As a modular, hyper-redundant, and limbless machine inspired by biological snakes, the serpentine bionic robot exhibits remarkable stability, multimodal locomotion, and superior adaptability to complex environments. These attributes make it a compelling platform for applications ranging from military reconnaissance and disaster response to exploration in hazardous terrains. A critical maneuver for such a robot in any operational scenario is turning—the ability to adjust its heading to circumvent obstacles or align with a target direction. Unlike wheeled or legged robots, turning for a snake-like mechanism is inherently coupled with its whole-body locomotion dynamics, presenting a unique and complex control challenge. This work delves into the intricacies of turning motion control for bionic robots based on the foundational serpentine curve, proposing novel methods and establishing a framework for their evaluation.
The locomotion of a biological snake, particularly lateral undulation, has been extensively studied to generate efficient robotic gaits. A seminal mathematical model for this motion is the *serpentine curve* proposed by Hirose. This curve describes the backbone shape of a snake moving through lateral force generation against environmental irregularities. The curvature \(\rho\) along the arc length \(s\) of this curve is given by:
$$ \rho(s) = -\alpha b \sin(bs) $$
Here, \(\alpha\) is the amplitude or winding angle, and \(b\) is a constant that determines the spatial frequency of the wave. For a robotic implementation with \(n\) links of length \(l\), this continuous curve is discretized into joint angles. The angle \(\phi_i\) for the \(i\)-th joint as a function of time \(t\) is derived as:
$$ \phi_i(t) = A \sin(\omega t + (i – 1)\beta) \quad \text{for} \quad i = 1, 2, \ldots, n $$
where \(A = -2\alpha \sin(bl)\) is the joint amplitude, \(\omega\) is the temporal angular frequency, and \(\beta = 2bl\) is the phase difference between adjacent joints. This equation forms the cornerstone for generating efficient and continuous traveling waves along the body of the bionic robot, enabling its primary forward propulsion.
While straight-line motion is governed by a constant set of parameters \((A, \omega, \beta)\), executing a turn requires a deliberate modulation of the body wave. The fundamental parameters available for this modulation are the offset (or center value), the phase, and the amplitude of the serpentine wave. However, not all modulation strategies are equally effective. To systematically evaluate and compare turning control methods for the serpentine bionic robot, we propose four key criteria:
- Adherence to the Serpentine Curve: An optimal control method should maintain the intrinsic sinusoidal wave pattern during and after the turn. Abruptly abandoning this pattern can disrupt gait efficiency, induce instability, and force the control system into complex recovery maneuvers.
- Continuity of Joint Angles: The commanded joint angles must be continuous in time. Discontinuities cause sudden torque spikes, excessive load on actuators, mechanical stress, and undesirable body jerking, compromising the robot’s smooth operation and longevity.
- Turning Efficacy and Precision: The method should enable accurate, rapid, and well-defined turns. Precise heading change minimizes the need for post-turn corrections and facilitates reliable path following in cluttered environments.
- Magnitude of Joint Angle Variation: During the turn, the peak-to-peak variation in joint angles should be bounded. Excessively large angles can lead to increased motion inertia, interference between adjacent modules, and potential loss of traction with the ground.
Based on these criteria, we first analyze three conventional parameter-modulation strategies for turning the bionic robot.
Conventional Turning Control Methods
1. Offset (Center Value) Control Method
This method introduces a linearly growing offset to the serpentine curve, effectively biasing the entire body wave. The modified curve and corresponding joint angle are described by:
$$ \theta(s) = \alpha \cos(bs) + c(s – s_z) $$
$$ \phi_i(t) = A \sin(\omega t + (i-1)\beta) + C(t) $$
where \(c\) is a constant controlling the turn rate, \(s_z\) is the arc length at which the turn initiates, and \(C(t)\) is the time-dependent offset proportional to \(c\). The resulting heading change \(\varphi\) is directly proportional to the integrated offset.
Analysis: This method produces a smooth, continuous turn and can even achieve circular trajectories for constant \(c\). The turn is pronounced and effective. However, as shown in the joint angle profiles, the offset addition distorts the pure sinusoidal wave, meaning the locomotion deviates from the optimal serpentine curve. This reduces propulsion efficiency. Furthermore, the turning radius is often large, the heading change is not directly and precisely specified by a single parameter, and the joint angles can reach large magnitudes.
2. Phase Control Method
This method applies an instantaneous or rapid phase shift \(\sigma\) to the body wave at the moment of turning. The mathematical expression is:
$$ \theta(s) = \alpha \cos(bs) + \sigma \quad \text{for} \quad s > s_z $$
$$ \phi_i(t) = A \sin(\omega t + (i-1)\beta + \Sigma) $$
where \(\Sigma\) is the equivalent temporal phase shift. The new heading \(\varphi\) is directly equal to the applied phase shift \(\sigma\).
Analysis: Phase control is highly effective and widely used. It provides direct and accurate control over the turning angle (\(\varphi = \sigma\)), results in a very small turning radius, and crucially, maintains the perfect serpentine wave shape before, during, and after the turn. Its major drawback is the discontinuity in the joint angle command. At the instant of the phase jump, the joints are commanded to instantly move to a new angle on the sine wave, causing a jerky motion, high actuator loads, and potential instability, as evidenced in angle-time plots.
3. Amplitude Control Method
This method asymmetrically modulates the amplitude of the serpentine wave, typically by increasing it on one side of the body. A common implementation uses a signum function:
$$ \theta(s) = \alpha \left[1 + \Delta_\alpha \cdot \text{sgn}(\cos(bs))\right] \cos(bs) $$
where \(\Delta_\alpha\) is the asymmetry factor. The resulting turn angle \(\varphi\) is a function of \(\Delta_\alpha\) and can be approximated by analyzing the net displacement over a cycle.
Analysis: Amplitude control preserves the wave continuity and the sinusoidal form. The turn is smooth and allows for continuous modulation. However, the achievable turning angle is strictly limited by the physical range of the joints (the amplitude cannot exceed mechanical limits). Furthermore, turns generated this way tend to have a larger radius and take longer to complete compared to phase control, as the robot must undergo a full or partial wave cycle to effect the heading change.
The properties of these three fundamental methods for the bionic robot are summarized in the table below:
| Control Method | Advantages | Disadvantages |
|---|---|---|
| Offset Control | Continuous angles, pronounced turn, enables circular motion. | Deviates from serpentine curve, inefficient, imprecise angle, large radius, large joint angles. |
| Phase Control | Accurate, small radius, fast, maintains perfect serpentine curve. | Discontinuous joint angles cause jerk and high actuator load. |
| Amplitude Control | Continuous angles, maintains serpentine curve, allows modulation. | Turning angle limited by amplitude range, large radius, slow. |
Optimized Turning Control Methodologies
The analysis reveals complementary strengths and weaknesses. To overcome the critical issue of joint angle discontinuity in phase control while retaining its precision, we propose the Tangent Control Method.
Tangent Control Method
The discontinuity in phase control arises because the wave is shifted to a point where the sine value is different. The tangent method enforces continuity by ensuring that at the transition point \(s_z\), the body wave’s tangent angle (which dictates the robot’s head direction) is equal on both sides of the transition. For a desired turn angle \(\varphi\), we require:
$$ \alpha \cos(bs_1) = \alpha \cos(bs_2) + \varphi $$
Solving for the new phase parameter \(s_2\):
$$ s_2 = \frac{1}{b} \arccos\left(\cos(bs_1) – \frac{\varphi}{\alpha}\right) $$
For a real solution to exist, we must have \(\left| \cos(bs_1) – \varphi/\alpha \right| \le 1\), which implies \(\varphi \le 2\alpha\). In practice, for a turn initiated from the wave’s midline, the condition simplifies to \(\varphi \le \alpha\). This is a significant limitation: the turn angle is bounded by the wave amplitude.
Analysis: The tangent method successfully eliminates joint angle discontinuity, providing a smooth transition. It retains the high precision and small turning radius of phase-based methods and perfectly maintains the serpentine curve. Its sole drawback is the turn angle limitation. To overcome this, one can first execute an amplitude increase when the joint angle rate is zero (to maintain continuity), thereby enlarging \(\alpha\), and then apply the tangent turn. However, this two-step process (increase amplitude, turn, decrease amplitude) increases the total maneuver time and temporarily operates the bionic robot at high, potentially less stable, joint angles.
Combined Control Method
To achieve large, precise, smooth, and fast turns without prolonged operation at high amplitudes, we synthesize a Combined Control Method. This method intelligently sequences amplitude and phase-shift actions within a single control cycle. Let the initial amplitude be \(\alpha_1\). The maneuver consists of three seamless stages within one locomotor cycle:
- Amplitude Boosting: At an instant \(s_f\) where the wave tangent is zero (i.e., \(\cos(bs_f)=0\)), the amplitude is instantaneously increased to \(\alpha_2 = \zeta \alpha_1\) (\(\zeta > 1\)). Since the transition occurs at a point where the sine function’s derivative is maximal and the value is zero, the joint angle command remains continuous: \(\alpha_1 \cos(bs_f) = \alpha_2 \cos(bs_f) = 0\).
- Guided Turning: The robot moves with the boosted amplitude \(\alpha_2\). When the head direction (wave tangent) aligns with the desired final heading \(\varphi\), the turn is triggered. At this instant \(s_z\), we have \(\alpha_2 \cos(bs_z) = \varphi\).
- Turn Execution & Recovery: Simultaneously at \(s_z\), the amplitude is reverted to the original \(\alpha_1\), and a phase shift is applied to lock in the new heading. The post-turn joint angle is given by:
$$ \theta(s) = \alpha_1 \cos\left(b(s – s_z) + \frac{\pi}{2}\right) + \varphi \quad \text{for} \quad s > s_z $$
The term \(\pi/2\) is a specific phase shift that ensures continuity. It can be verified that:
$$ \lim_{s \to s_z^-} \theta(s) = \lim_{s \to s_z^+} \theta(s) = \varphi $$
ensuring a perfectly smooth transition.
The complete piecewise function for the body curve in the combined control method is:
$$
\theta(s) =
\begin{cases}
\alpha_1 \cos(bs), & 0 < s < s_f \\[6pt]
\zeta \alpha_1 \cos(bs), & s_f \le s \le s_z \\[6pt]
\alpha_1 \cos\left(b(s – s_z) + \dfrac{\pi}{2}\right) + \varphi, & s \ge s_z
\end{cases}
$$
Analysis: The Combined Control Method is a superior strategy for the bionic robot. It satisfies all four evaluation criteria excellently:
- Serpentine Curve Adherence: The body follows a sinusoidal wave in all stages.
- Joint Angle Continuity: Angles are continuous at both transition points (\(s_f\) and \(s_z\)).
- Turning Efficacy & Precision: The turn is accurate (\(\varphi\) is directly achieved), has a small radius, and completes very quickly—within a fraction of a locomotor cycle.
- Bounded Joint Angles: The high amplitude \(\alpha_2\) is used only briefly. The robot quickly returns to its nominal, efficient amplitude \(\alpha_1\), promoting stability.
This method effectively decouples the turning angle range from the nominal operating amplitude, enabling large turns without sacrificing steady-state performance.
Experimental Validation on a Bionic Robot Prototype
The proposed control methods were implemented and tested on a modular serpentine bionic robot prototype. The mechanical structure consists of identical joint modules connected by universal joints, each providing two degrees of freedom (yaw and pitch) actuated by DC motors with position feedback. The skin features passive wheels to ground the lateral undulation. The control system employs a distributed architecture with a central planner computing joint trajectories and sending commands via a real-time bus to local controllers on each module.
Experiments were conducted on a flat surface. The central planner generated joint angle commands based on the different turning algorithms: Offset, Phase, Amplitude, Tangent, and Combined control. The results were consistent with theoretical analysis:
- The Offset Control bionic robot turned with a noticeable curve but exhibited body sway and imprecise final heading.
- The Phase Control bionic robot executed sharp, accurate turns, but visible body jerks were observed at the turn initiation and completion due to joint angle discontinuities.
- The Amplitude Control bionic robot turned smoothly but through a wide, gradual arc, taking more time and space.
- The Tangent Control bionic robot performed smooth, precise turns for angles within its amplitude limit, showing no jerky motion.
- The Combined Control bionic robot successfully executed large-angle turns (e.g., 45°) that were beyond the tangent method’s limit for the nominal amplitude. The turn was rapid, accurate, and executed with smooth body motion throughout, confirming the method’s practical superiority.
Conclusion
Turning control is a vital capability for the autonomous operation of serpentine bionic robots in unstructured environments. This work established a clear set of criteria—Adherence to Serpentine Curve, Joint Angle Continuity, Turning Efficacy, and Bounded Joint Variation—for evaluating such control methods. Through analysis of conventional parameter modulation techniques, we identified their inherent trade-offs. To address the critical flaw of joint angle discontinuity in the otherwise effective phase control method, the Tangent Control Method was introduced, ensuring smooth transitions at the cost of a limited turn angle. Finally, the Combined Control Method was synthesized, merging amplitude boosting and phased recovery within a single cycle. This optimized method enables large, precise, rapid, and smooth turns while fully preserving the efficient serpentine gait and maintaining joint angle continuity. Experimental validation on a physical bionic robot prototype confirmed the analytical findings and demonstrated the practical effectiveness of the proposed combined control strategy, marking a significant step forward in the locomotion control of these versatile hyper-redundant machines. Future work will focus on integrating these turning strategies with environment-sensing feedback for fully autonomous navigation in complex obstacle fields by the bionic robot.