1. Objective 

2. Onelegged running robot 
1) ModelBased on energy analysis, we proposed a novel stabilization controller for a planar onelegged robot shown in Fig. 1.
Fig. 1 Onelegged passive running robot
2) Passive running gaitIt is impossible to obtain periodic running gaits analytically. Therefore we developed numerical search algorithm using NewtonRaphson method, based on the exact nonlinear hybrid dynamics. It involves calculation of fundamental matrix solution of variational equation at each phase (including discrete jump phase). All the calculations are carried out in terms of Poincare Map . Since the nontrivial passive running gaits are unstable periodone gaits (linealized Poincare Map has eigenvalues larger than unity), this robot cannot run without stabilizing controller. The instability increases as the forward speed does. 3) ControllerWe proposed two kinds of controllers: First, we derived a linear state feedback controller using only two control inputs that move the unstable poles to ZERO. This local feedback controller stabilizes running gaits, but its region of attraction is found to be very small, and large steadystate error remains. Secondly, we derived a new simple controller. Instead of depending on some precalculated periodic solutions, or target (desired) dynamics, here we tried to utilize the natural dynamics of the original nonlinear hybrid system to generate (unknown) natural running gaits. For this purpose, an energypreserving control strategy has been proposed, in which the controller tries to minimize the dissipative energy as much as possible. The most important reason why we use this strategy is; if the system energy is preserved, the system is expected to generate natural periodic gaits, just as some class of Hamiltonian systems exhibit do. This strategy led to a new touchdown controller at flight phase. Based on the energy dissipation analysis, the controller deadbeat the state to ensure the energy preservation at touchdown. 4) Simulation
Simulation results shows that the robot can hop from wide set of initial conditions, and the generated running gaits are found to be quasiperiodic orbits, which can be seen in Hamiltonian system (Fig. 2). Since the controlled running gaits exist for every admissible energy level, they have some robustness against disturbances (Fig. 3).
It is shown that simple adaptation laws, which is similar to delayed feedback controllers for chaotic systems, can asymptotically stabilize quasiperiodic gaits to the periodic ones having desired period for some cases (Fig. 4). In particular, for 1periodic gait, the robot eventually hops without any control inputs (Fig. 5).
Animation
Fig. 6 Successive threesteps running at 5 m/s


3. Biped running robot 
1) ModelWe consider a highlynonlinear planar biped model having massive legs and torso (Fig. 7)
2) ControllerUnfortunately, complete passive running gaits cannot be found by numerical search algorithm. Nevertheless, we can stabilize this system. The controller of passive onelegged hopper is extended to the biped robots. Specifically, we derive deadbeat controller at flight phase based on the energypreserving strategy to preserve mechanical energy at touchdown. Then, we combine this with a posture controller at stancephase to obtain stable running gaits. 3) Simulation
Fig. 9 Orbital stabilization to oneperiodic gait Although the running gaits seem to have symmetricity, there are a little asymmetricity in the torso motions, as recognized from the right top graph. This asymmetricity become more significant when the feedback gains of body attitude control is smaller. Without attitude control, the robot falls down after a few steps.
Animation
The planar controller is partially extended to a 3D biped model, where a rotor rotating around yawaxis of torso was introduced.


4. Quadruped running robot 
1) ModelThe model of our robot is shown in Fig. 10. The legs are connected to the body through revolute joints. Each leg includes upper and lower sections, which are connected with a linear spring. The distance between the toe and hip changes because of the sliding motion between the lower and the upper section of the legs, so each leg has one linear passive degree of freedom. The first research is to explore the properties of passive running, no control input assembles are configured in our profile. In replace, we set two springs in hips. This setting is expected to save the leg swinging energy. To summarize, each leg in our robot has two degree of freedom: a rotational one and a linear one and both of them are completely passive.
Fig. 10 Planar biped running robot with torso
2) Passive running gaitIn a complete bounding cycle, a full stride of the robot can be divided in four different phases. These phases are:
Fig. 11 Phase transition of planar quadruped running gait Unlike the other related researches, we take the mass of the legs into account. Therefore our approach can capture not only the body's oscillatory pitch motion, but also reveal the properties of leg swinging. The nonzero mass assumption complexes the system by introducing impact and the formidable equations which can not be expressed well symbolically. To solve these problems, we apply Hybrid System Theory. As we do not assume a specific running sequence for this algorithm, the propagation of the system follows no specific order. Only the gaits which tend to converge will exhibit a repetition of ordered phases.
3) SimulationThe following figures show different gaits of different profiles.
Animation
Different from other related studies, all the passive running gaits we found are proved to be unstable.
Although the motion starting from the fixed points can continue for some steps (25 steps as maximum), it falls finally.
Extensive investigation of characteristic multipliers corresponding to the trajectories revealed that they all have eigenvalues whose magnitude is bigger than unity.
Another property we found from the passive running gaits is the symmetricity involved in the running pattern. Our simulations reveal that all the gaits of both Gait 1 and Gait 2 exhibit symmetrical behavior. Our ongoing task is to develop orbital stabilization controllers. 

4. Publication 
