2.1. Bipedal robot walking model with impulsive actuation

Unlike constant torques drive (see Figure 1(a)), the impulsive actuation (see Figure 1(b)) applies a pulse torque $u$ to the hip joint and a pulse thrust $ P=a(\dot \theta _{s}^-)^2+b\dot \theta _{s}^-+c$ to the heel of the stance leg at the heel strike moment to provide power for the bipedal robot to walk on level ground.

For the 2-DOF bipedal robot with impulsive actuation, the robot has two rigid straight legs of length l and equal mass m, with the mass M of the upper body and trunk concentrated at the hip joint, $\theta _{s}$ and $\theta _{ns}$ denote the angles of stance leg and the swing leg with respect to the ground normal, respectively, and $\alpha$ is the angle between the two legs.

The gait cycle of bipedal robots comprises a swing phase, an impact phase, and an alternation between the support leg and swing leg. The swing motion of each step continues until the occurrence of impact condition, which is given as follows:

(1) \begin{equation} \Omega = \left \{ {({\theta _s},\ \dot \theta _{s},\ {\theta _{ns}},\ \dot \theta _{ns} )\left | \theta _s(t) = -\frac {\gamma }{2}, \theta _{ns}(t) = \frac {\gamma }{2} \right .} \right \}\!. \end{equation}

Figure 1.

Schematic representation of a bipedal robot walking model: (a) constant torques $u_1$ and $u_2$ and (b) pulse torque $u$ and pulse thrust $P$ .

Figure 1 Long description

The image shows a schematic representation of a bipedal robot walking model. In part (a), the model is depicted with constant torques labeled as u1 and u2. The diagram includes angles alpha and theta2, a mass M at the top, and a mass m at the bottom, connected by a leg of length l. The angles and torques illustrate the forces and positions during walking. In part (b), the model is shown with a pulse torque labeled as u and a pulse thrust labeled as P. The angles alpha and gamma are equal, indicating a specific phase of the walking cycle. The diagram also shows the heel strike position, illustrating the transition between steps. The overall structure highlights the mechanical dynamics and control mechanisms of the bipedal robot's walking pattern.

Let

\begin{equation*} x=(x_{1}(t),\ x_{2}(t),\ x_{3}(t),\ x_{4}(t))^T=(\theta _{s}(t),\ \dot \theta _{s}(t),\ \theta _{ns}(t),\ \dot \theta _{ns}(t))^T. \end{equation*}

The walking model of a bipedal robot with impulsive actuation is obtained as follows:

(2) \begin{equation} \begin{array}{ll} \dot x=f(x), \ \ \ x\notin \Omega ,\\ x^+=H (x^-,\ P,\ u), \ x\in \Omega , \end{array} \end{equation}

where

\begin{align*} \left . \begin{array}{ll} f(x)=\left (x_{2}(t),\ x_{1}(t),\ x_{4}(t),\ x_{1}(t)-x_{3}(t)\right )^T,\\[4pt] \Omega = \left \{ {(x_1,\ x_2,\ x_3,\ x_4 )\left | x_1(t) = -\dfrac {\gamma }{2}, x_3(t)= \dfrac {\gamma }{2}\right .} \right \}\!,\\[8pt] H (x^-,\ P,\ u)= \big(x_{3}^-,\ x_{1}^-\cos \gamma +\dfrac {P}{lM} \sin \gamma , \ x_{1}^-,\ x_{4}^-\cos \gamma + \dfrac {P}{lM} \sin \gamma \cos \gamma +u \big)^T. \end{array} \right . \end{align*}

Let $\Omega$ be the Poincaré section, where $\Omega$ is shown in (1). Suppose a trajectory of system (2) reaches the point $A_{k}(\!-\frac {\gamma }{2},\ x_{2k},\ \frac {\gamma }{2},\ x_{4k})$ on $\Omega$ . The trajectory jumps from the point $A_{k}$ to the point $\bar {A}_{k}(\frac {\gamma }{2},\ \bar {x}_{2k},\ -\frac {\gamma }{2},\ \ \bar {x}_{4k})$ , which is on $\Gamma = \left \{ {(x_1,\ x_2,\ x_3,\ x_4 )\left | x_1(t) = \frac {\gamma }{2}, x_3(t)= -\frac {\gamma }{2}\right .} \right \}\!.$ This trajectory originating from the initial point $\bar {A}_{k}$ reaches the point $A_{k+1}(\!-\frac {\gamma }{2},\ \ x_{2(k+1)},\ \frac {\gamma }{2},\ \ x_{4(k+1)})$ at $t=\tau$ , where $A_{k+1} \in \Omega$ and $\tau$ is called a step period.

These two points $A_k$ and $A_{k+1}$ are used to construct the following two-dimensional map:

\begin{align*} \left \{ \begin{array}{ll} x_{2(k+1)}=cos\gamma {x}_{2k}+\dfrac { sin\gamma (a x^2_{2k}+b x_{2k}+c )}{lM},\\[9pt] x_{4(k+1)}=\dfrac {0.75\gamma (cos\tau +1)}{sin\tau }+0.5\left ( cos\gamma {x}_{2k}+\dfrac { sin\gamma (a x^2_{2k}+b x_{2k}+c )}{lM} \right )\!, \end{array} \right . \end{align*}

where $x_{4(k+1)}$ is determined by $x_{2k}$ . The first sub-map of the above map, which is rewritten as the following Poincaré map by assuming $x_{2k}=y_{k}$ and setting $A=\frac {sin\gamma }{lM}$ :

(3) \begin{equation} y_{k+1}=Aa y^2_{k}+(Ab+\cos \gamma )y_k+Ac=g(y_{k}, b), \end{equation}

where $b$ is treated as a parameter.

For a given fixed point $y_{0}$ of the Poincaré map (3), the point $A^{*}(\!-\frac {\gamma }{2},\ y_{0},\ \frac {\gamma }{2},\ \frac {0.75\gamma (\cos \tau +1)}{\sin \tau }+0.5y_{0})$ corresponds to one period- $1$ gait of (2), where the pulse thrust $P$ , the pulse torque $u$ , and the step period $\tau$ are given as follows:

(4) \begin{equation} P=ay_{0}^2+by_{0}+c,\ u=\frac {0.75\gamma (1+\cos \tau )}{\sin \tau }+(0.5-\cos \gamma )y_{0},\ \tau =\ln \frac {2y_{0}-\gamma }{2y_{0}+\gamma }. \end{equation}

For more details on the bipedal robot model (2) with impulsive actuation and the Poincar $\acute {\textrm { e}}$ map (3), refer to Ref. [Reference Long, Hui and Jiang38] and related references cited therein. In the bipedal robot model (2), angular momentum conservation is adopted to model the impact phase, primarily because it is highly consistent with the Lagrangian dynamics used for the swing phase and naturally accommodates the leg role exchange without requiring pointwise impact force analysis. Additionally, the nonlinear impulse thrust introduced in this paper, which is applied at the ankle of the stance leg and directed toward the hip, can be conveniently integrated into the impact mapping via angular momentum conservation, whereas Newton’s impact law would necessitate extra modeling of the coupling between thrust and contact forces. It is acknowledged that Newton’s impact method was employed for systematic modeling of multi-link bipedal robots, which offers good scalability for multi-rigid-body collisions [Reference Fazel, Shafei and Nekoo39]. In future work, a comparison of both approaches is planned to further validate the applicability and robustness of the proposed modeling strategy.

The construction of Poincaré maps plays a crucial role in studying the walking dynamics of bipedal robots. For instance, an explicit-form Poincaré map is developed to discuss passive dynamic walking [Reference Znegui, Gritli and Belghith40], a controlled Poincaré map is designed to analyze the stabilization of the passive walking dynamics, thereby deriving an effective state-feedback control law in Ref. [Reference Znegui, Gritli and Belghith41]. A key distinction to note is that the control term in the controlled Poincaré map in Ref. [Reference Znegui, Gritli and Belghith41] is a continuous torque $u$ applied during the swing phase, while in this study, no continuous torque is considered. Instead, the control term in the developed Poincaré map (3) is related to the pulse thrust $P$ , which acts during the instantaneous collision-switching phase.

The walking model with impulsive actuation (2) can be practically implemented through spring-based impulsive mechanism. Compression springs or torsional springs can be used to store energy during the swing phase and release it as an impulsive thrust at heel strike. This approach enables energy recovery from the robot’s own dynamics and has been successfully implemented in passive dynamic walkers. By adjusting the spring stiffness and pre-compression, the theoretical pulse thrust parameters $(a,\ b,\ c)$ are mapped to actual control variables, thereby regulating the pulse thrust $ P=a(\dot \theta _{s}^-)^2+b\dot \theta _{s}^-+c$ . The pulse thrust must be precisely synchronized with heel strike, requiring high-bandwidth force/torque sensors at the foot.

It is worth noting that $y_0\lt 0$ due to the fact that the angular velocity of the stance leg $\dot \theta _{s}\lt 0$ in actual walking. Therefore, one negative fixed point of the Poincaré map (3) corresponds to one period- $1$ gait of (2) and vice versa.

2.2. Controlled gaits

For the pulse thrust $ P=a(\dot \theta _{s}^-)^2+b\dot \theta _{s}^-+c$ , first assume that the parameters $a\gt 0$ and $c\geq 0$ . Map (3) has two negative fixed points:

(5) \begin{equation} y_{ 0 1} = \frac {-(Ab+\cos \gamma -1)- \sqrt {\Delta _1 }}{2Aa},\ y_{ 0 2} =\frac {-(Ab+\cos \gamma -1)+ \sqrt {\Delta _1 }}{2Aa}, \end{equation}

for $b\gt b_1$ , where $ \Delta _1=(Ab+\cos \gamma -1)^2-4A^2ac$ and

(6) \begin{equation} b_1=\frac {2A\sqrt {ac}+1-\cos \gamma }{A}. \end{equation}

The fixed point $ y_{ 0 2}$ , which eigenvalue $ \lambda _{03}=1+\sqrt {\Delta _1 }\gt 1,$ corresponds to an unstable period- $1$ gait of (2).

The eigenvalue of the fixed point $ y_{ 0 1}$ is

\begin{equation*} \lambda _{01}=g'(y_{k}, b)|_{y_{k}=y_{ 0 1}}=2Aa y_{0}+Ab+\cos \gamma =1-\sqrt {\Delta _1 }. \end{equation*}

It is easy to calculate that $\lambda _{01}=1$ at $b=b_1$ while $\lambda _{01}=-1$ at

(7) \begin{equation} b=b_2=\frac {2\sqrt {1+A^2ac}+1-\cos \gamma }{A}. \end{equation}

Hence, a fold bifurcation occurs at $b=b_1$ and a flip bifurcation occurs at $b=b_2$ . For $b\in (b_1,\ b_2)$ , $|\lambda _{01} | \lt 1$ and the fixed point $ y_{ 01}$ corresponds to a stable period- $1$ gait of (2).

In Model (2), the parameters for the robot are set as $l=1m$ , $\gamma =0.4 rad$ , and $M=10kg$ . For the pulse thrust $ P=a(\dot \theta _{s}^-)^2+b\dot \theta _{s}^-+c$ , set $a=12$ and $c=8$ . It follows from (6) and (7) that $b_1=21.6230$ and $b_2=56.9972$ . Figure 2 shows that as parameter $b$ varies ( $b\gt b_1=21.6230$ ), flip bifurcations occur, consequently giving rise to multi-periodic gaits ( $P-n,\ n=2,\,4,\ \cdots$ ) and chaotic gaits.

Figure 2.

The bifurcation diagram of periodic gaits with respect to $b$ for system (2) with $a=12$ and $c=8$ .

Figure 2 Long description

A line graph showing the bifurcation diagram of periodic gaits with respect to b for a system with a equals 12 and c equals 8. The x axis represents the variable b ranging from negative 70 to 80. The y axis represents the variable x 2 k ranging from negative 10 to 6. The graph includes a red dashed line indicating unstable P minus 1 gait, a blue solid line indicating stable P minus 1 gait, and red arrows pointing to fold bifurcation and flip bifurcation. Key points on the x axis are marked at b equals negative 17.5688, b 1 equals 21.6230, and b 2 equals 56.9972. The graph shows complex behavior with regions of stability and instability. All values are approximated.

The stable period- $1$ gait is the most ideal walking mode for bipedal robots. There exist multi-period gaits and chaotic gaits for $b\gt b_2$ in the case of $a\gt 0$ and $c\geq 0$ , which need to be controlled to period- $1$ gait. There is no fixed point in (3) for $b\in (\!-17.5688,\ b_1)$ , and there are only positive fixed points in (3) for $b\lt -17.5688$ ; hence, there is no periodic gait for $b\lt b_1$ in the case of $a\gt 0$ and $c\geq 0$ , which leaves room for us to explore the existence of period- $1$ gait. Thus, the gaits of walking model (2) in the case of $a\gt 0$ and $c\geq 0$ are designated as the target gaits for control to discuss walking gait control.

To study the gait control of bipedal robots, it is also necessary to consider the initial conditions for gait simulation. All numerical simulations in this study employ initial conditions selected according to the following systematic methodology.

Phase space analysis: Initial states are chosen based on the system’s phase portrait and Poincaré map analysis. For chaotic gait simulations, points are selected from within the chaotic attractor.

Physical consistency check: Initial states are chosen based on practical joint limits $|\theta _{s}|,\ |\theta _{ns}|\lt \frac {\pi }{3}$ and angular velocities limits $\pm 10 rad/s$ .

Bifurcation-informed selection: For period- $1$ gait analysis, initial conditions are taken as fixed point of the Poincaré map (3) and its nearby points. For period- $2,4,\cdots$ gaits analysis, points are selected from corresponding periodic orbits and their neighborhoods.

Robustness verification: Multiple initial conditions from different regions of the state space are tested to ensure control strategies are not overly sensitive to specific starting points.

Different from the BVP-based initial condition determination method for passive compass-gait bipedal robots [Reference Safartoobi, Dardel, Ghasemi and Daniali15], the above strategy is specifically developed for the impulse-driven bipedal robot model in this study, which fully considers the dynamic characteristics of impulsive actuation at heel strike and the bifurcation behavior of the system. It not only ensures the physical rationality of initial states but also guarantees the relevance of initial conditions to the controlled gait types (chaotic, multi-period, and period-1), which is essential for the validity of the subsequent gait control simulation results.

3.1. OGY control

Assume that system (2) is chaotic at $b=b_0$ (see Figure 3(a)). The period- $1$ gait, which corresponds to the fixed point $y_{01}$ for $b=b_0$ in (5), is unstable and embedded within the attractor. Now calculate the small disturbance about parameter $b$ .

At $(b_0,\ y_{01})$ , it follows from (3) that

\begin{equation*} \begin{array}{cc} & \dfrac {\partial g} {\partial y_k}(b_0,\ y_{01})= 2Aa y_{01}+Ab_0+\cos \gamma ,\\ & \dfrac {\partial g} {\partial b}(b_0,\ y_{01}) = Ay_{01}. \end{array} \end{equation*}

By linearizing the Poincaré map (3) around $(b_0,\ y_{01})$ , we derive the perturbation parameter as:

(8) \begin{equation} \bar {b}=-\dfrac {\dfrac {\partial g} {\partial y_k}}{\dfrac {\partial g} {\partial b}}(b_0,\ y_{01})=\frac {2Aa y_{01}+Ab_0+\cos \gamma }{Ay_{01}}=\frac {\sqrt {\triangle _1}-1}{Ay_{01}}. \end{equation}

Figure 3.

For system (2) with $a=12$ and $c=8$ , (a) the bifurcation diagram of periodic gaits with respect to $b$ and (b) the maximum Lyapunov exponent with respect to $b$ .

Figure 3 Long description

The image contains two graphs. The first graph, labeled (a), is a bifurcation diagram showing the relationship between the variable x2k and the parameter b. The diagram features a blue line representing the system's behavior and a red dashed line indicating a specific threshold or boundary. The second graph, labeled (b), displays the maximum Lyapunov exponent as a function of the parameter b. This graph also includes a blue line representing the Lyapunov exponent values and a red dashed line marking a critical threshold. Both graphs highlight a vertical dashed line at b = b0, indicating a significant point of interest. The bifurcation diagram illustrates the stability and transitions of periodic gaits, while the Lyapunov exponent graph shows the system's sensitivity to initial conditions and the presence of chaotic behavior. The graphs collectively demonstrate the dynamic behavior of the bipedal robot system under varying parameter conditions.

The controlled system is

\begin{equation*} \left \{ \begin{array}{ll} \dot x=f(x), \ \ \ x\notin \Omega ,\\ x^+=H_{1}x^-+H_{2}\dfrac {\bar {P}}{lM}+H_{3}, \ x\in \Omega , \end{array} \right . \end{equation*}

where

\begin{equation*} \bar {P}=a(x_2(t)^-)^2+(b+ \mu \bar {b}) x_2(t)^- +c, \end{equation*}

(9) \begin{equation} \mu =\left \{ \begin{array}{ll} 1, \ \ \ \textrm {if}\ x_1(t)= - \dfrac {\gamma }{2}\ \textrm {and}\ |x_2(t)-y_{01}|\lt \sigma ,\\ 0, \ \ \textrm {else}, \end{array} \right . \end{equation}

$\mu$ is the switching function and $\sigma$ is the threshold value for state perturbation. Hence, the following result regarding chaotic gait control using the OGY control method is obtained.

For the parameters set $a=12$ , $c=8$ , and $b=b_0=73$ identified as chaotic in Figure 3(a), compute the maximum Lyapunov exponent to quantify the system’s sensitivity to initial conditions. The maximum Lyapunov exponent $\lambda _{max}$ is calculated using the following standard algorithm (see [Reference Yang and Wu42, Reference Gritli and Belghith43] and related references cited therein):

\begin{equation*} \lambda _{max}=\lim _{t \to \infty } \frac {1}{t} \frac {|\delta X(t)|}{|\delta X(0)|}, \end{equation*}

where $\delta X(t)$ is the separation between initially close trajectories. Numerical computation yields the maximum Lyapunov exponent $\lambda _{max}=0.2735\gt 0$ for $b_0=73$ (see Figure 3(b)), confirming the chaotic gait.

In view of (5) and (8), one can calculate that

\begin{equation*} y_{01}=-5.7995,\ \ \bar {b}= -7.3343. \end{equation*}

Hence, the controlling pulse thrust is

(10) \begin{equation} \bar {P}=12(x_2(t)^-)^2+(73- 7.3343 \mu ) x_2(t)^- +8, \end{equation}

where

\begin{equation*} \mu =\left \{ \begin{array}{ll} 1, \ \ \ \textrm {if}\ x_1(t)= - \dfrac {\gamma }{2}\ \textrm {and}\ |x_2(t)+5.7995|\lt 0.9,\\ 0, \ \ \textrm {else}, \end{array} \right . \end{equation*}

Consider a chaotic gait starting from the point $(0.2000,\ -5.1495,\ -0.2000,\ 5.1417)$ , which is selected based on the requirements outlined in Section 2, including the chaotic attractor, joint limits, and angular velocity constraints.

Proposition1 provides a theoretical prediction for stabilizing the above chaotic gait through parameter perturbation of $\bar {b}$ . To validate this prediction, the controlling pulse thrust (10) with $\sigma =0.9$ is added to the bipedal robot at the $k$ -th step by using OGY control method ( $k_{OC}$ ). As shown in Figure 4(a), after implementing control at step $k_{OC}=16$ , the chaotic gait converges to the target period-1 orbit. The phase-plane trajectory in Figure 4(b) intuitively depicts the dynamic evolution of the system state from the chaotic attractor to the stable period-1 limit cycle in the state space, directly reflecting the suppression of chaotic motion by the OGY control method. The phase-plane plot of the stabilized period-1 gait (Figure 4(c)) further characterizes the bounded and periodic dynamic behavior of the controlled system, while the time series of the stance leg angular velocity (Figure 4(d)) verifies the steady-state performance of the controlled gait in the time domain.

Figure 4.

For system (2) with $a=12$ , $c=8$ , $b=b_0=73$ , and the controlling pulse thrust (10) with $\sigma =0.9$ at $k_{OC}=16$ , (a) time series of $x_{2k}$ , (b) the chaotic gait is controlled to the target period- $1$ gait, (c) phase-plane plot of the stabilized period-1 gait in $x_1(x_3)$ - $x_2(x_4)$ space, and (d) time series of the stance leg angular velocity $x_2$ ( $deg/s$ ).

Figure 4 Long description

The image contains four graphs illustrating the control of chaotic gaits in a bipedal robot. The first graph (a) shows the time series of x2k, with a vertical axis labeled x2k in degrees per second and a horizontal axis labeled k. The second graph (b) presents a 3D plot comparing the chaotic gait and the target period-1 gait, with axes labeled x1, x2, and x3. The third graph (c) is a phase-plane plot of the stabilized period-1 gait in x1(x3) - x2(x4) space, with stance leg and swing leg trajectories marked in red and blue, respectively. The fourth graph (d) displays the time series of the stance leg angular velocity x2 in degrees per second over time t. The graphs collectively demonstrate the stabilization of chaotic gaits to a period-1 gait using controlling pulse thrust.

3.2. Bifurcation control

Suppose that there is a chaotic gait in (2) under the pulse thrust $P=a(x_2(t)^-)^2+b x_2(t)^- +c$ , where $b=b_0$ . Now suppress the occurrence of the first flip bifurcation and stabilize the chaotic gait to periodic gaits by adjusting the value of $c$ . Further suppose that the controlling pulse thrust is

\begin{equation*} \bar {P}=a(x_2(t)^-)^2+b x_2(t)^- +C. \end{equation*}

It follows from (7) that the first flip bifurcation occurs at $b=\bar {b}_2=\frac {2\sqrt {1+A^2aC}+1-\cos \gamma }{A}$ under the controlling pulse thrust $\bar {P}$ . If $\bar {b}_2\gt b_0$ , the robot’s gait is no longer chaotic but is converted into a period-1 gait. In view of the fact that the period- $1$ gait after control is stable for $b\in (\bar {b}_1,\ \bar {b}_2)$ , the following condition needs to hold:

\begin{equation*} \bar {b}_1\lt b_0\lt \bar {b}_2, \end{equation*}

where $\bar {b}_1=\frac {2A\sqrt {aC}+1-\cos \gamma }{A}$ . It follows that

\begin{equation*} \frac {2A\sqrt {aC}+1-\cos \gamma }{A} \lt b_0\lt \frac {2\sqrt {1+A^2aC}+1-\cos \gamma }{A}, \end{equation*}

\begin{equation*} c_1\lt C\lt c_2, \end{equation*}

where

(11) \begin{equation} c_1=\frac {(A b_0-1+ \cos \gamma )^2-4}{4A^2 a},\ c_2=\frac {(A b_0-1+ \cos \gamma )^2}{4A^2 a}. \end{equation}

Hence, the following result about chaotic gait control by using bifurcation control method is obtained.

As shown in the above, the robot’s gait is chaotic for $a=12$ , $c=8$ , and $b=b_0=73$ . In view of (11), one can calculate that

\begin{equation*} c_1=49.9884,\ \ c_2=104.9407. \end{equation*}

In this case, Proposition2 determines the stable parameter range for successfully achieving bifurcation control as $49.9884\lt C \lt 104.9407$ . Numerical simulations are presented by setting $C=52$ .

Figure 5.

The bifurcation diagrams of periodic gaits with respect to $b$ for system (2) with pulse thrust $P=a(x_2(t)^-)^2+b x_2(t)^- +c\ (a=12,\ c=8)$ and controlling pulse thrust $\bar {P}=a(x_2(t)^-)^2+b x_2(t)^- +C\ (a=12,\ C=52)$ .

Figure 5 Long description

The line graph shows the bifurcation diagrams of periodic gaits with respect to b for a system with pulse thrust P and controlling pulse thrust P bar. The x-axis represents the parameter b, ranging from 20 to 90. The y-axis represents x subscript 2k, ranging from -10 to 2. The graph includes two main data sets: one in blue for P equals a times x subscript 2t minus squared plus b times x subscript 2t minus plus c with a equals 12 and c equals 8, and another in red for P bar equals a times x subscript 2t minus squared plus b times x subscript 2t minus plus C with a equals 12 and C equals 52. The blue line shows a bifurcation at b equals 35.3742, while the red line shows a bifurcation at b equals 12.6837. The graph illustrates the transition from chaotic and multi-period gaits to period-1 gaits through bifurcation control and the expansion of stable parameter regions. All values are approximated.

Figure 6.

For system (2) with $a=12$ , $c=8$ , $b=b_0=73$ , and the controlling pulse thrust $\bar {P}=12(x_2(t)^-)^2+73 x_2(t)^- +52$ at $k_{BC}=16$ , (a) time series of $x_{2k}$ , (b) the chaotic gait (blue) is controlled to a period- $1$ gait (red), (c) phase-plane plot of the stabilized period-1 gait in $x_1(x_3)$ - $x_2(x_4)$ space, and (d) time series of the stance leg angular velocity $x_2$ ( $deg/s$ ).

Figure 6 Long description

The image contains four graphs illustrating the control of chaotic gaits in a bipedal robot. The first graph (a) shows the time series of x2k with a red dashed line indicating a threshold at kBC equals 16. The second graph (b) presents a 3D plot comparing the chaotic gait in blue dashed lines with the period-1 gait in red solid lines. The third graph (c) displays a phase-plane plot of the stabilized period-1 gait in the x1(x3) - x2(x4) space, with the stance leg in red and the swing leg in blue. The fourth graph (d) shows the time series of the stance leg angular velocity x2 in degrees per second, with red dashed lines for the stance leg and blue dashed lines for the swing leg. All values are approximated.

The controlling pulse thrust is

(12) \begin{equation} \bar {P}=12(x_2(t)^-)^2+73 x_2(t)^- +52, \end{equation}

and the bifurcation diagrams of periodic gaits before and after control are shown in Figure 5. The suppression of the first flip bifurcation confirms that chaos is eliminated at $b=73$ under the controlling pulse thrust (12). In Figure 6, the controlling pulse thrust $\bar {P}$ (12) is added to the bipedal robot at the $k$ -th step by using bifurcation control method ( $k_{BC}$ ), where $k_{BC}=16$ . It is observed that the chaotic gait is immediately controlled and gradually controlled to a period- $1$ gait after implementing the controlling pulse thrust. The phase-plane trajectory in Figure 6(b) clearly shows the smooth transition of the system state from chaotic motion to the stable period-1 limit cycle in the state space, which reveals the effective suppression of flip bifurcation and chaos by the proposed bifurcation control method. The phase-plane plot of the fully stabilized period-1 gait (Figure 6(c)) quantifies the stable limit cycle of the controlled system, offering direct visual support for the stability of the period-1 gait, and the time series of the stance leg angular velocity (Figure 6(d)) confirms the time-domain stability and consistency of the controlled gait.

It also can be observed that the length of the stable parameter interval for period-1 gait is 35.3742 for $c=8 (C=8)$ and reduces to 12.6837 for $C=52$ . The width of the window of steady gaits decreases with parameter $C$ in the controlling pulse thrust $ \bar {P}=a(x_2(t)^-)^2+b x_2(t)^- +C$ increasing. Similar result was obtained in the works [Reference Gritli and Belghith44, Reference Gritli and Belghith45].This fact will be quantitatively analyzed in Section 5.

3.3. Comparison of OGY and bifurcation control methods

Now compare the advantages and disadvantages of the two chaotic gait control methods mentioned above, namely OGY control (OC) method and bifurcation control (BC) method. The control effects of methods OC and BC on the chaotic gait, which starts from the initial point $(0.2000,\ -5.1495,\ -0.2000,\ 5.1417)$ under the pulse thrust $P=12(x_2(t)^-)^2+73x_2(t)^- +8$ , are presented in Figure 7 and Table I.

It can be observed that a notable advantage of the OGY control method lies in its efficient perturbation application – only minimal perturbations within the neighborhood of a chaotic attractor are required to stabilize the chaotic state onto unstable periodic orbits embedded in the chaotic attractor, resulting in relatively low control energy consumption. However, after implementing control on the chaotic gait starting from the step 16, the subsequent three steps are left uncontrolled, which may lead to the robot’s fall due to the lack of immediate control. Furthermore, the OC method requires real-time monitoring of the state trajectory relative to the target unstable period-1 orbit. It is highly sensitive to detection accuracy and imposes high demands on the controller design.

The bifurcation control method offers dual advantages for chaotic gait regulation: it ensures timely system response and employs a time-invariant incremental pulse control force (fixed at 44 units per step). This constant actuation input thereby renders the controller design notably straightforward. Its drawback, however, lies in the high thrust consumption and the slow process of controlling the system toward a period- $1$ gait.

Given the critical drawback that the OGY control method may cause the bipedal robot to fall, the bifurcation control method is thus selected for chaotic gait stabilization in practical applications, despite its higher pulse thrust consumption. Next, we continue to use bifurcation control method to control the multi-period gaits to the period-1 gait.

Table I.

Pulse thrust increments ( $\bar {P}-P$ ) for OC and BC methods.

Figure 7.

For system (2) with $a=12$ , $c=8$ , $b=b_0=73$ , time series of $x_{2k}$ under the BC method (blue) and OC method (red) starting from step 16.

Figure 7 Long description

A line graph compares two data series, BC and OC, over a range of values for n from 0 to 50. The x-axis represents the variable n, ranging from 0 to 50. The y-axis represents the variable x subscript 2k in degrees per second, ranging from -8 to -2. The BC data series is represented by blue downward-pointing triangles, while the OC data series is represented by red upward-pointing triangles. The BC series shows a pattern of fluctuations that stabilize around -5 degrees per second after n equals 20. The OC series also shows fluctuations but stabilizes around -6 degrees per second after n equals 20. All values are approximated.