1. What is Jacobian Matrix
The Jacobian matrix is a matrix of partial derivatives that describes the rate of change of a system with respect to its variables. In other words, it relates the change in input variables to the change in output variables. The Jacobian matrix is commonly used in multivariable calculus and in the fields of physics, engineering, and robotics.
For example, consider a robotic arm with n degrees of freedom. The state of the robot can be described by a set of joint angles q = (q1, q2, …, qn), and the position of the end-effector can be described by a vector x = (x, y, z). The Jacobian matrix J relates the joint angles to the end-effector position:
$$
\frac{\partial x}{\partial q_1} & \frac{\partial x}{\partial q_2} & \cdots & \frac{\partial x}{\partial q_n}\
\frac{\partial y}{\partial q_1} & \frac{\partial y}{\partial q_2} & \cdots & \frac{\partial y}{\partial q_n}\
\frac{\partial z}{\partial q_1} & \frac{\partial z}{\partial q_2} & \cdots & \frac{\partial z}{\partial q_n}
$$
2. Jacobian for Robot Arm
In robotics, the Jacobian matrix is used to calculate the end-effector velocity given the joint velocities. This is important for controlling the motion of a robotic arm, as it allows us to determine the joint velocities required to achieve a desired end-effector velocity.
To calculate the Jacobian matrix for a robotic arm, we need to know the kinematic structure of the arm. This can be described using a URDF (Unified Robot Description Format) file, which defines the joint types, joint limits, and link lengths of the arm. Once we have the URDF file, we can derive the jacobian matrix mannauly
3. Jacobian for Inverse Kinematics
Inverse kinematics is a challenging problem in robotics, as it requires solving a system of non-linear equations.
The Jacobian matrix can be used to solve this problem by providing a linear approximation of the relationship between the joint angles and end-effector position. Specifically, we can use the Jacobian transpose to convert the desired end-effector velocity into joint velocities:
$$\dot{q} = J^T(x) \cdot \dot{x}$$ where $\dot{q}$ is the joint velocity, $\dot{x}$ is the desired end-effector velocity, and $J^T$ is the transpose of the Jacobian matrix. By iteratively updating the joint angles using the calculated joint velocities, we can converge to a solution that achieves the desired end-effector position.