Advanced turning maneuver of a many-legged robot using pitchfork bifurcation

Our previous work [2, 3] revealed that when we used torsional springs with the same spring constant for all body-segment yaw joints (yaw joints 1–5) and changed the spring constant uniformly among the joints, the straight walk became unstable through Hopf bifurcation, which induced body undulations. This bifurcation was verified by a Floquet analysis, which investigates the stability of solutions of linear differential equations with periodic coefficients, with a simple physical model. In this study, we performed robot experiments of walking in a straight line, where we changed the body-axis flexibility in a way that was different from that in our previous work. Specifically, we used the same torsional spring for yaw joints 2–5 with $k_{i}=41$ Nmm/deg ($i=2,\dots,5$) and various torsional springs for yaw joint 1 with $k_{1}=15$, $17$, $21$, $28$, $41$, and $75$ Nmm/deg. We set all the body segments parallel to each other as the initial conditions. The leg yaw joints in the first module were fixed during the experiments and we did not attempt to control the walking direction, that is, the walking direction was an open loop.

When we used large spring constants for $k_{1}$, the robot kept walking in a straight line as expected, and the body segments were aligned, with all body-segment yaw joint angles being almost zero (Figs. 2A and E, see Movie 1). However, when $k_{1}$ was set to below a threshold value, the body-segment yaw joints showed non-zero angles with the same sign; that is, the body axis was curved and the robot walked in a curved line (Figs. 2B, F, and G, see Movie 1). Specifically, the robot walked in a curved line for $k_{1}=15$, $17$, $21$, and $28$ Nmm/deg, but not for $k_{1}=41$ and $75$ Nmm/deg. Furthermore, both left- and right-curved walking appeared. Figure 2C shows the angles of yaw joint 1 for $1/k_{1}$ averaged over 5 s during a curved walk (the angles for the other body-segment yaw joints are shown in Fig. 3). Although the angles slightly differ among yaw joints 1–5 partly due to feet slippage, these data indicate that the body axis shows a curved shape. Furthermore, while the fluctuation among the trials increases with $1/k_{1}$, the magnitude of these angles increases with $1/k_{1}$. These results suggest that the presence of pitchfork bifurcation depends on $k_{1}$. These angle data were fitted by the square root of $1/k_{1}$ [47]. The bifurcation point was estimated to be $k_{1}=34$ Nmm/deg ($1/k_{1}=0.030$ deg/Nmm).

The dependence of the body-segment yaw joint angles on $1/k_{1}$ (Fig. 2C and Fig. 3) indicates the change of the curved shape of the body axis for the curved walk. Figure 2D shows the radius of curvature $r$ of the body axis for $1/k_{1}$ calculated as $r=5L/\sum_{i=1}^{5}|\theta_{i}|$, where $\theta_{i}$ is the angle of yaw joint $i$ ($i=1,\dots,5$) and $L$ is the length of the body segments. This figure shows that we can control the curvature of the body axis to perform a curved walk by adjusting $k_{1}$ through pitchfork bifurcation.

The pitchfork bifurcation also depends on parameters other than $k_{1}$. To investigate this, we used different values for the spring constant of yaw joints 2–5, gait speed, and relative phase between the ipsilateral legs on adjacent modules. Specifically, we changed $k_{2-5}$ from $41$ to $28$ Nmm/deg, the distance between the AEP and the PEP in each leg (Fig. 1B) from $3$ to $1.8$ cm, or the relative phase from $2\pi/3$ to $\pi$, and performed the same experiments shown in Fig. 2. As a result, these values also induced a curved walk below a critical value of $k_{1}$. However, the estimated bifurcation point and radius of curvature of the body axis changed as shown in Fig. 4. Specifically, when the spring constant of yaw joints 2–5 and gait speed decreased, the estimated bifurcation point decreased from $k_{1}=34$ to 23 and 25 Nmm/deg ($1/k_{1}=0.030$ to 0.043 and 0.040 deg/Nmm), respectively. However, the radius of curvature remained almost unchanged. In contrast, while the relative phase did not change the bifurcation point as much ($k_{1}=31$ Nmm/deg, $1/k_{1}=0.032$ deg/Nmm), it achieved a smaller radius of curvature.

The robot experiments suggested that the presence of pitchfork bifurcation in the straight walk depends on the spring constant $k_{1}$ (Figs. 2C, 3, and 4). We verified this bifurcation from a theoretical viewpoint using a Floquet analysis with a simple physical model, as done in our previous work [2]. The model was simplified from the original high-dimensional mechanical model to extract the fundamentals of locomotion dynamics (Fig. 5A). In particular, the model was two-dimensional because the movements were designed to make the robot walk without up-and-down, roll, or pitch motions of the body segments. Furthermore, because an important role of legs in locomotion is to receive reaction forces from the floor, we neglected the inertial force of the legs and instead modeled the reaction forces at the leg tips based on the geometric conditions. Specifically, we assumed that the leg tips move relative to the body segments as designed and that the leg tips receive the friction forces during the straight line from the AEP to the PEP (Fig. 1B), which are proportional to the velocities relative to the floor.

The equations of motion of the simple model can be expressed as

where $q=[x\,y\,\theta_{0}\,\cdots\,\theta_{5}]^{\scriptsize{\mbox{T}}}$, $[x\,y]$ and $\theta_{0}$ are the position and yaw angle of the first module, respectively; $K(q)$ is the inertia matrix; $h(q,\dot{q})$ is the nonlinear term; $u(q,\dot{q})$ is the torque term of the torsional springs; and $\lambda(q,\dot{q},t)$ is the reaction force term. Because the leg tips move periodically relative to the body segments, the reaction force $\lambda$ becomes a function of time $t$. The detailed description and derivation of the equations of motion (1) are shown in our previous work [2]. During the straight walk of the model, we can write $\hat{q}=[vt+x_{0}\,y_{0}\,0\,\cdots\,0]^{\scriptsize{\mbox{T}}}$ and $\dot{\hat{q}}=[v\,0\,0\,\cdots\,0]^{\scriptsize{\mbox{T}}}$, where $x_{0}$ and $y_{0}$ represent the initial position of the first module and $v$ is the velocity of the leg tips relative to the body segments. For $z^{\scriptsize{\mbox{T}}}=[q^{\scriptsize{\mbox{T}}}\,\dot{q}^{\scriptsize{\mbox{T}}}]$, the linearization of the equations of motion (1) for a straight walk using $z=\hat{z}+\delta z$ gives

Because the movements of the leg tips are periodic with the gait cycle $\tau$, $A(t+\tau)=A(t)$ is satisfied. The detailed description and derivation of the linearized equation (2) are also shown in our previous work [2]. The fundamental solution matrix $Z(t)$ of the linearized equation with periodic coefficients (2) can be expressed as

where $\Phi(t+\tau)=\Phi(t)$ [19]. Because we can use an identity matrix for $Z(0)$ and $\Phi(0)$, the integration of (2) from $t=0$ to $\tau$ yields

The Floquet exponents (eigenvalues of the constant matrix $\Lambda$) and corresponding eigenvectors explain the behavior of the model. Specifically, when all real parts of the exponents are negative, the straight walk of the model is asymptotically stable. In contrast, if any real part of the exponents is positive, the straight walk becomes unstable. In our previous work [2], when all the spring constants of yaw joints 1–5 of the simple model were decreased uniformly, one pair of the Floquet exponents crossed the imaginary axis from the left-half plane and entered the right-half plane above a critical value of the spring constant, which indicates an oscillatory destabilization of the straight walk to produce body undulations and implied Hopf bifurcation (strictly speaking, this corresponds to Neimark-Sacker bifurcation when considering the periodicity of the gait cycle [26]). Furthermore, the relative amplitude and phase between the components of the destabilizing eigenvector were comparable to those of the yaw joint movements during the robot experiments. The Floquet analysis using a simple model is useful for verifying the bifurcation observed in the robot experiments.

Figure 5B shows the Floquet exponents of the simple model when $k_{1}$ was varied while the other spring constants $k_{i}$ ($i=2,\dots,5$) remained fixed, as done in the robot experiments. Except for the zero exponents, all exponents lie in the left-half plane for large $k_{1}$. However, with decreasing $k_{1}$, one exponent moves along the real axis and enters the right-half plane above $k_{1}=12$ Nmm/deg. Although this critical value is smaller than the bifurcation point estimated in the robot experiments in Section III-A, the components of the destabilizing eigenvector for yaw joints 1–5 at the critical point were $0.64$, $0.36$, $0.48$, $0.46$, and $0.16$, respectively, showing the same sign. This indicates that the straight walk is destabilized to produce a curved walk (positive disturbance induces right-curved walk and negative disturbance induces left-curved walk) and implies pitchfork bifurcation. Furthermore, the ratio between the components of the destabilizing eigenvector is consistent with the ratio between the yaw joint angles during the curved walk of the robot experiments (Fig. 3). Although this analysis does not estimate the radius of curvature after the bifurcation due to the limitation of the linear analysis, these results verify the destabilization of the straight walk and the emergence of a curved walk through pitchfork bifurcation, as observed in the robot experiments.

We also investigated the parameter dependence of the pitchfork bifurcation. Figure 5C shows how the critical value of $k_{1}$ changes as the spring constant of yaw joints 2–5, gait speed, and relative phase between the ipsilateral legs on adjacent modules change in the same way as in the robot experiments in Fig. 4. In addition, Fig. 5D shows the change in the critical value when the number of the body segments is increased. When the spring constant of yaw joints 2–5 and gait speed decrease, the critical value decreases. However, the relative phase does not change the critical value as much. These results are consistent with those in the robot experiments (Fig. 4). The number of body segments also does not change the critical value very much. Although we investigated the effects of only changes in the spring constant of yaw joints 2–5, gait speed, relative phase, and number of body segments, this Floquet analysis can investigate other parameters, such as the length and mass, as performed in our previous work [2] for Hopf bifurcation.

To investigate the maneuverability of the robot achieved with the aid of pitchfork bifurcation, we focused on a turning task in which the robot approached a target located on the floor in a direction different from that to which the robot was oriented, as performed in our previous work [3], which investigated the maneuverability with the aid of Hopf bifurcation. For a target at any location (relative angle $\psi$ and distance $R$), there exists a unique radius of curvature $\hat{r}$ of the curved walk with which the robot will approach the target (Fig. 6A). Because the radius of curvature $r$ of the body axis induced by the pitchfork bifurcation monotonically decreases with $1/k_{1}$ (Fig. 2D), $k_{1}=\hat{k}_{1}$ is uniquely determined so that $r=\hat{r}$. This means that when we use $k_{1}=\hat{k}_{1}$, the robot spontaneously approaches the target due to the pitchfork bifurcation characteristics, which is an optimal strategy for turning. However, this strategy is feedforward, depending on the initial conditions of the robot and target, that is, an open loop for the walking direction. In particular, the direction in which the robot turns (left or right) depends on the initial robot conditions because of the pitchfork bifurcation characteristics, and thus this strategy does not guarantee the success of the turning tasks. Therefore, we also used a supplementary turning controller developed in our previous work [3], which uses the laser range scanner of the first module to measure the relative target angle and manipulates the leg yaw joints of the first module to approach the target based on the measured target angle (see Appendix B). This supplementary controller is a closed loop for the walking direction and allows the robot to approach the target even when $k_{1}\neq\hat{k}_{1}$.

For the initial conditions, we used $\psi=45^{\circ}$ and $R=1.3$ m for the relative angle and distance between the first module and the target, respectively, which yielded $\hat{r}=0.88$ m and $\hat{k}_{1}=21$ Nmm/deg ($1/\hat{k}_{1}=0.048$ deg/Nmm), and set all body-segment yaw joint angles to zero (Fig. 6F). Figure 6B shows the trajectory of the first module on the floor during the turning task for three torsional spring constants, namely $k_{1}=15$ ($<\hat{k}_{1}$), $21$ ($\sim\hat{k}_{1}$), and $41$ Nmm/deg ($>\hat{k}_{1}$). Figures 6C and D show the time profiles of the target distance and relative target angle with respect to the walking direction, respectively, for these three spring constants. When the distance was less than 0.15 m, we assumed that the robot reached the target and this task was successfully completed. For $k_{1}=41$ Nmm/deg ($>\hat{k}_{1}$), the robot hardly changed walking direction and the first module trajectory bulged outward. As a result, the robot could not reach the target (Fig. 6G, see Movie 2). For $k_{1}=15$ Nmm/deg ($<\hat{k}_{1}$), although the robot could quickly change walking direction, it moved in directions away from the target due to the small radius of curvature created by pitchfork bifurcation and could not reach the target (Fig. 6I, see Movie 2). In contrast, for $k_{1}=21$ Nmm/deg ($\sim\hat{k}_{1}$), the robot reached the target through the optimal curved walk generated by pitchfork bifurcation (Fig. 6H, see Movie 2).

To quantitatively clarify the turning performance dependence on $k_{1}$, we employed three evaluation criteria, namely $\varepsilon_{1}$, $\varepsilon_{2}$, and $\varepsilon_{3}$. For the criterion $\varepsilon_{1}$, we used the distance of the target at 23 s (the earliest time at which the task is successfully completed) to evaluate how quickly and successfully the robot approached the target. For the criterion $\varepsilon_{2}$, we used the absolute value of the relative target angle with respect to the walking direction at 23 s to evaluate how quickly and successfully the robot was oriented to the target. For the criterion $\varepsilon_{3}$, we used the amount of the control input during the task from the supplementary turning control in the leg yaw joints of the first module (Appendix B) to evaluate how efficiently the robot performs turning. Specifically, $\varepsilon_{3}=\int(\hat{\psi}_{1}^{2}+\hat{\psi}_{2}^{2})dt$. Figure 6E shows the results for $1/k_{1}$. All criteria showed minimum values around $k_{1}=\hat{k}_{1}$, which means that the turning strategy using pitchfork bifurcation achieved the best performance and that the robot made the best use of the curved walk induced by pitchfork bifurcation to complete the turning task.

To verify the performance of the proposed controller using pitchfork bifurcation, we additionally performed the same experiment as that in Fig. 6B but using different initial conditions of the target, namely $\psi=40^{\circ}$ and $R=1.5$ m, which yielded $\hat{r}=1.2$ m and $\hat{k}_{1}=26$ Nmm/deg ($1/\hat{k}_{1}=0.039$ deg/Nmm). Figures 7A, B, and C show the evaluation criteria $\varepsilon_{1}$, $\varepsilon_{2}$, and $\varepsilon_{3}$, respectively, for $1/k_{1}$. All criteria show minimum values around $k_{1}=\hat{k}_{1}$, which means that the turning strategy using pitchfork bifurcation achieved the best performance, in the same way as shown in Fig. 6E. The results show similar trends, which verifies the performance of the proposed controller.

As demonstrated above, the robot achieved maneuverable and efficient turning to approach a target using the curved walk induced by the pitchfork bifurcation. However, when the robot starts the approach with the body axis straight, it takes some time for the convergence to a curved walk, as shown in Fig. 2B. In addition, because the initial straight walk corresponds to the unstable solution of the pitchfork bifurcation, it is unclear whether the robot will turn to the left or right, as shown in Fig. 2C (the disturbance determines the turning direction and the robot continues walking straight unless disturbed). Because the use of the optimal spring constant $\hat{k}_{1}$ does not necessarily guarantee the success of the approach to the target, we also used the supplementary turning controller in the leg yaw joints in the first module.

To clarify the contribution of this supplementary controller, we also performed experiments without using the supplementary controller for $k_{1}=21$ Nmm/deg ($\sim\hat{k}_{1}$). The experimental conditions were identical to those in Fig. 6B. Figure 8 compares the trajectory of the first module on the floor during the turning task with and without the supplementary controller. In all the trials with the supplementary controller, the robot successfully approached the target. In contrast, the robot without the supplementary controller failed in many trials, because the robot started walking in a straight line and took some time to converge to walking in a curved line, and it sometimes curved to the different direction from the target, as expected.

To examine how the turning performance was improved by pitchfork bifurcation, we also performed experiments using the turning strategy based on Hopf bifurcation used in our previous work [3] and compared the performance. For Hopf bifurcation, we used the same spring constant among the body-segment yaw joints and employed five spring constants ($k_{i}=8.7$, $11$, $15$, $21$, and $41$ Nmm/deg, $i=1,\dots,5$) to evaluate the turning performance for $k_{i}$, where the Hopf bifurcation point is about $k_{i}=18$ Nmm/deg ($1/\hat{k}_{i}=0.057$ deg/Nmm), as obtained in our previous work [3]. The experimental conditions were identical to those in Fig. 6E except for the spring constants of the body-segment yaw joints. Figures 9A, B, and C compare the turning performance in terms of the criteria $\varepsilon_{1}$, $\varepsilon_{2}$, and $\varepsilon_{3}$, respectively, between the strategies based on pitchfork and Hopf bifurcations. All criteria for Hopf bifurcation showed minimum values in the unstable region, as observed in our previous work [3]. However, the minimum values of pitchfork bifurcation are lower than those of Hopf bifurcation for all criteria. This means that the turning strategy based on pitchfork bifurcation created by tuning the body-axis flexibility is superior to that based on Hopf bifurcation, which was developed in our previous work [3].