Stability and control of a single or three-body constrained system are considered. Several different types of constrained motion are among them: the impact phase of a free body colliding with the ground, contact with a stationary or moving platform, movement on a frictionless surface or multiple rigid bodies connected by holonomic constraints, and moving as in the human arm. The single body constrained system is controlled by sliding mode control. The stability of the three-link arm at arbitrary equilibrium points and Lyapunov stability in the vicinity of the equilibrium point are formulated. The formulation and derivations are by computational tools, that is, state space analysis and matrices. The approach can easily be extended to larger systems with many rigid bodies such as skeletal systems. The formulation minimizes human labor in formulations and simulations. The sliding mode behavior of the model on a frictionless surface and the three link arm stability are demonstrated via simulation. Challenges for application to natural systems are outlined.H. HEMAMI AND V. I. UTKIN the computations can be inferred by simulation. The construction of the energy function in single or multirigid body systems is guided by the clear definition of the kinetic and potential energies in mechanics [10], in robotics [11][12][13], and human motion [14,15].Another issue worth discussing is how to combine the tools of analysis, computation, and experiment. The present formulation of contact and impact problems of a single or the three-link body system is intended to minimize human effort and relegate the bulk of the work to the computer.The dynamics of mechanical systems can be formulated by analytical methods [16][17][18], geometrical methods [9], and differential geometry. A variety of control methods may be used [8,19,20]. The performance of the controller can be evaluated by Lyapunov methods, and a number of methods exist for the construction of Lyapunov functions [21,22]. In the formulation here, the computation of constraint forces becomes simple and lends itself to better physical interpretation. This same formulation of the constrained forces leads to a better understanding of impact and contact with the ground. As stated before, the problems of impact as well as soft and hard contact and collision with objects and the environment are part of the everyday activities of humans. Impact and collision also arise in the fields of robotics [23] and mechanics [24][25][26]. Impact has been the subject of numerous studies in the past [17,27,28]. Rigid body collision [29,30], finite element methods [31], and the problem of propagation of waves in elastic or visco-elastic materials as a result of collision of the bodies [32] are notable studies. One approach in dealing with the impact problem is to assume a coefficient of restitution according to Newton and Poisson [17,33,34]. Another approach is based on elasticity and visco-elasticity [30,35,36]. Numerical methods and finite element methods are also utilized [37,38].The constrained ...