Mathematical Model of a Two-Joint Robotic Arm

The two-joint robotic arm is one of the most fundamental multi-body systems in robotics, commonly used for studying kinematic and dynamic characteristics of robotic manipulators. Its mathematical model typically includes core components such as kinematic equations, dynamic equations, and control strategies.

### Kinematic Modeling Kinematic analysis of a two-joint robotic arm primarily involves forward kinematics and inverse kinematics. Forward kinematics calculates the end-effector's position and orientation based on joint angles and link lengths, typically implemented using homogeneous transformation matrices. Inverse kinematics solves for joint angles given a desired end-effector position, often requiring numerical methods like Newton-Raphson or analytical solutions for simple configurations.

### Dynamic Modeling Dynamic models describe the relationship between forces/torques and motion during robotic arm movement. Common modeling approaches include Lagrange's equations and Newton-Euler equations. The Lagrangian method formulates dynamic equations through energy relationships, suitable for complex systems with symbolic computation capabilities. The Newton-Euler method, based on force and moment equilibrium, offers higher computational efficiency through recursive algorithms.

In MATLAB, the Symbolic Math Toolbox can derive dynamic equations symbolically using functions like `sym` and `diff`, while Simulink provides numerical simulation capabilities for analyzing motion responses through block diagram modeling.

### Control Strategies Common control methods include PID control, computed torque control, and adaptive control. PID control offers simple implementation using functions like `pidtune` for linearized systems. Computed torque control employs inverse dynamics compensation with `invdyn` calculations to enhance tracking performance. Adaptive control handles parameter uncertainties through online estimation algorithms like model reference adaptive control (MRAC).

MATLAB simulations enable verification of trajectory tracking performance and disturbance rejection capabilities through tools like Robotics System Toolbox, providing theoretical support for hardware implementation.