3.1 Lagrangian dynamics for robotic systems

Lagrangian dynamics offers a powerful approach to modeling robotic systems. By combining kinetic and potential energy, it provides a comprehensive framework for deriving equations of motion. This method is particularly useful for complex multi-link manipulators.

The Lagrangian formulation leads to a matrix equation that captures a robot's dynamic behavior. This includes mass properties, velocity-dependent forces, and gravity effects. Understanding these dynamics is crucial for designing effective control systems and planning trajectories.

Lagrangian Dynamics for Robotic Systems

Lagrangian equations of motion

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• Lagrangian formulation combines kinetic and potential energy $L = T - V$ to describe system dynamics
• Euler-Lagrange equation $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = \tau_i$ relates generalized coordinates to forces/torques
• Deriving equations of motion involves:
  1. Express energies in generalized coordinates
  2. Compute Lagrangian
  3. Apply Euler-Lagrange equation for each coordinate
  4. Simplify to obtain final equations

Energy expressions for robots

• Kinetic energy includes translational $T_{trans} = \frac{1}{2}mv^2$ and rotational $T_{rot} = \frac{1}{2}\omega^T I \omega$ components
• Potential energy considers gravitational $V_g = mgh$ and elastic $V_e = \frac{1}{2}kx^2$ (springs) effects
• Robot-specific factors: link masses, inertias, joint variables (angles, displacements), center of mass locations

Dynamic behavior analysis

• Mass matrix represents inertial properties, symmetric and positive definite
• Centripetal and Coriolis terms arise from velocity-dependent forces (Christoffel symbols)
• Gravity terms derived from potential energy expression
• Dynamic equation in matrix form $M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau$ captures system behavior
• Stability analysis uses Lyapunov theory to examine equilibrium points

Equations of motion for manipulators

• Process involves defining coordinates, deriving kinematics, computing Jacobians, formulating energies, constructing Lagrangian, applying Euler-Lagrange equations
• Two-link planar manipulator example: generalized coordinates $q_1$, $q_2$ (joint angles), energies $T = \frac{1}{2}m_1v_1^2 + \frac{1}{2}I_1\omega_1^2 + \frac{1}{2}m_2v_2^2 + \frac{1}{2}I_2\omega_2^2$, $V = m_1gy_1 + m_2gy_2$
• Numerical methods (Runge-Kutta, Euler integration) solve equations of motion
• Applications include trajectory planning, control system design, dynamic parameter identification