Velocity kinematics in 3D



In previous lectures, we talked a lot about the position and orientation of objects in the world. In this lecture, we're going to talk about the rate of change of position and the rate of change of orientation. We're going talk about velocities.

In this picture, we can see a ball moving through the air. Before we can say anything meaningful about the velocity or position of the ball, we need to introduce a reference coordinate frame. By now this is a familiar concept to us.

Now I can talk about the velocity of the ball and I describe that by this blue vector which I've labeled V. So V is the velocity vector and that describes the direction and the magnitude of the motion or the rate of change of position of the ball at this particular moment in time. The moment when this photograph was taken.

Now velocity is a vector quantity. It's got 3 components defined with respect to the coordinate frame and that is the rate of change of the X coordinate, that's x dot. The rate of change of the Y coordinate and the rate of change of the Z coordinate.

Now we can consider the rather more complex motion of a Frisbee flying through the air and once again before we can say anything meaningful about the position or velocity of this object, we need to define a reference coordinate frame. Now the Frisbee has got a translational of velocity. It's moving through the air from the person who threw it to the place where its going to land but the Frisbee also spins and we say the Frisbee has got an angular velocity and that's denoted here by the red vector that I've called omega.

Once again the translational velocity V and the angular velocity omega change continually as the object is moving. So these vectors represent translational and angular velocity at the instant that this picture was taken. The angular velocity, that's the red vector here, has got a direction and that is the axis about which the body is rotating at this particular instant in time and the magnitude of this vector is the rate of rotation around that axis. If the object was spinning very quickly, then the angular velocity vector would be very long. Once again, the angular velocity vector has got 3 components so we can take this red vector and we can describe components with respect to the X, Y, and Z axis of a reference coordinate frame.

Now it's not just Frisbee and balls that have got translational and rotational velocity. The end effector of a robot arm has a translational velocity and an angular velocity and often we need to specify what these are and in fact command the robot joints so that we get the desired translational and angular velocity of the robot’s end effector.

Typically we stack the elements of the translational velocity, the elements VX, VY and VZ and the elements of the angular velocity, omega x, omega y and omega z into one vector.

That's six elements long and we refer to this as the spatial velocity or the twist of the object. We use the greek letter nu which looks a bit like a V to denote this special quantity, this spatial velocity or twist which has got 6 elements. It's a six element vector.


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A body moving in 3D space has a translational velocity and a rotational velocity. The combination is called spatial velocity and is described by a 6-element vector.

Professor Peter Corke

Professor of Robotic Vision at QUT and Director of the Australian Centre for Robotic Vision (ACRV). Peter is also a Fellow of the IEEE, a senior Fellow of the Higher Education Academy, and on the editorial board of several robotics research journals.

Skill level

This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus and MATLAB programming.

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