![]() We can solve for delta-theta as J-inverse times x_d minus f-of-theta_i. If we ignore the higher-order terms, this simplifies to x_d minus f-of-theta_i equals J-of-theta_i times delta-theta. f-of-theta_d is equal to f-of-theta_i, where theta_i is the current guess at the solution, plus the Jacobian of f evaluated at theta_i times delta-theta, plus higher-order terms. To generalize the Newton-Raphson procedure to vectors of joints and endpoint coordinates, not just scalars, we can write the Taylor expansion of the function f-of-theta around theta_d, as shown here. If the initial guess were near the top of the plateau, the calculated slope would have been small, and the next iteration would be far away, where it may be difficult to converge to a solution. In general, the initial guess should be close to a solution to ensure that the process converges. If our initial guess theta_zero had been to the left of the plateau in the function x_d minus f-of-theta, then the iterative process may have converged to the root on the left. Now we can repeat the process, getting a new guess theta_2, and continue until the sequence theta_zero, theta_1, theta_2, etc., converges to the solution theta_d. Since it is not linear in general, theta_1 is only closer to a solution, not an exact solution. If the function x_d minus f were linear, theta_1 would be an exact solution. The change delta-theta in the guess is given by the expression in the figure. If we extend the slope to where it crosses the theta-axis, we get our new guess theta_1. Since we know the forward kinematics f-of-theta, we can calculate the slope of x_d minus f of theta. At that guess, we can calculate the value of x_d minus f-of-theta. ![]() We also make an initial guess at the solution, theta_zero. Now, with the benefit of hindsight, we designate one of the solutions as theta_d. In this example, two values of theta solve the inverse kinematics. The roots of this function correspond to joint values theta that solve the inverse kinematics. Here is a plot of the desired end-effector position x_d minus f-of-theta as a function of theta. In the case that theta and f-of-theta are scalars, the Newton Raphson method can be illustrated easily. To solve this problem, we will use the Newton-Raphson numerical root-finding method. Then the inverse kinematics problem is to find a joint vector theta_d satisfying x_d minus f of theta_d equals zero, where x_d is the desired end-effector configuration. For simplicity, we will start instead with a coordinate-based forward kinematics, where f of theta is a minimal set of coordinates describing the end-effector configuration. The forward kinematics maps the joint vector theta to the transformation matrix representing the configuration of the end-effector.
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