Inverse Kinematics with DH

To calculate inverse kinematics using either MATLAB or Python based on a DH (Denavit-Hartenberg) parameter list and the final position, the general approach involves solving the nonlinear equations that relate joint angles to the end-effector's position and orientation. Here’s how you can approach this in both MATLAB and Python:

1. DH Parameters and Inverse Kinematics Basics

Inverse kinematics aims to find joint angles (θ1, θ2, ..., θn) given the desired end-effector position and orientation. In the DH parameter method, each link of the manipulator is described by four parameters:

θ: Joint angle (for revolute joints)
d: Link offset
a: Link length
α: Link twist

For a serial manipulator, the transformation from the base to the end-effector is obtained by multiplying the individual transformations for each joint, constructed from the DH parameters.

The forward kinematics is: \(T_{0,n} = T_1 \cdot T_2 \cdot ... \cdot T_n\) Where each \(T_i\) is the transformation matrix for joint i, constructed using the DH parameters.

2. Inverse Kinematics Steps

Inverse kinematics involves solving the equation: \(T_{0,n}(\theta_1, \theta_2, ..., \theta_n) = T_{desired}\) Here, \(T_{desired}\) is the final position and orientation of the end-effector, and we solve for the joint angles.

3. MATLAB Implementation

MATLAB has built-in functions to handle inverse kinematics in robotics, particularly with the Robotics System Toolbox.

Method 1: Using Robotics Toolbox (If installed)

% Define robot using DH parameters
L(1) = Link([theta1, d1, a1, alpha1]);
L(2) = Link([theta2, d2, a2, alpha2]);
% Continue for other links...

% Create robot model
robot = SerialLink(L);

% Define desired end-effector position and orientation (4x4 matrix)
T_desired = [R p; 0 0 0 1];  % R is rotation matrix, p is position vector

% Solve inverse kinematics
q = robot.ikine(T_desired);

Method 2: Custom Calculation (if toolbox is not available)

% Define the DH parameters
theta = [theta1, theta2, theta3]; % Initial guesses
d = [d1, d2, d3];
a = [a1, a2, a3];
alpha = [alpha1, alpha2, alpha3];

% Define desired end-effector position and orientation
T_desired = [R p; 0 0 0 1];

% Solve using optimization (fmincon, fsolve, etc.)
options = optimoptions('fmincon', 'Display', 'off');
q = fmincon(@(q) norm(forward_kinematics(q) - T_desired), theta0, [], [], [], [], [], [], [], options);

% Define forward kinematics
function T = forward_kinematics(q)
    % Compute transformation using DH parameters for given q
    % Return T (4x4 matrix)
end

4. Python Implementation

In Python, you can use libraries like SymPy for symbolic math or SciPy for numerical solving. If you're using robotics libraries, pybullet, Klampt, or numpy are helpful.

Method 1: Using `numpy` and `SciPy`

import numpy as np
from scipy.optimize import minimize

# Define DH parameters
d = [d1, d2, d3]
a = [a1, a2, a3]
alpha = [alpha1, alpha2, alpha3]

# Define forward kinematics function
def forward_kinematics(q):
    # Calculate transformation matrix based on DH parameters and q (joint angles)
    T = np.eye(4)
    for i in range(len(q)):
        T_i = np.array([
            [np.cos(q[i]), -np.sin(q[i]), 0, a[i]],
            [np.sin(q[i]) * np.cos(alpha[i]), np.cos(q[i]) * np.cos(alpha[i]), -np.sin(alpha[i]), -np.sin(alpha[i]) * d[i]],
            [np.sin(q[i]) * np.sin(alpha[i]), np.cos(q[i]) * np.sin(alpha[i]), np.cos(alpha[i]), np.cos(alpha[i]) * d[i]],
            [0, 0, 0, 1]
        ])
        T = np.dot(T, T_i)
    return T

# Objective function to minimize
def objective(q, T_desired):
    T_current = forward_kinematics(q)
    return np.linalg.norm(T_current - T_desired)

# Initial guess for joint angles
q_initial = np.array([theta1, theta2, theta3])

# Desired end-effector position and orientation (4x4 matrix)
T_desired = np.array([[R p],
                      [0, 0, 0, 1]])

# Solve inverse kinematics
result = minimize(objective, q_initial, args=(T_desired,))
q_solution = result.x

Method 2: Using `SymPy` for Symbolic Solution

import sympy as sp

# Define symbolic variables
theta1, theta2, theta3 = sp.symbols('theta1 theta2 theta3')

# Define DH parameters
d = [d1, d2, d3]
a = [a1, a2, a3]
alpha = [alpha1, alpha2, alpha3]

# Define forward kinematics symbolically
def forward_kinematics_sym(theta):
    T = sp.eye(4)
    for i in range(len(theta)):
        T_i = sp.Matrix([
            [sp.cos(theta[i]), -sp.sin(theta[i]), 0, a[i]],
            [sp.sin(theta[i]) * sp.cos(alpha[i]), sp.cos(theta[i]) * sp.cos(alpha[i]), -sp.sin(alpha[i]), -sp.sin(alpha[i]) * d[i]],
            [sp.sin(theta[i]) * sp.sin(alpha[i]), sp.cos(theta[i]) * sp.sin(alpha[i]), sp.cos(alpha[i]), sp.cos(alpha[i]) * d[i]],
            [0, 0, 0, 1]
        ])
        T = T * T_i
    return T

# Solve symbolically
T_desired = sp.Matrix([[R p], [0, 0, 0, 1]])
q = sp.nsolve(forward_kinematics_sym([theta1, theta2, theta3]) - T_desired, [theta1, theta2, theta3], [theta1_0, theta2_0, theta3_0])

Summary of Steps:

Define the DH parameters for your robot.
Set up the forward kinematics function.
Define the desired end-effector position and orientation.
Use a solver (e.g., fmincon in MATLAB or minimize in Python) to compute the joint angles that match the desired end-effector position.