Robotics: Theory and Examples

2. Forward and Inverse Kinematics: Key Concepts for Controlling Robotic Manipulators

2.1. Introduction

Forward and inverse kinematics are fundamental concepts in robotics. They are used to describe the relationship between a robot’s joint angles and the position of its end-effector, which is the part of the robot that interacts with its environment. Understanding these concepts is crucial for controlling robotic manipulators and achieving precise motion.

In this article, we will explain forward and inverse kinematics, discuss their importance in controlling robotic manipulators, and illustrate how to compute joint angles and end-effector positions using the forward_kinematics and inverse_kinematics functions in Python. Please refer to the 2R pick-and-place robot simulation on GitHub.

2.2. Forward Kinematics

Forward kinematics is the process of determining the position and orientation of a robot’s end-effector given its joint angles. In other words, it is the mapping from joint space to task space (Cartesian coordinates). Forward kinematics is relatively straightforward because it involves a direct computation based on the robot’s geometry.

To demonstrate forward kinematics, let’s take a look at the forward_kinematics function in Python:

def forward_kinematics(theta1, theta2, offset=0):
    # Calculate the position of the first joint (x1, y1)
    x1 = L1 * np.cos(theta1)
    y1 = L1 * np.sin(theta1)

    # Calculate the position of the second joint (x2, y2)
    x2 = x1 + L2 * np.cos(theta2)
    y2 = y1 + L2 * np.sin(theta2) - offset

    return x1, y1, x2, y2

In this function, theta1 and theta2 represent the joint angles of a two-link planar robot arm, and offset is an optional parameter for accounting for additional height due to the object being manipulated. The function computes the positions of the first and second joints (x1, y1) and (x2, y2) using trigonometry.

2.3. Inverse Kinematics

Inverse kinematics is the reverse process of forward kinematics: given the desired position and orientation of a robot’s end-effector, it computes the required joint angles. Inverse kinematics is more challenging than forward kinematics because it involves solving nonlinear equations and may have multiple solutions or no solution at all, depending on the robot’s configuration and the desired end-effector position.

Let’s examine the inverse_kinematics function in Python:

def inverse_kinematics(x, y, offset=0):
    # Calculate the distance from the base joint to the target position
    d = np.sqrt(x**2 + (y + offset)**2)

    # Calculate the angle between the two links using the law of cosines
    a = np.arccos((L1**2 + L2**2 - d**2) / (2 * L1 * L2))

    # Calculate the angle of the first link with respect to the horizontal axis
    b = np.arctan2(y + offset, x) - np.arctan2(L2 * np.sin(a), L1 + L2 * np.cos(a))

    return b, a + b

In this function, x and y represent the desired end-effector position, and offset is an optional parameter for accounting for additional height due to the object being manipulated. The function computes the required joint angles theta1 and theta2 using the law of cosines and trigonometry. Note that this implementation assumes that the robot has a two-link planar arm and is set up for a specific configuration. More complex robotic systems may require different inverse kinematics solutions.

2.4. Importance in Controlling Robotic Manipulators

Forward and inverse kinematics are essential for controlling robotic manipulators because they allow us to map between joint angles and end-effector positions. This mapping is crucial for designing motion control algorithms, planning trajectories, and achieving precise, coordinated movements.

By understanding forward and inverse kinematics, we can:

Generate smooth and accurate motion trajectories for the end-effector.
Optimize the robot’s movement for energy efficiency or obstacle avoidance.
Implement control algorithms that consider joint constraints, such as joint limits and singularities.

2.5. Jacobian Matrix

The Jacobian matrix is a key concept in the field of robotics, particularly when it comes to controlling robotic manipulators. It falls under the topic of "Inverse and Forward Kinematics" as well as "Motion and Trajectory Planning." The Jacobian matrix helps us understand the relationship between the joint velocities and the end-effector velocities, which is essential for controlling the motion of robotic manipulators.

The Jacobian matrix is a matrix of partial derivatives that represents the relationship between the joint velocities (joint angular velocities for revolute joints or linear velocities for prismatic joints) and the end-effector’s linear and angular velocities in the workspace. The Jacobian matrix is widely used in various robotic applications, including motion planning, force control, and redundancy resolution.

In the context of forward and inverse kinematics, the Jacobian matrix helps us understand how the joint velocities affect the end-effector’s motion. When controlling a robotic manipulator, it is often necessary to determine how to move the joints to achieve a desired end-effector velocity or how a given set of joint velocities will influence the end-effector’s motion.

The Jacobian matrix has several useful properties and applications:

It can be used to compute the end-effector’s linear and angular velocities given a set of joint velocities.
It can be used to solve the inverse kinematics problem iteratively using techniques such as the Jacobian transpose method or the pseudoinverse method.
The Jacobian’s determinant, known as the manipulability measure, provides insights into the manipulator’s dexterity and ability to move in different directions.
Singularities occur when the determinant of the Jacobian matrix becomes zero, indicating a loss of control in certain directions or configurations.

To compute the Jacobian matrix, we use the partial derivatives of the end-effector’s position with respect to the joint angles. The Jacobian matrix, denoted by \(J\), can be represented as follows:

\[\begin{bmatrix} \frac{\partial x}{\partial \theta_1} & \frac{\partial x}{\partial \theta_2} \\ \frac{\partial y}{\partial \theta_1} & \frac{\partial y}{\partial \theta_2} \end{bmatrix}\]

Here, x and y represent the end-effector’s position in the workspace, and theta_1 and theta_2 represent the joint angles of the 2R robot arm. By calculating these partial derivatives, we can create the Jacobian matrix, which represents the relationship between the joint velocities and the end-effector’s linear and angular velocities.

Here’s an example of Python code that introduces the Jacobian matrix for a 2R robot arm:

import numpy as np

def jacobian_matrix(theta1, theta2, l1, l2):
    # Convert the angles to radians
    theta1_rad = np.radians(theta1)
    theta2_rad = np.radians(theta2)

    # Calculate the partial derivatives
    j11 = -l1 * np.sin(theta1_rad) - l2 * np.sin(theta1_rad + theta2_rad)
    j12 = -l2 * np.sin(theta1_rad + theta2_rad)
    j21 = l1 * np.cos(theta1_rad) + l2 * np.cos(theta1_rad + theta2_rad)
    j22 = l2 * np.cos(theta1_rad + theta2_rad)

    # Create the Jacobian matrix
    jacobian = np.array([[j11, j12],
                         [j21, j22]])

    return jacobian

# Example values
theta1 = 30
theta2 = 45
l1 = 1
l2 = 0.5

# Calculate the Jacobian matrix
jacobian = jacobian_matrix(theta1, theta2, l1, l2)
print("Jacobian matrix:")
print(jacobian)

In summary, the Jacobian matrix is a fundamental concept in robotics that plays a vital role in understanding and controlling the motion of robotic manipulators. It falls under the topics of "Inverse and Forward Kinematics" and "Motion and Trajectory Planning," as it bridges the gap between joint space and workspace, allowing for more effective control of robotic systems.

2.6. Conclusion

In this article, we have introduced the concepts of forward and inverse kinematics and explained their importance in controlling robotic manipulators. We have also shown how to compute joint angles and end-effector positions using Python functions. With a solid understanding of these concepts, you can begin to design and implement effective control strategies for robotic systems, ranging from simple planar arms to complex, multi-degree-of-freedom manipulators.

2.7. Exercise

Example 1

Given a 2-link planar robot manipulator with link lengths L1 = 4 units and L2 = 3 units, find the joint angles (theta1, theta2) for the inverse kinematics solution when the end-effector’s position is at (x, y) = (5, 2). Implement your solution using Python programming.

Solution:

To find the joint angles for the given end-effector position, we can use the inverse kinematics function provided in the example code. Here’s the Python code to find the joint angles:

import numpy as np

def inverse_kinematics(x, y, L1, L2):
    d = np.sqrt(x**2 + y**2)
    a = np.arccos((L1**2 + L2**2 - d**2) / (2 * L1 * L2))
    b = np.arctan2(y, x) - np.arctan2(L2 * np.sin(a), L1 + L2 * np.cos(a))
    return b, a + b

L1 = 4
L2 = 3
x = 5
y = 2

theta1, theta2 = inverse_kinematics(x, y, L1, L2)
theta1_degrees = np.degrees(theta1)
theta2_degrees = np.degrees(theta2)

print(f"Joint angles (theta1, theta2) in radians: ({theta1:.2f}, {theta2:.2f})")
print(f"Joint angles (theta1, theta2) in degrees: ({theta1_degrees:.2f}, {theta2_degrees:.2f})")

Example 2

Implement a Python function to compute the Jacobian matrix for a 2-link planar robot manipulator. The function should take the link lengths L1 and L2, and joint angles theta1 and theta2 as input parameters, and return the 2x2 Jacobian matrix. Use the function to compute the Jacobian matrix for a manipulator with L1 = 4 units, L2 = 3 units, and joint angles theta1 = 45 degrees and theta2 = 30 degrees.

Solution:

To compute the Jacobian matrix, we use the partial derivatives of the end-effector’s position with respect to the joint angles:

\[J = \begin{bmatrix} \frac{\partial x}{\partial \theta_1} & \frac{\partial x}{\partial \theta_2} \\ \frac{\partial y}{\partial \theta_1} & \frac{\partial y}{\partial \theta_2} \end{bmatrix}\]

Here’s the Python code to compute the Jacobian matrix:

import numpy as np

def forward_kinematics(L1, L2, theta1, theta2):
    x = L1 * np.cos(theta1) + L2 * np.cos(theta1 + theta2)
    y = L1 * np.sin(theta1) + L2 * np.sin(theta1 + theta2)
    return x, y

def jacobian(L1, L2, theta1, theta2):
    J = np.zeros((2, 2))
    J[0, 0] = -L1 * np.sin(theta1) - L2 * np.sin(theta1 + theta2)
    J[0, 1] = -L2 * np.sin(theta1 + theta2)
    J[1, 0] = L1 * np.cos(theta1) + L2 * np.cos(theta1 + theta2)
    J[1, 1] = L2 * np.cos(theta1 + theta2)
    return J

L1 = 4
L2 = 3
theta1 = np.deg2rad(45)
theta2 = np.deg2rad(30)

J = jacobian(L1, L2, theta1, theta2)
print(f"Jacobian matrix:\n{J}")

This code first defines a forward_kinematics function to calculate the end-effector position for given link lengths and joint angles. It then defines the jacobian function that calculates the Jacobian matrix for a 2-link planar robot manipulator. Finally, it computes and prints the Jacobian matrix for the given link lengths and joint angles.