A comparative study of leg kinematics for energy-efficient locomotion

It has been theorized that biological legs or the serial leg with a knee and hip joint has evolved over centuries for energy efficient locomotion and as such, has been adapted into a multitude of legged robots. However, recent success of legged robots with alternate leg morphologies without actuated knees, such as the parallel and symmetric five-bar link leg raises the question: which leg geometry is more energy-efficient and why? To answer this question, we created a minimal model of bipedal walking whose non-dimensionalised equations of motion have a single free parameter, the leg ratio and defined as the ratio of the distal to the proximal leg length. Then we performed an energy minimization for a given leg ratio and combination of speed and step length. When we optimized mechanical work, we found that all three legs have an identical efficiency, but the symmetric leg has the lowest peak torque. When we optimized a cost representative of an electric motor, we found that the serial leg is most energy efficient for all leg ratios, and the cost decreases as the leg ratio increases. For a leg ratio of 1, the parallel and symmetric leg have identical efficiencies. As the leg ratio increases, the efficiency of the symmetric leg approaches that of the serial leg while that of parallel leg decreases. However, the symmetric leg produces the least peak torque followed by the serial leg for leg ratios greater than 1. Our conclusion is that the symmetric leg rivals the serial leg by being easier to design and having smaller peak torques leading to smaller actuators at the cost of being slightly less energy-efficient.