I recently got into robot motion planning and came across the term 'inverse kinematics'. How exactly is inverse kinematics calculated, and what methods are typically used to determine joint angles from a desired end-effector position? What pitfalls are there, and what strategies do you recommend for stable solutions? Thanks for your input!
How does inverse kinematics work in robotics?
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Inverse kinematics is typically solved using analytical or numerical methods. For simple mechanisms, the solution can often be derived in closed form by combining the transformation matrices of individual links and then rearranging for the joint angles. For more complex robots—especially kinematically redundant systems—iterative methods are used, such as the Newton-Raphson method with the Jacobian matrix, Damped Least Squares (Levenberg-Marquardt), or the CCD method (Cyclic Coordinate Descent). These algorithms minimize the error between the desired end-effector pose and the current pose until an acceptable tolerance is reached.
Common pitfalls include singularities, where the Jacobian matrix becomes rank-deficient, making the solution unstable or non-unique. Joint angle limits can also cause issues if the numerical method proposes a solution outside the allowable range. Ambiguous solutions (e.g., elbow-up/down) require a selection strategy, often based on minimality criteria or predefined preferences, to avoid jumps in the motion sequence. The choice of damping factor in DLS is also critical—too large leads to sluggish movements, while too small risks the system approaching singularities again.
Now, consider this scenario: You have a 7-DoF arm with redundant degrees of freedom and want to simultaneously achieve the end-effector pose and a desired joint configuration (e.g., a specific elbow position)—which optimization strategy would you prefer, and how would you handle potential conflicts between placement and redundancy goals?
A few months ago, I encountered the same inverse kinematics challenge while conducting a grasping experiment with a six-degree-of-freedom robotic arm. The most straightforward approach is to first express the target pose (position + orientation) as a homogeneous transformation matrix, then reverse-engineer the forward kinematics equations of the robotic arm to derive the joint angle expressions. For common "spherical wrist" structures (where the last three rotational axes intersect at a single point), I used a closed-form solution: first determining θ1~θ3 for the first three joints using position data (via the law of cosines and projection), then decomposing the end-effector orientation onto the last three axes to solve for θ4~θ6. This method is fast and precise but requires the robotic arm to have well-defined geometric parameters; otherwise, singular poses (e.g., wrist alignment leading to non-unique solutions) or unreachable targets can easily occur.
In practical projects, I switched to numerical methods—iterative solutions using Jacobians (such as Damped Least Squares) combined with pseudoinverse matrices to avoid sudden singularities. Adding a damping term (λ) at each step significantly improves convergence stability, even allowing reasonable joint angles to be obtained for targets at the edge of the workspace. Another commonly used strategy is "segmented solving": first obtaining a rough solution via closed-form methods, then fine-tuning it with numerical optimization. This approach retains the analytical nature of the solution while compensating for the shortcomings of closed-form solutions in singular poses. Ultimately, the choice of method depends on the complexity of the robotic arm's structure and real-time requirements; combining multiple techniques when necessary can yield more robust inverse kinematics solutions.