Robotic Systems · Lecture 9

Holonomic constraints and stabilization

Simeon Nedelchev

Full notes, practice, and references

Constraint levels

ϕ(q)=0,Jv=0,Ja+J˙v=0\phi(q)=0,\qquad Jv=0,\qquad Ja+\dot Jv=0

Position, velocity, and acceleration consistency are distinct checks.

Multiplier dynamics

Ma=τ−c+J⊤fMa=\tau-c+J^\top f

The force sign determines the block-system signs.

Circle example

p=(1,0)p=(1,0), v=(0,2)v=(0,2)

p⊤a=−v⊤v=−4p^\top a=-v^\top v=-4

Centripetal acceleration points inward.

Drift

An acceleration constraint can hold while position error accumulates.

Integration accuracy still matters.

Baumgarte stabilization

Ja+J˙v=−2αJv−β2ϕJa+\dot Jv=-2\alpha Jv-\beta^2\phi

Large gains introduce fast error dynamics.

Practice

Compare consistent starts with deliberately inconsistent starts.

Report constraint error and residual separately.

Reading

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