Frame Transformation in rigid bodies

[Dunky]

I'm using rigid body dynamics/kinematics in robotics stuff but I don't have a background in mechanics, I'm interested in understanding the kinematics of frame transformations for rigid bodies.

Suppose we have two reference frames fixed on a rigid body, F_1 and F_2 and a transformation T which takes F_1 to F_2. Suppose we have the linear and angular velocities of the object wrt F_1, how do we get them wrt to F_2?

 

[Filip Larsen]

You will want to search for rigid-body kinematics. The topic is straight forward, but requires rigor in notation to avoid mixing up the multitude of ways the direction of transform can be interpreted, and this notation often varies wildly so be careful when comparing sources. This notation complexity is especially true if you also want to venture into dynamics.

However, with the right assumptions what you seek is simple enough.
Assume you have a constant coordinate change rotation $R^{B\gets C}$ for two Cartesian coordinate systems B and C then any vector $v$ (including linear and angular velocity and acceleration) coordinated in C as $v^C$ transforms like $v^B = R^{B\gets C} v^C$. In case you are using 4x4 matrices note you only need the 3x3 rotation part to transform vectors.

I note you said "a transformation T which takes F_1 to F_2" and this sounds to me like T could be an active transform and not a passive coordinate change (because the transform "moves" frames not changing coordinates). If so, the coordinate change you are looking for is the inverse of T. In a more rigorous notation that means $T^{F_2\gets F_1} = (T_{F_2\gets F_1}^{F_1})^{-1}$ where names in superscript indicate (passive) coordinate systems and subscript indicate (active) frames.

 

[A.T.]

...two reference frames fixed on a rigid body, F_1 and F_2 ... linear and angular velocities of the object wrt F_1

Since F_1 is fixed to the object, the velocities of the object wrt F_1 are always zero, which is probably not what you mean. As @Filip Larsen noted, you have to be very precise and explicit in describing what the transformations do.

 

[A.T.]

Assume you have a constant coordinate change rotation $R^{B\gets C}$ for two Cartesian coordinate systems B and C then any vector $v$ (including linear and angular velocity and acceleration) coordinated in C as $v^C$ transforms like $v^B = R^{B\gets C} v^C$.

@Dunky Just be careful if B and C have different handedness (left hand vs right hand system). Since angular velocity and angular acceleration are pseudo-vectors, they need to be negated when being transformed like that. This doesn't apply to linear velocity tough.

 

[AlexisBlackwell]

To obtain linear and angular velocities of an object relative to the second frame \(F_2\) based on data on velocities relative to the first frame \(F_1\), you can use formulas for changing kinematics. The linear speed \(v_2\) of the object relative to \(F_2\) is equal to \(v_2 = v_1 + \omega_1 \times r\), where \(v_1\) is the linear speed of the object relative to \(F_1\), \(\omega_1 \) is the angular velocity of the object relative to \(F_1\), and \(r\) is the distance vector from the beginning of \(F_1\) to the beginning of \(F_2\) in the system \(F_1\). The angular velocity of the object relative to \(F_2\), \(\omega_2\), is equal to \(\omega_2 = R \omega_1\), where \(R\) is the rotation matrix that transforms vectors from \(F_1\) to \( F_2\).