10.7. Numerical Methods for Solving Inverse Kinematics

Analytical solutions are not always feasible for complex robotic structures. Hence, numerical methods are employed.

In robotics, especially for complex structures, finding exact solutions (analytical solutions) can be very difficult or impossible. Therefore, we use numerical methods, which employ iterative calculations to approximate solutions. These methods can find joint angles that will achieve a given position for the robot’s end effector effectively.

Imagine trying to solve a jigsaw puzzle, but you don't have a picture to guide you. Instead, you try out different pieces (like testing joint angles) one by one until you find a combination that works. Numerical methods are like this trial-and-error approach, helping you find the right arrangement.

1. Newton-Raphson Method:

   • Linearizes the non-linear kinematic equations.
   • Update rule: $q =q+J^{-1}(q)(X−f(q))$
   • Converges quickly near the solution but requires good initial guess.

2. Gradient Descent Method:

   • Minimizes the cost function: E(q)= rac{1}{2} \\|f(q)−X \\|^{2}
   • Slower than Newton-Raphson but more stable in some cases.

3. Damped Least Squares (Levenberg–Marquardt Algorithm):

   • Handles singularities by adding damping: $Δq=¿$
   • Offers a balance between speed and stability.

Iterative methods are techniques used to approximate solutions over successive iterations. The Newton-Raphson method focuses on linearizing the kinematic equations and updating guesses based on how far off the current guess is from the desired output. The Gradient Descent method, on the other hand, reduces a defined error value step by step until a minimum is reached, making it stable and reliable, albeit slower. The Damped Least Squares method helps navigate situations where traditional methods struggle by adding an element of control to maintain convergence, especially near challenging configurations.

Think of it like climbing a hilly path to reach the top of a mountain (the solution). The Newton-Raphson method is like taking big, confident steps towards the peak, but it can stumble if you start too far off course. Gradient Descent is more like taking careful, smaller steps, ensuring you don’t overreach and slip. Damped Least Squares is akin to using hiking poles for stability to help keep you upright when the path gets rocky.

When the Jacobian is not square or invertible, use: $J^{+}=??$

• The joint velocities are estimated using: $q˙=J^{+}X˙$
• Common in redundant manipulators.

The Pseudo-Inverse Jacobian is a technique used when the Jacobian matrix derived from the robot’s kinematics cannot be inverted directly due to the robot having more joints than necessary for the given task (redundant manipulator). The pseudo-inverse allows us to still calculate joint velocities that correspond to the desired velocity of the end effector, facilitating smoother and more controlled movement even in complex situations.

Imagine trying to fit a square peg into a round hole. Sometimes, it just won't fit! The Pseudo-Inverse Jacobian approach is like using a flexible material that can adapt and stretch into the proper shape, allowing the joint movements to adjust to achieve the desired position without forcing it, just like creating a tailored fit for a specific scenario.

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