# Controllable Height Hopping of a Parallel Legged Robot
## 1 Introduction  
The most prominent advantage of a legged robot is its outstanding ability of highly dynamic locomotion strategies such as running and jumping, imitating real legged animals in field situations [1]. In general, two main elements, actuator output ability and motion controllability, place great demands on the systematic design and control methods of bio-inspired legged robots. To examine these requirements on robot design, continuous height-controlled hopping serves as a fair criterion for practical research because it is a type of simplified movement pattern. The capacity for explosiveness can be embodied in robot's maximum jumping height, which is heavily dependent on the actuator output. Most previous bipedal robot such as ASIMO [2] and ATRIAS [3], in order to support heavy trunk and realize accurate position control, using motors with relatively high-ratio reducers, have limited ability in robot jumping and running research because of the non-transparency in reducers. In other words, the reason for the weakness in dynamic motion is that the large reducer ratio brings increased friction and other nonlinear elements to the actuator. On the other side, most low-ratio or even direct-driven actuators can accurately control their output with high transparency, but they have limited output torque which can only be used in small-sized robot. To overcome this conflict, MIT Cheetah series robot [4; 5; 6; 7] used a high-torque-density brushless direct current (BLDC) motor with a large gap radius as its actuator [8; 9]. This type of actuator is typically a large-torque, low-speed motor that usually cooperates with alow-ratio gearbox to achieve high back-drivability. Nevertheless, this actuator is relatively large in size and heavy when used on robot systems with more degrees of freedom (DoFs), such as humanoid robots. Thus, the balance of output ability, transparency, and structure size for a legged robot is still a crucial problem to be solved. In terms of hopping height control, [PERSON] firstly realized controllable hopping of legged robot in 1986 [10]. Later, based on the measurement of real legged animal motion, the spring loaded inverted pendulum (SLIP) model [11] was constructed and became the universal model to describe dynamic motion such as hopping, running, and other manipulations [12]. However, because of the nonlinear characteristics of SLIP model, it is difficult to resolve its dynamics analytically for every step. To address this problem, some approximation methods were proposed, such as the research of [PERSON], who assumed a small angular sweep and spring compression [13]. Without SLIP, several other intuitive approaches to hopping height control are available. The simplest approach is to maintain a constant value, such as the actuator input voltage [14] and trunk velocity [15], and further use basic model to realize adaptive hopping control [16]. This type of control is not connected closely to dynamic model, although applicable and universal for most legged robots, lacking control accuracy and maneuverability for a specific robot system. For whole-body jumping control, there are other methods such as ZMP to realize hopping on legged robot [17; 18; 19]. Besides, virtual model control (VMC) is an efficient method in robot leg or arm control, which has also been validated in walking and jumping motion control [20; 21; 22]. Nevertheless, with different control target and robot structure, it should be combined with specific superior control strategy. To balance the requirements of both load supporting ability and dynamic locomotion function, an integrated legged robot system including hardware design and control strategy is plainly needed. In previous work [23], we proposed a parallel-elastic legged robot for loaded jumping.