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How Differential Kinematics and the Jacobian Explain Robot Motion

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Robots can be described in two ways: joint space, which tracks individual joint angles, and task space, which tracks the position and orientation of the end effector. Forward kinematics maps joint angles to end-effector pose, while inverse kinematics works in reverse to find the joint angles needed for a desired position. To describe motion rather than static poses, the forward kinematics equation is differentiated with respect to time, yielding the relationship ẋ = J(q)q̇. In this equation, the Jacobian matrix J(q) connects joint velocities to end-effector velocities, making it a central tool in robot manipulation. This mathematical framework allows engineers to plan and control how a robot moves through space in a precise and predictable way.

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