4  Differential Drive

Differential Drive Robot (Picture from LaValle (2006))

4.1 Dynamics

  • Parameters
    • \(r\): radius of the wheels [m]
    • \(L\): distance between the wheels [m]
  • State: \(\mathbf{x}= \begin{pmatrix}x, y, \theta\end{pmatrix}^\top\) [m, m, rad] (position and orientation of the robot in the world frame)
  • Action: \(\mathbf{u}= \begin{pmatrix} u_l, u_r \end{pmatrix}^\top\) [rad/s, rad/s] (angular velocity of the left and right wheels, respectively)
  • Dynamics: \[ \begin{aligned} \dot x &= \frac{r}{2} (u_l + u_r) \cos \theta \\ \dot y &= \frac{r}{2} (u_l + u_r) \sin \theta \\ \dot \theta &= \frac{r}{L} (u_r - u_l) \end{aligned} \]

4.1.1 Relationship to Unicycle

There is a linear relationship between the differential drive and the unicycle. One can substitute actions to \(v = \frac{r}{2} (u_l + u_r)\) and \(\omega = \frac{r}{L} (u_r - u_l)\) to get \[ \begin{aligned} \dot x &= v \cos \theta \\ \dot y &= v \sin \theta \\ \dot \theta &= \omega \end{aligned} \]

Then, the “original” actions can be computed as \[\begin{aligned} u_l &= \frac{2v-L\omega}{2r}\\ u_r &= \frac{2v + L\omega}{2r} \end{aligned} \]

\[ 2v = r (u_l + u_r) \tag{4.1}\]

\[ L\omega = r (u_r - u_l) \tag{4.2}\]

Adding Equation 4.1 and Equation 4.2:

\[ \begin{aligned} 2v + L\omega = r (u_l + u_r + u_r - u_l) = 2 r u_r\\ u_r = \frac{2v + L\omega}{2 r} \end{aligned} \tag{4.3}\]

Substituting Equation 4.3 in Equation 4.1: \[ \begin{aligned} 2v = r (u_l + u_r) = r (u_l + \frac{2v + L\omega}{2 r}) = r u_l + v + \frac{L\omega}{2}\\ u_l = \frac{2v - v - \frac{L\omega}{2}}{r} = \frac{2v - L \omega}{2 r} \end{aligned} \tag{4.4}\]

4.2 Differential Flatness

Pick flat outputs \(\mathbf{z}(t) = (x(t), y(t))^\top\), i.e., the position of the robot. Then we can compute all necessary variables if \(\mathbf{z}(t)\) is at least C2-continuous. \[ \begin{aligned} \mathbf{x}(t) &= g_x(\mathbf{z}, \dot{\mathbf{z}}) = \left(x, y, \arctan \left(\frac{\dot y}{\dot x}\right)\right)\\ \mathbf{u}(t) &= g_u(\dot{\mathbf{z}}, \ddot{\mathbf{z}}) = \left(\frac{2v-L\omega}{2r}, \frac{2v + L\omega}{2r}\right), \text{ where }\\ v &= {\color{red} \pm}\sqrt{\dot y^2 + \dot x^2}\\ \omega &= \frac{\dot x \ddot y - \dot y \ddot x}{\dot x^2 + \dot y^2} \end{aligned} \]

One can apply the same differential flatness result from the unicycle, followed by the linear mapping derived earlier to recover the actual actions \(g_u\).

4.3 Invariance

The dynamics are translation-invariant.

4.4 Controllers

4.4.1 Geometric Controller

This is the same controller as for the unicycle proposed in (Kanayama et al. (1990)), with the linear mapping applied.

  • Given reference state \(\mathbf{x}_r = \begin{pmatrix}x_r, y_r, \theta_r\end{pmatrix}^\top \in \mathcal{X}\) and reference action \(\mathbf{u}_r = \begin{pmatrix} u_{l,r}, u_{r,r} \end{pmatrix}^\top \in \mathcal{U}\)
  • \(K_x, K_y, K_\theta\in\mathbb R^+\) are tuning gains
  • Control law: \[\begin{aligned} v_r &= \frac{r}{2} (u_{l,r} + u_{r,r})\\ \omega_r &= \frac{r}{L} (u_{r,r} - u_{l,r})\\ x_e &= (x_r-x)\cos \theta + (y_r-y)\sin \theta\\ y_e &= -(x_r-x)\sin \theta + (y_r-y)\cos \theta\\ \theta_e &= \theta_d \ominus \theta \\ v &= v_r \cos \theta_e + K_x x_e\\ \omega &= \omega_r + v_r (K_y y_e + K_\theta \sin \theta_e)\\ u_l &= \frac{2v-L\omega}{2r}\\ u_r &= \frac{2v + L\omega}{2r} \end{aligned} \]

4.4.2 Action Mixing

Geometric controllers might output actions that are outside the nominal action space \(\mathcal{U}\) (saturation limits). To remedy this, a QP can be used that prefers \(\omega\) over \(v\) using a tuning parameter \(\lambda\):

\[ \begin{aligned} \min_{v^*, \omega^*} & \, (\omega^* - \omega) ^2 + \lambda (v^* - v)^2 \\ \text{s.t.} & \, \begin{pmatrix} \frac{2v^*-L\omega^*}{2r}, \frac{2v^* + L\omega^*}{2r} \end{pmatrix}^\top \in \mathcal{U} \end{aligned} \]

followed by converting the result back to the desired control values:

\[\begin{aligned} u_l &= \frac{2v^*-L\omega^*}{2r}\\ u_r &= \frac{2v^* + L\omega^*}{2r} \end{aligned} \]

4.5 Useful Parameters

4.5.1 pololu-3piplus-hyper

Commercially off-the-shelves (Cots) robot: Pololu 3pi+ Hyper Edition

\[\begin{aligned} L &= 0.089\\ r &= 0.016\\ \mathcal{U}&= [-157.08, 157.08] rad/s \times [-157.08, 157.08] rad/s \end{aligned} \]