Robot Dynamics¶
Overview¶
Robot dynamics describes the relationship between forces/torques and robot motion. It answers the question: "What joint torques are needed to produce desired motion?"
Lagrange-Euler Formulation¶
The Lagrangian \(L = K - P\) (kinetic energy minus potential energy) leads to:
\[\tau = M(q)\ddot{q} + C(q,\dot{q})\dot{q} + g(q)\]
Where: - \(M(q)\): Mass/Inertia matrix - \(C(q,\dot{q})\): Coriolis and centrifugal forces - \(g(q)\): Gravity vector - \(\tau\): Joint torques
Newton-Euler Formulation¶
Recursive formulation that propagates velocities and accelerations from base to tip, then propagates forces back.
Algorithm Steps:¶
- Outward recursion: Compute \(\omega_i, \dot{v}_i\) for each link
- Inward recursion: Compute \(\tau_i\) for each joint
Manipulator Equations of Motion¶
\[M(q)\ddot{q} + N(q,\dot{q}) = \tau\]
Where \(N\) represents all nonlinear terms including Coriolis, centrifugal, and gravity effects.
Dynamic Parameters¶
Key parameters affecting robot dynamics: - Link masses - Moments of inertia - Center of mass locations - Friction coefficients
Practical Considerations¶
Friction Models¶
- Coulomb friction
- Viscous friction
- Stiction (static + kinetic)
Actuator Dynamics¶
- Motor constants
- Gear ratios
- Backlash
