The simplest object we can study in kinematics is the point, also known as particle in classical mechanics. We can describe a point in the Euclidean space by providing a vector of coordinates in the reference frame .
The three vectors form an orthonormal basis of . We can use it to express the Cartesian coordinates of , which consist of:
- The signed distance along the x-axis.
- The signed distance along the y-axis.
- The signed distance along the z-axis.
On a side note, the figure above introduces a color convention from computer graphics where the x-axis (sagittal) is red, the y-axis (lateral) green and the z-axis (vertical) blue. This convention is common in off-the-shelf robotics software such as RViz and OpenRAVE.
Cylindrical coordinates represent the position of a point relative to an axis of interest, typically the vertical z-axis, and a direction of interest, typically the sagittal x-axis. They consist of three numbers:
- The distance from to the z-axis.
- The angle , called azimuthal angle (or just the azimuth, which sounds way cooler), between the vector from to and .
- The axis coordinate .
Mathematically, we can map the vector of cylindrical coordinates to Cartesian ones by:
These coordinates are sometimes useful to analyze quantities that act on a body frame of a car or mobile robot. For example, when an external disturbance is applied to a humanoid robot, looking at the direction and amplitude of the disturbance in the horizontal plane can be more insightful than looking at individual x- and y-coordinates.
Spherical coordinates represent the position of a point by:
- Its distance to the origin of the reference frame.
- The angle , the azimuth™, identical to the one used in cylindrical coordinates.
- The angle , called polar, corresponding to the amount of rotation to apply in the resulting plane to go from the "pole" to the desired point.
Mathematically, we can map the vector of spherical coordinates to Cartesian ones by:
Difference between the object and its coordinates¶
These three coordinate systems can be used equivalently to represent the position of our point in the reference frame . Note how we have made, and will consistently make, the distinction between the point itself and the vector (of coordinates (with respect to a given reference frame)) used to represent this point mathematically. The same point can for instance be written in another reference frame translated with respect to . This mental gymnastics of keeping track of both the object and its representation will become all the more important as we move on to linear velocities, angular velocities, twists, etc.
Omitting the reference frame¶
When there is no ambiguity, the reference frame exponent is often dropped and the position of point is abbreviated as . Yet, physical quantities are always expressed with respect to a reference frame, so implicitly there is always an underlying frame to any vector of position coordinates.
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