5.4.1. The Unicycle Model

Dealing with the displacement and velocities of the two wheels of a differential drive robot is messy. A preferred model is that of a unicycle, where we can think of the robot as having one wheel that can move with a desired velocity (V) at a specified heading (\phi). Having the equations to translate between the unicycle model and our wheel velocities is what allows us to simplify the robot with the unicycle model. We have seen how to take measured wheel displacements to calculate the new robot pose. Now we do the reverse and calculate the desired wheel velocities from the unicycle model. Note the temporary assumption of holonomic robot movement, which we will soon correct for differential drive robots.

../../_images/unicycle.png

Fig. 5.4 The Simplified Unicycle Robot Model

Note

We represent the measured forward velocity and robot orientation with variables V_x and \theta. The desired speed and direction are represented with the unicycle model using variables V and \phi.

In the global coordinate frame, we can represent the velocity of the unicycle robot as:

(5.1)\mathcal{U} = \left[ \begin{array}{c} \dot{x} \\ \dot{y} \\ \dot{\phi} \end{array} \right] = \left[ \begin{array}{c} V \!\cos\,\phi \\ V \!\sin\,\phi \\ \omega \end{array} \right]

5.4.1.1. The Jacobian

The kinematics of mechanical systems are often described in terms of Jacobian matrices. A Jacobian defines how movement of parts of a system affect the movement of the whole or a larger part of the system. Let V_s be the velocity vector of the system, \mathcal{P} be the pose of the system, and P_q be the pose vector of certain parts of the system.

\mathbf{J} = \frac{\partial \mathcal{P}}{\partial P_q}

V_s = \dfrac{\mathrm{d}\mathcal{P}}{\mathrm{d}t} = \frac{\partial \mathcal{P}}{\partial P_q} \frac{\partial P_q}{\partial t} = \mathbf{J} \frac{\partial P_q}{\partial t}

We can calculate the robot velocity relative to the velocities of the left and right wheels. Recall from Updating the Robot Position that the kinematics of directional drive systems gives us the forward and rotational displacement of the robot within a short time interval based on the displacement of the left and right wheels. The forward and rotational velocities can also be computed using the velocity of the left and right wheels. The forward velocity is the average of the wheel velocities.

V_x = \frac{R}{2} \,(v_r + v_l)

Note

Remember that v_r and v_l have units of radians/second, while V_x has units of meters/second. Multiplying by R, which is the radius of the wheels (meters/radian) does the unit conversion.

The rotational velocity is the difference of the wheel velocities divided by the radius of rotation. In robotics literature, the radius of rotation is L, or the distance between the wheels. One way to think of this is to consider the case when the left wheel is stopped while the right wheel moves forward. The robot will rotate about the left wheel making an arc with radius of L.

../../_images/robot_rotation.png

\omega = \frac{R}{L}\,(v_r - v_l)

In the local coordinate frame, the three components of the robot velocity are its forward velocity (V_x), its perpendicular velocity (V_p) and its rotational velocity (\dot{\theta} = \omega) in radians per second. Differential Drive robots can not move perpendicular to the direction of wheel rotation, so V_p is always zero. This is the non-holonomic constraint of a differential drive system.

(5.2)\spalignvector{V_x; V_p; \omega} = \spalignmat{R/2, R/2; 0, 0; R/L, -R/L} \spalignvector{v_r; v_l} \\ \spalignvector{V_x; \omega} = \spalignmat{R/2, R/2; R/L, -R/L} \spalignvector{v_r; v_l}

Thus,

\mathbf{J} = \frac{R}{2 L}\,\spalignmat{L, L; 2, -2}.

In the global coordinate frame, the velocity vector is the time derivative of the pose.

\dot{\mathcal{P}} = \left[ \begin{array}{c} \dot{X} \\ \dot{Y} \\ \dot{\theta} \end{array} \right] = \spalignmat{\cos \theta, 0; \sin \theta, 0; 0, 1} \spalignvector{V_x; \omega}

Note

  • Another gotcha to pay attention to is the units used when expressing velocities.
    • Since our Optical Encoders do not know the size of wheels on the robot, they report displacement or velocities in terms of radians. Wheel velocities (v_r and v_l) are in terms radians per second. Forward velocities in the Unicycle and Jacobian equations use linear measurements.
    • Meters or meters per second are the most common linear measurements used.
    • In equations, the radius of the wheels is usually represented with the letter R.
  • When we talk about the rotational velocity, we are describing its velocity towards turning, so the unit of radians here is the distance between the directional drive wheels. It is sometimes called the base length and in equations, it usually represented with the letter L.

5.4.1.2. Calculating Wheel Velocities

Starting with the desired forward and rotational velocities, we can calculate the desired wheel velocities. Using the above equations for V_x and \omega, we can use either the inverse of the Jacobian or equation substitution with some simple algebra to come up with the needed equations.

\spalignvector{V_x, \omega} = \mathbf{J} \,\spalignvector{v_r, v_l}

\spalignvector{v_r, v_l} = \mathbf{J}^{-1}\,\spalignvector{V_x, \omega}

\spalignvector{v_r, v_l} = \frac{1}{2 R}\,\spalignmat{2, L; 2, -L} \spalignvector{V_x, \omega}

v_r = \frac{2\,V_x + \omega\,L}{2\,R}

v_l = \frac{2\,V_x - \omega\,L}{2\,R}

A desired forward velocity (V) is an expected input for steering the robot towards a goal, but we don’t normally think of a steering heading in terms of a rotational velocity (\omega). So we need a controller, either a Proportional Control (P – regulator) or The Point Forward Steering Controller.