The pursuit of advanced mobility in unstructured environments has consistently driven research towards legged systems, with the bionic robot standing as a prime example of this endeavor. Inspired by the biological principles of animal locomotion, these machines offer superior terrain adaptability compared to their wheeled or tracked counterparts. A central challenge in the design of high-performance legged robots lies in the inherent trade-off between speed, payload capacity, and control complexity inherent in leg mechanism design. Traditional serial-chain legs, while enabling agile and fast motions, often suffer from limited structural stiffness and payload capacity under high dynamic loads. Conversely, purely parallel mechanisms offer exceptional rigidity and force-bearing capabilities but are typically constrained by a smaller workspace and more complex kinematics, which can limit stride length and walking speed.
This work focuses on the comprehensive dynamic modeling and analysis of a novel quadruped bionic robot featuring a hybrid serial-parallel leg architecture. This design strategically combines a planar parallel mechanism with a serially attached lateral swing mechanism, aiming to synthesize the benefits of both topologies. The core objective is to achieve a bionic robot capable of rapid walking, bearing significant payloads, and exhibiting good lateral decoupling for stable and efficient omnidirectional motion. A precise and computationally efficient dynamic model is paramount for the successful development of such a system. It forms the foundation for critical tasks such as actuator sizing, the design of buffer structures to manage impact forces, model-based controller design, and energy-efficient gait planning. This article details the kinematic analysis, derives the explicit dynamic equations using the Lagrangian formulation, validates the model through simulation, and investigates the influence of gait parameters on driving forces and overall energy consumption.
The leg mechanism of the studied quadruped bionic robot is a 3-degree-of-freedom (3-DOF) system. The primary motion in the sagittal plane (forward/backward stepping and lifting) is generated by a closed kinematic chain. This chain integrates a planar five-bar linkage (O-A-B-C-D) with a dual-rhombus (or pantograph) mechanism (O-F-G-E and G-H-I-J). The dual-rhombus acts as a motion amplifier; small displacements from two linear actuators within the five-bar linkage result in significantly larger displacements at the foot endpoint (I). This design is key to enabling rapid walking with relatively compact actuator strokes. The entire planar assembly is then attached to the robot’s torso via a serial four-bar linkage (M-K-N-Q), which is driven by a third linear actuator. This serial stage provides pure rotation of the leg assembly about an axis parallel to the robot’s forward direction, enabling independent lateral swinging motions and thus granting the leg its side-stepping capability and improving lateral decoupling. All linear actuators are servo-hydraulic cylinders, chosen for their high power density, large force output, and good dynamic response, which are essential for a high-performance, load-carrying bionic robot.
The kinematic analysis begins by defining coordinate frames and geometrical parameters. A fixed frame {O-XYZ} is attached to the robot’s torso at the hip joint, while a moving frame {I-UVW} is attached to the foot. The position of the foot point I in the fixed frame is a function of the geometric angles and link lengths of the dual-rhombus mechanism:
$$ p_X = -4l\cos\theta\sin\psi $$
$$ p_Y = 4l\cos\theta\cos\psi\sin\phi $$
$$ p_Z = -4l\cos\theta\cos\psi\cos\phi $$
Here, $l$ is the side length of the rhombus links, $\theta$ is half the acute angle within the rhombus, $\psi$ is the orientation angle of the planar mechanism’s output link, and $\phi$ is the lateral swing angle provided by the serial four-bar stage. The inverse kinematics, determining the required actuator lengths ($l_2$, $l_4$, $l_8$) for a desired foot position ($p_X$, $p_Y$, $p_Z$), is solved geometrically. First, $\phi$, $\theta$, and $\psi$ are extracted from the foot position. Then, these angles are used within the equations governing the planar five-bar and the lateral four-bar linkages to compute the necessary actuator displacements. The reachable workspace of the leg, considering practical joint limits and actuator stroke, is substantial, particularly in the vertical (Z) direction, demonstrating the effectiveness of the motion-amplifying dual-rhombus design.
Velocity analysis is crucial for dynamics. The linear velocity Jacobian matrix $\mathbf{J}_v$ is derived, which maps the foot tip’s Cartesian velocity vector $\mathbf{v} = [v_X, v_Y, v_Z]^T$ to the rates of change of the actuator lengths $\dot{\mathbf{q}} = [\dot{l}_2, \dot{l}_4, \dot{l}_8]^T$:
$$ \dot{\mathbf{q}} = \mathbf{J}_v \mathbf{v} $$
The explicit form of $\mathbf{J}_v$ is obtained through differentiation of the kinematic constraint equations. Furthermore, the angular velocities $\boldsymbol{\omega}_i$ and translational velocities $\mathbf{v}_{C_i}$ of the center of mass for every significant component (each link, cylinder rod, and cylinder barrel) are derived as linear functions of the foot tip velocity $\mathbf{v}$. These relationships are expressed as:
$$ \boldsymbol{\omega}_i = \mathbf{G}_i \mathbf{v}, \quad \mathbf{v}_{C_i} = \mathbf{G}_{C_i} \mathbf{v} $$
where $\mathbf{G}_i$ and $\mathbf{G}_{C_i}$ are configuration-dependent matrices. This explicit formulation of all velocities in terms of $\mathbf{v}$ is a critical step that lays the groundwork for an efficient derivation of the dynamics using energy-based methods.
| Symbol | Description |
|---|---|
| $p_X, p_Y, p_Z$ | Foot endpoint coordinates |
| $\phi, \theta, \psi$ | Leg orientation angles (lateral swing, rhombus, plane) |
| $l_2, l_4, l_8$ | Actuator lengths (planar upper, planar lower, lateral) |
| $\mathbf{v}$ | Foot endpoint linear velocity vector |
| $\mathbf{J}_v$ | Linear velocity Jacobian matrix |
| $\boldsymbol{\omega}_i$ | Angular velocity of link $i$ |
| $\mathbf{v}_{C_i}$ | Linear velocity of the center of mass of link $i$ |
The dynamic model is developed using the Lagrangian formulation, which is well-suited for complex mechanical systems as it systematically handles constraint forces through the use of generalized coordinates. The foot point position $\mathbf{p} = [p_X, p_Y, p_Z]^T$ is chosen as the set of generalized coordinates for the leg. The corresponding generalized forces are the forces $\mathbf{F} = [F_X, F_Y, F_Z]^T$ applied at the foot point. The Lagrangian $L$ is defined as the difference between the total kinetic energy $T$ and total potential energy $U$ of the system:
$$ L = T – U $$
The total kinetic energy $T$ is the sum of the translational and rotational kinetic energy of all $n$ rigid bodies (links and actuator components) in the leg:
$$ T = \frac{1}{2} \sum_{i=1}^{n} \left( \mathbf{v}_{C_i}^T m_i \mathbf{v}_{C_i} + \boldsymbol{\omega}_i^T \mathbf{I}_i \boldsymbol{\omega}_i \right) $$
where $m_i$ is the mass and $\mathbf{I}_i$ is the inertia tensor of body $i$ about its center of mass. Substituting the velocity relations $\boldsymbol{\omega}_i = \mathbf{G}_i \mathbf{v}$ and $\mathbf{v}_{C_i} = \mathbf{G}_{C_i} \mathbf{v}$ transforms the kinetic energy into a quadratic form in terms of the generalized velocity $\dot{\mathbf{p}}=\mathbf{v}$:
$$ T = \frac{1}{2} \mathbf{v}^T \mathbf{M}(\mathbf{p}) \mathbf{v} $$
Here, $\mathbf{M}(\mathbf{p})$ is the configuration-dependent generalized inertia matrix of the leg mechanism, assembled from the mass properties and the $\mathbf{G}$ matrices. The potential energy $U$ is due to gravity:
$$ U = g \sum_{i=1}^{n} m_i {Z}_{C_i} $$
where $g$ is gravity and ${Z}_{C_i}$ is the vertical height of the center of mass of body $i$.
The Lagrange equations for this system are:
$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\mathbf{p}}} \right) – \frac{\partial L}{\partial \mathbf{p}} = \mathbf{F} $$
Performing the differentiation and rearranging terms yields the equation of motion in a standard form:
$$ \mathbf{M}(\mathbf{p}) \dot{\mathbf{v}} + \mathbf{C}(\mathbf{p}, \mathbf{v}) \mathbf{v} + \mathbf{G}(\mathbf{p}) = \mathbf{F} $$
In this equation, $\mathbf{C}(\mathbf{p}, \mathbf{v}) \mathbf{v}$ represents Coriolis and centrifugal forces, and $\mathbf{G}(\mathbf{p})$ is the gravity force vector. The force $\mathbf{F}$ at the foot includes both the inertial/gravitational forces of the leg itself and any external contact force $\mathbf{Q}$ from the ground: $\mathbf{F} = \mathbf{F}_{\text{leg}} + \mathbf{Q}$. For the swing phase, $\mathbf{Q}=0$. For the stance phase, $\mathbf{Q}$ is the ground reaction force.
The relationship between the generalized force $\mathbf{F}$ at the foot and the actuator forces $\boldsymbol{\tau} = [\tau_2, \tau_4, \tau_8]^T$ is found using the principle of virtual work, which leads to:
$$ \boldsymbol{\tau} = \mathbf{J}_v^{-T} \mathbf{F} $$
Therefore, the inverse dynamics solution—calculating the required actuator forces for a desired motion and external load—is given by:
$$ \boldsymbol{\tau} = \mathbf{J}_v^{-T} \left( \mathbf{M}(\mathbf{p}) \dot{\mathbf{v}} + \mathbf{C}(\mathbf{p}, \mathbf{v}) \mathbf{v} + \mathbf{G}(\mathbf{p}) – \mathbf{Q} \right) $$
This explicit model allows for the computation of joint forces directly from a planned trajectory of the foot $\mathbf{p}(t)$, its derivatives $\mathbf{v}(t)$, $\dot{\mathbf{v}}(t)$, and the known external force $\mathbf{Q}(t)$.
The dynamic model was validated against a simulation conducted in ADAMS multi-body dynamics software. A lateral walking gait was defined for the leg. The foot trajectory during the swing phase (first half of the cycle) was defined using a parabolic function for the X (forward) and Y (lateral) directions and a cycloidal-like function for the Z (vertical) direction to ensure smooth lift-off and touch-down. During the stance phase (second half of the cycle), a constant vertical ground reaction force of 1200 N was applied at the foot to simulate supporting the robot’s weight. The theoretical actuator forces computed from the derived Lagrangian model were compared with the forces output by the ADAMS simulation. The results showed excellent agreement, with force error magnitudes within approximately ±2 N throughout the gait cycle, confirming the accuracy of the analytical dynamic model for this hybrid serial-parallel bionic robot leg.
A key observation from the simulation is that the maximum force in each actuator occurs precisely at the moment of foot impact with the ground at the beginning of the stance phase. This peak is governed by the need to decelerate the leg’s mass and manage the impact impulse, and its magnitude is largely independent of the chosen swing height. This insight is critical for designing buffer mechanisms, such as the foot-mounted spring system mentioned in the context, to mitigate shock loads and protect the actuation system.
The influence of step length on the peak actuator forces was analyzed for two gait types. For straight-line walking (where the lateral actuator force is zero), as the step length increases, the peak forces in the two planar actuators show a monotonic increasing trend. For lateral walking, the peak force in the lateral swing actuator increases approximately linearly with lateral step length. Crucially, across all hybrid motions involving lateral stepping, the lateral swing actuator consistently experiences the highest force demand among the three actuators. This is due to its serial connection to the entire planar leg assembly, effectively making it a force-amplifying lever in certain configurations. This finding directly informs the hydraulic system pressure rating and the structural design priority for this specific joint in the bionic robot.
| Gait Type | Actuator | Trend vs. Step Length | Key Insight |
|---|---|---|---|
| Straight-line Walk | Planar Upper ($\tau_2$) | Monotonically increases | Peak force dictates planar actuator sizing. |
| Planar Lower ($\tau_4$) | Monotonically increases | ||
| Lateral Walk | Lateral Swing ($\tau_8$) | ~Linearly increases | Dominant force; critical for system pressure & joint design. |
Energy efficiency is a major concern for autonomous legged bionic robot systems. Based on the inverse dynamics model, a mobility energy consumption performance index $\eta$ is established to evaluate and compare the efficiency of different gait patterns:
$$ \eta = \frac{P_E}{m g L_1} $$
where $P_E$ is the total mechanical energy output by all actuators on all four legs over one complete gait cycle, $m$ is the total mass of the bionic robot including payload, $g$ is gravity, and $L_1$ is the distance traveled by the robot’s body in one cycle (i.e., the stride length). This index represents the energy consumed per unit weight per unit distance traveled, with a lower value indicating higher locomotion efficiency.
The total actuator energy $P_E$ is computed from the dynamic model as:
$$ P_E = \sum_{i=1}^{4} \sum_{j=1}^{3} \int_{0}^{T} \tau_{ij}(t) \cdot v_{ij}(t) \, dt $$
where $\tau_{ij}(t)$ and $v_{ij}(t)$ are the force and velocity of the $j$-th actuator on the $i$-th leg, and $T$ is the gait period.
Analysis using this index reveals clear trends for optimizing gait parameters of the quadruped bionic robot:
- Straight-line Walking: The energy consumption index $\eta$ decreases monotonically with increasing step length and increases monotonically with increasing leg lift height. Therefore, to minimize energy consumption during straight-line motion, the bionic robot should use a gait characterized by long strides and low foot clearance.
- Lateral Walking: The relationship between $\eta$ and lateral step length is non-monotonic, with an optimal efficiency found at a moderate step length (around 100 mm for the analyzed robot geometry). As with straight-line walking, $\eta$ increases with higher lift height. Thus, the optimal energy-saving strategy for lateral motion is to use a moderate lateral step length combined with a low lift height.
These guidelines provide a concrete basis for the offline optimization of gait parameters and for the online planning of energy-efficient trajectories for the hybrid quadruped bionic robot.
| Gait Parameter | Straight-line Walk Effect on $\eta$ | Lateral Walk Effect on $\eta$ | Recommended for Efficiency |
|---|---|---|---|
| Increasing Step Length | Decreases | Decreases then increases (has an optimum) | Straight: Long step. Lateral: Use optimal step (~100mm). |
| Increasing Lift Height | Increases | Increases | Minimize lift height for both gait types. |
In conclusion, this work has presented a complete framework for the dynamic modeling and analysis of a quadruped bionic robot employing a novel hybrid serial-parallel leg mechanism. The explicit kinematic velocity relationships and the Lagrangian-derived dynamic model provide an accurate and computationally tractable tool for system analysis. Validation confirms the model’s fidelity. The analysis yields practical design insights: the lateral swing actuator is the most heavily loaded component during hybrid motions, and impact forces at touch-down are a primary design driver for buffer systems. Furthermore, the establishment of an energy consumption index and the analysis of gait parameters offer clear strategies for enhancing the operational efficiency of the bionic robot. Specifically, adopting long, low steps for straight-line travel and optimized moderate, low steps for lateral movement will minimize energy expenditure. This comprehensive dynamic understanding is fundamental for advancing the development of robust, efficient, and high-performance legged bionic robot systems capable of operating in demanding real-world environments.