· Hankyu Kim · Robotics · 3 min read
FABRIK
FABRIK(A Fast Iterative Solver for the Inverse Kinematics Problem) is a heuristic, iterative inverse kinematics solver that avoids complex matrix operations and singularities, providing smooth motion and fast convergence for robotic chains.
Introduction
Traditional approaches to solving inverse kinematics (IK) often rely on computing the Jacobian matrix, which linearly approximates the relationship between joint angles and the motion of the end-effector. In other words, the Jacobian provides a linear mapping of how small changes in joint angles move the robot’s links and end-effector. Popular Jacobian-based methods include:
- Jacobian Transpose
- Damped Least Squares (DLS)
- DLS with Singular Value Decomposition (SVD-DLS)
- Selectively Damped Least Squares (SDLS)
However, these methods are computationally intensive, especially when handling singularities or complex chains. Newton-based solvers, like Powell or BFGS methods, approach IK as a minimization problem, producing smooth motions but requiring heavy computation and careful implementation. Cyclic Coordinate Descent (CCD) is faster but can produce discontinuous motions and struggles with robots with multiple end-effectors.
Heuristic approaches like Sequential Monte Carlo and particle filters have also been explored.
The Articulated Body Model
A multibody system consists of rigid links connected by joints, forming a chain starting from a root. If constraints are properly defined, the system’s poses can be represented within a limited feasible set. This chain structure allows iterative approaches to efficiently propagate changes along the links.
Algorithm
FABRIK introduces a new heuristic for IK that avoids complex matrix calculations. The core idea is simple: update joint positions iteratively by moving forward and backward along the chain, minimizing positional error.
Algorithm Overview
- Backward Reaching: Start from the end-effector and move toward the root. Adjust each joint slightly to reduce the distance to the target.
- Forward Reaching: Move from the root toward the end-effector. Adjust joints along the way to maintain link lengths and further reduce the error.
A single iteration consists of:
- Computing the distance from the root to the target. If reachable (distance < total link length), proceed.
- Moving the end-effector to the target, then repositioning each preceding joint along the line connecting it to its successor at the correct distance.
- Once the root is displaced, a forward pass repositions joints from the root to the end-effector while maintaining link constraints.
This process repeats until the end-effector reaches or sufficiently approaches the target. Because FABRIK operates using only points, lines, and distances, it avoids heavy matrix computations, converges quickly, and does not suffer from singularities. It also produces smooth, continuous motion.
Optimization
Using Conformal Geometric Algebra (CGA), one can represent spheres, lines, and planes algebraically, enabling predictive updates that accelerate convergence. Another optimization involves projecting unreachable targets along feasible lines toward the chain.
Performance
- FABRIK is generally faster than Jacobian-based methods, especially for long chains or when handling multiple end-effectors.
- It produces smooth, continuous motion without discontinuities.
- Convergence is typically robust, and singularities are naturally avoided.
Limitations
While FABRIK handles most chain types efficiently, its main limitations involve joint type flexibility. Ball-and-socket and hinge joints are well-supported, but more complex joint constraints may require additional modifications. Nevertheless, the original authors report no severe limitations in practice.
