); depth = min over axes of (half + margin − |local|) − margin

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Positional correction

  Pushes two bodies apart along a normal by a given distance, sharing it by their mass and inertia.

  w = Σ (1 ÷ m + (r × n) · I⁻¹ (r × n)); λ = c ÷ w; x += n λ ÷ m; q += ½ (Δω, 0) q, made unit

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Step size and frame count

  How many frames are recorded and how long one substep is.

  frames = max(1, round((to − from) × fps) + 1); h = 1 ÷ fps ÷ substeps

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Integration of one substep

  Advances each moving body's speed, position and turn by one substep under gravity.

  v += g h; x += v h; q += ½ h (ω, 0) q, made unit

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Contacts and static friction

  Pushes each touching point out of the surface and holds it still where friction can.

  μ = √(μ_a μ_b); held when slide < μ |λ| × 1.5 + 1e-9

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Velocities from positions

  Works each body's speed and spin back from how far it moved in the substep.

  v = (x − x_prev) ÷ h; ω = 2 ÷ h × vector part of (q × q_prev⁻¹)

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Dynamic friction and bounce

  Slows sliding at each contact and sets the speed a body leaves a surface with.

  friction impulse = min(μ |λ| ÷ h, sliding speed ÷ inverse mass); leaving speed = −e × min(incoming normal speed, 0); e = 0 when |incoming| < 2 |g · n| h

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Settled test

  Says whether every body has come to rest at the end of the run.

  settled when every |v| < 0.02 and every |ω| < 0.05

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• 4×4 matrix product

  The product of two column-major 4×4 matrices.

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• 4×4 matrix inverse

  The inverse of a 4×4 matrix by cofactors.

  M⁻¹ = adjugate(M) ÷ det(M)

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• A bone's rest frame

  The matrix from a bone's own frame to the mesh: Y from head to tail, Z as near the mesh's +Z as it can be, turned by the roll.

  y = (tail − head) made unit; z = (up − y (up · y)) made unit; x = y × z; roll turns x and z about y

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• Posed bone and skin matrices

  Each bone's posed frame and the matrix that moves rest points to the pose.

  W = W_parent × R_parent⁻¹ × R × local; skin = W × R⁻¹

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• Skin matrices as dual quaternions

  Each bone's turn and move as a unit dual quaternion, for dual-quaternion skinning.

  dual part = ½ × (t, 0) × q

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• A bone's pose from a drag

  The location or rotation a bone gets when it is dragged in the view.

  turn: angle swept round the bone's head on screen, about the view axis carried into the bone's frame; move: pixels × pixel size, along the camera's right and up, carried into the bone's frame

  Bone frames and pose matrices anim/rig-view.js

  • ModelCraft
  64-bit

• The four strongest bones per vertex

  Keeps each vertex's four largest weights and scales them to add up to 1.

  w_q = w_q ÷ Σ of the kept weights

  Skin weights and skinning anim/rig.js

  • ModelCraft
  32-bit in part

• Linear blend skinning

  Moves each vertex to the weighted sum of its positions under its bones' matrices.

  p′ = Σ wᵢ × (Mᵢ × p)

  Skin weights and skinning anim/rig.js

  • ModelCraft
  32-bit in part

• Dual quaternion skinning

  Blends the bones' dual quaternions by weight, makes the result unit and moves the vertex with it.

  (r, d) = Σ ±wᵢ (rᵢ, dᵢ); both divided by |r|; p′ = r p r⁻¹ + 2 d r⁻¹

  Skin weights and skinning anim/rig.js

  • ModelCraft
  32-bit in part

• Distance from a point to a bone

  The distance from a vertex to the segment from a bone's head to its tail.

  k = clamp(((p − a) · (b − a)) ÷ |b − a|², 0, 1); d = |a + (b − a) k − p|

  Skin weights and skinning anim/weights.js

  • ModelCraft
  64-bit

• Edge weights of the mesh

  For each vertex, its neighbors and a weight for each edge, used to spread heat and to smooth.

  edge weight = 1 ÷ max(1e-12, length²)

  Skin weights and skinning anim/weights.js

  • ModelCraft
  64-bit

• Automatic weights

  Each bone's weight on each vertex, by bone heat or by distance, with faint weights dropped and each vertex's weights adding up to 1.

  heat: solve (L + H) w = H p per bone, H = 1 ÷ nearest distance²; envelope: w = 1 ÷ d⁴ for bones within twice the nearest distance; drop w < 0.01; divide by the sum

  Skin weights and skinning anim/weights.js

  • ModelCraft
  32-bit in part