Falling cat problem

Abstract: How does a falling cat change her orientation in midair without violating the angular momentum constraint? This has become an interesting question to both control engineers and robotists. In this paper, we address this problem together with a constructive solution. First, we show that a falling cat problem is equivalent to the constructive nonlinear controllability problem. Thus, the same principle and algorithm used by a falling cat can be used for space robotic applications such as reorientation of a satellite using rotors, attitude control of a space structure using internal motion and for other robotic tasks such as dextrous manipulation with multifingered robotic hands and nonholonomic motion planning for mobile robots. Then, using ideas from Ritz Approximation Theory we develop a simple algorithm for motion planning by a falling cat. Finally, we test the algorithm through simulation on two widely accepted models of a falling cat. It is interesting to note that one set of simulated trajectories is close to trajectories used by a real cat.

Abstract: Udwadia–Kalaba equation is a simple, aesthetic and thought-provoking description of the world at a very fundamental level. It is about the way systems of bodies move. We creatively apply the Udwadia–Kalaba approach to study falling cat’s movements. The cat is modeled as a constrained discrete dynamical system. In an alternative way, Udwadia–Kalaba formulation is used for analysis of the falling cat’s dynamics. With this novel approach, we can easily obtain the dynamical model and get the explicit analytic form of the general equations of motion of the falling cat. The surprise phenomenon (that a cat when dropped at rest with its feet pointing up can always manage to right itself and land safely on its feet) is observed through numerical simulation based on the constructed dynamical model. Unmatched ease, clarity and elegance of the Udwadia–Kalaba formulation for solving the falling cat problem (constrained discrete dynamical system or multibody system) are presented.

Abstract: We introduce a specific four-particle, four degree-of-freedom model and calculate the rotation that can be achieved by purely internal torques and forces, keeping the total angular momentum zero. W ...

Abstract: The attitude control of space structures is an important problem. There has been considerable research in this area that has focussed on the use of momentum exchange devices. In this paper, we propose to control the attitude of space structures using a serial three-link PUMAtype manipulator that can be mounted on the space structure. This unconventional method of attitude control exploits the nonholonomic nature of the constraints that arise due to the conservation of angular momentum. We adopt a surface integral approach for the motion planning of the manipulator that will reorient the space structure in any desired way. The snlient features of our algorithm are: (a) It is possible to mathematically prove the controllability of the system, (b) The motion of the manipulator can be planned amidst additional constraints like joint limits of the manipulator, and (c) the algorithm can be easily extended for application to flexible space structures. I . I n t r o d u c t i o n The attitude control of space structures has been studied over the years by various researchers. Such research has been motivated by the fact that structures in space are required for high precision tasks. The research is pertinent because the space environment causes a drift in the orientation of free flying bodies. The change could take place due to differential gravitational forces or effects due to solar radiation. In the case of a space station, the change could take place rather rapidly due to dynamic interaction between the space station and robots performing tasks on board the space station. Large changes in the orientation of the space station could take place due to docking with the shuttle, or due to the operation of booster rockets required for orbit maintenance. An extensive literature survey of attitude control of space structures can be carried out. This survey will lead us to believe that the best wav for attitude correction or stabilization would be to use momentum exchange devices. Though control momentum gyroscopes are regarded as one of the most desirable devices for attitude control, they have certain dis:rdvantages. The gyroscopes dissipate energy at a constant rate to overcome the friction in the bearings. This steady power consumption is significant in the space environment where the energy is available from rechargeable solar batteries. The system is also ~11sceptible to failure because of the presence of mechanical components constantly in motion. In the event of failure, replacement or rep:~ir is going to be a diflicult task. Furthennore, the gyroscopes consisting of the flywheels and the motors add significantly to the weight of the system. In this paper we propose to use a three-link manipulator for the attitude correction of the space structure. h4anipul:rtors with three or more links are expected to be present onboard spacecrafts or at the site of space stations. If these rn:~nipulirtors can effectively reorient space structures, they will serve the d u d purpose of attitude control and automation in space. In manned space explorations, astronallts have been expected to reorient themselves in space during cxtrtr-veliicul:~r activity. 111 1072, experimental investigations of an astronaut m:rncl~vering sclicme wis carried out [I]. Various linih motions were stl~tlietl t l ~ a t produced nearly pure rotations about each of tho threc niutually perpentlic~~lar axes fixed at the torso of the astronaut. M'hile istronauts h:lve trictl to learn thc process of self-induced rotations, it h:s heen seen tlitrt cats perform this kind of maneuver with ense. They always land on their feet when dropped from a height with an arbitrary orientation. The cat performs the maneuver through a cyclic motion in which it first bends its spine forward, then to one side, then to the other side, and finally forward again. In 1969 a dynamical explanation of the falling cat problem was provided [2]. The falling cat problem [2] and the astronaut maneuvering scheme [I] are simple examples, where a system in the absence of angular momentum, can undergo a change in orientation through internal motion. In 1991, Nakamura and Mukherjee [4] proposed a bi-directional approach to the motion planning of free-flying space robots. It was shown that by utilizing the nonholonomy, the vehicle orientation in addition to the joint variables of the manipulator can be controlled by actuating only the joint variables. This work was motivated by the work of Vafa and Dubowsky in 1987 15) where cyclic motion of the joint variables were proposed to reorient the space vehicle. Later, Fernandes et.al. [3] proved the controllability of a space robot system using a three link manipulator. If mechanical manipulators were to replace momentum exchange devices for attitude control, an important question pertaining to the controllability of the system needs to be answered. In simple words, this question can be posed as follows. Is it possible to change the orientation of a space structure from any initial orientation to :my desired orientation ? As an answer to this question, we show in this paper that a three link PUMA-type manipulator can indeed pcrforni this task. Furthermore, this task can be accomplished amidst :rdditional constraints like joint limits of the manipulator. This paper is organized as follows. In section 2 we review two theorems on line antl surface integrals and analyze the properties of nonholonomic systems. In section 3 we develop an algorithm for the reorientation of a sp:m structure in the plane using a two link manipulator. We extend this algorithm for the three dimensional case in section 4, and in section 5, we provide simulation results. 2. M a t h e m a t i c a l p r e l i m i n a r i e s a n d b a c k g r o u n d 2.1 L i n e and s u r f a c e i n t e g r a l s T h e o r e m 1: Green's T h e o r e m 161 Let S be :r closed bounded region in the x-y plane whose boundary C consists of finitely many smooth curves. Let v l ( s ,y ) antl v z ( s , y ) be functions which are continuous and have continuoes partial derivatives Dvl/i)?/ and Bv2/8:c everywhere in some some domain cont;rining S . Then the integration being taken along the enlire boundary of C of S such that S is on the left as one advances in the direction of integration (please refer to Fig.1). Another theorem that is important for our'analysis deals with the path independence of line integrals. This theorem is formally stated next [GI. T h e o r e m 2: Let v = v l i + v.2 j + vs k , antl let v,, v2, and vs be continuous functions of x, y, and z iu a domain D of space. Then the line integral is independent of path if and only if the differential form under the integral sign is exact in D, or equivalently the integral is zero for every simple closed path in D, or equivalently