Part 1 - How to Solve Inverse Kinematics of a 4 Leg Robot

Summary

In this video, the process of calculating the inverse kinematics (IK) for a robot's leg is explained, along with how to apply these calculations to the entire robot's body. The script covers determining key angles for joint movement, rotating coordinate axes, and using Python functions for efficient calculations. The speaker emphasizes breaking down the problem into manageable steps, starting with a single leg and expanding to the full-body kinematics. The explanation includes geometric principles, trigonometric formulas, and the use of rotation matrices, all critical for programming robotic movement.

Takeaways

• 😀 In this video, the goal is to explain how to calculate the inverse kinematics for one leg of a robot and extend it to the whole body using Python.
• 😀 The tutorial starts by focusing on just one leg to simplify the process before moving to the full body inverse kinematics.
• 😀 The calculation of angles such as theta 1, alpha 2, alpha 3, and others begins with defining a coordinate system for the robot's leg.
• 😀 To calculate theta 1, the length 'a' is first determined using the square root of x squared plus y squared, or by using a Python norm function.
• 😀 Alpha 2 is calculated using a sine rule, while alpha 3 is found by subtracting alpha 2 and a constant angle 'phi' from pi.
• 😀 Depending on which side of the robot the leg is on, either alpha 3 is subtracted or added to alpha 1 to correctly orient the robot.
• 😀 After finding theta 1, the focus shifts to calculating theta 2 and theta 3 by rotating the z and y axes to simplify the problem.
• 😀 The rotation of the coordinate system is achieved by using a transformation matrix, which is created using the sine and cosine of a rotation angle 'r'.
• 😀 The new x and z prime planes are used to calculate the angles beta 1, beta 2, and beta 3, with beta 1 derived from the tangent of the transformed coordinates.
• 😀 Theta 2 and theta 3 are then calculated from the beta angles, where theta 2 equals beta 1 minus beta 2, and theta 3 equals pi minus beta 3.
• 😀 The video concludes by explaining how the single-leg inverse kinematics can be extended to calculate the full body kinematics of the robot.

Q & A

What is the primary objective of the video?

The primary objective is to explain how to calculate the inverse kinematics of one leg of a robot and then extend this to the full body inverse kinematics using Python.

Why is the focus initially on just one leg of the robot?

The focus is on one leg to simplify the problem, making it easier to understand the principles of inverse kinematics before extending the solution to the full body.

What is the significance of the angles `alpha1`, `alpha2`, and `alpha3` in the calculation?

These angles define the geometry of the robot’s leg in the coordinate system and are used to determine the necessary joint movements to reach a target position.

How is `theta1` calculated in the script?

The angle `theta1` is calculated by first finding the length `a` using the formula `sqrt(x^2 + y^2)` and then adjusting the angle based on `alpha3`, depending on whether the leg is on the left or right side of the robot.

What role does the constant angle `phi` play in the calculation?

The constant angle `phi` represents a fixed value based on the robot's configuration (in this case, 90 degrees), which is used to adjust the geometry and calculations for the angles of the leg joints.

What does the transformation matrix do in the process of solving for `theta2` and `theta3`?

The transformation matrix is used to rotate the coordinate axes, which simplifies the calculations of the angles `theta2` and `theta3` by changing the reference frame to a new plane.

How are `beta1`, `beta2`, and `beta3` calculated?

The angles `beta1`, `beta2`, and `beta3` are calculated using trigonometric rules, specifically the tangent and cosine rules. Adjustments are made to ensure the results are within the range of 0 to 2π radians.

What is the importance of using the inverse tangent function (`atan2`) in the calculations?

The inverse tangent function (`atan2`) is important because it returns the angle in the correct range (0 to 2π) and accounts for the correct quadrant of the angle, making it more reliable than a simple inverse tangent calculation.

How are `theta2` and `theta3` determined once the `beta` angles are known?

Once `beta1`, `beta2`, and `beta3` are calculated, `theta2` is found by subtracting `beta2` from `beta1`, and `theta3` is determined by subtracting `beta3` from π (180 degrees).

Why is the calculation of inverse kinematics important for robot motion?

Inverse kinematics is crucial for determining the joint angles required to move a robot's end-effector (like a foot or hand) to a desired position in space, ensuring precise control over the robot's movement.