The above methods required brakes at the passive joints for positioning. However, the nonholonomic constraints on such manipulators indicate the possibility of the positioning using no brakes. The nonholonomic constraints on manipulators with passive joints have different characteristics from those of wheeled vehicles and space robots etc., which have been studied as typical nonholonomic systems. These examples have kinematic constraints, which geometrically limit the direction of mobility, or dynamic constraints caused by conservation of angular momentum. These constraints can be represented as a differential equation including the generalized coordinates and the generalized velocities. On the other hand, the dynamic constraints introduced by passive joints are also functions of the generalized accelerations and are called second-order nonholonomic constraints. These constraints make it difficult to directly apply the mathematical techniques so far developed for first-order nonholonomic systems.
It should be noted that the linear approximation of this manipulator is not controllable, while that of an underactuated mechanism in vertical plane, e.g. an inverted pendulum, is controllable due to gravity. The system in this video is equivalent to a planar rigid body sliding on a horizontal frictionless plane, a point of which a force of arbitrary direction and magnitude can be applied. The same dynamics are found in pushing manipulation of a object, a hovercraft, and an omnidirectional tractor connected to a ball-caster trailer. The control method in the video can also be applied to these systems.
It is obvious that no angular acceleration occurs at the free link when the passive joint is accelerated or decelerated in the direction of the center of gravity of the link. The link can be translated without rotation by this motion. When the joint is rotated around the center of percussion (C.P.) of the link with regard to the joint, the C.P. stays still. The link can then be rotated without translation. (The C.P. is marked as a white circle in the video.)
The trajectory from an arbitrary initial configuration to an arbitrary desired configuration can be composed of these trajectory segments. The positioning between the initial configuration and the desired configuration is as follows. First, the joint moves along the rotational trajectory around the C.P. of the link. Then the joint is accelerated towards the center of gravity and the link moves straight along the translational trajectory. Finally, the joint rotates along the rotational trajectory to the desired configuration.
If the initial and the desired orientations of the link are not parallel, the positioning trajectory can be constructed in a different way. First, the link moves along the translational trajectory. Next, the link rotates around the C.P. along the rotational trajectory. Finally, the joint moves along the translational trajectory. Thus, the link can reach the desired configuration from the initial configuration.
A horizontal manipulator with passive joints cannot be asymptotically stabilized to an equilibrium point by any smooth state feedback, based on Brockett's theorem. In other words, no state feedback law can stabilize the position and orientation of free link simultaneously to the desired configuration. Here, the objective of the feedback control is changed to stabilize the free link to the desired trajectory, instead of the desired point.
The nonlinear feedback control used in the video is based on the following idea. In the case of a translational trajectory, the acceleration of the passive joint in the direction of the desired trajectory gives rise to the inertial force on the free link in the opposite direction. Then the link has the same dynamics as a pendulum or an inverted pendulum in a virtual gravity field. Feedback to stabilize the orientation and position of the link to the desired trajectory can be designed using the acceleration normal to the desired trajectory as the input. Exact linearization of the system and linear state feedback are employed for stabilization. In case of the rotational trajectory, feedback can be constructed by treating the centrifugal force as virtual gravity.