Programming tunable active dynamics in a self-propelled robot

A distinctive feature of a polar active particle is its finite self-propulsion speed. In general, it can have rotational as well as translational noise. We first study the case without an explicit source of translational noise. In that case, it is often modeled as an ABP, which is equivalent to a biased random walk in two dimensions where the particle always moves in the direction of its instantaneous orientation. Consequently, the ABP moves with a constant speed but its direction changes randomly in the plane. The dynamics of an ABP of speed $v$ is governed by the following equations of motion:

where $\zeta$ is the instantaneous rotational speed of the ABP; it is a white noise with a zero mean and is often modeled as white Gaussian noise. Here, we construct an active Brownian robot with $\zeta$ having a uniform distribution between $[-\eta,\eta]$. In order to produce an ABP motion of given $\eta$ and $v$ in our robot, we compare (2) and (1) to obtain the expressions of $v_{l}$ and $v_{r}$, which yields

The values of $v_{l}$ and $v_{r}$ are updated in the robot after every $\epsilon$ seconds, termed as delay time. In experiments, the accessible values of parameters $\eta$ and $\epsilon$ lie between 1 - 6 rad/s and 0.1 - 0.9 s, respectively.

When implemented in the robot, it mimics the ABP motion quite well. A typical trajectory of the robot is shown in Fig. 2(a) and SI movie S1. To get this unbounded free particle trajectory, we use boundary wall detection using IR sensors to remove the confinement effects (See Appendix B). The zoomed-in part of the trajectory ascertains the absence of translational noise at short time scales. The mean squared displacement (MSD) shows a transition from ballistic to diffusive behavior, a hallmark feature of the ABP dynamics (see Fig. 2(b)). Using a simple theoretical calculation, we obtain the following expression for the MSD [42, 43]

where $D_{r}=\epsilon\left\langle\zeta^{2}\right\rangle/2=\epsilon\eta^{2}/6$ (see Appendix C). The solid red line in Fig. 2(b) represents analytically obtained MSD, which shows an excellent agreement with our experimental data. We further plot angular MSD defined as $\left\langle\Delta\theta^{2}(\tau)\right\rangle$ for varying $\epsilon$ and $\eta$ independently (See Figs. 2(c) and  2(d)). Interestingly, we find that it scales linearly with $\tau$ only for $\tau>\epsilon$, whereas for $\tau<\epsilon$, the plot is ballistic with $\tau^{2}$ scaling. This happens because during the time duration $\epsilon$, the robot always rotates with a constant angular velocity of magnitude $\lvert(v_{l}-v_{r})/d\rvert$, resulting in $\Delta\theta$ scaling linearly with $\tau$. For given $\epsilon$ and $\eta$, the experimental value of $D_{r}$ is measured by line-fitting $\left\langle\Delta\theta^{2}(\tau)\right\rangle$, using the formula $\left\langle\Delta\theta^{2}(\tau)\right\rangle$ = $2D_{r}\tau$. We then plot $D_{r}$ as a function of $\epsilon$ and $\eta$ in the insets of Figs. 2(c) and  2(d), respectively, showing $D_{r}$ indeed scaling according to the expression $D_{r}=\epsilon\eta^{2}/6$, as predicted by the theory.

We also measure spatial velocity autocorrelation function, $C(l)=\left\langle\mathbf{v}(r).\mathbf{v}(r+l)\right\rangle_{r}/\mathbf{v^{2}}$, where $r$ is the distance traveled by the robot, with respect to $l$ for different values of $\eta$ keeping $\epsilon$ constant at 0.25 s as shown in Fig. 2(e) inset. We find that $C(l)\propto e^{-l/L_{s}}$, where $L_{s}$ is the persistence length. Using the above relation, we extracted $L_{s}$ for different $\eta$. We find that $L_{s}$ decreases algebraically with $\eta$ with an exponent of “-2”. This also agrees with the theory since $L_{s}=v/D_{r}\propto\eta^{-2}$.

In general, at small time scales, a polar particle may have translational noise along as well as normal to its orientation. In this case, therefore, $v$ in (2) is not a constant and rather fluctuates around its mean value. We achieve this by generating $v_{l}(t)$ and $v_{r}(t)$ from two statistically independent random uniform distributions in a range of $2V$ with a finite mean $v_{a}$, i.e., in the interval $[v_{a}-V,v_{a}+V]$. Here $V$ quantifies the strength of the translational noise. Interestingly, since our robot, at any instance, can only move along the direction of its wheels, this noise features only in the velocity component, which is parallel to its orientation, while noise normal to its orientation is forbidden. As a result, both $\dot{\theta}(t)$ and $v(t)$ become random variables with zero and a non-zero mean (= $v_{a}$), respectively.

Fig. 3(a) and SI movie S2 show a typical trajectory of the robot following noisy ABP dynamics. At long time scales, the trajectory qualitatively looks similar to the ABP without translational noise (Fig. 2(b)) but differs significantly at short time scales as depicted by frequent back-and-forth events in the zoomed-in trajectory. We plot experimentally measured MSD in Fig. 3(b), which shows scaling with $\tau$ at both short and very long time scales as reported in past studies [43]. We solve the Langevin equation for this case and obtain the following expression of the MSD

where $D_{t}=\epsilon V^{2}/24$ and $D_{r}=\epsilon V^{2}/3d^{2}$ are the translational and the rotational diffusion constants, respectively (see Appendix C). Again the experimental data agree well with the analytical MSD expression (solid red line) over the entire range of time scales (Fig. 3(b)). Clearly, in the limits $\tau\rightarrow 0$ and $\tau\rightarrow\infty$, (5) reduces to $\left\langle\Delta r^{2}(\tau)\right\rangle_{\tau\rightarrow 0}=4D_{t}\tau$ and $\left\langle\Delta r^{2}(\tau)\right\rangle_{\tau\rightarrow\infty}=4[D_{t}+v_{a}^{2}/(2D_{r})]\tau$, respectively, showing an agreement with our experimental results.

A Brownian particle (BP) in two dimensions can move in any direction with an equal probability. However, quite obviously, the wheels of our robot can move only along its direction of orientation, not normal to it. Here we show that it is possible for our robot to perform BP dynamics with limited movement of its wheels. Notably, the ABP motion with noise described in the previous section reduces to the BP dynamics when $v_{a}$ = 0. As a result, both $v(t)$ and $\dot{\theta}$ become random variables with zero mean. Fig. 3(c) and SI movie S3 show the trajectory of the robot following the BP model with $V$ = 7 cm/s and $\epsilon$ = 0.2 s. The resulting MSD shows that at short times ($<\epsilon$) the motion is ballistic, and at longer times it is diffusive (see Fig. 3(d)), similar to $\left\langle\Delta\theta^{2}(\tau)\right\rangle$ observed in Figs. 2(c) and  2(d). Again, the ballistic regime is a consequence of finite velocity events experienced by both left and right wheels over a time interval of $\epsilon$. This is further verified by the fact that the crossover from ballistic to diffusive regime happens exactly after $\epsilon$.

Moreover, since $\left\langle\Delta r^{2}(\tau)\right\rangle=4D\tau$, we extract the diffusion constants $D$ from the MSD plots for various values of $V$ and $\epsilon$. We plot this value with the analytical result, which predicts $D=\epsilon\left\langle v^{2}\right\rangle/4=\epsilon V^{2}/24$ in Fig. 3(d) inset. Clearly, we find an excellent agreement with the theory (solid line indicating the equality).

We now turn our attention to the RTP models, where an individual particle moves in a straight line interrupted by frequent tumble events that randomize its run direction (Fig. 4(a)) [44, 45]. Such motion is widely observed in micro-organisms like bacteria [46] and even in active granular particles [47]. In the case of our robot, it runs when the left and right wheels have the same velocities, i.e., $v_{l}=v_{r}=v_{\text{run}}$, and it tumbles to a new direction when $v_{l}=-v_{r}$. A typical trajectory of the robot performing RTP motion is shown in Fig. 4(b) and SI movie S4 with its instantaneous speed plotted in Fig. 4(c). Here, the robot runs with a constant speed $v_{\text{run}}$ for a duration $\tau_{r}$ (run time), after which it tumbles over a time $\tau_{t}$ (tumble duration). It is important to highlight that $\tau_{t}$ can never be zero in our experiments since the robot can never tumble instantaneously. Inspired by the experimental observations in bacteria [46], we choose $\tau_{r}$ from an exponential distribution $P(\tau_{r})=\gamma e^{-\gamma\tau_{r}}$, whereas $\tau_{t}$ is kept constant for all tumble events. This is achieved by scaling the tumble speed with the tumble angle, which is chosen randomly between 0 to 360 degrees. The RTP trajectory shown in Fig. 4(b) has $v_{\text{run}}=5$ cm/s, $\gamma=3$ s^-1, and $\tau_{t}=0.5$ s. Later, we perform experiments for various $\tau_{r}$ distributions characterized by $\gamma$ at fixed $\tau_{t}$ = 0.5 s. The resulting MSD is shown in Fig. 4(d), showing a ballistic ($\sim\tau^{2}$) to diffusive ($\sim\tau$) transition with transition time decreasing as we increase $\gamma$. However, when we vary $\tau_{t}$ while keeping $\gamma$ constant (Fig. 4(e) for $\gamma$ = 1 $\text{s}^{-1}$), the transition time remains independent of $\tau_{t}$. Regardless, the MSD measured experimentally shows an excellent agreement with the theoretical expression given by [48]

The solid red lines in Fig. 4(d) and (e) show the fit, which is in excellent agreement with the experimental results.

Finally, we demonstrate that the robot can be programmed to switch between the various dynamics discussed above using light intensity as an external parameter. Since our robot is equipped with two LI sensors at its front and back (Fig. 1(a) $\&$ (b), it continuously measures the value of ambient light intensity, which is set using the overhead projector. We load multiple programs onto the robot’s microcontroller and assign light intensity values for each program. For example, we can program the robot to display ABP dynamics when the measured light intensity is between $I_{1}-I_{2}$, BP between $I_{3}-I_{4}$, etc., where $I_{1}$, $I_{2}$, $I_{3}$ $\&$, $I_{4}$ are arbitrary. We summarize this result in SI movie S5, where a robot is programmed to display BP, ABP, and RTP motion as a function of increasing light intensity.

In experiments on modeling the robot dynamics, we cover the robot’s components (except the light intensity sensors) with a white cap with two black circles drawn on top of it, one bigger than the other, along the robot’s orientation. We then use MATLAB’s blob analysis to extract their position coordinates. The vector connecting them provides the instantaneous orientation of the robot, and the midpoint of the line connecting them provides the center-of-mass coordinate. The frame rate at which movies are captured varies between 10-100 depending on the robot’s speed. The movies are captured for a duration of 1 hour for each model.

Theoretical models of ABP, RTP, and BP dynamics consider an unbounded motion of the particle. In our experiments, we use the following protocol to remove confinement effects caused by boundary walls. Once the robot detects the boundary through its IR sensors, it moves 40 cm in the reverse direction while maintaining its orientation. Later, we remove these straight trajectories from the experiments and stitch all the stochastic tracks together by shifting the coordinates appropriately (See Fig.2 in SI). This creates a continuous free particle motion that is effectively unconfined.