Robotics & Artificial Intelligence

Inverse Kinematics and Inverse Dynamics

Jun 10, 2024
9 minute read

Robo-segway

Understanding when to use inverse kinematics and when to use inverse dynamics depends on the specific requirements of the robotic task at hand. Here’s a detailed explanation of each and their appropriate use cases:

Inverse Kinematics

Purpose: Inverse kinematics is used to determine the joint angles (positions) required to place the robot’s end-effector at a desired position and orientation in space.

Typical Use Cases:

  1. Path Planning: When you need the end-effector of the robot to follow a specific path or reach a specific point, such as in pick-and-place tasks.
  2. Motion Control: When you want to control the position and orientation of the robot’s end-effector directly.
  3. Animation: In computer graphics and animation, inverse kinematics is used to position characters’ limbs accurately.

Example:

  • Moving the robot’s hand to a specific point to grasp an object.
  • Controlling a robot arm to draw a shape by following a sequence of points.

Inverse Dynamics

Purpose: Inverse dynamics is used to calculate the required joint torques or forces to achieve a specific motion, considering the dynamics of the robot, such as mass, inertia, and external forces.

Typical Use Cases:

  1. Force Control: When you need to apply specific forces at the end-effector, such as in tasks that involve interaction with the environment (e.g., assembly tasks, surface finishing).
  2. Trajectory Tracking: Ensuring the robot follows a desired trajectory with the correct dynamics, such as in high-speed or high-precision applications.
  3. Compliance Control: In scenarios where the robot needs to be compliant or adaptive to external forces, such as collaborative robots working alongside humans.

Example:

  • Calculating the torques needed to move a robot arm along a desired trajectory while accounting for gravity, inertia, and other dynamic factors.
  • Ensuring a robotic manipulator applies a consistent force while polishing a surface.

Key Differences

  1. Focus:
    • Inverse Kinematics: Focuses on the geometric aspect of positioning the end-effector.
    • Inverse Dynamics: Focuses on the physical forces and torques required to achieve a motion.
  2. Inputs and Outputs:
    • Inverse Kinematics:
      • Input: Desired end-effector position and orientation.
      • Output: Joint angles.
    • Inverse Dynamics:
      • Input: Desired joint trajectories (positions, velocities, accelerations).
      • Output: Joint torques/forces.
  3. Complexity:
    • Inverse Kinematics: Generally simpler, dealing with position calculations.
    • Inverse Dynamics: More complex, involving the robot’s physical properties and motion equations.

Practical Example

Consider a robotic arm performing a pick-and-place operation:

  1. Inverse Kinematics:
    • Determine the joint angles needed to move the end-effector to the location of the object to be picked up.
    • Move the end-effector to the drop-off location.
  2. Inverse Dynamics:
    • Calculate the necessary torques to move the arm smoothly and quickly to the object’s location, considering the arm’s inertia and any payload effects.
    • Ensure the arm applies the correct force to grip the object without damaging it and moves it to the drop-off location efficiently.

Combining Both

In many robotic applications, both inverse kinematics and inverse dynamics are used in conjunction:

  • Step 1: Use inverse kinematics to determine the joint configurations needed to reach target positions.
  • Step 2: Use inverse dynamics to calculate the torques/forces required to move through these configurations smoothly and accurately.

Yes, it is possible to generate a dataset for a two-link robot arm using ridge regression to determine the joint angles (theta) for point-to-point movements, and then use a neural network to verify and optimize the trajectory. Here’s a step-by-step approach to achieve this:

Assume the two-link robot arm has lengths \(l_1\) and \(l_2\). The forward kinematics can be described as:

\(x = l_1 \cos(\theta_1) + l_2 \cos(\theta_1 + \theta_2)\) \(y = l_1 \sin(\theta_1) + l_2 \sin(\theta_1 + \theta_2)\)

Given a target point \((x, y)\), the inverse kinematics can be derived to find the joint angles \(\theta_1\) and \(\theta_2\).

Multiple Solutions

Inverse kinematics for robot arms often have multiple solutions due to the periodic nature of trigonometric functions. For instance, a given end-effector position can correspond to multiple sets of joint angles.

When you can only apply force to the end-effector, and you need to determine the corresponding joint torques and forces to achieve a desired motion, you are dealing with a problem typically solved using the Principle of Virtual Work or Jacobian Transpose method. Here’s how you can approach this:

Principle of Virtual Work

The principle of virtual work states that the work done by the external forces applied to the end-effector is equal to the work done by the internal forces (joint torques).

Jacobian Transpose Method

This method involves using the transpose of the Jacobian matrix, which relates the velocities of the joints to the velocities of the end-effector. The transpose of the Jacobian can be used to map end-effector forces to joint torques.

Steps to Calculate Joint Torques from End-Effector Forces

  1. Determine the Jacobian Matrix: The Jacobian matrix \(J\) relates the joint velocities to the end-effector velocities.
  2. Apply the Force to the End-Effector: Represent the force applied to the end-effector as a vector.
  3. Calculate Joint Torques: Use the transpose of the Jacobian matrix to calculate the joint torques from the end-effector force.

Detailed Example

Consider a two-link planar robot arm:

  1. Forward Kinematics: \(\begin{aligned} x &= l_1 \cos(\theta_1) + l_2 \cos(\theta_1 + \theta_2) \\ y &= l_1 \sin(\theta_1) + l_2 \sin(\theta_1 + \theta_2) \end{aligned}\)

  2. Jacobian Matrix: The Jacobian matrix \(J\) for a two-link planar robot arm is: \(J = \begin{bmatrix} -l_1 \sin(\theta_1) - l_2 \sin(\theta_1 + \theta_2) & -l_2 \sin(\theta_1 + \theta_2) \\ l_1 \cos(\theta_1) + l_2 \cos(\theta_1 + \theta_2) & l_2 \cos(\theta_1 + \theta_2) \end{bmatrix}\)

  3. Apply Force to the End-Effector: Let \(F\) be the force vector applied to the end-effector: \(F = \begin{bmatrix} F_x \\ F_y \end{bmatrix}\)

  4. Calculate Joint Torques: The joint torques \(\tau\) can be calculated using the transpose of the Jacobian matrix: \(\tau = J^T F\)

Implementation in Python

Here’s the Python code to calculate the joint torques from the applied end-effector force:

import numpy as np

# Robot arm lengths
l1 = 1.0
l2 = 1.0

# Joint angles (example values)
theta1 = np.pi / 4
theta2 = np.pi / 3

# End-effector force (example values)
F = np.array([10.0, 5.0])

# Forward kinematics function
def forward_kinematics(theta1, theta2, l1, l2):
    x = l1 * np.cos(theta1) + l2 * np.cos(theta1 + theta2)
    y = l1 * np.sin(theta1) + l2 * np.sin(theta1 + theta2)
    return np.array([x, y])

# Jacobian matrix
def jacobian_matrix(theta1, theta2, l1, l2):
    J = np.array([
        [-l1 * np.sin(theta1) - l2 * np.sin(theta1 + theta2), -l2 * np.sin(theta1 + theta2)],
        [l1 * np.cos(theta1) + l2 * np.cos(theta1 + theta2), l2 * np.cos(theta1 + theta2)]
    ])
    return J

# Calculate the Jacobian matrix
J = jacobian_matrix(theta1, theta2, l1, l2)

# Calculate the joint torques using the Jacobian transpose method
tau = J.T @ F

print("Joint torques:", tau)

Explanation

  1. Forward Kinematics Function: Computes the position of the end-effector based on the joint angles.
  2. Jacobian Matrix Function: Computes the Jacobian matrix for the given joint angles.
  3. Joint Torques Calculation: Uses the transpose of the Jacobian matrix to calculate the joint torques from the applied end-effector force.

Use Case

  • Manual Guidance: If a human operator applies force to guide the robot, this approach can calculate the required joint torques to move the robot as desired.
  • Force Control: Ensuring the robot applies specific forces during tasks such as assembly, welding, or interacting with compliant surfaces.

By using the Jacobian transpose method, you can effectively translate the forces applied to the end-effector into the corresponding joint torques, allowing for precise control of the robot’s motion and interaction with its environment.

Certainly! Here is a detailed explanation of inverse kinematics and inverse dynamics for a two-link robot arm, including the relevant equations and Python code. Additionally, we will cover how the Jacobian is used to convert forces at the end-effector to joint torques.

Forward Kinematics

The forward kinematics equations for a two-link planar robot arm are:

\[\begin{aligned} x &= l_1 \cos(\theta_1) + l_2 \cos(\theta_1 + \theta_2) \\ y &= l_1 \sin(\theta_1) + l_2 \sin(\theta_1 + \theta_2) \end{aligned}\]

Where:

  • \(l_1\) and \(l_2\) are the lengths of the two links.
  • \(\theta_1\) and \(\theta_2\) are the joint angles.

Inverse Kinematics

The inverse kinematics involves finding the joint angles \(\theta_1\) and \(\theta_2\) for a given end-effector position \((x, y)\). The equations are:

\[\begin{aligned} \theta_2 &= \arccos\left(\frac{x^2 + y^2 - l_1^2 - l_2^2}{2 l_1 l_2}\right) \\ \theta_1 &= \arctan2(y, x) - \arctan2\left(\frac{l_2 \sin(\theta_2)}{l_1 + l_2 \cos(\theta_2)}\right) \end{aligned}\]

Inverse Dynamics

Inverse dynamics calculates the joint torques \(\tau_1\) and \(\tau_2\) needed to achieve a desired motion. The equations involve mass \(m\), inertia \(I\), and other physical properties:

\[\begin{aligned} \tau_1 &= I_1 \ddot{\theta}_1 + I_2 \left(\ddot{\theta}_1 + \ddot{\theta}_2\right) + m_2 l_1 c_2 \left( \ddot{\theta}_1 + \ddot{\theta}_2\right) + m_2 l_1 l_2 \cos(\theta_2) \ddot{\theta}_2 - m_2 l_1 l_2 \sin(\theta_2) \left( \dot{\theta}_2 \left( 2 \dot{\theta}_1 + \dot{\theta}_2 \right) \right) + (m_1 l_1 + m_2 l_1 + m_2 l_2 \cos(\theta_2)) g \cos(\theta_1) \\ \tau_2 &= I_2 \left( \ddot{\theta}_1 + \ddot{\theta}_2 \right) + m_2 l_1 l_2 \cos(\theta_2) \ddot{\theta}_1 + m_2 l_1 l_2 \sin(\theta_2) \left( \dot{\theta}_1^2 \right) + m_2 l_2 g \cos(\theta_1 + \theta_2) \end{aligned}\]

Where:

  • \(I_1\) and \(I_2\) are the moments of inertia.
  • \(\ddot{\theta}_1\) and \(\ddot{\theta}_2\) are the joint accelerations.
  • \(m_1\) and \(m_2\) are the masses of the links.
  • \(g\) is the acceleration due to gravity.

Jacobian Matrix

The Jacobian matrix \(J\) relates the end-effector velocities to the joint velocities. For a two-link arm, it is given by:

\[J = \begin{bmatrix} -l_1 \sin(\theta_1) - l_2 \sin(\theta_1 + \theta_2) & -l_2 \sin(\theta_1 + \theta_2) \\ l_1 \cos(\theta_1) + l_2 \cos(\theta_1 + \theta_2) & l_2 \cos(\theta_1 + \theta_2) \end{bmatrix}\]

Converting End-Effector Forces to Joint Torques

To convert end-effector forces \(F\) to joint torques \(\tau\), use the transpose of the Jacobian matrix:

\[\tau = J^T F\]

Summary

  • Use Inverse Kinematics when the primary concern is the position and orientation of the end-effector.
  • Use Inverse Dynamics when the task requires precise control of forces and torques to account for the physical behavior of the robot during motion.

By understanding the specific requirements of your robotic application, you can decide whether inverse kinematics, inverse dynamics, or a combination of both is needed to achieve your goals.