Geometric vs. Analytical Jacobian

几何雅可比与解析雅可比Advanced

Both map joint velocity to end-effector velocity; they differ in whether orientation is expressed as angular velocity or as coordinate derivatives.

The Jacobian matrix describes the linear relationship between joint velocity q̇ and end-effector velocity, and it comes in two versions depending on how end-effector velocity is expressed. The geometric Jacobian outputs the end-effector's linear velocity together with its angular velocity ω (this corresponds to the space/body Jacobian in Modern Robotics; Siciliano and colleagues' textbook defines it slightly differently, and terminology isn't fully standardized). The analytical Jacobian first describes the end-effector pose with a minimal set of coordinates, such as position plus Euler angles, then differentiates those coordinates directly, giving ẋ = J_a q̇. Angular velocity is not the same as the derivative of the Euler angles — the two differ by a transformation matrix that depends on the orientation representation — and that matrix can become non-invertible at certain orientations (such as gimbal lock), which makes the analytical Jacobian fail there, even when the arm itself isn't at a kinematic singularity. Velocity control and statics (τ = JᵀF) generally use the geometric Jacobian, while error feedback or trajectory optimization done directly in coordinates like Euler angles uses the analytical one.

ExampleDescribing end-effector orientation with ZYX Euler angles hits gimbal lock at a pitch angle of ±90°: some directions of angular velocity can no longer be expressed through the finite derivatives of the Euler angles, so the analytical Jacobian becomes non-invertible there, while the geometric Jacobian remains perfectly well-behaved at the same orientation.

Also called: Geometric Jacobian, Analytic Jacobian
Related: Jacobian Matrix · Differential Kinematics · Euler Angles · Gimbal Lock · Twist · Singular Configuration (Kinematic Singularity)
Sources: Lynch & Park, Modern Robotics（预印本 PDF，5.1.5 节 Alternative Notions of the Jacobian） (Chinese)