Robotic Systems · Lecture 10

Gauss least constraint and soft engine constraints

Simeon Nedelchev

Full notes, practice, and references

Gauss's principle

min⁡a12(a−a0)⊤M(a−a0)\min_a\frac12(a-a_0)^\top M(a-a_0)
Ja=bJa=b

The mass matrix defines the correction metric.

Multiplier solution

a=a0+M−1J⊤fa=a_0+M^{-1}J^\top f
JM−1J⊤f=b−Ja0JM^{-1}J^\top f=b-Ja_0

Unequal masses

M=diag⁡(1,4)M=\operatorname{diag}(1,4), a0=(1,0)a_0=(1,0)

Constraint: a1+a2=0a_1+a_2=0

Physical projection: (0.2,−0.2)(0.2,-0.2)

Udwadia–Kalaba

a=a0+M−1/2(JM−1/2)+(b−Ja0)a=a_0+M^{-1/2}(JM^{-1/2})^+(b-Ja_0)

Rank and compatibility determine interpretation.

Soft constraints

Compliance permits modeled constraint error.

Solver convergence and physical stiffness are different questions.

Experiment

Vary compliance while holding numerical settings fixed.

Then refine timestep at fixed compliance.

Reading: MuJoCo computation

Reading

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