Formula Sheet
Rotation matrix based on ZYX Euler angle rotation,
𝐶𝛼𝐶𝛽
(𝛼,
𝑅𝑍𝑌𝑋 𝛽, 𝛾) = [𝑆𝛼𝐶𝛽
−𝑆𝛽
𝐶𝛼𝑆𝛽𝑆𝛾 − 𝑆𝛼𝐶𝛾
𝑆𝛼𝑆𝛽𝑆𝛾 + 𝐶𝛼𝐶𝛾
𝐶𝛽𝑆𝛾
𝐶𝛼𝑆𝛽𝐶𝛾 + 𝑆𝛼𝑆𝛾
𝑆𝛼𝑆𝛽𝐶𝛾 − 𝐶𝛼𝑆𝛾]
𝐶𝛽𝐶𝛾
where 𝛼, 𝛽, 𝛾 are the angles rotated with respect to Z, Y and X axis.
Rotation matrix based on ZYZ Euler angle rotation,
𝐶𝛼𝐶𝛽𝐶𝛾 − 𝑆𝛼𝑆𝛾
𝑅𝑍𝑌𝑍 (𝛼, 𝛽, 𝛾) = [𝑆𝛼𝐶𝛽𝐶𝛾 + 𝐶𝛼𝑆𝛾
−𝑆𝛽𝐶𝛾
−𝐶𝛼𝐶𝛽𝑆𝛾 − 𝑆𝛼𝐶𝛾
−𝑆𝛼𝐶𝛽𝑆𝛾 + 𝐶𝛼𝐶𝛾
𝑆𝛽𝑆𝛾
𝐶𝛼𝑆𝛽
𝑆𝛼𝑆𝛽 ]
𝐶𝛽
where 𝛼, 𝛽, 𝛾 are the angles rotated with respect to Z, Y and Z axis.
DH Matrix
𝐶𝜃
𝑆𝜃
𝑇=[
0
0
−𝑆𝜃𝐶𝛼
𝐶𝜃𝐶𝛼
𝑆𝛼
0
𝑆𝜃𝑆𝛼
−𝐶𝜃𝑆𝛼
𝐶𝛼
0
𝑎𝐶𝜃
𝑎𝑆𝜃
]
𝑑
1
Transform Derivatives Formulae
Prismatic Joint
𝑗−1
𝜕𝑇𝑗
𝜕𝑞𝑗
𝑗−1
𝜕2𝑇𝑗
𝜕𝑞𝑗2
Revolute Joint
0
0
[
0
0
0
0
0
0
0
0
0
0
0
0
]
1
0
0 −1 0
1 0 0
[
0 0 0
0 0 0
0
0
𝑗−1
] × 𝑇𝑗
0
0
0
0
[
0
0
0
0
0
0
0
0
0
0
0
0
]
0
0
−1 0 0 0
0 −1 0 0
𝑗−1
[
] × 𝑇𝑗
0
0 0 0
0
0 0 0
𝑁
𝑀𝑛𝑗 = ∑ 𝑡𝑟𝑎𝑐𝑒(𝑈𝑖𝑛 𝐽𝑖 𝑈𝑖𝑗 𝑇 )
𝑁
𝑁
𝑖=1
𝑁
𝑁
∑ 𝑀𝑛𝑗 𝑞̈ 𝑗 + ∑ ∑ 𝐶𝑛𝑗𝑘 𝑞̇ 𝑗 𝑞̇ 𝑘 + 𝐺𝑛 = 𝑄𝑛
𝑗=1
𝐶𝑛𝑗𝑘 = 𝐶𝑛𝑘𝑗 = ∑ 𝑡𝑟𝑎𝑐𝑒(𝑈𝑖𝑛 𝐽𝑖 𝑈𝑖𝑗𝑘 𝑇 )
𝑗=1 𝑘=1
𝑖=1
𝑁
𝐺𝑛 = ∑ −𝑚𝑖 𝒈𝑇 𝑈𝑖𝑛 𝒓̅𝒊
𝑖=1
[ℳ(𝒒)]{𝑞̈ } + {𝓒(𝒒, 𝒒̇ )} + {𝓖(𝒒)} = {𝑸}
𝑀11
[ℳ(𝒒)] = [ ⋮
𝑀𝑁1
…
⋱
𝑀1𝑁
]
𝑀𝑁𝑁
𝒞1
{𝓒(𝒒, 𝒒̇ )} = { ⋮ }
𝒞𝑁
𝐶𝑛11
𝒞𝑛 = 𝒒̇ [ ⋮
𝐶𝑛𝑁1
𝐺1
{𝓖(𝒒)} = { ⋮ }
𝐺𝑁
𝑄1
{𝑸} = { ⋮ }
𝑄𝑁
1
[−𝐼𝑥𝑥 + 𝐼𝑦𝑦 + 𝐼𝑧𝑧 ]
2
𝐽=
[
…
⋱
𝐶𝑛1𝑁
] 𝒒̇
𝐶𝑛𝑁𝑁
𝐼𝑥𝑦
𝐼𝑥𝑧
𝑚𝑥̅
𝐼𝑥𝑦
1
[𝐼 − 𝐼 + 𝐼 ]
2 𝑥𝑥 𝑦𝑦 𝑧𝑧
𝐼𝑦𝑧
𝑚𝑦̅
𝐼𝑥𝑧
𝐼𝑦𝑧
𝑚𝑥̅
𝑚𝑦̅
𝑚
Cuboid
𝐼𝑥𝑥 = 12 (𝐿𝑦 2 + 𝐿𝑧 2 )
Cylinder
𝐼𝑥𝑥 = 𝑚𝑟 2
1
2
𝑇
1
[𝐼 + 𝐼 − 𝐼 ] 𝑚𝑧̅
2 𝑥𝑥 𝑦𝑦 𝑧𝑧
𝑚𝑧̅
𝑚
𝐼𝑦𝑦 = 12 (𝐿𝑥 2 + 𝐿𝑧 2 )
𝐼𝑦𝑦 = 𝐼𝑧𝑧 =
𝑚]
𝑚
𝐼𝑧𝑧 = 12 (𝐿𝑥 2 + 𝐿𝑦 2 )
𝑚
(3𝑟 2 + ℎ2 )
12
Trigonometric identities
sin2 𝛼 + cos 2 𝛼 = 1
sin (𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
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sin (𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽
cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽
Calculus
Product rule
𝑑
𝑑𝑢
𝑑𝑣
(𝑢𝑣) = 𝑣 + 𝑢
𝑑𝑡
𝑑𝑡
𝑑𝑡
Chain rule
𝑑
𝑑𝑓 𝑑𝑥
(𝑓(𝑥)) = 𝑑𝑥 ⋅ 𝑑𝑡
𝑑𝑡
Matrix determinant
𝑎
det ([
𝑐
𝑏
]) = 𝑎𝑑 − 𝑏𝑐
𝑑
Matrix inversion
𝑎
𝑖𝑛𝑣 ([
𝑐
1
𝑏
𝑑
]) =
[
𝑑
𝑎𝑑 − 𝑏𝑐 −𝑐
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−𝑏
]
𝑎
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