Dynamics

While kinematics and differential kinematics describe how a robot moves, dynamics describes why: it relates the forces and torques applied at the joints to the resulting motion of the configuration. For a manipulator, this is the bridge between a control signal sent to the motors and the trajectory the end-effector actually traces.

The classical approach using Newton’s laws becomes unwieldy for multi-link mechanisms — it requires an inertial frame, explicit handling of constraint forces, and one $\vec{F}=m\vec{a}$ equation per rigid body in three dimensions. The energy-based Lagrangian formulation sidesteps all three problems and is the standard tool in robotics.

Notes

• 6.1. Lagrangian Dynamics — Generalized coordinates, action and the Lagrangian $L=T-V$, the Euler–Lagrange equations, procedure for deriving equations of motion
• 6.2. Kinetic Energy of a Rigid Body — Decomposition into translational and rotational parts, the inertia tensor $\mathcal{I}$, the generalized inertia matrix $\mathcal{M}$, body-frame twist
• 6.3. Manipulator Equations of Motion — Standard form ${\rm Y}=M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)$, the system inertia matrix, Coriolis matrix, and gravity vector
• 6.4. Wrenches — $6\times 1$ generalized force, power pairing $P={\mathcal{F}}^{T}V$, joint-torque duality ${\rm Y}=J^T\mathcal{F}$