Objective
The objective of this project is to create a highly adaptable legged robot with multifunctional capabilities, capable of autonomously morphing into diverse bioinspired forms. This robot aims to seamlessly adapt to a range of tasks across different project phases, ensuring maximum efficiency and versatility.
The proposed robot incorporates multiple innovative mechanisms and symmetrical designs that optimize the utilization of minimal parts while enabling continuous transformation across a wide array of forms. The adjustment of the torso’s length will be facilitated by GSA linear slide actuators from Tolomatic, which provide a maximum stroke length of 500 mm and a load-bearing capacity of up to 182 kg. This enables the torso length to flexibly vary within the range \( ( d_b \in [0.5, 1.0] ) \) m, while keeping the center of mass (CoM) fixed at the same location. The separation between each pair of limbs can be independently adjusted through an actuated slider-crank mechanism, enabling the distances \(w_1\) and \(w_2\)between the limbs to vary within the range \([0.1, 0.6] \) m. Each limb is uniformly designed with a gimbal joint, featuring an adjustable knee configuration and variable leg length for enhanced adaptability. The gimbal joint, which connects to the torso, is controlled by three GO-M8010-6 motors with joint angles \(q_1, q_2, q_3\), providing a wide range of motion as depicted by the colored regions. The knee joint’s movement is facilitated by actuators near the torso through a timing belt, allowing \(q_4\) for bidirectional bending up to \(\pm160^\circ\), mirroring the design principles seen in the Solo and Mini-Cheetah robots. Adjustments to the leg shank length are achieved using ERD electric cylinders from Tolomatic, with a force capacity of up to 890 N, allowing for variations of length \(d_l \in [0.25, 0.5]\) m. The joint variables of the \(z\)-th leg of the TLR are represented by a row vector \( q^{z} = [ q^z_1, q^z_2, q^z_3, q^z_4, d_l^z ] \). They are measured in the leg’s own body coordinate frame, where \(z \in {\text{FR}, \text{FL}, \text{RR}, \text{RL}}\) denotes the front right, front left, rear right, and rear left legs, respectively. As a result, a specific shape \( \mathcal{R}_k\) of the TLR can be uniformly defined as a single comprehensive column vector \( r := [ d_b, w_1, w_2, q^{ \text{ FR}}, q^{\text{ FL}}, q^{\text{ RR}}, q^{\text{ RL}} ]^\top \in \mathcal{R}_k \subset \mathcal{R}\).
Proposed Approach
Symmetry in legged robots manifests in various forms: a legged system may exhibit consistent gaits from any spatial starting point, maintaining identical leg movements on either side of the torso in phase; some systems also perform forward and backward movements that are so alike they seem to reverse time. In general, a symmetry of a legged robot can be defined as an invertible energy-preserving transformation that connects equivalent states across different regions of the phase space (Wieber, 2006). By applying group theory, each symmetry \( s \) is identified as an element of a group \( \mathbb{S} \), which is closed under the operations of composition \( \circ: \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{S} \) and inversion \( (\cdot)^{-1}: \mathbb{S} \rightarrow \mathbb{S} \). From a practical perspective, the value of studying symmetries is underscored by their ability to relate the dynamics of a state \( (g, \xi, r, \dot{r}) \) with those of the symmetry-transformed state \( s(g, \xi, r, \dot{r}) \). This relationship suggests that effective modeling and control near any given state also enable control across all symmetrically equivalent states.
