Abstract – Conservation of angular momentum that introduces nonholonomic behavior, underactuation and time dependence, makes the trajectory planning of gymnastic robots difficult. By defining appropriate values for the initial angular momentum, a method is developed that can lead a mechanism to a desired final configuration from an initial given one, in prescribed time. This method is optimization-based and fully exploits the initial mechanism angular momentum. Obstacle avoidance during flight is achieved by setting additional constraints. The method results in smooth, small in magnitude, and therefore easily applicable joint torques. Index Terms – Gymnast robots, underactuated, nonholonomic planning, non-zero angular momentum. I. INTRODUCTION esearch in gymnastic robots and related subjects has a number of purposes, such as the development of gymnast robots and the understanding of human torque requirements to execute a gymnastic movement. Designing gymnast robots improves our capabilities in developing humanoid robots and can provide us with useful knowledge about the importance of motion-related parameters. Also, the study of gymnastic motions is needed because it is difficult for an athlete to realize a priori the amount of torques required for a figure, and this, in turn, can result in injuring himself or herself. Gymnastic motions include various tasks, such as jumping, dismounting from a bar, somersault, diving, back flips and so on. A number of researchers have dealt with these problems and have produced some interesting results. In the early 80’s, Raibert achieved jumping, somersault control and dynamic stabilization of a 3D biped robot, [1]. Spong has succeeded in swinging up the acrobot and stabilizing it in its upright position, [2]. Mita et al. have introduced an analytical time optimal solution for an acrobot with non-zero initial momentum, [3]. They have shown that time optimal trajectories can be obtained using singular control and switching controllers. Grizzle et al. have Manuscript received September 15, 2006. This work was supported by the General Secretariat for Research and Technology EPAN Cooperation Program 4.3.6.1.b (Greece-USA 035). Additional support provided by public funds (European Social Fund 75% and GSRT 25%) and private funds, (Zenon SA), within 8.3 of Op. Pr. Comp., 3rd CSP - PENED 2003. E. Papadopoulos is with the Department of Mechanical Engineering, National Technical University of Athens, (NTUA) 15780 Athens (tel: +30- 210-772-1440; fax: +30-210-772-1455; e-mail: egpapado@central.ntua.gr). I. Fragkos was with the Department of Mechanical Engineering, NTUA, 15780 Athens (e-mail: mc01027@mail.ntua.gr). I. Tortopidis is with the Department of Mechanical Engineering, NTUA, 15780 Athens (e-mail: itor@central.ntua.gr). constructed a scalar function, similar to the one in [3], by partially integrating the angular momentum equation and implemented it in a two-link structure undergoing ballistic motion, [4]. Trajectory planning has been studied by using analytical formulas subject to a switching condition in the neighborhood of a singular point. It is worth noticing that, till now, there are no analytical expressions for 3-link structures like these in [3,4]. Some researchers have given numerical solutions in gymnastic problems, like Godhavn et al. [5] or Kamon et al. [6] for somersault motions. Also related is the work of Sang-Ho Hyon et al., where they simulated a 4-link mechanism in jumping, back handspring and somersault using target dynamics-based control, [7]. They focused on controlling physical quantities such as the center of mass (CM) motion or the angular momentum. Furthermore, they have constructed a 4-link model to implement the results of their work. In this paper, a novel planning technique for planar rigid multibody systems undergoing ballistic motions is implemented. The method is extended so that the mechanism can also perform obstacle avoidance. This problem tackled is modeled after the gymnastic high jump problem, see Fig. 1. Fig. 1. A four link robot imitates the high jump movement of an athlete. The main problem that is presented and solved in this paper is the construction of a trajectory that transfers the robot from a given initial configuration to a desired final one, in given time. This is a common phase for many gymnastic problems like the ones mentioned above, while it is also related to the famous cat problem, [8]. The main characteristic in this system’s dynamical model is that the conservation of angular momentum poses a velocity-type constraint to it. At this point, the problem’s nonholonomic nature and all the subsequent difficulties in trajectory planning are revealed: we are capable of controlling explicitly n actuated dof while we intent to affect the behavior of n + 1 dof. Therefore, as we have fewer actuators than dof, the system is underactuated. Leading the system to the desired final configuration is all but trivial. Another important characteristic is that base orientation does not affect the robot kinetic energy and the variable representing the orientation is called cyclic. The planning On Robot Gymnastics Planning with Non-zero Angular Momentum Evangelos Papadopoulos, Senior Member IEEE, Ioannis Fragkos, and Ioannis Tortopidis R 2007 IEEE International Conference on Robotics and Automation Roma, Italy, 10-14 April 2007 WeE4.3 1-4244-0602-1/07/$20.00 ©2007 IEEE. 1443