Motion Model using Quaternions & Angular Velocity

In summary, to track the motion of each chain element in a robot arm, you can use forward or inverse kinematics algorithms to calculate the position and orientation of each joint, and then use control algorithms to update the joint angles over time.
  • #1
oracle3001
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I would be extremely grateful if somebody could help me out with the following setup.

I have a robot arm of x chains, where each chain element, i, at time t, is defined by length l_i(t) and quaternion q_i(t) (where the quaternion is a rotation to rotate from a previous chain element to current, for the first element it is in relation to a global up vector).

I want to track the motion of each chain element in the robot arm, and think that using a constant angular velocity model would be the an ideal starting assumption. What I am looking for is how to construct his i.e

how to predict q_i(t+1) from q_i(t) and w(t) (the angular velocity).
and how to update w(t+1) from w(t)

i.e am looking for the quaternion / angular velocity (or even acceleration) equivalent of the for simple position / velocity model

x(t+1) = x(t) + v

but I am definitely looking to track the quaternions rotations not the positions of the end points of the chains if possible (due to an existing framework).
 
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  • #2
Any help would be greatly appreciated. The answer to your question depends on the type of robot arm you are using. If it is a serial kinematic chain, then you can use the forward kinematics equations to calculate the position and orientation of each joint in the chain from the current joint angles and link lengths. Then, you can use a velocity controller to update the joint angles (and hence the position and orientation) over time.You can also use an inverse kinematics algorithm to calculate the joint angles required to reach a desired position and orientation, given the current joint angles and link lengths. This can be used to control the motion of the robot arm by specifying a desired end effector pose and letting the inverse kinematics algorithm calculate the necessary joint angles.Finally, if your robot arm is a parallel kinematic chain, then you will need to use a different approach to control the motion. In this case, you may need to use control algorithms such as hybrid position/force control or model predictive control.
 
  • #3


Thank you for reaching out for assistance with your motion model setup. I am happy to help you understand how to construct and update your model using quaternions and angular velocity.

First, let's briefly discuss what quaternions and angular velocity are. Quaternions are a mathematical representation of rotations in three-dimensional space. They have four components (x, y, z, w) and are often used in robotics and computer graphics due to their efficiency in representing rotations. Angular velocity, on the other hand, is a vector that describes the rate of change of an object's orientation in space.

Now, to construct your motion model using quaternions and angular velocity, you can use the following equations:

1. To predict q_i(t+1) from q_i(t) and w(t), you can use the quaternion integration formula:

q(t+1) = q(t) + 0.5 * q(t) * w(t) * Δt

Where q(t) is the quaternion at time t, w(t) is the angular velocity at time t, and Δt is the time step. This formula takes into account the rotation of the previous time step and updates it based on the angular velocity.

2. To update w(t+1) from w(t), you can use the angular velocity integration formula:

w(t+1) = w(t) + a(t) * Δt

Where w(t) is the angular velocity at time t, a(t) is the angular acceleration at time t, and Δt is the time step. This formula takes into account the change in angular velocity over time and updates it accordingly.

These equations will allow you to track the motion of each chain element in your robot arm using quaternions and angular velocity. You can use them iteratively to predict and update the rotations and velocities at each time step.

Additionally, if you need to track the positions of the end points of the chains, you can use the quaternion to rotation matrix conversion to obtain the rotation matrix at each time step. From there, you can use the rotation matrix to calculate the positions of the end points.

I hope this helps you with your setup. Please let me know if you need any further clarification or assistance. Best of luck with your project!
 

1. What is a motion model using quaternions and angular velocity?

A motion model using quaternions and angular velocity is a mathematical representation of the movement of an object in three-dimensional space. It combines the use of quaternions, a type of complex number, and angular velocity, which measures the rate of change of an object's orientation, to accurately describe the position and orientation of the object.

2. How does a motion model using quaternions and angular velocity differ from other motion models?

Unlike traditional motion models that use Euler angles or rotation matrices, a motion model using quaternions and angular velocity offers several advantages, including a more compact representation, better numerical stability, and the ability to avoid gimbal lock, a phenomenon where the rotation of an object becomes restricted due to the limitations of a particular representation.

3. What are the applications of a motion model using quaternions and angular velocity?

A motion model using quaternions and angular velocity is commonly used in computer graphics, robotics, and aerospace engineering applications. It is particularly useful for tasks such as simulation, motion planning, and control of complex systems that require accurate and efficient representation of 3D motion.

4. How is a motion model using quaternions and angular velocity implemented in practice?

The implementation of a motion model using quaternions and angular velocity involves the use of mathematical formulas and algorithms to calculate the position and orientation of an object based on its initial state, angular velocity, and time. This can be done using programming languages such as C++, Python, or MATLAB, and specialized libraries and tools are also available to assist with the implementation.

5. What are some limitations of a motion model using quaternions and angular velocity?

While a motion model using quaternions and angular velocity offers several advantages, it also has some limitations. These include the potential for computational errors due to the use of floating-point arithmetic, the need for careful handling of quaternion normalization to avoid numerical instabilities, and the complexity of the model, which may require some mathematical expertise to understand and implement.

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