CS4610: Lecture 3 - Differential Drive Kinematics

Diff Drive: Coordinate frames and Transformations

a diff drive robot can have any position and orientation in the plane
- three degrees of freedom, which can be defined as
  - two perpendicular translation components that give the location of an identified point on the robot relative to an identified point in the environment
  - the relative angle between an identified vector on the robot and an identified vector in the environment
- together, we can call the position and orientation of the robot its pose
define a world frame with basis (i.e. perpendicular unit vectors in axis directions)

and origin
define a robot frame that moves with the robot. Put the origin of robot frame at the point on the ground midway between the contact points of the two wheels. Define the robot frame basis so that points forward on the robot and points to the robot’s left.
the actual numbers of measure where the robot currently is relative to world frame
- is a point in world frame
- and are unit vectors in world frame
- 6 total numbers, but we only wanted 3
- actually not any choice of 6 numbers for is allowed, because and must be perpendicular unit vectors
- these three constraints reduce the space of possible numeric assignments to from 6 to 3 dimensions
- we can choose any and then set to denote the robot’s position
- we can then choose any and set to denote the robot’s
- this makes the CCW angle from to (measured in radians)
- is also determined by : (remember your trig identities?)
in matrix form, the robot frame basis is thus
those that have studied linear algebra will recognize that the constraints make this matrix orthogonal, which also means that it is a rotation matrix
we can use to transform vectors from robot frame to world frame:
for example, if is a unit vector pointing straight ahead with coordinates measured in robot frame

is the same vector but with coordinates measured in world frame (note in this case )
to transform a point (i.e. a location) in robot frame to world frame, first consider to be a vector relative to , transform it as above, and then add :
this can be made more convenient by defining a homogenous transformation matrix
or, using ,
then extend from to by appending a third homogenous coordinate equal to 1, and we can write the whole transformation from robot to world coordinates as a single matrix multiplication:
what about going the other direction, i.e. transforming a vector or point from world to robot frame?
- since is orthogonal, it is trivial to invert: the inverse of any orthogonal matrix is its transpose:
- we can then derive
- and
- putting the latter into homogenous form,
  with

Diff Drive: Forward Velocity Kinematics

we want to drive the robot around!
we can tell the motor drivers to supply power to the motors, which will make the left and right wheels independently rotate forward or backward at some rotational velocities (let these be measured in radians per second, and define them so that the robot moves forward when they are positive)
but how fast does this make the robot move?
this is a question of forward kinematics: we know the motions of the actuators; we want to predict the motion of the robot
over a short time interval, the robot’s position and orientation will change a little
in the limit as this time interval approaches zero, we can define
as the robot’s translation velocity in world frame
as the robot’s rotation velocity in world frame
observe that in this limit, the constraints of the wheel interaction with the ground (the wheels cannot go directly sideways) imply that is parallel to
so we can define a scalar robot translation speed where
we can thus summarize the robot speed as the pair
in robot frame:
- the translation velocity is
  
  i.e. a vector pointing straight forward with magnitude
- the rotation velocity is still
how are determined as a function of ?
at any instant, it is not hard to see that the robot speed is the average of the wheel speeds on the ground :
- what are given ?
- remember that the length of a circular arc with radius and angle in radians is (derivation: , , )
- so
finally we get
to find , notice that only the difference between the wheel speeds matters
- consider a moving coordinate frame with origin at the point of contact of the left wheel with the ground
- in this frame, the right wheel speed is
- let the distance between the wheels be
- then the point of contact of the right wheel on the ground moves along a circular arc in the moving frame, centered at the left wheel’s point of contact, with radius
- if the robot rotates by over a short time interval, the length of this arc will be
- the arc length is also , where is the time interval:
- differentiating,
the two key equations can be summarized in matrix form:
Assume the robot drives at a constant linear speed and angular speed for a time interval . What trajectory will be driven? (Ans: an arc with radius , subtending angle , and with arc length .)

Diff Drive: Inverse Velocity Kinematics

it is also clearly going to be useful to calculate given
we can apply the analytic inverse of a matrix

to invert the matrix in the final equation for the forward velocity kinematics:
multiply both sides of the forward velocity kinematics equation by this and by to get
done!

Diff Drive: Forward Position Kinematics

if we assume the wheels do not slip on the ground, we should be able to keep track of the robot pose or simply in world frame, relative to its starting value , just by keeping track of the wheel rotations
another way to put it is that, given the time histories, also called trajectories of as functions of time, from until “now” at time , we should be able to theoretically predict the corresponding pose trajectory from until
this process, which gives us an estimate of the robot’s position at any time, is called odometry
note again that is only known relative to ; in practice a very common approach is just to define world frame to coincide with robot frame at
one way to get is to take the derivative of to get , then apply the forward velocity kinematics transformation above to get , and then integrate
our development follows the nice writeup here
starting with orientation, recall that we had derived
- this can be directly integrated to give
  
  where ( if using the convention that the robot starts out aligned to world frame)
moving on to position, recall we had defined , where is the world-frame robot velocity, is the forward speed of the robot in robot frame, and is a world-frame vector that points forward on the robot depending on its current orientation
- we had also derived the relationship
- putting these three together gives
- if we assume that the wheel velocities remain constant, then (using integration by substitution) we can integrate this to give
  
  where ( if using the convention that the robot starts out aligned to world frame)
- This equation can also be derived by a purely geometric analysis, without calculus — the constant velocity assumption means that over any time interval the robot will travel along a well-defined arc path.
our assumption that the wheel velocities remain constant will be violated, of course
but over short enough time intervals, the violation is minimized
this suggests an approach to use these equations for incremental odometry
adapt the equation for to

where are the changes in the two wheel angles from the beginning to the end of the time interval
and adapt the equation for to
where is the estimated robot orientation at the start of the interval
- be careful: what happens to this equation when approaches zero? It becomes the indeterminate form
- in that case l’Hôpital’s rule says to replace each component with the deriviative of the numerator over the derivative of the denominator
- the result is the same as the point and shoot approximation below, with
then update the odometry estimate by

where is the estimated robot position at the start of the interval, and and are the updated estimates of the robot position and orientation at the end of the interval
sometimes people take an even simpler approach and model all the rotation change to happen instantaneously at the start of the interval
- the equation for stays the same
- now estimate as
- this is sometimes called the point and shoot approximation
- no divide by zero problem when

Diff Drive: Inverse Position Kinematics

One way to frame position inverse kinematics is to find wheel rotation trajectories corresponding to a given state trajectory .
We could follow a similar approach as above: differentiate to get , apply the inverse velocity kinematics transformation, and then integrate.
However this is rarely done. More frequently, a variation of the problem is considered where only the start and final poses are given.
The position IK problem is not well defined then, because there are infinitely many trajectories to get from one pose to any other.
We can seek trajectories that minimize certain quantities of interest, for example the total travel time, or the total amount of wheel rotation. Perhaps surprisingly, even these seemingly straightforward problems have been subjects of recent research.
Other simple approaches include the turn-drive-turn strategy:
1. turn in place to face the goal location
2. drive to the goal location
3. turn in plaece to the goal orientation
Or driving in two equal-radius arcs which meet at a common tangent, as demonstrated in this matlab script.
All of this assumes that there are no obstacles in the environment. It is usually more practical to consider obstacles, in which case the problem is called (local or global) path planning or navigation.