Understanding Numerical Inverse Kinematics: A Simple Guide

Introduction

Numerical inverse kinematics is a powerful method to compute joint configurations that bring the robot’s end-effector to a desired position when analytical solutions are difficult. By leveraging the Jacobian matrix and iterative updates, the algorithm minimizes the difference between the current and target end-effector positions using a gradient descent-like approach. This method can handle singularities and is adaptable to more complex kinematic chains.

Forward Kinematics

Forward kinematics maps joint angles to an end-effector position. For a planar arm with two serially connected links of lengths $L_1$ and $L_2$ and joint angles $\theta_1$ and $\theta_2$ :

x = L_1 \cos(\theta_1) + L_2 \cos(\theta_1 + \theta_2)

y = L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2)

This provides the current end-effector position, which is compared to the target.

Jacobian Matrix

The Jacobian describes how small changes in joint angles affect the end-effector’s position:

J = \begin{bmatrix} -\!L_1 \sin\theta_1 - L_2 \sin(\theta_1 + \theta_2) & -L_2 \sin(\theta_1 + \theta_2) \\ L_1 \cos\theta_1 + L_2 \cos(\theta_1 + \theta_2) & L_2 \cos(\theta_1 + \theta_2) \end{bmatrix}

It is used to translate position errors into joint angle updates.

Loss Function and Gradient

The error vector is defined as:

e = \begin{bmatrix} x_{\text{target}} - x_{\text{current}} \\ y_{\text{target}} - y_{\text{current}} \end{bmatrix}

The squared norm of this vector serves as a loss function:

L = \| e \|^2

Gradient descent updates the joint angles proportionally to the Jacobian pseudo-inverse multiplied by the error:

\Delta \theta = \alpha J^+ e

Where $J^+$ is the pseudo-inverse of the Jacobian and $\alpha$ is the learning rate.

Algorithm Workflow

Initialize: Start with an initial guess for the joint angles.
Iterative Update:
- Compute current end-effector position via forward kinematics.
- Calculate the error vector and loss.
- Compute the Jacobian and its pseudo-inverse.
- Update joint angles using $\Delta \theta = \alpha J^+ e$ .
Convergence: Stop when the error norm is below a threshold or after a maximum number of iterations.

Example Implementation in Python

import numpy as np

# Forward kinematics
def forward_kinematics(q, L1, L2):
    x = L1 * np.cos(q[0]) + L2 * np.cos(q[0] + q[1])
    y = L1 * np.sin(q[0]) + L2 * np.sin(q[0] + q[1])
    return np.array([x, y])

# Jacobian computation
def jacobian(q, L1, L2):
    J11 = -L1 * np.sin(q[0]) - L2 * np.sin(q[0] + q[1])
    J12 = -L2 * np.sin(q[0] + q[1])
    J21 = L1 * np.cos(q[0]) + L2 * np.cos(q[0] + q[1])
    J22 = L2 * np.cos(q[0] + q[1])
    return np.array([[J11, J12], [J21, J22]])

# Numerical inverse kinematics
def inverse_kinematics_jacobian(target_pos, initial_q, L1, L2, max_iterations=1000, tolerance=1e-4, learning_rate=0.1):
    q = np.array(initial_q, dtype=float)
    for i in range(max_iterations):
        current_pos = forward_kinematics(q, L1, L2)
        error = target_pos - current_pos
        if np.linalg.norm(error) < tolerance:
            print(f"Converged in {i} iterations.")
            return q
        J = jacobian(q, L1, L2)
        J_pseudo_inverse = np.linalg.pinv(J)
        q += learning_rate * np.dot(J_pseudo_inverse, error)
    print("Did not converge within maximum iterations.")
    return q

Practical Insights

Using the Jacobian pseudo-inverse allows the algorithm to handle singular configurations gracefully.
Proper initialization of joint angles and tuning the learning rate are essential for convergence.
The method generalizes to manipulators with more degrees of freedom and more complex kinematic chains.
This approach is particularly useful when analytical solutions are unavailable or cumbersome to derive.